Software Package This book is accompanied by a special edition of the software, PicWave from Photon Design (www.photond.com/products/picwave.htm), provided to simulate live many of the examples found in this book. By showing the device structures in detail and providing additional results, it will help to gain additional insight into the examples presented in this book. The software runs on any modern PC with Windows-2000 or later installed, with 1GB of memory or more. It can be downloaded from the books companion Web site, www.elsevierdirect.com/companions/9780123725790 free of charge to owners of this book. PicWave takes a rather different time-domain travelling wave (TDTW) approach to the frequency-domain based theory presented in this book, and illustrates how similar results can be obtained in the time domain. PicWave is a circuit model and as such is capable of modelling not just linear fiber components but also complex fiber devices such as fiber couplers, splitters and amongst others. Features illustrated by the software include: . Behavior of a fiber-Bragg grating, including transmission, reflection, group delay, group velocity dispersion (after Chapter 4) . Simulation of multi-mode effects, such as grating assisted co-directional coupling from a fiber core to a cladding mode (after Chapter 4) . Effect of apodization on FBG characteristics (after Chapter 5) . Simulation of fiber band pass filters, including devices based on single fibers, Mach-Zehnder interferometer circuits and in-coupler gratings (after Chapter 6) . Behavior of chirped fiber Bragg gratings (after Chapter 7) . Transmission of digital bit patterns through examples, showing the distortion of signals in the time domain (eye diagrams). The user is able to run the chosen examples, inspecting all the results available within PicWave, including optical transmission and reflection spectra, group delay, dispersion, time signals and more. This special version of Picwave is limited to modelling only the passive fibre devices covered in this book. However the full PicWave package is capable of modelling other non-linear and active devices such as laser diodes and SOAs as discussed in Chapter 8.
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[email protected]. You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Application submitted British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN: 978-0-12-372579-0 For information on all Academic Press publications visit our Web site at www.elsevierdirect.com Printed in the United States of America 09 10 9 8 7 6 5 4 3 2 1
Dedicated to the memory of my parents, Vimla and Kedar Nath Kashyap.
Preface Despite the lapse of a decade since the previous edition of this book was published, fiber Bragg gratings continue to flourish and their applications expand. As has been the experience with optical fibers in the past, new discoveries have continued to remain a driver for technological developments. In this respect, the past decade has seen further activity in the poling of glass, fiber Bragg grating sensors, high-power fiber lasers, and the opening of a new research on femtosecond (fs) laser processing, which was just beginning to grow when the first edition came out in print. To reflect these developments, this edition has three new chapters that touch on the topics of sensing, fs laser writing of fiber Bragg gratings (FBGs), and poling of glass and optical fibers. It is hoped that these chapters will bring the book into the mainstream of topical research interest. The basis of the FBG, the refractive index change induced by ultraviolet or fs laser pulses, now stands around a record 0.1, having met the prediction made in 1999. Truly broadband mirrors spanning 300 nm are now possible in fiber with high reflectivity (99%), challenging thin-film technology. In fact, some of the periodic nanostructured gratings formed by fs laser pulses have a glass-air boundary, which leads to the possibility of miniaturizing devices still further with the large refractive index contrast of 0.45. The fs laser has allowed the writing of strong gratings in materials that have traditionally been nonphotosensitive, such as pure silica and ZBLAN glass. The use of high-intensity pulses enables multiphoton absorption to occur, and these pulses also literally rip the electrons out of their orbits to the conduction band, inducing plasmas and carrier heating. The optical damage that results has interesting applications in strong gratings for high-temperature sensing. Indeed, sapphire fiber gratings for high-temperature turbine measurements would not have been possible without fs lasers. High-power lasers have suddenly become commonplace at unusual wavelength, fueled by the downturn in the telecommunications and the rise of the multibillion-dollar biophotonics and sensing industries. FBGs have found their place in peculiar applications such as in the investigation of strain in the human lumbar column. Glass poling, too, has evolved, even though the goal of the 10 pm-V 1 electrically induced nonlinearity remains elusive. New polarization controllers, fiber-based Q-switches, and other tunable FBG devices have come of age with optical fiber poling. The low-loss optical fibers for telecommunications made of a fused silica cladding and a germania-doped core still xv
xvi
Preface
maintain their pride of place in optical fiber technology. Rare earth dopants in silica and other glasses have made many more applications possible. The advent of photonic crystal fibers are now demonstrating a way to increase the powerhandling capacity of optical fibers, although high-quality gratings remain difficult to implement in these fibers. Gratings are being applied to reduce the impact of nonlinearities in fibers, pulse shaping and compression, and signal processing. The mechanisms contributing to photosensitivity continue to be debated, although major advances have been made in this area. There are a number of methods of the holographic inscription of Bragg gratings using ultraviolet radiation or infrared fs pulses, with the phase-mask technique holding a prominent position. These methods have multiplied, with several techniques demonstrated for the fabrication of ultralong gratings. New areas just on the brink of breakthroughs, such as random lasers, are highly compatible with the FBG. It is impossible to cover the massive advances made in this field in a book of this size (even though the second edition is now vastly expanded), a field in which the number of applications has exploded. The book therefore continues to be an introduction to the extremely rich area of the technology of fiber gratings, with a view to providing an insight to some of the exciting prospects, including the principles of fiber Bragg gratings, the photosensitization of optical fibers, Bragg grating fabrication, theory, properties of gratings, specific applications, sensing technology, glass poling, femtosecond processing of glass, and FBG measurement techniques.
Acknowledgments Writing a book is like planting a tree. One sees it grow and develop branches and roots, leading to connections that permeate throughout the world, with the hundreds of researchers providing the nourishment. At the end, the tree should flourish to shade the ones who nourished it, and those yet to come. Therefore, I am grateful to the scientific community at large for providing the data for this book, now in its much-expanded form. The writing of the second edition poses some problems, as the written data are often still valid and the new must be integrated into the old. The choice has been a difficult one, as the field is now very large, and it is often based on the examples that provide the information required. The book is not therefore intended to be a bibliography of all the research and applications that have been published in the area of laser-induced fiber gratings, for there are too many. Instead, we focus on the technology with the goals of guiding the reader on how to fabricate, use, and implement systems with fiber gratings and shedding light on recent advances in the field. I am deeply grateful to Walter Margulis for the major contribution he made by writing the chapter on glass poling. Choosing the right person to prepare that chapter was a difficult decision to make until I took the step of asking him, and since, it was to be the best decision I could have made. His dedication and lightening response is evident in the extremely thorough chapter he has written. Without his help, the book would still be somewhere in cyberspace. I am grateful, too, to my students and researchers, in particular Runnan Liu, ¨ issa Irina Kostko, Mathieu Gagne´, Jerome Poulin, Francis Guay, Julie Baron, A Harhira, and the numerous others who spent time in my labs for their research in the several areas of FBGs. Included among these are Jessica Chauve, Cedric Pruche, Lucien Bojor, and John Machlecler. Galina Nemova’s contribution on surface plasmons is most appreciated. I am indebted to Jacques Albert, Re´al Valle´e, Sidarath Ramachandran, Ian Bennion, and numerous others who have all generously contributed material included in the new edition. James Brennan and Bertrand Poumellec are gratefully acknowledged for their painstaking review of sections of the first edition and their constructive comments, which I have tried to incorporate in this edition. Fiber Bragg Gratings may not have made such progress without the help of Dr. Ju¨rgen Bartschke, who was instrumental in bringing to life the first CW intra-cavity 244 nm laser source in my lab at BT Laboratories in 1989. xvii
xviii
Acknowledgments
His recent visit from Xiton Photonics has renewed an old friendship and perhaps new innovations in gratings. I hope that the theft of his passport did not spoil an otherwise good visit to Montre´al! Finally, I would like to thank Hannah and Monika, whose patience was not only tested to the limits of exasperation, but whose caring and infinite capacity to see the light at the end of the tunnel kept me on the straight and narrow. Raman Kashyap Montre´al August 2009
Chapter 1
Introduction It is clear from the revolution of the year 2000 that optical fibers have made an enormous impact on modern telecommunication systems. The capacity of optical fiber systems is forever moving upward and distances longer, fueled by the exploding demand for the Internet. As we move into the next millennia, the need for interactive long-distance communications will increase, pressured by the need to reduce the demand for energy and the effects of global warming. Our perception of changing working habits and practices must include questions about the energy demands of the millions of servers and personal computers that remain awake 24 hours a day, 7 days a week, and 365 days a year. Despite changing work practices and the demand for more efficient transport systems, the equation is stacked in favor of increased entropy. Monitoring the planet’s health and the health of its people, wildlife, resources, rivers, glaciers, freshwaters, oceans, and land will come increasing under the microscope, if humans are to survive in peace, even for the next decade. Efficiency and conservation are thus the names of the game for the current century, not the least driven by the excesses of the previous decade. Therefore, technologies must carry the slogans “green” and “mean” if they are to succeed as next-generation solutions. It is with this in mind that one evokes the wonderful properties of glass, known to humans since the beginning of time. Recent events have shifted our focus on communications to the abundant applications in photonics, for which we owe much to the success of optical fibers; their success lies in their near ideal properties of low transmission loss, a high optical damage threshold, and low optical nonlinearity. The combination of these properties has enabled long-distance communication to become a reality. At the same time, the long lengths enabled the optical power to interact with the small nonlinearity to give rise to the phenomenon of optical solitons, overcoming the limit imposed by linear dispersion. The market for optical fiber continues to grow, despite the fact that major trunk routes and metropolitan areas have already seen a large deployment of fiber. The current stage in the field of communication is the mass delivery of integrated services, such as home banking, shopping, Internet services, and entertainment using video on demand, among others. Although the bandwidth available on single-mode fiber should meet the ever increasing 1
2
Introduction
demand for information capacity, the geographically dependent bottlenecks leading to variability in the speed of Internet services make it clear that architectures for future networks need to exploit technologies that have the potential to drive down costs and make services economically viable at higher speeds. There is tremendous scope for improvement, and it is only recently, after years of fits and false starts, that some networking companies are beginning to provide fiber to the home (FTTH) at 10 GB/s. The deployment of such a technology is highly cost sensitive and location dependent, with green-field sites being the preferred choice for providing such services. The die is cast, but the tremendous advantages of a fast Internet can only be realized by upgrading the entire network, end to end. Optical fiber will have to compete with other transport media such as radio, copper cable, and satellite. Short-term economics and long-term evolutionary potential will determine the type of technology likely to succeed in the provision of these services. But it is clear that optical fibers in communication systems of the future are secure for years to come. The technological advances made in the field of photosensitive optical fibers, although relatively recent, have made a significant impact on this growth. Optical fiber amplifiers would not have been possible without fiber Bragg gratings, as virtually every semiconductor pump laser has one. An increasing number of fiber devices based on this technology are already in the marketplace, with large volumes of fiber Bragg grating (FBG)–based dispersion compensators being sold. It is believed that there will be an increased use of such devices in wavelength-division-multiplexed (WDM) systems, channel selection, deployment of transmitters in the upstream path in a network, and viable routing schemes, amongst others. The fascinating technology of a simple in-line, all-fiber optical filter, with a vast number of applications to its credit, is based on the principle of photosensitive fiber. Not only does the FBG have applications in communications systems, but one of the first happened to be in the area of sensing. This has now become more mainstream, with oil and gas exploration leading the demand for fiber-based sensors. Other developments in photonics have shed light on biomedical applications, in-body sensing, tumor detection and treatment, and post-trauma care. FBGs are finding applications in these and other unusual areas, predominantly fueled by the low-cost optical devices cast off by the successes of optical communications. Thus, we find the field of sensing to be a growth area and as yet to be commercialized further.
1.1 HISTORICAL PERSPECTIVE The photosensitivity of optical fiber was discovered at the Canadian Communications Research Center in 1978 by Ken Hill et al. [1] during experiments using germania-doped silica fiber and visible argon ion laser radiation. It was
Historical Perspective
3
noted that as a function of time, light launched into the fiber was being increasingly reflected. This was recognized to be due to a refractive index grating written into the core of the optical fiber as a result of a standing wave intensity pattern formed by the 4% back reflection from the far end of the fiber and forward propagating light. The refractive index grating grew in consort with the increase in reflection, which in turn increased the intensity of the standing wave pattern. The periodic refractive index variation in a meter or so of fiber was a Bragg grating with a bandwidth of around 200 MHz. But the importance of the discovery in future applications was recognized even at that time. This curious phenomenon remained the preserve of a few researchers for nearly a decade [2,3]. The primary reason for this is believed to be the difficulty in setting up the original experiments and also because the observations were thought to be confined to the one “magic” fiber at the Canadian Communications Research Center. Further, the writing wavelength determined the spectral region of the reflection grating, limited to the visible part of the spectrum. Researchers were already experimenting and studying the even more bizarre phenomenon of second-harmonic generation in optical fibers made of germania-doped silica, a material that has a zero second-order nonlinear coefficient responsible for second-harmonic generation. The observation was quite distinct from another nonlinear phenomenon of sum-frequency generation reported earlier by Ohmori and Sasaki [4] and Hill et al. [5], which was also curious. Ulf Osterberg and Walter Margulis [6] found that ML-QS infrared radiation could “condition” a germania-doped silica fiber after long exposure such that second-harmonic radiation grew (as did Ken Hill’s reflection grating) to nearly 5% efficiency, and was soon identified to be a grating formed by a nonlinear process [7,8]. Julian Stone’s [9] observation that virtually any germania-doped silica fiber demonstrated a sensitivity to argon laser radiation reopened activity in the field of fiber gratings [10,11] and for determining possible links between the two photosensitive effects. Parent et al. [12] had pointed out the two-photon absorption nature of the phenomenon from the fundamental radiation at 488 nm. The major breakthrough came with the report on the holographic writing of gratings using single-photon absorption at 244 nm by Gerry Meltz et al. [13]. They demonstrated reflections gratings in the visible part of the spectrum (571–600 nm) using two interfering beams external to the fiber. The scheme provided the much-needed degree of freedom to shift the Bragg condition to longer and more useful wavelengths, predominantly dependent on the angle between the interfering beams. This principle was extended to fabricate reflection gratings at 1530 nm, a wavelength of interest in telecommunications, also allowing the demonstration of the first fiber laser operating from the reflection of the photosensitive fiber grating [14]. The ultraviolet (UV)-induced
4
Introduction
index change in untreated optical fibers was 104. Since then, several developments have taken place, which have pushed the index change in optical fibers up a hundred fold, making it possible to create efficient reflectors only a hundred wavelengths long. Lemaire and coworkers [15] showed that the loading of optical fiber with molecular hydrogen photosensitized even standard telecommunication fiber to the extent that gratings with very large refractive index modulation could be written. Pure fused silica has shown yet another facet of its curious properties. Brueck et al. [16] reported that at 350 C, a voltage of about 5 kV applied across a sheet of silica, a millimeter thick, for 30 minutes resulted in a permanently induced second-order nonlinearity of 1 pm/V. Although poling of optical fibers had been reported earlier using electric fields and blue-light and UV radiation [17–19], Wong et al. [20] demonstrated that poling a fiber while writing a grating with UV light resulted in an enhanced electro-optic coefficient. The strength of the UV written grating could be subsequently modulated by the application of an electric field. More recently, Fujiwara reported a similar photo-assisted poling of bulk germanium-doped silica glass [21], and many other glass systems have been studied. However, the silica–germanium system has produced further surprises in new photonic crystal fibers [22]. Developments in kW power lasers would not have been possible without the use of FBGs [23], and the establishment of compact sources for the generation of an ultra-broadband continuum would have been nearly impossible without the use of fiber lasers [24]. All these photosensitive processes are linked in some ways but can differ dramatically in their microscopic detail. The physics of the effect continues to be debated, although the presence of defects plays a central role in more than one way, and our understanding has improved dramatically. The field remains an active area for research, moving much more into the realms of the many applications of Bragg gratings, poled fiber devices, and glass.
1.2 MATERIALS FOR GLASS FIBERS Optical fiber for communications has evolved from early predictions of lowest loss in the region of a few dB km1 to a final achieved value of only 0.2 dB km1. The reason for the low optical loss is several fortuitous material properties. The band-gap of fused silica lies at around 9 eV [25], whereas the infrared vibrational resonances produce an edge at a wavelength of around 2 microns. Rayleigh scatter is the dominant loss mechanism with its characteristic l4 dependence in glass fibers, indicating a near perfect homogeneity of the material [26]. The refractive index profile of an optical fiber is shown in Fig. 1.1.
Materials for Glass Fibers
5
Core Cladding
r
Refractive Index Silica AIR −r
−a +a
+r
Figure 1.1 Cross-section of an optical fiber with the corresponding refractive index profile. Typically, the core to cladding refractive index difference for single-mode telecommunications fiber at a wavelength of 1.5 mm is 4.5 103 with a core radius of 4 mm.
The core region has a higher refractive index than the surrounding cladding material, which is usually made of silica. Light is therefore trapped in the core by total internal reflection at the core-cladding boundaries and is able to travel tens of kilometers with little attenuation in the 1550 nm wavelength region. One of the commonly used core dopants, germanium, belongs to the Group IVA, as does silicon, and it replaces the silicon atom within the tetrahedral, coordinated with four oxygen atoms. Pure germania has a bandedge at around 185 nm [27]. Apart from these pure material contributions, which constitute a fundamental limit to the attenuation characteristics of the waveguide, there may be significant absorption loss from the presence of impurities. The OH ion has infrared (IR) absorptions at wavelengths of 1.37, 0.95, and 0.725 mm [28], overtones of a stretching-mode vibration at a fundamental wavelength of 2.27 mm. Defect states within the ultraviolet and visible wavelength band of 190–600 nm [29] also contribute to increased absorption. The properties of some of these defects will be discussed in Chapter 2. The presence of phosphorus as P2O5 in silica, even in small quantities (0.1%), reduces the glass melting point considerably, allowing easier fabrication of the fiber. Phosphorus is also used in fibers doped with rare earths such as Yb and Er for fiber amplifiers and lasers. In high concentration, rare earth ions tend to cluster in germanium-doped silicate glasses. Clustering causes ion– ion interaction, which reduces the excited state lifetimes [30]. Along with aluminum (Al2O3 as a codopant in silica) in the core, clustering is greatly reduced, enabling efficient amplifiers to be built. Phosphorus is also commonly used in
6
Introduction
planar silica on silicon waveguide fabrication, as the reduced processing temperature reduces the deformation of the substrate [31]. Fluorine and trivalent boron (as B2O3) are other dopants commonly used in germania-doped silica fiber. A major difference between germanium and fluorine/boron is that while the refractive index increases with increasing concentration of germanium, it decreases with boron/fluorine. With fluorine, only modest reductions in the refractive index are possible (0.1%), whereas with boron, large index reductions (> 0.02) are possible. Boron also changes the topology of the glass, being trivalent. Boron and germanium together allow a low refractive index difference between the core and cladding to be maintained with a large concentration of both elements [32]. On the other hand, a depressed cladding fiber can be fabricated by incorporating boron in the cladding to substantially reduce the refractive index. The density of the boron-doped glass may be altered considerably by annealing, thermally cycling the glass, or by changing the fiber drawing temperature [33]. Boron-doped preforms exhibit high stress and shatter easily unless handled with care. The thermal history changes the density and stress in the glass, thereby altering the refractive index. The thermal expansion of boron-silica glass is 4 106 C1, several times silica (7 107 C1) [34]. Boron-doped silica glass is generally free of defects, with a much-reduced melting temperature. Being a lighter atom, the vibrational contribution to the absorption loss extends deeper into the short wavelength region and increases the absorption loss in the 1500 nm window. Boron with germanium doping has been shown to be excellent for photosensitivity [32, 35–39].
1.3 ORIGINS OF THE REFRACTIVE INDEX OF GLASS The refractive index, n, of a dielectric may be expressed as the summation of the contribution of i oscillators of strength fi each, as [40] n2 1 4p e2 X fi ; ð1:1:1Þ ¼ n2 þ 2 3 me0 i o2i o2 þ iGi o where e and m are the charge and mass of the electron, respectively, oi is the resonance frequency, and Gi is a damping constant of the ith oscillator. Therefore, refractive index is a complex quantity, in which the real part contributes to the phase velocity of light (the propagation constant), whereas the sign of the imaginary part gives rise to either loss or gain. In silica optical fibers, far away from the resonances of the deep UV wavelength region that contribute to the background refractive index, the loss is negligible at telecommunications
Origins of the Refractive Index of Glass
7
wavelengths. However, the presence of defects or rare-earth ions can increase the absorption, even within in the transmission windows of 1.3 to 1.6 microns in silica optical fiber. Gi can be neglected in low-loss optical fibers in the telecommunications transmission band, so that the real part, the refractive index, is [35–40] n2 ¼ 1 þ
X Ai l 2 : 2 2 i l li
ð1:1:2Þ
With i ¼ 3, we arrive at the well-known Sellmeier expression for the refractive index, and for silica (and pure germania), the li (i ¼ 1!3) are the electronic resonances at 0.0684043 (0.0690) and 0.1162414 (0.1540) mm, and lattice vibration at 9.896161 (11.8419) mm. Their strengths, Ai, have been experimentally found to be 0.6961663 (0.8069), 0.4079426 (0.7182), and 0.8974794 (0.8542) [41,42], where the data in parentheses refer to GeO2. The group index, N, is defined as N ¼nl
dn ; dl
ð1:1:3Þ
Refractive Index
which determines the velocity at which a pulse travels in a fiber. These quantities are plotted in Fig. 1.2, calculated from Eqs. (1.1.2) and (1.1.3). We note that the refractive index of pure silica at 244 nm is 1.51086 at 20 C. The data for germania-doped silica may be found by interpolation of the data for the molar concentration of both materials, although they apply to the equilibrium state in bulk samples and may be modified by the fiber fabrication process. The change in the refractive index of the fiber at a wavelength l may be calculated from the observed changes in the absorption spectrum in the ultraviolet using the Kramers–Kronig relations [35–40,43], 1.6 Group Index 1.58 1.56 1.54 1.52 1.5 1.48 n 1.46 1.44 0.2 0.7
1.2
Wavelength, microns Figure 1.2 The refractive index, n, and the group index, N, of pure silica at 20 C.
1.7
8
Introduction
DnðlÞ ¼
1 ð2pÞ2
XZ
l1
i
l2
Dai ðl0 Þ l2 2 d l0 ; l l0 2
ð1:1:4Þ
where the summation is over discrete wavelength intervals around each of the i changes in measured absorption ai. Therefore, a source of photo-induced change in the absorption at l1 l0 l2 will change the refractive index at wavelength l. The refractive index of glass depends on the density of the material, so that a change in the volume through thermally induced relaxation of the glass will lead to a change, Dn, in the refractive index, n, as Dn DV 3n e; n V 2
ð1:1:5Þ
where the volumetric change, DV, as a fraction of the original volume, V, is proportional to the fractional change, e, in the linear dimension of the glass. We now have the fundamental components, which may be used to relate changes in the glass to the refractive index after exposure to UV radiation. Other interesting data on the fused silica are its softening point at 2273 C and that it probably has the largest elastic limit of any material, 17% at liquid nitrogen temperatures [44].
1.4 OVERVIEW OF CHAPTERS The book begins with a simple introduction to the photorefractive effect as a comparison with photosensitive optical fibers, presented in Chapter 2. The interest in electro-optic poled glasses is fueled from two directions: an interest in the physics of the phenomenon and its connection with photosensitive Bragg gratings, as well as the practical need for devices that will overcome many of the fabrication problems associated with crystalline electro-optic materials, of cutting, polishing, and in–out coupling. A fiber-compatible device is an ideal that is unlikely to be abandoned. The fiber Bragg grating goes a long way in that direction. However interesting the subject of poled glasses and second-harmonic generation in glass optical fibers and nonlinear behavior of gratings, this topic is not covered until Chapter 12. With this connection left for later in the book, we simply point to the defects that are found to be in common with the process of harmonic generation, poling of glass, and Bragg gratings. The subject of defects alone is a vast spectroscopic minefield. Some of the prominent defects generally found in germania-doped fused silica that have a bearing on Bragg grating formation are touched upon. The nature and detection of the defects are introduced. This discussion is followed by the process of photosensitizing
Overview of Chapters
9
optical fibers, including reduced germania, boron–germanium codoped fibers, Sn doping, and hydrogen loading. The different techniques and routes used to enhance the sensitivity of optical fibers, including that of rare-earth-doped fibers, are compared in a summary at the end of Chapter 2. Chapter 3 is on fabrication of Bragg gratings. It deals with the principles of holographic, point-by-point replication and the technologies involved in the process. Various arrangements of the Lloyd and mirror interferometers, phase-mask, along with the fabrication of different types of Bragg and longperiod gratings, chirped gratings, and ultralong gratings are explored. The attributes of some of the laser sources commonly used for fabrication are introduced in the concluding section of the chapter. New developments in the writing of fiber Bragg gratings (FBGs) and long-period gratings (LPGs) using a number of different techniques for both short and ultralong grating are introduced in the new edition. Signal processing applications in the future will require highquality long gratings of arbitrary profile. With this in mind, several techniques have been presented and principles discussed. Chapter 4 begins with wave propagation in optical fibers, from the polarization response of a dielectric to coupled mode theory, and formulates the basic equations for calculating the response of uniform gratings. A section follows on side-tap gratings, which has special applications as lossy filters. Antenna theory is used to arrive at a good approximation of the filter response for the design of optical filters. Long-period gratings and their design follow, as well as the physics of rocking filters. The last section deals with grating simulation. Here, two methods for the simulation of gratings of arbitrary profile and chirp, based on the transfer–matrix approach and Rouard’s method of thin films, are described. Recent developments in simulation, synthesis, and reconstruction of FBGs and LPGs since the publication of the first edition have been included to finish the chapter. Chapter 5 looks in detail at the different methods available for apodization of Bragg gratings and its effect on the transfer characteristics. These include the use of the phase mask, double exposure, stretching methods, moire´ gratings, and novel schemes that use the coherence properties of lasers to self-apodize gratings. Chapter 6 introduces the large area of band-pass filtering to correct for the “errant” property of the Bragg grating: as the band-stop filter! We begin with the distributed-feedback (DFB) structure as the simplest transmission Bragg grating, followed by the multisection grating design for the multiple band-pass function, chirped grating DFB band-pass filters widening the gap to address the Fabry–Perot structure, and moving on to the superstructure grating. Other schemes include the Michelson interferometer-based filter, Mach–Zehnder interferometer, properties, tolerances requirements for fabrication, and a new device based on the highly de-tuned interferometer, which allows multiple band-pass
10
Introduction
filters to be formed using chirped and un-chirped gratings. An important area in applications is the optical add–drop multiplexer (OADM), and different configurations of these are considered, along with their advantages and disadvantages. The special filter based on the in-coupler Bragg grating as a family of filters is presented. Simple equations are suggested for simulating the response of the Bragg reflection coupler. Rocking and mode-converting filters are also presented, along with the side-tap radiation mode and long-period grating filter as band-pass elements. New devices, primarily for dispersion compensation based on Gires– Tournois and Fabry–Perot interferometers, tunable filters, long-period gratings, and higher-order mode fiber devices have been added to the new edition. Chirped gratings have found a niche as dispersion compensators. Therefore, Chapter 7 is devoted to the application of chirped gratings, with a detailed look at the dispersive properties related to apodization and imperfect fabrication conditions on the group delay and reflectivity of gratings. Further, the effect of stitching is considered for the fabrication of long gratings, and the effect of cascading gratings is considered for systems applications. Systems simulations are used to predict the bit-error-rate performance of both apodized and unapodized gratings. Transmission results are also briefly reviewed, but the reader is directed to Chapter 6 for some of the outstanding results of grating-assisted dispersion compensation, based on band-pass filters. The applications of gratings in semiconductor and fiber lasers can be found in Chapter 8. Here, configurations of the external cavity fiber Bragg grating laser and applications in fiber lasers as single and multifrequency and wavelength sources are shown. Gain flattening and clamping of erbium amplifiers is another important area for long-haul high bit-rate and analog transmission systems. Finally, the interesting and unique application of the fiber Bragg grating as a Raman oscillator is shown. This chapter has been extended to include recent developments in high-power fiber coupling to laser diodes and high-power lasers and amplifiers. Chapter 9 deals with measurements and testing of Bragg gratings. This includes basic measurements, properties of different types of gratings and measurement parameters. Progress in this area is also included briefly. Life-testing and reliability aspects of Bragg gratings conclude the chapter. Chapter 10 addresses an important applications area of sensing with FBG. This application will continue to increase in importance, as FBGs are key elements for monitoring the state of health of civil infrastructure and the planet’s environment. Thus, the chapter introduces the fundamental concepts of optical sensing – specifically using optical fiber Bragg gratings, the fibers used for these, their properties, and their fascinating diversity – limited only by the ingenuity of the thousands of researchers in the field today. The technologies of systems used for measuring distributed strain and temperature, thermal compensation, and applications in a number of important areas are reviewed.
References
11
The entirely new Chapter 11 focuses on an area closely related to photosensitivity, the subject of femtosecond laser writing of structures in glass. This relatively new area has made important contributions to FBGs and LPGs, and it is poised to make an impact on several areas of photonics, such as microfluidics, microengineering, and ultrastrong gratings. The physics is not fully understood; however, major progress has been made in the technological processes that allow high-quality waveguides and strong gratings to be fabricated without prior photosensitization in virtually any dielectric. This chapter examines how glass is damaged and the influence of plasma generation, self-focusing, limiting, and optical breakdown. Techniques for writing FBGs and LPGs are also introduced. Finally, Chapter 12, also new, and written by Dr. Walter Margulis, reviews the other area strongly related to photosensitivity and charge transport: poling of glass, fibers, and their applications. The process of electric field poling, which is a key element in breaking the symmetry of glass leading to a second-order nonlinearity, allows glass to morph into yet another magical form: possessing characteristics highly desirable in waveguides and the ability to modulate light by the application of an electric field. This field was driven by the desire to induce a large nonlinearity and to integrate this property in probably the best waveguide system available today – the optical fiber. Several glass systems are explored and their properties presented with a view of finding the right “sauce” and “recipe” for enhancing the modest, though usefully observable, effect. The physics of the problem was complex to unravel and has been an area of fervent activity. Regrettably, the large nonlinearity remains elusive: Either it can be large but short lived or small and longer lived, depending on the glass type. Transport in glass, therefore, comes under the microscope and different models have led to a better but not complete understanding of the process. However, research has led to several spin-offs, including fiber-optic modulators, tunable gratings, electric field sensors, Q-switches, polarimeters, and others, examples of which conclude the chapter.
REFERENCES [1] K.O. Hill, Y. Fujii, D.C. Johnson, B.S. Kawasaki, Photosensitivity in optical waveguides: Application to reflection filter fabrication, Appl. Phys. Lett. 32 (10) (1978) 647. [2] J. Bures, J. Lapiere, D. Pascale, Photosensitivity effect in optical fibres: A model for the growth of an interference filter, Appl. Phys. Lett. 37 (10) (1980) 860. [3] D.W.K. Lam, B.K. Garside, “Characterisation of single-mode optical fibre filters,” Appl. Opt. 20 (3) (1981) 440. [4] Y. Ohmori, Y. Sasaki, Phase matched sum frequency generation in optical fibers, Appl. Phys. Lett. 39 (1981) 466–468.
12
Introduction
[5] Y. Fujii, B.S. Kawasaki, K.O. Hill, D.C. Johnson, Sum frequency generation in optical fibers, Opt. Lett. 5 (2) (1980) 48–50. [6] U. Osterberg, W. Margulis, Efficient second harmonic in an optical fiber, in: Technical Digest of XIV Internat. Quantum Electron. Conf., paper WBB1, 1986. [7] R.H. Stolen, H.W.K. Tom, Self-organized phase-matched harmonic generation in optical fibers, Opt. Lett. 12 (1987) 585–587. [8] M.C. Farries, P.J.St. Russell, M.E. Fermann, D.N. Payne, Second harmonic generation in an optical fiber by self-written w(2) grating, Electron. Lett. 23 (7) (1987) 322–323. [9] J. Stone, Photorefractivity in GeO2-doped silica fibres, J Appl Phys 62 (11) (1987) 4371. [10] R. Kashyap, Photo induced enhancement of second harmonic generation in optical fibers, in: Topical Meeting on Nonlinear Guided Wave Phenomenon: Physics and Applications 1989, Technical Digest Series, Vol 2, held on February 2–4,1989, Houston, TX, Optical Society of America, Washington, DC, 1989, pp. 255–258. [11] D.P. Hand, P.J.St. Russell, Single mode fibre gratings written into a Sagnac loop using photosensitive fibre: transmission filters, IOOC, Technical Digest (1989) 21C3–21C4, Japan. [12] J. Bures, S. Lacroix, J. Lapiere, Bragg reflector induced by photosensitivity in an optical fibre: model of growth and frequency response, Appl. Opt. 21 (19) (1982) 3052. [13] G. Meltz, W.W. Morey, W.H. Glenn, Formation of Bragg gratings in optical fibres by transverse holographic method, Opt. Lett. 14 (15) (1989) 823. [14] R. Kashyap, J.R. Armitage, R. Wyatt, S.T. Davey, D.L. Williams, All-fibre narrowband reflection gratings at 1500nm, Electron. Lett. 26 (11) (1990) 730. [15] P. Lemaire, R.M. Atkins, V. Mizrahi, W.A. Reed, High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibres, Electron. Lett. 29 (13) (1993) 1191. [16] R.A. Myers, N. Mukherjee, S.R.J. Brueck, Large second order nonlinearity in poled fused silica, Opt. Lett. 16 (22) (1991) 1732–1734. [17] M.V. Bergot, M.C. Farries, M.E. Fermann, L. Li, L.J. Poyntz-Wright, P.J.St. Russell, A. Smithson, Opt. Lett. 13 (7) (1988) 592–594. [18] R. Kashyap, Phase-matched second-harmonic generation in periodically poled optical fibers, Appl. Phys. Lett. 58 (12) (1991) 1233. [19] R. Kashyap, E. Borgonjen, R.J. Campbell, Continuous wave seeded second-harmonic generation in optical fibres: The enigma of second harmonic generation, Proc. SPIE 2044 (1993) 202–212. [20] T. Fujiwara, D. Wong, S. Fleming, Large electro-optic modulation in a thermally poled germanosilicate fiber, IEEE Photon. Technol. Lett. 7 (10) (1995) 1177–1179. [21] T. Fujiwara, M. Takahashi, A.J. Ikushima, Second harmonic generation in germanosilicate glass poled with ArF laser irradiation, Appl. Phys. Lett. 71 (8) (1997) 1032–1034. [22] P.St.J. Russell, Review article of PCF, 1997. [23] V. Gapontsev, High Power Lasers, 1997. [24] J.R. Taylor, Super-continuum generation, 1997. [25] H.R. Phillips, Silicon dioxide (SiO2) Glass, in: E.D. Palik (Ed.), Handbook of Optical Constants of Solids, Academic Press, London, UK, p. 749. [26] M.E. Lines, Ultra low loss glasses, AT&T Bell Labs. Tech. Memo. TM 11535-85091633TM, 1985. [27] M.J. Yeun, Ultraviolet absorption studies in germanium silicate glasses, Appl. Opt. 21 (1) (1982) 136. [28] D.B. Keck, R.D. Maurer, P.C. Shultz, On the ultimate lower limit of attenuation in glass optical waveguides, Appl. Phys. Lett. 22 (7) (1973) 307–309. [29] See, for example, SPIE 1516, and articles therein.
References
13
[30] T. Georges, E. Delevaque, M. Monerie, P. Lamouler, J.F. Bayon, Pair induced quenching in erbium doped silicate fibers, IEEE. Optical Amplifiers and Their Applications. Technical Digest 17 (1992) 71. [31] F. Ladoucer and J.D. Love, Silica-Based Channel Waveguides and Devices, Chapman & Hall, London, UK, 1997. [32] D.L. Williams, B.J. Ainslie, J.R. Armitage, R. Kashyap, R.J. Campbell, Enhanced UV photosensitivity in boron codoped germanosilicate fibres, Electron. Lett. 29 (1993) 1191. [33] I. Camlibel, D.A. Pinnow, F.W. Dabby, Optical ageing characteristics of borosilicate clad fused silica core fiber optical waveguides, Appl. Phys. Lett. 26 (4) (1992) 1183–1185. [34] N.P. Bansal, R.H. Doremus, Handbook of glass properties, Academic Press, Orlando, 1986. [35] V. Fomin, A. Ferin, M. Abramov, D. Mochalov, V. Sergeev, V. Gaponstev, N. Platonov, Multikilowatt single mode lasers, 4th International Symposium on High-Power Fiber Lasers and Their Applications, St. Petersburg, Russia, paper TuFL-21, 24–26, June 2008. [36] Y. Jeong, J.K. Sahu, D.N. Payne, J. Nilsson, Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power, Opt. Express 12 (25) (2004) 6088–6092. [37] V. Gaponstev, N.S. Platonov, O. Shkurihin, I. Zaitsev, 400 W low-noise single-mode ytterbium fiber laser with an integrated fiber delivery, in: Proc. CLEO 2003, Baltimore, MD, June 1–6, 2003, post-deadline paper CPDB9. [38] P.St.J. Russell, Photonic-crystal fibers, J. Lightwave Technol. 24 (2006) 4729–4749. [39] S.V. Chernikov, Y. Zhu, J.R. Taylor, V.P. Gaponstev, Supercontinuum self-Q-switched ytterbium fiber laser, Opt. Lett. 22 (5) (1997) 298–300. [40] D.Y. Smith, Dispersion theory, sum rules and their application to the analysis of optical data, in: The Handbook of Optical Constants (Chapter 3). Academic Press, New York, 1985. [41] I.H. Malitson, Interspecimen comparison of the refractive index of fused silica, J. Opt. Soc. Am. 15 (10) (1965) 1205–1209. [42] J. Fleming, Dispersion in GeO2-SiO2 glasses, Appl. Opt. 23 (4) (1984). [43] D.P. Hand, P.J.St. Russel, Photoinduced refractive index changes in germanosilicate optical fibers, Opt. Lett. 15 (2) (1990) 102–104. [44] Data on fused quartz, Hareaus-Amersil Inc.
Chapter 2
Photosensitivity and Photosensitization of Optical Fibers We have seen in the last chapter that optical fibers have very good optical properties for light transmission. Electronic absorptions that lead to attenuation are in the deep UV wavelength regime, and the molecular vibrations are far removed from the optical fiber transmission windows of interest to telecommunications. We have briefly considered the possible link between the change in absorption and the effect on the refractive index. Another possibility for the refractive index change is via an electro-optic nonlinearity. However, the symmetry properties of glass prohibit the electro-optic effect [1]. If there is an electro-optic contribution to the changes in the refractive index as a result of exposure to UV radiation, then an internal symmetry would have to be created. This chapter considers aspects of defects connected with photosensitivity and techniques for photosensitization of optical fibers. We briefly compare in Section 2.1 the electro-optic effect [2] and how this may be invoked in glass. This aspect has received considerable attention worldwide and is now discussed in detail in Chapter 12. Section 2.2 introduces some of the defects that are linked to the UV-induced change in refractive index of glass. The hot debate on defects has continued for a number of years and there are a vast number of “subtleties” with regards to the same nominal defect state, as well as pathways to achieving transformations from one state to the other. Some of the defects cannot be detected by optical means and require sophisticated methods. The task is not made easy by the various nomenclature used in labeling, so that unraveling defects is made inaccessible to the layperson. A simple overview of the important defects is given and we point to the literature for a detailed discussion [3,4]. Section 2.3 looks at the evidence of photoexcitation of electrons and, in conjunction with Section 2.2, the methods for the detection of defects. The routes used to photosensitize and fabricate fibers are presented in the last section. The routes used to photosensitize and fabricate fibers are presented in Section 2.7. Chemical composition gratings are discussed in Section 2.8. 15
16
Photosensitivity and Photosensitization of Optical Fibers
2.1 PHOTOREFRACTIVITY AND PHOTOSENSITIVITY It is useful to distinguish the term photorefractivity from photosensitivity and photochromic effect. Photorefractivity refers to the sensitivity of a material property to light, often leading to a phenomenon usually ascribed to crystalline materials that exhibit a second-order nonlinearity by which light radiation can change the refractive index by creating an internal electric field [5]. Photosensitivity invariably refers to a permanent change in refractive index or opacity induced by exposure to light radiation with the internal field playing an insignificant role. The term traditionally applies to the color change in certain glasses with exposure to ultraviolet radiation and heat. Photochromic glass does not depend on the application of heat to change opacity, and the action is reversible. However, a combination of these properties is possible in glasses and is a novel phenomenon, which is currently being studied, not least because it is poorly understood. Considering the normal polarization response of materials to applied electric fields may provide a physical insight into the phenomenon of photorefractivity and poling of glass. The induced polarization, P, in a medium can be described by the relationship D ¼ e0 E þ P;
ð2:1:1Þ
where D is the displacement, E is the applied field, e0 is the free space permittivity, and P is the induced polarization. In a material in which the polarization is nonlinear, the polarization may be expanded in powers of the applied field as P ¼ e0 wð1Þ E þ e0 wð2Þ E 2 þ e0 wð3Þ E 3 þ . . . ¼ e0 wð1Þ E þ wð2Þ E 2 þ wð3Þ E 3 þ . . .
ð2:1:2Þ
and er ¼
D ¼ 1 þ wð1Þ ; e0 E
ð2:1:3Þ
where er ¼ 1 þ w(1) is the linear permittivity, w(2) is the first term of the nonlinear susceptibility (which can be nonzero in crystalline media), and w(3) is the thirdorder nonlinearity (nonzero in all materials). Using Eqs. (2.1.2) and (2.1.3), the perturbed permittivity under the influence of an applied electric field is D ¼ er þ wð2Þ E þ wð3Þ E 2 . . . e0 E ¼ er þ De ¼ e;
ð2:1:4Þ ð2:1:5Þ
and since the refractive index n is related to the permittivity as e ¼ n2 ¼ ðn0 þ DnÞ2 ; n20 þ 2n0 Dn
ð2:1:6Þ
Photorefractivity and Photosensitivity
17
from which immediately follows Dn ¼
1 ð2Þ ½w E þ wð3Þ E 2 . . . 2n0
ð2:1:7Þ
where Dn is the electric field-induced refractive index change. In photorefractive materials with an active w(2), an internal charge can build up due to trapped carriers released from defects. These give rise to an internal field, which modulates the refractive index locally via the first term in Eq. (2.1.7). The induced index changes result directly from the linear electrooptic effect (w(2)) and are in general quite large, 104. However, with w(2) being zero in glass, the induced refractive index with an applied field can only result from the nonzero third-order susceptibility, w(3). Even if an internal field could develop, the refractive index change is small, 107; however, as will be seen, if an internal field is possible in glass, it results in a modest nonlinearity [2]. We now assume the existence of an internal field Edc and apply an external field Eapplied. The induced index change is as follows: Dn ¼
1 ð3Þ w ðEdc þ Eapplied Þ2 2n0
2 2 þ 2Edc Eapplied þ Eapplied Þ: ¼ n02 ðEdc
ð2:1:8Þ ð2:1:9Þ
The first term in Eq. (2.1.9) indicates a permanent index change, whereas the third term is the usual quadratic nonlinear effect known as the dc-Kerr effect. We have used a prime on the n02 , to distinguish it from the optical Kerr constant ð2Þ n2 ¼ w2n0 . The interesting relationship is described by the remaining term, Dn ¼ 2n02 Edc Eapplied :
ð2:1:10Þ
This relationship is analogous to the linear electro-optic effect, in which the applied field operates on an enhanced nonlinearity, 2n02 Edc, due to the frozen internal field. If the internal field is large, then a useful nonlinearity is possible. This effect is believed to be partly the basis of poled glass [2]. In crystalline media with a large photorefractive response, the nonlinearity w(2) is several orders of magnitude larger than the next higher order coefficient, w(3) (and hence n02 ) in glass. From the first term in Eq. (2.1.9) we can calculate the required field for a change in the refractive index of 103. With a measured value of w(3) 1022 m2 V2 for silica, a large internal field of 109 V/m would be necessary, equivalent to n02 of 1 pm V1. These values have been exceeded in UV photoelectrically poled fiber, with the highest reported result of 6 pm/V [6]! Combined with the low dielectric constant of silica, it has a potentially large bandwidth for electro-optic modulation. Just how such a large field may develop has been debated. However, it has been suggested by Myers et al. [7,8] that the poling voltage is dropped across a thin layer (5 mm) within the glass, causing huge fields to appear.
18
Photosensitivity and Photosensitization of Optical Fibers
The electro-optic nature of UV photoinduced refractive index in Bragg gratings has not been reported, although the presence charges related to defects could indeed develop an internal field, as in the case of second-harmonic generation in glass [9]. In the next section, we consider some of the important defects, which are of interest in unraveling the mystery of photosensitivity of glass.
2.2 DEFECTS IN GLASS The nature of fabrication of glass is ideally suited to promoting defects. The chemical reactions that take place in a modified chemical vapor deposition (MCVD) [10] process are based on hot gases reacting to form a soot deposit on the inside of a silica support tube or on the outside in outside vapor phase deposition (OVD). The process allows the ratio of reactive gases such as silicon/germanium tetrachloride and oxygen to be easily changed to arrive at a nearly complete chemical reaction, depositing a mixture of germanium and silicon dioxides. It is not possible to have a 100% reaction, so the deposited chemicals have a proportion of suboxides and defects within the glass matrix. With sintering and preform collapse, these reaction components remain, although further alterations may take place while the fiber is being drawn, when bonds can break [11–13]. The end result is a material that is highly inhomogeneous on a microscopic scale with little or no order beyond the range of a few molecular distances. The fabrication process also allows other higher-order ring structures [14] to form, complicating the picture yet further. There is a possibility of incorporating not only a strained structure, but also one which has randomly distributed broken bonds and trapped defects. This is especially true of a fiber with the core dopant germanium, which readily forms suboxides as GeOx (x ¼ 1 to 2), creating a range of defects in the tetrahedral matrix of the silica host glass. Given this rich environment of imperfection, it is surprising that state-of-the-art germania-doped silica fiber has extremely good properties – low loss and high optical damage threshold – and is a result of better understanding of defects, which lead to increased attenuation in the transmission windows of interest. Among the well-known defects formed in the germania-doped silica core are the paramagnetic Ge(n) defects, where n refers to the number of next-nearestneighbor Ge/Si atoms surrounding a germanium ion with an associated unsatisfied single electron, first pointed out by Friebele et al. [17]. These defects are shown schematically in Fig. 2.1. The Ge(1) and Ge(2) have been identified as trapped-electron centers [18]. The GeE0 , previously known as the Ge(0) and the Ge(3) centers, which is common in oxygen-deficient germania, is a hole
Defects in Glass
19 Ge Si
Ge Si
Si
Ge Ge
Ge
Si
Si
Si
Si
Si
Si
Peroxy radical Ge(2)
Ge(1)
Ge
O O
Ge
Ge Ge Ge or Si
or Si Ge Ge
Ge Ge
NBOHC
Oxygen-deficient Ge (divalent bonding)
Figure 2.1 A schematic of proposed Ge (or Si) defects of germania-doped silica. The characteristic absorption of the Ge(1) is 280 nm (4.4 eV) [18] and is a trapped electron at a Ge (or Si) site; Ge(2) has an absorption at 213 nm (5.8 eV) and is a hole center. The peroxy radical has an absorption at 7.6 eV (163 nm) and at 325 nm (3.8 eV) [15,16]. NBOHC absorbs at 630 nm.
trapped next to a germanium at an oxygen vacancy [19] and has been shown to be independent of the number of next-neighbor Ge sites. Here an oxygen atom is missing from the tetrahedron, while the germania atom has an extra electron as a dangling bond. The extra electron distorts the molecule of germania as shown in Fig. 2.2. The GeO defect, shown in Fig. 2.2 (LHS), has a germanium atom coordinated with another Si or Ge atom. This bond has the characteristic 240-nm absorption peak that is observed in many germanium-doped photosensitive optical fibers [21]. On UV illumination, the bond readily breaks, creating the GeE0 center. It is thought that the electron from the GeE0 center is liberated and is free to move within the glass matrix via hopping or tunneling, or by two-photon excitation into the conduction band [22–24]. This electron can be retrapped at the original site or at some other defect site. The removal of this electron, it is believed, causes a reconfiguration of the shape of the molecule (see Fig. 2.2), possibly also changing the density of the material, as well as the absorption. It appears that the Ge(1) center is the equivalent of the germanium defects observed in a-quartz, known as the Ge(I) and Ge(II), but less well defined [23]. The differences between the absorptions of the Ge(1) and the Ge(2) defects have been discussed by Poumellec and Niay [24*].
Photosensitivity and Photosensitization of Optical Fibers
20
Ge/Si
Ge/Si
Absorption of photon
+
hν Ge
Spontaneous recombination
GeO defect
Ge
Ge(3) or GeE⬘ hole center
Figure 2.2 The GeO defect of germania-doped silica, in which the atom adjacent to germanium is either a silicon or another germanium. It can absorb a photon to form a GeE0 defect. The Ge(0) or Ge(3) is a GeE0 center [20]. The GeE0 defect shows the extra electron (associated with the Ge atom), which may be free to move within the glass matrix until it is retrapped at the original defect site, at another GeE0 hole site, or at any one of the Ge(n) defect centers.
Phosphorus forms a series of defects similar to those of germanium. However, the photosensitivity is limited at 240 nm and requires shorter wavelengths, such as 193-nm radiation [24]. Other defects include the nonbridging oxygen hole center (NBOHC), which is claimed to have absorptions at 260 and 630 nm, and the peroxy radical (P-OHC) [25], believed to absorb at 260 nm. Both are shown in Fig. 2.1.
2.3 DETECTION OF DEFECTS A considerable amount of work has been done in understanding defects in glass. Detection of defects may be broadly categorized into four groups: optically active defects can be observed because of their excitation spectrum or excitation and luminescence/fluorescence spectrum while optically inactive defects are detectable by their electron spin resonance signature, or ESR spectrum, together with optical emission spectrum. The model of the defects as shown in Fig. 2.2 suggests the liberation of electrons on absorption of UV radiation. It should therefore be possible to detect liberated charges experimentally; since silica has a high volume resistivity, it is necessary to choose a geometry that can directly enable the measurement of electric currents. Photosensitivity has been explored both indirectly, e.g., by etching glass exposed to radiation or using second-harmonic generation [9,26,27] as a probe, and directly, e.g., by measurement of photocurrent and electron trapping in germania-doped planar waveguides [28] and across thin films of bulk glass [29].
Photosensitization Techniques
21
It has been concluded that the photocurrent is influenced by the fluence of the exciting UV radiation; the photocurrent (probably by tunneling [29]) is a linear function of the power density for CW excitation [28], while for pulsed, high-intensity radiation, it takes on a two-photon excitation characteristic [29]. The paramagnetic defects of the Ge(n) type including the E0 center are detected by ESR. The GeE0 has an associated optical absorption at 4.6 eV [30].
2.4 PHOTOSENSITIZATION TECHNIQUES A question often asked is: Which is the best fiber to use for the fabrication of most gratings? Undoubtedly, the preferred answer to this question should be standard telecommunications fiber. Although techniques have been found to write strong gratings in this type of fiber, there are several reasons why standard fiber is not the best choice for a number of applications. Ideally, a compatibility with standard fiber is desirable, but the design of different devices requires a variety of fibers. This does open the possibility of exploiting various techniques for fabrication and sensitization. Here we look in some detail at the behavior of commonly used species in optical fiber and present their properties, which may influence the type of application. For example, the time or intensity of UV exposure required for the writing of gratings affects the transmission and reliability properties. This results in either damage (Type II gratings) [31] or the formation of Type I, at low fluence, and Type IIA gratings [32], each of which have different characteristics (see Section 2.4.1). The use of boron and tin as a codopant in germanosilicate fibers, hot hydrogenation and cold, high-pressure hydrogenation, and flame-assisted low-pressure hydrogenation (“flame-brushing”) are well-established photosensitization methods. The type of the fiber often dictates what type of grating may be fabricated, since the outcome depends on the dopants. The literature available on the subjects of photosensitivity, the complex nature of defects, and the dynamics of growth of gratings is vast [34]. The sheer numbers of different fibers available worldwide, further complicates the picture and by the very nature of the limited fiber set available within the framework of a given study and the complex nature of glass, comparisons have been extremely difficult to interpret. This is not a criticism of the research in this field, merely a statement reiterating the dilemma facing researchers: how to deal with far too many variables! In order to draw conclusions from the available data, one can simply suggest a trend for the user to follow. A choice may be made from the set of commonly available fibers. For a certain set of these fibers (e.g., standard telecommunications fiber) the method for photosensitization may be simply hydrogenation, or 193-nm exposure. It is often the availability of the laser source that dictates the approach.
Photosensitivity and Photosensitization of Optical Fibers
22
2.4.1 Germanium-Doped Silica Fibers Photosensitivity of optical fibers has been correlated with the concentration of GeO defects in the core [33,34]. The presence of the defect is indicated by the absorption at 240 nm, first observed by Cohen and Smith [35] and attributed to the reduced germania state, Ge(II). The number of these defects generally increases as a function of Ge concentration. Figure 2.3 shows the absorption at 242 nm in a preform with the germanium concentration [36]. The slope in this graph is 28 dB/(mm-mol%) of Ge before the preform sample is collapsed (dashed line). After collapse, the number of defects increases, and the corresponding absorption changes to 36 dB/(mm-mol%) (Fig. 2.3 continuous line). Increasing the concentration of defects increases the photosensitivity of the fiber. This can be done by collapsing the fiber in a reducing atmosphere, for example, by replacing oxygen with nitrogen or helium [36] or with hydrogen [37,49]. The 240-nm absorption peak is due to the oxygen-deficient hole center defect, (Ge-ODC) [38] and indicates the intrinsic photosensitivity. It can be quantified as [39] ð2:4:1Þ
k ¼ a242 nm =C;
where a242 nm is the absorption at 242 nm and C is the molar concentration of GeO2. Normally C lies between 10 and 40 dB/(mm-mol% GeO2). Hot hydrogenation is performed on fibers or preforms at a temperature of 650 C for 200 hours in 1 atm hydrogen [40]. The absorption at 240 nm closely follows the profile of the Ge concentration in the fiber [33], and k has been estimated to be large, 120 dB/(mm-mol% GeO2). The saturated UV-induced index change increases approximately linearly with Ge concentration after exposure to UV radiation, from 3 105 (3 mol% GeO%2) for standard fiber to 2.5 104 (20 mol% GeO2)
242 nm loss (dB/mm)
500 After collapse
400 300 200 100
Before collapse
0 0
5 10 Ge concentration (mol%)
15
Figure 2.3 Absorption at 242 nm in preform samples before and after collapse as a function of Ge concentration (after Ref. [36]).
Photosensitization Techniques
23
concentration, using a CW laser source operating at 244 nm [92]. However, the picture is more complex than the observations based simply on the use of CW lasers. With pulsed laser sources, high-germania-doped fiber (8%) shows an initial growth rate of the UV-induced refractive index change, which is proportional to the energy density of the pulse. For low germania content, as in standard telecommunications fiber, it is proportional to the square of the energy density. Thus, two-photon absorption from 193 nm plays a crucial role in inducing maximum refractive index changes as high as 0.001 in standard optical fibers [41]. Another, more complex phenomenon occurs in untreated germania fibers with long exposure time, in conjunction with both CW and pulsed radiation, readily observable in high germania content fibers [47]. In high-germania fiber, long exposure erases the initial first-order grating completely, while a second-order grating forms. This erasure of the first-order and the onset of second-order gratings forms a demarcation between Type I and Type IIA gratings. Increasing the energy density damages the fiber core, forming Type II gratings [31]. The thermal history of the fiber is also of great importance, as is the mechanical strain during the time of grating inscription. Significantly, even strains as low as 0.2% can increase the peak refractive index modulation of the Type IIA grating in high germanium content fiber [42,43]. High-germaniadoped (30%Ge) fibers drawn under high pulling tension show the opposite behavior [44], indicating the influence of elastic stress during drawing rather than the effect of drawing-induced defects [45]. Annealing the fiber at 1100 C for 1 hour and then cooling over 2 days reduces the time for the erasure of the Type I grating, as well as increasing the maximum refractive index modulation achievable in the Type IIA regime. With tin as a codopant in highgermanium fiber, the general overall picture changes slightly, but the dynamics are similar, except for reduced index change under strained inscription [46]. Thus, absolute comparison is difficult, and one may use the germania content as an indicator, bearing in mind the complex nature of the dynamics of grating formation in germania-doped silica fiber. Typical results for a high-germania fiber are shown in Fig. 2.4. The growth of the refractive index modulation as a function of time stops in the case of all three fibers shown, dropping to zero before increasing once again to form Type IIA gratings. Photosensitivity of fiber fabricated under reduced conditions as a function Ge concentration also increases, but it is not sufficient to interpret the data by the maximum index change. The reason for this is the induction of Type IIA gratings [47] in relatively low concentration of Ge. Measurements performed under pulsed conditions reveal that the onset of the Type IIA grating is almost certainly always possible in any concentration of Ge; only the time of observation increases with low concentrations, although for practical purposes this time
Photosensitivity and Photosensitization of Optical Fibers
24 0.002
Ge-Sn doped fiber L=2.5mm; lB=1535nm; I=26W/cm2
Amplitude of index modulation
B After anneal
C Strained fiber ΔL/L=2.10-3
0.001
A Before anneal 0.000
0
25
50 75 Exposure time (min)
100
Figure 2.4 The growth dynamics of the refractive index change in 20 mol% Ge: 1 mol% Sn. The three data are for A: pristine fiber, B: after annealing, and C: under strain of 0.2%. The Type IIA grating begins after the initial erasure (from: Douay M., Xie W.X., Taunay T., Bernage P., Niay P., Cordier P., Poumellec B., Dong L., Bayon J.F., Poignant H., and Delevaque E., “Densification involved in the UV based photosensitivity of silica glasses and optical fibers,” J. Lightwave Technol. 15(8), 1329–1342, 1997. # IEEE 1997).
may be too long to be of concern. Figure 2.5 shows data from the growth of the average index on UV exposure as a function of Ge concentration in fibers, which have been reduced. The maximum index should change monotonically; however, above a certain concentration, the onset of Type IIA reduces the observed maximum index change for point B (20 mol% Ge), since the grating being written slowly disappears before growing again. While the maximum reflectivity should increase to higher levels, within the time frame of the measurements this fiber appears to be less sensitive. A better indicator is the initial
Photosensitization Techniques
25 A
B
8.0E–04
1.2E–05
6.0E–04
8.0E–06
4.0E–04 4.0E–06
2.0E–04 0.0E+00
Initial dn/dt
Max dn (average)
1.0E–03
0.0E+00 3
8
13
18
23
Ge concentration (mol%) Figure 2.5 Concentration dependence of the maximum-index and its initial growth rate as a function of germania concentration in oxygen-deficient fibers. The two isolated points refer to unreduced samples (interpreted from Ref. [48]).
growth rate of the index change, since Type IIA grating is not observed for some time into the measurements. Figure 2.5 shows an approximately linear increase in the rate of growth of the UV-induced average refractive index. The data have been interpreted from Ref. [48], bearing in mind that for the initial growth rate in low germanium content fibers, there is a time delay before the grating begins to grow. Figure 2.6 shows the actual growth of the transmission dip (equivalent to the increase in reflection) for several reduced germania fibers [48]. Note in
30
20 mol% oxyg.-def. 11 mol% oxyg.-def. 9 mol% oxyg.-def. 11 mol% 3.5 mol% oxyg.-def.
Transmission dip [dB]
25 20 15 10 5 0 0
2000
4000 Total fluence
6000
8000
10000
[J/cm2]
Figure 2.6 Growth of the transmission dip with fluence for different types of reduced germania fibers. For the 20 mol% germania fiber, a reduction in the reflectivity is probably due to Type IIA grating formation (from Ref. [48]).
26
Photosensitivity and Photosensitization of Optical Fibers
particular the change in the transmission due to the onset of Type IIA grating. At this point, the Bragg wavelength shift is reduced [47], making the maximum average index measurement difficult. Measurement of the shift in the Bragg wavelength is a reasonable indicator for the UV induced index change for a fiber well below the start of saturation effects. With saturation, care needs to be taken, since the bandwidth of the grating increases, making it more difficult to accurately measure the wavelength shift. The ac index change should be calculated from the bandwidth and the reflectivity data along with the Bragg wavelength shift to accurately gauge the overall ac and dc components of the index change (see Chapters 4 and 9). The growth rate and the maximum index change are of interest if strong gratings are to be fabricated in a short time frame. This suggests that reduced germania is better than normal fiber on both counts. However, the maximum index change is still lower than required for a number of applications and the time of fabrication excessive. The use of hot hydrogen to reduce germania has the additional effect of increasing loss near 1390 nm due to the formation of hydroxyl species [49,50]. The absorption loss at 1390 nm is estimated to be 0.66, 0.5, and 0.25 dB/(m-mol%) at 1390,1500, and 1550 nm, respectively [40]. One major advantage of fibers that have been reduced is that they are rendered permanently photosensitive and require the minimum of processing, compared with hydrogenated fibers (see following sections). The incorporation of 0.1% nitrogen in germanium-doped silica fiber by the surface plasma assisted chemical vapor deposition (SPCVD) process [51] has been shown to have a high photosensitivity [52]. The effect on the 240-nm absorption is dramatic, raising it to 100 dB/mm/mol% GeO2, doubling it compared to the equivalent for germanium doping alone. The induced refractive index changes are reported to be large (2.8 103) and much larger (0.01) with cold hydrogen soaking of 7 mol%Ge;0 mol%N fiber. The Type IIA threshold is reported to increase by a factor of 6 over that in nitrogen-free, 20 mol% Ge fibers. However, there is evidence of increase in the absorption loss in the 1500-nm window with the addition of nitrogen. The next most photosensitive fibers are the germania–boron or tin-doped fibers.
2.4.2 Germanium–Boron Codoped Silicate Fibers The use of boron in soda lime and silicate glass has been known for a long time [53]. It has also been established that boron, when added to germaniadoped silicate glass, reduces the refractive index. The transformational changes that occur depend on the thermal history and processing of the glass. As such, it is generally used in the cladding of optical fibers, since the core region must
Photosensitization Techniques
27
remain at a higher refractive index. Compared to fluorine, the other commonly used element in the cladding (in conjunction with phosphorus), the refractive index modification is generally at least an order of magnitude larger, since more of the element can be incorporated in the glass. Thus, while the maximum index difference from fluorine can be approximately 103 with boron, the index change can be >j0.01j. This opens up many possibilities for the fabrication of novel structures, not least as a component to allow the incorporation of even more germania into glass while keeping a low refractive index difference between the cladding and core when both are incorporated into the core. One advantage of such a composition is the fabrication of a fiber that is outwardly identical in terms of refractive index profile and core-to-cladding refractive index difference with standard single-mode optical fibers, and yet contains many times the quantity of germania in the core. The obvious advantage is the increased photosensitivity of such a fiber with the increased germania. Indeed, this is the case with boron–germanium (B-Ge) codoped fused silica fiber [54]. The typical profile of a B-Ge preform is shown in Fig. 2.7. The raised refractive index dashed line shows the contribution due to the germanium concentration, while the negative refractive contribution is due to the boron, resulting in the continuous line positive refractive index profile. It should be noted that with boron and germanium, it is possible to selectively place a photosensitive region anywhere in the fiber, without altering the wave guiding properties. Other types of profiles possible are boron with highly doped germanium in a cladding matched to silica for liquid cored fibers [55], in-cladding gratings for lasers [56], and special fiber for side-tap filters and long-period gratings [57,58]. B-Ge codoped fiber is fabricated using MCVD techniques and a standard phosphorus–fluorine cladding matched silica tube with normal oxidizing
0.03 0.02
Ge Preform profile
Δn
0.01 0
Radial position
–0.01 –0.02
B
–0.03 Figure 2.7 The refractive index components due to germanium and boron (dashed and dotted lines) contributing to the resultant preform profile (continuous line) [54].
28
Photosensitivity and Photosensitization of Optical Fibers
conditions. The reactive precursor vapors are SiCl4, BCl3, and GeCl4, with oxygen as a carrier for the core deposition. For a composition equivalent to 16 mol% germanium, the photosensitivity in comparison with 20 mol% unreduced germanium fiber shows an improvement >3-fold in the UV-induced refractive index modulation as well as an order of magnitude reduction in the writing time. With respect to 10 mol% reduced germanium fiber, the improvement in the maximum refractive index modulation is 40% with a 6 reduction in the writing time. The maximum refractive index change is close to 103 for this fiber induced with a CW laser operating at 244 nm [54]. A point worth noting with B-Ge fibers is the increased stress, and consequently, increased induced birefringence [59]. The preforms are difficult to handle because of the high stress. However, the real advantages with B-Ge fibers are the shortened writing time, the larger UV-induced refractive index change, and, potentially, fibers that are compatible with any required profile, for small-core large NA fiber amplifiers, to standard fibers. B-Ge fibers form Type IIA gratings [60] with a CW 244-nm laser, as is the case with the data shown in Fig. 2.4. This suggests that there is probably little difference due to the presence of boron; only the high germanium content is responsible for this type of grating. There is a possibility that stress is a contributing factor to the formation of Type IIA [61]; recent work does partially indicate this but for germanium-doped fibers [44]. Typically, gratings written with CW lasers in B-Ge fiber decay more rapidly than low germanium doped (5 mol%) fibers when exposed to heat. Gratings lose half their index modulation when annealed at 400 C (B-Ge: 22:6.3 mol%) and 650 C (Ge 5 mol%) [46] for 30 minutes. A detailed study of the decay of gratings written in B-Ge may be found in Ref. [62]. The thermal annealing of gratings is discussed in Chapter 9. Boron causes additional loss in the 1550-nm window, of the order of 0.1 dB/m, which may not be desirable. For short gratings, this need not be of concern.
2.4.3 Tin–Germanium Codoped Fibers Fabrication of Sn codoped Ge is by the MCVD process used for silica fiber by incorporating SnCl4 vapor. SnO2 increases the refractive index of optical fibers and, used in conjunction with GeO2, cannot be used as B2O3 to match the cladding refractive index, or to enhance the quantity of germanium in the core affecting the waveguide properties. However, it has three advantages over B-Ge fiber: The gratings survive a higher temperature, do not cause additional loss in the 1500-nm window, have a slightly increased UV-induced refractive index change, and are reported to be 3 times larger than that of B-Ge fibers. Compared with B-Ge, Sn-Ge fibers lose half the UV-induced refractive index change at 600 C, similarly to standard fibers [63].
Photosensitization Techniques
29
2.4.4 Cold, High-Pressure Hydrogenation The presence of molecular hydrogen has been shown to increase the absorption loss in optical fibers over a period of time [64]. The field was studied extensively [65], and it is known that the hydrogen reacts with oxygen to form hydroxyl ions. The increase in the absorption at the first overtone of the OH vibration at a wavelength of 1.27 mm was clearly manifest by the broadband increase in loss in both the 1300-nm and, to a lesser extent, in the 1500-nm windows. Another effect of hydrogen is the reaction with the Ge ion to form GeH, considerably changing the band structure in the UV region. These changes, in turn, influence the local refractive index as per the Kramers–Kronig model. The reaction rates have been shown to be strongly temperature dependent [65]. It has been suggested that the chemical reactions are different on heat treatment and cause the formation of a different species compared to illumination with UV radiation. However, no noticeable increase in the 240-nm band is observed with the presence of interstitial molecular hydrogen in Ge-doped silica. The highest refractive index change induced by UV radiation is undoubtedly in cold hydrogen-soaked germania fibers. As has been seen, an atmosphere of hot hydrogen during the collapse process or hot hydrogen soaking of fibers enhances the GeO defect concentration [37]. The presence of molecular hydrogen has been known to induce increases in the absorption loss of optical fibers, since the early days of optical fibers [50]. Apart from being a nuisance in submarine systems, in which hydrogen seeps into the fiber, causing a loss that increases with time of exposure, cold high pressure hydrogen soaking has led to germanium-doped fibers with the highest observed photosensitivity [66]. Any germania-doped fiber may be made photosensitive by soaking it under high-pressure (800 bar) and/or high temperature (<150 C). Molecular hydrogen in-diffuses to an equilibrium state. The process requires a suitable high-pressure chamber into which fibers may be left for hydrogen loading. Once the fiber is loaded, exposure to UV radiation is thought to lead to a dissociation of the molecule, leading to the formation of Si–OH and/or Ge–OH bonds. Along with this, there is formation of the Ge oxygen-deficient centers, leading to a refractive index change. Soaking the fiber at 200 bar at room temperature for 2 weeks is sufficient to load the 125-micron diameter fiber at 21 C [66]. UV exposure of hydrogen-loaded standard single-mode fibers easily yields refractive index changes in excess of 0.011 [67] in standard telecommunications fiber, with a highest value of 0.03 inferred [68]. Almost all Ge atoms are involved in the reactions giving rise to the index changes. Figure 2.8 shows the changes in the refractive index profile of a standard fiber before and after exposure to pulsed UV radiation at 248 nm (600 mJ/cm2, 20 Hz, 60-minute exposure) [68]. The growth of gratings is slow with CW lasers (duration of 20 minutes for strong gratings with refractive index changes of 1–2 103). The picture is quite different with the growth kinetics when compared with non-hydrogen-loaded
Photosensitivity and Photosensitization of Optical Fibers
30 0.02
Refractive index
Untreated UV exposed
0.01
ΔnUV=0.011
0.00 –30
–20
–10
0
10
20
30
Fiber radius (mm) Figure 2.8 The refractive index profile of a 2.8% hydrogen-soaked standard fiber, before and after UV exposure with pulsed radiation at 248 nm (courtesy P. Lemaire from: Lemaire P.J., Vengsarkar A.M., Reed W.A., and Mizarhi V., “Refractive index changes in optical fibers sensitized with molecular hydrogen,” in Technical Digest of Conf. on Opt. Fiber Commun., OFC’94, pp. 47–48, 1994 [68]).
germania fibers. To date, Type IIA gratings have not been observed in hydrogenloaded fibers. There is also no clear evidence of the stress dependence of grating growth [44]. Whereas in Type IIA the average UV-induced refractive index change is negative, in hydrogen-loaded fibers the average refractive index grows unbounded to large values (>0.01). Heating a hydrogen-loaded fiber increases the refractive index rapidly, even in P2O5 and P2O5:Al2O3-doped multimode fibers [68], although pure silica is not sensitized. The dynamic changes that occur in the process of fiber grating fabrication are complex. Even with hydrogen-loaded fibers, there are indications that as the grating grows, the absorption in the core increases in the UV, as does the 400-nm luminescence [69]. Martin et al. [69] have found a direct correlation between refractive index change increase and luminescence. Further studies have shown a complex behavior of the dynamics of the luminescence in both hydrogen and non-hydrogen loaded fibers. Poumellec et al. [69*] have shown that the luminescence has a temporal signature. UV excitation has no effect initially, but probed a period of time after the exposure, the luminescence begins to grow once again. In non-hydrogen loaded fibers, it could be speculated that the regrowth is related to Type IIA grating formation. Figure 2.9 shows the transmission spectra of two gratings in hydrogenated standard fiber at different stages of growth, with the UV radiation at 244 nm CW switched on and off. With the UV switched on, the Bragg wavelength shifts 0.05 nm to longer wavelengths at a grating reflectivity of 1.4 dB. When the
Photosensitization Techniques
Transmission, dB
1520 0
1525
31 1530
1535
1540
1545
–5 –10 UV ON –15 –20
UV OFF Wavelength, nm
Figure 2.9 Shift in the Bragg wavelength as the UV radiation is switched on and off for two different strength gratings (after Ref. [70]).
grating has grown to 17 dB (different grating but same fiber), the shift is 1 nm, equivalent to an equilibrium temperature increase of the fiber of 80 C. At the start of grating growth (<1 dB), the shift is not noticeable. The formation of OH species with UV exposure increases the loss in the 1500nm window. There are two peaks associated with the formation of Si–OH (1.39 mm) and Ge–OH (1.42 mm) on UV exposure [71]. A concentration of 1 mol% of hydroxyl species increases the loss at 1.4 mm by 5 dB. This is avoided by soaking the fiber in deuterium, which shifts the first overtone OD of the water peak to 1.9 mm [74]. Another feature of hydrogen loading is the increased loss at wavelengths less than 1 mm after UV exposure of hydrogen-loaded fibers. The loss has a wavelength dependence at <0.95 mm of e4.6/l, where l is in microns [74].
Hydrogen Loading of Optical Fibers The loading of optical fibers with hydrogen is both temperature and pressure dependent. The diffusion coefficient of hydrogen is [50] D ¼ 2:83 104 eð40:19 kJ=molÞ=RT ; cm2 =s;
ð2:4:2Þ
where R ¼ 8.311 J/(K-mol) and T is the temperature in degrees Kelvin. The concentration of hydrogen in the fiber is calculated by solving the diffusion equation [72] as a function of the normalized radial position r and time t: ! nX ¼1 m 2 ð1 J0 ðmn rÞ n exp D t ’ðrÞrJ0 ðmn rÞdr; Cðr; tÞ ¼ 2C0 R J12 ðmn Þ n¼1 0 ! ð2:4:3Þ nX ¼1 m 2 J0 ðmn rÞ ¼ 2C0 exp D n t R m J ðm Þ n¼1 n 1 n where mn are the nth zeroes of the J0 Bessel function, D is the diffusion coefficient, and R is the radius of the fiber. C0 is the initial concentration, and ’(r) ¼ 1 if hydrogen is diffusing into the fiber and 1 for out-diffusion. The dynamics of
Photosensitivity and Photosensitization of Optical Fibers
32
Normalized hydrogen concentration
1
400h 300h 200h
0.75 0.5
100h 0.25 50h 0 12.5
0
25
37.5
50
62.5
Radial position (microns)
Normalized hydrogen concentration
Figure 2.10 In-diffusion profile of hydrogen at 200 bar pressure in a 125-mm diameter fiber as a function of time at 23 C (courtesy F. Bhakti, Ref. [73]).
1 10h
0.75 50h 100h
0.5 150h 200h
0.25
240h 0 0
12.5
25 37.5 Radial position (microns)
50
62.5
Figure 2.11 Out-diffusion profile of hydrogen in a 125-mm diameter fiber at 23 C (courtesy F. Bhakti, Ref. [73]).
the concentration have been modeled by Bhakti et al. [73] to study the effect on the drift of the resonance wavelength of a long-period grating. Under an ambient pressure of 200 bar, the in-diffusion of hydrogen is shown in Fig. 2.10 as a function of the radius of the radial position for different times. The out-diffusion of hydrogen is shown for different times as the fiber is removed from the highpressure chamber [74] in Fig. 2.11. The first overtone hydroxyl peak at 1.245 mm can be used to verify the concentration of hydrogen in the core [74,75]; the absorption at 1.245 mm is 3 dB/(mol% H2) (equivalent to 104 ppm in a mole of silica). The wavelength shifts of long-period gratings (LPGs, see Chapter 4) resonances can be in excess of 150 nm to the long wavelength as the hydrogen diffuses out of the core after fabrication of the grating, before returning to the original wavelength. Note that the resonance of the LPG is only dependent on
Photosensitization Techniques
33
the difference in the core and cladding refractive indexes, since it is the relative difference that is important (see Chapter 4). The effect on the refractive index of the fiber as the hydrogen out-diffuses is a complicated process, as shown by Malo et al. [76] using Bragg gratings. Once the fibers are removed from the chamber, the hydrogen begins to diffuse out but is fixed in the core by UV irradiation during grating fabrication. Depleted in the core, hydrogen in the cladding diffuses in before diffusing out. The stress changes the molecular polarizability of hydrogen [77], as well as the Bragg wavelength, first toward long, and then toward short wavelengths. The drift in the wavelength is found to be 0.72 nm for an initial pressure of 100 bar. The Bragg wavelength is sensitive to the net refractive index of the core, not to the difference between the core and the cladding as for the LPG. The dynamics of the coupling between the modes in LPGs have also been reported [78]. To prevent the fiber from out-gassing prematurely, it should be stored at low temperatures (70 C) until it is used. The diffusion time taken to reduce the hydrogen initial concentration, C0/e, in the core is shown in Fig. 2.12a as a function of fiber diameter, at room temperature (20 C). Also shown is the diffusion time for standard fiber as a function of temperature [Fig. 2.12b].
Diffusion time (hrs)
300 250 200 150 100 60
70
A
80 90 Fiber diameter (microns)
100
110
Diffusion time (hrs)
600
400
200
0 0
B
10
20 30 Temperature (C)
40
50
Figure 2.12 Out-diffusion time (1/e of initial concentration, C0 of hydrogen as a function of fiber diameter at 20 C (a), and the diffusion time for a standard fiber as a function of temperature (b) (courtesy F. Bhakti).
Photosensitivity and Photosensitization of Optical Fibers
34
2.4.5 Rare-Earth-Doped Fibers For a vast number of applications, such as fiber lasers and amplifiers, it is necessary to fabricate gratings in rare-earth-doped fibers. It is more difficult to write Bragg gratings in these fibers than in standard fibers. Worse, germanium is replaced by aluminum (Al2O3) to reduce the effect of quenching and lifetime shortening [79]. The lack of germanium reduces the photosensitivity of optical fibers, even with hydrogen sensitization, although gratings have been reported [46]. Gratings can be formed in most fibers, but the index changes remain weak (<104) in all cases with 240-nm irradiation except in hydrogenloaded Al/Ce or Al/Tb. With 193-nm irradiation, Al/Yb/Er fiber has shown index changes of 104 while hydrogen loading increases this figure to 103. The conclusion is that hydrogen loading improves the photosensitivity of germanium-free rare-earth-doped fibers. However, only a small subset of Er, Nd, and Ce and Tb-doped silicate fibers show reasonable photosensitivity (>104) [46]. The direct writing of gratings in RE fibers is limited to longish lengths, which suits the fabrication of narrow-band DFB fiber lasers. All gratings are of Type I or Type II; Type IIA has not been reported. A summary of the results on the most photosensitive fibers is listed in Table 2.1.
2.5 DENSIFICATION AND STRESS IN FIBERS There is increasing evidence that densification and stress increase in optical fibers contributes to the change in the refractive index [84–87]. Surface AFM scans of unetched samples show that the surface densifies in the UV illuminated regions [100]. In Fig. 2.13 is shown an atomic force microscope (AFM) scan of Table 2.1 League table of rare-earth-doped silica photosensitive fiber Core dopant with Al, Ge-free Undoped reference sample Eu2þ Ce3þ Yb3þ and Er3þ P and Ce3þ Ce3þ and H2 Tb3þ and H2 Er3þ and H2 Tm3þ and H2 Yb3þ: Er3þ and H2
UV source: pulsed (nm) 193 248 265 193 266 240 240 235 235 193
Dn (pk–pk) 5
5 10 2.5 105 3.7 104 104 1.4 104 1.5 103 6 104 5 105 8 105 5 104
Reference 46 80 43, 81 46 82 43 83 43 43 43
Summary of Photosensitive Mechanisms in Germanosilicate Fibers
35
mm 0.30 0.15 0.00 12 12 8 mm
8 mm 4
4
0 0 Experimental conditions : • Grating inscription: 8000 pulses at a fluence per pulse of 185 mJ/cm2, grating pitch = 0.453 mm, R ≈ Rmax. •• A.F.M. was performed after etching the fiber for 110 mn using buffered solution BOE mixed with 25% of HF. BOE is (3 vol. of [NH4F] 40%, 1 vol. of [HF] 49% diluted by 50% with saturated citric acid). Figure 2.13 AFM scan of a D-fiber surface. Surface etched after grating inscription to reveal the chemically modified structure (courtesy Marc Douay, Ref. [88].
the surface of an Andrew Corp. D-fiber. After a grating has been written, the fiber is etched in buffered solution (3 vol% NH4F 40%, 1 vol of HF 49% diluted with 50% saturated citric acid) with HF 25% for 110 minutes. The revealed pattern is indicative of structural modification in the glass, which influences the etch rate [88]. Stress measurement made optically show that the tensile stress increases in the core, reducing the induced refractive index change by as much as 30% [101] (see Chapter 9).
2.6 SUMMARY OF PHOTOSENSITIVE MECHANISMS IN GERMANOSILICATE FIBERS There appear to be three routes by which a photo-induced refractive index change occurs in germanosilicate optical fibers:
36
Photosensitivity and Photosensitization of Optical Fibers
1. Through the formation of color centers (GeE0 ) 2. Densification and increase in tension 3. Formation of GeH Broadly speaking, all three mechanisms prevail in optical fibers. The relative importance of each contribution depends on the type of optical fiber and the photosensitization process used. Most fibers, if not all, show an increase in the population of GeE0 centers (trapped hole with an oxygen vacancy) after UV exposure [89]. This is formed by the conversion of the electron-trapped Ge(I) center, which absorbs at 5 eV. The change in the population of the GeE0 centers causes changes in the UV absorption spectra, which lead to a change in the refractive index directly through the Kramers–Kronig relationship [90] [Eq. (1.1.4)]. This process is common to all fibers. The color center model, originally proposed by Hand and Russell [91], only explains part of the observed refractive index changes of 2 104 in nonhydrogenated optical fibers [92,93]. The second mechanism is a structural alteration in the mechanical nature of the glass and was pointed out several years ago by Bernardin and Lawandy [94]. In the model, a collapse of a higher-order ring structure was proposed as a possible effect of irradiation, leading to densification. The densification of silica under UV irradiation is well documented [95]. However, the picture is not as simple, since another mechanism opposes it: the relief of the internal stress frozen in during fiber fabrication, on UV exposure [96,97]. Stress relief can only remove the effect of the frozen-in stress and is therefore strongly dependent on the initial thermoelastic stress at fabrication. There is correlation between fiber drawing tension and the maximum induced index change for Type I gratings but reduced maximum index change for Type IIA [98]. The process of densification has been shown to occur in fibers as evidenced by scans using an atomic force microscope of the surface of D-shaped fibers and in etched fibers [99], and in preform samples that were drawn into a D-shaped fiber [100]. These observations are on the surface of the sample and are unable to replicate the stress profiles within the core of the fiber directly. Direct optical measurement of in-fiber stress has indicated that rather than the relief of the stress, tensile stress actually increases with an associated reduction in the average refractive index by 30% of the observed UV induced refractive index change in nonhydrogen loaded, high-germania-content fibers [101]. The changes in the stress profile of the fiber are consistent with the shift in the Bragg wavelength of a grating during inscription [102]. The third mechanism for the UV induction of the refractive index change is via the formation of Ge–H and the generation of GeE0 centers; it was proposed by Tsai and Friebele [89]. The concentrations of the GeE0 have been previously correlated with the presence of the precursor states of the Ge(I) and Ge(II) centers. However, the concentration of GeE0 centers continues to grow despite the saturation of both Ge(I) and Ge(II), indicating that the formation of the
Summary of Routes to Photosensitization
37
E0 centers has another route. The color center model for the changes in the refractive index is supported by the measurements made by Atkins et al. [103].
2.7 SUMMARY OF ROUTES TO PHOTOSENSITIZATION A method for photosensitizing silica optical fiber is based on the observation that increasing the 240-nm absorption enhances the effect [104]. Thus, reduced germania present as GeO have been shown to have a good photosensitive response. Other defect formers, such as europium [80], cerium [81], and thulium [43], also fall in this category, albeit with a smaller effect. Phosphorus, which is used extensively in the fabrication of germania-doped silica planar waveguides, shows a weak response with radiation around 240 nm. However, the situation changes with radiation at 193 nm, as has been demonstrated [41]. Silica optical fibers with cores doped with germania, phosphorus, or alumina all exhibit increased photosensitivity when fabricated in a reducing atmosphere of hot hydrogen [105,106]. The negative effect of this type of treatment is the increase in the absorption due to the presence of OH ions and also an increase in the refractive index of the core. Flame-brushing, i.e., heating optical fiber or planar silica waveguides using a flame in a hydrogen-rich atmosphere, has also been used for photosensitization [107]. Hydrogen is able to diffuse into the fiber rapidly at elevated temperatures. Germanosilicate optical fiber codoped with boron has been shown to be highly photosensitive [40]. Another advantage of the presence of boron is the large reduction in the background refractive index, allowing more germania to be added in the core for a given core-cladding index difference. It is believed that with boron physiochemical changes are responsible for the UV-induced index changes (approximately a few thousandths). Boron-codoped fiber gratings decay with temperature more rapidly than pure-germania-doped fibers, although there are ways to enhance their stability by using burn-in (see Chapter 9). The presence of boron also increases the absorption loss in the 1500-nm window by 0.1 dB m1. An alternative scheme to circumvent the problems associated with boron is to use Sn codoping with germania [63]. This combination is more difficult to fabricate, but has virtually no additional absorption in the 1500-nm window, has a similar photosensitivity to the B-Ge system, and exhibits better temperature stability. There is no reduction in the refractive index with Sn doping. Finally, the system that has demonstrated the largest photosensitive response in germanosilicate optical fibers is high-pressure cold hydrogen soaking [66]. It has been demonstrated that nearly every germanium ion is a potential candidate for conversion from the Ge–O to the Ge–H state [108], causing index changes as large as 0.01, although the ultimate magnitude of the index change is not known. Once hydrogenated, these fibers need to be stored at low temperatures
38
Photosensitivity and Photosensitization of Optical Fibers
to maintain their photosensitivity, since molecular hydrogen diffuses out just as readily as it can be introduced. A major disadvantage of writing gratings into cold-hydrogen-sensitized fiber gratings is the high loss of several decibels per meter at 1320 nm (OH absorption). Deuteration eliminates the absorption at 1320 nm, while maintaining the photosensitivity [68]. A summary of the all currently known techniques of photosensitizing fibers is listed in Table 2.2.
2.7.1 Summary of Optically Induced Effects The chemical reactions that take place in photosensitive optical fiber exposed to UV radiation are probably never going to be understood completely. However, several known factors influence the index change. 1. Bleaching of the 240-nm GeO band in reduced germania fibers. This has been measured and alludes to the following picture of chemical modification. The Ge–Si bonds break, liberating an electron, which may be retrapped at another defect site. What remains is the type of picture seen in Fig. 2.2 (GeE0 ). It is not known whether a volumetric change occurs as well. It is likely that, owing to the confinement of the photosensitive species within a massive cladding, any physical relaxation or contraction of molecular bonds will result in a stressed state. Recently, the stress change in fiber cores has been theoretically modeled and also measured using optical methods [110], with good agreement between the two. Douay et al. [99] have etched fibers and preforms previously exposed to interfering UV beams and found that a relief grating is revealed. Although this is not direct evidence that physical changes occur on UV exposure, it does show that the chemistry has indeed been altered. Riant et al. [100] performed atomic force microscopy (AFM) on the surface of D-shaped fibers in which gratings have been written. The surface of the fiber, a few microns above the photosensitive core of germania-doped silica, showed a surface relief directly indicating stress changes. Therefore, not only does the absorption in the UV change, but so does the density of oscillators with UV exposure, both of which alter the refractive index based on the Kramers–Kronig rules. However, it must be remembered that the molecules within the core are not free to change their shape, but are elastically coupled to the mass in the cladding via bonds that remain predominantly unchanged. This is the basis for the induced stress and for part of the index modification by the stress-optic coefficient. Russell et al. [111] have estimated that a strain of only 104 is necessary to induce an index change in the effective index of the mode by the same order of magnitude. It is well known that fiber drawing conditions and stress annealing can alter the refractive index of boron-doped glasses [112]. 2. There is a change in the position of the band edge of germania-doped silica in the deep-UV spectrum, which alters the refractive index after UV exposure.
Routes to photosensitization of optical fibers Fiber type
Fabrication process
Standard telecommunications germania-doped fiber 3 mol% High germania 10–30 mol%
Standard CVD/ PECVD
Reduced germania (10 mol%)
Production depleted oxygen atmosphere
Boron–germania codoped
Needs additional calibration since boron reduces core index Preform cooked at 750 C in hydrogen atmosphere Same as above
Hot hydrogenated germania-doped fiber Hot hydrogenated B–Ge codoped fiber
Standard CVD/ PECVD
Photosensitivity
Advantages
Disadvantages
Very low: Photosensitive index change, dn 1 106 (except one report of 1 103 [109]) Low photosensitive index change, slow growth of gratings, dn 1 104 Good photosensitivity, faster growth in index change, dn 5 104 Very good, faster increase in index, dn 8 104
Easy production, useful for low reflectivity gratings
Very low photosensitivity, low birefringence, goodquality fiber
Good quality fiber, easy production
Birefringent, photosensitivity not enough to be useful in untreated fiber Needs high germania (10 mol%)
Good photosensitivity, dn 8104
Needs higher germania than standard telecommunications fiber; easy to store. Can be made compatible with standard telecommunication fiber
Very good, dn 1 103
Low OH loss, circular cored fiber Low OH loss, can be compatible with standard telecommunication fiber
Summary of Routes to Photosensitization
Table 2.2
Difficult preform fabrication, highly stressed, increase in boron induced loss, elliptical core Slight increase in OH loss
High stress, difficult to fabricate as B–Ge fiber. Loss at 1500 nm Continued
39
40
Table 2.2 Routes to photosensitization of optical fibers—Cont’d Fiber type
High pressure cold deuterium-soaked germania-doped fiber High-pressure hydrogen soaking of phosphorus rareearth-doped fiber (Ge free)
Photosensitivity
Easy fabrication since all types of above fiber may be used, including standard telecommunication fiber. But requires high pressure (up to 800Bar) facility Same as above
Excellent photosensitivity, dn 1 102 in standard telecommunications fiber
Extremely versatile
OH loss with increase in index change. The loss can be 0.1 dB per grating. Limited shelf life unless stored at low temperatures to prevent outdiffusion of H2
Same as above
No increase in loss at 1300/ 1500 nm since OD(OH) overtones are not in second or third telecommunications window Intracore cavity fiber gratings possible for fiber lasers – no splice loss
Extremely expensive option. Also limited shelf life unless stored at low temperatures to prevent outdiffusion of D2 Requires 193-nm radiation for useful index changes and ease of writing
Same as above
Can photosensitize many types of doped silica fibers
Advantages
Disadvantages
Photosensitivity and Photosensitization of Optical Fibers
High-pressure cold hydrogen-soaked germania-doped fiber
Fabrication process
Chemical Composition Gratings
41
This effect is difficult to quantify owing to the problems associated with measurements in the vacuum ultraviolet and the low transparency of silica below a wavelength of 190 nm, although some measurements have been made [113], suggesting that there is no shift in the edge, just an increase in the absorption in the deep UV. 3. High-temperature hydrogen treatment reduces germania, producing an enhanced concentration of GeODC(II) [114]. Again bleaching of the absorption at 240 nm partly contributes to the index change. Other effects as in points 1 and 2 above prevail. Chemically, the reduction process may occurs as follows: Ge
O
Si
+
1/2H2
– Ge e + OH
Si
4. Molecular hydrogen. The suggested reaction is the formation of GeH and OH ions from a Ge(2) defect. The GeH, which has no absorption [114*], is responsible for the change in the refractive index via the Kramers–Kronig rule. The possible route may be as follows: H
Ge
O
Ge
+ H2
Ge e–
H O
Ge
It is not clear what the reactions with a pure alumina-doped silica core may be, but it is possible that a similar set of observations may occur; the addition of phosphorus in germania-doped fibers reduces the concentration of GeE0 centers, increasing with increased fabrication temperatures or reducing conditions [115]. Figure 2.14 compares a set of different fibers exposed to the same intensity of radiation at 262 nm for different times. The comparison shows the refractive index growth rate for fiber with different photosensitization treatments when exposed to UV radiation, per mol% Ge in the fiber, normalized to the fastest fiber: the H2:BGe, which writes an 87% reflective, 1.5-mm long grating in 10 seconds.
2.8 CHEMICAL COMPOSITION GRATINGS Poumellec and Kherbouche [116] have presented an excellent review of the photosensitivity of glass fibers. The photosensitivity of various fibers, doped and undoped, under different conditions of hydrogen loading, laser wavelengths, and glass for the fiber has been summarized in detail, and the reader is directed to this review for more information. However, another route to enhancing photosensitivity in silica fibers is through massive hydroxyl formation in hydrogen loaded fibers, which have been heated to 1000 C for a short
Photosensitivity and Photosensitization of Optical Fibers
42
Normalized UV induced index change per Ge mol% per second 10
100 Germania content (mol%)
Ge mol%
0.1
10
0.01
Normalized Δn
1
Normalized dn per Ge (mol%) per sec
0.001 H2-Soaked B-Ge
B-Ge Doped Fiber
Reduced Ge Fiber
H2: Standard Fiber
High Ge Fiber
Deeside Std. Fiber
Philips Match. Clad
1
Fiber type Figure 2.14 Comparison of time-averaged growth rate of the UV-induced refractive index change for different fibers compared to cold H2-soaked:B-Ge fiber on the right column. Maximum index change recorded for the H–BGe codoped fiber was 6.76 103. The UV source was a quadrupled diode pumped QS Nd:YLF laser operating at 262 nm with 60-mW writing power and used with a phase mask. This is a direct comparison of the induced index change in Type I gratings showing the sensitivity to UV radiation (adapted from Ref. [40]).
time [117]. Although, the photosensitizing treatment renders the fiber fragile, it is possible through careful packaging to protect the grating for temperature sensing at high temperatures. Recently, chemical composition gratings (CCGs) have been proposed by Fokine [118], in which the chemical nature of the glass is permanently altered by a number of chemical reactions in the regions that are exposed to UV radiation. In this system, as an example, a fluorine core-doped fiber is hydrogen loaded and exposed to a periodic UV interference pattern. The hydroxyl groups formed in the core as a result of the photolytic reaction in the UV-exposed regions are converted to HF and water after subsequent thermal treatment. The HF diffuses out of the core easily, leaving behind a fluorine deficient region. This two-step process renders the glass permanently altered in the bright fringe areas. The result is a “chemical” grating, in which the regions are materially different, modulating the refractive index as a function of the spatial coordinate along the axis of the fiber. The result is a highly temperature-stable grating, whose stability is determined by the diffusion coefficients at elevated temperatures of the different chemical constituents in the exposed and
Chemical Composition Gratings
43
unexposed regions. As the fluorine leaves the core, it is expected that the refractive index will increase. The chemical reactions and pathways initiated by the action of light and subsequent heating, proposed by Fokine, are X O X þH2 2 X OH X OH þ F X X F H2 O
hvuv
! X OH þ H X ;
hvuv
! X O X þH2 O;
hvuv
! X O X þHF;
hvuv
! X OH þ HF:
ð2:8:1Þ ð2:8:2Þ ð2:8:3Þ ð2:8:4Þ
where () are bonds connecting to oxygen, (–) is a single bond, and X is either silicon or germanium. Dalle et al. suggested Eq. (2.8.1), which is known to occur in hydrogen-loaded fibers [119]. Hydroxyl groups, which can be formed in fibers by thermal treatment or by UV exposure, have been investigated by several researchers because of their impact on transmission loss in the 1.3-mm band [120]. The concentration of molecular water is governed by the equilibrium between the hydroxyl groups and water molecules. Eq. (2.8.2) has been suggested to explain the diffusion of hydroxyl species in silica [121]. Equation (2.8.3) shows how there is depletion in the hydroxyl concentration in fluorinerich regions in preforms of MCVD silica [122]. The last equation is the result of the previous two, which describe an ion exchange process. Diffusion thus results in the modulation of the fluorine concentration in the core. Both germanium and phosphorus lower the activation energy for the formation of hydroxyl groups, although phosphorus increases the diffusion of fluorine even in the nonUV-exposed parts of the core [123], thereby reducing the thermal stability of the CCG [118]. The fluorine–germanium CCGs have excellent thermal stability at temperatures even higher than 1000 C. Gratings in standard single-mode fibers cannot survive such high temperatures [124]. The process of CCG formation is quite general and may be applied to any two or more species that have different diffusion rates, such that UV exposure and the history of treatment induces a differentiation in the chemical composition of the material. Thus, several dopants in silica fibers have been shown to lead to oxygen-CCGs [125], fluorine [124], boron [126], and their properties being reviewed [127,128]. Figure 2.15 shows the typical thermal behavior of a CCG in germanium– fluorine-doped fiber. Note the onset of the chemical reaction and the reduction in the reflectivity of the grating, and then the reemergence after 50 minutes of a smaller reflectivity but highly durable grating. Survival of such a CCG to 1100 C has also been reported. The CCG has great potential for high-temperature sensing applications, for example, down oil wells.
Photosensitivity and Photosensitization of Optical Fibers
44
1000 Initial grating
0.8
800
0.6
600 CCG
0.4
400
0.2
Temperature [°C]
Normalized reflectivity
1.0
200 0
0.0 0
10
20
30 40 Time [min]
50
60
70
Figure 2.15 The decay and emergence of the CCG in a germanium–fluorine-doped silica fiber. Note the stability at 1000 C. (Reproduced with permission from Michael Fokine, “Thermal stability of chemical composition gratings in fluorine–germanium-doped silica fibers,” Optics Lett. 27(12), 1016–1018. Copyright OSA 2002.)
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[83] T. Taunay, P. Bernage, G. Martenelli, M. Douay, P. Niay, J.F. Bayon, et al., Photosensitization of terbium doped aluminosilicate fibers through high pressure H2 loading, Opt. Commun. 133 (1997) 454–462. [84] M. Douay, D. Ramecourt, T. Taunay, P. Niay, P. Bernage, A. Dacosta, et al., Microscopic investigations of Bragg grating photowritten in germanosilicate fibers, in: Photosensitivity and Quadratic Nonlinearity in Glass Waveguides: Fundamentals and Applications, OSA Technical Series, vol. 22, paper SAD2, Optical Society of America, Washington, DC, 1995, pp. 48–51. [85] I. Riant, S. Borne, P. Sansonetti, B. Poumellec, Evidence of densification in UVwritten Bragg gratings in fibers, in: Photosensitivity and Quadratic Nonlinearity in Waveguides: Fundamentals and Applications, OSA Technical Digest Series, vol. 22, Optical Society of America, Washington, DC, 1995, pp. 51–55. [86] M. Douay, W.X. Xie, T. Taunay, P. Bernage, P. Niay, P. Cordier, et al., Densification involved in the UV based photosensitivity of silica glasses and optical fibers, J. Lightwave Technol. 15 (8) (1997) 1329–1342. [87] P.Y. Fonjallaz, H.G. Limberger, R.P. Salathe´, F. Cochet, B. Leuenberger, Tension increase correlated to refractive index change in fibers containing UV written Bragg gratings, Opt. Lett. 20 (11) (1995) 1346–1348. [88] Courtesy Marc Douay, Bertrand Poumellec and Pierre Sansonetti. The AFM scan was performed after a grating with a 0.453-mm pitch had been inscribed with 8000 pulses at 185 mJ/cm2 at 244 nm, doubled dye pumped by a XeCl laser, before etching. Details may be found in Ref. [86]. [89] T.E. Tsai, G.M. Williams, E.J. Friebele, Index structure of fiber Bragg gratings in Ge-SiO2 fibers, Opt. Lett. 22 (4) (1997) 224–226. [90] T.E. Tsai, E.J. Friebele, Kinetics of defect centers formation in Ge-SiO2 fibers of various compositions, in: Bragg Gratings, Photosensitivity and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, OSA Technical Digest Series, vol. 17, Optical Society of America, Washington, DC, 1997, pp. 101–103. [91] D.P. Hand, P.St.J. Russell, Photoinduced refractive index changes in germanosilicate optical fibers, Opt. Lett. 15 (2) (1990) 102–104. [92] D.L. Williams, S.T. Davey, R. Kashyap, J.R. Armitage, B.J. Ainslie, Direct observation of UV induced bleaching of 240 nm absorption band in photosensitive germanosilicate glass fibres, Electron. Lett. 28 (4) (1992) 369. [93] L. Dong, J.L. Archambault, P.St.J. Russell, D.N. Payne, Strong UV absorption in germanosilicate fiber preforms induced by exposure to 248 nm radiation, 20th European Conference on Optical Communications (ECOC 1994), Florence, Italy, Sept. 1994 997–1000. [94] J.P. Bernardin, N.M. Lawandy, Dynamics of the formation of Bragg gratings in germanosilicate optical fibers, Opt. Commun. 79 (1990) 194. [95] M. Rothschild, D.J. Erlich, D.C. Shaver, Effects of excimer irradiation on the transmission, index of refraction, and density of ultraviolet grade fused silica, Appl. Phys. Lett. 55 (13) (1989) 1276. [96] M.G. Sceats, P.A. Krug, Photoviscous annealing – dynamics and stability of photorefractivity in optical fibers, SPIE 2044 (1993) 113–120. [97] D. Wong, S.B. Poole, M.G. Skeats, Stress birefringence reduction in elliptical-core fibers under ultraviolet irradiation, Opt. Lett. 24 (17) (1992) 1773. [98] I. Riant, B. Poumellec, Influence of fiber drawing tension on photosensitivity in hydrogenated and nonhydrogenated fibers, in: Tech. Digest of Conf. on Opt. Fib. Commun., OFC’98, paper TuA1, 1998, pp. 1–2.
50
Photosensitivity and Photosensitization of Optical Fibers
[99] M. Douay, D. Ramecourt, T. Tanuay, P. Bernage, P. Niay, D. Dacosta, et al., Microscopic investigations of Bragg gratings photowritten in germanosilicate fibers, in: Photosensitivity and Quadratic Nonlinearity in Waveguides: Fundamentals and Applications, OSA Technical Digest Series, vol. 22, Optical Society of America, Washington, DC, 1995, pp. 48–51. [100] I. Riant, S. Borne, P. Sansonetti, B. Poumellec, Evidence of densification in UVwritten Bragg gratings in fibers, in: Photosensitivity and Quadratic Nonlinearity in Waveguides: Fundamentals and Applications, OSA Technical Digest Series, vol. 22, Optical Society of America, Washington, DC, 1995, pp, 51–55. [101] H.G. Limberger, P.Y. Fonjallaz, R.P. Salathe´, UV induced stress changes in optical fibers, in: Photosensitivity and Quadratic Nonlinearity in Waveguides: Fundamentals and Applications, OSA Technical Digest Series, vol. 22, Optical Society of America, Washington, DC, 1995, pp. 56–60. [102] M. Douay, W.X. Xie, T. Taunay, P. Bernage, P. Niay, P. Cordier, et al., Densification involved in the UV based photosensitivity of silica glasses and optical fibers, J Lightwave Technol. 15 (8) (1997) 1329–1342. [103] R.M. Atkins, V. Mizrahi, T. Erdogan, 248 nm induced vacuum UV spectral changes in optical fibre preform cores: support for a colour centre model of photosensitivity, Electron. Lett. 29 (4) (1993) 385. [104] D.L. Williams, B.J. Anslie, R. Kashyap, G.D. Maxwell, J.R. Armitage, R.J. Campbell, et al., Photosensitive index changes in germania doped silica glass fibers and waveguides, SPIE 2044 (1993) 55–68. [105] G.D. Maxwell, R. Kashyap, B.J. Ainslie, D.L. Williams, J.R. Armitage, UV written 1550 nm reflection filters in singlemode planar silica waveguides, Electron. Lett. 28 (22) (1992) 2106–2107. [106] G. Meltz, W.W. Morey, Bragg grating formation and germanosilicate fiber photosensitivity, SPIE 1516 (1991) 185–199. [107] F. Bilodeau, B. Malo, J. Albert, D.C. Johnson, K.O. Hill, Y. Hibino, et al., Photosensitization in optical fiber and silica on silicon/silica waveguides, Opt. Lett. 18 (1993) 953–955. [108] D.M. Krol, R.M. Atkins, P.J. Lemaire, Photoinduced second-harmonic generation and luminescence of defects in Ge-doped silica fibers, SPIE 1516 (1991) 38–46. [109] H.G. Limberger, P.Y. Fonjallaz, R. Salathe´, Spectral characterization of photoinduced high efficient Bragg gratings in standard telecommunication fibers, Electron. Lett. 29 (1) (1993) 47–49. [110] P.Y. Fonjallaz, H.G. Limberger, R. Salathe´, F. Cochet, B. Leuenberger, Tension increase correlated to refractive index change in fibers containing UV written Bragg gratings, Opt. Lett. 20 (11) (1995) 1346–1348. [111] P.St.J. Russell, D.P. Hand, Y.T. Chow, L.J. Poyntz-Wright, Optically-induced creation, transformation and organisation of defects and colour centres in optical fibres, SPIE 1516 (1991) 47–54. [112] I. Camlibel, D.A. Pinnow, F.W. Dabby, Optical aging characteristics of borosilicate clad fused silica core fiber waveguides, Appl. Phys. Lett. 26 (4) (1975) 185–187. [113] R.M. Atkins, V. Mizrahi, T. Erdogan, 248 nm induced vacuum UV spectral changes in optical fibre preform cores: Support for the colour centre model of photosensitivity, Electron. Lett. 29 (4) (1993) 385. [114] A. Iino, M. Kuwabara, K. Kokura, Mechanisms of hydrogen induced losses in silica based optical fibers, J. Lightwave Technol 8 (11) (1990) 1675–1679, 42. [114*] B. Poumellec, Pvt. Comm. (2003).
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[115] G.R. Atkins, S.B. Poole, M.G. Sceats, H.W. Simmons, C.E. Nockolds, The influence of codopants and fabrication conditions on germanium defects in optical fiber preforms, IEEE Photon. Technol. Lett. 4 (1) (1992) 43–46. [116] B. Poumellec, F. Kherbouche, The Photorefractive Bragg Gratings in the Fibers for Telecommunications, J. Phys. III France 6 (1996) 1595–1624. [117] M. Fokine, W. Margulis, Large increase in photosensitivity through massive hydroxyl formation, Opt. Lett. 25 (2000) 302–304. [118] M. Fokine, Formation of thermally stable chemical composition gratings in optical fibers, J. Opt. Soc. Am. B 19 (8) (2002) 1759–1765. [119] C. Dalle, P. Cordier, C. Depecker, P. Niay, P. Bernage, M. Douay, Growth kinetics and thermal annealing of UV-induced H-bearing species in hydrogen loaded germanosilicate fibre preforms, J. Non-Cryst. Solids 260 (1999) 83–89. [120] J. Stone, Interactions of hydrogen and deuterium with silica optical fibers: a review, J. Lightwave Technol. LT-5 (1987) 712–733. [121] R.H. Doremus, The diffusion of water in fused silica, in: J.W. Mitchell, R.C. DeVries, R.W. Roberts, P., Cannon (Eds.), Reactivity of Solids, Wiley, New York, 1969, pp. 667–673. [122] J. Kirchhof, S. Unger, H.J. Pissler, B. Knappe, Hydrogen-induced hydroxyl profiles in doped silica layers, in: Optical Fiber Communication Conference, OSA Technical Digest Series, vol. 8, paper WP9.22, Optical Society of America, Washington, DC, 1995. [123] J. Kirchhof, S. Unger, K.F. Klein, B. Knappe, Diffusion behaviour of fluorine in silica glass, J. Non-Cryst. Solids 181 (1995) 266–273. [124] M. Fokine, Thermal stability of chemical composition gratings in fluorine-germaniumdoped silica fibers, Optics Lett. 27 (12) (2002) 1016–1018. [125] M. Fokine, Thermal stability of oxygen-modulated chemical-composition gratings in standard telecommunication fiber, Opt. Lett. 29 (11) (2004) 1185–1187. [126] S. Bandyopadhyay, J. Canning, M. Stevenson, K. Cook, Ultrahigh-temperature regenerated gratings in boron-codoped germanosilicate optical fiber using 193 nm, Opt. Lett. 33 (16) (2008) 1917–1919. [127] M. Fokine, Underlying mechanisms, applications, and limitations of chemical composition gratings in silica based fibers. Section 2. Optical properties, J. Non-Cryst. Solids 349 (2004) 98–104, doi:10.1016/j.jnoncrysol.2004.08.208. [128] M. Fokine, Manipulating glass for photonics, Physica Status Solidi (a) 206 (5) (2009) 880–884.
Chapter 3
Fabrication of Bragg Gratings 3.1 METHODS FOR FIBER BRAGG GRATING FABRICATION This chapter reviews many of the schemes proposed for both holographic and nonholographic grating inscription and considers some of the salient features of the methods. This introduction excludes methods used for internal grating writing, traditionally known as “Hill” gratings, for which the reader is directed to other sources [1–12]. Fiber Bragg gratings, which operate at wavelengths other than near the writing wavelength (non-Hill gratings), are fabricated by techniques that broadly fall into two categories: those that are holographic [13] and those that are noninterferometric, based on simple exposure to UV radiation periodically along a piece of fiber [14]. The former techniques use a beam splitter to divide a single input UV beam into two, interfering them at the fiber; the latter depend on periodic exposure of a fiber to pulsed sources or through a spatially periodic amplitude mask. There are several laser sources that can be used, depending on the type of fiber used for the grating, the type of grating, or the intended application. The sources used for grating production are also discussed in this chapter.
3.1.1 The Bulk Interferometer The method for the side writing of fiber gratings demonstrated by Meltz et al. [15] is shown in Fig. 3.1. The interferometer is one encountered in standard holography [16], with the UV beam divided into two at a beam splitter and then brought together at a mutual angle of y, by reflections from two UV mirrors. This method allows the Bragg wavelength to be chosen independently of the UV wavelength as
53
54
Fabrication of Bragg Gratings UV radiation Compensating plate UV mirror 50% beam splitter Fiber q
rn
tte
e
nc
pa
re rfe
UV mirror te
In
Figure 3.1 UV interferometer for writing Bragg gratings in optical fibers. Note the use of an additional phase plate (mirror blank) in one arm to compensate for the path length difference.
lBragg ¼
neff luv ; y nuv sin 2
ð3:1:1Þ
where lBragg is the Bragg reflection wavelength, neff is the effective mode index in the fiber, nuv is the refractive index of silica in the UV, luv is the wavelength of the writing radiation, and y is the mutual angle of the UV beams. The essential difference between a “Hill” grating and one produced by external interference of two UV beams is that with the holographic technique the Bragg reflection wavelength depends on UV radiation wavelength and geometric factors. Since luv is around 240 nm, y lies between 0 and 180 , and assuming that the refractive index in the UV is approximately equal to the effective index, the Bragg wavelength is adjustable from one nearly equal to the UV source wavelength to infinity [see Eq. (3.1.1) with y ¼ 0]. The fiber is held at the intersection of the beams. This method was originally successfully used to write gratings at visible wavelengths. The interferometer is ideal for single-pulse writing of short gratings, and great care has to be taken in the design of the optical mounts. Mechanical vibrations and the inherently long path lengths in air can cause the quality of the interferogram to change over a period of time, limiting its application to short exposures. For low-coherence sources, the path difference between the two interfering beams must be equalized; a simple method is to introduce a mirror blank in one arm to compensate
Methods for Fiber Bragg Grating Fabrication
55
for the path imbalance imposed by the beam splitter, as shown in Fig. 3.1. Note that in arriving at the fiber, the beam that is transmitted through the beam splitter undergoes a 180 rotation so that they have different spatial profiles. This is an important factor for spatially incoherent beams. The interferometer shown in Fig. 3.1 has several beams paths in open air. It is important that these are shielded from turbulence, since the interference fringes formed at the fiber can drift if the paths of the two beams change during the inscription time. As is common with all holographic arrangements, it is not sensible to mount mirrors, beam splitters, or the fiber on flimsy platforms prone to disturbance, such as tall 10-mm diameter mounting posts. The interferometer needs to be built on a sturdy base, with stable optical mounts. This is especially true in cases that require long (minutes to hours) exposures. It is common practice to enclose the entire interferometer within a Perspex housing, which allows visual and physical access to the setup, at the same time protecting the interferometer from constant path-length variations and the operator from accidental exposure to UV radiation. Extreme care needs to be taken to minimize exposure of personnel to high-energy UV radiation or long-term exposure to low-power radiation. Adhering to safe operating practices is essential when using UV radiation. In principle, a diffraction grating used in reflection can replace the 50% beam splitter shown in Fig. 3.1. In this interferometer, two coherent beams are required, so that reflection from a diffraction grating to divide the input UV beam into two is equally feasible. However, a simpler component, the transmission phasegrating, otherwise known as the phase mask, is better suited to this application.
3.1.2 The Phase Mask A major step toward easier inscription of fiber gratings was made possible by the application of the phase mask as a component of the interferometer. Used in transmission, a phase mask is a relief grating etched in a silica plate. The significant features of the phase mask are the grooves etched into a UV-transmitting silica mask plate, with a carefully controlled mark-space ratio as well as etch depth. The principle of operation is based on the diffraction of an incident UV beam into several orders, m ¼ 0, 1, 2 . . . . This is shown schematically in Fig. 3.2. The incident and diffracted orders satisfy the general diffraction equation, with the period Lpm of the phase mask, Lpm ¼
mluv ; ym sin sin yi 2
ð3:1:2Þ
56
Fabrication of Bragg Gratings Incident UV beam
qi
Λpm Phase mask
m = –1
qt
Transmitted beam, m = 0
qm /2 Figure 3.2 A schematic of the diffraction of an incident beam from a phase mask.
where ym/2 is the angle of the diffracted order, luv the wavelength, and yi the angle of the incident UV beam. In instances when the period of the grating lies between luv and luv/2, the incident wave is diffracted into only a single order (m ¼ 1) with the rest of the power remaining in the transmitted wave (m ¼ 0). With the UV radiation at normal incidence, yi ¼ 0, the diffracted radiation is split into m ¼ 0 and 1 orders, as shown in Fig. 3.3. The interference pattern at the fiber of two such beams of orders 1 brought together by parallel mirrors (as in Fig. 3.1) has a period Lg related to the diffraction angle ym/2 by Lg ¼
Lpm luv ¼ : 2 sinðym =2Þ 2
ð3:1:3Þ
The period Lpm of the grating etched in the mask is determined by the Bragg wavelength lBragg required for the grating in the fiber (see Chapter 4) and using Eq. (3.1.3) to arrive at Lg ¼
NlBragg Lpm ; ¼ 2neff 2
ð3:1:4Þ
where N 1 is an integer indicating the order of the grating period. For nonnormal incidence of the UV radiation on the phase mask, intensities in the m ¼ 0 and 1 orders are not necessarily equal. However, for the visibility
Normally incident UV beam Λpm Phase mask Diffracted beam, m = +1
m = –1 qm /2 qm /2 m=0
Figure 3.3 Normally incident UV beam diffracted into two 1 orders. The remnant radiation exits the phase-mask in the zero order (m ¼ 0).
Methods for Fiber Bragg Grating Fabrication
57
of the interference pattern to be a maximum, the intensities must be equalized. This is important if gratings are to be inscribed efficiently. For a first-order (N ¼ 1) grating at a Bragg wavelength of 1.55 mm and a mode effective index neff 1.46, Lpm ¼ 1.06 mm, which is greater than the wavelength of the UV radiation used for grating inscription (0.193 to 0.360 mm). Therefore, more than a single diffracted order (m ¼ 0, 1, 2 . . .) exists. To suppress the positive orders and control the diffraction efficiency, and to equalize the power between the 1 order and the transmitted beam (m ¼ 0), one face of the etched grating walls may be coated with a metal film to form reflecting mirrors. This may be done by evaporating the metal on to the phase-mask plate at an angle so that only the walls facing the evaporation source are coated [17]. Another method uses a deeper etched grating in the phase mask [18] to suppress higher orders and control the relative intensities. However, it is easier and more cost-effective simply to use a phase mask at normal incidence. If necessary, an antireflection coating may be applied to the back facet of the phase-mask plate to reduce reflections, which can cause the quality of the interference fringes to be degraded [25]. The depth d of the etched sections of the grating is a function of the UV wavelength, but the period is dependent only on the Bragg wavelength and the effective index of the mode. However, in the case of UV writing of gratings, it is necessary to ensure that the intensity of the transmitted zero-order beam is minimized and, ideally, blocked from arriving at the fiber. To minimize the zeroth order from a UV beam normally incident on a phase mask, the smallest etch depth d of the relief grating in silica is dðnuv 1Þ ¼ luv =2:
ð3:1:5Þ
Equation (3.1.5) assumes monochromatic radiation with no divergence; however, for a practical nonmonochromatic source, the m ¼ 0 order cannot be eliminated. In practice, the zeroth order can be reduced to a level of a percent or so. At a wavelength of 244 nm, d 240 nm. The phase-mask zero order can be nulled only at a single wavelength. Changing the laser source wavelength will require a different phase mask, unless the zero order is physically blocked. For efficient diffraction onto the first orders, it is necessary for the relief grating to have a mark-space ratio of 1:1, or for the corrugations in the phase mask to be purely sinusoidal.
Fabrication of the Phase Mask The phase mask is normally fabricated by one of two methods: by exposure of a photoresist overcoated, silica mask plate to an electron beam to form the pattern [19,20], or by holographic exposure [21]. With the e-beam facility, a silica wafer, which has a bilevel resist comprising a 450-nm layer of AZI400-27 is hard baked at 190 C for 30 minutes, followed by a 200-nm layer of silicon-containing
58
Fabrication of Bragg Gratings
negative resist (SNR) baked at 85 C, also for 30 minutes. Charge dispersal during the e-beam exposure is effected by evaporating a thin layer of aluminum. After exposure to delineate the pattern, the A1 coating is removed in an alkaline solution and the SNR spray developed in MIBK for 35 sec, then rinsed in a 50:50 ratio of MIBK þ IPA solution for 5 sec, followed by 15 sec in IPA. The developed pattern is transferred to the AZI400-27 layer by reactive ion etching (RIE) at 10 m torr in oxygen and 50 W RIE. The resist is then used as a mask for etching into the silica plate using CHF3:Ar RIE. The final depth of 240 nm, for use at a UV wavelength of 244 nm, is achieved by a two-stage etch. A scanning electron microscope photograph of a phase-mask plate is shown in Fig. 3.4. Generally, the phase mask is fabricated in small fields, which are then stitched together to form a long grating. Common problems with phase masks processed by e-beams have to do with inaccurate stitching of the fields. The positioning accuracy of the e-beam and variation of the silica mask plate height cause phase steps to occur between fields, and the resolution of the photoresist causes random variations in the individual periods of the grating. Techniques have been developed to minimize these errors [46,20]; however, the random variation in the absolute positioning of the e-beam is a fundamental limitation. Typically, the writing of long phase masks by e-beam needs constant referencing and correcting owing to small temperature variations during the exposure period, which may last several hours. These problems are of greater importance as the length of the grating increases. Phase masks as long as 120 mm have been reported [22], although the quality of the fabricated Bragg grating has not been reported in detail. Stitching is not an issue when the alternative technique of holographic phase-mask fabrication is used. This technique is, in principle a superior method for phase-mask production. However, long phase masks have
Figure 3.4 SEM photograph of a high-quality phase mask used for grating inscription of 1060-nm Bragg gratings. (Courtesy Ian Lealman, BT Labs)
Methods for Fiber Bragg Grating Fabrication
59
been difficult to produce holographically, owing to problems with uniformity of illumination and the requirement for large mirrors. Lenses can be used to alter the phase front of one of the interfering beams to allow the fabrication of continuously chirped gratings, as opposed to step-chirped gratings by e-beam fabrication [23]. Since the fabrication of the holographic phase mask depends on geometrical alignment of interfering beams, the mass production of identical phase masks may remain a problem.
3.1.3 The Phase-Mask Interferometer UV lithographic replication has been used extensively to fabricate phase masks directly in silica plates using e-beam writing and plasma etching [24], to function as lenses and complex spatial elements. This technique has also been applied successfully to fiber Bragg grating inscription and reported in the literature by several laboratories at around the same time [25–28]. There are several methods of using the phase mask: As has been stated, it may perform the function of simple beam splitting in the interferometer in Fig. 3.1. So why is it such a useful element, when a far cheaper dielectric beam-splitter can be used instead? Its primary aim is to be used simply as a wavelength-defining element in an interferometer (as shown in Fig. 3.5); used as a beam splitter with the beamcombining mirrors (Fig. 3.5) to adjust the wavelength of the fiber grating.
Cylindrical lens UV beam Phase mask Zeroorder block q/2
Rotatable and laterally translatable mirrors
Fiber Rotation for blaze Figure 3.5 The phase mask used as a beam splitter in an interferometer for inscription of fiber gratings. The phase mask predefines the wavelength of the reflection grating, when the mirrors are at right angles to the axis of the fiber and the phase-mask plate (Talbot interferometer) [see Eqs. (3.1.3) and (3.1.4)]. In this scheme, the paths of the two interfering beams are identical, making the interferometer suitable for use with low-spatial-coherence sources.
2.0
0.25
1.8
0.2
1.5
0.15
1.3 0.1
1.0
0.05
0.8 0.5
Δlb /degree, mm/deg
Fabrication of Bragg Gratings
Bragg wavelength, mm
60
0 8
10
12
14 16 18 20 22 Angle, q/2, degrees
24
26
Figure 3.6 The Bragg wavelength (dashed curve) and the rate of change of the Bragg wavelength (continuous curve) as a function of the phase-mask diffraction angle. A change of 1.6 around a mean diffraction angle of 10 is equivalent to a change in the Bragg wavelength of 250 nm.
The change in the Bragg wavelength as a function of the change in the mutual angle between the two interfering beams as shown in Fig. 3.5 is found by substituting Eqs. (3.1.3) and (3.1.4) into Eq. (3.1.1) and differentiating with respect to y: Dl Dy y ¼ cot : lBragg 2 2
ð3:1:5aÞ
Figure 3.6 shows the Bragg wavelength in the fiber as a function of the halfwriting angle. With the diffraction angle fixed at 10 (phase mask for 1550 nm), a change of 5 alters the Bragg wavelength by 800 nm [29]. The enormous tunability of this interferometer, as well the ability to find a reference position for the phase-mask Bragg wavelength, makes it highly flexible. It requires a single-phase mask, which is used as both a beam splitter and a Bragg wavelength reference. It can also be used to replicate chirped phase masks, and a tunability of the Bragg wavelength of 250 nm has been demonstrated [29]. Alternatively, the fiber may be placed directly behind the phase mask for photo imprinting of the grating. In this scheme, there are two important issues. First, since the diffracted beams interfere in the region of overlap immediately behind the phase mask, the fiber core needs to be at the phase-mask surface for maximum overlap. The closest the phase mask can be placed to a fiber core is a distance equal to the fiber radius (unless a “D-fiber” is used), which means that there is no overlap of the two beams in a short region at either end of the grating. Second, the interference pattern generated at the fiber core is the sum of
Methods for Fiber Bragg Grating Fabrication
61
the interference of all the diffracted orders. For a pure sinusoidal pattern at the fiber core, it is important to allow only the two 1 orders to interfere with the zero-order suppressed. As has been observed with the phase mask in contact with the fiber, even with a zero-order nulled phase mask, the period of the imprinted grating depends strongly on the intensity of the writing UV beam. At low intensities, the period is half the phase-mask period [see Eq. (3.1.3)], but at high intensities, even a low zero-order intensity can interfere with the 1 orders to create a grating of the same period as the phase mask itself [120]. Tilting the fiber at an angle a behind the phase mask so that one end is farther away shifts the Bragg reflection to longer wavelengths as the inverse of cosine a, since the fringe planes are no longer orthogonal to the propagation axis. This method for tuning the Bragg wavelength has been demonstrated [30]; it should, however, be remembered that the grating length shortens with tilt, and not only does the reflectivity drop (due to limited coherence of the UV source), but radiation loss can increase [41] (see Section 3.1.4). The zero-order beam may be avoided by repositioning the mirrors. This is shown in Fig. 3.7a, where the grating is written at a point well removed from
Cylindrical lens UV beam Phase mask
Mirror m = 0
Fiber
A
Grating(m = ±1) Uniform UV irradiation (m = 0) Tilted mirror
m = ±1 Fiber
UV beam m= 0
B
Phase mask
2a
Figure 3.7 (a) Avoiding the zero order from the region of the grating by repositioning the interferometer mirrors. Alternatively, tilting the mirrors outward from their normally vertical positions, means that the beams no longer interfere in the plane of the zero order (b).
62
Fabrication of Bragg Gratings
the incident zero order. The path length of the two interfering beams remains identical. A similar result may be achieved by tilting the beam-folding mirrors by an angle a from the perpendicular to the horizontal plane. On reflection from the surfaces, the beams are directed at angles of 2a to the horizontal plane, out of the plane of the zero-order beam, as shown in Fig. 3.7b. It is usual to place a cylindrical focusing lens before the phase mask in the path of the UV beam so as to allow two stripes (within the plane of the paper in Fig. 3.7a) to overlap at the fiber. This has the advantage of focusing in one plane and increasing the power density, while leaving the length of the grating unaltered. Care needs to be taken in adjusting this interferometer, since the orientation of the phase mask determines whether the two beams will overlap at the fiber, while the cylindrical lens determines whether or not the overlapping stripes align along the fiber core. It is important to ensure that the path lengths from the phase mask to the fiber are identical so that the mutual coherence of the beams is maximized [31]. The fiber should be placed in the region of the fringes such that the propagation axis is normal to the fringe planes, since any angular misalignment increases radiation loss from the light propagating in the fiber (see Chapter 4) and shifts the Bragg wavelength. If mutual counterrotation of the mirrors is incorporated in the phase-mask interferometer, grating inscription becomes infinitely flexible with a single phase mask. Two modifications are required if Bragg wavelength tunability is required: The mirrors need to be rotated, and the distance of the fiber from the phase mask must be changed. Using a translation stage to hold the fiber in situ easily incorporates the latter. The alignment of the interferometer is simply and quickly carried out by using a borosilicate glass microscope coverslip to view the fluorescence of the individual UV beams. The coverslip is moved toward or away from the interferometer until the fluorescence from the two beams overlaps. The fiber simply replaces the glass slide for grating inscription. Figure 3.8 shows a photograph of the fully flexible interferometer in use at BT Laboratories. The polarization of the UV laser beam affects the inscription of the grating in the fiber [32–39]. To ensure that the inscribed grating has low polarization sensitivity, the polarization of the UV laser beam should be oriented parallel to the propagation axis of the fiber [40]. (Gratings inscribed with UV laser radiation polarized orthogonal to the propagation direction show significant birefringence due to effects of induced birefringence.) This may be achieved by placing an appropriately oriented half-wave plate before the phase mask. Replacement of the two mirrors in Fig. 3.5 by a rectilinear UV transmitting silica block [23] results in an extremely compact and stable interferometer. The diffracted UV beams enter a face of the silica block and are totally internally reflected by adjacent sides to emerge through the opposite face, interfering at the fiber. The beam paths are shown in Fig. 3.9.
Figure 3.8 The photograph shows the tunable phase-mask interferometer. The distance between the mirrors can also be altered without misaligning the interferometer. The fiber can also be rotated around the vertical axis to allow the inscription of slanted gratings (see Section 3.1.4).
Lg
UV beam Phase mask qm /2
Ws Ls
Rectilinear silica block
Fiber Figure 3.9 Replacement of the two mirrors in Fig. 3.5 by a UV-transmitting silica block. Only one set of diffracted UV beams is shown for simplicity.
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Fabrication of Bragg Gratings
The silica block is placed halfway between the phase mask and the fiber. The maximum grating length, which can be written in a fiber, is related to the dimensions of the block. The length Lg of the grating is a function of the length of the side of the silica block (mirror) as sinðym =2Þ Lg ¼ Ls tan sin1 ; ð3:1:6Þ ns where ns is the refractive index of the silica block at the writing UV wavelength. Assuming that sin ym/2 ym/2 for small angles, the maximum length of the grating, which may be written with a side of Lg is Lg
Ls ym Ls ym : 2ns 3
ð3:1:7Þ
For fiber Bragg gratings at a wavelength of 1500 nm, the angle ym/2 10 ; the length of the silica block side is then approximately 17 times the grating length. The minimum width of the silica block Ws is equal to the grating length Lg without in-line zero-order suppression. However, if the zero order is to be physically blocked by an opaque element, then the width Ws 3Lg. This width is reduced to 2Lg if the interferometer in Fig. 3.5 is used with the zero-order beam block halfway in between the mirrors. For long gratings the dispersion of the silica block is a limitation if the interferometer is used with a lowcoherence source. However, simplification of grating inscription makes this setup highly attractive. There are many advantages of using a phase mask. It allows the wavelength of the fiber grating to be defined precisely for replication. Matching fiber grating reflection wavelengths is made easier, since the phase mask is the interferometer itself. Mass production of identical fiber gratings is thus possible. Another advantage of the phase mask is that a predetermined function may be inscribed in it for replication into the fiber [23] (see Section 3.1.13). The phase mask forms a very stable interferometer since there are no adjustable parts, allowing long inscription times. It is also insensitive to the translation of the inscribing UV beam and tolerant to beam-pointing instability of the laser beam. The advantage of translation insensitivity allows long fiber gratings to be written by the scanning technique, discussed in Section 3.1.5. Disadvantages of using the phase mask nearly in contact with the fiber are the dangers of contamination and permanent damage of the phase mask. A different phase mask is required for each specific Bragg wavelength. This need not be a problem, since several gratings can be written on a single phase-mask plate, each at the required wavelength [41]. Alternatively, a tunable interferometer can be used with a single phase mask; however, it does require careful calibration and alignment.
Methods for Fiber Bragg Grating Fabrication
65
3.1.4 Slanted Grating If the fiber is tilted out of the plane of Fig. 3.5, the grating inscribed in the fiber will be slanted in the direction of propagation of the mode. This, however, requires the interfering beams to have a large cross-sectional area so that the beams may overlap, as shown in Fig. 3.10a. This is inconvenient for most interferometers, since the cylindrical lens focuses the beams in the plane of the figure, unless the unfocused beam intensity is already high. An alternative and simple method for inscribing slanted gratings is to tilt the fiber in the plane of the figure, as shown in Fig. 3.10b [41]. In this case, the coherence properties of the laser will determine the visibility of the fringes at the fiber. Since the fiber is at an angle to the incoming beams, the inscription of the grating depends
Interference fringes a
A
Fiber
Cylindrical lens UV beam Phase mask
Diffraction orders Rotatable mirrors Zero-order block Fiber a
B Figure 3.10 (a) The fringes formed at the point of intersection need a large cross-sectional area for a slanted grating to be written in the fiber. Shown in (a) is a view of the fringes in a plane normal to the zero-order, and a is the rotation angle in that plane. (b) shows another method for writing slanted gratings, in the plane of the incoming beams [41]. The fiber is rotated by an angle a within the plane of the beams such that it overlaps with the interference fringes; the coherence properties of the UV source as well as the depth of fringes determine this.
66
Fabrication of Bragg Gratings Interfering beams D
Fringes W
Figure 3.11 The overlap of the two interfering beams forms a diamond figure, with a depth of the fringes D. The grating length varies depending on the placement of the fiber within the fringes.
on the overlap of the two beams and is slightly shortened; the depth D of the fringes for the interferometer shown in Fig. 3.10b is D
W ; tanðym =2Þ
ð3:1:8Þ
where W is the width of the normally incident UV beam and ym/2 is the diffraction angle shown in Fig. 3.3. Figure 3.11 shows the depth of the fringes and the overlap of the beams. For small tilt angles a, the period Ls of the slanted grating in the direction of propagation varies as Ls
Lg : cos a
ð3:1:9Þ
The fringe pattern shown in Fig. 3.3 is unchanged; however, since the fiber core is at an angle to the fringes, a grating, which is blazed with respect to the propagation direction of the mode, is formed. These gratings have special applications as lossy filters and are discussed in Chapters 4 and 8. In Fig. 3.11, the extent of the fringes formed in the overlap region is shown. The maximum possible fringe depth D is indicated in Fig. 3.11. Coherence, both temporal and spatial, limits D to less than this value, as described in Eq. (3.1.8). For a phase mask used in near contact with the fiber, the depth of the fringes is less than D/2.
3.1.5 The Scanned Phase-Mask Interferometer Figure 3.12 shows how the phase mask may be scanned for inscribing long gratings into fibers. This technique was first demonstrated by Ouellette et al. [42]. It was shown that 19-mm long gratings may be faithfully reproduced in fibers; a slight nonuniformity in the phase mask was also removed by applying a temperature gradient across the fiber length after writing the grating. Byron et al. [43] reported a 50-mm long grating in which the quality of the interferogram was varied by adjusting the intensity of the writing beam along the length
Methods for Fiber Bragg Grating Fabrication
67
Translation of UV beam Scanned UV beam Phase mask Fiber Diffraction orders
Figure 3.12 The phase mask used as a scanned interferometer is a powerful method of fabricating long-fiber gratings. The quality of the grating is dependent on the uniformity of the phase mask.
of the grating. This method allows the tailoring of the transfer characteristics of the fiber grating and will be discussed in Chapter 5. The length and the quality of the phase mask limit the scanning technique. With the best e-beam facility, the absolute positional accuracy is around 5 nm. This positional error sets the limit on the stitching of the fields. Although the error is random and the effects are averaged out over the length of the mask [44], the stitching errors are manifest in the transfer characteristics of the grating [45]. This causes multiple reflections and structure within the reflection envelope determined by the field size, while the reflection bandwidth is inversely dependent on the overall length of the grating. Techniques have to be applied to reduce the effects of stitching errors in phase masks by altering the field size of each subgrating processed by the e-beam. By overlaying N e-beam exposures with different field sizes, each with 1/N of the dose, the total dose required to imprint the grating pattern in the photoresist on the phase mask is maintained while averaging out the periodic nature of the stitching errors. Developing the resist dramatically reduces the effects of the stitch errors, which appear as multiple out-of-band reflections. This has been successfully demonstrated [46], and the effects on the reflection spectrum are shown in Fig. 3.13. Another technique, albeit used less successfully, monotonically increased the field sizes for a 14-mm long grating, from 100 mm to 200 mm in steps of 1.055 mm. Although many of the features were eliminated, field sizes around 200 mm produced a cluster in the reflection spectrum [46], since the fractional change in the field-size remains small. Stitching errors or undesirable chirp in a phase mask are replicated in a fiber grating. It is possible to use the technique of “UV trimming” [47] to adjust the local refractive index in the fiber to correct the transmission spectrum. Scanning a UV beam across a phase mask while also moving the fiber enables the chirp in the phase mask to be compensated for [48,49]. By adjusting the velocity of the fiber relative to the scanning UV beam at different positions along the phase mask, the induced refractive index can be changed, altering the local Bragg
1.4715 0
1.4725
1.4735
Measured Computed
–5 Reflection, dB
1.4745
–10 –15 –20 –25 –30 –35
A
–40 1.4715 0
1.4725
1.4735
1.4745
1.473
1.474
1.475
Reflection, dB
–5 –10 –15 –20 –25 –30 –35
B
–40 1.472 0
Reflection, dB
–5 –10 –15 –20 –25 –30 –35
C
–40 Wavelength, microns
Figure 3.13 Reflection spectra from imprinting with different phase masks as a function of the number of overlaid e-beams exposures of the phase mask. This grating is discussed in Section 3.1.13. (a) Single-pass field size of 65 mm; (b) three passes with 65.4, 55.3, and 59.26 mm; (c) as for (b) with additional 66.04 mm pass (from: Albert J., Theriault S., Bilodeau F., Johnson D.C., Hill K.O., Sixt P., and Rooks M.J. “Minimisation of phase errors in long fiber Bragg grating phase masks made using electron beam lithography,” IEEE Photon. Technol. Lett. 8(10), 1334–1336, 1996. # 1996 IEEE [46]).
Methods for Fiber Bragg Grating Fabrication
69
wavelength. If the phase mask has an unintended chirp, the fiber grating can be “trimmed.” This technique has been applied to reduce the chirp of a grating written by using a 100-mm long phase mask that had undesired chirp situated close to the middle of the mask. This was found by monitoring the growth of the reflection of a grating. The velocity is adjusted in steps (5 sec to 22 nm/sec) with the help of a piezoelectric stage, while the UV beam is scanned at a velocity of 250 mm/sec during fabrication of a second grating. The induced wavelength shift is directly related to the velocity of the scanning beam vuv and that of the fiber vf as Dl ¼ lvf/vuv. Removal of the chirp reduced the bandwidth of the grating from 0.23 nm to 0.1 nm [48]. The elimination of out-of-band ghosts is important for telecommunications, while the postfabrication repair of expensive phase masks is very useful for fabrication.
3.1.6 The Lloyd Mirror and Prism Interferometer In the double mirror arrangement of the interferometer shown in Fig. 3.1 the incident beam is split into two, and fringes form from interference between identical copies of the incident radiation. Any phase distortion across the input beam is automatically compensated for at the fringe plane, or can be compensated for by the introduction phase plates, so that local visibility of the fringes remains close to unity and chirp in the period of the fringes reduced to zero. Figure 3.14 shows an arrangement for an interferometer based on a single mirror known commonly as the Lloyd mirror. A parallel beam incident at the UV Gaussian-beam
qm /2 Wuv
Foldmirror
Fiber Wuv /2 Intensity of fringe pattern Figure 3.14 The Lloyd mirror interferometer, showing the intensity of the fringes formed at the fiber with a folded Gaussian beam [50].
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Fabrication of Bragg Gratings
surface of the mirror at a shallow angle is reflected across the path of the beam. Interference occurs in the region of overlap of the reflected and unreflected parts of the incident beams. The interferometer is therefore extremely simple and easy to use. However, since half of the incident beam is reflected, interference fringes appear in a region of length equal to half the width of the beam. Secondly, since half the beam is folded onto the other half, interference occurs, but the fringes may not be of high quality. In the Lloyd arrangement, the folding action of the mirror limits what is possible. It requires a source with a coherence length equal to at least the path difference introduced by the fold in the beam. Although a phase plate may be used over half the beam to compensate, experimentally this is not straightforward. Ideally, the intensity profile and coherence properties should be constant across the beam; otherwise, the fringe visibility will be impaired and the imprinted grating will be nonuniform. Since most sources tend to have a Gaussian beam profile, it is difficult to produce fringes which have a uniform transverse profile. The grating profile remains halfGaussian, unless the beam is expanded to provide a more uniform profile. The Gaussian intensity profile of the fringes introduces a chirp in the imprinted grating. Diffractive effects at the edge of the mirror may also cause a deterioration of the fringes closest to it. The Lloyd arrangement with the single mirror is easy to tune. However, the fiber axis should be placed orthogonal to the plane of the mirror so that the grating is not slanted. Replacement of the mirror by a prism in the Lloyd arrangement results in a more stable interferometer. This is shown in Fig. 3.15. The UV writing beam is now directed at the apex of a UV-transmitting silica right-angled prism such that the beam is bisected, as in the Lloyd mirror. Both halves of the UV beam are therefore refracted and no longer travel in air paths, which can change with time. The interferometer thus becomes intrinsically stable and is used to produce the first photo-induced fiber Bragg gratings in the 1500-nm wavelength window [51]. The interferometer has been used to demonstrate a distributed-feedback dye laser. Interfering the pump beams in the dye at the appropriate angle to create a Bragg reflector (absorption grating) causes the dye to emit laser radiation [52]. The advantages and disadvantages of this interferometer are similar to those of the Lloyd mirror. However, there are two further points of interest. Owing to the shallow angle subtended by the UV beam on the hypotenuse, the prism face has to be much larger than the beam width, but because of refraction, the side of the prism at which total internal refraction occurs is smaller than the length of the Lloyd mirror [see Section 3.1.3, similar to Eq. (3.1.6)]. For a given beam width, the prism interferometer expands the length of the grating, and this is
Methods for Fiber Bragg Grating Fabrication
71
UV beam bisected by apex of prism W qi a
qr qm /2 Fiber Lg Figure 3.15 The Lloyd prism interferometer. The interfering beam paths are within the bulk of the prism; however, there is a path difference introduced between them due to the refractive index of the prism. The UV beam must be spatially and temporally coherent with a uniform intensity for the production of high-quality gratings.
shown in Fig. 3.16. Using simple geometry, the length of the grating Lg may be shown to be W ym sin a þ cos a tan Lg ¼ ; ð3:1:9aÞ 2 2 cos yi
2.5
90 Theta, degrees
70
Lg/ W
2
50
1.5
30
1
Lg/ W
Angle of incidence qi, degrees
the parameters for which are defined in Fig. 3.15.
0.5
10 39
49 59 Prism apex angle a, degrees
Figure 3.16 Length of a grating normalized to the width of the UV beam as a function of prism apex angle a. The writing angle ym/2 is fixed at 10 for a Bragg wavelength of 1500 nm. Also shown is the external writing angle of incidence on the surface of the prism, yi. Note that since the beam is folded about the center, the grating length is approximately half the width for the Lloyd mirror, whereas with the prism, the grating can be longer.
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Fabrication of Bragg Gratings
At large angles of incidence (small apex angles), the grating length increases rapidly but reflection losses increase at the same time. The polarization useful for writing a grating with low birefringence (p-polarized) is reflected more than the unwanted polarization (s-polarized) [40] (see Section 3.1.8). Antireflection coating of the surface will naturally reduce this loss. Dispersion in the silica prism may also be an issue when writing gratings longer than a few millimeters. Other techniques for the production of gratings with a single prism in a slightly modified version of that shown in Fig. 3.15 have also been reported [53,54]. These methods require precision-fabricated prisms and are restricted in tunability of the Bragg wavelength but may prove to be useful in cases where a rudimentary inscription procedure is necessary.
3.1.7 Higher Spatial Order Masks According to Eq. (3.1.4) the period of the mask may be integer multiples of the Bragg wavelength. It is therefore possible to use a coarser grating period than the fundamental period required for the reflection wavelength. As the period gets larger, it is not practical to use the phase mask as a diffraction element, and it is necessary to use an amplitude mask for direct replication of the grating. The larger periods also allow the photoreduction of the mask using imaging to imprint the correct period grating in a fiber [55,56]. The latter scheme requires the higher-order mask to be M the length of the photo-reduced grating, where M is the demagnification factor, posing a problem for the production of fiber gratings that are longer than a few millimeters. The projection scheme enables the production of gratings with a single 20 ns pulse from a KrF excimer laser at a wavelength of 248 nm. One- to six-micron period gratings have been produced by projection of an amplitude mask with multilayer stacked high-reflectivity dielectric stripes as the pattern (5 to 120 mm wide stripes). An image demagnification of 1:10 was used with 0.3 NA optics to produce 4-mm from long gratings of 6th, 11th, and 12th order. Reflectivities as high as 70% were noted for the lower-order gratings with correspondingly lower reflectivities of 8 and 2% for the 11th and 12th orders, respectively. An advantage of the projection system is that the fluence at the fiber is increased by the demagnification, reducing the power density at the mask plate [56]. Also noted was the imprinting of a physical grating on the surface of the fiber cladding, penetrating some 2 mm into the cladding. A threshold for the production of the grating in the core at 0.8 J/cm2 was observed, while 1.4 J/cm2 was required for optimal production of the grating. It appears that the physical damage grating in the cladding produces a phase grating in the core due to heavy surface modification, causing light to be scattered out
Methods for Fiber Bragg Grating Fabrication Excimer laser 248nm aperture
73
Field lens Phase mask
X10 0.3NA Fiber to diagnostics XYZ stages Figure 3.17 The projection system used to photoinscribe gratings by the photoreduction of the phase-mask (from Rizvi N.H., Gower M.C., Godall F.C., Arthur G., and Herman P., “Excimer laser writing of submicrometre period fiber Bragg gratings using a phase-shifting mask projection,” Electron Lett. 31(11), 901–902, 1995, # IEE [55]).
of the core. Gratings formed by physical damage, known as Type II, will be discussed in Section 3.2. A similar technique of photoreduction can also be applied for the projection of a phase mask and is shown schematically in Fig. 3.17. Projection of the phase mask rather than the amplitude mask overcomes the problem of the Rayleighlimited resolution for the 0.3 NA UV transmitting lens used in a photoreducer. The limit with the amplitude mask is 0.6 mm. It therefore cannot be used for the production of first-order gratings [55] with periods of 0.5 mm. Disadvantages of the projection scheme for use with both the amplitude and phase mask are the requirement of large-scale high-quality UV-grade spherical optics, and the use of large-area amplitude and phase masks. The additional cost and complexity of the projection system may well offset the cost advantage of using coarser features. A 10.66-mm period phase mask has been photoreduced by 10 to generate first-order gratings in a Ge-doped fiber at a Bragg wavelength of 1530 nm. It was reported that the production of damage Type II gratings was more reproducible using the phase-mask projection scheme [55]. Observation of higher-order interactions is possible with a first-order phase mask. This is simply governed by Eq. (3.1.4). A grating written for a Bragg wavelength lBragg in first order will also operate at wavelengths l ¼ lBragg/N. Additionally, if the grating imprinted in the fiber has a nonsinusoidal amplitude
74
Fabrication of Bragg Gratings
profile, for example, by effects of saturation of the refractive index modulation or physical damage (e.g., square wave modulation amplitude), then the grating will have Fourier frequency components at multiples of the first order, as L ¼ Lg/m (m ¼ 1, 2, 3 . . .). These will in turn affect the efficiency of the reflections at shorter wavelengths, but function as first-order gratings for the Bragg wavelengths matching each of the spatial harmonic frequencies. High-intensity UV printing through a phase mask results in multiple-order reflections, not least by the interference of the zero-order beam through the phase mask with the 1 orders, but also due to the formation of a damage grating which is no longer sinusoidal in amplitude [120].
3.1.8 Point-by-Point Writing The period of a reflection grating operating at 1.5 mm is 0.5 mm, as per Eq. (3.1.3). Since a diffraction-limited spot size of radiation at 244 nm is 0.25 mm, it is possible in principle to form a periodic refractive index grating by illuminating a single spot at a time using a point-by-point writing scheme. Technically, using positioning sensors linked to an interferometer, a grating of such periods can be written. This is only suitable for short gratings, since it is difficult to control translation stage movement accurately enough to make point-by-point writing of a first-order grating routinely practical. Other methods, including the use of a phase mask or the multiple-printing in-fiber-grating scheme, are better suited to writing long first-order gratings. However, highquality high-order gratings have been demonstrated for N ¼ 3 and 5 [57]. While excellent reflection gratings can be written using other schemes, pointby-point writing is extremely useful for fabricating gratings of long periods (>10 mm). These gratings couple light from one propagating polarization mode to another in the backward [58] or forward direction as in a rocking filter [59–61] (discussed in Chapters 4 and 6), to forward-propagating radiation modes (also see Chapter 4), or from one guided mode to another [62–64]. Figure 3.18 shows the technique used for point-by-point writing of reflection gratings and polarization couplers. For the simple reflection grating, the fiber is illuminated by a tightly focused spot through a mask for the required duration before being translated by a motorized micropositioner for the next illumination. In this way, a reflection grating of any order may be written. Naturally, the method benefits from the use of a pulsed laser, since the motion of the fiber can be stepped without the need to control the operation of the laser as well. However, the method is most useful for long-period gratings, which do not require such a demanding positional accuracy.
Methods for Fiber Bragg Grating Fabrication
75
Pulsed UV radiation Mask
A
Rollers
Fiber Induced index grating LP01x LP01y [n(LP01x) - n(LP01y)]/l = N/Λ Mask
Fiber
B
Blazed grating LP01
Angled slit LP11
[n(LP01) - n(LP11)]/l = N/Λ
Figure 3.18 Point-by-point writing of fiber gratings. (a) shows a uniform grating being written as the fiber is pulled forward. A focused beam from a pulsed laser illuminates the fiber through a slit. The fiber pulling speed and the laser pulse rate determine the period of the grating. (b) Blazed gratings are written with a slanted mask. Coupling between dissimilar modes is therefore possible (after Ref. [57]).
3.1.9 Gratings for Mode and Polarization Conversion Polarization mode converters may be fabricated in birefringent fibers using this scheme. In this case, it is necessary to orient the birefringent axes of the fiber at 45 to the illuminating beam. Two methods have been demonstrated to form such polarization converters. In order to use a nonbirefringent standard telecommunication fiber for a polarization mode converter, it must be made birefringent. The fiber can be wrapped on a cylinder of an appropriate diameter to induce a specific birefringence [65]. The beat length of the modes in the fiber is a function of the bend-induced birefringence and may be changed by altering the diameter of the cylinder. The induced birefringent axes are along the radius and parallel to the surface of the cylinder, with the fast axis in the radial direction. The induced birefringence B is [66] dfiber 2 ; ð3:1:10Þ B ¼ nfast nslow ¼ a Dcylinder where dfiber and Dcylinder are the diameters of the fiber and cylinder, respectively, and the fast and slow axis refractive indices n are indicated by the subscripts. a is a constant that is dependent on the photoelastic properties of the fiber material, 0.133 for fused silica. Typically, the induced birefringence in fibers wrapped around the smallest practicable diameter cylinders (Dcylinder 25 mm), based on consideration of mechanical strength, is of the order of 2 106. While this value is large
76
Fabrication of Bragg Gratings
compared to the intrinsic birefringence of standard telecommunications fiber, it is well below that of birefringent optical fibers (maximum birefringence Bmax 0.4 Dn2, where Dn is the core-cladding index difference [67]; for Dn ¼ 0.04, Bmax 6.4 104). If a grating with a period, Lg ¼ l/B is written in the fiber, then coupling between the two polarizations will occur. However, the grating has to be written oriented at 45 to the fast and slow axes. This may be done simply by arranging the UV illumination in a direction of the axis of the cylinder but rotated at an angle of 45 to the surface of the cylinder. As the fiber moves across the slit, the laser is switched on so that half the period of the grating is exposed to UV radiation before it is switched off for the second half. The process is repeated until the required number of periods is written [59]. Parameters that may be varied are the angle of inclination C (known as the rocking angle), as shown in Fig. 3.19, and the mark-space ratio Ag of the grating given by the ratio of the length of fiber exposed to UV radiation and the grating period, in any one grating period. A transversely uniform grating will promote coupling between modes of the same order whereas coupling of different-order modes requires a blazed grating [68] or, equivalently, a grating that is transversely nonuniform. A blaze may be imparted by rotating the mask slit so that the exposed region makes an angle to the propagation axis in the fiber, as shown in Fig. 3.18b. The blaze angle and the mark-space ratio Ag ¼ Lexposed/Lg are again parameters that may be adjusted to alter the performance of the filter. Blaze also enhances coupling to radiation modes of the fiber, inducing transmission loss, as has been mentioned in Section 3.1.4. However, the period of the grating Lg required to couple between modes is generally much longer than that required to couple to the radiation modes in the counterpropagating direction. This is no longer true for coupling to copropagating radiation modes, and care needs to be taken to ensure that the periods of the gratings are not identical, by choosing an appropriate fiber. Intermodal coupling has been demonstrated using internally [62,69], as well as externally written gratings for different order mode coupling [61], as well as similar order modes [64]. The functioning of these devices is discussed in Chapter 6.
UV radiation ψ Birefringent fiber cross-section Figure 3.19 The rocking angle C may be varied to alter the coupling coefficient for the filter.
Methods for Fiber Bragg Grating Fabrication
77
3.1.10 Single-Shot Writing of Gratings Single-shot writing of fiber gratings has been demonstrated using pulses from an excimer laser [118–121]. Higher reflectivity gratings have also demonstrated in boron–germanium codoped optical fiber [123]. Although the quality of these gratings has not been comparable with those written with other methods, the principle has led to a novel scheme of writing grating in the fiber while it is being drawn from a preform. The process of writing a grating in an optical fiber generally requires the stripping of the protective polymer coating, which is opaque to short-wavelength UV radiation. Stripped fiber is weakened owing to mechanical processing (see Chapter 9) during grating inscription and should ideally be recoated. However, grating inscription on a fiber-drawing tower enables the fiber grating to be coated immediately after fabrication. Further, an array of gratings may be fabricated sequentially in a length of fiber by stepping the Bragg wavelength after each inscription. Figure 3.20 shows a schematic of
Grating
Pulsed UV beam
Fiber draw tower
Polymer coating bath
Figure 3.20 The scheme of writing gratings in a fiber while it is being drawn from a preform: 20-ns pulses at a wavelength of 248 nm were used to inscribe a sequential array of 50 gratings stepped at 0.1 nm. Also demonstrated were 120 gratings written at the same wavelength. The wavelength stability was good, but the peak reflectivity was 3%. Gratings with a wavelength spacing of 5 nm were also reported [122].
78
Fabrication of Bragg Gratings
grating fabrication during fiber drawing that was originally proposed by Askins et al. [122] and subsequently demonstrated [123]. Stepped-wavelength gratings were demonstrated using an interferometer that was tuned to a different Bragg wavelength between pulses from the excimer laser [122]. Although the gratings can be written during fiber drawing, the quality and repeatability remain poor, owing to problems of beam uniformity, mechanical alignment, and stability. For some applications in which a simple reflection is required, the quality of the grating may not be a critical parameter. Advances in polymer coating materials have resulted in perfluorinated polymers that are essentially transparent in the longer UV wavelength regions of 266–350 nm [70,71]. Grating inscription at these wavelengths has been demonstrated as well, indicating that high-quality and high-reflectivity gratings may be possible without stripping the coating off the fiber. The fabrication of gratings in fiber with the primary coating may well be restricted to the use of CW UV sources, since high-power pulsed sources are likely to damage the polymer coating unless tight focusing is used to reduce the intensity at the surface [119].
3.1.11 Long-Period Grating Fabrication Long-period gratings have found applications as lossy filters. These gratings couple the guided and the radiation modes in the copropagating direction, and typically require periods in the region of 100 to 500 mm. LPGs are therefore most conveniently fabricated by UV exposure through a shadow mask [72]. These gratings tend to be longer than reflective Bragg gratings (tens of millimeters) since the periods are a few hundred times longer. However, shadow masks are easily available, and the inscription requires simple contact printing as with the phase mask, but without the complexity of interferometry. Point-by-point writing is also possible. Long-period amplitude masks have been patterned on dielectric mirrors to reduce the problem of optical damage with excimer lasers. In this application, a dielectric mirror coated with photoresist was first exposed to an amplitude pattern using a UV laser. The photoresist was developed to expose the dielectric mirror, which was then etched in 5% HF solution in deionized water. The resultant mirror had stripes at the required period for the LPG. Exposure through this mirror only allows the UV radiation to be transmitted through the regions where the mirror has been etched away. These mirrors were shown to withstand 200 mJ/cm2 per pulse over several pulses, thus making them better suited for use than chrome-coated silica masks with an average damage threshold of 50–100 mJ/cm2 [131].
Methods for Fiber Bragg Grating Fabrication
79
3.1.12 Ultralong-Fiber Gratings To circumvent the limitations of the finite length of the phase mask, several techniques have been proposed to fabricate gratings of arbitrary length >200 mm [49,73–75]. The simplest method is to sequentially inscribe gratings in a fiber from a phase mask of length L, to result in a grating with a length equal to the number of sequential inscriptions L [74]. This is a powerful technique, which has special applications in the fabrication of long chirped gratings and is discussed in Section 3.1.15. A technique based on the principle of inscribing small, more elementary gratings to create a longer one has also been reported [73]. The principle of inscription may be understood as follows: A short (4 mm) interference pattern is printed periodically in a continuously but slowly moving fiber. Using a pulsed laser (20-ns pulses), a 4-mm long section is imprinted in the fiber at any one time. The velocity of the fiber is such that within the pulse width, it may be regarded as being stationary. When the fiber has moved a few integral numbers of grating periods, a second pulse arrives, imprinting yet another grating partially overlapped with the previous grating but adding a few extra periods to the length. A mini-Michelson interferometer operating at 633 nm is attached to the moving fiber platform to track its position relative to the interference fringes. The latter is undertaken by the use of serrodyne control [73]. The resulting gratings have the narrowest reported bandwidths of 0.0075 nm, although the quality of the grating was not perfect. It is important to eliminate stitching errors between the imprinted fields, as in the case of the phase mask, requiring a positional accuracy of better than 0.1L over the length of the grating. As an example, this implies maintaining an overall positional accuracy of less than 50 nm over the entire length of the 200-mm long grating, a demanding task. Figure 3.21 shows the apparatus used for writing long gratings by the multiple printing in fiber (MPF) technique [73]. The fiber is held in a glass V-groove along its entire length and translated along the pulsed interference fringes in synchrony with the pulses. By moving the fiber at a constant velocity with a linear motor, the vibrations common in stepper-motor-driven systems are eliminated. Mounting the fiber carriage on an air bearing further helps this. The critical features of the technique are the requirement of a long precision glass V-groove to hold the fiber in position with submicron accuracy, high beam quality of the pulsed laser, an accurate control system for fiber translation, and above all, stability of the pulsed source for imprinting gratings of a welldetermined index of modulation. Since the fabrication process imprints overlapping gratings, it is possible to change the period or the modulation index locally or continuously along the length of the grating. A detail of the printing process
80
Fabrication of Bragg Gratings UV-interferometer
Consecutively north-south flipped magnets in stator part of linear drive
“Rotor” part of linear drive
Retro-reflector prisms Polarization beamsplitter
Fiber holder
Detectors Air-cushion born movable carriage
Frequency stabilized He-Ne
Beam expander l/2-plates
Polarization beamsplitter Pockels cell Beamsplitter
Figure 3.21 The multiple printing in fiber (MPF) grating technique. The substantial carriage is potentially capable of movements of up to 1 meter (courtesy R. Stubbe).
is shown in Fig. 3.22. This system is flexible and is able to cater to any type of grating, including those with phase steps, chirp, and apodization (see Section 3.1.9 and Chapter 5). A slight modification of the MPF scheme is shown in Fig. 3.23. Here not only the fiber is allowed to move, but also the interferometer [49]. In this case, the interferometer is the phase mask that moves at a velocity of vpm, while the fiber moves with a velocity of vf. The change Dl in the period Lg of the grating is a function of the relative velocities, as vf : ð3:1:11Þ Dl ¼ LBragg vpm
Syncronized pulses of 240nm light in a two-beam interferometer controllable angle
Grating sequence
Interferometrically controlled motion of fiber Figure 3.22 A detail of the MPF technique showing the imprinting of a mini overlapping grating to produce the required profile or chirp; the movement of the fiber is interferometrically controlled, and the imprinting is synchronized with the arrival of the UV pulses (courtesy R. Stubbe).
Methods for Fiber Bragg Grating Fabrication
81
UV beam Rollers Moving phase mask/UV
Moving fiber, vf Δl = ΛB v /v f
pm
vpm
Apodization and chirp, but lower modulation index
Figure 3.23 A technique based on the moving phase mask and fiber. If the phase mask is not moved, it is identical to the MPF method, but can only be used to write unchirped gratings using a pulsed source.
Thus, a chirp may be programmed in the control computer by altering the relative velocities locally. A limitation is the need for a small spot size if a large chirp is being imparted in the grating in a long fiber, since the maximum change in the period is a single period over the length of the UV writing spot, as Dl ¼
LBragg ; 2w
ð3:1:12Þ
where w is the radius of the spot used for writing the short section of grating. Combining Eqs. (3.1.11) and (3.1.12) gives the following interesting relationship: vpm : ð3:1:13Þ 2w ¼ vf Equation (3.1.13) suggests the use of a minimum spot size related to the relative velocities. Further, it should be noted that at any one time, an entire grating of spot size 2w is written with a constant amplitude and period. In the limit, this method trades in some of the refractive index modulation for chirp, but can at best imprint a quasi-stepped function instead of a continuous one, especially when the grating is being apodized.
3.1.13 Tuning of the Bragg Wavelength, Moire´, Fabry–Perot, and Superstructure Gratings The effective index of a propagating mode in a fiber is both temperature and strain sensitive. The functional dependence of the mode index is given by the relationship @neff ¼
@nneff @nneff DT þ Ds; @T @s
ð3:1:14Þ
where @n/@T is the temperature coefficient of refractive index, DT is the change in temperature, @n/@s is the longitudinal stress optic coefficient, and Ds is the
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Fabrication of Bragg Gratings
applied longitudinal stress. Since the Bragg wavelength is a function of neff [see Eq. (3.1.4)], the simplest method of altering the transfer characteristics of a fiber grating is to impose a temperature or strain profile along the length of the grating. However, prestraining a fiber during grating fabrication alters the Bragg grating wavelength in the relaxed state [7]. It is also possible to multiplex several gratings at the same location to form Moire´-type gratings [7,77]. It should be noted that the Bragg wavelengths of all multiplexed gratings written at the same location shift to longer wavelengths as each grating is superimposed. The shift in the wavelength of the gratings is dependent on the overall change in the index of modulation, resulting in a change in the period averaged neff of the mode in the fiber. The shift DlBragg in the Bragg wavelength, lBragg as the UV induced index change dn increases can be shown to be DlBragg ¼ lBragg
dn ; neff
ð3:1:15Þ
where < 1, is the overlap of the guided mode and the distribution of the refractive index modulation (see Chapter 4). Thus, when a grating is superimposed on an already-written grating, both gratings move to longer Bragg wavelengths. By altering the angle of the interfering beams, several gratings may be written at a single location using the prism interferometer or the Lloyd mirror arrangement discussed in Section 3.1.6. These gratings show interesting narrow bandpass features with uniform period [77] or chirped gratings [78], and are discussed in Chapter 6. If the temperature distribution along the length of a uniform grating is a linear function of length, then the Bragg wavelength, too, will vary linearly with length. The grating will demonstrate a linear chirp. This means that the different wavelengths within the bandwidth of the grating will not be reflected from the same physical location and the grating will behave as a dispersive component. The temperature profile (or the strain profile) may be altered to change the functional property of the grating [79]. On the other hand, prestraining or imposing a temperature profile along a fiber prior to writing a fiber grating will also result in a chirped fiber grating once it is written and the stress/temperature profile is removed [80,81]. However, the chirp in a grating fabricated in such a way will have the opposite sign of a grating chirped by the application of a temperature or strain profile after it has been manufactured. During fabrication of the grating at an elevated temperature Tw, the Bragg wavelength will be defined by the period, Lg of the grating. After fabrication, when the temperature is returned to a final temperature Tf, the Bragg wavelength will be
Methods for Fiber Bragg Grating Fabrication
dneff Tf Tw ; lBragg ¼ 2Lg ð1 þ a½Tf Tw Þ neff þ dT
83
ð3:1:16Þ
where a is the thermal expansion coefficient of the fiber and dneff/dT is the temperature coefficient of the mode index; to the first approximation, this is merely the change in the refractive index of the fiber core as a function of temperature. Equation (3.1.16) may be simplified by expanding and rearranging to dneff ðTf Tw Þ : ð3:1:17Þ lBragg ¼ 2Lg neff ½1 þ aðTf Tw Þ þ dT The thermal expansion coefficient of silica a is approximately þ5.2 107, whereas dn/dT þ1.1 105 C1; the contribution of the thermal expansion coefficient term is approximately 10% in comparison. Equation (3.1.17) is further simplified to dn0 ðTf Tw Þ ; ð3:1:18Þ lBragg 2Lg neff þ dT where the combined effect of the thermal expansion and the refractive index change is included in n0 , so that the change shift in the Bragg wavelength is simply LlBragg 2Lg
dn0 ðTf Tw Þ: dT
ð3:1:19Þ
Typically, dn0 /dT 0.5 to 1.0 105 C1. At a wavelength of 1500 nm the change in the Bragg wavelength with temperature is 1 to 2 102 nm C1 [82,7]. With long uniform gratings, a thin heating wire suitably placed below a point in the grating can result in a distributed feedback (DFB) structure, with a doublepeaked reflection spectrum. The transfer characteristics of each half of the grating are identical; however, a l/4 phase difference induced by the heating wire causes a hole to appear within the band stop [83]. Such a grating in rareearth-doped fiber can be used in DFB lasers, which require the suppression of one of the two lasing modes to force the laser into single-frequency operation, and in narrow band-pass filters. A number of methods have been reported for fabricating DFB structures in fibers, including postprocessing a uniform grating to locally induce a “gap” in the center of the grating [84]. Alternatively, two gratings may be written on top of each other, each with a slightly shifted wavelength to form a Moire´ phase-shifted grating, opening a bandgap once again [77]. Radic and Agrawal [85] reported that using an additional quarter-wave phase shift within a grating opens up yet another gap. An extension of this principle directly leads to the superstructure grating, which has been extensively used in tunable semiconductor-laser design [86]. A schematic of the superstructure grating is shown in Fig. 3.24. The composite grating consists of a number
84
Fabrication of Bragg Gratings Lg dl
ΔL
Figure 3.24 A schematic of a superstructure grating. This is constructed by blanking (N – 1) sections of length dl in a long continuous grating of length Lg. The superstructure grating is a collection of cascaded Fabry–Perot interferometers.
of subgratings of length DL (but not necessarily of identical lengths), which are separated by “dead” zones of length dl (these lengths may be different). The superstructure grating was first demonstrated in an optical fiber by Eggleton et al. [87], produced by a phase mask. A more general problem of stitching errors in phase masks has been addressed by Ouellette et al. [88]. Multiple reflections occur within the bandwidth of a single subgrating; each reflection has a bandwidth defined by the length of the grating without the gaps, i.e., NDL ¼ Lg – (N – 1)dl. The Fourier components of the grating shown in Fig. 3.24 basically have a fundamental component with a uniform period Lg and a fundamental modulation envelope of period Le ¼ dl þ DL. Thus, the reflection spectrum will have components at the sum and difference frequencies. The new reflection waveþ lengths, l Bragg and lBragg , are calculated from Eq. (3.1.4): 1 1 1 ¼ þ L Le l 2n 2n av g av Bragg
ð3:1:20Þ
1 1 1 ¼ ; L Le 2n 2n lþ av g av Bragg
ð3:1:21Þ
and
where nav is the average index of the mode. It follows, therefore, from and Lþ Eqs. (3.1.20) and (3.1.21) that the new grating periods L g g corresponding to the superscripted Bragg wavelengths are 0 1 L gA @ L g Lg 1 Le ð3:1:22Þ 0 1 L gA @ : Lþ g Lg 1 þ Le From Eqs. (3.1.4) and (3.1.22), it follows that the spacing Dl between the reflected wavelengths in any such superstructure is
Methods for Fiber Bragg Grating Fabrication
Ll ¼ 2nav
85
L2g : Le
ð3:1:23Þ
Depending on the shape of the composite structure (and thus the magnitudes of the individual Fourier components), higher order components can appear [87]. Superstructure gratings represent a number of types of gratings: the DFB with a single phase step f < 2p [85,86]; the grating Fabry–Perot with f 2p resulting in multiple high-finesse transmission peaks [89]; cascaded grating Fabry–Perots in which the phase steps f 2p [87], leading to replicated multi-band-pass transmission spectra; gratings with multiple-flat-top reflection spectra, fabricated with unequal grating lengths as well as phase steps [90]; and truly flat-top reflection gratings fabricated by introducing regular p/2 phase shifts (l/4) at equal intervals, while altering the strength of the grating amplitude in each section to mimic a sinc function [91]. There are several methods of making these gratings. In order to introduce a specific phase step, the simplest and most reliable is via the replication of a phase mask with the appropriately prerecorded phase steps [92]. Another method successfully used for introducing a p/2 phase step in a grating is by UV postprocessing [84]. After the grating has been written, the small central section is illuminated with UV radiation to introduce a phase shift. As the refractive index of the exposed region of length dl increases by dn, the transmission spectrum of the grating is monitored to stop the exposure when dn dl ¼ l/4. A disadvantage of post-UV exposure is that it not only changes the phase between the two halves of the grating, but also alters the local neff of the fiber. This in turn shifts the Bragg wavelength of the already-written grating exposed to UV radiation to a longer wavelength. The effect of the shift is a slightly broader overall reflection spectrum. The use of an amplitude mask in conjunction with a phase mask allows the precise printing of a superstructure grating [93]. Of course, mini-gratings may be printed by precise translation of the fiber between imprints [91,73]. This method has been used to write a sinc function grating with remarkably good results. However, it is difficult to write a continuous sinc function. Approximating the sinc function in a limited number of steps creates additional side bands, which limits the out-of-band rejection in the reflection spectrum. Combing the sinc function grating with apodization results in an improved transfer function, increasing the depth of the out-of-band rejection [91]. Chirped gratings are useful for many applications. There are a number of ways of chirping gratings, including writing a uniform period grating in a tapered fiber [94], by application of varying strain after fabrication [43,79,95], by straining a taper-etched fiber, by fabrication by a step-chirped [96] or continuously chirped phase mask, or by using one of the several schemes
86
Fabrication of Bragg Gratings
of writing a cascade of short, varying-period gratings to build a composite, long grating. These methods for writing chirped gratings are discussed in Section 3.1.14. The properties of many of these gratings along with their applications may be found in Chapter 6.
3.1.14 Fabrication of Continuously Chirped Gratings Short, continuously chirped gratings are relatively straightforward to fabricate; longer (>50 mm) ones become more difficult. One of the simplest methods is to bend a fiber such that a continuously changing period is projected on it. This is shown in Fig. 3.25 in which the fiber is bent either in the fringe plane or orthogonal to it. Altering the lay of the fiber may change the functional dependence of the period on position, so that either linear or quadratic chirp may be imparted. Figure 3.26 shows a curved fiber with a radius of curvature R in a fringe plane. At any point of arc a distance S from the origin O where the fiber axis is normal to the fringe planes, the local period of the grating can be shown to be Ls ¼
Lg ; cosðS=RÞ
ð3:1:24Þ
so that even with a large radius of curvature, the grating may be substantially chirped. Gratings with bandwidths of 7.5 nm and peak reflectivity of 99%, as well as 15-nm bandwidth with a peak reflectivity of 5%, have been reported with this technique [98].
Interfering beams
Fringes
Fiber
Figure 3.25 Writing of a continuously chirped grating by bending the fiber in the fringe plane. Note that the bending also causes the grating to be blazed with respect to the fiber axis.
Methods for Fiber Bragg Grating Fabrication
87
q R
Fiber bend radius = R
Λg
Fringes
Λs
S O
Figure 3.26 A fiber curved with a radius R in the fringe plane. The period of the grating seen by the fiber varies continuously along its length S.
As in the case of fiber tilted with respect to the fringe planes, bending has a similar effect of imparting a blaze and consequently radiation loss. The loss, which manifests itself on the short-wavelength side of the Bragg wavelength even in unblazed gratings, is increased by blazing and may not be desirable; for a chirped grating, this can be a serious problem, if the radiation loss spectrum lies within the chirped bandwidth of the grating. Loss due to coupling to cladding modes in chirped gratings can be reduced by using fibers that are strongly guiding, but cannot be entirely eliminated. It may be substantially reduced, however, by the use of special fibers with a photosensitive cladding [97] (see Chapter 4). A constant strain along the length of a fiber while a grating is imprinted merely shifts the Bragg wavelength on strain release. In order to impart a chirp, a nonuniform strain profile has to be used, and there are several practical methods for implementing this. If an optical fiber is tapered (e.g., by etching) such that the outer diameter varies smoothly in the region of grating, the application of a longitudinal force leads directly to strain that is nonuniformly distributed along its length. The local strain el may be computed from the local crosssectional area Ai as [80] ei ¼ ei1
Ai1 ; Ai
ð3:1:25Þ
where the subscript (i – 1) refers to the previous section of the grating. Thus, a linear change in the cross-sectional area leads to a linear chirp. Applying positive strain to a uniform-period grating in a tapered fiber has three effects: The
88
Fabrication of Bragg Gratings
mean Bragg wavelength of the entire grating shifts to longer wavelengths, and the grating reflection spectrum broadens, while being reduced in reflectivity. Alternatively, the tapered fiber may be strained before writing of the grating. On strain release, the grating becomes chirped, but now shifts to a shorter wavelength. If a chirped grating is written in a strained tapered fiber such that the longer wavelength is inscribed in the narrower diameter, releasing the strain has the effect of unchirping the grating. If the relieved strain is too large, the grating becomes chirped once again but with an opposite sign of chirp: i.e., the narrower diameter end has a shorter wavelength than the larger diameter end. The method is capable of high reproducibility, is simple to implement, and allows nonlinear chirps to be programmed into the grating [80]. There are two factors that affect the Bragg wavelength in strained etched fibers: the change in the physical length of the fiber, and the effective index of the mode through the stress-optic effect, Lljs¼0 ¼ 2Lg @neff þ 2neff Lg ei ;
ð3:1:26Þ
where the change in the Bragg wavelength is Dl under zero stress, after the grating has been written under local strain of ei, and dneff is the change in the effective index of the mode due to the strain-optic coefficient. The two quantities on the RHS in Eq. (3.1.26) have opposite signs, with the strain term being much larger than the stress-optic coefficient. Nevertheless, when a grating is written in a tapered fiber under strain, it appears chirped, although the period of the grating is uniform. Because of the stress-optic effect, the local effective index of the mode is not uniform along the length of the grating. On removal of the strain after the grating has been inscribed, the effective index of the mode becomes uniform, but the period is altered because of the change in local strain, and the grating becomes uniform at some lower strain value and chirped with the opposite sign when fully relaxed. Etched linear tapers produce nonlinear chirp, since the chirp is inversely proportional to the square of the radius of the section according to Eq. (3.1.25). Since the Bragg wavelength is directly proportional to the applied strain [see Eq. (3.1.14)], the induced chirp becomes nonlinear. To compensate for this disparity, a nonlinear etching profile [99] can be used, resulting in a linear chirp. In order to fabricate a predetermined etch profile, the time of immersion of the fiber in the etching solution (usually buffered hydrofluoric acid) must be controlled, since the etch rate at constant temperature is highly reproducible. A three-section vessel with a layer A of a mixture of decahydronaphthalene and dichlorotoluene (10%) floating above layer B of 32% HF and with a third layer C of trichloroethylene below it may be used. This allows the acid to come into contact only with a small section of the fiber when it is immersed vertically into it. With the fiber remaining in position, the vessel is lowered at a
Methods for Fiber Bragg Grating Fabrication
89
programmed rate to expose another part of the fiber in section B, while the top layer A immediately stops the fiber from etching any further. Using this method, highly repeatable tapers have been produced and linear chirps of 4.8 nm demonstrated [100]. Using the same method, different structures have been fabricated, such as a grating with a p/2 phase shift in the middle. A fiber in which the taper in one half has been etched more than the other was stretched before a grating was written. Relaxing the fiber introduced a phase shift and therefore a band pass in the center of the transmission spectrum of the otherwise unchirped grating [99]. A somewhat less flexible method relies on the tapering of the fiber core [94]. Tapering the core affects the local effective index neff of the mode. To the first approximation, the effective index varies linearly with decreasing diameter for an initial fiber V-value of approximately 2.4, but varies more slowly, asymptotically approaching the cladding index, as the core diameter goes to zero. Thus, a uniform-period grating written in tapered section will be chirped. The maximum chirp Dl achievable in a fiber may be calculated from Dl lBragg
Dn ; neff
ð3:1:27Þ
where Dn is the difference between the mode index and the cladding refractive index, Eq. (3.1.27) translates to a maximum chirp on the order of 30 nm (Dn ¼ 0.03); however, it would be difficult for practical reasons to achieve more than 10 nm of chirp. A chirp of 2.7 nm for a 10-mm long grating was reported for a fiber tapered by 50 mm over that length [94]. Local heating and stretching may fabricate a tapered fiber. Note, however, that a fiber with a large taper will have a lower reflectivity for the shorter wavelengths (with a uniform period grating), since the mode power spreads to the nonphotosensitive cladding, reducing the efficiency of the grating. Another technique that overcomes the problem associated with the coretaper method described above is based on expanding the core by thermal out-diffusion of the photosensitive core dopant [101]. The important difference between the two methods is that while the tapering of the core reduces the V-value of the fiber, the out-diffusion of the photosensitive core leaves the V-value unchanged [102]. This may be understood by remembering that the reduction in the core index as the dopant out-diffuses is compensated for by the increase in the core radius. The fractional power in the core remains unchanged (due to the fixed V-value), but since the core index is reduced, so is the mode index. Heating the fiber locally by an oxyhydrogen flame for 2 minutes resulted in the mode field diameter expanding from 7.8 to 16.8 mm. Subsequent writing of
90
Fabrication of Bragg Gratings
a 10-mm long grating in the tapered core region of a hydrogenated sample resulted in a chirped grating with a bandwidth of 6.0 nm [101]. Stretching a fiber prior to writing a grating shifts the Bragg wavelength in the relaxed state [7]. Byron and Rourke [103] applied the stretch–write technique to form a chirped grating with a scanned phase mask. As the UV beam (2 mm long) was stepped across the phase mask (and the fiber), the prestrained (0.6%) fiber was also relieved of strain in 15 steps of 0.04%. A chirped grating with a bandwidth of 10 nm for a grating length of 30 mm was demonstrated. Care has to be taken with this method, since the fiber can easily slip when under tension. As will be appreciated with the above methods, the bandwidth of the chirp is generally small; in order to increase the chirp, it is necessary to write a chirped grating in the first place. Continuously chirped gratings with larger chirp values can be fabricated with two beams with dissimilar phase fronts. If one parallel beam is interfered with a second diverging beam, the resulting interference pattern will have a period that varies with spatial position in the fringe plane. Figure 3.27 shows the scheme for writing chirped gratings with two diverging, two converging, or a combination of interfering beams. The advantage of using lenses as chirp-adjusting elements is that any chirp bandwidth is possible, limited only by the photosensitive response of the fiber. Using such an interferometer, chirp bandwidths of 44 nm have been demonstrated with a reflectivity of
UV beam 50% UV beam
Cylindrical lens F1 and F2
Beam steering mirror Cylindrical lens F3 with focal plane in plane of beams
Fiber
Chirped grating
Figure 3.27 Nonuniform wave fronts used in the interferometer to produce chirped gratings [104]. Two cylindrical lenses with focal lengths F1 and F2 create a chirped interference pattern at the fiber. The third cylindrical lens with focal length F3 focuses the interfering beams into a stripe at the fiber.
Methods for Fiber Bragg Grating Fabrication
91
80%, covering the entire erbium amplifier gain band. The mechanical and geometrical positioning of the lenses makes the interferometer easy to use, although the repeatability may be not be so good. A disadvantage of this method is the strong curvature of the fringe pattern inscribed in the fiber, which results in coupling of light to the radiation modes on the blue side of the grating transmission spectrum [104].
3.1.15 Fabrication of Step-Chirped Gratings Gratings that are chirped in discrete steps are known as step-chirped. The concept was introduced by the fabrication of phase masks, which were not continuously chirped [23]. Figure 3.28 shows the principle of this type of grating. The grating of length Lg is split into N sections, each of length dl and uniform period Ln (1 < n < N), differing from the previous one by dL, with dL ¼
DLg dl ; L
ð3:1:28Þ
where DLg is the total chirp of the grating and Lg ¼ dl N. If the changes in the period and the sections are sufficiently small, then the grating becomes continuously chirped. The important choice is the number of sections required to build in the chirp. This has been analyzed [105], and it was shown that the length dl of the uniform period section should be such that its bandwidth is 50% greater than the chirp DLg of the grating, irrespective of length of the grating. Making this choice results in a deviation of the characteristics of the grating that differ <1% from those of a continuously chirped grating. There must be an integer number of periods in each subsection to ensure that phase mismatch does not occur. A result of this requirement is that the lengths of the subsections are only approximately equal. The step-chirped grating is ideally implemented in the phase mask using e-beam lithography [23]. Splitting a grating into small fields is exactly how the
dΛ = ΛN – ΛN–1 dl
dl
dl
dl
dl
Λ1 Λ2
Λ
Λ4
ΛN
Figure 3.28 The step-chirped phase mask. It is important to ensure that each section of the grating has an integral number of grating periods, so that the section lengths dl are only nominally identical [23].
92
Fabrication of Bragg Gratings
e-beam process works [20], so that it is naturally suited to the fabrication of step-chirped phase masks. Wide and narrow bandwidth (1–50 nm) phase masks have been fabricated with linear and quadratic step chirps [23] and used for pulse recompression of femtosecond pulses transmitted over optical fiber [106]. Step-chirped phase masks 100 mm long, useful for dispersion compensation in telecommunications transmission links [107], have also been demonstrated [108,74]. There are other techniques that mimic the step-chirped gratings, for example, with a limited chirp capability, using stretch and write [103] discussed in Section 3.1.14. Riant and Sansonetti have also shown that by the use of a focusing lens and a phase mask, small sections with different wavelengths can be built up to create a step-chirped grating [109]. Focusing or defocusing a beam at a phase mask changes the wavelength of the inscribed period in the fiber immediately behind the phase mask. Thus, stepping the spot along the mask while adjusting the focus allows the inscribed wavelength to be altered at each step. Using this method, 50-mm long gratings have been demonstrated with chirp values of 1 nm [109]. There is great advantage in the use of the step-chirped phase-mask, since it allows not only the definition of the grating wavelength, but also any value of chirp rate. The minimum chirp bandwidth possible is, of course, limited by the natural bandwidth of a grating of length Lg. An extension of the step-chirped principle is the concept of superstep-chirped gratings [75]. This technique allows even longer gratings to be assembled using a set of short step-chirped phase masks. This overcomes a major limitation of e-beam fabrication: The writing of masks longer than 100 mm is difficult and expensive. An alternative is to fabricate several gratings on a single-phase-mask plate, each with a fixed chirp. Each phase mask is designed to begin at a wavelength dl longer than the wavelength at which the last phase mask finished. The gratings on the phase mask are aligned one above the other with exactly the correct lengths. After a single grating is scanned into the fiber, the fiber is translated along accurately, and the next phase-mask grating moved vertically into position for writing the succeeding grating. Any small inaccuracy in the placement (stitching) of the phase-mask grating can be “trimmed” [110], using a single UV beam exposure to adjust the phase between adjacent sections [112]. Using this method, gratings 2 meters long have been written in a single contiguous piece of fiber, with a chirp bandwidth of 15 nm [111], and a 1.3-meter long grating with a bandwidth of 10 mn has been used for multichannel dispersion compensation [75,112]. The detailed characteristics of step-chirped and super-step-chirped gratings are discussed in Chapter 4.
Methods for Fiber Bragg Grating Fabrication
93
3.1.16 Techniques for Continuous Writing of Fiber Bragg Gratings There are several techniques for the continuous writing of gratings to increase both the complexity and length of the FBGs. These may be divided into two methods. The first method relies on the monitoring of either the grating as it is written or the grating just written, sequentially in small steps. The second one depends on the blind writing of the grating continuously. These methods rely on the movement of the fiber as a key controlling parameter for the rollout of the grating. Figure 3.29 shows a technique for writing long gratings with complex reflection characteristics [132]. This method relies on the writing of small sections of gratings as for writing super-step-chirped gratings [74], but with phase control of the grating sections so that the positions of the subsequent sections remain aligned. After a grating is written, it is physically moved with a translation stage and fine-tuned with a piezoelectric stage, while monitoring the side-diffracted light to align the periods of the grating. Using this method, several gratings have been demonstrated up to 70 mm long. A similar method was demonstrated earlier by Douay et al. [133], in which the grating was monitored as it was being written, using self-diffraction. The positioning error in this system was reported
Mirror UV laser SL HeNe laser PM PBS HWP Mirror SL Fiber HWP SL BC CCD Figure 3.29 Sequential writing of gratings using an interferometer to adjust the position of the following grating. PM: phase mask, SL: spherical lens, HWP: half-wave plate, BC: beam combiner, PBS: polarizing beam splitter, and CCD: charge-coupled camera display. (Adapted from reference [132].)
94
Fabrication of Bragg Gratings Mirror
CW UV laser
AOS SL
High-voltage driver PM Square wave signal Function generator
Mirror Position encoder triggered Air-bearing motor
Figure 3.30 An acousto-optic shutter (AOS) is opened by a square-wave generator and used to control the exposure of a moving fiber to write small gratings sequentially. The positional accuracy of the translation stage ensures synchronism with the grating sections. (Adapted from reference [134].)
to be 4 nm, setting a lower limit to the accuracy of the grating period of 0.25%. Another technique that seems promising is based on a similar system, but with an acousto-optic shutter to control the illumination of the fiber as it moves at position of the fringes [134]. Figure 3.30 shows the schematic of this technique. Complex gratings are possible with this scheme, although limited length, high-quality gratings were demonstrated in reference [134]. A third technique, which writes small gratings sequentially but continuously, was reported by Petermann et al. [135]. In this method, a continuously moving fiber is exposed to moving fringes using a piezoelectric transducer on one of the fold mirrors of the Lloyd interferometer, and is shown in Fig. 3.31. The piezoelectric transducer is driven by a ramp generator, which is synchronized to the movement of the fiber over the distance of a fringe and is then quickly returned to allow the fringes to line up in phase. This method therefore has the potential of writing truly long gratings with arbitrary profiles. However, the mechanical nature of the movement of the mirror renders this system to relatively slow writing, as inertial forces limit the frequency applied to the piezoelectric transducer to a few tens of Hz. Electro-optic phase-modulators (EOPM) may also be used to shift the fringes continuously. In this technique, two EOPMs are used to induce opposite phaseshifts in the two arms of a Talbot interferometer, using a ramp generator. This scheme is similar to the previous one, but with a major difference: The frequency
Methods for Fiber Bragg Grating Fabrication Shutter
Mirror
95
CW UV laser
PM
Piezoelectric crystals mounted mirrors
CL HeNe interferometer Air-bearing motor Figure 3.31 Continuously moving fringe technique for writing long FBGs. The piezotransducer moves the fiber to shift the fringes in synchrony with the movement of the fiber.
at which the fringes may be moved is significantly higher and can be in the kHz region. This allows for faster inscription and with greater flexibility, as no inertial or mechanical resonances are involved in the phase-shifting process. With precise movement of the fiber, the EOPMs can be driven in synchrony while the grating is printed endlessly. A schematic of the system is shown in Fig. 3.32. Mirror 266 nm pulsed laser
SL High-voltage amplifier PM EOPM
EOPM Mirror
Ramp signal Function generator
Motor Figure 3.32 A schematic of the push-pull phase shifting interferometry for writing fiber gratings continuously. SL: spherical lens, EOPM: electro-optic phase-modulator. (Adapted from reference [136].)
96
Fabrication of Bragg Gratings
The simple setup allows the Bragg wavelength to be set by altering the frequency of modulation, as well as apodization by altering the applied ramp voltage. The interferometer may be tuned over a wide wavelength range of more than 2500 nm, simply and with no additional moving parts [137]. The first visual indication of moving UV fringes (see Fig. 3.33 was made possible by using a self-homodyne technique as shown in Fig. 3.33a [137]. The phase-modulated beams are reflected to retrace their path through the phase mask, and they are sampled at a photodiode or simply by a piece of paper, which fluoresces in the UV light. As the ramp voltage is applied to the EOPMs, the fringes can be seen to move continuously. The reset of the ramp is too fast to be seen. By changing the overlap between the two recombined reflected beams, the fringe pattern may be oriented in any direction: forward, backward, sideways, or moved with a tilt. Thus, a different type of grating can be written, by adjusting the overlap of the two modulated UV beams. The velocity of the fringe pattern has to be matched to the velocity of the fiber, which is given by v¼
MlB f 2neff
ð3:1:29Þ
where M is the number of grating periods moved in the period of the modulation, f is the frequency of the ramp, and neff is the effective index of the fiber. lB is the required Bragg wavelength. The induced refractive index change is a function of both the applied frequency and the voltage; however, the period of the grating is not dependent on the applied voltage. The refractive index change can be shown to be [137] 1=f ð
Dn / 0
PD
0
2px 2p 0 1 cos ½ðL þ dLÞf Lf t dt; 0 þ L L
ð3:1:30Þ
UV Beam BS
PM
EOPM
Mirrors
A
B
Figure 3.33 (a) The technique of self-homodyne, which demodulates the phase modulation to reveal the UV fringes (b). From reference [137].
Methods for Fiber Bragg Grating Fabrication
97
1
Visibility
0,8 0,6 0,4
e=0.7 e=1 e=1.3
0,2 0 200
400
600
800
Frequency (Hz) Figure 3.34 The visibility of the fringe pattern as a function of frequency for different ratios of V/Vp.
where f 0 is the applied frequency and f is the frequency that matches the movement of the fiber calculated from Eq. (3.1.29), L is the period of the grating, L0 ¼ (Lf 0 /f) is the period resulting for a Doppler shift, and dl ¼ (Vp/V – 1)L. The visibility function, which determines the modulation depth of the grating, is dependent on the applied frequency and the ratio, e ¼ V/Vp and is shown in Fig. 3.34 for a synchronous frequency of 500 Hz. The visibility can be restored by altering the applied frequency if the ratio, e, is altered. This may be seen in Fig. 3.34. On the other hand, the visibility is also a function of the applied frequency for a given value of e, as shown in Fig. 3.35. Finally, we can see the tuning curve for the interferometer as a function of frequency in Fig. 3.36. The tuning of the Bragg wavelength of more than 2500 nm for a change in the applied frequency of 600 Hz is possible. Using this interferometer, it is possible to write long gratings, and an example of a grating with many continuous phase stitches is shown in Fig. 3.37
1
Visibility
f=600 Hz f=500 Hz f=400 Hz
0.5
0 0
1
2
3
V/Vpi Figure 3.35 The visibility of the fringe pattern as a function of different ratios of V/Vp for different applied frequencies at a synchronous frequency of 500 Hz.
Fabrication of Bragg Gratings Bragg wavelength shift (nm)
98 2500 2000 1500 1000 500 0 –500 –1000 200
400 600 Frequency (Hz)
800
Figure 3.36 Tuning curve for the EOPM interferometer, showing the ultrawide tunability by the simple changing of the ramp frequency.
1580
1590
1600
1610
1620
Reflection, dB
–5 –15 –25 –35 –45 –55 Wavelength, nm Figure 3.37 The reflection spectrum of a 20-cm-long random grating, showing the effect of Anderson light-localization [137].
[137,138]. This grating has a mean period, which is modified stochastically along its length via random phase shifts during the writing process. This results in the equivalent of what is observed in solids – Anderson localization [140]. A reflection spectrum shown in Fig. 3.38 has the features of the reflection spectrum decaying exponentially as a function of detuning. This type of a grating is of great topical interest and leads to the formation of 1D random lasers as has been demonstrated recently. In this type of a laser, the random grating is written into a doped fiber. Figure 3.38 shows the reflection spectrum of a 30-cm-long continuous random grating written into hydrogen-loaded polarization maintaining erbium-doped fiber. When pumped at a wavelength of 980 nm, the Er-doped fiber demonstrates a very low lasing-threshold of approximately
Methods for Fiber Bragg Grating Fabrication 1525 0
1530
1535
99 1540
1545
–5 Reflection (db)
–10 –15 –20 –25 –30 –35 –40 Wavelength (nm) Figure 3.38 Refletion spectrum of a 30-cm-long random fiber grating written into an erbiumdoped fiber. The almost continuous random phase noise in the writing process results in this reflection spectrum.
Laser power (mW)
1 mW, as well as random mode jumping on a fast timescale [141]. The lasing characteristics of this laser are shown in Fig. 3.39. A compact technique based on a Sagnac loop interferometer using rotating wave plates as differential optical phase shifters, polarizers, and a phase mask has also been demonstrated for writing high-quality grating [142]. Brennan et al. [143] demonstrated another technique for making ultralong gratings, which is shown in Fig. 3.40. An amplitude modulator is used to modulate a beam in a point-by-point writing technique. The fiber is positioned in a spiral groove on a large drum and rotated synchronously with the pulsation of the focused spot. A precision movement allows the rotation of the drum at 0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0 0
1
2
3 4 5 6 Pump power (mW)
7
8
9
Figure 3.39 Lasing characteristics of the 30-cm-long random fiber laser, showing almost no threshold for lasing. (From reference [141].)
100
Fabrication of Bragg Gratings Amplitude modulator Mirror UV Laser High precision rotation of drum
Fiber in high precision groove
Lens
Screw driven movement Figure 3.40 Continuous writing of ultralong fiber Bragg gratings using a fiber wrapped around a drum. (From reference [143].)
a highly controlled speed using a phase-locked loop. With this method, gratings as long as 10 meters have been written with a chirp covering the entire C-band (1535 nm–1565 nm), with a dispersion slope of 0.5 ns/nm. As with all chirped gratings, the ripple in the group delay slope remains at a level of few ps. Circular phase masks and amplitude masks were demonstrated to write ultralong gratings. In this scheme, a circular phase mask with a grating 1 mm wide is fabricated by the e-beam at a diameter of 50 mm to make FBGs. Alternatively, a titanium mask with slots cut out at a similar position was also used to make long-period gratings. A schematic of this method is shown in Fig. 3.41.
Rotation
Ceramic cylinder
V-groove
UV beam
Fiber
Lens Axle
Circular phase or amplitude mask
Side view Fiber
Slot or Grating
Top view
Figure 3.41 A technique for writing gratings of infinite length using circular phase or amplitude masks. (From reference [144].)
Tunable Phase Masks
101
Transmission, dB
0 –10 –20 –30 –40 1450
1500 1550 Wavelength, nm
1600
Figure 3.42 Transmission spectrum of a 1.3-m long LPG made with a circular amplitude mask. (From reference [143].)
The fiber is located in a precision groove machined into a ceramic cylinder and wrapped once around it, immediately behind the circular phase/amplitude mask. The fiber is held under tension by the action of a vacuum chuck on the far end of the stripped fiber. As a motor draws the fiber, the circular mask and fiber move in synchronism. With a stationary focused UV beam incident on the fiber, continuous FBGs or long-period gratings can be written. In the demonstration, 150-mm-long FBGs and long-period gratings were written (equivalent to one turn of the phase mask). High-quality long-period gratings (LPGs) are easily produced by this method; however, higher-quality phase masks are required for good-quality gratings; the e-beam writing of the phase mask is based on an x–y movement, which induces phase stitches every quadrant of the circle. This problem could be eliminated with a radial movement during writing with the e-beam. Figure 3.42 shows the transmission spectra of a 1.3-m-long LPG.
3.2 TUNABLE PHASE MASKS The problem of using a phase mask is that the wavelength of the Bragg gratings inscribed by it is more or less decided by its period, unless used with an interferometer. One way to overcome this limitation is to use the step-by-step writing technique, which can alter the period of the inscribed grating by altering the time between steps. Although this technique is very flexible, it cannot usually be used to write first-order gratings, unless UV radiation is used. An alternative method is to use a tunable phase mask. A UV transparent polymer phase mask has been used to inscribe Bragg gratings using UV radiation [144]. The polymer, polydimethylsiloxane (PDMS), is highly transparent at 266 nm, and a replica made of a silica phase mask of the appropriate period and etch depth allows a low zero-order diffraction. This elastomeric phase mask not only makes it is
102
Fabrication of Bragg Gratings
possible to write gratings with the phase mask as part of the fiber coating, but it also allows the Bragg grating to be tuned to the desired wavelength before inscription by stretching the mask. Figure 3.43 shows a PDMS phase mask integrated with an optical fiber. When exposed to UV radiation, a grating appears without having to align the phase mask. An example of the reflection and transmission spectrum of a 15-mm-long grating written through the polymer phase mask is shown is Fig. 3.44. The simulation is also shown on the same figure. High-quality gratings are possible using this technique without destroying the polymer, provided a modified technique is used to stabilize the phase mask by low-intensity exposure to UV radiation, before the inscription at the normal intensity. The preexposure removes some of the methyl groups from the PDMS, making it less susceptible to dynamic changes during inscription [145,146]. Because the PDMS masks are cheap and easily reproduced, they may be disposed of after single use. The additional advantage of the PDMS phase mask is that chirp can be easily accommodated into the mask by casting a slave mask in PDMS with a
PDMS Phase Mask
Fiber
Figure 3.43 A photograph of the PDMS phase mask integrated into an optical fiber [144].
–2
Reflection (dB)
–4 –6 –8
1530.7
1530.8
1530.9
1531
1531.1
1531.2
1531.3 0 –2
Reflection (Meas)
–4
Rx (sim) Transmission (Meas)
–6
Tx (sim)
–8
–10
–10
–12
–12
–14
–14
–16
–16
–18
–18
Transmission (dB)
1530.6 0
–20
–20 Wavelength, nm
Figure 3.44 Measured (meas) reflection and transmission spectra of a 15-mm-long grating written with an on-fiber phase mask, along with the simulated data (sim). A small chirp of 0.07 nm was included in the simulation.
Tunable Phase Masks
103
slightly linear-tapered thickness. Applying strain to the slave mask introduced a precise chirp. Complex PDMS aperiodic phase masks can thus be implemented with a uniform grating silica or PDMS mask. A PDMS phase mask is useful for writing into planar waveguide structures, as the contact of the phase mask does not pose a threat of damage to either the planar circuit or the phase mask. The soft mask easily attaches to the surface of the planar circuit, allowing several gratings to be written using one exposure.
3.2.1 Fabrication of Long-Period Gratings Long-period gratings (LPGs) can be fabricated by using a number of techniques. These include the exposure to an amplitude mask with the appropriate period using UV radiation as shown in Fig. 3.41 [142], or simply with a linear amplitude mask [147], through heating with a laser [148], an electric arc [149], or a CO2 laser [150]. Because heating is localized to 10 s of microns with a focused laser beam, the use of the CO2 laser also allows ultrashort devices such as microcouplers only 200-mm long, long-period gratings, and transversely excited whispering gallery mode ring resonator devices to be fabricated in pretapered fibers [150]. Many of these systems have been used for LPGs with periods of tens to hundreds of microns. However, LPGs with gratings of greater than a millimeter have also been demonstrated. These depend on higher-order grating coupling and include coupling of the core-fundamental mode to a large band of cladding modes with cross-coupling up to 7 dB [151]. The advantage of using ultralong LPGs is that another free parameter is available: Mode coupling between the same modes (e.g., the core mode LP01 ! LP06 in the cladding for the second- and fourth-order interactions) can occur at the same wavelength, or at entirely different wavelengths, because the slopes of the phase-matching curves are different and cross at some period of the grating. At the same time, it was also demonstrated that first-order resonances coupling in the 1.1–1.7 mm region occur for grating periods less than 1200 mm, whereas resonances for other higher harmonic orders do not. Thus, there is no overlap between the orders. The properties of the coupling with resonances of different order at the same wavelength are also dissimilar. More important, the temperature and strain response for a second-order coupling from the fundamental core mode to the fundamental cladding mode and a fourth order LP01 ! LP06 are radically different [151]. The temperature dependences were measured to be 0.31 and 0.43 nm- C1, and 0.89 and 0.05 nm-me1 for the two mode interactions, respectively. Indeed it should be remembered that these values are strongly dependent on the type of fiber used, and the values have to be calculated for each fiber. However, the general principles remain valid.
104
Fabrication of Bragg Gratings
3.3 TYPE II GRATINGS Fiber gratings formed at low intensities are generally referred to as Type I. Another type of grating is a damage grating formed when the energy of the writing beam is increased above approximately 30 mJ [119]. Physical damage is caused in the fiber core on the side of the writing beams. The definite threshold is accompanied by a large change in the refractive index modulation. It is therefore possible to write high-reflectivity gratings with a single laser pulse. Above 40–60 mJ, the refractive index modulation saturates at around 3 103. Energy of the order of 50–60 mJ can destroy the optical fiber. The sudden growth of the refractive index is accompanied by a large short-wavelength loss due to the coupling of the guided mode to the radiation field. The gratings generally tend to have an irregular reflection spectrum due to “hot spots” in the laser beam profile. By spatially filtering the beams, gratings with better reflection profiles have been generated but with a much reduced reflectivity [113]. These gratings decay at much higher temperatures than Type I, being stable up to 700 C. Some of the properties of these gratings are outlined in Chapter 9.
3.4 TYPE IIA GRATINGS Yet another type of grating is formed in non-hydrogen-loaded fibers. These may form at low power densities or with pulsed lasers after long exposure [114]. The characteristics of a Type IIA grating are the growth of a zero-order (N ¼ 1) grating, and its erasure during which a second-order grating grows at approximately half the initial Bragg wavelength, followed by the growth of the N ¼ 1 grating. The final grating is stronger than the original and is able to withstand a higher temperature. These gratings form in most fibers, although they have yet to be observed in hydrogen-loaded fibers. The gratings are reviewed in Chapters 2 and 9.
3.5 SOURCES FOR HOLOGRAPHIC WRITING OF GRATINGS There are several UV laser sources that may be used for inducing refractive index changes and for fabricating gratings in optical fibers. Methods for generating UV or deep UV radiation require the use of excimer lasers, nonlinear crystals for frequency mixing of coherent visible/infrared radiation, or linenarrowed dye laser radiation. UV laser sources may be categorized into two types – low spatial/temporal coherence or spatially coherent sources. Sources
Sources for Holographic Writing of Gratings
105
in the first category have been used extensively for grating fabrication but need to be used with care for high-quality grating production. The primary advantage is in the high average and peak power available in the UV region. On the other hand, some frequency mixing methods produce UV radiation of high temporal and spatial coherence and are ideally suited for grating formation. Typically, these lasers demonstrate high average power capability but have the disadvantage of lower peak power densities. Nevertheless, these have also been shown to be highly successful for inducing large index changes while maintaining excellent grating quality.
3.5.1 Low Coherence Sources The first source reported for use in grating formation was the excimer-laser pumped frequency doubled oscillator–amplifier dye laser operating in the 240-nm window [13]. This source can produce approximately 0.1 W average tunable radiation between 240 and 260 nm using a b-barium borate (BBO) crystal by doubling the (CHRYSL 106) dye-laser output. The laser must be line-narrowed to increase the coherence length, and longitudinal pumped dye lasers are preferable since they produce a higher quality beam compared to side-pumped dyes which produce a triangular beam profile. An alternative pump laser for the dye is the frequency-mixed output at 355 nm from a YAG laser operating at a wavelength of 1064 nm. Similar output may be achieved with a maximum of around 120 mW average at 244 nm for a 20-Hz system with 12-ns pulses [115]. The frequency-doubled tunable dye-laser source can be line-narrowed to increase temporal coherence, helping maintain interference despite path-length differences. However, the UV beam may need to be spatially filtered if the beam-quality is not good so as to be able to write uniform reflection gratings. High-quality gratings have been reported using this method [116], however, by multiple pulse writing. With low transverse coherence, it is still possible to write good gratings, provided the spatial nonuniformity averages out over the time period required to write the grating. However, the pulse-to-pulse transverse beam variation affects single-pulse writing of gratings [117–121] or, for example, during fiber drawings [122,123], as was originally suggested by Askins et al. [118]. The power density variation manifests itself in two ways: First, the reflectivity varies from grating to grating, and second, the peak reflection wavelength, which is a function of the induced index change, varies between gratings. Beam profile nonuniformity has a rather more serious deleterious effect on the grating reflection spectrum, causing multiple peaks and chirp. The spectral shape is discussed in more detail in Chapters 4 and 9. Multiple pulses can ensure that
106
Fabrication of Bragg Gratings
each section of the grating can be driven into saturation (for a given UV flux), although if the beam profile has a Gaussian average intensity profile, the grating will appear chirped. The regions with low flux will see a smaller index change and hence a smaller change in the effective index of the mode than the central region of the beam. Since the Bragg wavelength is proportional to the effective index of the mode, the beam profile imparts a Bragg wavelength profile proportional to the intensity profile, leading to chirp [124]. It is for this reason that there is a concern over the use of a laser with “hot-spots” in the beam. The grating may show fine structure in the reflection spectrum due to the nonuniform refractive index modulation of the inscribed grating. This has so far not been reported in the literature but may well be a problem for long gratings. Another aspect of low coherence sources, which must be taken into account when designing an interferometer, is the effect of the coherence length. These laser sources are generally better suited to direct printing using a phase mask with the fiber immediately behind the phase plate. Alternatively, a one-to-one imaging system may be used to form the interferometer such that the paths of the two beams are equalized and overlapped. If, however, the paths are not equal or the beams do not overlap, degradation in the visibility can result in poor grating reflectivity and/or spectral profile. The limiting factor for all lasers is the divergence of the beam, which determines how far away the fiber may be placed from the phase mask as (see Fig. 3.1) dl df l2
cosðym =2Þ ; 2L
ð3:4:1Þ
where dl is the source bandwidth, df the source angular divergence, l the source wavelength, ym/2 the half diffraction angle, and L the distance of the fiber from the phase mask. The physical significance of Eq. (3.4.1) is that as the diffracted beams are brought together, the divergence causes a dephasing of the interfering beams, reducing the visibility. The contact method is therefore ideally suited for use with low-coherence sources. However, it must be remembered that the phase mask is more likely to be damaged owing to contamination from the fiber, i.e., dust, etc., using high-intensity pulses. High peak-power laser sources do allow the writing of Type II gratings, which depend on physical damage to the core region [125]. This aspect is discussed in Section 3.3.
3.5.2 High Coherence Sources Lasers with good spatial and/or temporal coherence fall in this category. Examples include CW intracavity frequency doubled argon-ion lasers operating at 257 nm/244 nm [26,126], QS frequency quadrupled YLF [127], spatially
Sources for Holographic Writing of Gratings
107
filtered, line-narrowed frequency doubled dye lasers [13]. The first laser has excellent spatial and temporal coherence, being derived from the argon ion laser mode. Even with this laser, there can be fine structure in the transverse beam profile of the UV mode due to the low walk-off angle in the frequency doubling crystal, BBO. The beam is elliptical, but this is an advantage because the shape is suited to writing fiber gratings. The latter two lasers can be made to have good temporal coherence by line narrowing, but require good spatial filtering to generate a satisfactory Gaussian mode profile. The pinholes used for filtering require careful alignment and have a tendency to burn after a short time owing to the high peak power; they require frequent replacement. The quadrupled Nd:YLF and the intracavity frequency doubled argon ion lasers belong to the class of turnkey systems. These are reliable lasers and need little attention other than routine maintenance; for example, mirrors need occasional cleaning. The frequency doubling crystals tend to be KTP for the YLF followed by a BBO crystal enclosed in an airtight holder with silica windows. Although the transverse mode profile may not be uniform shot-to-shot for this laser, it does allow the use of cylindrical lenses for focusing into the BBO crystal. This scheme overcomes the transverse walk-off problem in BBO associated with spherical lens focusing. However, it is not possible to use cylindrical focusing in the intracavity frequency doubling in BBO. A schematic of the Z-folded resonator is shown in Fig. 3.45. With careful adjustment of the fold and focusing mirrors, this laser can be operated at a stable CW output in excess of 300 mW at 244 nm. A cylindrical lens can be used at the output to circularize the beam. Typically, the beam diameter is approximately 2 mm. The coherence length of the argon-ion laser operating in the single-frequency mode is more than adequate for use in a beam splitting interferometer even with path length differences approaching several centimeters, although the path length difference is usually not so large. This laser is also suited for use in the scanned phase-mask interferometer, or for writing blazed gratings, as discussed in Section 3.1.4. Recent work on writing grating with near-UV wavelength radiation [128– 130] means that other high-quality laser sources may be used for grating Argon ion laser
Fold mirror
Spherical 488nm HR/244nm HT
Spherical HR Spherical focusing mirror
Brewster cut BBO
UV Out
Figure 3.45 A schematic of the intracavity frequency-doubled, argon ion laser. 488-nm wavelength radiation is doubled to generate 244-nm UV emission. Angle tuning does phase matching of the BBO crystal. It is not necessary to temperature stabilize the BBO crystal [26].
108
Table 3.1 Common options of UV laser sources for grating formation
Pump laser
UV wavelength (nm)
UV power/energy
Beam shape
Application Mass production, fiber drawing tower grating fabrication, Type II gratings Planar devices, non-Ge fibers/planar, direct writing Hydrogen loaded fibers, planar Ge/P-doped silica, þ RE-doped silica, Type II gratings UV-induced index change studies, grating writing, material studies, Type II grating formation, scanned phase mask Same as above
KrF
248
1 J/100 Hz, QS
Few cm diameter
Low spatial/temporal coherence: requires injection locking
ArF
193
1 J(100 Hz) QS
Few cm diameter
Low spatial/temporal coherence
Quadrupled Nd: YAG (1064 nm)
266, KDP/BBO frequency doublers
>100 mW 5 ns QS pulses at >1k Hz
1 mm diameter
10 mm coherence length, low transverse beam coherence
Nd:YAG (355 nm via frequency mixing)
220–260 tunable extra cavity BBO frequency doubled
>100 mW, 4 ns QS
2 mm diameter, not Gaussian, flattopped, or triangular
Low coherence, unless linenarrowed
XeCl (308 nm) pumped dye laser
Same as above
Same as above
Same as above
Same as above
Fabrication of Bragg Gratings
Beam quality
257, 244, intracavity frequencydoubled in KDP/ BBO
Up to 2 W; up to 1 W CW
1 mm, elliptical TEM00
High coherence, can be operated singlefrequency but not necessary
Diode pumped quadrupled QS Nd:YLF (1048 nm) Argon ion laser at 302 nm
262, frequencydoubled externally in KTP and BBO Fundamental wavelength
100 mW QS 100 ns pulses
1 mm elliptical TEM00
Medium coherence, good beam quality
200 mW
1 mm TEM00
Excellent coherence: tube lifetime?
Krypton ion laser at 647 nma
Frequency-doubled in BBO to 323.5 nm
>1 W
1 mm TEM00
Ideal near-UV source; long lifetime
a
Source with an excellent beam quality for grating fabrication; contact and noncontact phase mask/ interferometer but not ideal for Type II gratings Good source for Types I and II grating writing in fibers and planar Writing through the jacket of D/H2 loaded fibers Through the jacket inscription of D/H2 loaded fibers
Sources for Holographic Writing of Gratings
Argon ion laser, 514/488 nm
Proposed by author; unused as yet.
109
110
Fabrication of Bragg Gratings
fabrication. An argon laser operating at 302 nm is one option, which allows the inscription of gratings directly through the silicone resin polymer jacket [71] or the use of a novel polymer at a wavelength of 257 nm [70]. The lifetime of the argon ion laser operating in the UV is probably an issue for the fabricator. However, other sources as yet not demonstrated, e.g., the intracavity krypton ion laser operating at 647 nm, frequency doubled to 323.5 nm, would be an attractive option, with plenty of power available at the UV wavelength. Table 3.1 lists a summary of the different types of lasers operating in the UV, used for grating fabrication. Gratings, with a refractive index modulation of 5 104, have been written with the 3rd harmonic of the Nd: YAG laser with a writing time of 30 minutes and a writing power of 1.3 W [152]. This source has the potential of being used without stripping the coating of the fiber. A new source for grating writing is the compact 213 nm wavelength laser – the fifth harmonic of the Nd: YAG laser from Xiton Lasers Gmbh. This laser has a similar coherence and beam quality as the quadrupled Nd: YLF (262 nm) or the Nd: YAG, (266 nm) but a lower Q-switched power of 110 mW. Gratings can be written with this laser with considerable ease, as this is a turnkey, small footprint laser. With only 3 mW of 213 nm radiation, it is possible to induce 104 refractive index change in borongermanium co-doped fiber [153]. The transmission spectra of an 8 mm long grating written in non-hydrogen loaded B-Ge doped fiber with 110 mW of 213 nm wavelength radiation immediately behind a 213 nm optimized phasemask in 50 seconds, is shown in Fig. 3.46. The reflectivity of this grating is 99.85%. 90% gratings have also been written in non-H2hydrogen loaded standard telecommunications fibers. This laser has excellent potential for writing strong gratings in non-hydrogen loaded fibers and may be a good alternative for use with phosphorus doped fiber as well. 1553
1554
1555
Transmission (dB)
0 −4 −8
−12 −16 −20 −24 −28 Wavelength, nm
Figure 3.46 Transmission spectrum of an 8 mm long Bragg grating inscribed in 50 seconds non-hydrogen loaded Redfern B:Ge doped fiber, using 110 mW of 213 nm, 15 kHz Q-switched radiation.
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111
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[116] H.G. Limberger, P.Y. Limberger, R. Salathe´, Spectral characterization of photoinduced high efficient Bragg gratings in standard telecommunication fibers, Electron. Lett. 29 (1) (1993) 47–49. [117] R. Kashyap, G.D. Maxwell, unpublished (1991). [118] C.G. Askins, T.E. Tsai, G.M. Williams, M.A. Puttnam, M. Bashkansky, E.J. Friebele, Fibre Bragg reflectors prepared by a single excimer pulse, Opt. Lett. 17 (11) (1992) 833. [119] J.L. Archambault, L. Reekie, P.J.St. Russell, High reflectivity and narrow bandwidth fibre gratings written by a single excimer pulse, Electron. Lett. 29 (1) (1993) 28. [120] B. Malo, D.C. Johnson, F. Bilodeau, J. Albert, K.O. Hill, Single-excimer-pulse writing of fiber gratings by use of a zero-order nulled phase mask: grating spectral response and visualization of index perturbations, Opt. Lett. 18 (15) (1993) 1277. [121] J.L. Archambault, L. Reekie, P.J.St. Russell, 100% reflectivity Bragg reflectors produced in optical fibres by single excimer pulses, Electron. Lett. 29 (5) (1993) 453. [122] C.G. Askins, M.A. Putnam, G.M. Williams, E.J. Friebele, Stepped-wavelength opticalfiber Bragg grating arrays fabricated in line on a draw tower, Opt. Lett. 19 (2) (1994) 147–149. [123] L. Dong, J.L. Archambault, L. Reekie, P.J.St. Russell, D.N. Payne, Single pulse Bragg gratings written during fibre drawing, Electron. Lett. 29 (17) (1993) 1577. [124] V. Mizrahi, J.E. Sipe, Optical properties of photosensitive fiber phase gratings, Lightwave Technol. 11 (10) (1993) 1513–1517. [125] L. Dong, J.L. Archambault, L. Reekie, P.J.St. Russell, D.N. Payne, Single pulse Bragg gratings written during fibre drawing, Electron. Lett. 29 (17) (1993) 1577. [126] H. Patrick, S.L. Gilbert, Growth of Bragg gratings produced by continuous-wave ultraviolet light in optical fiber, Opt. Lett. 18 (18) (1993) 1484. [127] J.R. Armitage, Fibre Bragg reflectors written at 262 nm using frequency quadrupled Nd3þ: YLF, Electron. Lett. 29 (13) (1993) 1181–1183. [128] E.M. Dianov, D.S. Starodubov, Microscopic mechanisms of photosensitivity in germanium-doped silica glass, SPIE Proc, 2777 (1995) 60–70. [129] D.S. Starodubov, E.M. Dianov, S.A. Vasiliev, A.A. Frolov, O.I. Medvedkov, A.O. Rybaltovskii, et al., Hydrogen enhancement of near-UV photosensitivity of germanosilicate glass, SPIE Proc. 2998 (1997) 111–121. [130] D.S. Starodubov, V. Grubsky, J. Feinberg, T. Erdogan, (1997) Near-UV fabrication of ultrastrong Bragg gratings in hydrogen loaded germanosilicate fibers, in: Proc. of CLEO’97, post-deadline paper CDP24. [131] H.J. Patrick, C.G. Askins, R.W. McElhanon, E.J. Friebele, Amplitude mask patterned on an excimer laser mirror for high intensity writing of long period fibre gratings, Electron. Lett. 33 (13) (1997) 1167–1168. [132] K.C. Hsu, L.G. Sheu, K.P. Chuang, S.H. Chang, Y. Lai, Fiber Bragg grating sequential UV-writing method with real-time interferometric side-diffraction position monitoring, Opt. Express 13 (2005) 3795–3801. [133] M. Douay, BGPP, Monterey, USA, 2003. [134] Y. Liu, J.J. Pan, C. Gu, L. Dong, Novel fiber Bragg grating fabrication method with high-precision phase control, Opt. Eng. 43 (2004) 1916–1922. [135] I. Petermann, B. Sahlgren, S. Helmfrid, A.T. Friberg, P.Y. Fonjallaz, Fabrication of advanced fiber Bragg gratings by use of sequential writing with a continuous-wave ultraviolet laser source, Appl. Opt. 41 (2002) 1051–1056. [136] M. Gagne´, L. Bojor, R. Maciejko, R. Kashyap, Novel custom fiber Bragg grating fabrication technique based on push-pull phase shifting interferometry, Opt. Express 16 (26) (2008) 21550–21557.
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[137] M. Gagne´, L. Bojor, R. Maciejko, R. Kashyap, Novel long fiber Bragg gratings fabrication technique based on push-pull phase-shifting interferometry, ICOOPMA 2008. [138] N. Liza´rraga, N.P. Puente, E.I. Chaikina, T.A. Leskova, E.R. Me´ndes, Single-mode Erdoped fiber random laser with distributed Bragg grating feedback, Opt. Express 17 (2009) 395–404. [139] P. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109 (1958) 1492–1505. [140] M. Gagne´, R. Kashyap, Demonstration of a 1 mW threshold Er-doped fiber random laser based on a unique fiber Bragg grating, Submitted to Optics Express, August 2009. [141] C. Knothe, E. Brinkmeyer, Reset-free phase shifter in a Sagnac-type interferometer for control of chirp and apodization of Bragg gratings, in: Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, Technical Digest, Optical Society of America, 2003, paper TuB3. http://www.opticsinfobase.org/abstract.cfm?URI¼BGPP-2003-TuB3. [142] J. Brennan, et al., Bragg Grating, Photosensitivity and Poling in Glass Waveguides, Optical Society of America, Technical Digest, Sep. 23–25, pp. 36/ThD2-1–37/ThD2-3. 1999. [143] R. Kashyap, Infinite length fibre gratings, Electron. Lett. 35 (21) (1999) 1871–1872. [144] J. Poulin, R. Kashyap, Novel widely tuneable phase masks for fibre Bragg gratings, Optics Exp. 13 (12) (2005) 4414–4419. [145] J. Poulin, R. Kashyap, (2009) Novel fabrication of Fibre Bragg Gratings using imprinted silicone rubber phase-mask stamp on the cladding, accepted in J. Lightwave Technol. ´ tude des proprie´te´s optiques et me´canique du (poly) dime´thylsiloxane sous [146] J. Poulin, E radiation ultraviolette pour l’e´criture de re´seaux de Bragg et la fabrication de microstructures photoniques accordables, M.Sc.A. Thesis, Dept. Physics. Eng., E´cole Polytechnique de Montre´al, Montre´al, Canada, 2007. [147] A.M. Vengsarkar, P.J. Lamaire, J.B. Judkins, V. Bhatia, T. Erdogan, J.E. Sipe, Longperiod gratings as band-rejection filters, J. Lightwave Technol. 14 (1996) 58–65. [148] U.C. Paek, Appl. Opt. 13 (1974) 1383. [149] C.H. Huang, H. Luo, S. Xu, P. Chen, in: Digest of Optical Communication Conference (OFC), 1999. [150] G. Kakarantzas, T.E. Dimmick, T.A. Birks, R. Le Roux, P. Russell, Miniature all fiber devices based on CO2 laser microstructuring of tapered fibers, Opt. Lett. 15 (26) (2001) 1137–1139. [151] X. Shu, L. Zhang, I. Bennion, Fabrication and characterization of ultra-long-period fibre gratings, Opt. Commun. 203 (2002) 277–281, 15 March. [152] J. Blows and D.Y. Tang, Gratings written with tripled output of Q-switched Nd:YAG laser, Electron. Lett. 36 (22) (2000) 1837–1839. [153] M. Gagne, J. Bartschke and R. Kashyap, Strong gratings written in non-hydrogen loaded optical fibers with a 213 nm wavelength Q-switched laser radiation, submitted to Optics Lett.
Chapter 4
Theory of Fiber Bragg Gratings Fashionable gratings are in.
Wave propagation in optical fibers is analyzed by solving Maxwell’s equations with appropriate boundary conditions. The problem of finding solutions to the wave-propagation equations is simplified by assuming weak guidance, which allows the decomposition of the modes into an orthogonal set of transversely polarized modes [1–3]. The solutions provide the basic field distributions of the bound and radiation modes of the waveguide. These modes propagate without coupling in the absence of any perturbation (e.g., bend). Coupling of specific propagating modes can occur if the waveguide has a phase and/or amplitude perturbation that is periodic with a perturbation “phase/amplitudeconstant” close to the sum or difference between the propagation constants of the modes. The technique normally applied for solving this type of a problem is coupled-mode theory [4–9]. The method assumes that the mode fields of the unperturbed waveguide remain unchanged in the presence of weak perturbation. This approach provides a set of first-order differential equations for the change in the amplitude of the fields along the fiber, which have analytical solutions for uniform sinusoidal periodic perturbations. A fiber Bragg grating of a constant refractive index modulation and period therefore has an analytical solution. A complex grating may be considered to be a concatenation of several small sections, each of constant period and unique refractive index modulation. Thus, the modeling of the transfer characteristics of fiber Bragg gratings becomes a relatively simple matter, and the application of the transfer matrix method [10] provides a clear and fast technique for analyzing more complex structures. Another technique for solving the transfer function of fiber Bragg gratings is by the application of a scheme proposed by Rouard [11] for a multilayer dielectric thin film and applied by Weller-Brophy and Hall [12,13]. The method relies on the calculation of the reflected and transmitted fields at an interface between two dielectric slabs of dissimilar refractive indexes. Its equivalent 119
120
Theory of Fiber Bragg Gratings
reflectivity and phase then replace the slab. Using a matrix method, the reflection and phase response of a single period may be evaluated. Alternatively, using the analytical solution of a grating with a uniform period and refractive index modulation as in the previous method, the field reflection and transmission coefficients of a single period may be used instead. However, the thin-film approach does allow a refractive index modulation of arbitrary shape (not necessarily sinusoidal, but triangular or other) to be modeled with ease and can handle effects of saturation of the refractive index modulation. The disadvantage of Rouard’s technique is the long computation time and the limited dynamic range owing to rounding errors. The Bloch theory [14,15] approach, which results in the exact eigenmode solutions of periodic structures, has been used to analyze complex gratings [16] as well. This approach can lead to a deeper physical insight into the dispersion characteristics of gratings. A more recent approach taken by Peral et al. [17] has been to develop the Gel’Fand–Levitan–Marchenko coupled integral equations [18] to exactly solve the inverse scattering problem for the design of a desired filter. Peral et al. have combined the attributes of the Fourier transform technique [19,20] (useful for low reflection coefficients, since it does not take account of resonance effects within the grating), the local reflection method [21], and optimization of the inverse scattering problem [22,23] to present a new method that allows the design of gratings with required features in both phase and reflection. The method has been recently applied to fabricate near “tophat” reflectivity filters with low dispersion [24]. Other theoretical tools such as the effective index method [25], useful for planar waveguide applications, discrete-time [26], Hamiltonian [27], and variational [28], are recommended to the interested reader. For nonlinear gratings, the generalized matrix approach [29] has also been used. For ultrastrong gratings, the matrix method can be modified to avoid the problems of the slowly varying approximation [30]. The straightforward transfer matrix method provides high accuracy for modeling in the frequency domain. Many representative varieties of the types and physical forms of practically realizable gratings may be analyzed in this way.
4.1 WAVE PROPAGATION The theory of fiber Bragg gratings may be developed by considering the propagation of modes in an optical fiber. Although guided wave optics is well established, the relationship between the mode and the refractive index perturbation in a Bragg grating plays an important role on the overall efficiency and type of scattering allowed by the symmetry of the problem. Here, wavepropagation in optical fiber is introduced, followed by the theory of mode coupling.
Wave Propagation
121
We begin with the constitutive relations D ¼ e0 E þ P
ð4:1:1Þ
B ¼ m0 H
ð4:1:2Þ
where e0 is the dielectric constant and m0 is the magnetic permeability, both scalar quantities; D is the electric displacement vector; E is the applied electric; B and H are the magnetic flux and field vectors, respectively; and P is the induced polarization, ð1Þ
P ¼ e0 wij E :
ð4:1:3Þ
ð1Þ
The linear susceptibility wij is in general a second-rank tensor with two laboratory frame polarization subscripts ij and is related to the permittivity tensor eij with similar subscripts as ð1Þ
eij ¼ 1 þ wij :
ð4:1:4Þ
Assuming that the dielectric waveguide is source free, so that r D ¼ 0;
ð4:1:5Þ
and in the absence of ferromagnetic materials, r B ¼ 0; the electric field described in complex notation is i 1h E ¼ EeiðotbzÞ þ EeiðotbzÞ ; 2
ð4:1:6Þ
ð4:1:7Þ
and the induced polarization vector is also similarly defined. Using Maxwell’s equations, @B @t
ð4:1:8Þ
@D þ J; @t
ð4:1:9Þ
rE ¼ rH ¼
where J is the displacement current, and using Eq. (4.1.1) in Eq. (4.1.9) and with J ¼ 0, we get @ rH ¼ ð4:1:10Þ e0 E þ P : @t Taking the curl of Eq. (4.1.8) and using Eqs. (4.1.2)–(4.1.5) and the time derivative of Eq. (4.1.10), the wave equation is easily shown to be r2 E ¼ m0 e0
@2E @2P þ m : 0 @t2 @t2
ð4:1:11Þ
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Theory of Fiber Bragg Gratings
Using Eq. (4.1.3) and (4.1.4) in (4.1.11), we arrive at i @2 h ð1Þ r2 E ¼ m0 e0 2 1 þ wij E ; @t
ð4:1:12Þ
or r2 E ¼ m0 e0 eij
@2E : @t2
ð4:1:13Þ
4.1.1 Waveguides The next step in the analysis is to introduce guided modes of the optical fiber into the wave equation. The modes of an optical fiber can be described as a summation of l transverse guided mode amplitudes, Am(z), along with a continuum of radiation modes, Ar(z) [2], with corresponding propagation constant, bm and br, m¼l i X ð r¼1 1 Xh Am ðzÞxmt eiðotbm zÞ þ cc þ Ar ðzÞxrt eiðotbr zÞ dr; ð4:1:14Þ Et ¼ 2 m¼1 r¼0 where xmt and xrt are the radial transverse field distributions of the mth guided and rth radiation modes, respectively. Here the polarization of the fields has been implicitly included in the transverse subscript, t. The summation before the integral in Eq. (4.1.14) is a reminder that all the different types of radiation modes must also be accounted for [e.g., transverse electric (TE) and transverse magnetic (TM), as well as the hybrid (EH and HE) modes]. The following orthogonality relationship ensures that the power carried in the mth mode in watts is |Amt|2: ð þ1 ð þ1 ð þ1 ð þ1 bm ^ez ½xmt xvt dxdy ¼ 1=2 1=2 x x dxdy ¼ dmv : om0 1 1 mt vt 1 1 ð4:1:15Þ Here, ^ez is a unit vector along the propagation direction z. dmv is Kronecker’s delta and is unity for m ¼ v, but zero otherwise. Note that this result is identical to integrating Poynting’s vector (power-flow density) for the mode field transversely across the waveguide. In the case of radiation modes, dmv is the Dirac delta function which is infinite for m ¼ v and zero for m 6¼ v. Equation (4.1.15) applies to the weakly guiding case for which the longitudinal component of the electric field is much smaller than the transverse component, rendering the modes predominantly linearly polarized in the transverse
Wave Propagation
123
direction to the direction of propagation [1]. Hence, the transverse component of the magnetic field is rffiffiffiffiffiffiffiffi e0 er @ ^ez xt : Ht ¼ ð4:1:16Þ m0 @z The fields satisfy the wave Eq. (4.1.13) as well as being bounded by the waveguide. The mode fields in the core are J-Bessel functions and K-Bessel functions in the cladding of a cylindrical waveguide. In the general case, the solutions are two sets of orthogonally polarized modes. The transverse fields for the mth x-polarized mode that satisfy the wave equation (4.1.13) are then given by [2]
r cos mf ð4:1:17Þ x x ¼ Cm J m u sin mf a rffiffiffiffiffi e0 ð4:1:18Þ x ; Hy ¼ neff m0 x and the corresponding fields in the cladding are
r Jm ðuÞ cos mf Km w xx ¼ Cm sin mf Km ðwÞ a rffiffiffiffiffi e0 x ; Hy ¼ neff m0 x
ð4:1:19Þ ð4:1:20Þ
where the following normalized parameters have been used in Eqs. (4.1.17)– (4.1.20): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pa ð4:1:21Þ n2core n2clad v¼ l ffi 2pa qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4:1:22Þ u¼ n2core n2eff l
and neff
w 2 ¼ v2 u 2 ;
ð4:1:23Þ
ncore nclad ¼ nclad b þ1 ; nclad
ð4:1:24Þ
where neff is the effective index of the mode and b¼
w2 : u2
ð4:1:25Þ
124
Theory of Fiber Bragg Gratings
Finally, assuming only a single polarization, the y-polarized mode, xy ¼ Hx ¼ 0:
ð4:1:26Þ
The choice of the cosine or the sine term for the modes is somewhat arbitrary for perfectly circular nonbirefringent fibers. These sets of modes become degenerate. Since the power carried in the mode in watts is |Am|2, from the Poynting’s vector relationship of Eq. (4.1.15), the normalization constant Cm can be expressed as " #1=2 pffiffiffiffiffiffiffiffiffiffiffi m0 =e0 2w
Cm ¼ ; ð4:1:27Þ av neff pem Jm1 ðuÞJmþ1 ðuÞ where em ¼ 2 when m ¼ 0 (fundamental mode) and 1 for m 6¼ 0. Matching the fields at the core–cladding boundary results in the waveguide characteristic eigenvalue equation, which may be solved to calculate the propagation constants of the modes: u
Jm1 ðuÞ Km1 ðwÞ ¼w : Jm ðuÞ Km ðwÞ
ð4:1:28Þ
4.2 COUPLED-MODE THEORY The waveguide modes satisfy the unperturbed wave Eq. (4.1.13) and have solutions described in Eqs. (4.1.17) through (4.1.20). In order to derive the coupled-mode equations, effects of perturbation have to be included, assuming that the modes of the unperturbed waveguide remain unchanged. We begin with the wave Eq. (4.1.11) r2 E ¼ m0 e0
@2E @2P þ m0 2 : 2 @t @t
ð4:2:1Þ
Assuming that wave propagation takes place in a perturbed system with a dielectric grating, the total polarization response of the dielectric medium described in Eq. (4.2.1) can be separated into two terms, unperturbed and the perturbed polarization, as P ¼ P unpert þ P grating ;
ð4:2:2Þ
P unpert ¼ e0 wð1Þ E m :
ð4:2:3Þ
where
Equation (4.2.1) thus becomes, r2 Emt ¼ m0 e0 er
@2 @2 Emt þ m0 2 Pgrating;m ; 2 @t @t
ð4:2:4Þ
Coupled-Mode Theory
125
where the subscripts refer to the transverse mode number m. For the present, the nature of the perturbed polarization, which is driven by the propagating electric field and is due to the presence of the grating, is a detail which will be included later. Substituting the modes in Eq. (4.1.14) into (4.2.4) provides the following relationship: 2 3 m¼l X ð r¼1 X iðotbm zÞ iðotbr zÞ 2 41 ½Am ðzÞxmt e þ cc þ Ar ðzÞxrt e dr5 r 2 m¼1 r¼0 2 3 m¼l X ð r¼1 @ 2 41 X ½Am ðzÞxmt eiðotbm zÞ þ cc þ Ar ðzÞxrt eiðotbr zÞ dr5 m0 e0 er 2 @t 2 m¼1 r¼0 ¼ m0
@2 Pgrating;m : @t2 ð4:2:5Þ
Ignoring coupling to the radiation modes for the moment allows the lefthand side of Eq. (4.1.13) to be expanded. In weak coupling, further simplification is possible by applying the slowly varying envelope approximation (SVEA). This requires that the amplitude of the mode change slowly over a distance of the wavelength of the light as @ 2 Am @Am ; bm @z2 @z
ð4:2:6Þ
so that
2 3 m¼l 1 X4 @Am 2 iðotbm zÞ iðotbm zÞ r Et ¼ x e 2ibm bm Am xmt e þ cc5: @z mt 2 m¼1 2
ð4:2:7Þ
Expanding the second term in Eq. (4.2.5), noting that o2 m0 e0 er ¼ b2m , and combining with Eq. (4.2.7), the wave equation simplifies to m¼l X @Am @2 xmt eiðotbm zÞ þ cc ¼ m0 2 Pgrating;t : ibm ð4:2:8Þ @z @t m¼1 Here, the subscript t on the polarization Pgrating,t reminds us that the grating has a transverse profile. Multiplying both sides of Eq. (4.2.8) by xm and integrating over the wave-guide cross-section leads to 2 3 m¼l ð þ1 ð þ1 X 4ibm @Am xmt x eiðotbm zÞ þ cc5dxdy mt @z 1 1 m¼1 ð4:2:9Þ ð þ1 ð þ1 @2 m0 2 Pgrating;t xmt dxdy: ¼ @t 1 1
126
Theory of Fiber Bragg Gratings
Applying the orthogonality relationship of Eq. (4.1.15) directly results 2 3 m¼l X @A m iðotb zÞ m 42iom0 e þ cc5 @z m¼1 ð4:2:10Þ ð þ1 ð þ1 @2 ¼ m0 2 Pgrating;t xmt dxdy: @t 1 1 Equation (4.2.10) is fundamentally the wave propagation equation, which can be used to describe a variety of phenomena in the coupling of modes. Equation (4.2.10) applies to a set of forward- and backward-propagating modes; it is now easy to see how mode coupling occurs by introducing forward- and backward-propagating modes. The total transverse field may be described as a sum of both fields, not necessarily composed of the same mode order: 1 Et ¼ ðAv xvt eiðotbv zÞ þ cc þ Bm xmt eiðotþbm zÞ þ ccÞ 2
ð4:2:11Þ
1 Ht ¼ ðAv Hvt eiðotbv zÞ þ cc Bm Hmt eiðotþbm zÞ ccÞ: 2
ð4:2:12Þ
Here the negative sign in the exponent signifies the forward- and the positive sign the backward-propagating mode, respectively. The modes of a waveguide form an orthogonal set, which in an ideal fiber will not couple unless there is a perturbation. Using Eqs. (4.2.11) and (4.2.12) in Eq. (4.2.10) leads to 2 3 2 3 @A @B v m 4 eiðotbv zÞ þ cc5 4 eiðotþbm zÞ þ cc5 @z @z ð4:2:13Þ ð ð i þ1 þ1 @ 2 ¼þ Pgrating;t xm;vt dxdy 2o 1 1 @t2
4.2.1 Spatially Periodic Refractive Index Modulation In a medium in which the dielectric constant varies periodically along the wave-propagation direction, the total polarization can be defined with the perturbed permittivity, De(z) and the applied field as P ¼ e0 ½er 1 þ DeðzÞEm :
ð4:2:14Þ
The terms within the parentheses are equivalent to w(1), and er is the relative permittivity of the unperturbed core. The constitutive relations between the permittivity of a material and the refractive index n result in the perturbation modulation index being derived from n2 ¼ er so that
Coupled-Mode Theory
127
½n þ dnðzÞ2 ¼ er þ DeðzÞ:
ð4:2:15Þ
Assuming the perturbation to be a small fraction of the refractive index, it follows that DeðzÞ 2ndnðzÞ: Defining the refractive index modulation of the grating as n o v dnðzÞ ¼ Dn 1 þ ei½ð2pN=LÞzþfðzÞ þ cc ; 2
ð4:2:16Þ
ð4:2:17Þ
where Dn is the refractive index change averaged over a single period of the grating, v is the visibility of the fringes, and the exponent term along with the complex conjugate cc describe the real periodic modulation in complex notation. An arbitrary spatially varying phase change of f(z) has been included. L is the period of the perturbation, while N is an integer (– 1 < N < þ1) that signifies its harmonic order. The period-averaged change in the refractive index has to be taken into account since it alters the effective index neff of a mode. Combining Eqs. (4.2.15) and (4.2.17), the total material polarization is n h io v Em ; ð4:2:18Þ P ¼ e0 n2 1 þ 2nDn 1 þ ei½ð2pN=LÞzþfðzÞ þ cc 2 where the first term on the RHS is the permittivity, the second term is the dc refractive index change, and the third term is the ac refractive index modulation. Finally, defining a new modulation amplitude by incorporating the visibility, Dn i½ð2pN=LÞzþfðzÞ þ cc ; ð4:2:19Þ e dnðzÞ ¼ 2n Dn þ 2 with Dn ¼ nDn as the amplitude of the ac refractive index modulation. Equation (4.2.19) describes the UV-induced refractive index change due to a grating written into the fiber core. Figure 4.1 shows the refractive index modulation for a uniform grating on a background index of the core of the fiber for different visibilities. Also shown is the effect on the average core index. Note that the change in the average index in the core is constant, irrespective of the visibility of the fringes, although it remains a function of dn. In the example shown, however, both the average index and the refractive index modulation dn increase with UV exposure time. The perturbed polarization can now be related to refractive index change shown in Eq. (4.2.19) to give Dn i½ð2pN=LÞzþfðzÞ þ cc Em : ð4:2:20Þ e Ppert ¼ 2ne0 Dn þ 2
128
Theory of Fiber Bragg Gratings 0.01 0.009 0.008
ncore - nclad
0.007 0.006 0.005 Visibility = 0.2
0.004
Visibility = 0.5
0.003
Visibility = 1 0.002
Average (ncore - nclad)
0.001
Unperturbed core to cladding index difference
0 0
0.5
1
1.5
2
Z, microns Figure 4.1 Refractive index modulation in the core of a fiber for different visibilities of the fringe pattern. Also shown is the average refractive index change in the core (dashed line). The unperturbed core-to-cladding refractive index difference is 5 103, while the maximum refractive index modulation for unity visibility is 2 103.
Including Eq. (4.2.20) in Eq. (4.2.13) results in 2 3 2 3 @A @B v m 4 eiðotbv zÞ þ cc5 4 eiðotþbm zÞ þ cc5 @z @z ð þ1 ð þ1
h i @2 iðotbv zÞ iðotþbm zÞ xm;vt dxdy þ cc dnðzÞ A e x þ B e x v m vt mt 2 1 1 @t 2 3 ð þ1 ð þ1 Dn 4Dn þ ¼ inoe0 Av ðei½ð2pN=LÞzþ’ðzÞ þ ccÞ5xvt eiðotbv zÞ xm;vt dxdy 2 1 1 2 3 ð þ1 ð þ1 Dn 4Dn þ inoe0 Bm ðei½ð2pN=LÞzþ’ðzÞ þ ccÞ5xmt eiðotþbm zÞ xm;vt dxdy þ cc: 2 1 1
¼þ
ie0 2o
ð4:2:21Þ On the LHS of Eq. (4.2.21), the rate of variation of either Av or Bm is determined by the mode order m or v of the electric field xm;vt chosen as the multiplier according to the orthogonality relationship of Eq. (4.1.15). This was shown in
Coupled-Mode Theory
129
Eq. (4.2.9) for the case of the single field. Once the term on the LHS has been chosen, the next question is the choice of the terms on the RHS. Before this is examined, we consider the terms on the RHS in general. The RHS of Eq. (4.2.21) has two generic components for both A and B modes as ð þ1 ð þ1 Dnxmt xmt dxdy RHS ¼ inoe0 Bm eiðotþbm zÞ 1
inoe0 Av eiðotbp zþfðzÞÞ
1
ð þ1 ð þ1 1
Dn xvt xmt dxdy þ cc; 1 2
ð4:2:22Þ
where the first exponent must agree with the exponent of the generated field on the LHS of Eq. (4.2.21) and has a dependence on the dc refractive index change, Dn. The reason is that any other phase-velocity dependence (as for other coupled modes) will not remain in synchrony with the generated wave. The second term on the RHS has two parts. The first one is dependent on the phasesynchronous factor, bp ¼
2pN bv : L
ð4:2:23Þ
The mode interactions that can take place are determined by the right-hand sides of Eqs. (4.2.21) and (4.2.22). Two aspects need to be taken into account: First, conservation of momentum requires that the phase constants on the LHS and the RHS of Eq. (4.2.22) be identical [Eq. (4.2.23)] and so influences the coupling between copropagating or counterpropagating modes. Secondly, the transverse integral on the RHS of Eq. (4.2.22), which is simply the overlap of the refractive-index modulation profile and the distributions of the mode fields, determines the strength of the mode interactions. Let us first consider the conservation of momentum, otherwise known as phase matching.
4.2.2 Phase Matching We begin with Eq. (4.2.23) in which the phase factor is the sum or difference between the magnitude of the driving electric-field mode propagation constant bv and the phase factor of the perturbation. The resultant bp is the phase constant of the induced polarization wave. This is the propagation constant of a “boundwave” generated by the polarization response of the material due to the presence of sources. For there to be any significant transfer of energy from the driving field amplitude Av to the generated fields on the LHS of Eq. (4.2.22), the generated and the polarization waves must remain in phase over a significant distance, z. For continuous transfer of energy, bm ¼ bp :
ð4:2:24Þ
130
Theory of Fiber Bragg Gratings Collective polarization response of phase-matched dipoles
Λ
Radiating dipoles
Figure 4.2 The principle of phase matching. The polarization wave grows in synchronism with the driving field. The radiating dipoles are shown to be spatially distributed with a period of L, allowing the radiated wave to remain in phase with the driving field. This schematic applies to guided or radiation-mode coupling.
Equation (4.2.24) then describes the phase-matching condition. A phase mismatch Db is referred to as a detuning, Db ¼ bm bp :
ð4:2:25Þ
Including Eq. (4.2.23) in (4.2.25), we get, Db ¼ bm bv
2pN : L
ð4:2:26Þ
If both bv and bm have identical (positive) signs, then the phase-matching condition is satisfied (Db ¼ 0) for counterpropagating modes; if they have opposite signs, then the interaction is between copropagating modes. Identical relationships for co- and counterpropagation interactions apply to radiation mode phase matching. A schematic of the principle of phase matching is shown in Fig. 4.2. Finally, energy conservation requires that the frequency o of the generated wave remains unchanged.
4.2.3 Mode Symmetry and the Overlap Integral The orthogonality relationship of Eq. (4.1.15) suggests that only modes with the same order m will have a nonzero overlap. However, the presence of a nonsymmetric refractive index modulation profile across the photosensitive region of the fiber can alter the result, allowing modes of different orders to have a
Coupled-Mode Theory
Amplitude
–2 1
131
–1
0.5
r/a 0
Field(LP01)
1
2 Refractive index modulation profile
0 –0.5
Field(LP11)
LP01-LP11 field overlap in core
–1 Figure 4.3 A cross-section of the fiber showing the fields of the LP01 and the LP11 modes, along with the transverse refractive index modulation profile. The overlap of the two fields with the profile of the index modulation [as per Eq. (4.2.22)] changes sign across the core but does not have the same magnitude. The field overlap is therefore nonzero. The transverse profile of the refractive index thus influences the symmetry of the modes allowed to couple. The transverse profile of the perturbation is equivalent to a “blaze” across the core (tilted grating), which benefits coupling to odd-order radiation modes as well. The two dashed lines indicate the core boundary.
nonzero overlap integral. The reason for this fundamental departure from the normalization of Eq. (4.1.15) is the nonuniform transverse distribution of sources, giving rise to a polarization wave that has an allowed odd symmetry. This is graphically displayed in Fig. 4.3: A driving fundamental mode (LP01, m ¼ 0) electric field, xv, interacts with a modulated permittivity that has a uniform transverse profile. Also shown is a polarization field that is in the LP11 mode (v ¼ 1). Examining the transverse overlap (which is proportional to the product of the field amplitudes and the refractive-index profile) on the left half of the core, we find that magnitude is the same as on the right half, but they have the opposite signs, resulting in a zero overlap. The orthogonality relationship holds and exchange of energy is not possible between the different order modes. If, however, the refractive index profile is not uniform across the core (Fig. 4.3), then although the signs of the overlap in the two halves (around a plane through the axis of the fiber) are different, the magnitudes are no longer identical. Thus, the overlap is now not zero, allowing a polarization wave to exist with a symmetry (and therefore, mode order) different from that of the driving mode. The selection rules for the modes involved in the exchange of energy are then determined by the details of the terms in the integral in Eq. (4.2.22) and apply equally to radiation mode orders. The consequence of the asymmetric refractive index perturbation profile may now be appreciated in Eq. (4.2.21). On the RHS, the integrals with the electric
132
Theory of Fiber Bragg Gratings
fields of the driving field xmt and the polarization wave xvt along with the asymmetric profile of the refractive index modulation are nonzero for dissimilar mode orders, i.e., m 6¼ v. The magnitude of the overlap for a particular mode combination will depend on the exact details of the perturbation profile.
4.2.4 Spatially Periodic Nonsinusoidal Refractive Index Modulation Note that in Eq. (4.2.21), the refractive-index perturbation can have a sign in the exponent. This is a direct result of the Fourier expansion of the permittivity perturbation. However, since it is equivalent to an additional momentum, which can be either added to or taken away from the momentum vector of a driving field, it may be viewed as a factor that can be included, as already discussed. In the general case when the refractive index modulation is not simply sinusoidal but a periodic complex function of z, it is more convenient to expand dn in terms of Fourier components as " # X Dn N¼þ1 i½ð2pN=LÞzþfðzÞ aN ðe þ ccÞ ; ð4:2:27Þ dnðzÞ ¼ 2n Dn þ 2 N¼1 where aN is the Fourier amplitude coefficient of the Nth harmonic of the perturbation. Differently shaped periodic functions have their corresponding aN coefficients, which in turn influence the magnitude of the overlap integral, and hence the strength of the mode coupling.
4.2.5 Types of Mode Coupling The phase-matching condition is defined by setting Db in Eq. (4.2.26) to zero. Therefore, 2pN bv ¼ bm : ð4:2:28Þ L Equation (4.2.28) states that a mode with a propagation constant of bm will synchronously drive another mode Av with a propagation constant of bv, provided, of course, the latter is an allowed solution to the unperturbed wave Eq. (4.1.28) for guided modes and its equivalent for radiation modes. The guided modes of the fiber have propagation constants that lie within the bounds of the core and the cladding values, although only solutions to the eigenvalue Eq. (4.1.28) are allowed. Consequently, for the two lowest order modes of the fiber, LP01 and LP11, the propagation constants bv and bm are
Coupled-Mode Theory
133
the radii of the circles 2pnv/l and 2pnm/l. A mode traveling in the forward direction has a mode propagation vector K LP01 that combines with the grating vector K grating to generate K result . Since the grating vector is at an angle yg to the propagation direction, and the allowed mode solution, K LP11 is in the propagation direction, the phase-matching condition reduces to
ð4:2:29Þ Db ¼ bLP þ jbLP j K grating cos yg ; 01
11
Under these circumstances, the process of phase matching reverses in sign after a distance (known as the coherence length lc) when Dblc ¼ p:
ð4:2:30Þ
Consequently, the radiated LP11 mode (traveling with a phase constant of bm) propagates over a distance of lc before it slips exactly half a wavelength out of phase with the polarization wave (traveling with a phase constant bv). In order to understand the various phase-matching conditions, we shall begin with the dispersion diagram of modes. The propagation constants of modes and their dispersion are crucial to the understanding of phase matching. To facilitate an insight into the properties of modes, we use the approximate analogy between rays and modes, since the visual aspect of rays is easier to understand. In Fig. 4.4 we see a section of an optical fiber with a ray incident at the angle at which it is refracted out of the fiber core to exit in a direction parallel to the z-axis. The propagation direction is indicated as the z-axis while the transverse direction is the x-axis. The angle ycritical ¼ sin1(nclad/ncore) is marked as the critical angle for that ray. The ray propagation angle is ycutoff. Thus, all ray angles below ycutoff are allowed, but only those that form standing waves [2] exhibit mode properties, with a specific effective propagation index neff. We note an important relationship in the ray picture: Since the effective index of a mode at cut-off is the cladding refractive index, the effective index of a mode is the cutoff index of a mode propagating in a waveguide with a cladding refractive index of neff.
qcritical Ray
Critical ray 90° ncore
qcutoff
nclad Fiber
Figure 4.4 Ray propagation in a waveguide.
core Cladding z
134
Theory of Fiber Bragg Gratings q rcritical
nmeff
Gmb
Gmf Rb
q mcritical
A Loci of neff of guided modes (backward) Loci of neff of radiation modes (backward)
Rf
nmeff
D C
qr
cut off
Gm
B +z
q mcut off O q m Vacuum n=1 Radiation zone nclad Guided zone
ncore
Loci of neff of guided and radiation modes (forward)
Figure 4.5 Generalized dispersion diagram for guided and radiation modes and radiation field for waveguides.
We now transfer this picture to the one shown in Fig. 4.5. Three circles with radii n0, nclad, and ncore form the boundaries for the waveguide. Figure 4.5 shows the generalized dispersion diagram for an optical fiber. The outer circle has a radius of ncore, the middle circle has a radius of nclad, and the shaded circle represents free space and has a radius of unity. It is based on the ray diagram shown in Fig. 4.4, so that the critical angle for the backward-propagating guided modes is marked as ym critical between the dotted and m the dashed lines at point Gm b , with a similar angle at Gf for the forwardpropagating modes. The two vertical dotted lines are tangential to the cladding and inner circles, respectively. All guided modes have their cutoff at ym critical . The equivalent cutoff angles in the propagation direction for all guided modes are also marked at the origin, as ym cutoff . For the radiation (cladding) modes the equivalent angles are yrcritical and yrcritical , subtended by the dashed lines to points Rb and Rf. The dashed lines in Fig. 4.5 mark these. We note that all guided modes have effective indexes lying within the region bounded by the outer two circles. A forward-propagating m guided mode has an effective index of nm eff , which lies on a circle of radius neff m (part of a dashed circle is shown) and on a vector OG , propagating at an angle ym to the z-axis. It is easy to show that the loci of all the effective indexes of the (forward- and backward-propagating) modes lie on circles (shown as the outer two dashed circles). The length of the vector from the origin to the intercept m with the nm eff circle subtends the ray angle y for that mode. The point at which
Coupled-Mode Theory
135
these dashed circles meet the nclad circle defines the cutoff of the guided modes. A similar set of circles intersects the free space shaded inner circle to define the cutoff of all cladding modes. Beyond this point and into the inner shaded circle is the radiation field region. If the cladding were extended to infinity, the middle circle would become the locus of all cladding space modes (continuum). In the present situation, the inner circle remains the locus of the free space modes, which are the cladding modes beyond cutoff. Having defined the phase space for all the modes, we can proceed to the phase-matching diagram, shown in Fig. 4.6. Here we see a forward-propagating mode, with an effective index of ncore cos ym f , phase matched to a counterpropagating mode with an effective index of ncore cos ym b (point Gpm) with a grating that has an “effective index” of ng cos yg. The grating period Lg ¼ l/(ng cos yg). When yg ¼ 0, we have the normal Bragg condition. We can now see the effect of detuning this interaction to shorter wavelengths. The point Gm moves down toward B, dragging the grating vector ng with it. This action carves out a phase-matching curve on the LH side of the figure, marked by the dashed curve. Since the grating angle, yg is fixed, the phase-matching point on the loci of mode coupling rotates clockwise as shown by the dotted line towards point E. Phase matching is lost since there is no intersection with the outer or inner circles. There is a gap in the spectrum, in which no phase matching is possible. At some point the arrowhead meets the vertical dotted line for the radiation Cutoff for radiation modes
Cutoff for guided modes
Tuning to shorter wavelengths
G pm qg Coupling to radiation modes begins A
ng
Gm a
b
+ D C
E q mcutoff
O
B
q rcutoff q r q m q b b
m f
+z dn
Vacuum n=1
Loci of phase matching point as wavelength is tuned
nclad
ncore
Figure 4.6 Guided mode and radiation mode/field phase-matching diagram for the slanted Bragg grating (counterradiating coupling).
136
Theory of Fiber Bragg Gratings
modes on line a, and phase matching to the radiation modes begins. This will couple to the lowest-order modes. With an infinite cladding, free-space radiation mode phase matching occurs. As the wavelength becomes even shorter, the angle of the radiation modes increases, and only when the vector b meets the nclad circle is radiation mode coupling at an angle of yrb . After this point, the angle of the radiation mode increases beyond yrb . We now note that the change in the mode index is dn from the RH side of the figure, so that we can calculate the wavelength at which the radiation loss starts to occur. Figure 4.6 shows the phase-matching diagram for coupling to the guided and radiation modes and fields with a tilted grating, known as side-tap-grating (STG, also see Chapter 6). This grating has a period similar to Bragg gratings but does not have its grating planes normal to the fiber axis, and it is tilted at an angle, yg. The diagram specifically deals with the case of coupling to counterpropagating fields. In the first interaction with ng, we have Bragg reflection at lBragg. We assume that the grating angle yg ¼ 0, and that when the wavelength is tuned, the effective index of the mode is nstart eff at the point indicated by a on Fig. 4.6, so that mathematically, this is simply phase matching to a mode with the cladding index as 2pnstart eff lstart
Bragg 2pnstart 2pN 4pneff clad þ ¼ ¼ ; lstart lBragg Lg
ð4:2:31Þ
where start indicates the wavelength at which the radiation mode coupling begins. Bragg Rearranging and using the approximation neff nstart eff neff , it follows that lBragg nclad 1þ lstart ¼ : ð4:2:32Þ 2 neff Therefore, radiation loss begins at a wavelength slightly shorter than the Bragg wavelength, governed by the ratio in the parentheses in Eq. (4.2.32). For example, in a fiber with a large core–cladding index difference with a tightly confined Bragg wavelength (1550 nm) mode (neff ¼ 1.475), the start wavelength will be at
1537 nm, some 13 nm away. We can estimate the maximum angle for the radiation by observing the point E on the phase-matching curve in Fig. 4.6. The tangent to this point on the phase matching at E intersects the cladding circle at the “þ” point. This point subtends the largest radiation mode angle for this particular grating, at the origin. The maximum angle of the radiation for an untilted grating is at the shortest wavelength and is easily shown to be 2neff ncore ; ð4:2:33Þ yrmax ¼ cos1 nclad
Coupled-Mode Theory
137
which is maximum if neff ¼ nclad. For a core-to-cladding refractive index difference of 0.01 in a silica fiber, yrmax 6:7 . It should be remembered that phase matching to specific radiation modes will only occur if a cladding mode exists with the appropriate mode index. However, with an infinite cladding, coupling to a continuum of the radiation field occurs so that the spectrum is continuous. There is another possibility for coupling to radiation modes. We begin with yg ¼ 0 and the condition for Bragg reflection from, for example, the forward to the counterpropagating LP01 mode. If the grating is tilted at an angle yg, it is shown simply as a rotation of ng around the pivot at Gm. Following the mathematical approach taken for Eq. (4.2.33), we find that at some angle yg0 the radiation mode is at the Bragg wavelength, i.e., the start wavelength moves toward the Bragg wavelength, until they coincide. At this point, there is strong coupling to the radiation modes. Referring to Fig. 4.6, the angle is easily found by changing the tilt of the grating. This directly leads to Kg Kg cos yg0 ¼
4p ðneff nclad Þ; lBragg
ð4:2:34Þ
where neff is the effective index of the mode at the Bragg wavelength of the untilted grating (when yg ¼ 0), so that nclad : ð4:2:35Þ cos yg0 ¼ neff Again, the tilt angle of the grating for this condition to be met increases with neff and is a maximum when neff ¼ ncore. We can calculate that for a standard fiber, with an neff nclad þ bDn (dn ¼ 4.5 103) [see Eq. (4.1.24)] and b ¼ 0.4 at 1550 nm [2], the angle at which the Bragg wavelength equals the radiation wavelength is yg0 2:85 . It is clear from Eq. (4.2.35) that the angle becomes larger with increasing core–cladding index difference. Finally there is a set of unconfined radiation modes at a continuum of angles subtended at O, but with vector lengths within the space of the radiation zone. Making the grating “effective index” ng small so that the arrowhead remains on the RH side of Fig. 4.6, one can see that phase matching will occur between copropagating modes, or to radiation modes in the forward direction. This is better shown in Fig. 4.7. The following points should be noted regarding the phase-matching diagram. The guided mode propagation constants have discrete values and lie on the loci for the particular mode propagation constants. The grating vector can have any angle yg to the propagation direction, as can the radiated field, provided the cladding is assumed to be at infinity. If, however, a cladding boundary is present as shown by the innermost dashed circle, then the radiated modes only have allowed b-values. This radiation may be viewed as the modes of a waveguide with a core of refractive index nclad and a radius equal to the fiber-cladding
138
Theory of Fiber Bragg Gratings ncore B nclad
qr
nr
ng A
+z Vacuum
nLP01
nLP01 qLP01
Radiation zone
Figure 4.7 The phase-matching diagram for copropagating modes with radiation mode coupling with a long-period grating.
radius surrounded by an infinite cladding of air/vacuum refractive index. The diagram then acquires a set of circles with radii nvacuum < nr < nclad representing discrete cladding modes, similar to those for guided modes, at the points of intersection with the dispersion curves. The radiation mode fields are slightly modified by the presence of the high-index fiber core. Coupling is also possible to the forward-radiating modes and fields. This requires a different grating, known as a long-period grating, which has a much longer period than a Bragg grating, since the momentum of the mode does not change sign (as in forward-to-backward coupling). The phase matching for the generalized case of the tilted grating for copropagating coupling is shown in Fig. 4.7. The form of the diagram is similar to Figs. 4.5 and 4.6. For phase matching, the movement of the ng arrowhead for the LPG is opposite to that of the STG. We begin with an LP01 mode with the propagation index arrow nLP01 pointing in the þz direction. The grating ng starts at the tip of the guided mode arrow, inclined at yg ¼ 0 to the fiber axis. The wavelength at which radiation is first emitted is when the tip of the grating vector from point A intersects the tangent to the cladding mode circle (dashed vertical line). This point represents the longest-wavelength LP01 mode that has a propagation constant equal to the cladding index and has the lowest angle. Light is coupled to radiation modes within the radiation zone as ng is moved to the left and the LP01 mode is “cut off” at the radiation angle, yr. Therefore, yr is the angular spread of the radiated fields. Mode coupling is only possible if there is phase matching to specific modes. Note that this wavelength approaches 1, since
Coupling of Counterpropagating Guided Modes
139 Reflection grating guided mode
A Ey
B
Reflective polarization/mode and radiation mode couplers
Ex
Er
Copropagating polarization and mode couplers: rocking filters
C Er
D
Long period grating forward radiation mode coupler
Figure 4.8 Types of Bragg gratings categorized by action of coupling. The schematics show various gratings in the core of an optical fiber. All gratings are shown to be transversely uniform. A nonuniform transverse refractive index modulation profile enhances coupling to either different mode orders of guided or to the radiation field. (a) shows a guided mode reflection grating. (b) shows a reflecting guided mode polarization coupler, mode converter, or radiation mode coupler (“side-tap” grating). (c) is a polarization coupler for copropagating modes, also knows as a “rocking filter” [31]. (d) is the copropagating guided-mode to radiation-mode coupler, also known as the “long-period grating.”
the fundamental guided mode effective index approaches nclad. The wavelength versus angle has the opposite dependence of the STG, i.e., long wavelengths exit at the largest angle in the LPG, while it is the shortest wavelengths in the STG. The first Bragg wavelength reflection (very weak) is at the short wavelength side of the LPG radiation loss spectrum, while it is on the long wavelength side of the STG radiation loss spectrum. The spectrum of the LPG is “reversed” around the Bragg wavelength. Figure 4.8 shows various types of phase-matched interactions possible with different types of gratings.
4.3 COUPLING OF COUNTERPROPAGATING GUIDED MODES The simplest form of interaction is between a forward-propagating and an identical backward-propagating mode. However, for a general approach, dissimilar modes are considered for the counterpropagating (reflected) mode phase matching with Eq. (4.2.21) rewritten as
140
Theory of Fiber Bragg Gratings
@Bm iðotþbm zÞ þ cc ¼ inoe0 Bm e @z
ð þ1 ð þ1
Dnxmt xmt eiðotþbm zÞ dxdy
1 1 ð þ1 ð þ1
þ inoe0 Av
1
Dn i½ð2pN=LÞzþfðzÞ xvt xmt eiðotbv zÞ dxdy þ cc: e 1 2
ð4:3:1Þ By choosing the appropriate b value for identical modes (m ¼ v) but with opposite propagation directions in Eq. (4.3.1) and dividing both sides by exp[i(ot þ bmz)], ð þ1 ð þ1 @Bm ¼ inoe0 Bm Dnxmt xmt dxdy @z 1 1 ð þ1 ð þ1 ð4:3:2Þ Dn ið½ð2pN=LÞbv bm zþfðzÞÞ xvt xmt dxdy; þ inoe0 Av e 1 1 2 which leads to the following simple coupled-mode equations by choosing the appropriate synchronous terms, @Bm ¼ ikdc Bm þ ikac Av eiðDbzf½zÞ ; @z
ð4:3:3Þ
with Db ¼ bm þ bv and the dc coupling constant, kdc ¼ noe0
ð þ1 ð þ1 1
1
2pN ; L
Dnxmt xmt dxdy;
while the ac coupling constant kac includes the overlap integral, ð þ1 ð þ1 Dn kac ¼ noe0 xvt xmt dxdy 1 1 2 v ¼ kdc ; 2
ð4:3:4Þ
ð4:3:5Þ
ð4:3:6Þ
if m ¼ v. The change in the amplitude of the driving mode may also be derived from Eq. (4.2.21) as @Av ¼ ikdc Av ikac Bm eiðDbzfðzÞÞ : @z
ð4:3:7Þ
Equations (4.3.3) and (4.3.7) are the coupled-mode equations from which the transfer characteristics of the Bragg grating can be calculated.
Coupling of Counterpropagating Guided Modes
141
To find a solution, the following substitutions are made for the forward (reference) and backward propagating (signal) modes [32]: R ¼ Av eði=2Þ½DbzfðzÞ S ¼ Bm eði=2Þ½DbzfðzÞ :
ð4:3:8Þ
Differentiating Eq. (4.3.8) and substituting into Eqs. (4.3.3) and (4.3.7) results in the following coupled-mode equations: dR 1 dfðzÞ ð4:3:9Þ þ i kdc þ Db R ¼ ikac S dz 2 dz dS 1 dfðzÞ ð4:3:10Þ i kdc þ Db S ¼ ikac R: dz 2 dz The physical significance of the terms in brackets is as follows: kdc influences propagation due to the change in the average refractive index of the mode, as has already been discussed. Any absorption, scatter loss, or gain can be incorporated in the magnitude and sign of the imaginary part of kdc. Gain in distributed feedback gratings will be discussed in Chapter 8. There are also two additional terms within the parentheses in Eqs. (4.3.9) and (4.3.10), the first one of which, Db/2, is the detuning and indicates how rapidly the power is exchanged between the “radiated” (generated) field and the polarization (“bound”) field. This weighting factor is proportional to the inverse of the distance the field travels in the generated mode. At phase matching, when Db ¼ 0, the field couples to the generated wave over an infinite distance. Finally, the rate of change of f signifies a chirp in the period of the grating and has an effect similar to that of the detuning. So, for uniform gratings, df/dz ¼ 0, and for a visibility of unity for the grating, kac ¼ kdc/2. The coupled-mode Eqs. (4.3.9) and (4.3.10) are solved using standard techniques [33]. First the eigenvalues are determined by replacing the differential operator by l and solving the characteristic equation by equating the characteristic determinant to zero. The resultant eigenvalue equation is in general a polynomial in the eigenvalues l. Once the eigenvalues are found, the boundary values are applied for uniform gratings: We assume that the amplitude of the incident radiation from –1 at the input of a fiber grating (of length L) at z ¼ 0 is R(0) ¼ 1, and that the field S(L) ¼ 0. The latter condition is satisfied by the fact that the reflected field at the output end of the grating cannot exist owing to the absence of the perturbation beyond that region. These conditions result in the following analytical solution for the amplitude reflection coefficient: r¼
S ð0Þ kac sinhðaLÞ ¼ ; Rð0Þ d sinhðaLÞ ia coshðaLÞ
ð4:3:11Þ
142
Theory of Fiber Bragg Gratings
where d ¼ kdc þ and a¼
1 dfðzÞ Db 2 dz
ð4:3:12Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kac 2 d2 :
ð4:3:13Þ
A few points regarding Eqs. (4.3.9)–(4.3.13) are worth mentioning. First, for reflection gratings that have a constant period L, the variation in the phase df(z)/dz ¼ 0. Second, at precise phase matching, Db ¼ 0, and the ac coupling constant kac is a real quantity. Finally, the power reflection coefficient is |r|2, jrj2 ¼
jkac j2 sinh2 ðaLÞ jkac j2 cosh2 ðaLÞ d2
;
ð4:3:14Þ
in which Eq. (4.3.13) has been used to simplify the result. Noting from Eq. (4.3.14) that a can be real or imaginary, the following regimes may be identified: 1. a is real when |kac| > d and Eqs. (4.3.11) and (4.3.14) apply. 2. a is zero when |kac| ¼ d. 3. a is imaginary when |kac| < d and Eqs. (4.3.11) and (4.3.14) transform to r¼
kac sinðaLÞ ; ia cosðaLÞ þ d sinðaLÞ
jkac j < d;
ð4:3:15Þ
jkac j < d:
ð4:3:16Þ
and jrj2 ¼
jkac j2 sin2 ðaLÞ d2 jkac j2 cos2 ðaLÞ
;
4.4 CODIRECTIONAL COUPLING In a multimode fiber, coupling can occur between orthogonally polarized modes of the same order, or to cladding modes (LPG) if the transverse profile of the refractive index perturbation is uniform. However, as has been described by the general mode-coupling constants of Eqs. (4.2.5) and (4.2.6) dissimilar mode orders that normally cannot couple owing to the orthogonality relationship [Eq. (4.1.15)] are allowed to couple when the transverse profile of the
Codirectional Coupling
143
refractive index is nonuniform. This applies equally to copropagating modes. In this case, coupling may normally occur between 1. Copropagating orthogonal polarizations, e.g., ðHE11 Þx;y $ ðHE11 Þy;x (LP01,x and the LP01,y). A uniform grating profile is necessary for good efficiency. To allow coupling between these modes, the grating is written at 45 to the principle birefringent axes of the fiber (see Section 4.5 and Chapter 6). 2. ðLP01 Þx;y $ ðLPvm Þx;y . Here, the transverse profile of the grating strongly influences the strength of the coupling. With a uniform profile, the coupling is zero for v 6¼ 0. 3. Coupling to the radiation field Er (as with LPGs). Since the radiation field is evanescent in the core of the fiber and oscillatory in the cladding, coupling can be strongly influenced if a grating extends into the cladding as well. The latter diminishes the overlap integral between the guided lowest-order mode and the radiation modes, while an asymmetric transverse grating profile can enhance the interaction with odd modes. Following the analysis developed in Section 4.2 and 4.3, the mode coupling equations for copropagating modes are @Bm ¼ ikdc;m Bm ikac;vm Av eiðDbzf½zÞ ; @z
ð4:4:1Þ
but with the phase-mismatch factor Db ¼ bv bm
2pN ; L
and the dc self-coupling constant for each of the modes, ð þ1 ð þ1 Dnxmt xmt dxdy kdc;m ¼ noe0 kdc;v ¼ noe0
1
1
1
1
ð þ1 ð þ1
Dnxvt xvt dxdy:
ð4:4:2Þ
ð4:4:3Þ ð4:4:4Þ
The cross-coupling constant kac remains the same as for contradirectional coupling as ð þ1 ð þ1 Dn ð4:4:5Þ kac;vm ¼ noe0 xvt xmt dxdy: 1 1 2 The amplitude of the input mode evolves as @Av ¼ ikdc;v Av ikac;mv Bm eiðDbzfðzÞÞ : @z
ð4:4:6Þ
Notice that the dc coupling constants may be different for the evolution of the input and coupled modes. To resolve this problem, we introduce new
144
Theory of Fiber Bragg Gratings
variables, R and S as before, but slightly modified, to result in a common coupling factor: Av ¼ Reiððkdc;m þkdc;v Þ=2Þz eði=2Þ½DbzfðzÞ
ð4:4:7Þ
Bm ¼ Seiððkdc;m þkdc;v Þ=2Þz eði=2Þ½DbzfðzÞ :
ð4:4:8Þ
The subscripts m and v on the dc coupling constants kdc are specific to each mode and is defined by Eq. (4.2.5) for identical modes. Differentiating R and S, collecting terms, and substituting into Eqs. (4.4.1) and (4.4.6) leads to dR i dfðzÞ ð4:4:9Þ kdc;m kdc;v þ Db þ R ¼ ikac S dz 2 dz dS i dfðzÞ ð4:4:10Þ Db þ kdc;v kdc;m S ¼ ikac R: dz 2 dz The phase-mismatch factor Db is now proportional to the difference in the propagation constants of the two modes as shown in Eq. (4.4.2). The cross-coupling constant kac is defined by Eqs. (4.2.6) and (4.4.5) as kac,mv for identical or nonidentical modes. Note that the coupling constant is real so that kac;mv ¼ kac;vm ¼ kac . The grating transmission function comprises two modes – in the simplest case, two orthogonal modes of the same order. However, the general case includes nonidentical modes (including a radiation mode) with the same or orthogonal polarization. The details of the coupling constants kac and kdc need to be evaluated numerically. Radiation modes are considered in Section 4.7, while coupling between different polarizations is presented in Section 4.5. The solutions to the coupled-mode Eqs. (4.4.9) and (4.4.10) are found by applying the boundary values as in the case of the reflection grating. However, for the transmission grating, the input fields, R(–L/2) ¼ 1 and S(–L/2) ¼ 0. The power couples from R to S so that the transmission in the uncoupled state is jRðL=2Þj2 2
jRðL=2Þj
¼
d2 sin2 ðaLÞ þ cos2 ðaLÞ; a2
ð4:4:11Þ
and the transmission in the coupled state (also known as the crossed state) is jSðL=2Þj2 jRðL=2Þj
2
¼
k2ac sin2 ðaLÞ: a2 :
In Eqs. (4.4.11) and (4.4.12), a ¼ (|kac|2 þ d2)1/2, and 1 dfðzÞ d ¼ kdc;v kdc;m þ Db : 2 dz
ð4:4:12Þ
Polarization Couplers: Rocking Filters
145
1 Tx (kL = pi/2, 2L) C
Transmission
0.8
Tx (kL = pi, L) B
A
Tx (kL = pi/2,L) A
0.6 0.4
B C
0.2 0 –0.02
–0.01
0.00
0.01
0.02
Normalized detuning, ( lB - l )/lB Figure 4.9 Cross-coupled transmission for codirectional coupling. The data shown is for coupling constants of p/2 with a grating length of L (curve A) and length 2L (curve C). The bandwidth is halved for the longer grating. Also shown is data for kL ¼ p and grating length L (curve B). Note that the grating overcouples at zero detuning. The transmission spectra of LPGs are identical to that shown above for each of the cladding modes to which the guided mode couples.
The difference between reflection as in contradirectional coupling and codirectional mode coupling is immediately apparent according to Eqs. (4.2.14) and (4.4.12). While the reflected signal continues to increase with increasing aL, the forward-coupled mode recouples to the input mode at aL > p/2. Therefore, a codirectional coupler requires careful fabrication for maximum coupling. Figure 4.9 demonstrates the optimum coupling to the crossed state with kacL ¼ p/2 (curve A) as the coupling length doubles, the transmission band becomes narrower (C), while B shows the situation of kacL ¼ p, when the light is coupled back to the input mode.
4.5 POLARIZATION COUPLERS: ROCKING FILTERS Equations (4.4.9) and (4.4.10) also govern coupling of modes with orthogonal polarization. An additional subscript is used to distinguish between the laboratory frame polarizations. However, there are differences in the detail of the coupling mechanism. In order to couple two orthogonally polarized modes, the perturbation must break the symmetry of the waveguide. This requires a source term, which can excite the coupled mode. In perfectly circular fibers, any perturbation can change the state of the output polarization. Nondegenerate orthogonally polarized modes can only exist in birefringent fibers and so require a periodic perturbation equal to the beat length.
146
Theory of Fiber Bragg Gratings
Generically, the polarization coupler behaves in a similar way to the intermodal coupler, except that the coupling is between the two eigenpolarization states of the fiber rather than two different order modes. As a result, gratings that have a uniform refractive index modulation across the core are used rather than blazed (or tilted) gratings. Coupling between two dissimilar order modes occurs when symmetry is broken by slanting the grating in the direction of propagation; for coupling between the eigenpolarization states of the same order, symmetry is broken by orienting the grating at 45 to the polarization axes of the fiber. “Slanting” the grating azimuthally at an angle of 45 to the birefringent axes “rocks” the birefringence [34] of the fiber backward and forward, with a period equal to the beat length, 2p ¼ bx by ; Lb
ð4:5:1Þ
where Lb is the beat length, so that the rocking period Lr is Lr ¼
Lb : N
ð4:5:2Þ
N is the order of the grating, and the detuning parameter is Db ¼ bx by
2Np : Lr
ð4:5:3Þ
We assume that the dielectric constants of the principal axes are ex and ey. A UV beam incident at an angle y to the x-axis and orthogonal to the propagation direction induces a new set of orthogonal birefringent axes with a change Dex0 and Dey0 in the dielectric constants. Figure 4.10 shows the incident UV beam on the cross-section of the fiber. The major and minor axes of the ellipse are the birefringent axes of the fiber. The beam is incident at an angle y. As a result, the birefringence changes locally, inducing a rotation in the birefringent axes of the fiber. The rotation angle f is related to the change in the birefringence as [35] tan 2f ¼
dDe sin 2y ; De þ dDe cos 2y
ð4:5:4Þ
y
ey Fiber core
y⬘ UV beam
Δey⬘ θ
Fiber cladding
Δex⬘
ex x
Figure 4.10 Birefringence induced by the incident UV beam in a birefringent fiber for the formation of a rocking filter.
Polarization Couplers: Rocking Filters
147
where dDe ¼ Dex0 – Dey0 and De ¼ ex – ey. For the case when the induced birefringence is much less than the intrinsic birefringence, then the rotation angle f is small, and it follows that 2f ¼
dDe sin 2y: De
ð4:5:5Þ
Remembering that dDe/De ¼ 2nav(dDn0 )/(2navDn), with dDn0 ¼ Dnx0 – Dny0 , f¼
d Dn0 sin 2y; 2 B
ð4:5:6Þ
where B is the difference in the refractive index of the principle axes. For the special case of y ¼ p/4, Eq. (4.5.6) simplifies to f ¼ dDn0 /(2B). A rotation that changes sign over one beat length implies a change in the rotation of 2f radians per beat length so that the coupling constant, kac, kac Lb ¼ 2f ¼
dDn0 ; B
ð4:5:7Þ
and remembering that B ¼ l/Lb leads to kac ¼
dDn0 ; l
ð4:5:8Þ
where l is the resonance wavelength. The coupler length Lr is given by the distance at which the input polarization is rotated by p/2, from which it follows that p ð4:5:9Þ kac Lr ¼ : 2 Substituting Eq. (4.5.8) into Eq. (4.5.9), we get the rotation length for 100% polarization conversion as Lr ¼
pl : 2dDn0
ð4:5:10Þ
In order to calculate the bandwidth between the first zeroes of the transmission spectrum, we note the argument of Eq. (4.4.12), aLr ¼ p, which leads to 2
Dl pLb : ¼2 Lr l
ð4:5:11Þ
Using typical figures for the reported changes in the birefringence [36,37,35,31], at a wavelength of 1550 nm, we find that the rocking filter has a length of 0.5 m. Note that the coupler length is only dependent on the wavelength of operation and the induced birefringence, but not the intrinsic birefringence of the fiber. If however, the “duty cycle” of the UV-exposed region is varied so that less than half of a beat length is exposed per beat length, then the effective rocking angle per beat length will be reduced, as with the change in y.
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Theory of Fiber Bragg Gratings
4.6 PROPERTIES OF UNIFORM BRAGG GRATINGS Quantities of interest are the bandwidth, Dl, reflectivity, transmissivity, the variation in the phase f, and the grating dispersion D as a function of detuning. For the purpose of illustration, Fig. 4.11 shows the reflection spectrum of two Bragg gratings with different coupling constants kacL of 2 and 8 calculated from Eq. (4.3.16). Note that the central peak is bounded on either side by a number of subpeaks. This feature is characteristic of a uniform-period grating of finite length, with a constant fringe visibility. The abrupt start and end to the grating is responsible for the side structure. In the weak grating limit (R < 0.2), the Fourier transform of the variation in the index modulation results in the reflection spectrum [5]. Thus, a uniform period and index-modulation grating (“top hat”) will produce side lobes. However, for stronger gratings, a Fourier transform for the fundamental component of the refractive index modulation alone is no longer appropriate and gives rise to increasing errors. For the uniform grating, df/dz ¼ 0, the peak reflectivity occurs at a wavelength at which d ¼ 0 (and therefore, a ¼ kac), and Eq. (4.3.16) leads to kdc þ
Db ¼ 0: 2
ð4:6:1Þ
At the phase matching wavelength, the reflectivity reduces to jrj2 ¼ tanh2 ðkac LÞ:
–2.5E−04 0
–1.5E−04 –5.0E−05
Reflectivity (dB)
kL = 8 –5
5.0E−05
ð4:6:2Þ
1.5E −04
2.5E −04
kL = 2
–10 –15 –20 Normalized detuning, ( lb - l )/lb
Figure 4.11 Reflectivity of two gratings with coupling constants kawL of 2 and 8, as a function of normalized detuning. Note that for the weaker reflection grating (kawL ¼ 2, dashed curve), the bandwidth to the first zeroes (between the main reflection peak and the next subpeaks) is much narrower than for the stronger grating (kawL ¼ 8, continuous curve). The side-mode structure increases rapidly for stronger gratings.
Properties of Uniform Bragg Gratings
149
For identical forward- and counterpropagating modes, it is simple to show by using the orthogonality relationship of Eq. (4.1.15) in Eqs. (4.3.5) and (4.3.6) that kdc ¼
4pDn ; l
ð4:6:3Þ
where the overlap integral 1 for identical modes, and it therefore follows that the peak of the Bragg reflection is at Dn l max ¼ lB 1 þ : ð4:6:4Þ n The Bragg wavelength lB is defined at the phase-matching point Db ¼ 0 for the general case of dissimilar modes, 2p 2pneff ;v 2pneff ;m þ ; ¼ l l L
ð4:6:5Þ
with the result L¼
l : neff ;v þ neff ;m
ð4:6:6Þ
For identical forward- and counterpropagating modes or nearly identical mode indexes, Eq. (4.6.6) reduces to L¼
lBragg : 2neff
ð4:6:7Þ
The reason why the reflection peak is at a longer wavelength than the Bragg wavelength is because the average refractive mode index Dn continuously increases with a positive refractive index modulation. For nonidentical modes, the integral in Eq. (4.3.5) has to be integrated numerically. However, the integral is simply a weighting factor, 0 < < 1, dependent on the mode and refractive index profiles. It is for this reason that has been introduced in Eq. (4.6.3), normalized to unity for identical modes. There are several definitions of bandwidth. However, the most easily identifiable one is bandwidth between the first minima on either side of the main reflection peak (with reference to Fig. 4.11). This may be calculated by equating the argument aL in Eq. (4.3.11) to p, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4:6:8Þ aL ¼ k2ac d2 L ¼ ip: Therefore,
2 kac d2 L ¼ p2 ;
ð4:6:9Þ
150
Theory of Fiber Bragg Gratings
which, after rearranging, becomes d¼
1 L
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2ac L2 þ p2 :
ð4:6:10Þ
From Eq. (4.3.12), assuming kdc ¼ 0 and df/dz ¼ 0 (no chirp in the grating), we get d¼
Db ; 2
so that the detuning from the peak to the first zero is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DbL ¼ 2 k2ac L2 þ p2 :
ð4:6:11Þ
ð4:6:12Þ
For identical modes, m ¼ v, using Eq. (4.3.4) we get, Db ¼
4pneff ðlB lÞ 4pneff Dl ; llB l2
ð4:6:13Þ
where the bandwidth from the peak to the first zero is Dl. Combining Eqs. (4.16.12) and (4.6.13), and noting that the bandwidth between the first zeroes is twice the bandwidth between the peak and the first zero, leads to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 ðkac LÞ2 þ p2 : ð4:6:14Þ 2Dl ¼ pneff L From Eq. (4.6.14) it follows that if ðkac LÞ2 p2 , then the bandwidth is an inverse function of the grating length as 2Dl
l2 ; neff L
ð4:6:15Þ
while if the reverse is true, ðkac LÞ2 p2 , then the bandwidth is independent of the length of the grating and is proportional to the ac coupling constant, 2Dl
l2 kac ; pneff
ð4:6:16Þ
so that increasing kac increases the bandwidth. Zeroes in the reflection spectrum of the grating can be evaluated by using a similar analysis, to occur at aL ¼ iMp;
M ¼ 1; 2; 3; . . .
from which the corresponding detuning follows, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 Mp 2 Dl ¼ kac kac þ : pneff L
ð4:6:17Þ
ð4:6:18Þ
Properties of Uniform Bragg Gratings
151
It is also useful to note the approximate position of the side-lobe peaks at 1 aL ¼ i M þ p; M ¼ 1; 2; 3; . . . ; ð4:6:19Þ 2 which leads to
dlsl ¼
l2 pneff
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 32ffi u 2 u 1 u Mþ p u 6 2 7 ukac k þ 6 7 : ac t 4 5 L
ð4:6:20Þ
4.6.1 Phase and Group Delay of Uniform Period Gratings Figure 4.12 shows the phase response of the two gratings in Fig. 4.11, as a function of detuning, in units of radians per meter of grating. In the region outside of the band stop of the grating, the phase of the light changes according to the unperturbed material refractive index. Into the band stop, the phase velocity slows down with increasing strength of the grating. In Fig. 4.13 is the group delay of the same grating as in Fig. 4.12 in units of psec/m. This group delay per meter is t 1 l2 df ¼ : Lg Lg 2pc dl
ð4:6:21Þ
Total phase (rad/m)
Close to the edge of the band stop, strong dispersion can be seen with increasing strength of the grating. However, this dispersion is limited to a small bandwidth. At the center of the band, the group delay is a minimum and is approximately 1/(2kacLg) for gratings with a reflectivity close to 100%. –2.5E−04 0 –50 –100 –150 –200 –250 –300 –350
–1.5E−04 –5.0E−05
5.0E −05
1.5E −04
2.5E −04
kL = 8 kL = 2 Normalized detuning, (lb–l)/lb
Figure 4.12 Total phase change in reflection from a uniform-period grating as a function of detuning for kawL ¼ 2 (dashed line) and kawL ¼ 8 (solid line). The stronger gratings behaves as a point reflector, since the phase change on reflection is large and almost constant for the same bandwidth when compared with the weaker grating.
Theory of Fiber Bragg Gratings
Normalized delay (ps/m)
152 –2.5E−04 –1.5E−04 –5.0E−05 0 –10000
5.0E−05
1.5E −04
2.5E −04
kL = 2
–20000 kL = 8 –30000 –40000 Normalized detuning, (lb -l) /lb
Figure 4.13 The normalized delay of a uniform period grating as a function of normalized detuning. For the larger coupling constant, the group delay in the center of the band is constant, while at the edges it increases rapidly, but is confined to a small spectrum.
4.7 RADIATION MODE COUPLERS
4.7.1 Counterpropagating Radiation Mode Coupler: The Side-Tap Grating These gratings couple from a forward-propagating guided mode to a backward-propagating cladding mode, or a continuum of radiation modes. In this section, a new and simple theory is presented to gain a physical insight into the scattering of a guided mode into the counterpropagating radiation modes. The theory for radiation mode coupling has been presented elsewhere by various authors: Marcuse [2], Erdogan and Sipe [38], Mizrahi and Sipe [39], Erdogan [40], and Morey and Love [41]. These articles explain the phenomenon of radiation mode coupling using coupled mode theory and have successfully described the short-wavelength radiation loss from unblazed gratings [39], as well as the fine detailed spectrum observed under cladding mode resonance in tilted gratings [38,40]. Surprisingly little work has been reported on the application of these gratings. The term “side-tap” is appropriate for such gratings since the radiation is tapped from the side of the fiber, as happens when bending a fiber. A particular difference from bending is the reduced bandwidth and wavelength selectivity of the loss induced by such blazed gratings. Important properties of blazed gratings are their stability and low intrinsic temperature sensitivity, which may be reduced even further by appropriate design of fiber or coating [42,43]. Although the reflectivity into the counterpropagating guided mode is not generally zero, for acceptable
Radiation Mode Couplers
153
performance in practical applications, it can be made very low by careful design of the filter. The intention of this section is to provide a physical insight into the functioning of blazed gratings with the purpose of intentionally designing filters, which predominantly exhibit only radiation loss. The potential applications are numerous, e.g., in-fiber noninvasive taps, spectrum analyzers [44], and gain flattening of optical amplifiers using a single blazed grating [45] and multiple blazed gratings [46] and mode converters [47]. Figure 4.14 shows a schematic of the blazed “side-tap” grating written in the core of an optical fiber. The guided mode shown on the LHS of the figure can couple to the radiation field or to a “supermode” of a composite waveguide formed by the cladding and air interface. These are shown as a field distribution leaving the core at an angle and as a mode of the waveguide formed by the cladding, respectively. While the radiated fields form a continuum if unbounded [see Eq. (4.2.5)], they evolve into the bound supermodes of the composite waveguide in the presence of a cladding. The power in the radiated field and the radiated bound mode may grow provided the overlap of the interacting fields and the transverse distribution of the “source” (refractive index perturbation) is nonzero (see Section 4.2.3). The exchange of energy between the core mode and the radiated bound supermode is determined by the prevailing phase-matching conditions discussed in Section 4.2.5 and is solely a coherent interaction; the coupling to the unbound continuum of the radiation field is, however, only partly governed by this requirement. Physically, the radiated field exiting from the fiber core at a nonzero angle is spread away so that the distance over which it
Radiation field (infinite cladding)
Counter propagating cladding mode
Fiber Cladding Core
Guided mode qg
Blazed grating
Bragg phase-matching condition: 2pneff /l + 2pncl /l = 2πN cos qg /Lg Figure 4.14 Schematic of counter-propagating radiation field and bound cladding mode coupling from a forward propagating guided mode with a blazed grating.
154
Theory of Fiber Bragg Gratings
Transmission, dB
1543.5 0
1544.5
1545.5
1546.5
1547.5
1548.5
–5 –10 –15 –20 –25 Wavelength, nm
Figure 4.15 Cladding mode resonance in untilted gratings. On the right is the LP01 ! LP01 guided mode reflection, while the others are to the LP0n cladding modes (n ¼ 2 to 10). See cover picture for example of cladding modes.
is coupled to the driving field is limited. This may be understood by the following: The driving mode field amplitude, which is assumed to be spatially constant, overlaps with a radiation field that is spreading rapidly away from the guided mode. For a coherent interaction, the fields must overlap over a distance with the correct phases. While the phases may remain synchronous, the radiated field spreads away, reducing the overlap as a function of propagation distance. With the cladding present, the field forms a mode, which propagates in the cladding and is then strongly coupled to the guided mode. This type of coupling is similar to simple Bragg reflection to discrete modes of the cladding. The transmission spectrum of a grating, which demonstrates this effect, is shown in Fig. 4.15. In this case, coupling is to modes of the HE1n order (LP0n), with a blaze angle close to zero. The cladding resonances are clearly visible. Also shown in Fig. 4.14 is the equation describing the phase-matching condition for coupling to counterpropagating radiation at zero angles. This radiated field is at the longest wavelength at which coupling to the radiation field is possible, and only to zero-order modes, i.e., LP0m. Note that the period of the grating, Lg, is dependent on the sum of the propagation constants of the guided driving mode and the radiated field (see also Fig. 4.5). As a consequence, any change in the cladding mode index only weakly affects the radiated field, but does change the coherent coupling to the supermodes. For a radiation mode tap, it is useful to consider the coupling to the unbounded radiation field. Another point to note is the angle of the radiated field, which is always slightly more than twice the tilt angle of the grating, apparent from Fig. 4.6. However, this angle is reduced when the overlap to the radiated field is taken into account. In Fig. 4.16 is shown the practical case of side illumination of optical fiber with UV irradiation. The absorption in the fiber causes the refractive index to
Radiation Mode Couplers
155
UV radiation
Index change profile
Core
UV penetration
Mode field
Depth Figure 4.16 Effect of side illumination of a fiber core with UV radiation, giving rise to a tilted grating.
be highly asymmetric [48]. This asymmetry is like a blaze, since one side of a propagating mode experiences more of a perturbation than the other. Consequently, such an asymmetric grating breaks the symmetry to allow coupling to odd, l ¼ 1, order modes, i.e., LP1m. This is also true of blazed gratings, which are uniform across the core with the same effect on the guided mode. A few points should be noted about scattering from a blazed grating. It is known that scattering of light from bulk blazed gratings [49] is directional and the phase-matching conditions easily derived for scattering, in thin and thick holograms [32]. The general approach taken in the next section is similar in so far as the scattering element is considered nonlocal and all the scattering events summed to arrive at the final unbounded coupling to the radiation field.
Theoretical Model for Coupling to the Radiation Field The STG is a useful device for filter applications when used to couple the guided mode to the radiation field, rather than a mode. It forms a narrow band stop, whose spectral width is not dependent on the length of the grating in the same way as a Bragg grating. The wavelength and bandwidth are easily adjusted by choice of fiber, and the properties of the grating are as robust as those of the Bragg grating in terms of temperature sensitivity and strain. This allows the design of filters that depend only on the properties of the guidance of the core. The side-tap grating is modeled as a periodic collection of uniform inclined planes of perturbed refractive index across the core. The mode fields are defined by the wave-guiding parameters, but it is assumed for the grating that the boundary of the cladding is absent; i.e., the grating is written in an infinite
156
Theory of Fiber Bragg Gratings
medium, although it is itself confined to the core. A consequence of this approximation is that refraction at the core–cladding boundary may be ignored, but can be accounted for later, to form modes. The assumption allows the design of filters in the same way in which the Bragg grating can be modified. The physical phenomenon of scattering is treated as Fraunhofer diffraction, with the amplitude of the scattering obeying the laws of conservation of energy. Figure 4.17 shows the schematic of the blazed grating. The mathematical description that follows shows that this type of a grating is equivalent to an infinite sum of small gratings written perpendicular to the axis of the fiber, but has an azimuthal dependence that makes it possible to couple to a particular set of radiation modes. In other words, the z-dependence of the grating due to the inclination of the planes is translated into a transverse variation in refractive index modulation, with the result that it immediately connects with the idea of the mode overlap integral, while separating the issue of phase matching. The grating inclination angle is yg with respect to the transverse axis, x, in the x–z plane. The refractive index perturbation of the grating, dn(x,y,x), simply described as a product of a grating of infinite extent and a “window” function Wgrating, which takes account of the transverse variation in the amplitude of the grating, as
2pN ð4:7:1Þ dnðx; y; zÞ ¼ 2nDnWgrating ðrÞ cos ðx sin yg z cos yg Þ : Lg
R
Spherical surface x
j
z
qg Blazed grating enclosed in cylinder
Figure 4.17 Scattering of power from a blazed grating entirely embedded in a cylinder.
Radiation Mode Couplers
157
Converting Eq. (4.7.1) into cylindrical coordinates leads to the grating function dnðr; f; zÞ ¼ Wgrat 0 1 8 ðrÞ2nDn !9 1 > > X > > 2pNz cos y g > m @ A J0 ðgrÞ þ 2 ð1Þ J2m ðgrÞ cosð2mfÞ > > > > > > cos > < = Lg m¼1 0 1 ; > > 1 > > 2pNz cos yg A X > > m > > @ 2 ð1Þ J2mþ1 ðgrÞ cosðð2m þ 1ÞfÞ > þ sin > > > : ; Lg m¼1
ð4:7:2Þ
where g ¼ 2pN sin yg/Lg. Equation (4.7.2) requires explanation, since it has real physical significance for the process of mode coupling. Each term on the RHS is responsible for coupling from the guided mode (here the fundamental) to a different set of radiation modes. Terms in the Bessel function Jm couple to modes with an azimuthal variation of cos(mf), i.e., to even-order radiation modes, while the J0 terms lead to the guided mode back-reflection from the grating. Similarly, odd modes couple via the remaining set of terms within the curly brackets. Immediately obvious is the dependence of the back-reflection on g, which periodically reduces the reflection to zero as a function of yg. We refer to Fig. 4.17, in which a grating blazed at angle yg is shown entirely within a cylinder. The scattered total power at a wavelength l impinging on a surface of radius R can be shown to be due to radiation from a current dipole situated at the grating [3] as ð 2p ð p R2 Sscatter ðR; f; ’; lÞ sin ’d’df; ð4:7:3Þ Pscatter ðlÞ ¼ f¼0 ’¼0
where f is the angle between projection of the radius vector R and the x-axis. The Poynting vector is rffiffiffiffiffi 1 e0 ð4:7:4Þ jEscatter ðR; f; ’; lÞj2 : Sscatter ðR; f; ’; lÞ ¼ ncl m0 2 By integrating the scattered contributions from each part of the grating separated by dR, the scattered field, E(R, f, ’, l) may be derived by neglecting the angular dependence on ’ and f and follows as ððð k2 eibclad R eibclad dR deðx; y; zÞEincident ðx; y; zÞdxdydz: Escatter ðR; f; ’; lÞ ¼ 4pR ð4:7:5Þ The above result is consistent with Fraunhofer diffraction theory [3], and we note that it is in the same form as scattering due to the polarization response of a material. We note that far away from the grating, dR x cos f sin ’ y sin f sin ’ þ z cos ’:
ð4:7:6Þ
158
Theory of Fiber Bragg Gratings
The result in Eq. (4.7.5) neglects secondary scattering, so that it is implicitly assumed that the incident radiation is the primary cause for the radiation. This may be justified for STGs, since it is the aim of the exercise to consider radiation loss to the exclusion of reflection by proper choice of blaze angle, and because the radiation field is only weakly bound to the core. We are now in a position to calculate the propagation loss of the incident radiation. The power scattered as a function of length of the grating described in Eq. (4.7.1) and into even azimuthal mode orders can be described as Pscatter ðzÞ ¼ a0 Pincident ðzÞ cos2 ð2pz cos yg =Lg Þdz:
ð4:7:7Þ
0
a is a loss coefficient, which is dependent on the wavelength, the transverse profile of the grating, and the incident field and is equivalent to the overlap integral of Eq. (4.3.6). The incident field therefore decays as Pincident ðzÞ ¼ Pincident 8 2 ð0Þ 0 193 = 0 < a z L 4pz cos y g g A 5: exp4 sin@ 1þ ; 4pz cos yg Lg 2 :
ð4:7:8Þ
The contribution due to the oscillating term within the exponent becomes insignificant for large z, and the power decays as 0
Pincident ðzÞ Pincident ð0Þeða z=2Þ :
ð4:7:9Þ
From Eq. (4.7.9) follows the approximate decay of the incident electric field, Eincident ðx; y; zÞ E0 ðx; yÞeðibf aÞz ;
ð4:7:10Þ
where a ¼ a0 /4 is a function of wavelength only, and bf is the propagation constant for the incident fundamental mode. The longitudinal component of the guided mode field is small and has been neglected in Eq. (4.7.10). The physical analogy of the STG as a distributed antenna is particularly useful, equivalent to an infinite sum of mirrors, each contributing to the light scattered from the fiber core. For small lengths, we have to include the oscillating term in quadrature in Eq. (4.7.8), but with z Lg , the electric field for the fundamental mode decays approximately as it would for constant attenuation per unit length. The attenuation constant depends on wavelength and the transverse distribution of the grating and the incident field, but not on z. This approximate result suggests that the filter loss spectrum should be independent of the length of the grating, which is indeed the case. To calculate the scattered power and the spectrum of the radiation, we use Eq. (4.7.6) in Eq. (4.7.5) and include the grating function Wgrating to arrive at ðð G Wgrating ðx; yÞE0 ðx; yÞeibclad ðx cos f sin ’y sin f sin ’Þ Escatter ðR; f; ’Þ ¼ R ð4:7:11Þ IL ðx; Lg Þdxdy;
Radiation Mode Couplers
159
where Lg is the length of the fiber grating, the constant G is given G¼
bclad Dneibclad R ; l
ð4:7:12Þ
and IL(x, Lg) is obtained by integration with respect to z, IL ðx; Lg Þ ¼
eigx eðiDbb aÞLg 1 eigx eðigbf aÞLg 1 þ ; 2 2 iDbb a iDbf a
ð4:7:13Þ
where g was defined in (4.7.2), and Dbf and Dbb are the forward and backward phase mismatch factors, Dbb ¼ bf þ bclad cos ’
2p cos yg Lg
Dbf ¼ bf þ bclad cos ’ þ
2p cos yg ; Lg
ð4:7:14Þ
where bclad ¼ 2pnclad/l, and the signs are consistent with the measurement of the angle, ’. The forward scattering process can easily be included if necessary but is ignored for now. For the backward phase-matching condition, the radiation angle at resonance, ’L, is given by the Dbf ¼ 0, as has been seen in Section 4.2.5, so that bf þ bclad cos ’L ¼ 2p cos yg =Lg :
ð4:7:15Þ
The last result is a longitudinal phase-matching condition, which is exactly the same as normal Bragg reflection. It requires that the path difference between light scattered from points that are both on a line parallel to the optical axis of the fiber, and on adjacent fringes of the grating, should be exactly l (Fig. 4.18).
EA
E0 Incident field
EB
C jL
jL Λg
A
B
qg Figure 4.18 Scattered light from the fringe planes of the gratings adds up in phase when the resonance condition for longitudinal phase matching is met. AB þ BC ¼ Nl, at resonance.
160
Theory of Fiber Bragg Gratings
Ignoring the forward scatter, we find the scattered counterpropagating power from Eqs. (4.7.11), (4.7.3), and (4.7.4) as sffiffiffiffiffi 1 e0 aL Pscatter ðlÞ ¼ nclad G Ge m0 2 ð 2p ð p ð4:7:16Þ sinh2 ðaL=2Þ þ sin2 ðDbb L=2Þ jIcore ðg; ’; fÞj2 sin ’ d’df; Db2b þ a2 f¼0 f¼0 where the overlap integral over the profile of the grating, which we refer to as the transverse phase-matching condition, is Icore ð ðg; ð ’; fÞ ¼
Wgrating ðx; yÞE0 ðx; yÞei½xðgbclad cos f sin ’Þbclady sin f sin f dxdy:
ð4:7:17Þ
In understanding the physics of the scattering, we consider separately the two components of the integral, the transverse phase-matching term (Eq. 4.7.17) and the longitudinal phase-matching (pm) term which depends on the detuning, Dbb. In the low-loss regime ða Dbb Þ, the longitudinal pm term is simply like the Bragg matched reflection condition, but now as a function of ’. For all practical purposes, this term is like a delta function that is only significant at very small angles of radiation ð’ 1∘ Þ. The integral has a term dependent on cos ’, which becomes broader and asymmetric in its angular bandwidth as ’ ! 0 and which is also inversely dependent on the length of the grating. For typical filter lengths of a few millimeters, we find the angular bandwidth to be 1 . The asymmetry and broadening at small phase-matching angles have been observed in phasematched second-harmonic generation with periodic structures [50]. In the high-loss regime, we find that the delta function broadens but has a width similar to that of the low-loss case. We can therefore choose to consider the dependence of the scattered power on the longitudinal phase matching as a very narrow filter at a given angle. Comparison of the longitudinal term with the transverse pm condition of Eq. (4.7.17) shows that the angular dependence of the radiation for the transverse case varies much more slowly and may be approximated to be a constant over the region of the longitudinal bandwidth. Figure 4.19 shows the dependence of the longitudinal and the transverse pm as a comparison for standard fiber and a uniform grating profile, Wgrating ¼ 1. The longitudinal response for a blaze angle of 5 and the transverse response for three blaze angles are shown. The analytical result for the loss coefficient a has been shown to be [51], ð 2p jIcore ðg; ’; fÞj2 df k2 a2 bclad Dn2 f¼0 ð1 : ð4:7:18Þ a 32p2 rE02 ðrÞdr r¼0
Radiation Mode Couplers
161
Normalized output
1.2 1.0
0 dB filter loss
0.8
20 dB
0.6
40 dB
0.4 0.2 0.0 9.8
10.0 Output angle, j (degrees) Blaze Angle
1.0 Normalised output
10.2
4 degrees 6 degrees 8 degrees
0.8 0.6 0.4 0.2 0.0 0
2
4
6 8 10 12 14 Output angle, j (degrees)
16
18
Figure 4.19 (a) shows the longitudinal integral and (b) is the transverse integral for different blaze angles.
By normalizing the radius as r ¼ r/a (a is the core radius), 1 X I02 þ 2 Im2
a
pk2 a2 bclad Dn2 ð1 4
r¼0
m¼1
rE02 ðrÞdr
;
ð4:7:19Þ
and E0(r) is the field distribution of the fundamental mode. The integrals I0 and Im are defined as ð1 I0 ¼ Wgrat ðrÞE0 ðrÞJ0 ðgraÞJ0 ðxL raÞrdr R¼0 ð4:7:20Þ ð1 Im ¼ Wgrat ðrÞE0 ðrÞJm ðgraÞJm ðxL raÞrdr: r¼0
In Eq. (4.7.20), we remind ourselves that g is the transverse grating momentum that allows the mode to couple out of the core and is a function of the grating period as well as the blaze angle,
162
Theory of Fiber Bragg Gratings
g¼
2p sin yg : Lg
ð4:7:21Þ
xL is the transverse momentum of the mode, depending on the output radiation angle of the scattered light, ’L, at a given wavelength, and is xL ¼ bclad sin ’L :
ð4:7:22Þ
In Fig. 4.20 is shown the calculated and measured loss spectrum of fibers with nominally the same v-value, but different core radii. The agreement between the measured loss and the calculated loss spectrum is quite good for two fibers. The blaze angle for the grating is 8 . The results also show that the loss spectrum due to scattering into the radiation modes is independent of the fiber length, and, indeed, this has been confirmed by experimental observations [51]. The reflection coupling constant for a tilted grating [38] with an arbitrary profile is ð1 rWgrating ðrÞJ0 ð2pa sin yg r=LÞE02 ðrÞdr r¼0 ð1 : ð4:7:23Þ kac / rE02 ðrÞdr r¼0
This integral has been plotted in Fig. 4.21 and shows that zero Bragg reflection into the guided mode occurs at a lower blaze angle if the grating is moved 1.0 0.9
Radiation loss
0.8 0.7 0.6 0.5 0.4 0.3
Theory: core dia = 7 um Core dia = 9 um Core dia = 12 um Rad. loss meas 1 Measurement 2
0.2 0.1 0.0 1510
1520
1530
1540
1550
1560
1570
1580
1590
Wavelength, nm Figure 4.20 Measured radiation loss from large core weakly guiding fibers with radii of 7, 9, and 12 microns and a v-value of 1.9. Two measurements on 12-micron core-diameter fibers are also shown (after Ref. [52]).
Radiation Mode Couplers
163
Back Reflection (dB)
0
standard telecommunication fiber photosensitive cladding
–20
–40
–60 0
2
4
6
8
10
12
Writing angle (degrees) Figure 4.21 Comparison of back reflection from two fibers: Both have nominally the same v-values, but one has a photosensitive cladding only (after Ref. [52]).
outward from the core. For comparison, the back reflection from two fibers has been shown, one with a grating situated entirely in the cladding and the other with a standard telecommunications fiber core. We note that the first back reflection minimum occurs at 3 external writing angle for the photosensitive cladding fiber, compared with 8 for the standard fiber. This has an additional benefit of reducing the bandwidth over which radiation loss occurs, as seen from the phase-matching diagram in Section 4.2.5. In Fig. 4.22 is shown the filter response for coupling to radiation modes for the photosensitive cladding fiber. The benefit of making the cladding 1.515 12
Filter loss, dB
10
1.520
1.525
1.530
1.535
1.540
1.545
1.550
1.555
Filter response (dB), 4um 3.4 deg, calculated
8
3.0 deg, measured
6
Filter response (dB), 3.4um/3.6 deg, calculated
4 2 0 Wavelength, um
Figure 4.22 The loss spectrum (calculated and measured) for a photosensitive cladding fiber. The ripple in the loss spectra is a measurement artifact (after Ref. [52]).
164
Theory of Fiber Bragg Gratings
photosensitive is clear, since it reduces the bandwidth at the zero reflection writing angle (measured at 3 and calculated for the fiber to be 3.6 ). The core radius of this fiber is 3.4 mm, and the photosensitive cladding extends from a to 4a. The agreement between the theoretical and experimentally observed properties of tilted fiber Bragg gratings is extremely good [38] using the complete theory presented by Erdogan [40,38]. In particular, the measured peak visible at 1545 nm in Fig. 4.22 is shown to be due to leaky mode coupling. The polarization dependence of tilted Bragg gratings in fibers with a core radius of 2.6 mm and a core-to-cladding refractive index difference of 5.5 103 becomes obvious as the tilt angle exceeds 6.5 [38]. Above this angle, the p-polarization scatters less efficiently than the s-polarization. Below a tilt angle of 6.5 , the radiation loss is predominantly due to coupling to even-azimuthal order radiation modes, giving rise to a sharp narrow-bandwidth peak. Above 6.5 , the coupling is to odd-azimuthal order modes and becomes much broader. By making angles for the back-reflection small (Fig. 4.21), one benefits from both low polarization sensitivity and a narrow-loss spectrum. In Fig. 4.23 is shown the design diagram for STG filters as a function of the core-to-cladding refractive index difference, assuming an infinite cladding. Two important parameters, the FWHM bandwidth and the tilt angle for zero back reflection into the fundamental mode for step index fibers for different v-values, are shown. The trend is as follows. Small core-cladding index difference and
9
45 V 1.6
FWHM bandwidth, nm
35 30
8 7
2 2.4
6
Tilt angle
25
5
20
4
15
3
10
2
5 0 0.0E+00
1
Bandwidth 2.0E–03
Tilt angle, q
40
4.0E–03
6.0E–03
Core-cladding Δn Figure 4.23 The design diagram for tilted STG filters with infinite cladding.
0 8.0E–03
Radiation Mode Couplers
165
large v-value give the smallest radiation loss bandwidths, as well as the smallest tilt angles and accordingly the lowest polarization sensitivity. The penalty is the increased bend loss sensitivity.
4.7.2 Copropagating Radiation Mode Coupling: Long-Period Gratings These gratings couple light from forward-propagating guided modes to the forward-propagating cladding modes (as with an LPG) and the radiation field. A schematic of the interaction and the phase-matching condition for coupling to forward-propagating radiation modes is shown in Fig. 4.24. The mode-coupling equations for forward coupling are given in Section 4.4 [Eqs. (4.4.11) and (4.4.12)]. The overlap integrals governing the interaction are shown in Eqs. (4.4.14)–(4.4.17) with the appropriate phase-matching terms. This type of coupling is similar to counterpropagating interactions, so far as the overlap of the modes is concerned. However, the power is exchanged between the radiated and guided modes periodically, as shown in Fig. 4.9, so that the filter length governs the bandwidth of the coupling to the radiation mode (as it does to the individual cladding modes in the counterpropagating direction). The fundamental, LP01 guided mode can only couple to the even-order cladding modes of the same order, LP0n [53]. Only if there is an asymmetry in the transverse profile will modes of different order couple. For example, in depressed cladding fibers that support a leaky LP11 mode, coupling to the LP16 mode is possible because of the very large overlap of the fields in the core [53], almost as large as the LP01 ! LP01 modes. For fibers that support only the LP01 mode,
Radiation field (infinite cladding)
Co-propagating cladding mode
Fiber Cladding Guided mode
Core Blazed grating
Bragg phase-matching condition: 2πneff / l +2pN cos qg /Λg = 2pncl / l Figure 4.24 Schematic of co-propagating radiation field and bound cladding mode coupling from a forward propagating guided mode with a blazed grating.
166
Theory of Fiber Bragg Gratings
a tilt in the grating allows coupling to copropagating (and counterpropagating) modes of different order. Erdogan has shown that the coupling constants to the radiation modes of the order v and the core mode (LP01) follow the definitions of Eqs. (4.4.3) and (4.4.4). They are [40] ða ð noe0 DnðzÞ 2p ð4:7:24Þ df Erclad Ercore þ Efclad Efcore rdr: kv!01 ðzÞ ¼ 2 0 0 The eigenvalues and the field distributions for the cladding modes may be calculated by field matching at the boundaries as for the core for the low-order LP0n modes, using a procedure similar to the guided core modes of the fiber [54]. Only coupling to the radiation modes with the azimuthal order l ¼ 1 (LPl–1,v type) has a nonzero integral. The equations that describe the overlap integral, Eq. (4.7.24) of the modes for a transversely uniform grating are involved and cumbersome [40]. For the v ¼ 2, 4 modes, the field in the core is very low, and therefore contributes little to the coupling. However, the field for the odd-numbered v modes has high intensity and these fields dominate the coupling for the lower-order modes. In Fig. 4.25 is shown the calculated coupling constants for a set of 168 cladding modes for a fiber at 1550 nm, normalized to the refractive index modulation, Dn(z). The important point is that coupling to the low-even-order modes is weak compared to the odd modes. For v > 40, both even and odd order modes have almost identical coupling constants, but remain <20% of the maximum possible for the odd modes.
Coupling constant /s(z) (μm–1)
0.5 v = Odd v = Even
0.4
0.3
0.2
0.1
0.0 0
50
100
150
Cladding mode number v Figure 4.25 The coupling constants for the fundamental guided mode to 168 even- (l ¼ 2) and odd-order cladding modes with azimuthal order 1 (type LP0n) (from: Erdogan T, “Cladding mode resonances in short and long period fiber grating filters,” Opt. Soc. Am. A 14(8), 1765, 1997.).
Radiation Mode Couplers
167
Therefore, for many applications, it is necessary only to take account of a maximum of first 20–30 cladding modes, especially when computing the loss spectrum of an infinite cladding fiber (pure radiation loss). A major difference between the STG and the LPG is shown in the phasematching Eq. (4.7.14). We note that the detuning Dbf for an LPG is sensitive to the difference in the propagation constants of the guided and radiation modes. Any UV-induced change in the core index will result in a shift in the propagation constants, and thereby strongly affect the resonance wavelength. This aspect does not affect the counterpropagating resonance strongly, since the percentage change in the sum of the propagation constants is small. The change in the resonance condition for the STG in which only the core mode is affected can be calculated as [55] dlSTG dbUV dbUV ¼ ; lSTG bcore þ bclad 2b0
ð4:7:25Þ
whereas the resonance condition for the LPG changes as dlLPG dbUV dbUV : ¼ lLPG bcore bclad Db
ð4:7:26Þ
b0 is the average propagation constant of the core and cladding modes, and dbUV is the effect of the additional UV-induced detuning over and above the initial mismatch, Db between the modes, and is the overlap of the field within the core. A comparison between the two leads to dlLPG
2ncore dlSTG : dncore!clad
ð4:7:27Þ
In Eq. (4.7.27) the average effective index has been replaced by ncore and the difference in the mode propagation constants by the core-to-cladding index difference. Therefore, for a typical fiber, the LPG is between 100 and 1000 more sensitive than the STG to the changes between the propagation constants of the core and the cladding modes. The transmission spectra of a typical LPG are shown in Fig. 4.26. A number of resonances beginning with the coupling of the fundamental guided mode to the cladding n ¼ 2, 3, 4, 5, and 6 modes can be seen. It should be noted that the transmission loss for each mode depends on the strength of the coupling constant kac and kdc. The former indicates the length of the grating required for 100% coupling, while the latter causes the resonance wavelengths to shift [see Eq. (4.7.26)]. This requires a grating period to be adjusted according to the conversion efficiency and the required resonance wavelength. The bandwidth of a single resonance of an LPG filter is approximately [56] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2resonance 4kv!01 ð4:7:28Þ Lg ; Dl Dneff Lg p
168
Theory of Fiber Bragg Gratings Wavelength, nm 1250
1150 0
1350
1450
1550
1650
1750
LP02 Transmission, dB
LP03 –5 LP04 –10 LP05 –15
LP06
–20
Figure 4.26 Transmission spectra of a ten mm long LPG in standard single mode-type fiber (Corning SMF 28), with a period of 450 mm. Coupling is shown from the fundamental core mode to the odd (v ¼ 1), n ¼ 2 ! 6 cladding modes (LP0n).
where coupling from the fundamental guided mode to the appropriate 1u cladding mode has the core-to-cladding mode effective index difference of Dneff. The resonance wavelength is determined by Eq. (4.6.4). Long-period gratings for coupling a guided mode to the cladding may be designed for standard single-mode fibers by the data in Figs. 4.27 and 4.28. The period of a grating normalized to resonance wavelength lresonance is shown as a function of the difference of guided mode effective index from the cladding index. For example, a resonance peak at 1550 nm requires a grating with a period 100 1.55 ¼ 155 microns for a mode-effective index difference of 0.01.
Normalized grating period, Λ/l resonance
500 400 300 200 100 0 0
0.005
0.01
0.015
0.02 Δ Neff
0.025
0.03
0.035
0.04
Figure 4.27 Normalized approximate resonance wavelength for phase-matched radiation mode coupling for LPG.
Radiation Mode Couplers
169
LPG Period, microns
675 625
01-02 01-03
575
01-04
525 01-05 modes
475 425 375 1350
1400
1450
1500
1550
1600
Wavelength, nm Figure 4.28 The computed grating periods for coupling of the LP01 core mode to the first four LP0n cladding modes in standard (Corning SMF) fiber.
We note that the change in the resonance wavelength of a long-period grating is influenced by the change in the core refractive index as a grating is being written [Eq. (4.7.30)]. As the coupling constant increases, the coupling loss increases until [Eq. (4.4.12)] p ð4:7:29Þ aLg ¼ ; 2 and at the resonance wavelength, a ¼ kac. To maintain the maximum loss, the length of the grating or the coupling constant has to be adjusted for a given bandwidth. Figs. 4.27 and 4.28 may also be used to show how the period of the grating varies as a function of the change in the effective index of the mode. The desired resonance wavelength for an LPG may be calculated for the final mode effective index after the grating has been fabricated, using the data in Fig. 4.28 for the real modes of standard SMF fiber. For a required transmission loss, the refractive index modulation can be calculated to arrive at the final mode effective index. With this data, and from the required bandwidth of the grating, the appropriate choice for the grating period can be made. Figure 4.29 shows the effect of immersing a long-period grating in oil. The resonance condition for coupling to the forward-propagating cladding mode is destroyed as the cladding is index matched. With further increases in the index of the oil, we note another resonance due to partial reflection at the cladding surface to form leaky modes. The loss (to 1/e2 of the input) by leakage at the boundary of the cladding and the oil is [57] 1 ð4:7:30Þ L 1þ Lb ; lnr where r is the Fresnel amplitude reflection coefficient at the cladding surface, and Lb is the distance between the successive reflections at the cladding surface.
170
Theory of Fiber Bragg Gratings
n = 1.000
n = 1.408
Transmission (dB) (25 dB/div)
n = 1.402
n = 1.448
n = 1.450
n = 1.486
n = 1.574
n = 1.661
n = 1.734 1450
1475
1500
1525
1550
1575
Wavelength (nm) Figure 4.29 Radiation loss from a long-period grating as a function of the index of the oil that surrounds the fiber (from: Stegall D. B. and Erdogan T., “Long period fiber for grating devices based on leaky cladding modes,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17, OSA Technical Digest Series (Optical Society of America, Washington, DC, 1997), paper BSuB2, pp. 16–18).
Grating Simulation
171
4.8 GRATING SIMULATION
4.8.1 Methods for Simulating Gratings Many of the techniques for simulating fiber Bragg gratings were introduced at the beginning of the chapter [10,58,9,19]. All the techniques have varying degrees of complexity. However, the simplest method is the straightforward numerical integration of the coupled-mode equations such as Eqs. (4.3.9) and (4.3.10). While this method is direct and capable of simulating the transfer function accurately, it is not the fastest. Another technique is based on the Gel’Fand–Levitan–Marchenko inverse scattering [58] method. This is again a powerful scheme based on integral coupled equations but has the primary disadvantage of obscuring the problem being solved. It has, however, the advantage of allowing a grating with particular characteristics to be designed. Perhaps the most attractive method is based on techniques developed for the analysis of metal waveguides by Rouard [11] and carries his name as a result. This technique, extended by Weller-Brophy and Hall [12], works on the principle that the waveguide may be segmented into subwavelength thin films. Standard thin-film techniques for calculating the amplitude and phase of the transmitted and reflected electric fields at each interface are propagated backward from the output end of the grating. The method is slow, but is one of the few that offers complete piecemeal control of the spatial variation in the refractive index modulation of the grating. For example, the transfer function of a grating with a sawtooth modulation is analyzed equally as well as a square or sinusoidal profile. The influence on the phase and amplitude response of the grating cannot be fully characterized if the Fourier coefficient of the phase-synchronous term for phase matching as shown in Eq. (4.2.27) alone is used. Thus, it is a laborious and time-intensive computation; however, the results accurately simulate the characteristics of complex-shaped grating periods with reasonable reflectivities, being limited by the rounding errors in the computation. A fast and accurate technique is based on the T-matrix (transfer) for calculating the input and output fields for a short section dl of the grating [10]. The outputs of the first matrix M1 are used as the input fields to the second matrix, M2, not necessarily identical to M1. The process is continued until an entire complex profile grating is modeled. This method is capable of accurately simulating both strong and weak gratings, with or without chirp and apodization. It has the advantage of handling a single period of grating as the minimum unit length for the matrix in the case when the period or amplitude is a slowly varying function of length. In the following section, two methods, Rouard’s and the T-matrix, will be presented for simulating gratings of arbitrary profile and chirp.
172
Theory of Fiber Bragg Gratings
4.8.2 Transfer Matrix Method An analytical solution for a grating of length Lg, with an arbitrary coupling constant k(z) and chirp L(z), is desirable but no simple form exists. The variables cannot be separated since they collectively affect the transfer function. In the T-matrix method, the coupled mode equations [for example, Eq. (4.3.9)] are used to calculate the output fields of a short section dl1 of grating for which the three parameters are assumed to be constant. Each may possess a unique and independent functional dependence on the spatial parameter z. For such a grating with an integral number of periods, the analytical solution results in the amplitude reflectivity, transmission, and phase. These quantities are then used as the input parameters for the adjacent section of grating of length dl2 (not necessarily ¼ dl1). The input and output fields for a single grating section are shown in Fig. 4.30. The grating may be considered to be a four-port device with four fields: input fields R(–dl1/2) and R(dl1/2) and output fields S(–dl1/2) and S(dl1/2). A transfer matrix T1 represents the grating amplitude and phase response. For a short uniform grating, the two fields on the RHS of the following equation are transformed by the matrix into the fields on the LHS as Rðdl1 =2Þ Rðdl1 =2Þ ¼ ½T 1 : ð4:8:1Þ Sðdl1 =2Þ Sðdl1 =2Þ The boundary conditions applied to Eq. (4.8.1) lead directly to the reflectivity and transmissivity of the grating. These conditions depend on whether the grating is a contradirectional or a codirectional coupler.
Reflection Grating For a reflection grating, the input field amplitude R(–dl1/2) is normalized to unity, and the reflected field amplitude at the output of the grating S(dl1/2) is zero, since there is no perturbation beyond the end of the grating. R(–dl /2)
+2n Δn
R(dl /2) z
S(–dl /2)
Λ
z=0
–2n Δn S(dl/2)
Figure 4.30 Refractive index modulation in the core of a fiber. Shown in this schematic are the fields at the start of the grating on the LHS and the fields at the output on the RHS. The modulated refractive index is 2nDn about a mean index.
Grating Simulation
173
Writing the matrix elements into Eq. (4.8.1) and applying the boundary conditions leads to T11 T12 1 Rðdl1 =2Þ ¼ ; ð4:8:2Þ T21 T22 0 Sðdl1 =2Þ in which, the superscript for T1 has been dropped for clarity. The transmitted amplitude is easily seen to be Rðdl1 =2Þ ¼
1 : T11
ð4:8:3Þ
The reflected amplitude follows from Eqs. (4.8.2) and (4.8.3) as Sðdl1 =2Þ ¼
T21 : T11
ð4:8:4Þ
Consequently, these are now the new fields on the RHS that can be transformed again by another matrix, T2 and so on, so that for the entire grating P after the Nth section, where L ¼ N j¼1 dlj , is RðL=2Þ RðL=2Þ ¼ ½T N . . . ½T 3 ½T 2 ½T 1 ; ð4:8:5Þ S ðL=2Þ S ðL=2Þ in which the field amplitudes on the RHS are the same as the those expressed in Eq. (4.8.1). Replacing the N multiplied 2 2 matrices in Eq. (4.8.5) by a single 2 2 matrix, we get the transfer function of the whole grating, RðL=2Þ RðL=2Þ ¼ ½T ; ð4:8:6Þ S ðL=2Þ S ðL=2Þ where the matrix T is ½T ¼
N Y
½T j :
ð4:8:7Þ
j¼1
Now the transmissivity t of the whole grating follows from Eqs. (4.8.3) and (4.8.4), t ¼ ð1 rÞ ¼
RðL=2Þ 1 ; ¼ RðL=2Þ T11
ð4:8:8Þ
and reflectivity r, r¼
S ðL=2Þ T21 : ¼ RðL=2Þ T11
ð4:8:9Þ
174
Theory of Fiber Bragg Gratings N
L =Σ dlj j =1
R (–L /2) S(–L /2)
RN
SN
R2
fN aN dlN
S2 f2 a2 dl2
S1
R1 f1 a1 dl1
R(L/ 2) S(L/ 2)
Figure 4.31 The concatenation of several short reflection gratings with constant parameters to form a composite grating. The phase fN is the phase of the grating in each section.
From the solution to the coupled-mode Eqs. (4.3.9) and (4.3.10), the transfer matrix elements for the jth section are T11 ¼ coshðadlj Þ
id sinhðadlj Þ a
ð4:8:10Þ
T22 ¼ coshðadlj Þ þ
id sinhðadlj Þ a
ð4:8:11Þ
T12 ¼ T21 ¼
ikac sinhðadlj Þ a
ikac sinhðadlj Þ : a
ð4:8:12Þ ð4:8:13Þ
A schematic of the sectioned gratings with the relevant parameters is shown in Fig. 4.31.
Codirectional Coupling For transmission gratings, different boundary conditions have to be used; R(–L/2) is again normalized to unity. However, the field S(L/2) on the LHS in Fig. 4.30 is copropagating and has an amplitude of zero. At the output, S(L/2) is also a copropagating mode. Codirectional coupling is shown in Fig. 4.32. Applying boundary conditions, one arrives at 1 T11 T12 Rðdl1 =2Þ : ð4:8:14Þ ¼ Sðdl1 =2Þ T21 T22 0 The uncoupled [R(dl/2)] and cross-coupled [S(dl/2)] amplitudes are then simply derived from Eq. (4.8.14) as T11 R þ T12 S ¼ 1 T21 R þ T22 S ¼ 0;
ð4:8:15Þ
Grating Simulation
175
R(–dl /2) +2n Δn
R(dl/2) z
S(–dl/2)
Λ
–2n Δn
z=0
S(dl /2)
Figure 4.32 For codirectional coupling, the direction of propagation of the S-fields is reversed on both ends of the grating; compare with the reflection grating.
so that the crossed state amplitude is S¼
1 ; T12 T11 ðT22 =T21 Þ
ð4:8:16Þ
T22 : T21 T11 T22
ð4:8:17Þ
and the uncoupled amplitude, R¼
Following the solutions for the codirectional coupled-mode Eqs. (4.4.9) and (4.4.10), the T-matrix elements for the jth section are id sinðadlj Þ a
ð4:8:18Þ
id sinðadlj Þ a
ð4:8:19Þ
T11 ¼ cosðadlj Þ þ T22 ¼ cosðadlj Þ T12 ¼
ikac sinðadlj Þ a
ð4:8:20Þ
T21 ¼
ikac sinðadlj Þ : a
ð4:8:21Þ
Equations (4.8.18)–(4.8.21) complete the analysis for guided-mode interactions.
Phase Shifts within a Grating It is often useful and necessary to incorporate phase jumps within a distributed grating structure. The phase jump opens up a bandgap within the reflection bandwidth, creating a narrow transmission band. This procedure has been applied to distributed feedback (DFB) lasers to allow stable singlemode operation [59]. A phase shift is accomplished in the T-matrix by multiplying the reflectivity of the jth section by matrix elements containing only phase terms. On this basis, the transfer matrix takes on the form RðL=2Þ RðL=2Þ ; ð4:8:22Þ ¼ ½T N . . . ½T 3 ½Tps ½T 1 S ðL=2Þ SðL=2Þ
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Theory of Fiber Bragg Gratings
where Tps is the new phase-shift matrix for a reflection grating, 0 eif=2 Tps ¼ : 0 eif=2
ð4:8:23Þ
For codirectional coupling we replace the phase-shift matrix [Eq. (4.8.23)] by its conjugate. The phase factor, f/2, is any arbitrary phase and may be as a result of a change in the neff or a discontinuity within the grating. The phase change could have arisen from a region in which the grating was not present or from exposure to uniform UV radiation, thereby changing the local neff. In either case this phase f¼
4p ðneff þ @nÞDL; l
ð4:8:24Þ
where @n is the local phase change over a length DL. The calculation of the reflectivity and transmissivity proceeds in the same manner as without the phase step.
General Conditions and Restrictions for the T-Matrix Method The transfer matrix method requires that certain conditions be met for accurate simulation of grating response [10]. First, when the grating parameters are a function of z, the minimum length of the section dlj Lj K, where K is a suitably large number. The actual factor K is a consequence of the slowly varying approximation. The magnitude of K depends on chirp rate dDl/dL where Dl is the total chirp bandwidth in the grating. This value determines the upper limit to the number of sections allowed in any simulated grating. Figure 4.33 shows the effect on the reflectivity of reducing the number of sections from 37 periods to one period per section of a 2-mm-long grating with a chirp of 6 nm. The refractive index modulation is 2 103.
Reflectivity, dB
1.0 0.8
100 sec
0.6 0.4 3729 sec
0.2 0.0 1550.0
1552.0
1554.0 1556.0 Wavelength, nm
1558.0
1560.0
Figure 4.33 The effect of too many sections on the simulation of a 6-nm chirped 2-mm-long grating with 3729 (1 period/section) and 100 (37 periods/section) sections. The 100-section grating accurately models the grating, while the assumptions for the simulation are violated for too many sections.
Multilayer Analysis
177
Second, care must be taken to ensure that each section j has an integer number of grating periods in order to have a smooth transition between sections. An abrupt change in the grating modulation is equivalent to a phase jump and hence the formation of a Fabry–Perot cavity, as has been explained in Section 3.1.13. A consequence of not maintaining this condition over several sections is that it can lead to a deleterious effect outside of the bandwidth of the grating, by forming a superstructure of cascaded Fabry–Perot cavities. Third, adequate care must also be taken to smooth any spatial variation in the refractive index modulation. Sections of constant kac but different from adjacent ones inevitably form a superstructure [60,61]. Superstructure can be responsible for replicas of the main Bragg reflection peak at far-removed wavelengths, causing spurious cross-channel interference in filters. The smaller the section, the wider the wavelength band over which the superstructure replicas may appear. Thus, there is a lower limit to the number of grating sections that must be used to reduce this detrimental effect. However, superstructure gratings, when carefully designed, can be used to perform useful functions [62] and are covered in Chapters 6 and 7. Fourth, when simulating long chirped gratings, care must be taken to allow adequate spectral resolution in order to calculate the group delay accurately.
4.9 MULTILAYER ANALYSIS
4.9.1 Rouard’s Method Simulation using this method relies on sectioning the grating into multilayers and replacing the layer by an interface with a complex reflectivity, which includes the phase change through the layer. To accurately model the refractive index profile of the grating, the period may be subdivided into further sections. A recursive technique is then applied to calculate the reflectivity of the composite period of the grating. Thus, the problem is reduced to calculating the amplitude reflectivity r of each single period. The process is repeated for N single-period sections, each with any local function for the refractive index modulation, period, or phase steps. It is easy to realize that any type of grating, microns or meters long, is then easily modeled. Alternatively, for certain types of pure sinusoidal refractive index modulation, the analytical solution for the constant period grating can be used [Eq. (4.3.11)] so long as the conditions described in Section 4.8.2 are adhered to. The power of this technique is, however, restricted by the computational errors when calculating the reflectivity and transmission of a large number of thin films. Despite this restriction, many types of gratings are adequately realized, provided the maximum reflectivity is limited to values 99.99%. With care and appropriate computational algorithms, better
178
Theory of Fiber Bragg Gratings
results may be possible. The basic analysis is similar to the T-matrix approach; however, the reflectivity is simply calculated from the difference in the refractive index between two adjacent layers.
4.9.2 The Multiple Thin-Film Stack Figure 4.34 shows a thin film on a substrate with light propagating at normal incidence and with transverse field components. The refractive index of each section is indicated. The reflectivities, r1 and r2, at each interface depends purely on the refractive indexes of the two dielectric materials on either side and are also shown. The field in each region Ej is the sum of the forward Rj and backward, Sj, traveling fields: E j ¼ Rj þ Sj :
ð4:9:1Þ
Applying continuity of the transverse field components (which are tangential to the interface) at the bottom layer, 1, and assuming propagation in a nonmagnetic medium, we get, R1 þ S 1 ¼ R 2 þ S 2
ð4:9:2Þ
n1 ðR1 S1 Þ ¼ n2 ðR2 S2 Þ:
ð4:9:3Þ
Equations (4.9.2) and (4.9.3) can be expressed by algebraic manipulation in the following matrix form: 1 1 r1 R2 R1 ¼ : ð4:9:4Þ S1 S2 t1 r1 1
r1
S3
R3 r2 R2
S2
n3 n2
r2 dl2
n3 R1
A
Substrate
S1
B
Figure 4.34 (a) Two layers on a substrate. The refractive index of each layer is also indicated. The left-hand figure is transformed into the problem shown in (b), with a complex reflectivity r2 at the interface with the substrate, replacing the layer n2.
Multilayer Analysis
179
The reflectivity r1 at the substrate interface is n1 n2 ; r1 ¼ n1 þ n2
ð4:9:5Þ
and the transmission coefficient, t1, t1 ¼
2n1 : n1 þ n2
ð4:9:6Þ
Once again applying continuity at the second interface from the bottom, the reflected and transmitted fields are described as i’2 1 R2 R3 e r2 ei’2 ¼ : ð4:9:7Þ i’2 i’2 S2 S3 r e e t2 2 The phase ’j ¼ bjdlj ¼ 2pnjdlj/l. As before for the matrix method, combining Eqs. (4.9.3)–(4.9.7) leads to 1 R R1 ½T1 ½T2 3 : ¼ ð4:9:8Þ S1 S3 t1 t2 Equation (4.9.8) may be simplified as for the T-matrix method as, R R1 ¼ ½T 3 ; S1 S3
ð4:9:9Þ
from which the matrix elements follow as T11 ¼
1 i’2 ðe þ r1 r2 ei’2 Þ t1 t2
T12 ¼
1 ðr2 ei’2 þ r1 ei’2 Þ t1 t2
T21 ¼
1 ðr1 ei’2 þ r2 ei’2 Þ t1 t2
T22 ¼
1 ðr1 r2 ei’2 þ ei’2 Þ: t1 t2
ð4:9:10Þ
Applying the boundary condition in the region n3, which is infinite, leads to S3 ¼ 0 as applied to Eq. (4.8.2). The simple complex reflectivity for the layer n2 follows r2 ¼
T21 r1 þ r2 ei2’2 ¼ ; T11 1 þ r1 r2 ei2’2
ð4:9:11Þ
and the transmission coefficient t2 is given by t2 ¼
1 t1 t2 ¼ : T11 ei’2 þ r1 r2 ei’2
ð4:9:12Þ
180
Theory of Fiber Bragg Gratings
Replacing the second section by a single interface as shown in Fig. 4.34b and adding another layer with fields R4 and S4 above section 3, the composite reflectivity r follows from i’3 1 R1 R4 e r2 ei’3 ¼ ; ð4:9:13Þ S1 S4 t2 r2 ei’3 ei’3 so that in a similar manner to Eqs. (4.9.11) and (4.9.12), r¼
T21 r3 þ r2 ei2’3 ¼ : T11 1 þ r3 r2 ei2’3
ð4:9:14Þ
It is now straightforward to appreciate that a single period of a grating may be divided into j sections and the composite reflectivity computed using this piecewise linear method for any complex shape for the grating.
4.10 GRATING DESIGN An interesting method for the design of gratings is based on the concatenation of several FBGs written with a single phase mask and no phase stitches, but with the strength and the length of each section as free parameters [63]. As the multiple FBGs are arranged to have different nominal lengths and reflectivities, it is easy to see that the phases and amplitudes of the reflected spectra will add up to result in a designed spectrum. The method relies on the fact that longer exposure not only increases the reflectivity but also the wavelength. Thus, by increasing the length of exposure of a section, the same reflectivity as a shorter section may be achieved. However, the peak reflectivity can be at either the same wavelength (with dc UV exposure) or at a different wavelength. The maximum UV-induced wavelength shift as a result of photosensitivity is 2 nm in hydrogen-loaded fibers, so that the design of the final grating is limited to a total bandwidth of 2 nm. In this scheme it is necessary to have good control of the refractive index change with UV exposure and requires a precharacterization of the response of the fiber to UV exposure. Fabrication of a grating with M-shaped reflection spectra has been presented and demonstrated [63] with reasonably good agreement with the design. Another fast technique [64] uses a genetic algorithm [65,66], which seeks a solution based on a difference between an intermediate and final solution. The difference between the desired and achieved solution is used to decide if the set of “chromosomes” used for the solution are good, and the chromosomes are then mutated and propagated to arrive at a better set. In the technique proposed, Carvalho et al. [64] show that by using a random set of wavelength
Grating Design
181
intervals, it is possible to achieve a closer match to the final solution with fewer samples in a significantly shorter time than with nonstochastic wavelength samples.
4.10.1 Phase-Only Sampling of Gratings A superstructure on an otherwise uniform long-period grating results in a multiple band-pass spectrum [67], or broadband WDM dispersion-compensating reflector in a chirped grating [68]. This method requires the use of long gaps between successive gratings and is therefore wasteful of the “real-estate” of the grating periods (i.e., it reduces the number of periods in a given length of a grating as the periods are sacrificed for gaps). A larger refractive index modulation is therefore required to achieve the same reflection coefficient, as the length of each grating is reduced. To overcome this problem, a phase-only superstructure is an efficient technique to reduce the demand on the refractive index modulation, which comes at a premium. Thus, applying periodic phase-only stitches to a grating conserves the length and therefore the strength of the grating for a multichannel filter [69]. The refractive index modulation required for this type of a grating is substantially reduced compared to the amplitude superstructure grating. In essence, the two schemes for the superstructure gratings are identical, except that the phase-only sampling point is either localized to one period or can be spread over a section by inducing a larger dc-refractive index change over a sampling period. In a similar manner, the amplitude-only sampling also may be spread over a sampling period (i.e., reducing or increasing the ac-refractive index modulation over the sampling period). To understand the formation of the multichannel FBG through phase-only sampling, the refractive index modulation may be described as 2pz i2pz þ fg ðzÞ S exp ; ð4:10:1Þ nðzÞ ¼ n0 þ Re DnðzÞ exp i Lg Ps where n0 is the effective index of the optical fiber, Lg is the central period of the grating, f(z) is the of the single channel of the grating, and S(z) is the sampling function with a period Ps. The channels that are generated by the sampling period are separated in frequency by Dn ¼ c/(2ngPs), where ng is the group index of the mode. In the Born approximation, the Fourier transform, Sˆ(n) of the sampling function, S(z), well represents the sampled spectrum. A rectangular function will thus generate N sinc spectrum channels as z z SðzÞ ¼ comb rect ð4:10:2Þ Ps Ps =N
182
Theory of Fiber Bragg Gratings
and ^ ðnÞ ¼ S
2 n n P sinc ; comb N Dn NDn
ð4:10:3Þ
where the comb function is a summation of an infinite number (m ¼ 1) of Dirac delta functions d(z m) and is convolved with the rectangle function, rect(z) ¼ 1 for z 0.5, and null outside. The advantage of phase-only sampling is that the refractive index modulation amplitude scales as √N rather than as N for equal reflectivity of each channel, a significant advantage over the amplitude-only sampling, because the refractive index change may not be possible for large channel counts with amplitude sampling. It should be noted that to attain similar reflectivities in the adjacent channels, optimization techniques can be used in conjunction with grating synthesis, for example, inverse-scattering theory [70]. To ensure that the channel spectra has low spurious content in adjacent channels, apodization functions have to be used [71]. The details of the design of such gratings may be found in reference [69]. Using phase-only sampling, it is possible to produce very high-count (>32) dispersion compensating modules using single 100-mm-long chirped gratings. Higher-order dispersion may also be corrected in the same grating [72].
4.10.2 Simulation of Gratings Grating synthesis and reconstruction are useful techniques to study defects and the detailed structure of gratings, as well as for designing the parameters that give the properties such as the desired reflection spectra and dispersion. As we saw earlier, the Fourier transform method is only really applicable to weak gratings in the Born approximation, as the refractive index modulation is nonlocal. As grating synthesis can require long computation times, efficient algorithms are required. The reflection and dispersion characteristics of a given grating structure are relatively easy to compute using the transfer matrix or Rouard’s method, as already explained. However, determining the grating parameters from a measured reflection and dispersion spectrum requires special techniques, such as the numerical method based on the Gel’fand-Levitan-Marchenko (GLM) inverse scattering technique [73], and has been used to design dispersion compensation gratings [74]. The integral GLM technique using a fast inversion procedure, which benefits from the symmetry properties of the Toeplitz–Hermitian matrix, has been suggested [75] to increase the computational speed. This algorithm appears to be slightly better for strong FBGs (30-dB reflectivity)
Grating Design
183
compared to the discrete layer peeling technique, but as is a general issue for ultrastrong gratings, the problem is intractable, as the reflection takes place close to the entry point and the field decays exponentially toward the rear end of the grating. For such gratings, it may be better to use the double-ended grating characteristics [76], as more information is then available for synthesis. However, for long and strong gratings, the synthesis is a difficult problem, although good results have been reported by Sherman et al. [77]. A general iterative approach has also been proposed by Peral et al. [17]. Poladian [76] suggested the use of solving the coupled-mode equations while at the same time evaluating a simple integral to calculate the grating strength. The process is analogous to the grating writing process being rolled backward in time and has a nice intuitive approach. More recently, simulated annealing has been used effectively to synthesize a grating [66]. The technique uses a cost-reduction parameter to minimize the errors between a parent and the solution sought. Sherman et al. [77] used an interferometric measurement technique with a tunable laser in a Michelson interferometer, along with the measurement of the light transmitted through the grating with an inverse scattering technique to reconstruct the grating. The grating is in one arm of the Micheleson interferometer, with a mirror in the other. The grating spectra can be obtained from the Fourier transform of the interferogram [78,79]. This is a high-coherence tunable laser version with a fixed reference mirror, of the optical low coherence technique proposed by Limberger et al. [80]. The transmitted light greatly enhances the ability to reconstruct strong gratings (>99.9%); however, it is important to isolate the interferometer from environmental effects (vibration and temperature drifts) and correct for signal to noise by measuring the grating from both sides. The spectral resolution of the measurement determines the maximum length of the grating, which can be measured by this technique, as Lmax ¼ l2B/(4Dlnavg). With a resolution of 1 pm, a maximum grating length measurable is 400 mm. The integral inverse scattering technique, based on solving the integral GLM equation in a layer-peeling procedure, has been discussed in detail for ultrastrong gratings [81] and shown theoretically to work of reflectivities of 1010. Reconstruction of a 99.99% grating was successfully demonstrated, translating to a 4-mm-long grating with a large refractive index modulation of 1.35 103. Reconstruction from noisy grating spectra is also possible [82], and in the presence of loss [83]. To reconstruct long-period gratings (LPGs), Rosenthal et al. [84] have shown that using an additional grating for recoupling the previously outcoupled light allows the Mach–Zehnder interferometer so formed to produce interference fringes as a function of wavelength, and the Hilbert transform is used with amplitude function for phase retrieval to reconstruct the grating. It should be
184
Theory of Fiber Bragg Gratings
remembered that the reconstruction of ultrastrong LPGs may be difficult as the minimum phase requirement must be fulfilled. Thus, only a weak second grating (15% coupling) is used with a 50% reflection first grating.
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[45] R. Kashyap, R. Wyatt, R.J. Campbell, Wideband gain flattened erbium fiber amplifier using a blazed grating, Electron. Lett. 24 (2) (1993) 154–156. [46] R. Kashyap, R. Wyatt, P.F. McKee, Wavelength flattened saturated erbium amplifier using multiple side-tap Bragg gratings, Electron. Lett. 29 (11) (1993) 1025. [47] D. Johlen, P. Klose, H. Renner, E. Brinkmeyer, Strong LP11 mode splitting in UV side written tilted fiber gratings, in: Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, 17, OSA Technical Digest Series, paper BMG12, Optical Society of America, Washington, DC, 1997, pp. 219–221. [48] P.J. Lemaire, A.M. Vengsarkar, W.A. Reed, D.J. DiGiovanni, Thermal enhancement of UV photosensitivity in H2 loaded optical fibers, in: Tech. Digest of Conf. on Opt. Fib. Commun., OFC’95. paper WN1, pp. 158–159. [49] See, for example, R.A. Syms, J. Cozens. in: Optical Guided Waves and Devices, McGrawHill, London, 1992. [50] R. Kashyap, Figure 15, Inset in phase-matched periodic-electric-field-induced secondharmonic generation in optical fibres, J. Opt. Soc. Am. B 6 (3) (1989) 313–328. [51] M.J. Holmes, R. Kashyap, R. Wyatt, R.R. Smith, Ultra narrow-band optical fiber sidetap filter, ECOC’98 (1998) 137–138. [52] M.J. Holmes, R. Kashyap, R. Wyatt, R.P. Smith, Development of radiation mode filters for WDM, in: Proc. of IEE Symposium on WDM Technology, IEE. 1998, pp. 16–17. [53] S.J. Hewlett, J.D. Love, G. Meltz, T.J. Bailey, W.W. Morey, Cladding mode resonances in Bragg fibre gratings depressed and matched-cladding index profiles, in: Photosensitivity and Quadratic Nonlinearity in Glass Waveguides: Fundamentals and Applications, vol. 22, 1995 OSA Technical Series, Optical Society of America, Washington, DC, 1995, pp. PMC2(235–238). [54] M. Monerie, Propagation in doubly clad single mode fibers, IEE Trans. Microwave Theory and Techniques MTT-30 (4) (1982) 381–388. [55] E.M. Dianov, A.S. Kurkov, O.I. Medvedkov, Vasil’ev, A new method for measuring induced refractive index change in optical fiber core, in: Photosensitivity and Quadratic Nonlinearity in Glass Waveguides: Fundamentals and Applications, vol. 22, 1995 OSA Technical Series. Optical Society of America, Washington, DC, 1995, pp. SuB4-(104–107). [56] D.G. Hall, Theory of waveguides and devices, in: L.D. Hutchinson (Ed.), Integrated Optical Circuits and Components, Marcel Dekker, New York, 1987. [57] D.B. Stegall, T. Erdogan, Long period fiber grating devices based on leaky cladding modes, in: Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, 17, OSA Technical Digest Series, paper BSuB2, Optical Society of America, Washington, DC, 1997, pp. 16–18. [58] J.E. Roman, K.A. Winnick, Waveguide grating filters for dispersion compensation and pulse compression, IEEE J. Quantum Electron. 29 (3) (1993) 975. [59] H.A. Haus, Y. Lai, Theory of cascaded quarter wave shifted distributed feedback resonators, IEEE J. Quantum Electron. 28 (1) (1992) 205–213. [60] V. Jayaraman, D.A. Cohen, L.A. Coldren, Demonstration of broadband tunability of a semiconductor laser using sampled gratings, Appl. Phys. Lett. 60 (19) (1992) 2321–2323. [61] M. Ibsen, B.J. Eggleton, M.G. Sceats, F. Ouellette, Broadly tunable DBR fibre using sampled Bragg gratings, Electron. Lett. 31 (1) (1995) 37–38. [62] F. Ouellette, P.A. Krug, T. Stephens, G. Doshi, B. Eggleton, Broadband and WDM dispersion compensation using chirped sampled fibre Bragg gratings, Electron. Lett. 31 (11) (1995) 899–901. [63] A. Quintela, J.M. La´zaro, M.A. Quintela, C. Jauregui, J.M. Lo´pez-Higuera, Fabrication of FBGs with an arbitrary spectrum, IEEE Sens. J. 8 (7) (2008) 1287–1291.
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[64] J.C.C. Carvalho, M.J. Sousa, C.S. Sales Jr., J.C.W.A. Costa, C.R.L. Franceˆs, M.E.V. Segatto, A new acceleration technique for the design of fibre gratings, Opt. Exp. 14 (22) (2006) 10715–10725. [65] J. Skaar, K.M. Risvik, A genetic algorithm for the inverse problem in synthesis of fiber gratings, IEEE J. Lightwave Technol. 16 (1998) 1928–1932. [66] P. Dong, J. Azana, A.G. Kirk, Synthesis of fiber Bragg grating parameters from reflectivity by means of a simulated annealing algorithm, Opt. Commun. 228 (2003) 303–308. [67] B. Eggleton, P.A. Krug, L. Poladian, F. Ouellette, Long periodic superstructure Bragg gratings in optical fibers, Electron. Lett. 30 (1994) 1620–1622. [68] F. Ouellette, P.A. Krug, T. Stephens, G. Dhosi, B. Eggleton, Broadband and WDM dispersion compensation using chirped sampled gratings, Electron. Lett. 31 (1995) 899–901. [69] H. Li, Y. Sheng, Y. Li, Member, and Joshua E. Rothenberg, Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation, IEEE J. Lightwave Technol. 21 (9) (2003) 2074–2083. [70] R. Feced, M.N. Zervas, M.A. Muriel, An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings, IEEE J. Quantum. Eletron. 35 (1999) 1105–1115. [71] J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Wang, R.B. Wilcox, et al., Dammann fiber Bragg gratings and phase-only sampling for high channel counts, IEEE Photon. Technol. Lett. 14 (2002) 1309–1311. [72] Y. Painchaud, A. Mailoux, H. Chotard, E. Pelletier, M. Guy, Multi-channel fiber Bragg gratings for dispersion and slope compensation, in: Paper. THAA5, Tech. Dig. Optical Fiber Communication Conf., 2002. [73] G.H. Song, S.Y. Shin, Design of corrugated waveguide filters by the Gel’fand-LevitanMarchenko inverse-scattering method, J. Optical Soc. Am. 2 (1985) 1905–1915. [74] J.E. Roman, K.A. Winick, Waveguide grating filters for dispersion compensation and pulse compression, IEEE J. Quantum Electron. 29 (1993) 975. [75] O.V. Belai, L.L. Frumi, E.V. Podivilov, D.A. Shapiro, Efficient numerical method of the fiber Bragg grating synthesis, J. Opt. Soc. Am. B 24 (7) (2007) 1451–1457. [76] L. Poladian, Simple grating synthesis algorithm, Opt. Lett. 25 (2000) 787–789; and errata: Opt. Lett. 25, 1400. [77] A. Sherman, A. Rosenthal, M. Horowitz, Extracting the structure of highly reflecting fiber Bragg gratings by measuring both the transmission and the reflection spectra, Opt. Lett. 32 (5) (2007) 457–459. [78] S. Keren, M. Horowitz, Interrogation of fiber gratings using low-coherence spectral Interferometry, Opt. Lett. 26 (2001) 328–330. [79] S. Keren, A. Rosenthal, M. Horowitz, Measuring the structure of highly reflecting fiber Bragg gratings, IEEE Photon. Technol. Lett 15 (4) (2003) 575–577. [80] H.G. Limberger, P.Y. Fonjallaz, P. Lambelet, R.P. Salathe´, C. Zimmer, H.H. Gilgen, Fiber grating characterization by OLCR measurements, European Conference on Optical Fibre Communications (2) (1993) 45 (paper MoP2.1). [81] A. Rosenthal, M. Horowitz, Inverse scattering algorithm for reconstructing strongly reflecting fiber bragg gratings, IEEE J. Quantum Electron. 39 (8) (2003) 1018–1026. [82] A. Rosenthal, M. Horowitz, Reconstruction of a fiber Bragg grating from noisy reflection data, J. Opt. Soc. Am. A 22 (1) (2005) 84–92. [83] A. Rosenthal, M. Horowitz, Inverse scattering algorithm for reconstructing lossy fiber Bragg gratings, J. Opt. Soc. Am. A 21 (4) (2004) 552–560. [84] A. Rosenthal, M. Horowitz, Reconstruction of long-period fiber gratings from their core-to-core transmission function, J. Opt. Soc. Am. A 23 (1) (2006) 57–68.
Chapter 5
Apodization of Fiber Gratings Light þ Light does not always give more light, but may in certain circumstances give darkness. —Max Born
Interestingly enough, apodization1 is a word often encountered in filter design; a word that flows easily off the tongue. Yet many readers are not aware of the exact meaning of the term. Etymologically, the word has its roots firmly in Greek, a podos, meaning “private-foot,” in other words, hidden foot – footless. Curiously, any of the approximately 150 species of the amphibian order Gymnophiona, known as caecilian, were formerly known as Apoda. They are burrowing or swimming, secretive animals, without limbs but with an elongate body length between 100 and 1500 mm, occurring in the Western Hemisphere [1]. Not unlike fiber gratings. . . . So what does the word mean when applied to fiber grating filter design? Fiber gratings are not infinite in length, so they have a beginning and an end. Thus, they begin abruptly and end abruptly. The Fourier transform of such a “rectangular” function immediately yields the well-known sinc function, with its associated side-lobe structure apparent in the reflection spectrum. The transform of a Gaussian function, for example, is also a Gaussian, with no side lobes. A grating with a similar refractive modulation amplitude profile diminishes the side lobes substantially. The suppression of the side lobes in the reflection spectrum by gradually increasing the coupling coefficient with penetration into, as well as gradually decreasing on exiting from, the grating is called apodization. Hill and Matsuhara [2,3] showed that apodization of a periodic waveguide structure suppresses the side lobes. However, simply changing the refractive index modulation amplitude changes local Bragg wavelength as well, forming a distributed Fabry–Perot interferometer [4], which causes structure to appear on the blue side of the reflection spectrum
1
Etymology: Greek. a, private, podos, a foot. Source: Chambers 20th Century Dictionary.
189
190
Apodization of Fiber Gratings
of the grating, although side-lobe amplitudes are reduced [5]. To avoid this complication, the key is to maintain an unchanging average refractive index throughout the length of the grating while gradually altering the refractive index modulation amplitude. The alternative approach for generating a reflection spectrum that has a constant reflectivity over a certain band and zero outside of it is to write a sin x/x refractive index modulation envelope. From the Fourier transform analogy, it is apparent that the grating reflection spectrum will be a “tophat” function. The problem, however, is to incorporate the grating in such a way that the fringes have the appropriate phase relationship on either side of the zeroes of the sinc function. Since the induced refractive index change is proportional to the square of the electric field amplitude (intensity), it is always positive. The phase change can be physically incorporated either by including a l/4 dead zone in which no grating exists at each zero or by changing the phase of the grating abruptly, for example, in a phase mask [6] or slowly over the length of the section [7]. Strictly speaking, the Fourier transform analogy is only applicable to weak gratings, as mentioned in Chapter 4. However, the principle of using the space–frequency transform does allow the techniques to be used for the design of gratings. The beneficial effects of apodization are not manifest only in the smoothness of the reflection spectrum, but also in the dispersion characteristics. There are many techniques, as there are appropriate profile functions (shading) for the refractive index modulation amplitude to achieve the end result. However, they all rely on a single principle: keeping the sum of the dc index change and the amplitude of the refractive index modulation constant throughout the grating. In the following section, several of these techniques and types of “shading” functions used for apodization are reviewed.
5.1 APODIZATION SHADING FUNCTIONS In filter and information theory, there are well-established functions for capturing a signal with a given bandwidth for the required signal-to-noise ratio [8]. Generally, these are known as Hamming and Hanning functions. The distinguishing feature between the two is whether or not the function forces the filter parameter to zero at infinite “detuning.” For example, if the window function profiling the refractive index modulation amplitude reduces it to zero at either end of the grating, it is known as a Hanning, and otherwise a Hamming function. There are various types of functions, each of which results in a compromise between roll-off of the filter and the useable bandwidth. For the numerical analysis commonly used for the computation of the response and also
Apodization Shading Functions
191
to define the functions for apodization and chirp, it is convenient to define the grating function as DnðzÞ ¼
Dnmax Dnmax fA ðzÞfg ðzÞ þ e ; 2 2
ð5:1:1Þ
where the maximum ac index change is indicated by the subscripted variable n, fA(z) and fg(z) are the apodization envelope and periodic refractive index modulation functions, and 0 e 2 is a parameter that controls the level of the background, dc index change. In general the function, fg(z) includes chirp as 0 1 B C 2pz m C fg ðzÞ ¼ cosB @ A; Ng z L0 þ Fg Trunc dL L
ð5:1:2Þ
where L0 is the period at the start of the of the grating, and Fg is a function that describes the spatial variation of the grating period with a power dependence m and chirp step dL ¼ DL/Ng (where DL is the total chirp of the grating). The grating is composed of Ng discrete sections. The apodization factor fA(z) is described by a number of commonly used functions Fg(y) and has an argument of the form NA z y ¼ 2p Trunc þ fA ; ð5:1:3Þ LA where NA is the number of sections in the entire grating, LA is the apodization envelope period, and fA is a starting phase. Commonly used functions are as follows: 1. Raised cosine: fA ðzÞ ¼ cosn ðyÞ 2. Gaussian:
"
y fA ðzÞ ¼ exp G 2p
ð5:1:4Þ 2 # ð5:1:5Þ
3. Tanh: a y fA ðzÞ ¼ 1 þ tanh T 1 2 ; p
ð5:1:6Þ
where the phase offset has been set to zero. 4. Blackman: fA ðzÞ ¼
1 þ ð1 þ BÞ cosðyÞ þ B cosð2yÞ 2 þ 2B
ð5:1:7Þ
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Apodization of Fiber Gratings
5. Sinc:
" A
fA ðzÞ ¼ sin
B # y 0:5 p
ð5:1:8Þ
6. Cauchy: fA ðzÞ ¼
1 ðy=pÞ2 1 ðCy=pÞ2
ð5:1:9Þ
:
Three of these apodization functions, the tanh, raised cosine, and cosine, are shown in Fig. 5.1 for a grating that has a full chirped bandwidth of 0.8 nm and is 100 mm long. The reflectivity has been adjusted to be 90%, with a peak-topeak refractive index modulation of 8 105. The beneficial effect of apodization is in the removal of a strong ripple in the group delay. The corresponding effect on the relative group delay as a function of wavelength, of the apodization profiles shown in Fig. 5.1 is demonstrated in Fig. 5.2. The role of the ripple is considered in Chapter 7.
Reflectivity, dB
1550.0 0
1550.2
1550.4
–10
1550.6
1550.8
1551.0
Tanh Raised cosine Cosine
–20 –30 –40
Wavelength, nm
Relative group delay, ps
Figure 5.1 The reflectivity spectrum of a 100-mm-long, 0.8-nm bandwidth chirped grating, designed to compensate the dispersion of 80 km of standard fiber, with tanh, cos2, and cosine apodization envelopes.
1550.0 50
1550.2
1550.4
1550.6
1550.8
1551.0
Tanh
30 10 –10
Cosine
Raised cosine
–30 –50 Wavelength, nm
Figure 5.2 The effect on the relative group delay of the apodization profiles used for Fig. 5.1. Although the tanh profile has a flatter characteristic, it also has more residual ripple.
Basic Principles and Methodology
193
5.2 BASIC PRINCIPLES AND METHODOLOGY In the discussion that follows, the aim of the exercise is to ensure that the effective index of the grating remains constant, even though the coupling constant becomes a slowly varying function of grating length. The approaches taken to solve this problem are either optical or mechanical, i.e., to program the variation in the coupling constant of the grating at a point, or a combination of both. Optical methods include the use of coherence properties of the UV source and the stamping of short overlapping gratings to build a composite. Mechanical techniques rely on physically blurring out the fringes in a controlled manner, by physically stretching the fiber or shaking it. Finally, a combination of the two may also be used to make complex gratings. Also discussed in the following sections is a specific example of the “top-hat” grating, one with the ideal filter characteristics: a reflectivity that is constant in-band and zero out-of-band, being another form of apodization. This type of a function is highly desirable for a vast number of applications in telecommunications but is restricted to unchirped gratings. Chirping apodizes a uniform refractive index modulation grating as well and may be used for broadband reflectors. Chirped gratings have an associated dispersion that may not be desirable for high-speed applications, unless the grating length is much less than the length of the pulse in the fiber, in which case the grating becomes a point reflector.
5.2.1 Self-Apodization Figure 5.3 shows the region of overlap of two monochromatic UV beams with intensities I1 and I2 with a mutual angle ym. The intensity I(z) at any point z along the z-axis varies according to the phase-difference between the two beams and can be shown to be pffiffiffiffiffiffiffiffi 2p ym IðzÞ ¼ I1 þ I2 þ 2 I1 I2 cos sin ð5:2:1Þ ðLg 2zÞ ; 2 lUV where the term in brackets is the mutual phase difference between the two beams, and Lg is the length of the overlap, where a fiber grating can form. The function is periodically modulated, and its visibility is v¼
Imax Imin ; Imax þ Imin
ð5:2:2Þ
determined by the maximum and minimum intensities in the fringes. The visibility is unity along the entire length Lg only if the radiation is monochromatic and the amplitudes of the two beams are identical.
194
Apodization of Fiber Gratings
I1
θm I2
z
Figure 5.3 The overlap of two beams in the z direction.
If the UV radiation is composed of two monochromatic frequencies, then the interference pattern is the sum of Eq. (5.2.1) for the frequencies present in the source. This leads to a situation in which the visibility of the fringes becomes a function of z. There are two sets of fringes with different periods according to Eq. (5.2.1), one for each frequency. For this simple case, the form of the visibility function is simply the “beat” envelope, with the fringe visibility vanishing at positions z from the center of the overlap region at which the fringes from one frequency get out of phase with those from the other. The envelope is described by the function 2pZg 2pZg Dl cos ; ð5:2:3Þ IðzÞ ffi 2 þ 2 cos lUV lUV lUV where
ym Zg ¼ sin ðLg 2zÞ: 2
ð5:2:4Þ
In Eq. (5.2.3), the interference term has been retained from Eq. (5.2.1), and equal, unity UV intensity of the interfering beams has been assumed. The first term on the RHS is identical to that of Eq. (5.2.1). However, notice that the fringes are now modulated by a slowly varying function with the argument dependent on Dl, the difference in the wavelength between the two frequencies of the source. For the general case of a source with a Gaussian spectral content, the visibility function becomes an integral over the bandwidth. The points at which the visibility vanishes are determined by the bandwidth of the source. For a particular length of the illuminated region, the fringes vanish at the edges, replaced by constant UV illumination. For a uniform intensity beam profile of the laser beam, the fringes are self-apodizing [9]. With the induced index change being
Fringe amplitude
Basic Principles and Methodology
195
1 0.5 0 –0.5 –1
0
20
40 60 z, Microns
80
100
Figure 5.4 The self-apodized fringe profile of a two-wavelength source. The parameters have been chosen for illustration purposes only [9]. The grating length must be chosen to exactly match a single period of the envelope. Since this process has not modified the beam intensity profile, the change induced in the effective index of the mode by the UV dose is the same all along the fiber.
proportional to the fringe amplitude, the interference pattern is written into a fiber core when exposed at this position. Figure 5.4 shows the fringes including the self-apodizing envelope for a 100-micron-long grating. The fringe period has been chosen for illustration purposes only. For real laser systems, it is possible to apodize grating lengths approaching 50 mm [9]. The principle of apodization described above is based on the moire´ effect. Apodization occurs in the presence of two gratings of different periods, without affecting the total UV dose. The envelope of such a Moire´ grating is a cosine function, and to alter it, for example, to a Gaussian, a laser with the appropriate spectral shape may be used. This could be a broadband frequency-doubled dye laser source. There are other methods of generating moire´2 patterns for apodization. An optical wedge placed in the path of one of the beams of the UV interferometer will change the Bragg wavelength of the grating being written. Removal changes the Bragg wavelength; if the wedge angle is chosen such that the wavelength difference between the two Bragg periods is exactly one period more or less over the length of the grating, then apodization will occur. In this case, the difference in the Bragg grating period is given by Dl
L2Bragg : Lg
ð5:2:5Þ
The wedge required for this purpose must impart a p phase change from one end of the grating to the other. Thus, for 10-mm-long gratings, a wedge angle of the order of 50 to 100 of arc is required at a Bragg wavelength of 1550 nm [9].
2
Etymology: Fr., watered silk; referring to pattern formed on it.
196
Apodization of Fiber Gratings
Now that the principles have been set out whereby apodization can be performed without altering the intensity of illumination across the length of the grating, other practical implementations are discussed in the following sections.
5.2.2 The Amplitude Mask Amplitude shading of the intensity profile of the interference pattern (for example, the natural Gaussian profile of a laser beam) helps reduce the side lobes in the reflection spectrum. However, a symmetric “chirp” is also incorporated in the grating such that the blue part of the reflection spectrum acquires an ugly structure (see Chapter 9). This is not good for systems in which many such filters may be required to isolate tightly packed channels. Clearly, simple amplitude shading is not in itself useful for apodization. However, amplitude masks may be used in conjunction with corrective measures to alter the waveguide parameters to result in a constant effective index of the mode. The method developed to apodize gratings relies on a double exposure: the first to precondition the fiber with an amplitude mask, followed by the inscription of the grating again with another amplitude mask in conjunction with a phase mask [10]. The dose in the preconditioning exposure is adjusted to allow for the inscription of the grating with a symmetric fringe intensity profile. In Fig. 5.5 the preconditioning and grating illumination profiles are shown along with the period-averaged UV intensity. The envelopes of the precondition and the fringes are orthogonal functions. The result of
1 Preconditioning profile
UV intensity
0.75 Period averaged
0.5
Fringe profile
0.25 0 0
0.25 0.5 0.75 z, Normalized grating length
1
Figure 5.5 The preconditioning UV intensity shaped by the amplitude mask, the fringe profile, and the period-averaged UV intensity is shown. Since the average UV intensity is constant as a function of position, so is the effective index.
Basic Principles and Methodology
197
0
Reflectivity (dB)
10
20
30
40 1550.9
1551.1 1551.3 Wavelength (nm)
1551.5
Figure 5.6 The reflection spectrum of an unapodized grating and a cos2 fringe envelope profile apodized grating written by the double exposure method. The grating length is 10 mm (from: Malo B., The´riault S., Johnson D.C., Bilodeau F., Albert J., and Hill K.O., “Apodised in-fibre Bragg grating reflectors photoimprinted using a phase mask,” Electron. Lett. 31(3), 223–225, 1995). # IEE 1995.
apodization on the reflection spectrum due to the double exposure is shown in Fig. 5.6. A clear reduction of 20 dB in the side lobes is apparent over the unapodized grating of the same strength, with a reflectivity of 90% and a FWHM bandwidth of 0.24 nm. At 0.4 nm away from the peak of the reflection, the reflection is less than 40 dB relative to the peak. The fiber used for fabricating the grating was standard Corning SMF-28, which was hydrogen loaded prior to processing. The refractive index modulation profile chosen for this grating was a cos2 function, which was closely reproduced in the resultant grating. This type of a filter is difficult to fabricate using any other technology and demonstrates the immense signal discrimination available with properly fabricated fiber gratings. While the process of double exposure can produce excellent results as already demonstrated, it requires a careful study of each and every type of fiber to be used for the fabrication of apodized gratings. The final result depends not only on the photosensitivity and composition of the fiber, but also on the type of exposure, whether hydrogen loaded or not, use of a pulsed or CW source, as well as the wavelength of the UV radiation. A further complication may occur due to the effects of “incubated” grating formation [11], in which nonlinear growth of a grating occurs. It is therefore expected that other methods may be easier to use routinely, requiring less processing.
198
Apodization of Fiber Gratings
5.2.3 The Variable Diffraction Efficiency Phase Mask A phase mask with a variable diffraction efficiency has been used for the fabrication of apodized gratings [12]. There are two methods of fabricating such a mask. The diffraction efficiency into the þ1 and –1 orders is maximized for a 1:1 mark–space ratio of the grating and the zero order minimized for a specific groove depth (see Chapter 3). Therefore, there are two degrees of freedom to alter the diffraction efficiency. The mark–space ratio of the grating etched in the phase mask or the groove depth may be varied. In the technique reported, a variable diffraction efficiency phase mask, was fabricated by direct exposure of a silica plate to an ion beam of silicon. In this direct write method the ion beam was focused to a spot diameter of 100 nm and scanned across the plate to delineate the grooves. Wet etching in a 1-mol% solution of hydrofluoric acid in water was used to develop the mask. It was demonstrated that the etching rate is dose dependent. Groove widths between 100 and 550 nm and depths from 7.5 to 300 nm could be achieved by varying the ion dose from 0.5 to 4 1014 ions/cm2. The etch rate is faster for regions exposed to higher doses. The diffraction efficiency into the 1 and 0 orders was measured as a function of the dose; the diffracted orders were shown to be a linear function up to a dose of 2.25 1014 ions/cm2. Thus, a variable diffraction efficiency phase mask was fabricated, using a Gaussian profile dose of (2.25 1014 ions/cm2)exp(–x2/ (0.420)2), where x is measured in mm from the center of the 1-mm-long grating, with a period of 1.075 mm. One of the difficulties of fabricating such a grating is the stepped movement of the ion beam. As a result, the Gaussian phase-mask profile can only be approximated, and altering the dose in 40 dose steps did this. Subsequently, apodized gratings were imprinted in standard telecommunications hydrogen loaded fiber and shown to have reduced the first set of side lobes by approximately 14 dBs for a 10% reflectivity grating. These results are not as good as those from the double exposure method (see Section 5.2.2) in which a reduction in the side lobes of 20 dB was achieved for a reflectivity 10 greater. It is anticipated that the fabrication of longer gratings is not only likely to be difficult owing to uniformity of the grating but also very expensive and timeconsuming. Phase masks fabricated with stepped sections have been demonstrated [13]. However, one of the problems with e-beam fabrication of phase masks is the step size, which can be programmed to allow a variable mark–space ratio of the grooves. An alternative to this approach is to write a moire´ grating on the photoresist of the phase-mask plate. Writing two gratings of different wavelengths so that at the edges the two patterns are exactly half a period out of phase as shown in Section 5.2.1 easily does this. The dose delivered by the e-beam for each overlaid grating is half that required for the resist to be fixed.
Basic Principles and Methodology
199
With two exposures, the correct dose is delivered [14] and on developing the mask, the moire´ grating is revealed [15]. In this phase mask, the diffraction efficiency varies continuously along its length, and it is possible to fabricate phase masks for long gratings. While it is attractive and convenient to replicate apodized gratings directly using apodized phase masks, there are issues that need to be addressed, which are definite drawbacks in the fabrication and use of such a mask. The feature size of the mask near the edges of the grating becomes infinitesimal, and as a result, the features have no strength. Nor is there any guarantee that they will survive the phase-mask fabrication process in a repeatable fashion. The fragility of such a phase mask makes it impossible to handle. It is likely to be damaged easily, either optically or mechanically, during the process of fiber grating replication. There is also the additional problem of the removal of contamination from such a fragile phase mask. The better option for an apodized phase mask is to alter the etch depth while keeping the mark– space ratio constant. This ensures the strength and eases handling and cleaning processes. The next section explores two techniques based on the application of a combination of optical and mechanical methods. Both are highly flexible and capable of producing a variety of gratings, apart from simple apodization.
5.2.4 Multiple Printing of In-Fiber Gratings Applied to Apodization The multiple printing in fiber (MPF) method has been discussed in Chapter 3. In this section, the particular attributes and requirements for the fabrication of (a) apodized gratings and (b) top-hat reflection gratings are discussed. The principle of this method is to write short (4-mm) gratings that are overlapped, so that at each printing only a few new periods are printed [16]. This is possible with a pulsed UV laser system but requires extreme precision in positioning the fiber. To overcome this problem, the fiber is supported over the entire length of the grating to be fabricated in a long glass vee-groove. The vee-groove is fabricated using two pieces of glass assembled together with a small gap at the apex of the vee. This allows a vacuum system to be used to hold the fiber precisely in position as it is translated. Figure 3.21 shows the overall fabrication equipment. Other important issues are the smoothness of the fiber translation system and the precise timing of the laser pulse. The former problem is overcome by translating the fiber continuously at a constant speed on an air bearing, during fabrication, using a linear motor capable of long translation (500 mm). The location of a point on the fiber carriage is measured continuously by an interferometer, which is modulated by a Pockels cell. This tracking interferometer has a
200
Apodization of Fiber Gratings
First pulse fringes
Position of fringes for third pulse: –d position
Fiber core
Movement direction Position of fringes for second pulse : +d position
Composite after 3 pulses
Figure 5.7 The sequence of pulses is staggered so that a fringe is spread symmetrically around the mean position of the fringe already printed in the first printing.
resolution of 0.3 nm over the translation distance. Part of the tracking signal is fed back to maintain a constant speed and to compensate for vibrations. The position of the interferometer is fixed, and the fiber is translated across the fringes, both backward and forward. The process is entirely controlled by a computer, which is programmed to generate a particular function in the firing sequence of the laser, movement direction, and speed of translation. The schematic of the printing is shown in Fig. 5.7. Overlapping fringes are shown in a sequence. The center shows the first printing of a grating; the bottom shows the position of the second printing relative to the first arranged to arrive just ahead of a fringe maximum already printed by a distance þd, and the top is the position of the third printing, which arrives immediately after the second pulse to write on top of the same fringe maximum but delayed by a distance –d. Also shown is the fringe in the fiber core at the third pulse, being a combination of the fringes due to the second and the third pulses only. As can be seen, the fringe spreads symmetrically around the original maximum. By altering the sequence of pulses, the fringes can be filled in so that the grating gradually disappears toward the edges. Using this method, unapodized gratings with an FWHM bandwidth of 4.6 pm (Lg ¼ 200 mm) and apodized gratings with a bandwidth of 27 pm (Lg ¼ 50 mm) have been demonstrated [17], both close to theoretically predicted values for low-reflectivity gratings (2–3%). The method naturally allows any type of apodization to be programmed in. There are some points that need to be considered in the use of this method. The minimum step size determines how smoothly the grating can be apodized. The step size is of the order of 0.01 mm, so that typically in a grating length of 100 mm, full symmetric apodization can be achieved with a maximum of
Basic Principles and Methodology
201
10,000 steps, which is more than adequate for a smooth profile. These parameters are discussed in Section 5.3. This technique has been applied to the fabrication of apodized chirped gratings as well. In this case, not only does the mark–space ratio have to be changed but also the period.
5.2.5 Position-Weighted Fabrication of Top-Hat Reflection Gratings The MPF method is suited to writing gratings with a “top-hat” (TH) reflection spectrum, shown in Fig. 5.8, along with the spatial profile of the refractive index modulation of the grating required for these characteristics (Fig. 5.9). For a perfect TH, an infinite number of cycles of a rect or sinc function are needed. This is not possible for practical gratings, since the continuous sinc function can only be approximated in a discrete number of steps. This task is demanding in any case, but a good approximation is possible with a few cycles. In order to invert the phase between the sections, additional UV exposure is given to induce a l/2 phase shift. The MPF scheme caters for both the changes in the amplitude of the refractive index modulation and the phase change. For the former, overlaid subgratings with the appropriate phase step between each printing reduce the fundamental component of the amplitude of the refractive index modulation. By the same technique, a phase change of p can also be introduced by shifting the fiber by the appropriate distance prior to the printing of the next subgrating. This method allows a very high degree of flexibility in the fabrication of gratings [17].
Reflectivity
1.0 0.8 0.5 0.3 0.0 1550
1552
1554 1556 Wavelength, nm
1558
1560
Figure 5.8 Simulated reflection spectrum of a sinc function approximated by including 22 zeroes for a grating 4 mm long. The peak-to-peak index difference is 6 103, and the grating has 1000 sections. These gratings demand better step resolution than is possible with the MPF scheme.
202
Apodization of Fiber Gratings 1
Amplitude
0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.5
–0.25
0 Position, z
0.25
0.5
Figure 5.9 The refractive index modulation profile showing 31 zeroes along the length of the grating. Figure 5.8 shows a top-hat reflection spectrum for a grating with 22 zeroes. The modulation amplitude must have positive and negative components in order to faithfully reproduce a top-hat spectrum. Truncation leads to a limiting out-of band rejection of 20 dB.
The positions for the phase steps are shown in the refractive index amplitude profile in Fig. 5.10. In trying to reproduce faithfully the TH function by an approximate method, two difficulties are encountered. The truncated sinc function throws up additional frequency components, which create outof-band reflections, since exact cancellation of the phases is no longer possible. Secondly, the approximate envelope of each period of the sinc function introduces additional phase shifts, which has a deleterious effect in the out-of-band spectrum. Typically, the background reflection remains just below 20 dB over a wide out-of-band frequency spectrum. A measured response of a 100-mmlong truncated sinc, TH grating is shown in Fig. 5.11. The achieved results are close to those of the simulation. The bandwidth is 20 GHz, with a rolloff of 4 dB/GHz at the band edges, and a peak in band reflectivity of 55%.
1 Amplitude
0.8 0.6
p phase shift at each zero
0.4 0.2 0 –0.5
–0.3
–0.1 0.1 Position, z
0.3
0.5
Figure 5.10 The actual refractive index modulation amplitude written in the fiber. In order to introduce the change in the sign of the modulation, a phase change can be placed at the zero.
Basic Principles and Methodology
203
Reflection, %
60
40
20
0 0
10
20
30
40
50
60
Frequency, GHz Figure 5.11 The reflection spectrum of a 100-mm-long TH grating. The grating is a result of a truncated sinc function made with the MPF method (from: Story H., Engan H.E., Sahlgren B., and Stubbe R., “Position weighting of fiber Bragg gratings for bandpass filtering,” Opt. Lett. 22(11), 784–786, June 1, 1997 (Ref. [18]).
The out-of-band rejection was not ideal, being only 16 dB below the in-band reflection, but can be improved by making the grating longer. Combining other shading functions with the sinc profile can reduce the outof-band reflection to below 50 dB [19]. Although it is possible to fabricate such gratings with the MPF method, the steep edges of the TH spectrum are degraded, and it is not clear whether or not a strong grating apodized using another simple function may be a better option.
5.2.6 The Moving Fiber/Phase-Mask Technique The MPF technique relies on the fiber being translated across an interference fringe pattern in synchrony with the arrival of the UV writing pulse. With the use of a CW beam, this is not possible since the grating would be washed out. This is exactly the principle of the moving fiber/phase-mask (MPM) writing scheme: The fiber is moved along with the phase mask in front of a stationary UV beam, or with the UV beam scanned across a fixed phase mask, with the fiber moving slowly relative to the phase mask. Figure 5.12 demonstrates the principle of scanning the UV beam across the phase mask. The fiber is mounted on a holder that can be moved in its entirety (as with the MPF method) but using a precision piezoelectric translator. Depending on the position of the UV beam, the fiber is dithered in the scanning direction backward and forward, to wash out the grating being inscribed by the required amount. Thus, if the dither amplitude is a linear function of position of the UV beam, with zero movement at the center of the phase mask, an apodized grating is written. Note that the smearing of the grating
204
Apodization of Fiber Gratings
244 nm UV beam
Mirror
Motorized translation stage Fiber
Phase mask
Piezoelectric-dither unit
Apodised grating 35dB
Reflection spectrum Figure 5.12 The moving phase-mask/fiber method for apodizing gratings as well as inducing chirp. The lower inset shows the achieved reflection spectrum of an unchirped grating, with side lobes 35 dB below the peak (after Ref. [20]).
occurs over the entire length of the writing beam width, so it is essential that the spot size be kept small. The result of apodization is also shown in the inset in Fig. 5.12. The side lobes have been suppressed by approximately 13 dB below the side modes of a uniform period grating [20,21]. If the velocity of the fiber is vf, the scanning UV beam moves at vsc and the width of the beam is W, then the normalized amplitude of the refractive index modulation Dn varies as a sinc function: Dn ¼
sinðpWvf =Lg vsc Þ : ðpWvf =Lg vsc Þ
ð5:2:6Þ
From Eq. (5.2.6) it follows that the modulation index goes to zero for the jargumentj ¼ p radians, so that vf ¼
Lg vsc : W
ð5:2:7Þ
Equation (5.2.7) shows the obvious result that if the width of the beam is equal to the period, then vf ¼ vsc to wash out the refractive index modulation. In general, the beam width W 100 Lg , so that the velocity of the fiber is
1% of the scanning velocity.
Basic Principles and Methodology
205
The relative movement of the fiber with respect to the scanning beam changes the Bragg wavelength of the grating, which is easily calculated as DL vf ¼ : vsc Lg
ð5:2:8Þ
Combining Eqs. (5.2.7) and (5.2.8) directly leads to the chirp as a function of the width of the UV beam, DL Lg : ¼ W Lg
ð5:2:9Þ
Therefore, the maximum wavelength shift is inversely proportional to the width of the beam. This condition is similar to the one encountered in the MPF technique: The maximum is equivalent to the fiber moving one period during the time it takes the UV beam to move a distance equal to its width at the scanning velocity. Using Eq. (5.3.9) in Eq. (5.3.7), and recalling the relationship between the grating period and the Bragg wavelength, leads to Dn ¼
sinðDbW Þ ; ðDbW Þ
Db ¼
2peff Dl lBragg lBragg
ð5:2:10Þ
where
is the equivalent of “phase detuning” between the Bragg wavelength and the grating that is being written over the width of the beam. Note here that at constant fiber velocity (and scanning beam), the wavelength shifts; if the fiber velocity changes during the scan, the result is a chirped grating. This is especially useful, since apodization and chirp can be programmed in at the same time. A parameter that needs to be attended to while fabricating a chirped grating is the loss in the amplitude of the refractive index modulation. This must be compensated for, since otherwise the grating will have a varying reflectivity as a function of wavelength. There are two possibilities. The first one is to slow down both vf and vsc while maintaining the ratio so that a stronger grating results as the grating is chirped. Alternatively, the intensity of the writing beam may be increased to take account of the reduction in the amplitude of the modulation index. There is no published data available on the choice of either approach [20]. It is useful to consider the application of this technique in the fabrication of longer, chirped apodized gratings. Very much in the spirit of the sinc profile TH reflection grating and the superstructure grating, another approach to the production of long chirped gratings uses a simple analogy in Fourier transforms. A grating with a uniform period, modulated by a low spatial frequency, pure
206
Apodization of Fiber Gratings
sinusoidal envelope of period Le, will produce two side bands only. This grating has the following refractive index amplitude modulation profile: 2pN 2pM DnðzÞ ¼ 2nDn0 cos cos ; ð5:2:11Þ Lg Le where N and M are integers indicating the orders of the periods involved, and 2nDn0 is the UV induced index change. Simplifying Eq. (5.2.11) directly leads to the resultant spatial frequencies, 8 0 2 3 1 < 2pN ML g5 A 41 þ DnðzÞ ¼ nDn0 cos@ z : NLe Lg 0 2 3 19 = 2pN ML g 41 5 zA : þ cos@ ; NLe Lg
ð5:2:12Þ
There are only two spatial frequencies present, at the sum and difference frequencies. Note that in Eq. (5.2.12) the amplitude of the index modulation for each spatial frequency has been halved and that two Bragg reflections will occur. Note also that there can be higher order terms according to the ratio of N and M. The next reflection will occur at roughly half the fundamental Bragg wavelength, for N ¼ M ¼ 2, and at shorter wavelengths for other orders, predicted here but not as yet reported in the literature. The new reflections occur at a wavelength separation of Dl ¼
l2Bragg lBragg lBragg : 2neff Le le
ð5:2:13Þ
In Eq. (5.2.13), the denominator is approximately the Bragg wavelength le “phase matched” to the period of the envelope, so that the fractional change in the fundamental Bragg wavelength is the same as the ratio of the two wavelengths. As in the case of the TH grating, a phase shift of p radians has to be introduced at each zero crossing, shown in Fig. 5.13. With a chirped grating, the bandwidth and the envelope period may be chosen so that the side bands are adjacent to each other. Ibsen et al. [22] demonstrated such a grating by incorporating a continuous chirp of 2.7 nm over a grating length of 1 meter, as well as an envelope period of 291 m m. Approximately 3500 individual periods were printed with as many p-phase stitches. The grating was apodized using a raised cosine envelope over 10% of the length of the grating on each edge. The reflection and delay spectrum are shown in Fig. 5.14. The dispersion of each section was reported to be 3.630 nsec/nm (short wavelength) and 3.607 nsec/nm (long wavelength), respectively, with a total delay of 9.672 ns. These gratings are designed to compensate the dispersion of 200 km of standard fiber with a dispersion of 17 ps/nm/km.
Basic Principles and Methodology
207
Λe Fringe amplitude
p phase shift
Short wavelength
Position along grating
Long wavelength
Figure 5.13 The modulated fringe profile of the moire´ chirped grating with periodic p phase shifts.
1529
1531
1533
–5
1537 12000 10000
–10 Reflectivity, dB
1535
8000
–15 6000 –20 4000
–25
Group delay, ps
1527 0
2000
–30
0
–35 –40
–2000 Wavelength, nm
Figure 5.14 Reflectivity and delay characteristics of the chirped moire´ grating (from: Ibsen M., Durkin Michael K., and Laming R.I., “Chirped Moire´ gratings operating on twowavelength channels for use as dual-channel dispersion compensators,” IEEE Photon. Technol. Lett. 10(1), 84–86, 1998).
Typically, to produce side bands at 2 nm away from the Bragg matched wavelength, the period of the envelope will be in the region of 300–400 microns. This is roughly the period required to couple a guided mode to a copropagating radiation mode (long-period gratings, see Chapter 4), so that at some wavelength (not necessarily within the chirped bandwidth), it is predicted that strong radiation loss will be observed. The radiation loss will be due to the stitches and not the envelope, since for the latter the average index remains unchanged. This prediction has not yet been confirmed. In order to avoid this problem, it is necessary to choose the bandwidth of the grating and the envelope period judiciously.
208
Apodization of Fiber Gratings
5.2.7 The Symmetric Stretch Apodization Method The methods for apodization that are most flexible also require active management under the control of a computer. This inevitably means synchronization of the grating inscribing UV pulse and the position of the fiber (cf. MPF as well as the MPM methods). While the flexibility is desirable for a number of applications, simpler methods such as the apodized phase mask are intrinsically faster and probably better suited to mass production. However, as has been discussed, the apodized phase mask is fragile in its reported implementation and perhaps less predictable in fabrication. It also has the severe limitation of allowing an apodized grating that is only as long as the phase mask. This requires a selection of phase masks, unless a tunable interferometer is used; this, unfortunately, counters the argument for using the phase mask, since it defines the wavelength for mass replication. Thus, a number of apodized phase masks may be required, each of a different wavelength and length. Apodization requires that the refractive index modulation at the edges of the grating gradually disappear. As described in Section 5.2.1, a moire´ grating is composed of two individual gratings, which leads to apodization. The Bragg wavelength of a grating can be changed by stretching the fiber prior to writing [23,24]. Therefore, two gratings written at the same location but differing in wavelength by exactly one period will be apodized. The problem is, how can the two gratings be overlaid such that they have the correct relative phase between them? One possibility is to use symmetric fiber stretching during the inscription of a grating [25]. This poor man’s apodization technique – the symmetric stretch apodization method (SAM) – is not only simple to operate, but also applicable to any type of grating that needs to be apodized. Figure 5.15 shows the schematic of the principle of inscription by symmetric stretching of the fiber. The technique can be understood as follows: A grating is first written into a fiber in its relaxed state (Fig 5.15b), for example, by scanning +/– Half-period stretch, relaxed state
A B
Zero stretch
C
Composite grating
Figure 5.15 A schematic of the symmetric stretch scheme for apodizing gratings. See text for explanations.
Basic Principles and Methodology
209
a phase mask, although the method of inscription is unimportant. The fiber is then stretched by straining it in opposite directions by exactly one period of the grating in the fiber, and another grating written on top of the first. Since the fiber is stretched, the inscribed grating is one period longer than the first (Fig. 5.15a) and also symmetrically overlaid (Fig. 5.15c). The central part of the grating periods is overlaid in phase, while farther away from the center they become increasingly out of phase, until the edges, where they are p out of phase. The difficulty of ensuring that both gratings receive the same UV dose is overcome by stretching the fiber back and forth continuously, e.g., by using two piezoelectric transducers, oscillating out of phase. If the fiber is periodically stretched at a high enough frequency, a perfectly apodized grating will result. The apodization function has a pure halfsinusoidal period as an envelope. For a scanning phase-mask interferometer, it is necessary to ensure that the scan speed is such that each point of the fiber is exposed to the UV beam for at least a single stretching cycle, for each scan. This is easily achieved by adjusting the scan speed to be slow enough, depending on the frequency of the stretcher. For a given UV beam width, WUV, and scan speed, VUV m/sec, the frequency f of the stretching oscillator is f¼
VUV : WUV
ð5:2:14Þ
Apodization works for a variety of situations: If the UV beam is static, the stretching scheme frequency is really not that important, so long as the UV power is low enough to enable the grating to form in a time frame much greater than a single period of the oscillator frequency. This condition is generally met unless the grating is written in a single shot from a pulsed laser. A certain amount of care does need to be taken if the apodization is to be performed with a pulsed laser source. It is important that the grating be inscribed over many pulses so that pulse averaging takes place, as well as that every possible position of the stretch of the fiber be inscribed with a grating. One exception is if the grating is inscribed in two uniform pulses of identical intensity, one for each extreme position of the stretch. Apodization of the grating is continuous and not stepped, since each part of the fiber is stretched exactly the correct amount for apodization. This is not true of the MPM technique, in which a whole subgrating length is “smeared” out by the same amount, so that only quasi-continuous apodization is possible. The same applies to the MPF technique. The reason both techniques work is because some of the index change is sacrificed over the finite length of the subgrating. In the case of the MPF scheme, each subgrating tries to print the new grating on what was printed before, but slightly altered. In the MPM, it continuously
210
Apodization of Fiber Gratings
builds on the regions that have been “wrongly” printed, the result of a finite length of subgrating. There is normally enough refractive index change available for this limitation not to be a problem. The apodization scheme is independent of the length of the grating, the only requirement being that the fiber be stretched by half-a-period in each direction, so that for a chirped grating, one end of the fiber is stretched slightly more than the other, by adjusting the stretch on that side. For uniform period gratings, no adjustment is necessary when changing the wavelength of the phase mask in the same spectral window (e.g., 1500 nm). Figure 5.16 shows the experimental setup of the equipment used in SAM. The required stretch to form perfectly apodized gratings as a function of length is shown in Fig. 5.17. Even for relatively short gratings, the strain is easily applied. Another possibility with this method is to write two gratings under static strain to form moire´ gratings.
UV Movement
Movement Phase mask Piezo elements
Fiber
Vacuum chuck Oscillator
~
Figure 5.16 The symmetric stretching apodization method (SAM). The phase mask may be replaced by any interferometer. The displacement of the piezoelectric stretchers is monitored by position sensors to set the required stretch [25].
Strain %
0.12
0.08
0.04
0 0
5 10 15 Length of grating, mm
20
Figure 5.17 The strain applied to a fiber for perfect apodization as a function of grating length for the SAM technique.
Basic Principles and Methodology Join
First grating with apodized LH end
211 Other unapodized gratings
Unapodized next grating
Last grating with apodized RH end
Figure 5.18 The super-step chirped grating, apodized on each end. With a uniform period phase mask, the chirp is zero so that a long, unchirped apodized grating can be written. Careful alignment can eliminate the stitch error at the join, or it may be UV “trimmed” [26].
By switching off one stretcher, the grating will be apodized only on the stretched side. As a result, left- and right-hand-end apodization may be performed independently. For super-step-chirped gratings [26], this feature allows apodization of each end of the grating. For the first, shortest-wavelength grating, the short-wavelength end is apodized; other intermediate gratings are printed sequentially without apodization, except for the last, longest-wavelength grating, in which the right-hand, long-wavelength end is apodized by switching on the RH piezoelectric stretcher. A schematic of this principle of making ultralong gratings is shown in Fig. 5.18. With greater stretch of the fiber, a larger number of cycles of the apodization profile may be written, for example, a bowtie profile. Turning off both stretchers may alter the apodization profile after a single pass. The next over laid grating is left unapodized, thereby building a modified cosine refractive index modulation profile. Increasing the stretch further forms a single cosine envelope, shown in Fig. 5.19. The stretch method has the same effect as the dual frequency moire´
Fringe amplitude
1 0.5 0 –0.5 –1
0
20
40 60 z, Microns
80
100
Figure 5.19 A single-period cosine envelope moire´ grating formed by stretching the fiber by twice the required amount for perfect apodization (as in Fig. 5.15). The arrow indicates the position of the automatically introduced p phase shift in the fringes, equivalent to a p/2 phase shift at the Bragg wavelength. The length of the grating has been chosen to be deliberately short to show the occurrence of the phase shift.
212
Apodization of Fiber Gratings
grating apodization. At the zero crossing of the envelope, a p phase change occurs between the two sections of the fringes, as can be seen in Fig. 5.19. This effect can be used to automatically introduce multiple, regularly spaced p/2 phase shifts at the Bragg wavelength for the fabrication of a top-hat reflection spectrum and multiple band-pass filters (also see Section 5.2.6). The difference between stretching or the dual-frequency multiple-period moire´ gratings and the MPF technique for writing a similar grating is that in the latter, a deliberate phase shift has to be written in, whereas in the former, the phase shift is automatically introduced.
5.3 FABRICATION REQUIREMENTS FOR APODIZATION AND CHIRP As has been demonstrated in Section 5.2.6, the maximum chirp that can be written using the two mini-grating replication methods (MPF and MPM) is dependent on the length of the subgrating; in the case of the 1-mm-long subgrating for the MPF method, it is only possible to write a grating with a chirp of 1 nm, since it is equivalent to a change of one period in 1 mm. In order to write larger chirps and apodize the grating at the same time, a smaller subgrating must be written. The maximum chirp that can be produced from a subgrating length of dLg is Dl ¼ 2neff
Lg : dLg
ð5:3:1Þ
A grating with a bandwidth of 10 nm near the Bragg wavelength of 1.55 mm requires a subgrating length of 300 mm. The second point to remember is that the refractive index modulation remains almost unchanged with the chirp induced using the MPF scheme, which can only be used with a pulsed laser. A reduction in the refractive index modulation occurs when a chirp is induced with the MPM method. It can be compensated for to some extent by adjusting the irradiation intensity or by changing the movement velocities of the fiber or phase mask and UV beam, but it requires a CW writing beam. Both methods allow the inscription of long gratings but do require a movement stage with a translation capability at least as long as the grating to be written. The complication of saturation effects [27,28] has not been addressed in the case of strong gratings written using either of these methods, although certain fibers show a linear response to the time of exposure to UV radiation at a longer wavelength of 334 nm and a much increased writing time [29]. Undoubtedly these will play an important role as the requirements for the types of gratings become more demanding. The effect of linearity of the photosensitive response
References
213
of the fiber to, for example, the change in the local intensity is as yet unknown. The further, more serious issue of the out-diffusion of hydrogen/deuterium from long gratings has also not been discussed. Out-diffusion causes a reduction in the refractive index of both the UV-exposed and the unexposed regions [30,31]. A differential change in the refractive index between the two regions will lead to a degradation in the transfer characteristics of long gratings, since the phase change accumulates over its entire length. Both the MPF and MPM methods are flexible and capable of apodizing gratings with arbitrary refractive index modulation profiles and are capable of the production of identical grating characteristics. The issues related to the other schemes, such as the apodized phase mask, although convenient, are limited flexibility allowing only the replication of the type of apodization programmed in the phase mask. There is a restriction on the maximum size of the phase mask as well as on the reproducibility of the apodized phase mask. Although, as with the step-chirped phase mask [6], it is possible to combine apodization and chirp in a phase mask. However, the symmetric stretch apodization method combines the two and is flexibly applied to any length of grating. SAM is excellent for long gratings, since the stretch is fixed by the period of the grating, and therefore the strain changes inversely with length. This is shown in Fig. 5.17. For a 1-mm grating, the fiber has to be strained by 0.1%, which is acceptable, but drops to an insignificant 0.01% for a 10-mm grating. It is, however, very important to ensure that the stretch is symmetric; otherwise, the apodization will not be satisfactory. Once the interferometer is aligned, then any length of grating may be apodized. If the grating is not symmetrically placed between the stretchers, then the piezoelectric movement can be adjusted to stretch one end of the fiber more than the other, easily compensating for the misalignment. Finally, certain types of fiber show a photosensitivity that is a function of applied strain [27]. However, the strain used for the apodization of fibers is only a small fraction of that reported in Ref. [27] and should not pose a problem for gratings longer than a millimeter. For long moire´ grating formation, the applied strain will also remain low enough to not cause nonlinearity in the photosensitivity.
REFERENCES [1] Caecilla, Encyclopedia Britannica, Micropaedia 2 715. [2] K.O. Hill, Aperiodic distributed-parameter waveguides for integrated optics, Appl. Opt. 13 (1974) 1853–1856. [3] M. Matsuhara, K.O. Hill, Optical-waveguide band-rejection filters: Design, Appl. Opt. 13 (1974) 2886–2888. [4] V. Mizrahi, J.E. Sipe, Optical properties of photosensitive fiber phase gratings, J Lightwave Technol. 11 (10) (1993) 1513–1517.
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Apodization of Fiber Gratings
[5] G. Meltz, W.W. Morey, W.H. Glenn, Formation of Bragg gratings in optical fibres by transverse holographic method, Opt. Lett. 14 (15) (1989) 823. [6] R. Kashyap, P.F. McKee, D. Armes, UV written reflection grating structures in photosensitive optical fibres using phase-shifted phase-masks, Electron. Lett. 30 (23) (1994) 1977–1978. [7] G. Pakulski, R. Moore, C. Maritan, F. Shepard, M. Fallahi, I. Templeton, G. Champion, Fused silica masks for printing uniform and phase adjusted gratings for distributed feedback lasers, Appl. Phys. Lett. 62 (3) (1993) 222. [8] See, for example, D.P. Morgan, Surface-Wave Devices for Signal Processing, Elsevier, Oxford, 1985. [9] H.G. Froehlich, R. Kashyap, Two methods of apodisation of fibre Bragg gratings, Opt. Commun. 157 (6) (1998) 273–281. [10] B. Malo, S. The´riault, D.C. Johnson, F. Bilodeau, J. Albert, K.O. Hill, Apodised in-fibre Bragg grating reflectors photoimprinted using a phase mask, Electron. Lett. 31 (3) (1995) 223–225. [11] P.E. Dyer, R.J. Farley, R. Giedl, K.C. Byron, D. Reid, High reflectivity fibre gratings produced by incubated damage using a 193 nm ArF laser, Electron. Lett. 30 (11) (1994) 860–862. [12] J. Albert, K.O. Hill, B. Malo, S. The´irault, B. Bilodeau, D.C. Johnson, et al., Apodisation of spectral response of fibre Bragg gratings using a phase mask with a variable diffraction efficiency, Electron. Lett. 31 (3) (1995) 222–223. [13] R. Kashyap, P.F. McKee, R.J. Campbell, D.L. Williams, A novel method of producing photo-induced chirped Bragg gratings in optical fibres, Electron. Lett. 30 (12) (1994) 996–997. [14] J. Albert, K.O. Hill, D.C. Johnson, F. Bilodeau, M.J. Rooks, Moire´ phase masks for automatic pure apodisation of fibre Bragg gratings, Electron. Lett. 32 (24) (1996) 2260–2261. [15] J. Albert, S. The´riault, F. Bilodeau, D.C. Johnson, K.O. Hill, P. Sixt, et al., Minimisation of phase errors in long fiber Bragg grating phase masks made using electron beam lithography, IEEE Photon. Technol. Lett. 8 (10) (1996) 1334–1336. [16] R. Stubbe, B. Sahlgren, S. Sandgren, A. Asseh, Novel technique for writing long superstructured fiber Bragg gratings, In: Photosensitivity and Quadratic Nonlinearity in Glass Waveguides: Fundamentals and Applications, vol. 22, 1995 OSA Technical Series. Optical Society of America, Washington, DC, 1995, pp. PD1-(1–3). [17] H. Story, Fibre Bragg gratings and fibre optic structural strain sensing. Ph.D. Thesis, Norwegian University of Science and Technology, NTUT, 1997. [18] H. Story, H.E. Engan, B. Sahlgren, R. Stubbe, Position weighting of fiber Bragg gratings for bandpass filtering, Opt. Lett. 22 (11) (1997) 784–786. [19] G.S. Kino, Acoustic Waves: Devices, Imaging, and Analog Signal Processing, Prentice Hall, New Jersey, 1987. [20] M.J. Cole, W.H. Loh, R.I. Laming, M.N. Zervas, S. Barcelos, Moving fibre/phase maskscanning beam technique for enhanced flexibility in producing fibre gratings with a uniform phase mask, Electron. Lett. 31 (17) (1995) 92–94. [21] M.J. Cole, W.H. Loh, R.I. Laming, M.N. Zervas, S. Barcelos, Moving fibre/phase maskscanning beam technique for writing arbitrary profile fibre gratings with a uniform phase mask, In: Photosensitivity and Quadratic Nonlinearity in Glass Waveguides: Fundamentals and Applications, vol. 22, OSA Technical Series. Optical Society of America, Washington, DC, 1995, pp. PD1-(1–3). [22] M. Ibsen, K. Durkin Michael, R.I. Lamming, Chirped Moire´ gratings operating on twowavelength channels for use as dual-channel dispersion compensators, IEEE Photon. Technol. Lett. 10 (1) (1998) 84–86.
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[23] R.J. Campbell, R. Kashyap, Spectral profile and multiplexing of Bragg gratings in photosensitive fibre, Opt. Lett. 16 (12) (1991) 898–900. [24] K.C. Byron, K. Sugden, T. Bircheno, I. Bennion, Fabrication of chirped Bragg gratings in photosensitive fibre, Electron. Lett. 29 (18) (1993) 1659. [25] R. Kashyap, H.G. Froehlich, A. Swanton, D.J. Armes, 1.3 m long superstep chirped fibre Bragg grating with a continuous delay of 13.5 ns and bandwidth 10 nm for broadband dispersion compensation, Electron. Lett. 32 (19) (1996) 1807–1809. [26] R. Kashyap, A. Swanton, D.J. Armes, A simple technique for apodising chirped and unchirped fibre Bragg gratings, Electron. Lett. 32 (14) (1996) 1226–1228. [27] P. Niay, P. Bernage, M. Douay, T. Taunay, W.X. Xie, G. Martinelli, et al., Bragg grating photoinscription within various types of fibers and glasses, In: Photosensitivity and Quadratic Nonlinearity in Glass Waveguides: Fundamentals and Applications, vol. 22, OSA Technical Series, paper SUA1. Optical Society of America, Washington, DC, 1995, pp. 66–69. [28] M. Douay, W.X. Xie, T. Taunay, P. Bernage, P. Niay, P. Cordier, et al., Densification involved in the UV-based photosensitivity of silica glasses and optical fibers, IEEE J. Lightwave. Technol. 15 (8) (1997) 1329–1342. [29] V. Grubsky, D.S. Starburodov, J. Feinberg, Wide range and linearity near-UV induced index change in hydrogen-loaded fibers: Applications for Bragg grating fabrication, In: Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, vol. 17, OSA Technical Digest Series, paper BME3. Optical Society of America, Washington, DC, 1997, pp. 156–158. [30] B. Malo, J. Albert, K.O. Hill, F. Bilodeau, D.C. Johnson, Effective index drift from molecular hydrogen diffusion in hydrogen-loaded optical fibres and its effect on Bragg grating fabrication, Electron. Lett 30 (5) (1994) 442–444. [31] F. Bhakti, J. Larrey, P. Sansonetti, B. Poumellec, Impact of hydrogen infiber and out-fiber diffusion on central wavelength of UV-written long period gratings, In: Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, vol. 17, OSA Technical Digest Series, paper BSuD4. Optical Society of America, Washington, DC, 1997, pp. 55–57.
Chapter 6
Fiber Grating Band-Pass Filters Let the band pass. . . .
For many applications, the transmission characteristics of a fiber Bragg grating are really the wrong way around: it is a band-stop rather than a band-pass filter. For example, tuning a radio enables the selection of a channel, not the rejection of it from a broad frequency spectrum. However, a Bragg grating works quite in reverse, and therefore cannot be easily used for channel selection. Optical transmission systems also require a “channel-dropping” function, in which a channel is selected from a large spectrum of designated channels. These optical channels are on a coarse grid of 100 GHz (multiples and submultiples of), which is currently being debated. A system based on wavelength or frequency sliced channels is a logical one and will prevail in future telecommunications networks. The advantage of such a standardized system is not in doubt, only the allocation of the channels, which is a matter for discussion by international standards committees around the world. In view of the future worldwide integration of telecommunication services, it is only a matter of time before an industry standard emerges. The immediate question that springs to mind is: Will fiber gratings play a role in emerging systems, given that their function is not the one naturally desired in a majority of applications in filtering? The answer to the question lies in their ability to invert the function to the desired one with a minimum of engineering and expense. The sales volume of gratings will crucially depend on how well and easily they fit this task. A problem needing a solution is ideal for creativity. To this end a number of options have appeared. None is ideal, but within the context of a wider technology, there are appropriate solutions for many applications, albeit at a cost. What are the options? These may be categorized into two types. First are those that work in reflection, as is normally the case with Bragg gratings. These are principally the following:
217
Fiber Grating Band-Pass Filters
218
1. 2. 3. 4. 5. 6.
The optical circulator with grating The single grating in one arm of a coupler (Possibly the most attractive) The in-coupler reflection band-pass filter The dual grating Michelson interferometer The dual grating Mach–Zehnder interferometer The super-structure grating
Those that work in transmission include most notably: 7. 8. 9. 10. 11. 12. 13. 14. 15.
The The The The The The The The The
distributed feedback (DFB) grating Fabry–Perot interferometer composite moire´ resonator chirped grating, or radiation loss with transmission window side-tap filter long-period copropagating radiation mode coupler polarization rocking coupler intermodal coupler in-coupler Bragg grating transmission filter
The above list may be subdivided into interferometric, which include devices 4–9, and noninterferometric. It is worth noting that although interferometric devices conjure up the image of sensitivity to external stimuli, it is not necessarily true of all in that category (devices 6, 7, and 9). By suitable design, devices 4, 5, and 8 have been rendered insensitive and demonstrated to be stable. All gratings are temperature and strain sensitive; however, the temperature sensitivity is low, <0.02 nm/ C, so that over a working temperature range of 100 C, the change in the operating wavelength is only 2 nm. As a fraction of the channel spacing (100 GHz or 0.8 nm), it is still too large and must be stabilized. Thus, schemes need to be developed to counter the effects of temperature. Passive packaging can isolate the grating from experiencing effects of strain. Both temperature and strain have been used along with grating-based band-pass filters to control novel functions such as optical add–drop multiplexers and demultiplexers. We now consider some of these filters in detail, along with their attributes and shortcomings.
6.1 DISTRIBUTED FEEDBACK, FABRY–PEROT, SUPERSTRUCTURE, AND MOIRE´ GRATINGS The general form of the band-pass filter described in this section is a grating or a combination of gratings physically written at the same location. The composite transmission spectrum of this band-pass filter can be a single or a series (one to many) of high transmission windows separated by bands that are rejected by reflection.
Distributed Feedback, Fabry–Perot, Superstructure, and Moire´ Gratings
219
6.1.1 The Distributed Feedback Grating The distributed feedback fiber grating is probably the simplest band-pass filter and comprises a phase-step within the length of the grating. It is a Fabry– Perot filter with a gap of less than one Bragg wavelength. The position and the size of the phase step determine the position and the wavelength of transmission band. A schematic of this grating is shown in Fig. 6.1. The DFB structure is used in semiconductor lasers to enable single frequency operation [1,2]. The single l/4 phase-shifted DFB [1] has a pass band in the middle of the stop band. The pass band has a very narrow Lorentzian line shape. While this narrow pass band is useful for filtering, the need exists for broader-bandwidth, high-finesse transmission filters. Cascading several such structures leads to an improved, broader transmission bandwidth [3–5], and has been generally well known in electrical filter design. The developments in fiber Bragg grating technology have made it possible to fabricate direct in-fiber analogs. Low-loss, high-finesse filters, using both a single l/4-shifted and cascaded phase-shifted DFB structures have been reported in the literature [6,7]. Rare-earth-doped fiber DFB lasers (see Chapter 8) have also been demonstrated [8]. Coupled-mode analysis developed in Chapter 4 leads directly to the fields in each grating. The matrix method provides a simple route to the transfer function of the DFB structure. Recalling Eq. (4.8.22), the transfer matrix of the DFB is TDFB ¼ T 2 T ps T 1 ;
ð6:1:1Þ
ps
where T is the phase shift matrix shown in Eq. (4.8.19) " # eiðf=2Þ 0 ps : T ¼ 0 eþiðf=2Þ
ð6:1:2Þ
Remembering that the transfer matrix elements of T1 and T2 are described by Eqs. (4.8.18)–(4.8.21) immediately leads to the solution for the DFB transfer function, TDFB: Grating length = Lg Fiber
dlgap = lBragg /4 Figure 6.1 The fiber DFB Bragg grating. A transmission peak appears in the center of the band stop when the gap is precisely l/4.
Fiber Grating Band-Pass Filters
220
2 T
DFB
¼4 2 ¼4
ps ps 2 1 2 1 T11 T11 þ T12 T21 T22 T11
ps ps 2 1 2 1 T11 T11 T11 þ T12 T22 T22
ps ps 2 1 2 1 T21 T11 T11 þ T22 T21 T22
ps ps 2 1 2 1 T21 T12 T11 þ T22 T22 T22
DFB T11
DFB T12
DFB T21
DFB T22
3 5 ð6:1:3Þ
3 5:
The transmitted power according to Eq. (4.8.8) is 2 1 2 1 1 DFB 2 jt j ¼ DFB ¼ DFB ∗ ¼ 2 1 ps ps ; 2 1 DFB T11 T11 T11 T11 þ T12 T21 T22 T11 T11 ps ∗ ps where * indicates the complex conjugate, and T22 ¼ T11 . From conservation of energy, we find that the reflectivity jrDFB j2 ¼ 1 jtDFB j2 :
ð6:1:4Þ
ð6:1:5Þ
Note that in general both grating sections need not be placed symmetrically around the phase step and that the gratings may have different bandwidths and refractive index modulation amplitudes. However, the simple band-pass filter has, in the center of a uniform grating, a p/2 phase step, which has the effect of introducing pass-band in the transmission spectrum. In this case, 2 ∗ a single 1 1 2 ¼ T12 , and T11 ¼ T11 . Figure 6.2 shows the band-pass spectrum of two T21 3-mm-long gratings, each with a quarter-wavelength step in the center. The amplitudes of the refractive index modulation are 5 10–4 and 10–3. Notice that although there is a very narrow transmission band in the center of the grating spectrum within a band stop of 1 nm, there are strong side lobes on either side. The band pass is a highly selective filter within a relatively narrow band stop. A uniform grating of the same length but without the phase step has a
Transmission
1.0 0.8 0.5 0.3 0.0 1550.0
1550.5
1551.0 Wavelength, (nm)
1551.5
1552.0
Figure 6.2 The calculated transmission spectrum of 3-mm-long DFB gratings, with a l/4 phase-step and refractive index modulation 5 10–4 for the dashed line and 10–3 for the continuous line, respectively.
Distributed Feedback, Fabry–Perot, Superstructure, and Moire´ Gratings
221
bandwidth approximately half that of the full band stop of the DFB grating. The DFB grating may be viewed as being composed of two single gratings, each half the length of a uniform one. The phase mask allows the replication of phase-shifted DFB structures into fibers in a simple and controlled manner [6]. This has been done successfully to produce a variety of band-pass DFB structures, and the transmission spectrum of one such grating is shown in Fig. 6.3. In this grating, a phase step of l/4 was introduced in the middle of the phase mask and replicated. This type of grating is a simple Fabry–Perot interferometer that has a band stop inversely proportional to 0.25 Lg. A feature to note is the second peak on the side of the main band pass, which is due to the birefringence of the fiber estimated from the separation to be 1 10–5. The combination of photoinduced and intrinsic birefringence becomes apparent with the extremely narrow band-pass structure of the DFB grating. The finesse of this DFB grating was 67. The transmission peak is very sensitive to losses within the grating structure. Note that although the band pass has not been fully resolved, there is OH– absorption loss in this grating because of hydrogenation, and consequently, the transmission peak is diminished. Phase shifts within a grating can be produced in several ways. Canning and Sceats [7] showed that postprocessing the center of a uniform grating with UV radiation results in a permanent phase-shifted structure. This method relies on the fact that the UV radiation changes the refractive index locally to produce an additional phase shift [53]. The UV-induced refractive index change required in a 1-mm-long fiber for a l/4 phase shift at a wavelength of 1530 nm is 3.8 10–4, which is easily achieved. The DFB structure in a rare-earth-doped fiber is useful for ensuring single-frequency operation [9], in much the same way as semiconductor DFB lasers. An inexpensive high-quality fiber-compatible laser exhibiting extremely low noise is particularly attractive for telecommunications.
Transmission
1 0.1 0.01 0.001 1531
0
15
31.5
1532
0
1532
5
1533
Wavelength,nm Figure 6.3 Transmission characteristics of an 8-mm-long DFB grating produced by replication of a phase mask with a l/4 phase shift in the center [6].
Fiber Grating Band-Pass Filters
222 1.0
Transmission
0.8 0.6 0.4 0.2 0.0 1550.0
A 1551.0
B
1552.0
1553.0
1554.0
Wavelength,nm Figure 6.4 Transmission spectra for two 3-mm-long DFB gratings with cosine (A) and raised cosine (B) apodization. The stronger apodization (B) reduces the effective length of the grating and therefore the reflectivity. The FWHM bandwidths of the transmission peaks are 0.1 nm (B) and 0.05 nm (A).
Initial laser diode pumped fiber DFB devices producing 1 mW single-frequency radiation have already shown the promise of low noise of 160 dB/Hz [9] (see Chapter 8). Apodization of the grating leads not only to a smoothing of the side mode structure but also to a reduction in effective length of the grating. The reflectivity is reduced and the band-pass bandwidth increases. There are several apodization envelopes to choose from (see Chapter 5), and they all have a similar effect in unchirped DFB gratings: a reduced reflectivity due to the reduction in the effective length of the grating. Figure 6.4 shows the effect of two such apodization envelopes: the cosine and raised cosine (exponent ¼ 2) for a 3-mm-long DFB with a refractive index modulation amplitude of 1 10–3 for both gratings. Compared to Fig. 6.2, the side-mode structure has been smoothed out. This effect is especially useful for concatenated fiber DFB lasers operating at different wavelengths [10]. The shape of the band pass is not ideal for many applications; a flat top is desirable. In order to increase the bandwidth of the band pass, several concatenated phase steps may be used within a single grating. The principle has been known in filter design [11], and its use was proposed by Haus for semiconductor DFB structures [3,4]. Figure 6.5 shows the practical implementation of such a grating. Agrawal and Radic [5] have shown that two phase steps placed in the grating open a second band pass. The position and size of the phase step shift the location of the band pass within the band stop of the grating spectrum. The transmission spectrum for a simple case of two symmetrically placed phase shifts of p/2 is shown in Fig. 6.6.
Distributed Feedback, Fabry–Perot, Superstructure, and Moire´ Gratings L1
L3
L2
L4
L5
223
L6
p /2 Identical phase-steps of p/2 at each gap between subgratings Figure 6.5 The schematic of a cascaded, quarter-wavelength shifted DFB filter, comprising gratings of identical length symmetrically placed around the center.
Transmission
1.0 0.8 0.5 0.3 0.0 1550.0
1551.0
1552.0 Wavelength,nm
1553.0
1554.0
Figure 6.6 A 3-mm-long grating with two p/2 phase shifts 1 mm apart. The refractive index modulation amplitude is 1 10–3.
Two peaks have appeared in the transmission spectrum. The two band-pass peaks can be made to coalesce or move apart by altering the ratio of the lengths of the grating sections. By choosing several phase shifts Zengerle and Leminger [4] demonstrated the ideal ratios of the lengths of the gratings for an optimized pass band. A quality factor s, defined as the ratio of the pass bandwidth at 10% and 90%, is 0.16 for a single phase-step DFB grating with a coupling constant kL 2.9. A length sequence of 1:2:2.17:2.17:2:1 appears to produce a ripple of only 0.3 dB within the pass band with a quality factor of 0.87. This ratio of the grating lengths will produce the quality factor irrespective of the overall length of the grating. The design was for a semiconductor grating, and the target was for a 0.4-nm band-pass filter. In Fig. 6.5, six gratings are shown with intermediate phase shifts. The lengths L1 ¼ L6 : L2 ¼ L5 : L3 ¼ L4 should therefore have the ratio of 1:2:2.17. Note the flat-top pass band. Wei and Lit [12] have examined symmetrical configurations of 3 and 4 phaseshifted structures. Unity transmission occurs for a symmetrical three-gratingsection filter with two p/2 phase shifts when the grating at either end is half the length of the grating(s) in between. Although this is not a unique condition for unity transmission at the Bragg wavelength (see, for example, Fig. 6.7) in a
Fiber Grating Band-Pass Filters
Transmission
224 1.0 0.8 0.6 0.4 0.2 0.0 1550
B A 1551
1552
1553
1554
Wavelength,nm Figure 6.7 Transmission spectrum of two gratings with 4 l/4 cascaded phase step and cosine apodized envelope. Five equal-length gratings of 1.50 mm each have been used, giving a total length 7.5 mm. The refractive index modulation amplitude is 5 10–4 (B) and 1 10–3 (A). The FWHM bandwidth of the central pass band is approximately 0.25 nm (B) and 0.1 nm (A); compare with the grating spectra in Fig. 6.4, which have the same individual grating lengths. The bandwidth of (A) above is the same as in Fig. 6.4B, but has a flat top and a squarer profile, with a quality factor of 0.65 (see text) for a kL of 4.
multisection grating, the three-section and four-section gratings (two phase shifts and three phase shifts) have been shown to have a near unity transmission at the Bragg wavelength if the ratio C of 2 is maintained; i.e., for the foursection grating, L1: L2: L3 : L4 1:2:2:1, for a variety of coupling constants, kL up to 2.5. It appears that the ratio for the sections C ¼ 2 is an asymptotic value for a flat-top band-pass filter. Increasing this ratio introduces a ripple in the band pass as a result of the individual band-pass peaks separating, while reducing it narrows the bandwidth of the pass band, also reducing the transmission. The ripple in the pass band may be kept between 1% and 5% for a ratio of 1.8 < C < 2.6. The bandwidth of the band pass is inversely dependent on the coupling constant. In order to maintain a reasonable bandwidth of the filter, coupling constants must remain low (kL 1), as should the grating section (L1 < 1 mm). For example, in the three-section grating, section lengths of 0.5 and 1 mm with a kL1 ¼ 1.4 will result in a bandwidth of 0.25 nm. These results change with larger number of sections. For example, Fig. 6.7A shows the transmission spectrum for a grating with a kL1 1.4, but with an FWHM bandwidth of 0.1 nm. The bandwidth is exactly the same as the single-phase-step DFB grating shown in Fig. 6.4B, but the square top shows that the roll-off is steeper for the larger number of sections [12]. A major concern is the trade-off between the bandwidth and extinction. For many filter applications, an extinction of >30 dB is necessary. This requirement immediately points to a kL > 4.16. Therefore, this filter may not be an ideal candidate, since both requirements may be difficult to achieve. Fiber Bragg gratings with multiple phase-shifted sections have been realized for band-pass applications. Bhakti and Sansonetti [13] have modeled the response of gratings with up to eight phase-shifted sections. The design strategy was for an optimized band-pass filter with a 0.8-nm bandwidth, as well as
Distributed Feedback, Fabry–Perot, Superstructure, and Moire´ Gratings
225
negligible in-band ripple. Increasing the number of sections was shown to make the pass band more rectangular, but reduced the stopped bandwidth. An asymptotic value for the band-stop bandwidth is approached with greater than 5 phase shifts and is twice the pass bandwidth. This is another severe limitation on the use of such filters. Phase masks with the appropriate quarter-wavelength shifts [6] were used to replicate a three-phase-shift grating. With careful UV illumination, the band pass was fully resolved and showed excellent agreement with theory [13]. The band-pass/stop widths were 0.88 nm/2.77 nm with a peak rejection of 13 dB. The optimized grating lengths were L1 ¼ L4 ¼ 0.22 mm and L2 ¼ L3 ¼ 0.502 mm, with a refractive index modulation amplitude of 1.5 10–3. While the principle of multiple phase shift within a single grating is useful, it has the additional effect of increasing the side-lobe structure despite apodization. The side lobes increase as a result of the formation of a super structure (see Chapter 3) and is discussed in the next section. Other methods need to be used to position the band pass and for a broader-bandwidth band pass and more controllable bandwidth of the band stop. A simple technique to accurately create a band pass at a particular wavelength is to introduce a phase step within a chirped grating. A start and a stop Bragg wavelength characterize a chirped grating. In a linearly chirped grating, the position of the local Bragg wavelength is uniquely known. Placing a p/2 phase step at that point results in a band pass at the local Bragg wavelength. Figure 6.8 shows the transmitted spectrum of two 10-mm-long gratings. Data A and B refer to the same spectrum, with B displayed on a 30 times expanded wavelength scale. A shows the effect of a single quarter-wavelength phase step in the center, while C shows the step at one-third the distance from the long-wavelength end of the grating. The effect of the stitch is localized in the reflected spectrum, and several more band-pass structures may be placed within this grating. For example, 1.0 Transmission
0.8 0.6 C
B
0.4 0.2 0.0 1535
A 1540
1545
1550
1555
1560
1565
Wavelength,nm Figure 6.8 The transmission spectrum of a grating with a single p/2 phase step in the center of a chirped bandwidth of 20 nm (A). A 30 expanded view of the band-pass spectrum is also shown (B). Also shown is the effect of placing the phase step at 2/3Lg (C); the band-pass peak shifts to the local Bragg wavelength. The grating is 10 mm long.
Fiber Grating Band-Pass Filters
226
Transmission, dB
0 –10 –20 –30 –40 1535
1540
1545
1550
1555
1560
1565
Wavelength,nm Figure 6.9 A 20 p/2 phase-step grating band-pass filter. Each band pass has an extinction of >30 dB, but with some side-lobe structure at –15 dB. Ghosts appear between the main pass bands as a result of the super structure of phase steps at –30 dB transmission.
a band-pass every 2 nm is easily achieved. However, the effects of the super structure become apparent in that additional peaks appear and the band-pass spectrum acquires side lobes. A 10-mm-long grating transmission spectrum with 20 phase steps is shown in Fig. 6.9. Note that within the pass band of the grating there is a small associated dispersion since the grating is chirped. Since the gratings are used as a band-pass, rather than in reflection, dispersion is less of an issue, other than at the band edges. Apodization only helps slightly; it also reduces the bandwidth and the extinction at the edges of the grating, so that it is of limited value. Blanking-off part of a chirped grating during fabrication instead of introducing phase steps is an effective way of creating a band-pass filter [14]. The net result is that part of the chirped grating is not replicated, thus opening a band gap. This principle is effective for a narrow-bandwidth band pass (1 nm) so long as kL < p. With a stronger reflectivity grating, the bandwidth of the band stop increases to encroach on the band pass from both sides, reducing the pass-band width. An alternative technique uses UV postprocessing first reported for UV trimming of the refractive index of photosensitive waveguides [53], to erase part of the chirped grating written in a fiber [15]. In order to fabricate a single band pass within a broad stop band (>50 nm), several chirped gratings may be concatenated alongside a chirped grating with the band gap. With care and choice of chirped gratings, single and multiple band-pass filters have been successfully demonstrated, with a pass/stop-bandwidth of 0.17 nm/11.3 nm and extinction of 10 dB. A four-channel filter evenly spaced over a stop bandwidth of 50 nm has also been reported [15]. Insertion loss of these types of chirped filters is a problem, since radiation loss on the blue-wavelength side affects the maximum transmission of the pass band. As a consequence of large kL and radiation loss, a maximum transmission of 75% was reported for these band-pass filters. Broader pass-bandwidth filters may be fabricated by the use of concatenated chirped gratings [16]. The effects of “in-filling” due to the use of large kL values
Distributed Feedback, Fabry–Perot, Superstructure, and Moire´ Gratings
227
are diminished by increasing the band-pass width. The arrangement for such a filter allows better extinction in the stop band (>30 dB) while permitting the placement of the band-pass at the required wavelength. Additionally, chirped gratings show reasonably smooth stop band edges. Concatenating two such gratings with a nonoverlapping band stop results in a band pass between the two band-stop regions. While this scheme has been applied to chirped gratings, Mizrahi et al. [17] have shown that two concatenated highly reflective gratings with a pass band in between the Bragg wavelengths can be used as a band-pass filter. Radiation loss within the pass band is avoided by using a strongly guiding fiber, which further blue-shifts the radiation loss spectrum from the long-wavelength stop band. The bandwidth of the pass band was 1.6 nm with an extinction in excess of 50 dB and a stop band of 6 nm.
6.1.2 Superstructure Band-Pass Filter It has been shown that placing more than a single l/4 phase step within the grating results in as many band-pass peaks appearing within the band stop [12]. This principle may be extended to produce the superstructure grating [18,22], but works in reflection. The reflection spectrum has several narrowbandwidth reflection peaks. The principle has been used in semiconductor lasers to allow step tuning of lasers. However, a badly stitched phase mask will produce similar results. Since a phase mask is generally manufactured by stitching small grating fields together, errors arise if the fields are not positioned correctly. These random “phase errors” are like multiple phase shifts within the grating, resulting in multiple reflection peaks, each with bandwidth inversely proportional to the overall length of the grating, and spaced at wavelength intervals inversely proportional to the length of the field size (see Chapter 3). Figure 6.10 shows the super structure on a 30-mm-long grating reproduced from a phase mask with stitching errors. Despite these errors, the grating reflection and phase response for the main peak are very close to being theoretically perfect [19]. The theory of superstructure gratings is discussed in Chapter 3. For filter applications, it is necessary to achieve the appropriate characteristics. Here we consider the spectra of short superstructure gratings, which may be conveniently fabricated with an appropriate phase mask. Figure 6.11 shows the reflection and transmission characteristics of a superstructure grating, comprising 11 0.182 mm long gratings, each separated by 1.555 mm. The overall envelope of the transmission spectrum (see Fig. 6.11b) has been shown in Chapter 3 to be governed by the bandwidth of the subgrating. Note in Fig. 6.11a that the bandwidth of the adjacent peaks becomes smaller. This is a function of the reflectivity at the edges of the grating. In order to use this filter as a band-pass filter, it is necessary to invert its reflection spectrum.
Fiber Grating Band-Pass Filters
228
Reflection spectrum of 30 mm long grating with stitching errors 1.547 0
1.548
1.548
1.549
1.549
1.55
Reflection, dB
–5 –10 –15 –20 –25 –30 –35 –40 Wavelength, microns Figure 6.10 Reflection spectrum of superstructure grating. The disadvantage of the superstructure grating – the reflection coefficient cannot be made the same for each reflection [29]. This limitation can be overcome by using a different type of moire´ grating [20], which has been discussed in Chapter 3.
Reflectivity, dB
1550 0
1552
1554
1556
1558
1560
–5
–10
–15 Wavelength,nm
Transmission, dB
A
B
1550 0 –5 –10 –15 –20 –25 –30
1552
1554
1556
1558
1560
Wavelength,nm
Figure 6.11 (a) The reflectivity spectrum of a superstructure grating with 9 222 micron grating sections separated by 1-mm gaps. Refractive index modulation amplitude is 10–3. (b) The transmission spectrum of the grating shown in (a). Also shown is the transmission spectrum of a single section of the grating of 0.181 microns long. The envelope has been normalized to fit the superstructure spectra.
The Fabry–Perot and Moire´ Band-Pass Filters
229
This may be done by using a fiber coupler. However, the input signal is split into two at the coupler. One half is reflected from the grating and suffers another 3-dB loss penalty in traversing the coupler once again. The reflected spectrum is therefore –6 dB relative to the input signal. A fiber circulator overcomes this loss penalty [21]. The insertion loss of a circulator is approximately 1 dB, so that an efficient multiple band-pass filter can be fabricated. In an interesting demonstration, a chirped superstructure grating has been used for multiple-channel dispersion compensation, since the repeat band stops have a near-identical chirp [22]. The advantage of such a scheme is that it requires only a single temperature-stabilized grating to equalize several channels simultaneously, although the reflection coefficient varies for each reflection.
6.2 THE FABRY–PEROT AND MOIRE´ BAND-PASS FILTERS The fiber DFB grating is the simplest type of Fabry–Perot (FP) filter. Increasing the gap between the two grating sections enables multiple band-pass peaks to appear within the stop band. The bandwidth and the reflectivity of the gratings determine the free-spectral range and the finesse of the FP filter. The grating FP filter has been theoretically analyzed by Legoubin et al. [23]. Equations (6.1.4) and (6.1.5) describe the transfer characteristics of the filter and have been used in the simulation of the gratings in this section. Figure 6.12 shows the structure of a Fabry–Perot filter. These filters work in the same way as bulk FP interferometers, except that the gratings are narrowband and are distributed reflectors. A broader bandwidth achieved with chirped gratings creates several band-pass peaks within the stop band. Control of the grating length L and the separation dl allows easy alteration of the stop-band and the free-spectral range. At zero detuning, the peak reflectivity of a FP filter with identical Bragg gratings is RFP ¼
4R ð1 þ RÞ2
Grating length = L/2
;
ð6:2:1Þ
Grating length L/2
Fiber dl Figure 6.12 A schematic of a Fabry–Perot etalon filter. In the simple configuration, the gratings are identical, although in a more complicated band-pass filter, a dissimilar chirped grating may be used.
Fiber Grating Band-Pass Filters
230
where R is the peak reflectivity of each grating. Since the gratings are not point reflectors, the free-spectral range (FSR) is a function of the effective length of the grating, which in turn is dependent on the detuning. For a bulk FP interferometer, e.g., a fiber with mirrors, the FSR is [23] FSR ¼
1 : 2dneff ðlÞ
ð6:2:2Þ
The distance between the mirrors is d, and the effective index of the mode neff is a function of wavelength. For an equivalent fiber-grating-based FP interferometer, the thickness d becomes a function of wavelength, and only at the peak reflectance is the FSR largest. The effective thickness is the separation between the inner edges of the gratings plus twice the effective length of the gratings. Off resonance, the penetration into the grating is greater than on-resonance, leading to a bigger thickness. Therefore, at the edges of the FP bandwidth, the FSR becomes smaller. The first in-fiber grating FP filter was reported by Huber [24]. A transmission bandwidth of 29 pm was reported. Further multi-band-pass in-fiber FP resonators have also been demonstrated [25]. In the latter report, a 100-mmlong FP interferometer was fabricated with two 95.5% reflecting gratings. A finesse of 67 was achieved with the free spectral range of 1 GHz and a pass bandwidth of 15 MHz. In order to measure the transmission spectrum of the FP, a piezoelectric stretcher was used to scan the fiber etalon in conjunction with a fixed frequency DFB laser source operating within the bandwidth of the grating band stop, at a wavelength of 1299 nm. A peak transmission of 86% of the fringe maximum was also noted. Figure 6.13 shows the transmission characteristics of a FP filter made with two gratings, each 0.5 mm long with
Transmission
1.0 0.8 0.6 0.4 0.2 0.0 1550.0
1551.0
1552.0
1553.0
1554.0
Wavelength,nm Figure 6.13 A FP filter with a 5-mm gap. Grating lengths are 0.5 mm with index modulation of 2 10–4. The arrows show where WDM channels may be placed within the band-pass filter for soliton guiding.
The Fabry–Perot and Moire´ Band-Pass Filters
Transmission, dB
1551.0 0
1551.5
1552.0
231 1552.5
1553.0
–5 –10 –15 –20 –25 –30 Wavelength,nm
Figure 6.14 A 4-mm-long grating FP filter with a 5-mm gap and a Dn of 5 10–4.
a 5 mm separation and a refractive index modulation of 2 10–4. The weak ripple within the band-stop of the filter is due to the poor finesse of the FP but is ideal in WDM transmission to control solitons. The shortest gratings in a FP filter reported to date are 0.3 mm long, separated by a similar distance [26]. The resulting multiple band pass, which was a shallow ripple of 50% transmission was used as a guiding filter in wavelength division multiplexed soliton transmission experiments to suppress Gordon–Haus jitter [27]. With stronger gratings, multiple band-pass filters with deeper band stops are easily possible. However, even slight loss in the grating (absorption due to OH– ions) can degrade the transmission peaks substantially. It is therefore advantageous to use deuterated fiber for this type of a filter. Figure 6.14 demonstrates a 4-mm-long grating with a gap of 5 mm in the center. This filter shows 30-dB extinction in the center of the band pass. Note that all these filters have a similar narrow band-pass response that plagues the highly reflecting DFB grating filter. Thus, applications for such a grating are likely to be in areas in which either high extinction or high finesse, or low extinction and large bandwidth are required. Figure 6.15 shows the measured transmission of a 0.6-mm-long FP filter with a 2.5-mm gap. The pass bands have been measured with a resolution of 0.1 nm and are therefore not fully resolved. The structure should be deeper and much narrower. Nevertheless, the dips in transfer characteristics match the theoretical simulation extremely well with the parameters shown. Typically, the best results for band-pass peaks for this type of FP filter, using either chirped or unchirped gratings with an extinction of 30 dB, is 70%. Wide-bandwidth (140-nm) fiber grating Fabry–Perot filters fabricated in boron–germanium codoped fibers have been demonstrated with a finesse of between 3 and 7 [28]. Two identically chirped, 4-mm-long gratings with a bandwidth of 150 nm and reflectivity of >50% were written in the fiber, displaced from each other by 8 mm. The resulting FP interference had a bandwidth of 0.03 nm and a free-spectral range of 0.09 nm. A larger free-spectral range was
Fiber Grating Band-Pass Filters
232 1.545 –45
1.547
1.549
1.551
1.553
1.555
Transmission, dB
–50 –55 –60 –65 –70 –75 –80 Wavelength, microns Figure 6.15 Measured transmission characteristics of a fiber FP filter. The length of each grating was 0.3 mm, a 2.5 mm gap and with a peak-to-peak refractive index modulation of 5 10–3. A theoretical fit to the data shows excellent agreement, although the peak transmission has not been fully resolved in the measurement [29]. A maximum extinction of >30 dB was measured.
obtained by overlapping the gratings with a linear displacement of 0.5 mm. These gratings had a bandwidth of 175 nm in the 1450–1650 nm wavelength window. A finesse of 1.6 with an FSR of 1.5 nm was demonstrated. These fiber-grating FP-like devices may find applications in fiber laser and WDM transmission systems. A further possibility of opening up a gap within the stop band is to write two gratings of slightly different Bragg wavelengths at the same location in the fiber [30] to form a moire´ fringe pattern. The physical reason why a band pass results may be understood by noticing that the phase responses of the gratings are not identical. Thus, at some wavelength, the phases can be out by p radians. If the wavelength difference is made larger, it is possible to create more than one band pass. The mechanics of producing such a band pass have been demonstrated by slightly altering the angle of the incoming beams in between the writing of the two gratings [30]. Unless the angle can be measured accurately, it may be difficult to reproduce the results with precision. Two gratings can be superimposed in a fiber by writing one grating with a chirped phase mask [31] and then stretching the fiber before writing the second [32,33]. The basic principle of moire´ grating formation is discussed in Chapter 5. However, for clarity, we consider the interference due to two UV intensity patterns to produce a grating with the refractive index profile 2pz 2pz cos ; ð6:2:3Þ DnðzÞ ¼ Dn 2 þ 2cos Le Lg
The Michelson Interferometer Band-Pass Filter
233
1.0
Transmission
0.8 0.6 0.4 0.2 0.0 1550.0
1551.0
1552.0 Wavelength,nm
1553.0
1554.0
Figure 6.16 Transmission spectrum of a moire´ grating with a single period cosine envelope of the modulation refractive index profile over the length of the grating. This grating is formed by colocating two chirped gratings with slightly different center wavelengths.
in which the slowly varying envelope with period Le is a result of the difference between the two grating periods, and the chirped grating period is Lg. If the envelope has a single cosine cycle over the grating length (the grating periods have been chosen to be such; see Chapter 5), then the effect of the zero crossing of the envelope is equivalent to a phase step of p/2 at the Bragg wavelength (see Chapter 5, Section 5.2.7). This grating is simple to simulate using the matrix method; the apodization profile of the grating can be specified to have n cycles of a cosine function, where n ¼ 1 is a single cycle of a cosine envelope (see Fig. 5.18). The computed transmission spectrum of this type of a band-pass filter is shown in Fig. 6.16. The experimentally achieved result is almost identical to that shown in Fig. 6.16 [33], apart from the short-wavelength radiation loss apparent just outside the band-stop spectrum in the measured result. The problem with this type of phase-shifted grating has already been discussed: There remains a trade-off between bandwidth and extinction, although it is a convenient method of producing a multiple-band-pass filter by increasing the number of cycles of the modulation envelope.
6.3 THE MICHELSON INTERFEROMETER BAND-PASS FILTER The Michelson interferometer (MI) may be used as a fixed-wavelength bandpass filter. Since the coupler shown in Fig. 6.17 splits the input power equally into the two ports, the light that is reflected from a single 100% reflection grating (HR1) is again equally split between ports 1 and 2. Thus, only 25% of
Fiber Grating Band-Pass Filters
234
Port 1: INPUT
HR1 lBragg Port 3: 50% Tx
50:50
Port 4: 50% Tx Port 2: OUTPUT 25% at lBragg Figure 6.17 A fiber coupler with a single grating in one arm. The output in port 2 is 25% of the input power in port 1. The transmitted signal at the Bragg wavelength at port 3 is (1 – R), where R is the grating reflectivity.
the light is available in the pass band at port 2. This arrangement works as an effective band-pass filter despite the loss. However, there are methods that can be used to eliminate the insertion loss of this filter. With two identical gratings, one in each arm of the MI, 100% of the reflected light can be routed to port 2. The principle of operation was originally proposed by Hill et al. [34], for a grating in a loop mirror configuration. A similar device is shown in Fig. 6.18. The light reflected from HR2 arrives at the input port p out of phase with respect to light from HR1. Light from HR1 and HR2 arrives in phase at the output port 2, so that 100% of the light at the Bragg wavelength appears at this port. The through light is equally split at ports 3 and 4, incurring a 3-dB loss. However, the phase difference between the reflected wavelengths arriving at the coupler has to be correct for all the light to be routed to port 2. The first demonstration of such a device in optical fibers was reported by Morey [35]. This all-fiber band-pass filter was made out of a standard fiber coupler with fiber gratings written into the two arms. Stretching the gratings showed limited tunability, but no data was available on stability of the filter. Since differential changes in the ambient temperature between the arms can detune the filter, it is essential that the two arms remain in close proximity and that the optical paths to and from the gratings be minimized.
Ai
R3
Port 1: INPUT Port 2: OUTPUT lBragg
R
50:50
HR1 lBragg Port 3: 50% Tx Port 4: 50% Tx
Bi
S4
S
HR2 lBragg
Figure 6.18 The Michelson interferometer band-pass filter. All the input light is equally split at the coupler into the output ports. The identical gratings in each arm reflect light at the Bragg wavelength, while allowing the rest of the radiation through.
The Michelson Interferometer Band-Pass Filter
235
The fiber Michelson interferometer has been used extensively for sensing applications with broadband mirrors deposited on the ends of the fiber [36]. The principle of operation of the grating-based filter is a simple modification of the equations that describe the broadband mirror device. We begin with the transfer matrix of the fiber coupler [37], R cosðkLc Þ isinðkLc Þ Ai ¼ ; ð6:3:1aÞ isinðkLc Þ cosðkLc Þ Bi S where R and S are the output field amplitudes at ports 3 and 4, Ai and Bi are the field amplitudes at ports 1 and 2 of the coupler, Lc is the coupling length of the coupler, and k is the coupling constant, which depends on the overlap of the electric fields E1 and E2 of the coupled modes, ð ð oe0 þ1 þ1 2 k¼ n ðx; yÞ n2cl E1∗ E2 dxdy; ð6:3:1bÞ 4P0 1 1 co where nco(x, y) is the transverse refractive index profile of the waveguides, ncl is the cladding index and, P0 is the total power. For a small difference in the propagation constants, Db is small compared to the coupling coefficient, k. Under these conditions, very nearly all the power can be transferred across from one fiber to the other [38]. With Db 0 and (nco ncl)/nco 1, the following expression for the coupling coefficient can be used [39]: k¼
l u2 K0 ½wðh=aÞ : 2pneff a2 v2 K12 ðwÞ
ð6:3:1cÞ
a is the core radius, and the normalized waveguide parameters u, v, and w are defined in Chapter 4, h is the distance between the core centers, and the modified Bessel functions of order 0 and 1, K0 and K1, are due to the evanescent fields of the modes in the cladding. Assuming that there is only a single field at the input port 1 of the coupler, Bi ¼ 0. Introducing gratings in ports 3 and 4 with amplitude reflectivities and phases, r1 exp[if1(l)] and r2 exp[if2(l)] described by Eq. (4.3.11), the normalized field amplitudes at the input to the coupler in ports 3 and 4 are 2 0 13 2pn L A3 f ðlÞ eff f 1 1 A5 þ R3 ¼ ¼ s1 r1 cosðkLc Þexp42i@ Ai l 2 ð6:3:2Þ 2 0 13 2pneff Lf 2 f2 ðlÞA5 B4 þ ¼ is2 r2 sinðkLc Þexp42i@ S4 ¼ ; Ai l 2 in which the path lengths from the coupler to the gratings are Lf1 and Lf2. The additional factors s1 and s2 include detrimental effects due to polarization and loss in arms 3 and 4. The output fields, R1 and S2, at ports 1 and 2 of the
236
Fiber Grating Band-Pass Filters
interferometer can be written down by applying Eq. (6.3.1a) again, with the input fields from Eq. (6.3.2). Therefore, R1 cosðkLc Þ isinðkLc Þ R3 ¼ : ð6:3:3Þ isinðkLc Þ cosðkLc Þ S2 S4 In Eq. (6.3.3) the normalized field amplitudes R1 and S1 fully describe the transmission transfer function of the Michelson interferometer band-pass filter. Substituting Eq. (6.3.2) into Eq. (6.3.3), and remembering that Re(R, S) ¼ 1/2 (R, S þ R*, S*), the transmitted power in each output port of the filter is the product of the complex field with their conjugate. By simple expansion and algebraic manipulation of the equations, the power transmittance at ports 1 and 2 can be shown to be
1 t1 ¼ 2s21 r21 cos4 ðkLc Þ þ 2s22 r22 sin4 ðkLc Þ s1 s2 r1 r2 sin2 ð2kLc Þcos d ð6:3:4Þ 2
1 t2 ¼ s21 r21 þ s22 r22 þ 2s1 s2 r1 r2 cos d sin2 ð2kLc Þ; ð6:3:5Þ 4 where the phase difference d between the reflections from the two gratings is f2 ðlÞ f1 ðlÞ 2pneff d¼2 Lf 2 Lf 1 þ : ð6:3:6Þ l 2 Equations (6.3.4) and (6.3.5) describe how the transmitted power at the output depends on the path-length difference, the reflectivities, and the Bragg wavelengths of the two gratings in the arms of the Michelson interferometer filter. For a 50:50 coupler, kLc ¼ p/4; ignoring loss and polarization effects, Eq. (6.3.5) simplifies to 1 1 2 ð6:3:7Þ t1 ¼ ðr1 þ r22 Þ r1 r2 cos d 2 2 1 1 2 2 t2 ¼ ðr þ r2 Þ þ r1 r2 cos d : 2 2 1
ð6:3:8Þ
Note that the power transfers is cyclic between the two ports, depending on the phase difference d and which is of paramount importance for the proper operation of the filter. This cyclic behavior is well known for unbalanced broad band interferometers, but in this device it is restricted to the bandwidth of the gratings. The choice of the gratings determines the wavelength at which the interference will occur. With the phase difference d ¼ 2Np (where N 0 is an integer), all the power is routed to port 1. This phase can to be adjusted mechanically [40], thermally [41], or permanently by optical “trimming” of the path using UV radiation [53].
The Michelson Interferometer Band-Pass Filter
237
1.00 Transmission
0 0.75
0.057
0.50
0.087
0.25 0.00 –0.0015
0.11 –0.0010
–0.0005
0.0000
0.0005
0.0010
0.0015
Normalized detuning, (l–l b)/l b Figure 6.19 The band-pass characteristics of a Michelson interferometer filter showing how the band-pass peak varies with differential path-length difference between the two gratings as a function of normalized detuning. For the calculations, the grating parameters used are: refractive index modulation index amplitude of 1 103 and a length of 2 mm [42]. The numbers on the chart refer to the detunings as a fraction of the Bragg wavelength. The figure shows just how critically the path difference needs to be controlled for efficient operation.
Detuning of the interferometer is an important issue for the acceptable performance of the band-pass filter. As such, there are two parameters, which are variables in a filter of this type. Assuming that the bandwidths of the two gratings are identical (nominally identical lengths and refractive index modulation amplitudes), the path difference may drift within the lifetime of the device, or the Bragg wavelengths may not be identical, or may change with time. Figure 6.19 shows how the band-pass at port 2 varies with changes in the path length for two identical 99.8% reflectivity gratings. At zero phase difference, the transmission is a maximum. The rapid decrease in the band-pass peak with detuning of less than half a wavelength shows that phase stability is critical for the long-term operation of this device. As expected, the transmission drops to zero with a phase difference of p/2. In most practical cases, it is nearly impossible to match gratings in the two arms exactly, both in reflectivity (and therefore bandwidth) and the Bragg wavelength. The mismatches set a limit to the performance of a band-pass filter. Figure 6.20 shows the reflectivity spectra of two raised cosine apodized MI gratings, which are offset in their Bragg wavelength by 1% of the FW bandwidth. For the gratings shown, this translates to 0.01 nm difference in the Bragg wavelengths and may well be at the limit of the technology for routine inscription and annealing of the gratings. The pass band of the filter generally does not suffer from such a small wavelength detuning. Since the reflectivity ensures that only the reflected signal appears in the pass-band with typical insertion losses of between 1 and 0.3 dB [43], it is the remnant signal that appears in the through port (in the Michelson, it is port 1) that is difficult to suppress.
Fiber Grating Band-Pass Filters
238 0 Reflectivity/ transmission, dB
–10
A&B
C
–20 E
–30 –40 –50 –60
D
–70 Normalized detuning, (l– lb)/lb Figure 6.20 Reflection spectra of two 4-mm-long raised cosine apodized gratings mismatched by 0.01 nm (A, B), each with a reflectivity of 85%. C is the transmission of the band-pass filter with a path-length difference of 0.006 lb. D shows the light that appears in port 1 of the Michelson. To achieve better than –30 dB extinction (for D), the Bragg wavelengths must be matched to within 0.01 nm. E shows the rejected light in port 1 for cosine apodized gratings with the same Bragg wavelength mismatch. The in-band rejection is never better than –28 dB. With stronger gratings, the Bragg wavelengths have to be even better matched.
The detuned Bragg wavelength of the gratings translates into an additional imbalance in the path difference of approximately 1 mradian. While UV trimming can balance the paths very accurately, the small difference in the Bragg wavelengths of the two gratings remains. Despite this error, a rejection of greater than 30 dB has been reported [43] and is certainly not an easy achievement, requiring the matching of the Bragg wavelengths to 0.01 nm. The theoretical spectra shown in Fig. 6.20 are in excellent qualitative agreement with the results of low-insertion-loss band-pass filters fabricated in optical fiber with matched gratings in the close proximity of a coupler [43]. Chirped gratings can be used in the Michelson to broaden the pass bandwidth [44]. With further mismatch in the Bragg wavelengths, the rejection in port 1 deteriorates. One can no longer have a high isolation, since there is less overlap between the bandwidths of the gratings. The conditions approach the case of a single grating in the coupler arm, when there is no overlap of the grating spectrum. Thus, at least 25% of the input power appears at both ports 1 and 2. While the rejection becomes poor, the band-pass suffers because the bandwidth decreases as a direct result of the limited overlap. Figure 6.21 shows the reflection spectra of gratings in a Michelson that have been detuned by one-quarter of the unapodized FWFZ bandwidth (to the first zeroes). The refractive index modulation amplitude is 1 10–3 and the gratings are 4 mm long. Since the interferometer has been detuned as a result of the difference in the Bragg wavelengths, at zero phase difference the band-pass output is not at its maximum
The Michelson Interferometer Band-Pass Filter
239
1.0 0.9 0.8
Reflectivity
0.7 0.6
RX G1 RX BP3
0.5
Output (pi phase diff) Output (0 phase diff) Output (pi/2 phase diff)
0.4 0.3 0.2 0.1 0.0 –6.0E–4
–4.0E–4
–2.0E–4
0.0E+0
2.0E–4
4.0E–4
6.0E–4
Normalized detuning, (l–l Bragg )/l Bragg Figure 6.21 The band-pass characteristics show that with slight detuning (0.25 FW bandwidth) between the two Bragg gratings, a slight reduction in the peak transmissivity occurs (crosses). However, there is an added benefit: reduction in the energy transmitted in the wings of the gratings, i.e., apodization occurs. Note that with a larger path length difference, “bat-ears” appear on either side. These normally appear in the rejected port 1 (squares).
(small crosses). As the path difference is changed to p/2 radians (triangles), a dip begins to appear in the band pass, and with p phase difference (squares), “batears” begin to appear, since at the edges of the band pass, there is little overlap of the reflected spectra. Note that at some Bragg wavelength detuning, the interference of the reflected light at the coupler forms moire´ fringes, and apodization of the band-pass spectrum begins to occur. In Fig. 6.21 (crosses), note the reduction of the side lobes as the dissimilar phases in the overlapped spectrum tend to cancel the formation of the side lobes. The filter rejection becomes worse when the detuning is one-half of the FW bandwidth, as shown in Fig. 6.22. In this case, the zero path difference is well off the optimum for band-pass operation (triangles), while squares show the band pass response at p phase difference between the gratings. Although this spectrum is still not the optimized output, note the strong apodization in the wings. The nonoverlapped high-reflectivity (100%) region (within the bandwidth of each grating) averages to approximately 25% of the input power, as in the case of the single-grating Michelson device.
Fiber Grating Band-Pass Filters
240 1.0 0.9 0.8
RX G1 RX G4 G4 + G1 (pi phase diff) G4 + G1 (pi/2 phase diff) G4 + G1 (0 phase diff)
Reflectivity
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 –6.0E–4
–4.0E–4
–2.0E–4
0.0E+0
2.0E–4
4.0E–4
6.0E–4
Normalized detuning, (l – lBragg ) / lBragg Figure 6.22 Two gratings detuned by approximately 0.5 bandwidth of the grating. The band-pass characteristics are sensitive to the path-length difference between the Bragg reflection peaks due to differential phase response of the gratings. With detuning, the optimum band-pass shifts from the normal zero phase difference for matched Bragg wavelength case and develops additional structure, although apodization occurs, reducing the reflection in the wings.
6.3.1 The Asymmetric Michelson Multiple-Band-Pass Filter Figure 6.19 showed how output power in port 1 varies with path difference dl. The reflected power within the entire grating spectrum is exchanged between port 1 and 2 so long as the detuning dl
l2 ; 2Dlg
ð6:3:9Þ
where Dlg is the FWFZ bandwidth of the grating. With larger path differences, neffDLf ¼ neff(Lf1 – Lf2), the phase variation d as a function of wavelength, according to Eq. (6.3.4), becomes substantial across the bandwidth of the grating. Thus, the single uniform band pass of the filter begins to split into a sinusoidal response with wavelength, restricted to the bandwidth of the grating [42]. Figure 6.23 shows the reflection spectrum of an apodized grating MI and the band-pass output of the filter with a path difference of 0.667 mm.
The Michelson Interferometer Band-Pass Filter
Reflectivity, dB
–0.0015 0
–0.001
–0.0005
0
241 0.0005
0.001
0.0015
–10 –20 –30 –40 –50 –60 Normalized detuning, (l Bragg –l)/l Bragg
Figure 6.23 The reflectivity and band-pass spectrum of the asymmetric Michelson interferometer. The one-way path imbalance is 0.667 mm and the apodized gratings are 4 mm long with a Dnmod of 1 10–3 [42].
Within the reflection spectrum of the grating, the band pass has three peaks. Each peak automatically has the maximum transmission possible for the band pass, i.e., determined by the reflectivities from the gratings. With the detuning shown in Fig. 6.24, nine peaks appear within the same bandwidth of approximately 0.0005 detuning. Being a nonresonant device, the output is simply equivalent to the interference between two beams as a function of phase difference, hence the sinusoidal variation, also the case with the low-finesse FP interferometer. The wavelength difference dl between the peaks of the band pass is dl ¼
–0.0015 0
–0.001
l2 ; 2neff DLf þ DfðlÞ
–0.0005
0
0.0005
ð6:3:10Þ
0.001
0.0015
Reflectivity, dB
–10 –20 –30 –40 –50 –60 Normalized detuning, (lBragg−Dl) /lBragg
Figure 6.24 The apodized reflection (dashed line) and band pass (continuous line) of the asymmetric Michelson interferometer, with a path difference of 2.67 mm. The output is sinusoidal as in the case of a low-finesse FP interferometer.
Fiber Grating Band-Pass Filters
242
where Df(l) is the differential phase of the two gratings, which becomes important when the gratings are dissimilar, for example, chirped. Using Eq. (6.3.10) one can calculate the exact number of pass bands within the bandwidth of the gratings. Note that with Bragg-wavelength detuned gratings, the resultant bandwidth for the pass bands is the difference between the individual bandwidths of the gratings. The measured response of such a filter is shown in Fig. 6.25. The extinction is 28 dB. There are three possible combinations for arranging chirped gratings in the Michelson interferometer: both gratings with the same sign of the chirp, either both positive or both negative, or with opposite chirp. In Fig. 6.26, the first two arrangements are shown (A and B). The difference between A and B is that the dispersion of the gratings has been reversed, and that a pair of such filters may be used to compensate for most of the dispersion in each filter.
Normalized reflectivity/ transmission
1540 1.0
1542
1544
1546
1548
1550 BP
0.8 Grating 1 & 2 0.5
0.3
0.0 Wavelength (nm) Figure 6.25 The reflection spectrum of each grating (1 and 2) measured in situ bending the fiber in the other port to induce loss: hence the 25% reflection. The normalized band-pass filter spectrum (BP) as a result of a path difference DLf of 1.33 mm [42].
ΔLf Port 1: INPUT
50:50
Chirped gratings Port 3: 50% Tx A
Port 2: OUTPUT lBragg
Port 4: 50% Tx
B Figure 6.26 Chirped gratings in configurations A and B, each with a path imbalance of DLf but with reversed sign of the chirp.
Reflectivity
The Michelson Interferometer Band-Pass Filter 1 0.8 0.6 0.4 0.2 0 1545
243
Output RX
1550
1555
1560
1565
Wavelength, nm Figure 6.27 Band-pass and reflection spectra with two identically chirped gratings (Lg ¼ 5 mm, chirp ¼ 10 nm, and DLf ¼ 1.724 mm).
The transmission band pass of the identical-sign, linearly chirped grating Michelson interferometer with DLf ¼ 1.724 mm is shown in Fig. 6.27. The gratings are 5 mm long with a chirped bandwidth of 10 nm. The unapodized gratings have a Dnmod of 1 10–3. The pass bandwidths of the channels are all identical and equal to the width of the channel spacing. The repeated pass bands can be adjusted by altering the path imbalance; in a demonstration, the paths were adjusted by stretching one arm of the interferometer to fit a grid of 1.1 or 2.2 nm using two 15-nm-bandwidth chirped gratings [45]. When the sign of the chirp of one of the gratings is reversed, the band-pass transmission characteristics change from regular repeated pass bands to a variable pass band. The path difference between the two gratings is reduced to zero at some wavelength. A gap opens at this point and the transmission is no longer uniform as in the previous cases. Ignoring dispersion, the phase difference between the light reflected from these two weakly reflecting, chirped gratings with a chirp of Dlg nm relative to the wavelength l0 in the center of the grating, is 0 1 0 1 4pneff Lg @l0 lA 4pneff Lg @l l0 A 4pneff DLf dðlÞ ¼ þ l l Dlg l Dlg 2 0 1 3 4pneff Lg 4 @l0 lA DLf 5 ¼ 2 þ ; l Lg Dlg
ð6:3:11Þ
where Lg is the length of the grating. It is apparent from Eq. (6.3.11) that the detuning d ¼ 0 when l ¼ l0
Dlg DLLf : 2Lg
ð6:3:12Þ
If DLf ¼ 0, d ¼ 0 at the wavelength l ¼ l0. For a fixed chirp bandwidth, the detuning can be reduced to zero at any wavelength within the bandwidth of the
Fiber Grating Band-Pass Filters
244
grating by tuning DLf. In Eq. (6.3.11) we have assumed that the lengths of the two gratings are identical. This need not be the case; it is enough that the bandwidths are identical, so that we have the extra parameter that can be adjusted. This also applies to the previous cases in which the sign of the chirp for both gratings was identical. The pass-band period is now chirped, since the variation in the detuning is no longer constant close to the phase-matching wavelength. Figures 6.28 and 6.29 demonstrate the effect of the counter-chirp of the gratings for the more general case of dissimilar length gratings. At the top of Fig. 6.28 is shown a schematic of the third combination for the chirped gratings (C) and the relative positions, orientations, and lengths of the gratings. The first case is for 10- and 5-mm-long gratings, each with a chirped bandwidth of 10 nm
Lg1 Lg2
C
Reflectivity
1
Output of asymmetric Michelson interferometer with 10nm, 5 & 10 mm long oppositely chirped gratings (03.448 mm path difference)
0.75
RX, grating Output port
0.5 0.25 0 1545
1550
1555
1560
1565
Wavelength, nm Figure 6.28 Above the chart is shown the arrangement of the gratings (C) used in this simulation of the asymmetrically placed grating Michelson interferometer BP with identical chirp bandwidth but dissimilar lengths (5 and 10 mm). Notice the chirp in the period of the pass bands. The reflectivity of one of the gratings is also shown.
Reflectivity
Output of asymmetric Michelson interferometer with 10 nm, 5 & 10 mm long oppositely chirped gratings (3.448 mm path difference) 1 0.75
RX, grating Output port
0.5 0.25 0 1545
1550
1555
1560
Wavelength, nm
Figure 6.29 The band-pass response with a 3.448-mm path difference.
1565
The Mach–Zehnder Interferometer Band-Pass Filter
245
and with a detuning of 0.344 mm (Fig. 6.28). The gap opens up close to one long-wavelength end of the grating, while with a larger DLf of 3.44 mm in the second case, the gap shifts to the short wavelength end (Fig. 6.29). In either case, moving away from the phase-matching point [Eq. (6.3.12)], the oscillations in the transmission spectrum become more rapid. There are important issues relating to the asymmetric grating Michelson interferometer band-pass filter. First of all, any interferometer is sensitive to differential temperature and strain. The asymmetric interferometer is especially so; however, with fiber leads to the gratings kept in close proximity, the only region, which needs stabilization, is the differential path. For sensing applications, the broad bandwidth of the chirped grating is a distinct advantage. Stabilization of the paths may be done in a number of ways, for example by the application of a special polymer coating [46,47], or use of a substrate to compensate for the thermal expansion [48]. The filter has very high extinction, but the stability is polarization sensitive, and the transfer characteristics are sinusoidal with wavelength. The period and chirp are easily designed into the filter. Applications may be found in signal processing, sensors, multiplexing, and spectral slicing within a well-defined bandwidth. Ideally, it would be suited to fabrication in fused fibers, close to the coupler, or in planar form.
6.4 THE MACH–ZEHNDER INTERFEROMETER BAND-PASS FILTER The dual-grating Mach–Zehnder interferometer band-pass filter (GMZI-BPF) overcomes the severe limitation of the Michelson interferometer filter – the loss of 50% of the through transmitted light – by recombining the output at a coupler, as shown in Fig. 6.30. The scheme was proposed by Johnson et al. [49], and using etched gratings in a semiconductor waveguide Mach–Zehnder interferometer, Ragdale et al. [50] were able to show a device operation. A major drawback of a device fabricated in this way is the high intrinsic losses due to scatter, absorption, and input/output coupling, although large bandwidths are possible due to the use of short gratings resulting from the large modulation index of the grating (air and semiconductor). Additionally, once the device has been fabricated, phase adjustment between the guides to balance the interferometer is difficult without active control. The first demonstration of a working band-pass device using the principle of the Mach–Zehnder interferometer (MZI) with two identical UV written gratings was in planar-Ge:silica waveguide form [53]. The device was an MZI with overclad ridge waveguides, which had been photosensitized using hot-hydrogen treatment [51]. “UV trimming” was used to balance the interferometer after the gratings were written, demonstrating this powerful technique also for the
Fiber Grating Band-Pass Filters
246
UV trimming for path equalization
Input
ADD 50:50 Drop
Identical reflection gratings 50:50
Mach−Zehnder interferometer Wavelength
Output BAND-STOP
Insert Wavelength
Figure 6.30 The Mach–Zehnder interferometer used as a band-pass filter. UV trimming of the paths has been shown to be a powerful tool for rebalancing the paths such that 100% of the light reflected from the gratings appears at the output port on the left [53,43]. By adjusting the phase difference at the coupler beyond the gratings, the output may be directed to either output port of the coupler.
first time [53]. This is shown Fig. 6.30. “UV trimming” relies on photoinduced change in the refractive index to adjust the optical path-length difference. The 6-dB insertion loss for a single-grating band-pass filter was overcome and reduced to 1.34 dB for the fiber pigtailed device, by UV trimming; much of it comprised coupling and intrinsic waveguide loss. The fiber gratings had a reflectivity of 15 dB each and were well matched in wavelength. Approximately 10% of the light was reflected into the input port. Although the insertion loss of this MZI-BPF was not as low as later devices, the planar MZI has the advantage of being extremely stable to environmental effects. Since the demonstration, several groups used this scheme of UV trimming in fiber-based MZIs to demonstrate band-pass filters with better extinction and insertion loss [43,52]. For proper operation, the output coupler needs to be balanced, requiring trimming on the RHSs of the gratings. Indeed, it is simple to observe that the device can be used as an add multiplexer at the dropped wavelength if the same wavelength is locally injected at the port marked “Insert” (RHS, bottom). This wavelength will be routed through to the Add port performing the basic Add–Drop multiplexer function. Many of these MZIs may be cascaded to perform a multiple-wavelength bandpass function. Cullen et al. [52] demonstrated a compact GMZI-BPF in fiber form. The device, based on two 50:50 splitting fused fiber couplers fabricated in boron– germania codoped fibers (core–cladding index difference Dn ¼ 7 10–3 and core diameter of 7 mm), with 1-meter tails. The two pieces of fiber were first tapered and fused to a constant diameter of 100 mm over a length of 20 mm. A 3-dB coupler was formed by further tapering one end of the fused region,
The Mach–Zehnder Interferometer Band-Pass Filter
247
until the desired splitting ratio of 50% was achieved. When the second coupler is made, if the path lengths in the two arms are identical, 100% of the light will appear in the crossed state, i.e., in port 4 when port 1 is excited. Allowing for fabrication loss and slight imbalance, between 95 and 99% of the light was available at port 4 after the second coupler was fabricated under the same conditions. The finished device had 5 mm of space in the parallel fiber section between the couplers for the inscription of the gratings and for UV trimming. The advantage of such a structure is the relative stability of the MZI, since the couplers and the fused fiber regions are so close together. Any ambient temperature fluctuations affect both fibers equally. This was established by a measured change in the output power of the MZI of <0.05 dB over a temperature excursion of –20 to þ60 C, with a wavelength window of 40 nm. It is necessary to mount the fibers on a mechanical support in order to proceed with grating inscription. A silica microscope slide is ideal for this application, since it enables the device to be supported, handled, and aligned in the interferometer. For this device, gratings of 3-mm length were written in both arms under identical conditions using an intracavity CW frequency-doubled argon ion laser. The grating reflectivity can be monitored accurately by the size of the dip from the transmitted level at a few nanometers on the long-wavelength side of the Bragg wavelength (to avoid the radiation loss region on the blue side, as well as the side lobes on the red side of the grating spectrum). Once the two gratings are written using identical conditions, the device can be balanced by examining the reflection in port 1. Ideally, either a coupler or a circulator may be used to monitor the reflection while optical trimming is undertaken in the region closest to port 1, to minimize the reflection at the Bragg wavelength [53]. Lastly, the output ports need to be balanced by trimming on the far side of the gratings. The power on the long-wavelength side of the Bragg wavelength can be steered to either port 3 or 4 as necessary. In their device Cullen et al. [52] reported an insertion loss of only 0.5 dB, and 0.35 dB for a standard fiber device. With 99% reflectors, strong radiation loss was noted on the blue side of the Bragg wavelength. The long-term stability of the GMZI-BPF depends on the stability of the substrate and the uniformity of the stress and temperature gradients. Silica for the substrate is a good choice since it is better matched to the properties of the fiber. However, the Bragg wavelength in Ge:doped fiber shifts by 16 pm/ C. Compensation of the drift can be countered by the use of packaging with an effective negative thermal expansion coefficient [54–56,47,48]. Athermalization using a b-eucrytite glass substrate has been shown to reduce the temperature sensitivity of fiber Bragg gratings to 0.0022 nm/ C, and the drift in the wavelength was measured to be 0.02 nm after thermal cycling 60 times over a temperature excursion from –40 C to þ85 C [56]. The glass is based on a stuffed
248
Fiber Grating Band-Pass Filters
derivative of the crystalline phase, b-quartz (LiO2:Al2O3:2SiO2). Normally, this phase has a large negative thermal expansion coefficient along the c-axis but forms a poor glass. Nucleated with TiO2 and adjusting the glass composition to form b-eucrytite results in a stable glass melt. Heat treatment at above 1200 C forms the glass ceramic with the appropriate crystalline micro-structure for a negative thermal expansion coefficient. The principle of athermalization of delay through a fiber using a tube of oriented liquid-crystalline polymer [54] has also been used to athermalize gratings. A measured dlBragg/dT of 0.01 nm/ C for the uncompensated FBG was reduced to 0.13 nm/100 C after compensation, with no significant hysteresis during the temperature cycling from –40 to þ80 C [47]. Use of such materials as substrates for the GMZI-BP should result in robust devices.
6.4.1 Optical Add–Drop Multiplexers Based on the GMZI-BPF Changing the phase in one of the arms of the MZI between the coupler and the gratings routes the reflected light to either the “drop” or the “input” port. Any method that can reliably alter the phase can be used to switch the OADM. The parameters that can be altered are temperature [40,41] or strain to alter the phase difference between the arms of the MZI, or the Bragg wavelength of the gratings. To alter output state, the phase in one arm can be tuned reliably and requires a 10-mm length of fiber to be heated by 13 C. With strain, a fiber extension of 1/4lBragg is required to switch the OADM. Temperature tuning of the gratings requires a 65 C change to shift the fiber Bragg grating wavelength by 0.8 nm. If the channels are spaced 1.6 nm apart, then a channel may be dropped by tuning the Bragg wavelength to match the channel wavelength and deselected by detuning. Tuning both gratings simultaneously by either strain or temperature while maintaining interferometric stability is not easy, so, this device requires careful engineering. The GMZI-BPF can be used as an optical add–drop multiplexer (OADM). If the phase in one of the arms can be controlled actively, e.g., by a piezoelectric stretcher, then a wavelength may be either switched to the drop port or reflected back to the source. The insert function is operated in a similar manner by the use of a second piezoelectric stretcher on the RHS of one of the gratings in the MZI. The “drop” and “add” ports have fiber-coupler taps to monitor the state of the output and to control the piezoelectric stretchers to switch the GMZI-BPF using phase-locked loops [40]. A disadvantage of this scheme is that it always blocks the transmission of the channel, whether it is dropped or not, and it must be reinserted for forward transmission.
The Mach–Zehnder Interferometer Band-Pass Filter ll , ... li ,lj , ... ln
249
Termination port
lj B1
A1
FG(5)
MZ-FG(2)
3dB CPL
li li
MZ-FG(3)
LINE B
LINE A
MZ-FG(1)
li MZ-FG(4)
B2
A2 FG(6)
lj
ll , ... li ,lj , ... ln lj
li
C2 Bragg grating (li)
LINE C
li
C1 Bragg grating (lj)
Figure 6.31 A two-wavelength cascaded GMZI-BPF with increased isolation between the add and drop ports (from: Mizuochi T. and Kitayama T., “Interferometric cross talk-free optical add/drop multiplexer using cascaded Mach–Zehnder fiber gratings,” in Technical Proc. of OFC ’97, pp. 176–177, 1997) [57].
Mizuochi and Kitayama [57] combined a set of GMZI-BPFs to perform a two wavelength OADM function, which is shown in Fig. 6.31. The basic element of the device is a double GMZI-BPF with four identical gratings in the two MZIs, as well as an additional highly reflecting grating as an “isolator” between the two MZIs, shown in the top half of Fig. 6.31. The function of the additional grating is to prevent light at the grating Bragg wavelength from crossing from one MZI to the other, increasing isolation. This is particularly important because light inserted into the OADM can cross from one MZI to the other (from left to right and the reverse) to cause in-band coherent beat noise [58] (see Section 6.5). Light arriving from the left in the top half at port A1 is dropped at the Bragg wavelength li and routed to the second GMZIBPF in the lower half of the figure, containing gratings at another wavelength, lj. The dropped wavelength li therefore appears at C1. Similarly, light injected in C2 at the wavelength li uses the “add” part of the top GMZI-BPF on the RHS and is routed to B2. All other wavelengths arriving at A1 simply go through the gratings and also appear at B1. Similarly, channels arriving at B2
Fiber Grating Band-Pass Filters
250
are routed to A2, except for the channel at lj, which is dropped at C1 and reinserted from C2, to be routed to A2. The inclusion of the “isolating” grating reduces the leakage between A1 and B1 from –26 to –71 dB. The improvement is evident in the add–drop function, by a reduction in the power penalty of 1 dB due to the elimination of coherent beat noise. The use of higher reflectivity gratings in the MZIs should eliminate the need for the “isolator” grating, although short-wavelength radiation loss will remain a problem.
6.5 THE OPTICAL CIRCULATOR-BASED OADM The optical circulator has become extremely important for applications with fiber gratings. While the use of a fiber grating with a circulator is an obvious method for converting the band stop to a band-pass filter, it is worthwhile to consider the benefits of such a configuration. The first reported use was as an ASE filter for an erbium amplifier [59]. The amplified signal is routed to the input of a circulator and reflected by a narrow-bandwidth grating in the second port. The grating filters the amplified spontaneous emission from the amplifier and routes the signal to the output port. The reduction in the out-of-band spontaneous emission can be considerable, but is determined by the quality of the reflectivity spectrum of the grating. Optical circulators have an insertion loss of only 1 dB, turning this very simple device into a superb band-pass filter. Several gratings with different Bragg wavelengths may be cascaded to form a multiple band-pass filter. The addition of a second circulator leads to a simple method of performing an optical circulator based add-drop multiplexing (OCADM) function using gratings and is shown in Fig. 6.32. Channels injected at the “input” port are reflected by the gratings in between the two circulators OPTICAL CIRCULATORS
INPUT l1,l2,l3
DROP l1....
Bragg gratings at l1.....................ln
OUTPUT l1,l2,l3
INSERT l1
Figure 6.32 An OC-ADM using an all optical circulator. This device allows several channels to be dropped or added according to the number of fiber gratings between the circulators. The signals at the Bragg wavelengths are reflected and appear at the drop port, while the same gratings may be used to insert the same channels for wavelength reuse from the insert port.
The Optical Circulator-Based OADM
251
Transmission, dB
and routed to the “drop” port. All other wavelengths continue to the “output” port. If the signals are injected at the “insert” port on the RHS of Fig. 6.32, the same gratings perform an insert function, routing the reflected channels to the “output” port, along with the rest of the channels from the “input” port. The low polarization sensitivity and extremely high return loss of the circulators are a distinct advantage for this type of function, despite their insertion loss of 1 dB. Since the circulator-fiber grating band-pass filter is not interferometric, it is intrinsically stable in its operation, but remains an expensive solution for some application. However, for amplified long-haul fiber communication routes with large capacity (e.g., submarine systems), the cost of a few of circulators is unlikely to be an overriding factor. The issues that need to be addressed with the OC-ADM are the channeldependent insertion loss, intra- and cross-channel cross-talk, and the dispersion penalty due to the bandwidth of the gratings. Channel-dependent loss is primarily due to the “blue-wavelength” radiation loss exhibited by all fiber Bragg gratings. Thus, reflected light from the shorter-wavelength gratings may experience loss either in the drop or in the add function, unless the gratings are fabricated with care. Figure 6.32 shows a sequence of gratings reflecting at l1, l2 . . . ln. The gratings are arranged so that the first grating reflects at the shortest wavelength and the last, ln, at the longest. A highly reflective grating transmission spectrum is shown in Fig. 6.33. The loss on the blue-wavelength side extends over a wide bandwidth. If each grating has similar transmission characteristics, spaced, say, 10 nm apart, the blue-wavelength radiation loss will increase with each additional grating. Light injected from the short wavelength side (e.g., from the “input” port) is reflected in sequence, so that the shortest wavelength is reflected first, and then the second shortest, and so on. Each wavelength ln has to traverse n – 1 gratings reflecting at a wavelength shorter than ln. However, since the loss in traversing the grating is always on the short-wavelength side of the Bragg wavelength, each wavelength is reflected without incurring radiation loss. 1520 0 –5 –10 –15
1530
1540
1550
1560
1570
1580
–20 –25 –30 Wavelength, nm
Figure 6.33 The measured transmission spectrum of a 7-nm full-bandwidth, highly reflecting chirped grating with the associated “blue”-wavelength loss extending over almost the entire gain bandwidth of an erbium amplifier.
252
Fiber Grating Band-Pass Filters
The situation is different if the order of reflection is reversed, as is the case when the light is injected into the “insert” port. The longest wavelength is reflected first and so does not incur loss. The next wavelength (in this example, at 1550 nm) to be reflected lies on the short-wavelength side of the reflection shown Fig. 6.32 and so suffers twice the radiation loss a of approximately 6–8 dB. Again, each wavelength ln has to traverse n – 1 gratings reflecting at longer wavelengths and suffers a loss of (n – 1) a dB. This loss causes severe skew in the wavelength response of the OADM. Although the radiation loss spectrum shown in Fig. 6.33 is very high and has a large bandwidth, the insertion loss argument applies to all such OADM devices. The insertion loss varies across the bandwidth but is always cumulative in one direction and can be more important for wavelengths that have not been dropped but lie on the short-wavelength side of the OADM. There are methods that can be used to reduce the radiation loss of the gratings, for example, by design of the fiber [60,61] to suppress radiation mode coupling, or by choosing the channel spacing such that none lies in the radiation loss region [17]. Intrachannel cross-talk occurs if the added channel is derived from the same source as the dropped channel and leaks through from the “insert” to the “drop” port. If a strong signal is added at the OADM, a small fraction leaks through to the drop port if the grating reflectivity is not very high. This small “breakthrough” can be of the same order of magnitude as the signal, which has been attenuated by the link loss and dropped at the OADM. If the added and dropped wavelengths are the same, signal beat noise occurs and degrades the bit error rate (BER). Alternatively, the breakthrough signal can cause simple cross-talk between adjacent channels that are dropped, in the same way it can be transmitted to the “add” port. The associated power penalty has been calculated to be pffiffi ð6:5:1Þ pcoherent ¼ 10 log 1 4 r for a ratio r of the leaked “add” and the dropped signal powers for coherent signals. This is much higher than for incoherent signals [62], for which pincoherent ¼ 10 logð1 rÞ:
ð6:5:2Þ
Dispersion due to the gratings is an issue, which depends on the signal and grating bandwidths. Gratings show strong dispersion at the edges of the reflection band (see Chapter 4). If the signal fills the grating or is severely limited by the grating bandwidth, apart from the simple pulse broadening (narrowed spectrum), additional dispersion from the band edges of the grating has the potential of causing severe pulse distortion. This area has only recently received attention [63]. Programmability of an OADM is often desirable, and there are several techniques available to integrate this feature. As has been seen with the
The Optical Circulator-Based OADM l1
IN
253 l3
l2
l4 ADD
DROP Circulator
l4
l2
l3
l1
Circulator
Heater controller Figure 6.34 Dynamic wavelength selective add/drop mux-demux using tunable gratings (after Ref. [64]).
GMZI-BPF, the most easily adjustable parameter is the path length difference from the coupler to the gratings, since tuning the Bragg wavelength of both gratings poses a difficult engineering problem. The OC-ADM is not an interferometric device, so that tuning of the gratings by stretching/compressing or heating is easily achieved. Quetel et al. [65] demonstrated this principle with a circulator and four gratings stretch tuned by piezoelectric (PZT) actuators, with a switching time of 40 ms and a voltage of only 50 V. Okayama et al. [64] proposed the use of a pair of identical gratings for each add/drop channel with two 4-port optical circulators for an OC-based tunable add–drop multiplexer (OC-TADM). Figure 6.34 shows the arrangement of the gratings for a four-channel OC-TADM. Channels arriving at the input port are reflected by the appropriate gratings, l1, l2, l4 in the top part of the circulator branch, group 1. Gratings 2 and 4 are tunable (by either temperature or strain); however, all channels are reflected and routed to the second set of gratings, l4, l2, l2, l1 in the bottom part of the circulator branch, group 2. In this section, gratings 3 and 1 are tunable. If each grating pair has identical Bragg wavelengths, all channels are routed to the drop port by reflections from group 2 gratings. However, since one of the gratings of the matched pair is tunable, it can be detuned, thus directing the channel at that wavelength to the through port. Since the device is symmetric, the channel insert function is performed in a similar manner: When injected into the “add” port, the wavelengths are routed to the “through” port. Poor peak and high side-lobe reflectivity cause cross-talk. The use of unapodized (–14 dB side-lobes), 95% peak reflectivity gratings resulted in poor cross-talk performance. With well-apodized, high-reflectivity gratings, low cross-talk performance is possible and the OC-TADM is appropriate for dense-WDM applications. In principle, piezoelectric stretchers can be used to make a fast OC-TADM, switchable in <1 ms [65].
Fiber Grating Band-Pass Filters
254
In a slightly simpler arrangement, Kim et al. [66] proposed the use of four identical gratings between two three-port circulators. The gratings are stretch tuned by piezoelectric stretchers, so that up to four channels can be dropped or inserted in any combination, when the Bragg wavelengths of the gratings are tuned to the channel wavelengths. Mechanical leverage designed into the grating mounts with the piezoelectric stretcher allows the Bragg wavelength of each grating to be tuned by 2.4 nm/120 V applied.
6.5.1 Reconfigurable OADM These devices are based on optical switches and circulators and overcome some of the limitations of the OC-TADM and the GMZI-BPF. A schematic of the reconfigurable OADM (ROADM) is shown in Fig. 6.35. Two fiber circulators sandwich a set of optical cross-connect switches connected in series. Fiber gratings at the optical channel wavelengths are connected between one of the output ports of the cross-connect switches. This way, the incoming signals may be switched to the grating or bypass it. When switched to the grating, the channel at the grating wavelength l1 is dropped and routed to the drop port. All other channels proceed to the next switch, where the choice is repeated for the other channels. If, however, l1 bypasses the first grating, it goes on to the through port. The same applies to the other channels. At each two-by-two optical cross-point switch, a channel (or more, depending on the number of gratings between the switches) may be dropped. The ROADM is extremely flexible, allowing the node to be programmed relatively fast (<50 ms). The insertion loss of the optical switches is low (0.7 dB), and the device has been demonstrated as a two-channel ROADM [67], with a total insertion loss of 3 dB. The equalizing filters shown in Fig. 6.35 are intended to compensate for the loss of the switches and the gratings so that all channels suffer the same insertion loss.
2 ⫻ 2 Optical switches Optical circulator Input
l1 Equalizing filter DROP PORT
Optical circulator
l2 Equalizing filter INSERT PORT
Figure 6.35 The reconfigurable optical add–drop multiplexer (ROADM) (after Ref. [67]).
The Polarizing Beam Splitter Band-Pass Filter
255
Through path Output
Insert Input
1563 nm 1549 nm 1552 nm 1557 nm
Drop port Figure 6.36 The RDC-ADM based on optical switches (after Ref. [68]).
A modification of the ROADM shown in Fig. 6.35 results in the simultaneous add–drop function, including automatic dispersion compensation of all channels, using dispersion compensating gratings (DCG) [68]. The channel dropping gratings are replaced by chirped dispersion compensating gratings. Uniform unchirped gratings can be used from either direction, whereas the dispersion of chirped gratings is reversed if it is turned around. Therefore, the chirped gratings cannot simply replace the unchirped gratings in the ROADM. Channels to be dropped are compensated for dispersion in the ROADM with DCGs. In order to dispersion compensate the “through” channels, the optical circuit has to be altered. The modified reconfigurable dispersion compensating ADM (RDC-ADM) is shown Fig. 6.36. The “through” channels are routed via the “through path” to the output circulator and reinjected into the cross-point switches. Each “through” channel is routed to the appropriate grating via the switches and dispersion compensated by reflection, to retrace the path to the circulator, and finally to the output. Channels are also inserted without reflection; they simply bypass the DCG allocated to their channel, so as to avoid the additional dispersion. The four-channel RDC-ADM was demonstrated with a 4 10 Gb/sec WDM system and used apodized 100mm-long gratings for each channel designed to compensate for the dispersion of 8 km of standard fiber (1312 psec/nm). An important feature of the configuration was the arrangement of gratings and amplifiers to equalize the insertion loss of the switches. The gratings are placed in the RDC-ADM such that the channel at the maximum gain of the amplifier is placed between the last pair of optical switches, and the lowest gain channel is reflected first [68].
6.6 THE POLARIZING BEAM SPLITTER BAND-PASS FILTER A filter based on an all-fiber polarization dividing coupler [69] changes the Michelson BPF into a noninterferometric device. The polarization splitting coupler allows only one of the orthogonally polarized eigenmodes to couple across
Fiber Grating Band-Pass Filters
256
to the other fiber. The coupler is fabricated to be an integral number of coupling lengths for one of the polarizations. Since the supermodes of the coupler have slightly different propagation constants, they have slightly different coupling lengths, lc(TE) and lc(TM), for the TE and TM polarized modes. By making the coupling region long, it can be arranged to be an odd number of coupling lengths for one polarization and an even number for the other. This ensures that only a single polarization couples across the coupler, while the other remains in the same fiber. Thus, the number of coupling lengths, Nc is Nc
lc ; Dlc
ð6:6:1Þ
where Dlc ¼ jlc(TE) – lc(TM)j, and lc lc(TE) lc(TM). As a result of overcoupling, the device is wavelength sensitive, with a cyclic coupling response shown in Fig. 6.37. It has excellent polarization isolation over a narrow band of wavelengths but may be designed to operate at any desired wavelength by adjusting the parameters at the time of fabrication. Figure 6.38 shows a schematic of the polarization splitting coupler in operation with Bragg gratings in ports 3 and 4. When operated at the correct wavelength, the polarization splitter will cross-couple only one of the two orthogonal polarization states in the input port 1, while the other propagates unaffected. If gratings are placed at the output ports of such a coupler, light at the Bragg
1460 0
1480
1500
1520
1540
1560
1580
1600
1620
1640
Transmission, dB
–5
–10
–15
–20
Normalized V-Pol, dB Normalized H-Pol, dB
–25 Wavelength, nm Figure 6.37 The transmission characteristics of the two orthogonal polarization states [72] at the output of the polarizing beam splitter. The extinction is well over 30 dB.
The Polarizing Beam Splitter Band-Pass Filter Ey
257 l/4PC
Ei
Port 3
Ex Input port 1
PBS Chirped gratings
Output port 2 E out
l/4PC
Port 4
Figure 6.38 The polarization dividing band-pass filter in operation. The dashed arrows indicate the direction of propagation. The circles with arrows indicate the circularity of the polarization (after Ref. [72]) PBS is a polarizing beam splitter. l/4 PC is a quarter-wave polarization controller.
wavelength is reflected and both polarizations are coupled back to the input port 1. If a quarter-wave plate is placed just before a grating and oriented such that at the output of the wave plate it is left-circularly polarized, on reflection from the grating it becomes right-circularly polarized. Traversing the wave plate once more the linearly polarized output of the wave plate is orthogonal to the incoming polarization. At the coupler, this state of polarization remains uncoupled and is routed to port 2 of the coupler. The same applies to the orthogonal polarization state, which also couples to port 2. Thus, as in the Michelson arrangement, the device operates as a polarization-independent band-pass filter [72]. Each reflected polarization is routed to the output port of the coupler, reconstituting the original polarization, but is rotated by p/2. The experimental arrangement using a cascade of four chirped gratings with two polarizing beam splitter band-pass filters (PBS-BPFs) is shown in Fig. 6.39. The quarter-wave plates are simply a few turns of fiber in a polarization controller [70]. In this dispersion-compensating PDBPF, 0.47-ps pulses at 1562 nm were 0.55 km fiber D = ~9ps/nm Port 1: INPUT τin = 0.47ps
Matched chirped gratings: bandwidth: 12 nm
PBS
Port 8:OUTPUT l Bragg τout = 0.50ps
Matched chirped gratings: bandwidth: 12 nm
Figure 6.39 Dispersion compensation with four cascaded 8-mm-long, 12-nm bandwidthchirped gratings and two PDBPF. The two polarization splitting band-pass filters shown here have been cascaded to enhance the dispersive effect of the chirped gratings [71,72].
Fiber Grating Band-Pass Filters
258
stretched to 60 ps after propagation through 0.55 km of partially dispersion-shifted fiber (dispersion of 9 ps/nm). A pair of identical 8-mm-long chirped gratings with a bandwidth of 12 nm in the PDBPF routed the pulses to a second, identical DCGPBS-BPF. The output port of the second PBS-BPF recovered pulses of 0.50 ps, a recompression ratio of 60/0.5 ¼ 109 [72]. The PBS-BPF is a noninterferometric device but relies on the coincidence of the arrival of the pulses at the PDS after reflection. A delay between the arrival of the pulses translates into polarization mode dispersion (PMD). A change in the path of 1 mm is equivalent to 10 ps PMD. For such short pulses, the paths were matched to <0.01 mm by stretching the fiber. Polarization variation in each arm causes an amplitude fluctuation. This device remains reasonably immune from physical disturbance, so long as the fibers after the PBS are not disturbed. Since all the components of this device are based on optical fiber, it has low insertion loss (0.2 dB, typically, for the splitter). The Michelson interferometer becomes the GMZI-ADM when a second coupler is included after the gratings. The same applies to the polarization dividing filter with a second PBS after the gratings. However, a major difference is the intrinsic stability of the latter device, since interferometer stability is no longer necessary. BER performance of transmission systems will degrade significantly as the PMD approaches half of the bit period. For a low BER at a transmission rate of 10 Gb/sec, a maximum path imbalance of a few millimeters would be required, which is easily achievable. The disadvantage of the filter is the relatively small bandwidth of the PBS (see Fig. 6.36), so that channels can only be spaced close to the nulls of the PBS. A PBS-OADM has been demonstrated for a single channel using seven wavelengths spaced at 0.8-nm intervals. The center channel was dropped with a cross-talk penalty of 0.3 dB when the same wavelength was added at a transmission rate of 2.5 Gb/sec. Heating part of one arm of the PBS-OADM by 65 C induced a change of 0.3 dB at the output [73]. However, it remains to be seen how this device will function under full environmental testing. Figure 6.40 shows a schematic of the PBS-OADM [73]. INPUT l1..ln Port 1
DROP lb Port 2
PC
lb
PC
PBS
PC
lb
PC
INSERT lb PBS Port 3
OUTPUT l1./ lb ...ln
Figure 6.40 The polarizing beam splitter OADM. Multiple gratings between the PCs allow more than a single channel to be dropped and added simultaneously (after Ref. [73]).
In-Coupler Bragg Grating Filters
259
6.7 IN-COUPLER BRAGG GRATING FILTERS Co- and contradirectional wavelength selective couplers have been known for a long time [74,75]. There are a number of ways that gratings in-couplers may be used to form band-pass filters. Figure 6.41 shows three different types of couplers, which include gratings to assist (grating-assisted coupler, GAC), to frustrate (grating-frustrated coupler, GFC) and to reflect (Bragg reflection coupler, BRC) light of a particular wavelength that meets the phase-matching requirements. The period of the refractive index-perturbation for codirectional gratingassisted coupling [GAC, Fig. 6.41 (i)] between two dissimilar fibers is determined
Port 1: INPUT lall
Port 2
Coupler
Port 3: THROUGH: lall other
Port 4: DROP: lassisted Long Period Grating-assisted coupler (GAC) (i ) A1100% Coupler T3 Port 3: DROP: lfrustratedA3 Lc R1 T4
Port 1: INPUT lall Port 2
R2
Lg Port 4: THROUGH: lall otherA4 A2 Grating-frustrated coupler (GFC) (ii ) Port 1: INPUT lall A1 B1 Port 2: DROP: lBragg
IN
Coupler with Port 3: ADD identical gratings in :lBragg each arm A2 z B2 I II III
Port 4: THROUGH: lall other Two Bragg reflecting coupler (TBRC) (iii ) Pre-taper L1 L2 L3
Taper
Grating DROP lBragg
THROUGH: lall other
Fused fiber Bragg grating coupler (FFBRC) (iv) Figure 6.41 Some examples of band-pass filters with in-coupler gratings.
260
Fiber Grating Band-Pass Filters
by the difference in the propagation constants of the two guides. This is generally small, and therefore the period is long. For weak overlap of the fields, the coupled-mode equations (see Chapter 4 on long-period gratings) describe the interaction between the modes. The coupling between the guides is sinusoidally periodic with the length of the grating-assisted region. The coupling has a relatively broad bandwidth (tens of nanometers) and therefore poor wavelength selectivity, unless the device can be made very long. A normally 100% coupler is strongly detuned by the dispersion of the grating and so fails to behave as a coupler near the Bragg wavelength, and is called a grating-frustrated coupler [GFC, Fig. 6.41 (ii)]. It works on the following principle: Two fibers with identical propagation constants will exchange power at all except the “grating-frustrated” wavelength. The in-fiber grating is a Bragg reflector at the frustrated-wavelength and is present in only one of the fibers. The far end of the input fiber becomes the “drop” port and is the one that does not contain the grating. The Bragg reflecting coupler [BRC, Fig. 6.41 (iii) and (iv)] requires a perturbation with a short period, as is the case for Bragg reflection, being dependent on the sum of the magnitudes of propagation constants of the two modes. This device can made with either two gratings in polished couplers [76,77] or a single one written into a fused coupler [78], since the perturbation has to be present in the entire cross-section of the coupling region. The dropped channel is reflected and routed to port 2 of the fiber coupler. In the following sections, the characteristics of the latter two devices are presented.
6.7.1 Bragg Reflecting Coupler OADM The BRC-OADM [76–78] is probably the most promising of all the OADM devices that rely on interference. Essentially, this is a new twist to a range of generic devices based on the grating-assisted coupling action. Apart from being a simple device and having very low insertion loss, the BRC has the potential of fulfilling the requirements for a high extinction at the “dropped” as well as the “through” ports, and low back-reflection into the input port. Schematics of the BRC in the assembled and fused forms are shown in Fig. 6.41 (iii, iv). In its fused form, it comprises two fibers tapered down to form a long coupling region in which the fibers are kept parallel, followed by a short grating and another long coupling region before the fibers separate. The principle of operation is probably the cleanest of all the different types of grating couplers and may be understood in the following phenomenological way: The light input into port 1 propagates adiabatically in the tapered region
In-Coupler Bragg Grating Filters
261
to excite the supermodes of the coupler. In this region, the two fibers merge into a single strand and become a glass rod without a core, surrounded by air. In a normal coupler without the grating, 100% of the light is transferred from one set of modes to the other to exit at port 4. If, however, a point reflector is placed at exactly half the coupling length, then the divided power between the modes travels backward toward ports 1 and 2, and the coupling process continues uninterrupted, apart from the p-phase change induced by the reflection in all the supermodes. Thus, instead of propagation in the positive z-direction, the supermodes travel in the negative z-direction and interfere at the exit of the coupler and are routed by symmetry into port 2. In the BRC, the Bragg grating replaces the point reflector, which is wavelength selective, and routes light only near the bandgap into port 2. This simple picture is surprisingly accurate, despite the fact that coupling continues within the grating region, due to light penetration. It is immediately apparent that a strong grating would be preferable, although complications arise since the presence of the grating detunes the coupling action. The BRC can also be considered to be a close analog to the Michelson/Mach–Zehnder interferometer with “zero-length” paths. This will become clear as one compares the theoretical performance of the BRC with that of the Michelson with detuned Bragg grating wavelengths. One essential difference between the two devices is the single grating as well as only one common path in the BRC, as opposed to two gratings and two paths in the Michelson interferometer. Many of the features of the BRC are easily understood by comparing it with the Michelson [79].
Theory of the BRC The theory of the BRC has been worked out using coupled-mode analysis [76,80,77]. We closely follow the nomenclature of Ref. [77]. Referring to Fig. 6.41 (iii), the fibers and A and B have unperturbed propagation constants ba and bb, respectively, and the grating with a coupling constant of kac exists in both fibers, evanescently coupled with a coupling constant k. In region II, there are co- and counterpropagating modes, which are coupled together. The presence of the grating introduces a detuning of the propagation constants in each fiber, which are p ð6:7:1Þ Dba ¼ ba Lg Dbb ¼ bb
p Lg
ð6:7:2Þ
Fiber Grating Band-Pass Filters
262
1 Dbab ¼ ðba bb Þ 2 1 ¼ ðDba Dbb Þ: 2
ð6:7:3Þ
Coupling between counterpropagating modes of different fibers (A1/B2 and A2/B1) is only significant if the fibers are very strongly coupled, so that coupling occurs over a distance of a few wavelengths (when k is of the order of 106m–1). Thus, these interactions can be ignored in a majority of cases. The mode fields propagate with the modified propagation constants and may be expressed as
~1 ðzÞeiDba z A1 ðzÞ ¼ A
ð6:7:4Þ
~2 ðzÞeiDba z A2 ðzÞ ¼ A
ð6:7:5Þ
~1 ðzÞeiDbb z B1 ðzÞ ¼ B
ð6:7:6Þ
~2 ðzÞeiDbb z : B2 ðzÞ ¼ B
ð6:7:7Þ
In region II, the coupled-mode equations take the matrix form 2 0 3 3 2 32 A ~ ik 0 iDba ikac A ~1 6 10 7 ∗ 6 ~2 7 6A 7 6 0 ik 7 7 6 ~20 7 ¼ 6 ikac∗ iDba 76 A 6 7 4 ik 54 B 0 iDb ik ~1 5; ac b 4B ~01 5 0 ik∗ ik∗ iDbb B ~2 ac B ~2
ð6:7:8Þ
where the prime indicates d/dz. Expressed as an eigenvalue equation, Eq. (6.7.8) results in the four eigenvalues (propagation constants) of the four supermodes of region II. These are arrived at by straightforward but tedious algebraic manipulation of Eq. (6.7.8) using standard techniques. The four eigenvalues are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Db þ Dbb 2 a1 ¼ jkac j2 jkj2 þ Db2ab þ a ð6:7:9Þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Db þ Dbb 2 jkj2 þ Db2ab a a2 ¼ jkac j2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Db þ Dbb 2 jkj2 þ Db2ab þ a a3 ¼ jkac j2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dba þ Dbb 2 2 2 2 jkj þ Dbab : a4 ¼ jkac j 2
ð6:7:10Þ
ð6:7:11Þ
ð6:7:12Þ
In-Coupler Bragg Grating Filters
263
These are the most general solutions for the case when the fibers have different propagation constants. The eigenmodes associated with these eigenvalues have spatial fields that are expressed as the sum of individual modes of each fiber. The initial boundary values determine how each individual field grows (or decays). The first part of the analysis recognizes the fact that an input at either A1 or B1 results in coupling between the fibers through simple coupler action. This is described by the equation for the transfer function of the coupler [Eq. (6.3.1)], but with the appropriate coupling length, Lc ¼ L1, with input fields A1(0) ¼ 1 and B1(0) ¼ 0. At the boundary to the grating, the two fields A1(L1) and B1(L1) become the input to the grating. With the assumption that A2(L1 þ L2) ¼ 0 and B2(L1 þ L2) ¼ 0, the amplitudes of the four super-modes in region II are evaluated using Eq. (6.7.8). Finally, the backward propagating field amplitudes at the input to the coupler are propagated in reverse through the coupler to arrive at the amplitudes of the fields A2(0) and B2(0). Ideally, with a point reflector (grating), the transmission through to the output ports would follow the equation for the coupler [Eq. (6.3.1a)], to route the entire out-off-band transmitted light to B2. However, because of the finite length of the grating and the additional coupling that occurs in region II, region III may no longer be equal to region I for optimum performance, since jkj(L1 þ L2 þ L3) ¼ np. As has been mentioned, a simple way of making such a device is to draw two identical fibers together to form a coupler and subsequently write a grating at the appropriate position, as shown in Fig. 6.41 (iv). With two fibers and two gratings, there is always a possibility of a small mismatch in the propagation constants after the fibers are polished and the device assembled. The refractive index mismatch may typically be 5 10–5, resulting in 95% coupling [38]. Gratings written into such fibers may therefore need to be written carefully in order to match the Bragg wavelengths. With a fused fiber coupler, the quality of the device can be very good with coupling approaching 100%, indicating the uniformity of the coupling region. Thus, assuming fibers with identical propagation constants, Dbab ¼ 0, and the detuning Dbb ¼ Db, simplifying Eqs. (6.7.9)–(6.7.12) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 ¼ jkac j2 ðjkj þ DbÞ2 ð6:7:13Þ a2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jkac j2 ðjkj DbÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jkac j2 ðjkj þ DbÞ2
ð6:7:15Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jkac j2 ðjkj DbÞ2 :
ð6:7:16Þ
a3 ¼ a4 ¼
ð6:7:14Þ
Fiber Grating Band-Pass Filters
264
Note that despite using phase-synchronous fibers, in Eqs. (6.7.13)–(6.7.16) the eigenvalues have been detuned from the exact phase-matching condition by k. To calculate the field at the input port 1 (return-loss) and the dropped port 2, the boundary conditions are applied. The dropped “transmission” in port 2 is B2 ð0Þ 1 k∗ k∗ ac sinhða1 L2 Þ ¼ e2if t2 ¼ A1 ð0Þ 2 jkj ia1 coshða1 L2 Þ þ ðd1 Þ sinhða1 L2 Þ
kac sinhða2 L2 Þ e2if ; ia2 coshða2 L2 Þ þ ðd2 Þ sinhða2 L2 Þ
ð6:7:17Þ
where the detuning, d1 ¼ – jkj – Db, d2 ¼ jkj – Db and f ¼ (jkj þ Db)L1. Similarly, the back-reflected amplitude in port 1 is deduced as A2 ð0Þ 1 k∗ ac sinhða1 L2 Þ ¼ e2if þ r1 ¼ A1 ð0Þ 2 ia1 coshða1 L2 Þ þ ðd1 Þ sinhða1 L2 Þ kac sinhða2 L2 Þ 2if e : ia2 coshða2 L2 Þ þ ðd2 Þ sinhða2 L2 Þ
ð6:7:18Þ
We note that in Eqs. (6.7.17) and (6.7.18), there are four terms of interest. The two terms in brackets can be immediately recognized to be identical to the reflectivity of two gratings at different Bragg wavelengths [see Eq. (4.3.11)], given by equating d1 and d2 to zero. Secondly, the phase term, 2f, has two components; the first is due to the difference in the propagation constants of two modes propagating through a fiber of length, L1, after being reflected by the grating. The second part, kL1, is simply the accumulated phase change due to the coupling action of the coupler. The equivalent reflectivity and phase factors of two gratings as in Eq. (6.3.2) can replace the two terms within the square brackets of Eqs. (6.7.17) and (6.7.18): r1 eif1 ðlÞ ¼
k∗ ac sinhða1 L2 Þ ia1 coshða1 L2 Þ þ ðd1 Þ sinhða1 L2 Þ
r2 eif2 ðlÞ ¼
kac sinhða2 L2 Þ : ia2 coshða2 L2 Þ þ ðd2 Þ sinhða2 L2 Þ
ð6:7:19Þ
We note that the magnitude of the reflectivity r1 ¼ r2, since it is the same grating with identical parameters, only different detuning. The dropped transmission in Eqs. (6.7.17) and (6.7.18) may be further simplified to t2 ¼
1 r ½ei½f1 ðlÞ2f ei½f2 ðlÞþ2f 2 1
r1 ¼
1 r ½ei½f1 ðlÞ2f þ ei½f2 ðlÞþ2f ; 2 1
ð6:7:20Þ
In-Coupler Bragg Grating Filters
265
from which the power transmittance T2 and back-reflectance R1 are T2 ¼
r21 ½1 cos d 2
ð6:7:21Þ
R1 ¼
r21 ½1 þ cos d ; 2
ð6:7:22Þ
where d ¼ f1 ðlÞ f2 ðlÞ þ 4 jkj L1 :
ð6:7:23Þ
Interestingly, Eqs. (6.7.21) and (6.7.22) are exactly the same form as Eqs. (6.3.7) and (6.3.8), which describe the transfer characteristics of the Michelson interferometer with identical reflectivities but Bragg detuned gratings. In the BRC, the detuning is implicit in the phase factors f1 and f2 and calculated by equating d1 and d2 to zero, so that p ; ð6:7:24Þ l1;2 Bragg ¼ 2neff Lg p Lg jkj where the sign in the denominator determines the perturbed Bragg wavelength of the slow, symmetric (negative sign) and fast (positive sign) antisymmetric supermodes. Note that for weak coupling, i.e., when jkj ! 0, the splitting in the Bragg wavelength tends to zero. The detuning is solely dependent on the coupling constant of the coupler. For a given detuning, 2Dl ¼ l2Bragg l1Bragg , we can calculate the coupling constant by solving Eq. (6.7.24) as jkj 2pneff
Dl lBragg
2 ;
ð6:7:25Þ
where lBragg is the unperturbed Bragg wavelength. Since the functional form of the properties of BRC band-pass filters are almost identical to that for the Michelson interferometer, the detuning that can be tolerated for low back reflection has been discussed in Section 6.3. For a back reflection of approximately –30 dB (requiring a detuning of 0.01 nm), we calculate the coupling constant k < 26 m–1. This low value of the coupling constant is necessary to suppress the reflections on each side of zero detuning, as shown in Fig. 6.20. In a practical device, there is additional “apodization” due to the variation in the coupling constant in the tapered or curved region of the coupler, which will also tend to reduce the back-reflected light. For minimum back reflection, we note that T1 in Eq. (6.7.20) is zero, so that the coupling length L1 may be calculated as L1 ¼
p þ f2 ðlÞ f1 ðlÞ : 4 jkj
ð6:7:26Þ
Fiber Grating Band-Pass Filters
266
Thus, for a low back reflection, the propagation constants of the fibers need to be matched carefully, as well as the Bragg wavelengths of the gratings in each fiber. Fusing two fibers together creates a highly uniform taper with excellent physical symmetry. A fused coupler with a grating therefore has the potential of functioning as a device with the required characteristics. A variant on the fused taper device is shown in Fig. 6.41 (iv), which relies on the propagation constants of the two fibers being different [78]. However, the overlap of the modes and the grating becomes very large, since the grating is in the entire waist region of the couple, and the fields are bounded by air. A tilted grating will therefore act as a mode converter when the Bragg matching condition is met, b1 þ b2 ¼
2p ; Lg
ð6:7:27Þ
where b1 and b2 are the propagation constants of the two odd and even modes. This device has been demonstrated by Kewitsch et al. [78] with two identical fibers, one of which is pretapered as shown in Fig. 6.41 (iv) to change the propagation constant. A grating written in the waist at 1547 nm “dropped” 98% of the light in a bandwidth of 0.7 nm with a reported insertion loss of 0.1 dB. One problem with a fused taper device is the coupling to radiation modes of the fiber on the short-wavelength side of the Bragg wavelength, which can cause both cross-talk and loss.
6.7.2 Grating-Frustrated Coupler The generic form of this device is shown in Fig. 6.41 (ii). The coupler consists of two fibers, which are assumed to be parallel, and a single Bragg grating is present in one waveguide alone. The grating-frustrated coupler can be modeled in several ways. These methods include supermodes of the structure [80,76] or using the coupled-mode theory developed by Syms [81]. Syms’s model applies to a grating in both regions of the coupler and so has to be modified. In the following, the latter approach has been taken to model this device. The analysis is in two stages as in the analysis of Bragg gratings: First, synchronous coupling is considered alone, i.e., at the Bragg-matched wavelength, and then the detuned case is analyzed. For synchronous operation, both fibers have identical effective mode indexes at the Bragg wavelength: We ignore the perturbation introduced by the grating. The evanescent fields of the two fibers overlap and an exchange of energy takes place.
In-Coupler Bragg Grating Filters
267
The coupler can be analyzed by considering four modes with amplitudes A1. . .4. Mode 1 travels in the positive z-direction (from left to right) in the upper fiber, from z ¼ 0 to z ¼ L. Mode 2 also propagates along the positive z-direction in the lower fiber, from z ¼ 0 to z ¼ L. Mode 3 propagates in the negative z-direction in the upper fiber, from z ¼ L to z ¼ 0. Last, mode 4 also propagates in the negative z-direction in the lower fiber, from z ¼ L to z ¼ 0. The evolution of the mode amplitudes of the four waves is then described by a set of four first-order coupled differential equations. There are three significant interactions that need to be considered: 1. Copropagating interactions between the modes of different waveguides (modes 1 and 2 as well as 3 and 4), as they do in a normal coupler without a grating. The overlap of the mode evanescent fields allows an exchange of energy to take place since the guides are phase matched. Because of symmetry considerations, a single coupling coefficient k [Eq. (6.3.1c)] can be used. 2. Counterpropagating interaction between the modes of the same fiber. Modes 1 and 3, and 2 and 4, interact because of presence of the Bragg grating. The general form of the refractive index modulation of the grating, which allows this coupling, is given by Eq. (4.2.27). The phase mismatch is Db ¼ b1 þ b2
2pN ; Lg
ð6:7:28Þ
where the moduli of the mode propagation constants for the forward- and counterpropagating modes 1 and 2 are b1 ¼ b2 ¼
2pneff : l
ð6:7:29Þ
The phase-matched condition determines the optimum coupling between the forward- and backward-propagating modes when Db ¼ 0, at the Bragg wavelength. The presence of the grating promotes coupling between modes 2 and 4 since the grating is only in the lower waveguide, while modes 1 and 3 remain uncoupled. The coupling coefficient kac for Bragg reflection is given by Eq. (4.3.6), kac ¼
pDn : l
ð6:7:30Þ
3. We assume that there is no coupling between modes 1 and 4, 4 and 1, 2 and 3, and 3 and 2. Taking the above considerations into account, the four coupled-mode equations which describe the coupling are
Fiber Grating Band-Pass Filters
268
dA1 ¼ ifkc A2 g dz dA2 ¼ ifkc A1 þ kac A4 g dz
ð6:7:31Þ
dA3 ¼ ifkc A4 g dz dA4 ¼ ifkac A2 þ kc A3 g: dz
A similar procedure is followed as in the BRC coupler, and applying the boundary conditions results in the outputs of the device [82]. The grating-frustrated coupler works best when port 1 is excited (the fiber without the grating) with the grating extending along the entire coupling length and beyond, and with a coupler kLc ¼ p/2. With this value of coupling, 100% of the power is transferred from one fiber to the other at a wavelength far removed from the Bragg condition, i.e., the device functions as a normal coupler. On-Bragg, the frustrated coupling depends on the strength of the grating coupling constant. The dependence of the transmission and reflection characteristics for port 1 excitation on the grating coupling strength is shown in Fig. 6.42. Figure 6.43 shows the theoretical performance of a device similar to the one reported by Archambault et al. [83], with approximately 70% in the dropped port. Note that with increasing kLc (over the coupling length), the on-Bragg
1.0 T2
0.6
T1
LG = LC
0.8
LG = 2LC 0.4
0.6 0.4
0.2 R1
0.2 R2 0.0 0
0.0 2
4
6 kLC
8
10
0
2
4 6 kLC
8
10
Figure 6.42 Transmission and reflection characteristics for the GFC as a function of the strength of the grating (courtesy Philip Russell from: Archambault J.-L., Russell P.St.J., Barcelos S., Hua P., and Reekie L., “Grating frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19(3), 180–182, 1994. Ref. [83]).
In-Coupler Bragg Grating Filters
269
Transmission/reflection
1.0 0.8 Port 3 (Tx) Port 4 (Tx) Port 1 (Refln) Port 2 (Refln)
0.6 0.4 0.2 0.0 1533.0
1534.0
1535.0
1537.0
1536.0
Wavelength Figure 6.43 Transmission and reflection characteristics of a GFC. Simulation for Lc ¼ 2.5 mm, Lg ¼ 4.5 mm, dn ¼ 3 10–4, neff ¼ 1.45. lBragg ¼ 1534.7 nm (from: Wolting S., “Grating frustrated couplers,” BT Research Laboratories Summer Studentship Report, 1995. Ref. [84]).
1.0
0.10
0.8
0.08
0.6
Port 4 (Tx) Port 3 (Tx) Port 1 (Refln) Port 2 (Refln)
0.4
0.04 0.02
0.2 0.0 1540
0.06
Reflectivity
Transmission
wavelength transmission approaches unity, and the reflected power tends to zero. In practice, in-fiber gratings are limited to index modulations of 10–3, resulting in kLc 10 for a coupling length of 4 mm. Thus, the performance of this device is likely to remain limited, however elegant the principle, since the return loss remains high and dropped power will suffer a loss of 1 dB, despite the insertion loss being intrinsically low (0.2 dB). Figure 6.44 shows the transmission characteristics of a GFC with a kacLc ¼ 9 (dn ¼ 2.75 10–3, and Lc ¼ 1.57 mm). The FWHM bandwidth is in excess of 4 nm, the pass-band transmission is >0.9, and the transmission in the stop band is still of the order of 0.05. As an OADM, this device is unlikely to have the performance necessary for telecommunications applications.
1545
1550
1555
0.00 1560
Wavelength, nm Figure 6.44 The transmission characteristics of the four ports of GFC with a kacLc ¼ 9, dn ¼ 2.75 10–3, and Lc ¼ Lg ¼ 1.57 mm (from Ref. [84]).
Fiber Grating Band-Pass Filters
270
6.8 SIDE-TAP AND LONG-PERIOD GRATING BAND-PASS FILTERS The theory of radiation mode coupling can be found in Chapter 4. Radiation mode coupling is generally used in applications requiring a lossy filter. For example, in flattening the gain spectrum of an erbium-doped fiber amplifier, a multiple side-tap grating (STG) filter [85] and the long-period grating (LPG) [86] have both been successfully used to eliminate the large variations within the gain bandwidth. The light “lost” from the fiber through radiation mode coupling can be substantial, if the grating is strong. Side-tap blocking filters can attenuate 100% of the light within a narrow band, which can be tailored to span 100 nm or more. While STGs in general allow coupling to all order modes (odd and even, LPmn), untilted LPGs couple guided-mode light to only m ¼ 0 order modes (LP0n). Choice of blaze angle and the v-value of the fiber easily tailor the loss spectrum and bandwidth of the STG. The bandwidth of the LPG is determined by the coherent interaction between the radiated cladding mode and the guided mode over the long grating length (centimeters), with a bounded cladding. With an unbounded cladding, the loss spectrum of the LPG becomes extremely wide (>100 nm) [87], since the dispersion in (neff – ncladding) is very weak. Figures 6.45a and b show light exiting from the side of a fiber by an STG and an LPG. The cladding mode has a better chance of interacting with the LPG. A ray exiting the core at an angle of 10 to the fiber axis will travel 0.4 mm before being reflected back toward the grating in a 125-mm diameter fiber. Shallower angle rays may miss the STG altogether after the first reflection at the cladding–air interface. This is less likely in the LPG, which may be 2–10 times longer than a typical STG. Therefore, there may be continual exchange of energy between the radiated mode llong
lshort lshort
llong
llong
lshort LPG
Light input Cladding
A
Fiber core
STG Reflection
B
Transmission
Figure 6.45 Light radiated (a) from the STG and (b) from the LPG.
Side-Tap and Long-Period Grating Band-Pass Filters
271
and the guided mode with the LPG, unless the cladding is made “infinite” by applying index-matching oil to the cladding. Instead of coupling to discrete radiation modes (approximately the same as the cladding modes), light is coupled to a continuum of the radiation field, so that a broadband loss spectrum is seen in transmission rather than a narrow bandwidth of the cladding mode [87]. Note that the angular distribution of the radiation for the LPG as a function of wavelength is reversed compared to the STG; i.e., the longest wavelengths exit at the largest angle (see Chapter 4). The basic principle of the coupling relies on the phase-matching conditions, and the overlap integrals determine the strength and the wavelength dependence of the loss. There are two bounds to the loss spectrum, one on the short- and the other on the long-wavelength side: Light radiated out of the fiber core subtends a wavelength-dependent angle y(l) to the counterpropagation direction (STG) and copropagating (LPG), which depends on the inclination of the grating and period. For the STG, this angle of the radiated light at wavelength l in the infinite cladding is easily shown to be 1 l Ncos yg neff ðlÞ ; ð6:8:1Þ cos½yðlÞ ¼ nclad Lg where yg is the tilt of the grating with respect to the propagation direction, and N is the order of the grating. The angle at which the light exits from the side of the fiber varies as a function of wavelength and therefore can be used as a bandpass filter. The bandwidth of the radiated light can be shown to be approximately Dl
Lg sin2 ðyg Þ : 2 cosðyg Þ
ð6:8:2Þ
However, the phase-matching condition alone does not determine the peak wavelength, in the general case when the grating is tilted; the overlap integral together with the phase matching alters the spectrum and shifts the wavelength of maximum loss (see Chapter 4). Typical transmission loss and reflection spectra for a strong STG are shown in Fig. 6.46a, and the transmission loss spectrum of an LPG, in Fig. 6.46b. Both gratings were written in the same fiber. The STG has been used as a spectrum analyzer by Wagener et al. [83]. A chirped grating blazed at 9 to the propagation direction was used to outcouple light from a fiber. The chirped grating had a decreasing period away from the launch end of the fiber. Since the angle y(l) subtended by the radiated light at a wavelength l becomes smaller with reducing pitch [Eq. (6.8.1)], the focus of the light coupled out at different points is a function of the wavelength.
Transmission loss, dB
1480 0 –5 –10 –15 –20 –25 –30 –35 –40 –45 –50
1500
1520
1560
1580 0 –5 –10 –15 –20 –25 –30 –35 –40 –45 –50
A
Wavelength, nm 1450 0
Transmission, dB
1540
Reflection, dB
Fiber Grating Band-Pass Filters
272
1475
1500
1525
1550
1575
1600
–5 –10 –15 –20 –25 –30 –35
B
Wavelength, nm
Figure 6.46 The transmission loss (and reflection) of (a) a 4-mm-long side-tap grating filter and (b) LPG filter with a 400-micron period, both written in a boron–germania codoped fiber.
The focal length is inversely proportional to the wavelength of the radiated light as [88] f ðlÞ ¼
L2g Lg sin½yðlÞ ; cosðyg Þ dlg l
ð6:8:3Þ
where Lg is the nominal period of the chirped grating, n is the refractive index of silica, and Lg and dlg are the length and the change in the period of the grating due to chirp (in nm), respectively. The radiated light was detected by a 256element photodiode array, the center of which was arranged to be at the focus of the light at lc, the wavelength radiated by the center of the grating. A schematic of the device is shown in Fig. 6.47. A weak, 20-mm-long grating (Lg ¼ 547 nm) with a chirp in the period of 1.92 nm ( 0.96 nm) was used to tap 5% of the light over a 35-nm bandwidth. An index-matched prism was placed in contact with a fiber of square cladding
Side-Tap and Long-Period Grating Band-Pass Filters
273
256 element detector array
Index-matching prism lshort
Input
lshort
llong
Core
Output
Chirped STG Square-cladding cross-section fiber Figure 6.47 A schematic of the in-fiber chirped STG spectrum analyzer (after Ref. [88]).
cross-section to promote good adhesion, to direct the light to the photodiode array. The effective focal length was 160 mm. The resolution of the spectrometer was demonstrated to be 0.12 nm with a measured bandwidth of 14 nm, an insertion loss of <1 dB, a wavelength accuracy of 0.03 nm, a power level accuracy of <0.25 dB, and a dynamic range of 35 dB. The spectrum of two resolved channels separated by 50, 100, and 200 GHz using this spectrometer is shown in Fig. 6.48 [88].
Power (dBm)
–10 50 GHz 100 GHz 200 GHz
–15
–20
–25
–30 1546 1548 1550 1552 1554 1556 1558 1560 1562 Wavelength (nm) Figure 6.48 The spectrum of two channels as the detuning is increased from 50 to 100 to 200 GHz (courtesy T. Strasser, from Wagener J.L., Strasser T.A., Pedrazzani J.R., and DeMarco J., “Fiber grating optical spectrum analyzer tap,” In Tech. Digest of ECOC’97, Publn. No. 448, Vol. 5, pp. 65–68. # IEE 1997 Ref. [88]).
Fiber Grating Band-Pass Filters
274
While this device is an excellent example of a band-pass filter, which is also tunable, the applications of the STG and the LPG in this area have not really been exploited. It is envisaged that supervisory operations of fiber amplifiers is another possible application in which the filter that attenuates the spectrum can allow the monitoring of the performance and control of the amplifier characteristics.
6.9 POLARIZATION ROCKING BAND-PASS FILTER Another type of band-pass filter is the polarization-converting filter. Much of the earlier work focused on the study of photoinduced polarization effects and fabrication of internally written filters [89–95]. The polarization sensitivity of fiber was recognized early in the development of photosensitive phenomena [96]. Birefringence can be photoinduced in Ge-doped fibers [97] because of preferential bleaching of defects [89]. Coupling between orthogonal polarizations was shown in elliptical core fibers due to dynamic changes in the birefringence [98]. A novel counterpropagating polarization coupling filter has also been reported, using internally written gratings [99]. It was shown that stress birefringence can be reduced in elliptical cored fibers illuminated with UV radiation [100]. Thus, in principle, the control of the birefringence by external means allows the fabrication of polarization coupling devices at any wavelength. In this section, we consider externally written filters, which can be fabricated for operation at any desired wavelength. The theory of coupling between polarization can be found in Chapter 4. The band-pass filter functions by rotating the polarization of the input mode by N p/2. In order to discriminate between the two states, a polarizing component has to be used. A schematic of the filter is shown in Fig. 6.49. Two types of polarization rocking filters can be fabricated. Birefringent polarization maintaining fiber can be used with a given predetermined beat
Ey
Lb
Polarizing beam splitter
Through, Ey (lall else)
Input Birefringent fiber with grating written at 45 degrees to principal axes
Figure 6.49 The polarization rocking “drop” filter.
Drop, Ex(lr)
Polarization Rocking Band-Pass Filter
275
100
Coupling ratio, %
80 60 40 20 0 1250
1270
1290 1310 Wavelength, nm
1330
1350 196/2
Figure 6.50 Transmission spectrum of an 85-step, polarization-rocking filter fabricated in a polarization maintaining fiber (from: Hill K.O., Bilodeau F., Malo B., and Johnson D.C., “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett. 27(17), 1548, 1991. # IEE 1991. Ref. [101]).
length [101], or nonbirefringent fibers, which are deliberately made birefringent, may also be used [102]. In the first case, Hill et al. [101] showed that a filter with only 85 periods had a polarization conversion efficiency of 89% at a wavelength of 1292 nm in a FW bandwidth of 8 nm. The filter length was reported to be 0.87 m, fabricated in an Andrew Corp. fiber with a birefringence B ¼ 1.27 10–4. The transmission spectrum of this filter is shown in Fig. 6.50. The narrow bandwidth is a direct result of the high birefringence of the fiber. Apodized filters have also been demonstrated using the same principle, but with a grating coupling constant which is weighted as a function of length [103]. For the filter to function properly, care needs to be taken, since the input polarization has to remain stable. In the second type of filter, a nonbirefringent fiber is wrapped on a cylinder to impart a known birefringence [104] (see Chapter 3). Given that the birefringence can be controlled, the beat length and therefore the rocking period is known. This method allows the fabrication of very long fiber filters, since the bend-induced birefringence is small (beat length 0.75 m at 1550 nm) even with the tightest tolerable bend in standard telecommunications fibers [102]. The reported filter had a polarization coupling efficiency of 100% at a peak wavelength of 1540 nm, with a FW bandwidth of 130 nm, in a filter that was only 17 beat lengths long (Lr ¼ 10.89 m), with 27% of each beat length exposed to UV radiation (duty cycle of 27%).
Fiber Grating Band-Pass Filters
276
The bandwidth of these filters between the first zeroes of the transmission spectrum follows from Eq. (4.4.12) as aL ¼ p;
ð6:9:1Þ
so that the bandwidth, 2Dl, from Eq. (4.5.11) 2Dl ¼
2plLb : Lr
ð6:9:2Þ
The fabrication of the rocking filter depends on the periodic exposure to UV radiation of half of the beat length of the fiber. This induces a refractive index modulation, which is only half of what it would be if the second half of the beat length was modulated by a negative index change, or if the rocking of the birefringence was truly f per beat length. This has the beneficial effect of halving the filter length Lr. This is particularly useful, because the rocking angle f saturates at approximately 0.4 –0.5 per beat length with exposure to many pulses [105]. In order to overcome this problem, Psaila et al. [105] used a double pass scheme to double the rocking angle per beat length. In the first pass, the fiber was exposed at 45 to the birefringent axes, half of each beat length of 14 mm. The stationary fiber was exposed to a moving, pulsed UV beam through a 0.5-mm slit, with the UV polarization orthogonal to the fiber propagation axis. On the second scan, the fiber was rotated by 90 around the propagation axis, and the other half of the beat length was exposed in a similar manner. The result is the doubling of the rocking angle per beat length, leading to a rocking filter with 98% conversion efficiency, only 33 beat lengths long, and with a FWFZ (full width to first zeroes) bandwidth of 20 nm. It should be noted that the polarization coupler is a band-pass filter in transmission and that it converts either input polarization to its orthogonal state. As a consequence of this, a concatenation of two such filters results in a Mach– Zehnder type interferometer [106]. A schematic of this filter is shown in Fig. 6.51. The transmitted output intensity can be shown to be [106]
Ey Input
Lb Rocking filter 1
Ls
Polarizing beam Rocking filter 2 Ex (lr)
Figure 6.51 The polarization rocking Mach–Zehnder band-pass filter fabricated in birefringent fiber with two gratings written at 45 to principal axes (after Ref. [106]).
Polarization Rocking Band-Pass Filter
I¼
k2ac 2 sin aLr a2
"
277
# Db2 sin2 aLr ½1 þ ð1 þ Ls ÞcosðkBÞ ; cos aLr þ 4a2 2
ð6:9:3Þ
where k ¼ 2p/l, Lr is the length of the rocking filter, Ls is the distance between the two filters, kac ¼ Dn/lr, and B is the birefringence, nx – ny. The detuning parameter Db ¼ bx – by – 2p/l, with the usual definition of a (see Chapter 4). If the filter separation Ls < LcohLb/l, where Lcoh is the coherence length of the source, then strong interference is visible at the output. Kannellopoulos et al. [106] fabricated the filter in Andrew Corp D-fiber with a polarization beat length of 4.35 mm (Lb) at a wavelength of 780 nm and B ¼ nx – ny ¼ 1.8 10–4. The core-to-cladding index difference was 3.3 10–2 with a higher mode cutoff of 710 nm. The rocking filters centered at 787 nm were 224.4 mm long (Lr) and separated by 320 mm (Ls). The period of the gratings was 4.4 mm, and they were written with 266-nm wavelength pulsed radiation, exposed to 2400 pulses at a peak intensity of 1.4 MW/cm2. Each filter had a coupling efficiency of 7.5% so that the peak refractive index modulation dDn0 ¼ 9.7 10–7. Figure 6.52 shows the transmission characteristics of the polarization coupling MZI. The output shows three peaks in transmission, each with a FW bandwidth of 8 nm. The temperature sensitivity of the resonant wavelength of the filter is high (0.5 nm/ C) compared to that of Bragg gratings (0.006 nm/ C), and so the filter is suited to sensing temperature. When one of the filters is heated, the resonant wavelength shifts; alternatively, heating the section between the couplers influences the output state, from which a phase sensitivity of 0.22 rad/ C was calculated over a temperature range of 150 C.
750
840 Wavelength (nm)
Figure 6.52 The transmission characteristics of the polarization coupling Mach–Zehnder interferometer (courtesy S. Kannellopoulos, from: Kanellopoulos S.E., Handerek V.A., and Rogers A.J., “Compact Mach–Zehnder fibre interferometer incorporating photo-induced gratings in elliptical core fibers,” Opt. Lett. 18(12), 1013–1015, 1993. [106]).
278
Fiber Grating Band-Pass Filters
6.10 MODE CONVERTERS Mode converters work on a principle similar to that of the polarization converter. There are several types of mode-converting filters: those that couple one guided mode to another co- or counterpropagating guided mode, as well as those which couple a guided (or cladding) mode to a co- or counterpropagating cladding mode (or guided mode). Coupling of a guided mode to a co- or counterpropagating cladding mode has been partially covered in Section 6.8 on sidetap and long-period grating band-pass filters, and also in Chapter 4. Guided– co- and counterpropagating mode coupling devices will be discussed in this section. Other devices based on concatenated couplers for mode reconversion will also be discussed.
6.10.1 Guided-Mode Intermodal Couplers The theory of guided-mode coupling to both copropagating and counterpropagating guided/radiation modes can be found in Chapter 4. Equations (4.3.16), (4.4.11), and (4.4.12) govern the coupling of counter- and copropagating guided modes. Coupling of the modes of the same order is always possible, provided the phase-matching condition is met. The overlap integral, Eq. (4.4.5), for the coupling of dissimilar modes is zero, unless the transverse distribution of the refractive index modulation is not constant, breaking the symmetry. Thus, a slanted grating will increase the overlap integral between modes of different order, since the transverse refractive index modulation profile is no longer uniform. This was first reported by Park and Kim [107] using a novel internal mode-beat grating, by exciting a bimoded fiber at a wavelength of 514.5 nm from a CW argon laser. The beating of the modes automatically creates a blazed grating with the intensity of the laser radiation varying transversely across the core with the mode-beat period. This intensity grating induced a photoinduced grating, which created the mode converter. This principle has been extended by Ouellette [108] by recognizing that the phase-matching condition is met by more than one set of modes due to the nondegeneracy of the modes, but at longer wavelengths. Thus, a grating written at 488 nm by exciting a chosen pair of modes allowed the “reading” of the holographic grating at around 720 nm. Further experiments have demonstrated the sensitivity of the mode beat length to the applied strain [109], since the propagation constants of the fundamental and higher-order modes change differentially. These types of converters have applications in strain sensing and as nonlinear optical switches [110]. A phasematching condition is also possible where the beat length Lb ! 1 [111,112], allowing very long period gratings to be used.
Mode Converters
279
A principle similar to the one reported by Park and Kim [107] has been used to fabricate mode converters at different wavelengths. The coupling between the LP11 and LP01 modes was made possible by slanting the grating using an external, point-by-point UV exposure technique [113], details of which may be found in Chapter 3. A slit (12 mm) at an angle of 2–3 was used to expose the fiber at the firstorder period of 590 m (N ¼ 1) mode beat length, 2p/[N(bLP01 – bLP11)], resulting in a mode converter at a wavelength of 820 nm in a Corning Telecommunications fiber with a higher-order mode cutoff of 1.1 mm [113]. The grating had typically 200–300 steps for 100% coupling, while overcoupled converters with 1000 periods were also reported. These convert the LP01 to the LP11 and then back again to the LP01. Unlike polarization rocking filters, mode-converting gratings can be very strongly coupled, so that the side lobes of the sinc function in Eqs. (4.4.11) and (4.4.12) become large with a large refractive index modulation, and multiple peaks can be seen in the transmission spectrum. A FWFZ of 5 nm was observed. A higher-order mode stripper has to be used to observe the loss spectrum of this filter. For the band-pass function, the higher-order mode has to be selected. Alternatively, the higher mode can be excited preferentially in the overcoupled filter and stripped at the output so that the “dropped” wavelength is in the fundamental mode, with the through state in the LP11 mode. Conversion from LP01 mode to the leaky counterpropagating LP11 mode has been observed in depressed-clad fibers in which tilted Bragg gratings were written [114]. In this case, the fiber supported the LP11 mode over short lengths, so that a “ghost” dip is observed in the transmission spectrum at a wavelength slightly shorter than the Bragg reflection wavelength. While the light is coupled into the LP11 mode, the reflection is not visible if the fiber on the input side of the gratings is mode stripped. Gratings with a tilt angle of 3 have been written in standard telecommunications fibers preexposed to raise the index to allow the LP11 mode to be supported [115] over the UVexposed length. The overlap between the LP01 mode and the LP11 mode is almost the same as the LP01 mode for this tilt angle, so that efficient reflective mode coupling is possible. The side illumination induces a nonuniform refractive index change across the core, breaking the LP11 mode degeneracy and causing two reflections to occur. Depressed-cladding fibers have been examined by Haggans et al. [116] with a view to reducing the coupling to radiation and cladding modes. Coupling to similar order modes does not require a transverse asymmetry in the grating (see Chapter 4). Therefore, the LP01 mode is coupled to the LP02 mode for both co- and contradirectional converters with a transversely uniform grating. The coupling constant kac [117] is
Fiber Grating Band-Pass Filters
280
pDn w w 201 02 2 kac ¼ 2 l v u01 u02
!
J0 ðu02 Þ J0 ðu01 Þ u02 u01 ; J1 ðu02 Þ J1 ðu01 Þ
ð6:10:1Þ
where the subscripts refer to the mode, and u, w, and v are the normalized waveguide parameters defined in Chapter 4. The interaction of the LP01 ! LP02 mode is particularly useful for narrow-band filters [118] and for broadband dispersion compensation [119]. In the latter application, gratings between 27 and 35 mm long have been fabricated with periods ranging from 116 to 135 mm in few-moded fiber. Up to 85% coupling is obtained with a FWHM bandwidth of 25 nm. The step-chirped technique [120] can also be used to broaden the bandwidth for longer mode-converter gratings. A 65-mm-long grating with a 0.3-mm chirp over four sections has been demonstrated to have a peak coupling efficiency of 90% at 1556 nm and a FWHM bandwidth of 14 nm.
6.11 SAGNAC LOOP INTERFEROMETER The optical fiber Sagnac loop is a self-referencing interferometer and is therefore extremely stable with respect to the environment. This is because the two counter propagating paths are identical and the light fields interfere at the coupler shown in Fig. 6.53a with a p/2 phase difference, and therefore all the light is reflected. However, if the loop is cut in the middle, the Sagnac
50:50
Output
50:50
Wavelength
A
Output
lB
Wavelength
B
Figure 6.53 (a) The configuration of a traditional Sagnac loop interferometer. (b) The GSLI band-pass filter. (From W. Margulis 1997, pvt. comm.)
Sagnac Loop Interferometer
281
turns into a Michelson interferometer, and the individual arms become independent; therefore the path lengths have to be stabilized for a predictable output. If a single highly reflecting FBG is spliced to each of the ends of the cut Sagnac to close the loop once again, the phase of the reflections arriving at the coupler depends on the exact position and chirp of the FBG. The output from the Sagnac interferometer has an identical response to the Michelson interferometer, which was discussed at length in Section 6.3, with identical gratings. With the uniform FBG offset by DL from the middle of the loop, the output is periodic like the Michelson interferometer, as shown in Fig. 6.53b. The transfer function of the grating-Sagnac loop interferometer (GSLI) can be simply calculated from the following slightly modified grating reflection Eq. (4.3.14): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 kac sinh k2ac d2 Lg 2pneff DL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2 ; ð6:11:1Þ O¼ l k2ac cosh2 k2ac d2 Lg d2 where the parameters used are defined in Section 6.3. DL is the physical difference in the reflection point within the grating, assuming that the coupling coefficient of the coupler is 50%. In the general case, Eq. (6.11.1) becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pneff DL 2 ð1 2aÞ k2ac d2 þ 2k2ac að1 aÞsinh k2ac d2 Lg cos l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; O¼ 2 2 2 2 2 kac d Lg d kac cosh ð6:11:2Þ where a is the splitting ratio of the coupler. The period of the band-pass peaks is given by Eq. (6.3.10), with the phase difference set to zero. In a chirped grating Sagnac loop interferometer (CGSLI), reflections become detuned even if the grating is spliced exactly in the middle, because the clockwise (CW) and counter-clockwise (CCW) paths at a given wavelength reflect in the CFBG from the same physical location, which changes with wavelength. Thus, in a symmetrically positioned linearly chirped grating, the CW and CCW reflections at the center of the band have the same path lengths. All other wavelengths detune linearly in phase. Wavelengths outside the bandwidth of the grating then experience a “normal” Sagnac interferometer and see no change in the path difference as a function of wavelength. The CGSLI is shown in Fig. 6.54. These devices have been studied for their application as sensor systems [121,122].
Fiber Grating Band-Pass Filters
282 lS
lL
Output
Output
50:50
Wavelength
Wavelength
Figure 6.54 The CGSLI is shown with the complementary outputs of the interferometer, which are now aperiodic. The transfer function is identical to the Michelson interferometer with identical chirped gratings in each arm but with one reversed in direction. See Section 6.3.1.
6.12 GIRES–TOURNOIS FILTERS The basic principle of the Gires–Tournois interferometer (GTI) is based on a Fabry–Perot (FP) etalon. The main difference is that the two mirrors of the GTI have very different reflectivity, which allows the finesse and dispersion to be tailored. Figure 6.55 shows the GTI. The grating at the input has a low reflectivity, r1, at a fixed wavelength, lB. The rear grating with a reflectivity of r2 is usually 100% and displaced from the input grating by a distance DL. The 100% reflectivity ensures an all-pass filter, whereas the input reflector determines the dispersion. The FP interferometer has a periodic reflection and transmission spectra as seen in Section 6.2. Figure 6.56 shows the reflection spectrum of a GTI with an input grating 0.2 mm long and a rear grating length of 0.8 mm. With a ΔL
Input / output
Tx Core r1
r2
Figure 6.55 In fiber GTI fabricated with two gratings of reflectives, r1(low) and r2(100%) separated by a distance DL. The gap determines the pass-bands of the filter and the device works in reflection with the dispersion being a function of wavelength.
Gires–Tournois Filters 1542
1541
1543 150
–5
100
–10
50
Group delay, ps
Reflection, dB
1540 0
283
0
–15 Wavelength, nm
Figure 6.56 The reflection and group-delay spectra of a 5 mm long CGTI, discussed in text.
refractive index modulation of 1.75 103 and DL ¼ 4.1 mm, an eight, near identical channel dispersion compensator can be fabricated. The dispersion of this device is suitable for applications in DWDM [123]. Several variations on this scheme have been reported. With CGTIs, it is possible to fabricate tunable multichannel dispersion compensators [124]. A pair of cascaded inline transmissive FP etalons may also be used for tunable dispersion, and dispersion slope compensation [125] on a 25-GHz grid using two pairs of chirped grating–based FP etalons separated by an inline isolator may be temperature or strained tuned to give any desired value of dispersion 0 to 750 ps-nm1, and a dispersion slope compensation of 617 ps-nm2 was achieved for 14 channels in the 1548–1552 nm window. It is simple to choose any wavelength band of interest for dispersion compensation with these types of filters. GTIs are compact, potentially low cost, and very flexible. The design of the GTI for dispersion compensation and slope compensation is discussed in references [126] and [127], respectively. Combining two GTIs for slope compensation requires the free spectral range (FSR) of each etalon to be identical [127]. If they are slightly mismatched, a beat frequency is generated, Df ¼ FSR1-FSR2 ¼ f1 – f2, which is approximately given by f1 f2 FSR2 jDf j jDf j
ð6:12:1Þ
and the channel number that has a different dispersion slope is given by N¼
f ; jDf j
ð6:12:2Þ
where f f1 ¼ f2. The dispersion slope can be calculated as s
1:4Df ðD1 þ D2 Þ : f2
ð6:12:3Þ
Fiber Grating Band-Pass Filters
284
In Eq. (6.12.3), D1 and D2 are the dispersions of the two GTIs, respectively. The group delay ripple of these devices is typically 2 4 ps. To tailor the response of the GTIs to a known group-delay slope, the reflection spectrum of the first mirror should have the following profile: 2T0 2 ; ð6:12:4Þ Rin ðlÞ ¼ 1 DtðlÞ where T0 ¼ 2neffL/c is the round-trip time of the GTI. The refractive index modulation of the gratings typically used by the GTI for dispersion compensation is approximately 3 103 with grating lengths of 11 mm. Tunable dispersion and dispersion slope compensation have been implemented by using a cascade of distributed GTI with two circulators. The first distributed CGTI is fabricated with three CFBGs; two weaker ones are overlaid with a slight separation and then a stronger CFBG is overlaid with a slight displacement as well. The second CGTI is a normal one with twin CFBGs overlaid with a small separation. A symmetric temperature gradient across one CGTI is used to alter the dispersion, without affecting the central wavelength. When combined, the net result of the two CGTIs is 60% greater tunability compared to a single cavity design [128]. A flat top multi-band-pass filter can be made with the Michelson interferometer shown in Fig. 6.57 using chirped gratings. If the second CFBG is replaced by a distributed CGTI, then the output of the filter becomes flattop in the bandpass, but at the cost of insertion loss [129], as increasing the front grating reflectivity in the CGTI compromises channel isolation; 50 GHz, 10 channel, with an insertion loss of 2.5 dB has been reported [129]. It should be noted that FP structures made on the same principle as the overlapped chirped longitudinally displaced FBG written in photosensitive Yb:Er doped fibers allow multichannel fiber lasers to be fabricated with excellent characteristics [130]. The channel spacing can be designed by adjusting the longitudinal displacement, chirp, and the length of the gratings. The structure of the laser is shown in Fig. 6.58. Two slightly displaced, co-located CFBGs in core of a SMF lL
lS
Input / output
Tx Core ΔL
Figure 6.57 The distributed CGTI showing two slightly displaced gratings with a weaker one on the left, laterally displaced by a distance DL. (From reference [123].)
Tunable Band-Pass Filters
285
Longitudinal displacement Yb:Er doped fiber
Two superimposed chirped FBGs 980nm pump laser
Yb:Er doped fiber
980/1550nm WDM coupler
ISO
OSA
Figure 6.58 Schematic of the multiwavelength fiber laser using chirped FP cavity. This laser demonstrates low cross-gain modulation owing to the separated FP resonances. (Adapted from reference [130].)
6.13 TUNABLE BAND-PASS FILTERS Following from the examples given in Figs. 6.1 through 6.9 for DFB filters, it may be noted that practical implementations of such filters are quite important. Tunability is also a great advantage in communication systems. One of the key requirements for a tunable band-pass filter is that it should be “hit-less” (i.e., that it must be tunable without scanning through all the available intermediate filter positions). The DFB filter can be implemented to meet this requirement. However, in practical configurations for tunable FBG band-pass filters, the phase step must be introduced in a controlled manner. Extremely high-quality and finesse tunable FGBs using a chirped grating can be made easily by inducing a local quarter-wavelength shift at the position of the appropriate Bragg wavelength in a chirped FBG. This is shown in Fig. 6.59. A chirped FBG is supported horizontally and translated below a very fine hot wire. The heat from the hot wire induces a phase shift in the FBG, which opens up a band gap at the local Bragg wavelength, as in a DFB filter. The extinction (reflectivity) of the grating determines the bandwidth of the pass band. However, it should be noted that the band-pass characteristics are Gaussian with a peak-transmission at the center of the band, as Figs. 6.2 and 6.4 illustrate. Figure 6.60 shows the band-pass for a position of the wire on the chirped grating, around 1540 nm. The reflectivity can be made very large for high discrimination. Multisection phase steps are required to alter the shape of the band-pass (see Section 6.1.1). This is difficult to implement in a chirped grating, however, some of the applications are well suited to the type of band-pass shape generated by
Fiber Grating Band-Pass Filters
286
Input
Hot wire
Power source
Chirped fiber Bragg grating
Fine movable wire Band-pass
Figure 6.59 A tunable band-pass filter using a DFB structure induced by a hot wire in a chirped FBG. (From “Winstom Filter,” Proximion Inc.)
Transmission, dB
–10 –20 –30
1530
1540 Transmission, dB
1520 0
1550
1560
1570
1539.5 1539.8 1540.0 1540.3 1540.5 0 –10 –20 –30 –40 –50 –60 Wavelength, nm
–40 –50 –60 Wavelength, nm
Figure 6.60 The effect of a phase step introduced at a wavelength of 1540 nm on a 35-nm chirped grating. The phase step introduced by a hot wire may be moved relative to the grating to shift the position of the band-pass wavelength to be anywhere in the bandwidth of the grating. The parameters of the grating are 35-nm chirp bandwidth, 50 mm long, with a refractive index modulation of 1.5 103 [131].
this grating filter. It should be noted that theoretically, the band-pass loss is zero, and it is possible to have below 0.1 dB loss in practical implementations with negligible polarization mode dispersion [131]. These filters are suitable for frequency discrimination, particularly in digital phase shift keying application (DPSK), in which a carrier needs to be suppressed. A simple configuration of this type of filter is thus easily implemented in hydrogen-loaded fiber.
LPG Filters 1552 0
287 1553
1554
1555
1556
1557
1558
–10 –100 –80–60 –40 –20 0 20 40 60 80 100 0
–15
Transmission, dB
Transmission, dB
–5
–5
–10
–20
–15 –20 –25
–25
Frequency, GHz
Wavelength, nm Figure 6.61 A DFB filter in a 2-mm-long FBG with a refractive index modulation of 103. The 3 dB bandwidth of this filter is 5 GHz. These filters have been fabricated in standard SMF without any measurable PMD [131].
However, strong gratings are needed with a large refractive index modulation (103), even after annealing. A high-quality band-pass filter with a quarterwave phase shift in the center of the FBG is shown in Fig. 6.61. The designed bandwidth of the filter is 5 GHz (3 dB) and can be used for suppressing a carrier 10 GHz away from the data. Compression tuning of FBGs for band-pass filtering has been demonstrated by Iocco et al. [132]. This scheme utilizes a FBG carefully inserted between two holders and compressed by applying a force on one plate. Tunability of around 45 nm was demonstrated at a tuning speed of 21 nm/ms; however, asymmetric forces limit the number of cycles of operation. Great care needs to be taken to overcome this problem. Recently, by inserting and potting a FBG in a polymer cylinder of large diameter (cm), compression tuning without failure over 80 nm, with an insertion loss of 0.28 dB, has been shown [133]. A very small bandwidth broadening was noted (see also Chapter 10) during tuning.
6.14 LPG FILTERS Long period gratings (LPGs) have several applications other than in sensing and gain flattening. The LPG is a device for coupling forward propagating modes, primarily the fundamental core mode to a higher-order mode in the core or the cladding. Therefore, the LPG has played an important role in the applications and the evolution of two-mode fibers. Ramachandran [134], who has contributed significantly to the understanding and design of novel LPGs, has provided an excellent review of their significance in applications, as well
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288
as their properties. In particular, a major application that was proposed by Poole et al. [135–137] based on two-mode fibers is in dispersion compensation. In this scheme, a grating is used to couple the dispersed fundamental mode to a higher-order mode (HOM) with the opposite sign of dispersion. Ramachandran’s pioneering work in this field further developed this technology to demonstrate its technological viability not only for dispersion compensation but also for dispersion slope compensation and band-pass filtering. Further, using a larger mode area of higher-order modes, the transmission of high-power pulses without suffering the detrimental effects of the fiber’s nonlinearity has been made possible. The reader is therefore directed to Ramachandran’s excellent review [134] for detailed information. This section covers only the properties of LPGs in band-pass applications. LPGs are highly selective mode couplers and have been shown to convert >99.99% of light in the fundamental mode into a higher-order mode (HOM). This strength and purity of coupling with extremely low insertion loss (<0.2 dB) make them ideal candidates for a number of applications. Figure 6.62 shows the LP05 (a) in SMF28 and the LP07 (b) mode excited by LPGs in an optical fiber with a standard SM core and with a deposited cladding diameter of 86 mm and an outer cladding of 125 mm. The reduced refractive index of the deposited cladding makes this fiber rather attractive for its integrity of the HOM. The clarity of the modes indicates the high-quality of mode coupling that is possible with LPGs. In general, band-pass filters (BPFs) fabricated with LPGs have larger bandwidths than FBGs. This is because fewer periods are used for coupling in
86 μm, and an outer cladding
A
B
Figure 6.62 (a) The near-field of the LP05 cladding mode in standard single mode optical fiber. (b) The near-field of the LP07 cladding mode excited in a low cladding NA fiber. (Reproduced with permission from: Ramachandran S., Nicholson J.W., Ghalmi S., Yan M.F., Wisk P., Monberg E., and Dimarcello F.V., “Robust, Single-Moded, Broadband Transmission and Pulse Compression in a Record Aeff (2100 mm2) Higher-Order-Mode Fiber”, ECOC-2005, Post-deadline Session, Paper No. 4.4.1. # IET 2005).
LPG Filters
289 Tx lB
Input Core LPG2
LPG1
Figure 6.63 The general case of the LPG-BPF, in which two LPGs are used to convert the fundamental mode light into a higher-order mode and then back into the core mode after some propagation distance. Of the broadband light coupled to the HOM, only the phase-matched wavelength is reconverted into the fundamental mode at the output using the LPG2. The short LPG1 at the input ensures broadband coupling. The output LPG2 filters out a narrowbandwidth signal. See the text for details on the ultra-broadband LPG1.
the forward direction compared to the FBG, as the period required for LPGs is 100 larger than for reflection gratings. Typically, the scheme for band-pass filtering works on the principle of a Mach–Zehnder interferometer. Figure 6.63 shows the simple configuration of the LPG-BPF. It is worth noting that the LPG period does not change with wavelength for certain fiber dimensions. This has been shown to be the case for specially designed fibers, in which the fundamental mode is surrounded by an additional region of lower refractive index in the cladding and a higher refractive ring farther out. The ring of higher refractive index in the cladding assists the LP02 mode to guide. At shorter wavelengths, the fundamental mode is better confined but begins to penetrate the higher refractive index ring at longer wavelengths, thereby reversing the change in the propagation constant. This feature introduces a turning point in the dependence of the period of the LPG with respect to the wavelength, resulting in an ultra-broadband LPG. This turning point grating (TPG) has been put to good use in designing a number of components, including LPG-BPFs and dispersion compensators [134]. Figure 6.64 shows the refractive index profile of such a fiber with the intensity profiles of the two Refractive index profile LP01
LP02
Figure 6.64 The evanescent field of the fundamental mode increasingly penetrates the outer higher refractive index region at longer wavelengths, thus reversing the change in the propagating constant. The result is broadband phase matching, increasing the bandwidth of LPG coupling to the HOM. (Adapted from reference [134].)
Fiber Grating Band-Pass Filters 123
119
0 –10 –20 –30
1450
1500 1550 1600 Wavelength, nm
LP01transmission, dB
121
63nm
Grating period, microns
290
1650
Figure 6.65 The phase-matching period for the interaction LP01 ! LP02 for a TPG-BPF implemented in a fiber with the profile shown in Fig. 6.61. (Adapted from reference [134].)
propagating and coupled modes, LP01 and LP02. A bandwidth of 63 nm has been reported for mode coupling LP01 ! LP02. The transmission spectrum of a TPG as well as the variation of the period of the phase-matching LPG is shown in Fig. 6.65. Around the designed wavelength of 1550 nm, there are two dips that match the phase-matching conditions on either side. The extinction of 20 dB is thus broadened to 63 nm. These filters have been shown to have very low polarization dependence loss, and dispersion is also low in these filters, a consequence of the broadband nature of the coupling [138]. The coupling between co-propagating modes is characterized by the following equation, 1 2p DbðlÞ dðlÞ ¼ ; ð6:14:1Þ 2 L where the propagation constant mismatch between the modes is given as Db(l), L is the period of the grating, and d(l) is the detuning. To calculate the bandwidth, it is necessary to find the variation of the period of the LPG as a function of the mismatch of the propagation constants when d(l) ¼ 0, so that with L¼
2p Db
ð6:14:2Þ
LPG Filters
291
and differentiating dL Db0 ¼ 2p 2 dlpm Db
ð6:14:3Þ
where, lpm is the phase-matching wavelength. Thus, the bandwidth of these gratings is inversely proportional to the slope of the propagation constant mismatch [139], Dlbandwidth /
1 : Db0
ð6:14:4Þ
However, the denominator on the RHS of Eq. (6.14.4) is proportional to the difference in the group delay between the modes Dtg ¼
l2 Db0 : 2pc
ð6:14:5Þ
Because this quantity cannot be derived from simply the phase-mismatch parameter, d(l), a Taylor series expansion of Eq. (6.14.1) is required, with higher-order dispersion terms of b. As the dispersion terms are linked to the physical parameters of the optical fiber, designing of the bandwidth of the gratings is possible through the design of the fiber. Using the higher-order terms, the following bandwidth relation is determined: lpm Dl / pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; DD Lg
ð6:14:6Þ
where DD is the difference in the dispersion of the two interacting modes. Clearly, when DD ¼ 0, higher-order terms become significant and must be taken into account. Tuning of turn around point (TAP)-LPGs can be achieved easily by either straining the LPG or by the application of heat. A significant difference between normal LPGs and TAP-LPGs is that the turning point determines the minimum period for phase matching of the interacting modes. In a normal LPG, if the period of the grating is not matched to a resonance at the designed wavelength, the resonance shifts to another wavelength. If, however, the period of the grating is smaller than this minimum period in the TAP-LPG, then the coupling has no resonance and no wavelength can match the phase-matching condition. Thus, a broadband dip is seen in the transmission spectra. Therefore, instead of the resonance wavelength shifting, the resonance peak merely diminishes as a function of detuning and turns out to be a powerful way of tuning the amount of coupling by either temperature or strain. This is demonstrated in Fig. 6.66 as a function of temperature and of strain. At some value of strain (or temperature), the TAP-LPG tunes into resonance, but the central wavelength does not shift substantially, nor does the bandwidth because of the broadband nature
Fiber Grating Band-Pass Filters
292
LP01intensity (dB)
0 –5 –10 –15 –20 –25 –30 1450
LP01intensity (dB)
A
1500 1550 1600 Wavelength (nm)
1650
–10 –20 –30
1450
B
Temp 30–80 C
Strain 0–0.027%
1500 1550 1600 Wavelength (nm)
1650
Figure 6.66 A TAP-LPG detuned a function of temperature (a) and strain (b). Note the location of the dip and the constant bandwidth of the device. A 4 C change in temperature or a 0.002% strain changes the transmission by 1 dB for the 1-cm-long TAP-LPG. From reference [134]. (Reproduced with permission from: Siddharth Ramachandran, “Dispersiontailored few-mode fibers: A versatile platform for in-fiber photonic devices”, J. Lightwave Technol. 23(11), 3426–3443, Nov. 2003. # IEEE 2003).
of the coupling. It is clear that these TAP-LPG gratings are extremely useful for sensing applications and have been used to measure the refractive index of a liquid surrounding the fiber using the LP01 ! LP0,14 coupling. An attenuation change of 25 dB was measured for a change in the refractive index of 5 103. However, as with other highly sensitive devices, the ambient refractive index over which this occurs is limited [140]. Another interesting application of these filters is in step tunable dispersion compensation [141]. In this scheme, a series of HOM fibers are used with each subsequent fiber with half the dispersion of the preceding fiber. Thus, if n such fibers are used, then 2n combinations of dispersion are possible, if each module can be switched in or out. Thus, with five such fibers starting with a kilometer length, three wavelengths, error and penalty-free transmission have been demonstrated with a dispersion tunable module, which is step adjustable from
References
293 1km
SLPG
500m
SLPG
SLPG
250m
SLPG
125m
SLPG
62m
SLPG
Figure 6.67 Step tunable dispersion module with switchable TAP-LPGs to switch in or out, sections of HOM fibers [141].
393 ps/nm to þ42 ps/nm (over 30 nm) for carrier suppressed return to zero (RZ) pulses at 40 Gb-s1 [141]. The schematic of the adjustable dispersion compensator is shown in Fig. 6.67. One of the key advantages of TAP-LPG HOM dispersion compensation compared to dispersion compensation fiber is that the mode areas are larger, and therefore nonlinear effects are reduced and should find application in high peak power transmission [142].
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Chapter 7
Chirped Fiber Bragg Gratings “Chirp” is the high-pitched varying sound emitted by certain birds and bats. Gratings that have a nonuniform period along their length are therefore known as chirped. Chirp in gratings may take many different forms. The period may vary symmetrically, either increasing or decreasing in period around a pitch in the middle of a grating. The chirp may be linear, i.e., the period varies linearly with length of the grating [1], may be quadratic [2], or may even have jumps in the period [3]. A grating could also have a period that varies randomly along its length [4], over and above a general trend; for example, uniform, linear chirp. Chirp may be imparted in several ways: by exposure to UV beams of nonuniform intensity of the fringe pattern, varying the refractive index along the length of a uniform period grating [5], altering the coupling constant kac of the grating as a function of position [6], incorporating a chirp in the inscribed grating [7], fabricating gratings in a tapered fiber [8], applying of nonuniform strain [9] etc., many of which have been covered in Chapter 3. All these gratings have special characteristics, which are like signatures and may be recognized as special features of the type of grating. Chirped gratings have many applications. In particular, the linearly chirped grating has found a special place in optics: as a dispersion-correcting and compensating device. This application has also triggered the fabrication of ultralong, broad-bandwidth gratings of high quality, for high-bit-rate transmission in excess of 40 Gb/sec over 100 km [10] or more [11] and in WDM transmission [12]. Some of the other applications include chirped pulse amplification [13], chirp compensation of gain-switched semiconductor lasers [14], sensing [15], higher-order fiber dispersion compensation [16], ASE suppression [17], amplifier gain flattening [18], and band-blocking and band-pass filters [19].
7.1 GENERAL CHARACTERISTICS OF CHIRPED GRATINGS It has been recognized for a long time that gratings can be used for correcting chromatic dispersion [20–22]. Winful [23] proposed the application of a fiber grating filter for the correction of nonlinear chirp to compress a pulse in 301
302
Chirped Fiber Bragg Gratings
transmission in a long grating. It was also suggested that since these gratings display negative group-velocity dispersion, their application in dispersion compensation could be tailored by chirping of the grating [23]. These are known as dispersion compensating gratings (DCGs). Nonlinear transmission is also a subject of renewed interest in long gratings [24–28]. Kuo et al. [29] measured the negative-group-velocity dispersion of Bragg gratings in the visible wavelength region. While the dispersion in these gratings in transmission is large, a limiting factor is the narrow bandwidth over which they can be used to correct for linear dispersion [30,31]. The drawback of such an arrangement is that the gratings have to be used on the edge of the band and consequently introduce a loss penalty, since some of the light is reflected, although it has been shown that strong, long, highly reflective gratings can be used to compensate for dispersion in communication links in transmission with negligible loss [32], by proper design of the grating. For high-bit-rate systems, higher-order dispersion effects become important, dissipating the advantage of the grating used in transmission. The criteria used for the design of the grating to compress pulses in a near ideal manner are a compromise between the reduction of higher-order dispersion and pulse recompression. Bandwidths are limited with this configuration by the strength of the coupling constant and length of a realizable uniform period grating. Uniform period DCGs may find applications in closely spaced WDM systems. If the coupling constant kac is ramped linearly as a function of grating length, the grating exhibits strong dispersion. Used in reflection, an 81-mm-long grating had a coupling coefficient that was varied from zero to 12 cm1 to achieve a dispersion of 3.94 nsec/nm [6] but over a bandwidth of only 0.2 nm. These gratings typically have >99% in-band reflectivity. Despite the large dispersion, unchirped gratings have symmetrical delay characteristics, so that the dispersion changes sign when detuning from one side of the bandgap to the other [33]. With the several possibilities of using unchirped gratings for the management of dispersion, chirped gratings are even more attractive for this application, despite their use in reflection. The application of reflective chirped gratings for dispersion compensation was originally suggested by Ouellette [34]. The group delay through a fiber is large in comparison with the dispersion of standard optical fibers at 1550 nm. A grating reflecting a band of wavelengths distributed over its length benefits from the large group delay in the fiber. We now assess the performance of chirped gratings as a dispersive filter specifically for the compensation of chromatic dispersion. Figure 7.1 shows a schematic of a chirped grating, of length Lg and chirped bandwidth Dlchirp. Referring to Fig. 7.1, we note that the chirp in the period can be related to the chirped bandwidth, Dlchirp of the fiber grating as
General Characteristics of Chirped Gratings Grating ΔΛchirp Λ0
Λshort
303
Λlong
Fiber Lg, Δλchirp Figure 7.1 The chirped grating.
Dlchirp ¼ 2neff ðLlong Lshort Þ ¼ 2neff DLchirp :
ð7:1:1Þ
The reflection from a chirped grating is a function of wavelength, and therefore, light entering into a positively chirped grating (increasing period from input end) suffers a delay t on reflection that is approximately tðlÞ
ðl0 lÞ 2Lg ; Dlchirp vg
for 2neff Lshort < l < 2neff Llong ;
ð7:1:2Þ
where l0 is the Bragg wavelength at the center of the chirped bandwidth of the grating, and vg is the average group velocity of light in the fiber. The effect of the chirped grating is that it disperses light by introducing a maximum delay of 2Lg/vg between the shortest and longest reflected wavelengths. This dispersion is of importance since it can be used to compensate for chromatic dispersion induced broadening in optical fiber transmission systems. At 1550 nm, the group delay t in reflection is 10 nsec/m. Therefore, a meter-long grating with a bandwidth of 1 nm will have a dispersion of 10 nsec/nm. An important feature of a dispersion-compensating device is the figure of merit. There are several parameters that affect the performance of chirped fiber Bragg gratings for dispersion compensation. These are the insertion loss (due to <100% reflectivity), dispersion, bandwidth, polarization mode-dispersion, and deviations from linearity of the group delay and group delay ripple. Ignoring the first and the last two parameters for the moment, we consider the performance of a chirped grating with linear delay characteristics, over a bandwidth of Dlchirp. Giallorenzi and Priest [35] have proposed a figure of merit for coherent communications, but taking into account only the dispersion and the bandwidth of the filter. This approach, while not entirely appropriate for chirped gratings owing to the larger parameter set, is nevertheless a guide in assessing the usefulness of the “ideal” chirped grating. It should be remembered that chirped gratings have a limited bandwidth over which the dispersion is useful, making them different from other truly broadband compensating devices, such as dispersion-compensating fiber [36]. We consider the propagation of an optical pulse in normalized units, in a frame of reference moving in the þz direction at a group velocity vg. The pulse
304
Chirped Fiber Bragg Gratings
amplitude A(z, t), with a normalized amplitude U(z, t), following Agrawal [37], is described as pffiffiffiffiffiffiffi ð7:1:3Þ Aðz; tÞ ¼ Pin Uðz; tÞeaz=2 ; in which a is the attenuation coefficient of the fiber, Pin is the input power, and the frame of reference normalized to the initial pulse width T0 is 1 vg t¼ : ð7:1:4Þ t z T0 For a Gaussian pulse with a 1/e-intensity half-width of T0, the normalized amplitude is [38] Uð0; TÞ ¼ eT
2
=2T02
:
ð7:1:5Þ
Transmission in a linearly dispersive system broadens the pulse, but does not change its shape, and is simply related to the input pulse width by [37] 2 T1 z ¼1þ ; ð7:1:6Þ T0 LD which is the well-known formula for linear pulse broadening, remembering that the dispersion length is a function of the spectral width of the pulse and the linear group velocity dispersion (GVD) of the fiber, b2, as LD ¼
T02 : b2
Combining Eqs. (7.1.6) and (7.1.7) leads to 2 T1 zb ¼ 1 þ 22 : T0 T0
ð7:1:7Þ
ð7:1:8Þ
Since the effect of dispersion in a linear system is additive, we may now modify the GVD parameter b2 by including the dispersion of the grating for the bandwidth of the pulse so that the pulse broadening is 2 bg T1 b ¼ 1 þ 22 z þ 2 Lg ; ð7:1:9Þ T0 T0 T0 where bg is the GVD of the grating of length Lg. We note that the dispersion Df of the fiber is related to the GVD by [37] b2 ¼
l2 Df : 2pc
ð7:1:10Þ
A similar expression relates the fiber grating dispersion Dg to the grating GVD, but with the sign depending on the sign of the grating chirp. We now find that the pulse broadening for the Gaussian pulse is
General Characteristics of Chirped Gratings
T1 T0
2 ¼ 1 þ 2pc
Dl2 ðDf z þ Dg Lg Þ; for Dl Dlchirp : l2
305
ð7:1:11Þ
In Eq. (7.1.11) we have used the relationship between a transform-limited pulse width and its spectrum (doT0 ¼ 1). Note the stipulation on the bandwidth of the pulse, since dispersion compensation is only valid for the bandwidth of the grating. If the pulse bandwidth is larger, then the pulse recovery is 2 T1 2pc ¼ 1 þ 2 ðDl2 Df z þ Dl2chirp Dg Lg Þ; for Dl > Dlchirp : ð7:1:12Þ T0 l For perfect recompression, Dfz ¼ –DgLg, and the pulse remains unaltered at the output of the fiber, so long as the bandwidth of the pulse is smaller than the bandwidth of the grating. We can now define a figure of merit (FOM) for the bandwidth of the grating, since the maximum compression ratio that can be achieved is 2 T1 2pc ¼ 1 þ 2 ðDl2chirp Dg Lg Þ ¼ 1 þ M 2 : ð7:1:13Þ T0 l We can redefine Eq. (7.1.13) by recognizing that the dispersion Dg of the grating is almost exactly 10 nsec/m/Dlchirp, so that M2 ¼
2pc ðDlchirp Lg 108 Þ: l2
ð7:1:14Þ
We note that the FOM is proportional to the square root of the length and the chirped bandwidth of the grating. Here, we remind ourselves that we have used the 1/e bandwidth of the grating. The conversion from the Gaussian 1/e width to its FWHM width, which is more commonly used, may be done by using the following relationship: 2 TFWHM ¼ 4 ln 2: T02
ð7:1:15Þ
It is clear from Eq. (7.1.13) that the pulse broadening, which can be compensated for, is DT 2 ¼
T12 T02 ¼ M2: T02
ð7:1:16Þ
As an example, a 1-meter-long grating with a bandwidth of 10 nm will have M ¼ 280. This means that an input pulse can undergo a pulse broadening of 280 times its initial pulse width and be recompressed. The dependence of the FOM on the grating bandwidth for maximum recompression is shown in Fig. 7.2. In the simple analysis just given, it is necessary to recognize that the chirped grating response is far from ideal. The actual reflection and detailed
306
Chirped Fiber Bragg Gratings
FOM / (meter ^ 0.5)
300 250 200 150 100 50 0 0
2
4 6 8 Chirped bandwidth, nm
10
12
Figure 7.2 The maximum pulse re-compression FOM per √(meter) of grating length. For optimum compression, the bandwidth of the pulse is the same as the chirped grating bandwidth. As the bandwidth gets smaller, the pulse width becomes larger, so that the figure of merit drops.
delay characteristics can have a profound influence on the performance, especially when the grating is to be used for compensation of large dispersion in ultrahigh-bit-rate systems. However, the FOM is a good indicator of the best possible performance of a grating and may be used to compare the performance achieved with gratings. Ultimately, the most important parameters that characterize a transmission link’s performance are the bit-error rate (BER), loss penalty, and error floor. The influence of deviations from ideal transfer characteristics on the BER and loss penalty is considered in Section 7.5.
7.2 CHIRPED AND STEP-CHIRPED GRATINGS We have seen the theory of fiber Bragg gratings in Chapter 4. Although it is possible to mathematically express the coupled modes in a way that exactly mimics the grating function, the methods of computation are numerical, since no suitable analytical solutions are available. The transfer matrix method (TMM) is ideally suited to chirped gratings, since the grating may be broken up into smaller sections of uniform period and/or refractive index profile. While there are other methods for extracting the reflection, transmission, and group delay response (see Chapter 4), in the following we use the TMM approach to evaluate the transfer characteristics of arbitrarily chirped gratings. There are naturally limitations to the application of the TMM. Since coupled mode analysis depends on the slow variation of the parameters of the grating as a function of the wavelength of light, e.g., chirp, refractive index modulation, and coupling constant kac, it is not possible to compute entirely “arbitrary” gratings (see Chapter 4, Grating Simulation). Apart from this limitation, there are other questions that need addressing: for example, in
Chirped and Step-Chirped Gratings
307 Uniform period grating
A Lg
Weakly chirped grating
B
Step-chirped grating d11 Λ1
C
d12 Λ2
d1N–1 ΛN–1
d1N ΛN
Chirped bandwidth, Δl
Figure 7.3 A uniform grating (a), a weakly chirped grating (b), and a step-chirped grating (c).
the synthesis of chirped fiber Bragg gratings, what constitutes a continuous chirp in view of the fact that most gratings exhibit quasi-continuous chirp, and the influence of apodization on the dispersion and reflection characteristics. First it is necessary to view the grating as a physical entity, in which coupling parameters are a weak function of space. Thus, it may be seen that a chirped grating, shown in Fig. 7.3, is merely a uniform period grating that has been slightly perturbed. How many sections do there need to be in grating (c) for it to be indistinguishable from a grating that is continuously chirped (b)? In other words, how small can the chirp parameter, df(z)/dz [Eqs. (4.3.9) and (4.3.10)] be for a given length of grating? In order to answers this question, we consider the bandwidth characteristics of the uniform period Bragg grating. We note that the bandwidth of an unchirped grating section dl is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 ðkac dlÞ2 þ p2 ; ð7:2:1Þ dl ¼ 2pneff dl from Eq. (4.6.14), and we have assumed that each section is identical in length dl. For most of the gratings of interest here, we assume that (kdl)2 p2. The phase matching condition for the section requires that lBragg ¼ 2Lg neff ;
ð7:2:2Þ
where Lg is the period of the grating section. The period is nearly constant for gratings with a small percentage chirp. Remembering that dl ¼ Lg/N, we get 2neff dl N ¼ : Lg pl2Bragg
ð7:2:3Þ
When N ¼ 1, the bandwidth of the grating is simply the bandwidth Dl0 of the unchirped grating of length Lg. For the chirped grating with a bandwidth >Dl0 ,
308
Chirped Fiber Bragg Gratings
made of sections, the bandwidth of each section can only be greater than the bandwidth of the unchirped grating (being shorter in length), but can equal the bandwidth of the chirped grating only if it is the appropriate length. Applying the relationship [Eq. (7.2.3)] for bandwidths greater than the unchirped bandwidth, Dl0 , we simply allow the bandwidth of each section to be identical to the bandwidth Dlchirp of the chirped grating, i.e., dl ¼ Dlchirp ;
ð7:2:4Þ
so that for a fiber Bragg grating at a wavelength of 1550 nm, N/Lg ffi 0.4Dlchirp steps/(mm-nm). Finally, we arrive at the relationship between the number of steps per unit length and the chirped bandwidth, N 2neff Dlchirp : ¼ L pl2Bragg
ð7:2:5Þ
Here lBragg is the central Bragg wavelength of the chirped grating. The simple relationships of Eqs. (7.2.4) and (7.2.5) are minimum requirements for the stepchirped grating and should approximate to a continuously chirped grating. It may be seen immediately that there is an intuitive feel about the conclusion – that the bandwidth of each step of the grating should be at least as large as the chirp of the whole grating. Increasing the number of steps, i.e., dl ! 0, approaches the continuously chirped grating. A more detailed analysis [39] bears out the simple conclusion; a more accurate (to 1%) fit increases the bandwidth of each step by only 40% [39]. Curiously, the comparison between gratings of the same chirp but different number of steps gets better if the total chirp is reduced by Dlchirp/N. The convergence to the asymptotic transfer characteristics is faster for all SCGs with the chirp bandwidth adjustment, although this is especially true for those with a few sections. For a large number of sections, this really does not matter. For all the following simulations, the bandwidth adjustment has been included. Figures 7.4 and 7.5 show the calculated reflectivity and delay, respectively, of an SCG (Lg ¼ 100 mm, Dlchirp ¼ 0.75 nm) grating with 200 sections (2 steps/mm) and an SCG with only 0.42 steps/mm [total steps ¼ 42, according to Eq. (7.2.5)]. It is immediately apparent that the agreement between the two spectra is good. The reflectivity spectrum has a ripple that is characteristic of unapodized gratings, as does the group delay. The strong ripple in the group delay plays an important role in the recompression of a dispersed pulse. While there is an average delay slope, the ripple frequency becomes smaller toward the end from which the reflection is measured. This is true for even continuously chirped gratings, and the influence of apodization is examined in Section 7.3. We now examine a 4-mm-long SCG with a chirp of 1 nm. The minimum number of sections of this grating according to Equation (7.2.5) is 2. Simulations of the reflectivity and delay are shown in Figs. 7.6 and 7.7, respectively, for two
Chirped and Step-Chirped Gratings
309
1.0 0.9 0.8
Reflectivity
0.7 0.6 0.5 0.4
Reflectivity, theory (200 sections)
0.3
RX (42 sections)
0.2 0.1 0.0 1561.0 1561.1 1561.2 1561.3 1561.4 1561.5 1561.6 1561.7 1561.8 1561.9 1562 Wavelength, nm Figure 7.4 Reflectivity of a 100-mm-long SCG with 200 sections as well as 42 sections. The reflectivity curves have been offset to allow easy examination.
1561.0 100
1561.2
1561.4
1561.6
1561.8
1562.0
Delay, ps
–400
–900 Theoretical unapodized time Delay(ps)(200 sections) Theoretical delay(42 sections) –1400 Wavelength, nm Figure 7.5 Comparison of the delay characteristics of the SCGs shown in Fig. 7.4. For the 200-section grating, N/Lg ¼ 2 steps/mm, while for the 42-section grating, N/Lg ¼ 0.42 steps/mm.
310
Chirped Fiber Bragg Gratings 1550.0 0
1550.5
1551.0
1551.5
1552.0
1552.5
1553.0
1553.5
1554.0
–5
Reflectivity, dB
–10 –15 –20 –25 –30 –35
10LOG(Rx) 2 sections
–40
10LOG(Rx) 3 sections 10LOG(Rx) 50 sections
–45 Wavelength, nm Figure 7.6 The reflectivity of 4-mm-long, 1-nm chirp SC gratings with 50 (dashed line: N/Lg ¼ 12.5 steps/mm), 2 (gray line: N/Lg ¼ 0.5 steps/mm), or 3 sections (continuous line: N/Lg ¼ 0.75 steps/mm).
1550.0 60
1550.5
1551.0
1551.5
1552.0
1552.5
1553.0
1553.5
1554.0
40 Time delay(ps) 2 sections
20
Time delay(ps) 3 sections
Delay, ps
Time delay(ps) 50 sections
0
–20
–40
–60 Wavelength, nm Figure 7.7 Theoretical delay of 4-mm-long, 1-nm chirp SC gratings with 50 (dashed line), 2 (gray line), and 3 (continuous line) sections. Notice that even with so few sections as determined by the simple relationship of Eq. (7.2.5), the characteristics are very similar.
Minimum number of sections/mm
Chirped and Step-Chirped Gratings
311
16 14 12 10 8 6 4 2 0 0
5
10
15 Chirp bandwidth, nm
20
25
30
Figure 7.8 Design diagram for step-chirped gratings (Reprinted from Kashyap R., “Design of step chirped gratings,” Optics Commun., Copyright (1997), 461–469, with permission from Elsevier Science. Ref. [39]).
and three sections. Along with these gratings the characteristics of a 50-section grating are also shown. The grating characteristics are surprisingly similar despite the few sections, especially noting the positions of the zeroes and the central part of the reflection spectrum. The agreement is equally valid for apodized fiber Bragg gratings [39]. For convenience, it may be simple to double the minimum number of calculated sections for good linearly chirped gratings. The design of quasi-linearly chirped gratings has been represented graphically in Fig. 7.8. This design diagram shows that a grating must be divided into a minimum number of sections per millimeter for a given chirp, irrespective of the length of the grating. The criterion used for the design is that the total group delay of a continuously chirped grating with the same coupling constant and length as the step-chirped grating should differ by less than 1% of its maximum value. For example, the maximum deviation in the delay ripple in a 100-mmlong unapodized grating should be less than 10/psec across 90% of the available bandwidth. This result is numerically evaluated and serves as a useful guide for a variety of chirped gratings [39]. We now consider the effect of having far fewer sections than the required minimum in a chirped grating. When this happens, the bandwidth of each section is no longer sufficient to overlap with the bandwidth of succeeding sections, so that the reflection spectra break up into several discrete peaks, each representing the effect of the single sections. However, the residual reflections due to the edges of the gratings (start/stop, see Section 7.4) do interfere with the others, causing the peaks to be altered from the smooth curves of uniform period gratings. The transmission spectrum of a 5-nm bandwidth, 8-mm-long grating is shown in Fig. 7.9 for two values of refractive index modulation:
312
Chirped Fiber Bragg Gratings 1.0 A
Transmission
0.8 0.6 0.4
B
0.2 0.0 1545
C 1547
1549
1551
1553
1555
Wavelength, nm Figure 7.9 The effect of too few sections in a grating of length 8 mm with a bandwidth of 5 nm. For 6 sections and a small refractive index (A, 0.75 sections/mm, Dn ¼ 4 104) the spectrum breaks up into individual peaks. The minimum number of sections for a continuously chirped grating is 3 sections/mm. For stronger refractive index modulation (B, Dn ¼ 1 103), the grating “appears” continuously chirped. However, neither the delay (not shown) nor the reflectivity spectrum matches those of the continuously chirped gratings, C.
A, Dn ¼ 4e-4 and B, Dn ¼ 1e-3, as well as a continuously chirped grating (C, Dn ¼ 1e-3). Note that while the increase in the reflectivity smoothes out the structure, the spectrum deviates from the continuously chirped grating spectrum. In particular, the side-lobe structure, which is absent in curve C, is obviously present in curve B. For the design of broadband reflectors, it is important to incorporate the correct number of sections; otherwise, the dispersion characteristics will suffer dramatically, as will the out-of-band reflectivity. The out-of-band reflection in continuously chirped gratings is generally apodized, since the distributed reflections from all the “different” sections may be thought to add in a way that averages out any coherent buildup. Within the band, it is an entirely different story. The edges of a grating have a large impact on the reflection ripple as well as the delay ripple, as seen in Figs. 7.4 and 7.5. Consequently even small broadband reflections can change the ripple in the delay spectrum.
7.2.1 Effect of Apodization In Chapter 5 we saw the effect of apodization on gratings; the immediate effect was the dramatic reduction in the side-lobe levels in the reflection spectrum. Chirped gratings tend to have lower side-mode structure in their reflection spectra to begin with, and the advantage of apodization is in the reduction of internal interference effects that cause the group delay to acquire a ripple. We consider here the properties of chirped gratings, which have reflectivities of
Chirped and Step-Chirped Gratings
313
the order of 10 dB, suited to the compensation of linear dispersion in fibers, and study the influence of apodization. While the details of both the reflection and group delay spectra change with the strength of the coupling constant of the grating, general observations remain essentially unchanged. The chirped grating is a continuously distributed reflector. Ideally, light entering into a chirped grating from one end should be dispersed in exactly the opposite way when entering from the other end. Early measurements on chirped gratings did show this feature [40]. However, the gratings were short, had a large bandwidth, and consequently had a small dispersion. Dispersion is not generally reversible with unapodized chirped gratings. To understand this phenomenon, we remind ourselves that light entering from the short-wavelength end of a highly reflective chirped grating is reflected such that only a small fraction of the short-wavelength light penetrates through to the other end of the grating, while the longwavelength light does. On entering from the long-wavelength end of the grating, exactly the opposite occurs. The detailed delay ripple is a result of the interference between the broadband reflection due to the edge of the grating and the distributed nature of the reflection of the grating [41]. As a crude comparison, when light enters from the long-wavelength end of the grating, the interference is predominantly due to the large long-wavelength reflection from the front of the grating and the small broadband reflection due to the front edge. Short-wavelength light penetrates the dispersive grating and is predominantly reflected from the rear end; it, too, interferes with the low broadband reflection from the front end. In neither case does the rear end of the grating play a strong role. Since the dispersion increases with greater penetration into the grating, the delay ripple frequency increases with decreasing wavelength (see Fig. 7.5). With the launch direction reversed, exactly the opposite occurs: The delay ripples increase in frequency with increasing wavelength. Therefore, light dispersed by reflection from the short-wavelength end of the chirped grating cannot be undone by reflection from the long-wavelength end! The simulated result of this asymmetry is shown in Fig. 7.10. The sign of the frequency chirp in the delay ripple is insensitive to the launch direction, i.e., the frequency of the chirp is always from a low to a high frequency (Fig. 7.10, B and C) when viewed from either end. The role played by the rear end of the grating is apparent when the coupling constant kac is apodized asymmetrically. In this example we consider a grating with a profile of the refractive index modulation as shown in Fig. 7.11a. The grating profile is half-cosine apodized so that the light launched from the long-wavelength end sees a gradually increasing coupling constant. The amplitude of the light reflected from the front end of the grating is now lower than in an unapodized grating, and long-wavelength light penetrates more deeply so that the amplitude at the rear end is higher than in the unapodized grating. The broadband reflection from the input of the grating has now been reduced significantly due to
1553.7 0 –2 –4 –6 –8 –10 –12 –14 –16
1553.9
1554.1
1554.3
1554.5
Launch direction B
C A Group delay D
1554.7 300 100 –100 –300 –500 –700 –900 –1100 –1300
Group delay, ps
Chirped Fiber Bragg Gratings
Reflectivity, dB
314
Wavelength, nm Figure 7.10 Reflection and delay spectrum of an unapodized 100-mm-long grating with a chirped bandwidth of 0.75 nm. The reflection spectrum remains unchanged when measured from either end (A). The group delays (B and C) have been computed for light launched in the direction shown, from the short-wavelength end (B) and the long-wavelength end (C). The net dispersion, which is the sum of the two, is shown as curve D. This residual dispersion prevents full recompression of a pulse dispersed by the grating.
A, B
Amplitude of kac
C, D
Lgrating
A 1553.8
1553.9
1554.0
1554.1
0
Group delay, ps
–200 –400 –600 –800
C D
1554.2
1554.3
Launch direction
1554.4
1554.5
1554.6
1554.7
A
Computed data
Computed (longwave launch) Computed (shortwave launch) Delay (SW launch) Delay (LW launch)
B
Measured data –1000 –1200
B
Wavelength, nm
Figure 7.11 Apodization profile of the half-cosine long wavelength-edge apodized chirped grating is shown above in figure (a). Measured and computed group delay (b), when measured in both directions.
apodization. Hence, the ripple should disappear on the long-wavelength end. However, we notice that the ripple is of the order of that of the unapodized grating, but now of higher frequency at the input end, Fig. 7.11b, A. This is indicative of interference from the reflection off the rear end of the grating. Note, too, that
Chirped and Step-Chirped Gratings
315
the ripple has the lowest frequency and disappears at the shortest wavelengths, quite the reverse of the unapodized grating, with reduced interference from the launch end (due to apodization). The corresponding measured result for this type of a grating is shown in Fig. 7.11b, B [42]. On the other hand, when light is launched into the short-wavelength end, the reflected delay ripple is almost identical to that of an unapodized grating (Fig. 7.11b, C and D). The apodized long-wavelength end does not play a role in generating the delay ripple. This result is of particular importance for long chirped gratings. When one half of a grating remains unapodized while the other half is cosine apodized, the results are even more dramatic, as shown in Fig. 7.12. Shown in curve A is the group delay of an unapodized grating, while B and C refer to the grating profiles shown above the figure in (B) and (C) with light launched in the directions shown for each grating. The group delay ripple measured from the long wavelength end, B, has all but disappeared for the long-wavelength apodized grating, and the residual ripple at the long-wavelength edge is again due to the interference from the short-wavelength end. For light launched into the long-wavelength end in the short-wavelength apodized grating (C), the group delay ripple C is as for the unapodized grating, A.
C
B Amplitude of kac ls
Lgrating /2
ll
Lgrating /2
ls
Lgrating /2
Lgrating /2
ll
C
B 1550.0
Delay, ps
–200
1550.2 B
1550.4
1550.6
1550.8
1551.0
A
–600 C –1000 –1400 Wavelength, nm
Figure 7.12 Apodization profiles (half-cosine over half the grating) and respective launch directions shown in (B) and (C). Group delay compared for long-wavelength launch into a grating, A: unapodized, B: short-wavelength apodized, and C: long-wavelength apodized (after Ref. [42]).
316
Chirped Fiber Bragg Gratings
The implication of the apodization is as follows: Long chirped gratings for dispersion compensation require apodization only on the long wavelength end of the grating. The type of apodization (see Chapter 5) will determine the bandwidth reduction in the reflectivity spectrum. The unapodized short wavelength end, provides extra bandwidth, with a small penalty on the long wavelength end due to the residual ripple. An important factor that influences the performance of chirped gratings in dispersion compensation is the deviation of the delay from linearity and group delay ripple. Symmetrically apodized gratings offer the prospect of excellent dispersion compensation [43]. The group delay differences from linear delay and reflectivity for commonly found cosine and raised cosine profile apodized gratings are shown in Fig. 7.13. The gratings have a peak reflectivity of 90% and are 100 mm long with a bandwidth of 0.75 nm (Dg ¼ 1310 psec/nm), designed for compensation of the dispersion of 80 km of standard telecommunications fiber (Df ¼ 17 psec/nm/km). The group delays have two features in common: The dispersion curves deviate from linearity slowly across the bandwidth of the grating, and they are flat within 5 psec. With higher-reflectivity gratings, the curvature worsens. Note, however, that the stronger, raised cosine apodization eliminates the delay ripple almost entirely, but reduces the available bandwidth. Roman and Winnick [44] have shown that using Gel’fand–Levitan– Marchenko inverse scattering analysis, it is possible to design a grating with a near perfect amplitude and quadratic phase response to recompress transform limited pulses. With asymmetric apodization as shown in Figs. 7.11 and 7.12, apodizing only one end of the grating has a beneficial effect of better bandwidth utilization than symmetrically apodized gratings, since less of the grating length is used in the apodization process. There is a slight increase in the peak-to-peak group delay ripple on the long-wavelength side, as seen in Fig. 7.12, but it is still <5 psec over a wider bandwidth. With a stronger coupling constant and a different apodization function (e.g., tanh), we note that less of the light reaches
–10.0 –20.0
B A
1550.2
1550.4
1550.6
1550.8 20
D
10 0 C
–30.0
–10
Group delay difference, ps
Reflectivity, dB
1550.0 0.0
–20
–40.0 Wavelength, nm
Figure 7.13 Reflectivity (A) and group delay difference (B) of cosine apodized grating as well as reflectivity (D) and group delay difference (C) for raised cosine apodized grating.
Chirped and Step-Chirped Gratings
Reflectivity, dB
–5
1550.2
1550.4
B
1550.6
1550.8
45
A
35
–10
25
–15 –20
1551.0
D 15
C
Δ (delay), ps
1550.0 0
317
5
–25 –30
–5 Wavelength, nm
Figure 7.14 Comparison between the delay and reflection spectra of asymmetrically (B, D), and symmetrically (A, C) apodized gratings.
the rear end of the grating, so that the ripple reduces still further. However, the curvature also increases. Figure 7.14 shows the reflection and group delay difference spectra of a symmetric and an asymmetric tanh apodized grating. The bandwidth of the asymmetric apodized grating, B, is now wider than that of the symmetric apodised grating, A, and so is the group delay difference, D. The peak reflectivity of B is 98%, whereas that of A is 90%. There is also a small ripple acquired in both the reflection and group delay spectra (B and D). With lower reflectivity, the ripple in the asymmetrically apodized grating increases, rather than decreasing as is the case with the symmetrically apodized grating [41].
7.2.2 Effect of Nonuniform Refractive Index Modulation on Grating Period During fabrication, it is necessary to ensure that the grating receives the correct UV dose along its length. If the dose varies, so does the effective refractive index modulation and therefore kac. A constant increase in the UV dose with length merely chirps the grating. However, random variations are generally common with pulsed lasers, since the UV radiation has hot spots across the beam, with the result that the grating is no longer uniformly exposed. While this may not be a problem for many filtering applications, it does degrade the performance of the group delay in chirped gratings, limiting performance. Ouellette [4] reported the effect of a noisy refractive index profile and dither in the period of the grating on the reflection and dispersion characteristics. It was found that, apart from a general increase in the out-of-band reflection, the group delay was also degraded. A period variation of 0.03 nm
318
Chirped Fiber Bragg Gratings
(5%) over a length scale of 1 mm degraded the delay spectrum substantially. This is a serious issue for the fabrication of high-quality gratings. Even with perfect phase masks, such factors as the random variations in the effective index of the mode, UV dose, or vibration during fabrication will cause deterioration in the quality of the grating. We consider the likely effect of a maximum variation of 10% of the refractive index modulation amplitude, Dn (7.5 105), but over different scale lengths of 50, 100, and 200 microns. These are expected to be typical regions over which the refractive index modulation varies. In order to model this behavior, we have assumed that each section of the scale length varies in the index modulation entirely at random with a maximum value of 1 105. This is realistic despite the averaging effect of multiple pulse exposure, since the peak UV intensities can fluctuate over several orders. The results of the simulations of the deviation from linearity of the delay and the transmission spectra are shown in Figs. 7.15 and 7.16. The uniformly exposed curve A is degraded rapidly on the 50 micron length scale with a high frequency ripple (C), to a slowly varying, more uniform, but larger-amplitude ripple across the spectrum with increasing length scale (C to D). The last curve D is consistent with observations of delay ripple variation [45]. It is very difficult to distinguish small differences from the reflection spectra of high reflectivity gratings [4]. We therefore choose to observe the grating transmission spectra, which can clearly resolve these differences. Figure 7.16 shows the transmission spectra, shifted vertically in order to highlight the small changes in the curves, using the same data as in Fig. 7.15. The noise on the spectra is apparent, although the energy in the reflected light is only slightly affected. The significant effect of the noisy profile is on the delay ripple. The variation
Group delay difference, ps
40
B C D
30 20 10 0 –10 1550.1
A 1550.3
1550.5
1550.7
1550.9
Wavelength, nm Figure 7.15 Group delay difference from a slope of 1310 psec/nm for cosine apodized 100-mm-long grating (90% reflector) with no variation (A) in the refractive index modulation, dDn; random variation of up to 1 105 over a length scale of 50-micron sections (B); 100-micron sections (C); and 200-micron sections (D).
Super-Step-Chirped Gratings
Transmission, dB
1550.0 3
1550.2
319 1550.4
1550.6
1550.8
1551.0
1 –1 –3
D C
–5
B A
–7 –9
–11
Wavelength, nm Figure 7.16 The transmission spectra of four cosine apodized gratings (Lg ¼ 100 mm, Dn ¼ 7.5 105, Dlchirp ¼ 0.75 nm) as a function of random variation in the refractive index modulation. A has dDn ¼ 0; B, C, and D have a maximum random variation dDn ¼ 1e 105 over length scales of 50, 100, and 200 microns, respectively, as in Fig. 7.15.
dlBragg in the Bragg wavelength as a function of the change in the refractive index Ddn and the grating period dLg is dlBragg ¼ 2Lg dDn þ 2neff dLg ;
ð7:2:6Þ
where is a core overlap factor of 0.9. The shift in the Bragg wavelength amounts to 10 pm at a wavelength of 1550 nm for an index change of 1 105. The same change in the wavelength occurs for a random variation in the Bragg wavelength period dLg of 3.42 pm.
7.3 SUPER-STEP-CHIRPED GRATINGS Chapter 3 introduces the fabrication of ultralong gratings. One of the methods of generating ultralong gratings is by stitching together a set of short chirped gratings, to form the super-step-chirped grating (SSCG) [46]. The structure is schematically shown in Fig. 5.18, Chapter 5. Here we consider the influence of an imperfect stitch between two sections of the SSCG. Figure 7.17 shows three step-chirped gratings with gaps in between. The first SCG (LHS) begins at l1 and finishes at l2, in N sections, each with an integral number of periods. The number of periods is adjusted so that each section is nominally the same length within the length of a period. The second SCG begins with a period l2 þ dl and ends at a period l2 þ (N – 1) dl. The two gratings are written sequentially, ideally with zero gap in between. In a perfect grating, the periods would simply follow each other, without a perturbation. In reality, unless care is taken, there is a small error dl/L2 at the join. This error will ultimately have an effect on the performance of the grating.
320
Chirped Fiber Bragg Gratings λ1
λ1 + 0.8nm
Step-Chirped Grating 1
λ1 + 1.6nm
Step-Chirped Grating 2
JOIN
λ1 + 2.4nm
etc
Step-Chirped Grating 3
JOIN
Figure 7.17 A schematic of the super-step-chirped grating (SSCG). Each grating is of nominally an identical integer number of periods [47].
We now compare this with a uniform period grating with a phase step. A phase step of a quarter wavelength, as we have seen in Chapter 6, introduces a bandgap in the reflection spectrum. The effect is similar in gratings, which have a small chirp, since the Bragg wavelengths of two adjacent sections are only slightly different, and the reflection spectrum is strongly overlapped. This is indeed the case with SCGs and is shown in Fig. 7.18. The reflection spectrum centered at the wavelength of the join shows the quarter wavelength dependence of the bandgap. However, the dependence changes with increasing gap, but the join is also invisible at a gap of 10.25 lBragg. The group delay is more sensitive but is only affected when light crosses the gap. This means that light launched from the long-wavelength end experiences a change in the group delay ripple on the short-wavelength side of the gap, and vice versa when traveling from the short-wavelength end. This has implications for SSCGs made with more than two SCGs. The grating nearest the launch end remains essentially unaffected, while the group delay ripple of subsequent gratings deteriorates. 1 Reflectivity
0.8 GAP/lambda A = 10.25 B = 0.25 C = 0.50 D = 0.75 E = 10.5
0.6 0.4 0.2 0 1560.9
1560.95
1561
1561.05
1561.1
Wavelength, nm Figure 7.18 The reflection spectrum of the join region of 2 100-mm-long SCGs as a function of the gap at the join in fractions of the Bragg wavelength. The gratings are 100 mm long with a chirped bandwidth of 0.75 nm each.
Super-Step-Chirped Gratings
Group delay, ps
1560.9 –400
321 1560.95
1561
1561.05
1561.1
GAP A = 0.25 × lambda
–600 –800 –1000 –1200 –1400
GAP B = 10.5 × lambda
–1600 Wavelength, nm Figure 7.19 The group delay at the join of the gratings shown in Fig. 7.18 for two values of the gap: 0.25 (A) and 10.5 (B) times the Bragg wavelength. The delay spike is localized to the join and is not apparent from the long-wavelength end of the grating. In this simulation, the light enters the long-wavelength end of the grating. The effect is reversed if the light is launched from the short-wavelength end.
Figure 7.19 shows the effect of the join on the group delay. At the join, there is a localized discontinuity, which becomes narrower as the gap gets larger, and almost disappears. Simulations have shown that large gaps (5 mm) tend to smooth out the effect of the join, but the delay through the gap does introduce a step change in the group delay on either side of the join. We see a relative increase in the noise from the long-wavelength end (launch end). In comparison, a 500-mm-long SSCG grating with random stitching errors at every 100 mm is shown in Fig. 7.20 [46]. Here, the stitching produces small spikes at the join, and also a general increase in the noise at the short-wavelength end. These are believed to be due to the cumulative effects of each join. However, as has been reported, these gratings may still be useful for dispersion compensation at 10 Gb/sec [48].
–5 –10
1557
1558
1559 0.1
0.2 0.35
0.2
–15
1560 1000 800 600 400
–20
200
–25
0
–30
Δ(Group delay), ps
Reflectivity, dB
1556 0
–200 Wavelength, nm
Figure 7.20 The reflectivity and group delay difference of a 500-mm-long SSCG with random stitching errors, shown as a fraction of the Bragg wavelength.
Chirped Fiber Bragg Gratings 1556.0 –10 –15 –20 –25 –30 –35 –40 –45 –50
1556.8
1557.6
1558.4
1559.2 4800 4200 3600 3000 2400 1800
mag_ab delay, ps
Delay, ps
Reflectivity, dB
322
1200 600 0
Wavelength, nm Figure 7.21 The measured characteristics of a 400-mm-long SSCG with a chirped bandwidth of 3 nm and random stitching errors. Compare with Fig. 7.20.
In Fig. 7.21 is shown the measured reflectivity and relative group delay characteristics of a 400-mm-long SSCG made in four sections with random stitching errors. We note that the characteristics of the relative group delay are better than the simulation suggests. This is probably due to small random variations in the chirp of each grating section. The computed reflectivity and group delay of a 1.3-meter-long SSCG [48] with a bandwidth of 10 nm and no stitching errors are shown in Fig. 7.22a. The overall reflectivity is similar to shorter gratings, but as the bandwidth is increased, the shorter-wavelength end group delay suffers less from the broadband reflection of the long-wavelength edge of the grating. Note that in this simulation, the resolution of the computation (1 pm) has lost the information on the group delay ripple. Figure 7.22b shows a high-resolution simulation (0.1 pm) of the reflectivity and group delay, to highlight the GDR of the first part of a 1-meter-long unapodized grating. A detail of the group delay ripple of 60 ps is shown in Fig. 7.22c. Long gratings with only a small amount of apodization on the longwavelength end (normally the input end for dispersion compensation) will remove a substantial part of the group delay ripple and should be useful for transmission rates in excess of 10 Gb/sec.
7.4 POLARIZATION MODE DISPERSION IN CHIRPED GRATINGS An issue that becomes important at high transmission bit rates (>10 Gb/sec) is the effect of polarization mode dispersion (PMD). As the transmission rate increases, the bit period reduces. If any component in the transmission path is birefringent, the different pulse arrival times of the two polarizations can
Polarization Mode Dispersion in Chirped Gratings 1554
1556
1558
1560 0
–1
–20
–2
–40
–3
–60
–4
–80 –100
–5
A
Wavelength, nm 1550.0 0.0
Reflectivity, dB
Δ (Delay), ps
1552
1550.2
1550.4 1550.6 1550.8
1551.0 200 150 100 50
–0.5 –1.0
0 –50
–1.5 –2.0
–100
B
Relative group delay, ps
Reflectivity, dB
1550 0
323
Wavelength, nm 1550.900 60
1550.902
1550.904
1550.906
GDR, ps
40 20 0 –20 –40
C
–60 Wavelength, nm
Figure 7.22 (a) The simulated reflectivity and group delay difference form a linear slope of 1310 psec/nm, for a 1.3-m-long SSCG with no stitching errors. (b) High resolution (0.1-pm) simulation of a 1-meter-long grating showing the reflectivity and the group delay ripple. (c) Detail of the relative group delay ripple of (b).
degrade the BER. In long transmission systems, these two polarizations mix stochastically, so that pulse broadening is not easy to compensate [49]. In a short grating component, however, PMD is not generally large in transmission, since the pulse arrival times are simply due to the difference in the propagation constants of the two polarization states times the length of the grating. In a chirped dispersion compensating reflection grating (DCG), the effect of birefringence is more severe, causing a large additional dispersion. PMD, or more correctly, birefringence induced PMD, in unapodized gratings is more of a nuisance than in unapodized gratings. In order to assess the impact on the PMD of birefringence in a fiber, whether intrinsic or due to the process of fabrication of a grating, we examine how the
324
Chirped Fiber Bragg Gratings
Bragg wavelength of a grating is affected by a change in the effective index of a mode. We assume that the central Bragg wavelength of a chirped grating is lBragg ¼ 2neff Lg ;
ð7:4:1Þ
so that the change in the reflection wavelength as a function of the change in the mode index becomes, dlBragg ¼ lBragg
dneff : neff
ð7:4:2Þ
Equating the change in the mode index to the birefringence in the fiber leads to dlBragg ¼ lBragg
B : neff
ð7:4:3Þ
For a DCG with a dispersion of Dg psec/nm, a change in the Bragg wavelength as a result of the change in the polarization induces a delay, tPMD ¼ doBragg Dg, from which we get the result tPMD ¼ lBragg B0 Dg ;
ð7:4:4Þ
Log(PMD), ps
where B0 is the normalized birefringence, B/neff. Equation (7.4.4) shows that the PMD is dependent on the dispersion and birefringence but not on the length of the grating. A chirped reflection grating with a perfectly linear dispersion of 1310 psec/nm and a birefringence of 105 at a wavelength of 1550 nm will have a PMD of 28 psec! This result is of the order that has been reported in apodized DCGs [50]. It is clear that even a small birefringence causes a severe PMD penalty. Figure 7.23 shows how the PMD changes with grating dispersion as a function of birefringence. For high dispersion values, it may be impossible to achieve the low birefringence needed for a low PMD value. For example, a
4 3 2 1 0 –1 –2 –3 –4
E, 1e-4
D, 1e-5 C, 1e-6 B, 1e-7
A, birefringence =1e-8
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Log(Dg), ps/nm Figure 7.23 PMD versus dispersion in a DCG for five values of birefringence, A (B ¼ 1 108), B (B ¼ 1 107), C (B ¼ 1 106), D (B ¼ 1 105), and E (B ¼ 1 104).
Systems Measurements with DCGs
325
dispersion of 5 nsec/nm and a PMD of 1 psec require a birefringence of 6 108, a value that may not be achievable even with gratings in standard fibers. The problem is compounded if the DCG is unapodized. From Eq. (7.4.3) we note that the change in the Bragg reflection wavelength is 0.02 nm for a birefringence of 1 105 at a wavelength of 1550 nm. Since there is a highfrequency ripple of period 0.01 nm on the short-wavelength side of the DCG shown in Fig. 7.10, on an overall average dispersion slope of 1310 psec/nm, large jumps in PMD may occur, even with very weak birefringence. These jumps could be of the order of the amplitude of the ripple (100 psec).
7.5 SYSTEMS MEASUREMENTS WITH DCGs Ultimately, the effectiveness of the DCG is determined by the bit error rate (BER). This measurement is an indicator of how many errors are received within a certain time window. Generally, a system is expected to achieve a minimum BER of 1 bit in 109 at the transmission rate, without and with the DCG. Some undersea systems require even lower BERs (e.g., 1015). However, in order to compare the transmission performance, the power has to be increased at the receiver to compensate for insertion loss and any nonlinear dispersion in the DCG. This is usually expressed as a penalty in decibels at the BER. There are several parameters that influence the BER. As has been seen, the DCG has an operating bandwidth that needs to accommodate the signal down to –20 dB, to reduce the dispersive effects of spectral filtering. Ideally, a filter matched to the signal bandwidth with perfect dispersion compensation is required with zero insertion loss. Other considerations, such as the effect of different types of apodization on the group delay ripple (GDR) as well as the ripple in the reflected signal of a DCG, PMD, insertion loss, and so on, cause an additional penalty. There have been many demonstrations of dispersion compensation using DCGs: from compensation of the chirp from a semiconductor laser using a half-Gaussian refractive index modulation induced chirped grating [2], to the measurement of dispersion in a grating [5], to the first report of DC in a transmission through a fiber [51,52]. In the last demonstration, 400-fsec pulses at a bit rate of 100 Gb/sec were transmitted through 245 m of standard telecommunications fiber and were recompressed with an 8-mm-long 12-nm bandwidth DCG to 450 fsec after dispersing to 30 psec, a compression ratio of 65. Since these demonstrations many more measurements of DC have been reported using a variety of gratings: notably, dispersion-tunable chirped gratings at 10- and 20-Gb/sec transmission rates with a 5-cm-long grating and 80 km of fiber using strain tuning [53], as well as 220 km at 10 Gb/sec and 100-mm-long temperature-tuned chirped gratings [54]. A novel offset core fiber grating has
326
Chirped Fiber Bragg Gratings
also been reported for strain tuned dispersion compensation of 270 km of standard fiber at 10 Gb/sec [55,56]. Fixed wavelength, 100-mm-long, chirped gratings have also been used at 10 Gb/sec transmission with up to 500 km of standard fiber [57,11]. Longer gratings, up to 400 mm [58], have been used at 40 Gb/sec over 109 km of fiber, and in excess of 1-m-long gratings with a bandwidth of 10 nm at 10 Gb/sec over 100 km of standard fiber in a WDM transmission system with up to 11 wavelengths simultaneously [48]. Other WDM experiments at 10 Gb/sec have shown DC at four wavelengths over 100 km using a single superstructure chirped grating (see Chapters 3 and 6) [59]. The long transmission lengths at high bit rates are possible with multiple chirped gratings, either lumped [60,61] or cascaded [12]. In the latter scheme, 8 20 Gb/sec transmission over 315 km used four 1-meter-long continuously chirped gratings at 3 80 km þ 1 75 km hops, and 8 10 Gb/sec over 480 km was demonstrated using six 1-meter-long continuously chirped, 6.5-nm bandwidth gratings [62,63] at 80-km hops. The demonstrated results showed near ideal operation at 10 Gb/sec, despite the 4 to 10 psec polarization mode dispersion of each grating, although at 20 Gb/sec there was some polarization dependence. A pseudo random bit sequence of 231 – 1 was used for the 10 Gb/sec bit stream (and multiplexed for the 20 Gb/sec) for each wavelength spaced at nominally 0.8 nm. The schemes used for lumped gratings are shown in Fig. 7.24. Either, bandpass filters can be used with identical pairs of gratings as in Fig. 7.24a, or circulators with four or more ports may be used. Fiber
Fiber
Input
Output
DCG DCG
DCG
A
DCG
Polarization independent chirped fiber dispersion compensating grating (DCG) transmission filter DCG Input
B
Four-port circulator
DCG Output
Figure 7.24 The lumped DCG in two configurations: (a) as a band-pass filter (after Ref. [64]) and (b) multiport circulator [65].
Systems Measurements with DCGs 1550.0 0
327
1550.2
1550.4
1550.6
1550.8
1551.0
Reflectivity, dB
1X –5 –10 10X –15 –20 Wavelength, nm Figure 7.25 The reflectivity of a single and 10 cascaded, 100-mm-long hyperbolic tanh apodized gratings (chirped bandwidth of 0.75 nm and peak–peak refractive index modulation of 7.5 105).
Cascading of chirped gratings reduces the available bandwidth and hence system tolerance in multihop routes. This effect is due to filtering at the edges of the grating and is seen clearly in Fig. 7.25. The reflectivity of a cascade of 10 identical hyperbolic-tanh profile gratings shows that the –10 dB (from the peak) signal bandwidth is reduced from 0.8 to 0.6 nm. For this simulation, the reflectivity was 90% at the peak for the single 100-mm-long grating with a chirped bandwidth (FW) of 0.75 nm. After the tenth reflection the incurred insertion loss was 5 dB at the peak. These figures indicate the ideal case for identical gratings. If, however, there is a variation in the bandwidth and the reflectivity, the penalty is worse. For system design, the signal bandwidth determines the bandwidth of the grating. The roll-off of the reflectivity (and therefore the type of apodization) will determine the bandwidth of each grating. Allowance also has to be made for drift of the grating wavelengths and the signal source. For the GDR, cascading of the gratings may have a beneficial effect if the ripple cancels; alternatively, it may increase where it is in phase. This is especially important for unapodized or imperfectly apodized gratings. Some of the large-amplitude high-frequency GDR generated in a cascade of identically apodized gratings is reduced with random variations in the grating profiles. Figure 7.26 shows the relative GDR of a cascade of three identical gratings of type B (in Fig. 7.15) and a for a cascade of all three gratings, B, C, and D.
7.5.1 Systems Simulations and Chirped Grating Performance The theoretical aspects of DCG in systems applications have been considered by several workers [30,66–70]. Of the many indicators of the performance of a grating, the receiver eye penalty is probably the most significant. As a number
Chirped Fiber Bragg Gratings
Relative group delay, ps
328 1550.1 140
1550.3
1550.5
1550.7
1550.9
3 × Identical
120 100 80
3 × Random
60 Wavelength, nm
Figure 7.26 Relative GDR for a sequence of three identical chirped gratings of type B in Fig. 7.15, and a mix of types B, C, and D.
alone, it is not very useful, since the properties of the DCG are not constant across the bandwidth. One clearly needs to know the effect of the insertion loss, the change in the reflected power as a function of detuning, and the influence of the nonlinear dispersion. Bungarzeanu [71] reported the computer-aided simulation of chirped gratings with a view to understanding the performance of DCGs by comparing the receiver eye closure penalty as a function of detuning across the bandwidth of different apodization profile gratings. This method of analysis is a direct approach to understanding the effect of the group delay ripple, nonlinear dispersion across the bandwidth of the grating, designed for a particular route length. By altering the bit rate, it is possible to map out the point at which the grating will be limited (a) by bandwidth limitation of the grating and (b) by the cumulative effects of the nonlinearity and GDR. The principle of the model is as follows: A 128-bit-long pseudorandom sequence is coded as a non-return-to-zero (NRZ) complex envelope, which is chirp-free. In principle, RZ and chirp may be added. The fiber has a linear dispersion and is a flattop band-pass filter. The receiver, with a 3-dB bandwidth of 0.75 bit rate is a fourth-order Bessel-type band-pass filter and is designed to meet the ITU-T guidelines [72]. The time domain output is analyzed and compared with that of an undispersed system to quantify the eye-closure penalty. Figure 7.27 shows the system used for the simulation. It is easy to see how the system can be extended for a more involved simulation. Figure 7.28a shows the result of the simulations for a transmission over 100 km of standard fiber at a bit rate of 10 Gb/sec, for a 150-mm-long DCG with a dispersion of 1.7 nsec/nm and a FW bandwidth of 107 GHz (0.86 nm). Shown are the results for an unapodized and cosine and hyperbolic-tanapodized profiles as a function of detuning. While the eye penalty is very low (<0.5 dB) close to the short-wavelength end of the unapodized grating, it steadily increases toward the long-wavelength end. Note that the bandwidth is
Systems Measurements with DCGs Dispersion of fiber
329 Detector (square law)
DCG
Low pass filter
Waveform analysis
NRZ coder Eye closure penalty
Pseudorandom bit
Figure 7.27 The block diagram of the simulation for the calculation of the eye penalty (after ref. [71]). 3
3
2
1
0 –60
A
Eye-penalty, dB
Eye-penalty, dB
cos tanh Unapodized
–40
–20
0
20
Frequency shift, GHz
40
2
1
0 –60
60
B
–40
–20
0
20
40
60
Frequency shift, GHz
Figure 7.28 The eye-penalty for unapodized and cosine and tanh-apodized, 150-mm-long gratings for a 107 – 1 NRZ pseudo-random bit sequence, at a bit rate of 10 Gb/sec (a) and 40 Gb/sec (b) (courtesy C. Bungarzeanu [71]).
similar to that of the tanh-, but larger than that of the cosine-apodized grating. Also shown in Fig. 7.28b are the simulations for 40 Gb/sec transmission. Here, too, we see that the penalty is only slightly higher than for the tanh-apodized grating (note that the bandwidth for a NRZ is half that of a RZ bit sequence), and the bandwidths are almost identical. Owing to the restricted bandwidth, the detuning is significantly narrower for the cosine-apodized grating. We note that at the longer-wavelength end of the unapodized grating, the eye penalty fluctuates because of the GDR frequency becoming closer to the transmission frequency, since the eye closes as a result of the additional dispersion. This becomes less important for 40 Gb/sec, since more of the grating bandwidth is being used for dispersion compensation, and the fluctuations on the longwavelength side also become smaller. This result has been further investigated by Garthe et al. [73] for long gratings. The conclusions are similar, in that the eye penalty has a maximum value depending on the position in the bandwidth,
Chirped Fiber Bragg Gratings 1558.75 0
1559.25
1559.75
Reflectivity, dB
–5 –10 –15
1560.25
1560.75 200 150 100 50 0 –50
–20
–100 –150
–25 –30
Relative group delay, ps
330
–200 Wavelength, nm
Figure 7.29 The reflectivity and relative group delay of a 200-mm SSCG (dispersion of 1310 psec/nm) [74].
because the GDR frequency induces satellite pulses that may coincide with adjacent time slots. Figure 7.29 shows the reflection and relative group delay of a 200-mm-long SSCG made with two 100-mm-long chirped gratings. Low-repetition-rate ultrahigh-speed measurements performed on this grating with a dispersion of 1310 psec/nm and bandwidth of 1.5 nm at the output of 77 km of standard telecommunications fiber [74] shows that a 3.8 psec input pulse sits on a wide lowlevel pedestal at the output. The pedestal, which is limited to approximately two bit periods, has the detrimental effect of causing the eye to close slightly when the entire bandwidth of the grating is used. However, systems transmission experiments performed on the same grating at 40 Gb/sec over 77 km of standard fiber indicate that the received eye remains open, as shown in Fig. 7.30, which without the DCG is completely closed.
7.6 OTHER APPLICATIONS OF CHIRPED GRATINGS As we have seen, pulses propagating in a fiber are broadened by dispersion. There is a broadening if the pulse is not transform limited and it is chirped. Thus, the individual parts of the spectrum arrive at different times and can skew the pulse. If the chirp is a time-varying function, then jitter is also introduced. On the other hand, in a dispersion-free system, neither the chirp introduced by any component nor the source bandwidth causes the pulse to be broadened. If a chirped grating compensates for the dispersion of a fiber, then the dispersion and jitter induced in chirped pulses is automatically compensated [75]. Further, pulses can be compressed or dispersed depending on the sign of the chirp, by launching the pulse in a fiber with either anomalous or normal
Other Applications of Chirped Gratings
20.0 ps/div
331
24.5977 ns
Figure 7.30 The received signal eye diagram at a transmission bit rate of 40 Gb/sec (1550 nm) after 80 km of optical standard fiber and a 200-mm-long unapodized SSCG dispersion compensating grating (courtesy D. Nesset, BT Laboratories).
dispersion. If a pulse is chirped, then it is possible to compensate for the chirp using a chirped fiber grating and to compress the pulse [14,76–80]. The grating may also be used for nonlinear pulse compression [81] of pulses that have been spectrally broadened and linearly chirped through self-phase modulation (SPM) and propagation in normally dispersive optical fiber [82]. For high-power use, chirped gratings are especially suitable for reducing nonlinear effects in the fiber by stretching pulses, reducing the peak power before amplification [13,83]. Adjustment of the chirp of the grating also allows the effects of higher-order dispersion to be canceled [10].
7.6.1 Pulse Shaping with Uniform Gratings The chirped fiber Bragg grating (CFBG) is a dispersive element in which the group delay is a function of wavelength. Thus, the CFBG may be used to manipulate the amplitude and phase of light, and therefore the shape of a pulse, as the amplitude interacts with the reflectivity and the phase with the dispersion. This interaction may be separated in the Born approximation regime, by manipulating the amplitude separately from the phase. The two-step process was implemented by Weiner [84] using bulk optical amplitude and phase filters, by first dispersing a pulse into its component wavelengths, manipulating the amplitudes and phases in the Fourier plane, and then recombining them to form the desired pulse shape.
332
Chirped Fiber Bragg Gratings
The FBG, being a versatile filter in which the amplitude and phase may be manipulated by the design of the grating, thus offers similar functionality. The first pulse shaping using the full properties of a FBG was demonstrated by generating a dark soliton from a train of bright-solitons [85]. For this application, it is necessary to invert the phase of the component frequency content of the pulses in such a way that the uniform phase across the soliton is inverted at the center of the frequency spectrum. This is equivalent to placing a phase element at the center of the frequency band so that the lower side band has the opposite phase to the upper side band frequencies. In addition, the amplitudes of each frequency component have to be adjusted so that the Fourier transform of the resulting spectrum does produce a propagating “light hole” of the dark soliton [86,87]. Figure 7.31a shows the amplitudes of the frequency components
Amplitude
Nn0
–4
–3
–2
A
–1
0
(N +1)n0
1
2
3
4
Detuning from nc
–3
–2
1.5 M(w) 1 0.5 –1 0 0.5 1
2
3
–1 –1.5
B 1.0
Amplitude
0.5 0.0 Bright soliton Dark soliton
–0.5 –1.0 195.42
C
195.43
195.44 THz
195.45
195.46
Figure 7.31 (a) The amplitude spectrum of a bright soliton. (b) The filter response required to invert the bright soliton into the dark soliton, as the dark has the opposite phase in one-half of the spectrum, shown in (c). (Adapted from reference [85].)
Other Applications of Chirped Gratings
333
Reflectivity/phase(radians)
1.6 1.2 0.8 RX PHASE
0.4 0.0 –0.4 –0.8 –1.2 –1.6 1556.65
1556.75
1556.95 1556.85 Wavelength, nm
1557.05
Figure 7.32 The amplitude and phase response of a uniform 30-mm-long grating (15 dB reflectivity) matched to turn the bright into a dark soliton. The arrows indicate the phases for the relevant frequency components on either side of the Bragg wavelength at 1556.85 nm. (Adapted from reference [85].)
in the bright soliton pulse, part (b) shows the filter’s amplitude and phase response, and part (c) shows the amplitudes of the bright and dark solitons. The reflection and phase spectrum of the grating required to match the dark soliton filter [Fig. 7.31b] are shown in Fig. 7.32. Reflecting the bright soliton from the grating filters the amplitude and reverses the phase of one-half of the spectrum turning the pulse into a dark soliton. The experimental arrangement is shown in Fig. 7.33. A laser was modulated at 6 GHz to match the frequency of the designed 3-mm-long reflection grating, amplified and transmitted through 7.7 km of standard optical fiber. The reflected light is displayed on an
30mm long dark soliton grating
6 GHz modulator
50:50
1556.9 nm laser
MOD Amp
Fiber
Intensity (a.u)
1
Bright solitons
0.8 0.6 0.4 0.2 0
0
100
200 Time (ps)
300
Oscilloscope: 100 ps dark solitons Figure 7.33 Experimental arrangement to generate the dark solitons using a matched filter. The oscilloscope shows the dark pulses on a bright background measure after transmission through 7.7 km of optical fiber. (Adapted from reference [86].)
334
Chirped Fiber Bragg Gratings
oscilloscope. Bright soliton pulses generated by the laser are converted into 33 ps dark soliton pulses [86]. The spectral shaping technique has been used successfully by Petropoulos et al. [88] to transform a soliton into rectangular pulses using a superstructure grating filter.
7.6.2 Optical Delay Lines If a CFBG is stretched, the point at which the reflection occurs for an incoming pulse changes. However, the pulse is also dispersed because of the dispersion in the grating. If this dispersed pulse is reflected from a second identical grating, it will be either be further broadened or compressed, depending on the sign of the chirp of the second grating. Bouncing the dispersed pulse from the other end of the chirped grating brings the pulse back to almost its original state (see Section 7.2.1). Thus, if two gratings are used in succession with the sign of the chirp reversed for the second grating, the pulse is recompressed. If, however, one of the gratings is stretched, the physical delay between the two gratings is altered and a time delay is introduced. It may be shown easily that the change in optical time delay through a chirped grating when it is strained is dt ¼
Dl0 ð1 Pe Þdl; L
ð7:6:1Þ
where, L is the length of the grating, l0 is the nominal central wavelength of the chirped grating, Pe is the stress optic coefficient, dl is the extension, and D is the dispersion of the grating given by D¼
2nL : c
ð7:6:2Þ
Therefore, it is clear that not only is the pulse dispersed, but it also undergoes a delay. If the dispersion is canceled by the use of another grating with a reverse chirp, the net group delay of the pulse is given by Eq. (7.6.1). The effect of the grating is to magnify the delay by a factor of G¼
l : Dl
ð7:6:3Þ
compared to simply stretching a fiber of the same length with a reflector at the end and without a grating. This may be understood by the fact that the reflection of a wavelength at a Bragg wavelength at the rear end of the grating will reflect from the front end if the grating is stretched enough to allow that to happen. The delay change would be equivalent to the length of the grating.
Other Applications of Chirped Gratings
335
INPUT 4-Port circulator
+ve chirped grating
Unstrained output
Dispersion
Δt Recompression –ve chirped grating
Strain
Strained output Figure 7.34 Schematic of a twin chirped grating delay line. Straining the lower grating introduces delay, whereas straining the top grating gives negative delay. There is an increasing mismatch between the overlap of the reflection spectra of the two gratings with increasing strain.
Delay lines with a delay of 3.1 mm in two 50 nm full width half maximum (FWHM) bandwidth, 40-mm-long chirped gratings have been demonstrated and used for optical coherence tomography [89]. Figure 7.34 shows the arrangement of the delay line. The first grating disperses the pulse and the second recompresses it. Applying strain to one or the other grating can produce positive or negative delay. Delay may also be achieved by applying a voltage on poled chirped gratings [90]. A large dispersion is possible if the chirped gratings are long. As an example, a 0.4% strain in a 100-mm-long grating with a bandwidth of 40 nm (D 25 ps/nm) results in a uniform group delay change of 76 ps, as shown in Fig. 7.35. It should be noted that the delay line works with pulses or with CW light.
Reflectivity, dB
–5
1540
1550
1560
1570
Strained G1: Reflectivity
–10 –15 –20
1580 1000 500
Strained G1: Group delay Net delay
Δt
0
–25 –30 –35
–500
Group delay, ps
1530 0
G2: Group delay
–40
–1000 Wavelength, nm
Figure 7.35 Simulation of the reflectivity, group delay, and time delay in a pair of oppositely chirped gratings, with one grating strained by 1%. The chirp in each grating is 40 nm and the length is 100 mm.
336
Chirped Fiber Bragg Gratings
7.6.3 Pulse Shaping with Chirped Gratings Pulse shaping by chirped FBGs was first investigated by Rotwitt et al. [91]. In the demonstration, a ps pulse reflected off a grating was shown to acquire a different form as a result of the dispersion in the grating. Pulse shaping has been demonstrated by several researchers using a number of different schemes, such as spectral encoding for code division multiplexing (CDMA) [92], real time spectral analysis [93], dispersion-less variable band-pass filtering [94], synthesis of pulses [95], and manipulating ps pulses [96]. Other applications include performing mathematical functions such as differentiation [97,98]. Cascaded gratings with different dispersion characteristics can thus be used for pulse shaping using a system similar to the one shown in Fig. 7.34. In this scheme, a first chirped grating is used as a pulse shaper and the second as the dispersion compensator [99]. It is possible to generate triangular and square pulses using this method.
7.6.4 Pulse Multiplication Linearly chirped FBGs may be used to multiply the repetition rate of a stream of pulses. The scheme is based on the temporal Talbot effect [100] and is otherwise known as temporal self-imaging [101,102,103]. If a high-repetition train of short pulses with a bandwidth matched to a chirped FBG is dispersed in reflection, the frequency components are dispersed in such a way that at a time in-between the incoming pulses, different frequency components from different pulses appear in the same time slot, thereby constituting an identical image of the original pulses. The condition for pulse multiplication has been shown to be [103] D¼
cT 2 ; ml2
ð7:6:4Þ
where D is the dispersion of the grating in ps-nm1, T is the pulse separation, and m is the multiplication factor. Eq. (7.6.4) may be rewritten as Total Group Delay ¼
T2 ; m t
ð7:6:5Þ
where t is the pulse width, and the integer m ¼ 2, 3, 4. . . . This is the general case, and the group delay may be calculated for any pulse width and pulse repetition time. However, rewriting Eq. (7.6.5) as Total Group Delay ¼
T T; m t
ð7:6:6Þ
Other Applications of Chirped Gratings
337
and if T ¼ m t;
ð7:6:7Þ
then the repetition time of the pulses equals the total group delay of the grating for pulse multiplication. This means that the pulse must arrive at the grating at a time equal to twice the pulse width for m ¼ 2, at three times the pulse width when m ¼ 3, and so on. Eq. (7.6.7) gives the closest packing for any multiplication rate. The temporal Talbot effect has also been proposed for multiplying multiwavelength optical pulse trains [101] using a superstructure chirped FBG. Figure 7.36 shows a circuit for the general case of the simultaneous pulse multiplication for many wavelengths.
7.6.5 Beam Forming
INPUT λ3 λ2 λ1
Super-structure CFBG
FBGs have been used in a number of microwave filtering applications [104–109], based on dispersive delay lines [110]. Yang et al. [111] have demonstrated the use of CFBGs for microwave photonic phased array antennas (PAAs). Using photonics to transmit optical signals is an efficient technique to distribute microwave signals with low loss. In this technique, optical fibers are used to transmit an RF-modulated high-power optical carrier frequency. When the light is detected with a square law detector, such as a photodiode, the signal is demodulated and RF current is generated, which can be transmitted via an antenna. In the steerable antenna application, several CFBGs are used, all with an identical bandwidth but with lengths that correspond to the bases of similar triangles (prisms), as shown in Fig. 7.37. Light from a
OUTPUT λ3 λ2 λ1
Circulator Figure 7.36 The general case of the temporal Talbot effect pulse repetition rate multiplier using a superstructure chirped FBG. (From reference [101].)
338
Chirped Fiber Bragg Gratings Tunable chirped FBG prism Splitter SMF l1 Electro-optic modulator
Beam forming antenna array
ltd
l1
q
ltd
Polarization controller Tunable laser
lc q1
qc Steering angle
qtd
Photodiodes & amplifier
Figure 7.37 The steerable beam antenna array using a tunable laser and a set of prism CFBGs. (Adapted from Jianping Yao’s courtesy slide, reference [112].)
tunable laser is modulated at a fixed RF frequency and is directed to the nCFBG via n-circulators. As the frequency of the tunable laser is tuned, the reflection from each CFBG is delayed according to the lines defining the sides of the prism at that wavelength. Thus, each reflection is delayed in time (and therefore phase). The light is then detected by individual photodiode/amplifier units and directed to antennas spaced equidistant, at lm/2 from each other. The RFs radiated from the antennas have a linear stepped phase difference between each element. At the central wavelength of the CFBG, all signals arrive at the antennas at the same time and are therefore in phase. The RF is thus radiated in the forward direction. When the laser is tuned, a linear delay, Dt, is introduced between each antenna; the phase front of the RF tilts and is radiated at an angle, y, given by Dt ¼
sin y ; 2fm
ð7:6:8Þ
where fm is the RF modulation frequency. The delay, Dt, is the time difference between the reflections from adjacent gratings and is given by Dt ¼ An ðlc lÞ;
ð7:6:9Þ
where An ¼
2ng Ln : Dl
ð7:6:10Þ
An and Ln are the dispersion and length, respectively, of the nth CFBG. ng is the group index at the wavelength, l, of the tunable laser, and lc is the center wavelength of the CFBGs of fixed bandwidth, Dl. It is clear that each CFBG has a different dispersion, An, defined in Eq. (7.6.10).
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The maximum of the radiation pattern is given by [111], 2ng Dd y0 ¼ sin1 ; d
339
ð7:6:11Þ
where Dd is the difference between the points at which the reflection at the given wavelength takes place for adjacent gratings. As these points are continuously tunable by tuning the wavelength, the antenna is also continuously tunable. However, the number of chirped gratings, n, determines the far-field width of the radiated pattern, with a large number of gratings providing a narrower beam. Using this technique, the radiated beam can be scanned over 4p, with no moving parts. The system is equivalent to a scanning radar and may be used as a directed beam antenna. Various versions of the true-time delay beam forming systems have been reported [112–114].
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[89] E. Choi, J. Na, S. Ryu, G. Mudhana, B. Lee, All-fiber variable optical delay line for applications in optical coherence tomography: feasibility study for a novel delay line, Opt. Express 13 (2005) 1334–1345. [90] R. Kashyap, Dispersion enhancement of the nonlinear electro-optical effect, Opt. Express 14 (23) (2006) 11012–11017. [91] R. Rottwitt, M.J. Guy, A. Boscovik, D.U. Noske, J.R. Taylor, R. Kashyap, Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fibre Bragg gratings, Electron. Lett. 30 (12) (1994) 995–996. [92] A.G. Jepsen, A.E. Johnson, E.S. Maniloff, T.W. Mossberg, M.J. Munroe, J.N. Sweetser, Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA, Electron. Lett. 35 (1999) 1096–1097. [93] J. Azan˜a, M.A. Muriel, Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings, IEEE J. Quantum Electron. 36 (2000) 517–527. [94] I. Littler, M. Rochette, B. Eggleton, Adjustable bandwidth dispersionless bandpass FBG optical filter, Opt. Express 13 (2005) 3397–3407. [95] J. Azan˜a, L.R. Chen, Synthesis of temporal optical waveforms by fiber Bragg gratings: a new approach based on space-to-frequency-to-time mapping, J. Opt. Soc. Am. B 19 (2002) 2758–2769. [96] S. Longhi, M. Marano, P. Laporta, O. Svelto, Propagation, manipulation, and control of picosecond optical pulses at 1.5 mm in fiber Bragg gratings, J. Opt. Soc. Am. B 19 (2002) 2742–2757. [97] N.Q. Ngo, S.F. Yu, S.C. Tjin, C.H. Kam, A new theoretical basis of higher-derivative optical differentiators, Opt. Commun. 230 (2004) 115129. [98] N.K. Berger, B. Levit, B. Fischer, M. Kulishov, D.V. Plant, J. Azan˜a, Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating, Opt. Express 15 (2007) 371–381. [99] M.A. Preciado, V. Garcı´a Mun˜oz, M.A. Muriel, Optical spectral pulse shaping by combining two oppositely chirped fiber Bragg grating, IEEE Photon. Technol. Letts. 10 (2007) 435–437. [100] H.F. Talbot, Philos. Mag. 9 (1836) 401. [101] J. Azan˜a, M.A. Muriel, Technique for multiplying the repetition rate of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings, Opt. Lett. 24 (23) (1999) 1672–1674. [102] J. Azan˜a, M.A. Muriel, Temporal self-imaging effects: theory and application for multiplying pulse repetition rates, IEEE J. Sel. Top. Quantum Electron. 7 (4) (2001). [103] C. Martijn de Sterke, B.J. Eggleton, Spectral Talbot effect: interpretation via band diagrams, Opt. Commun. 248 (1–3) (2005) 117–121. doi:10.1016/j.optcom. 2004.11.090. [104] G.A. Ball, W.H. Glenn, W.W. Morey, Programmable fiber optic delay line, IEEE Photon. Technol. Lett 6 (6) (1994) 741–743. [105] D.B. Hunter, R.A. Minasian, P.A. Krug, Tunable optical transversal filter based on chirped gratings, Electron. Lett. 31 (1995) 2205–2207. [106] D.B. Hunter, R.A. Minasian, Microwave optical filters using in-fibre Bragg grating arrays, IEEE Microwave and Guided Wave Lett. 6 (2) (1996) 103–105. [107] W. Zhang, J.A.R. Williams, I. Bennion, Optical microwave bandpass filtering using fiber Bragg grating arrays, in: T. Erdogan, E. Joseph Friebele, R. Kashyap (Eds.), Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, OSA TOPS vol. 33, # Optical Society of America, 2000, pp. 72–78. [108] W.S. Brinkmeyer, M.J. Wale, Proof-of-concept model of a coherent optical beamforming network, IEE Proc. J. 139 (1992) 301–304.
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Chapter 8
Fiber Grating Lasers and Amplifiers The arrival of the in-fiber Bragg grating could not have been at a better time for rare-earth-doped fiber. Research and development in the field of doped fibers was at such a stage that fiber amplifiers were already commercially available; thus, fiber Bragg gratings were a natural progression for combining the attributes of the fibers and amplifiers, as well as semiconductor devices. Their success in this area is due to the near-perfect reflection characteristics of the grating and the ultralow insertion loss, as well as the ease with which they can be tailored for a particular application. One of the most attractive features of the in-fiber Bragg grating is its transparency to wavelengths outside of the band stop. Apart from becoming integral components of narrow-band lasers, the side-tap filter and the long-period grating have also been particularly useful for equalizing the gain of optical fiber amplifiers. This chapter introduces some of the applications of fiber gratings in narrow-band, low-chirp lasers and modelocked, short-pulse sources, as well as in tailoring the gain and stability of amplifiers.
8.1 FIBER GRATING SEMICONDUCTOR LASERS: THE FGSL Semiconductor laser diode technology has been refined to produce lowthreshold, high-power, and narrow-linewidth devices (distributed feedback lasers) for a large number of applications [1]. The linewidth may be further narrowed by the use of a long external cavity. This technique has attracted much attention over the years [2]. Early work showed that while there was a benefit in line-narrowing from the linewidth enhancement factor of long external cavities, the instabilities were thought to be too difficult to overcome. In particular, reflections from intracavity elements such as the residual front facet reflections greatly affected the performance of the laser, except in the case of strong coherent feedback [3]. Much of the earlier work was limited to weak feedback due to 347
348
Fiber Grating Lasers and Amplifiers
in- and out-coupling losses to the external cavity. Wyatt and Devlin demonstrated line-narrowed (10 kHz) tunability over 55 nm in an InGaAsP 1.5-mm laser, using a bulk diffraction grating as the external cavity frequency selective mirror [4]. A semiconductor laser integrated with an etched Bragg reflector in a waveguide demonstrated the possibility of a compact configuration [5]. The fiber Bragg grating thus became the next obvious choice for an external reflector owing to its narrow bandwidth, with the possible advantage of defining the lasing wavelength and the potentially low coupling loss. Destroying the front facet reflectivity of a semiconductor laser and replacing it with a narrow-band fiber Bragg grating as an external cavity reflector is probably the worst possible configuration for making a high-quality laser. First of all, the semiconductor chip with cleaved facet mirrors forms an ultralow-loss cavity, since the mirrors are an integral part of the laser gain medium. Secondly, the laser semiconductor chip alone forms the shortest gain cavity, reducing the roundtrip time to a minimum and thereby allowing direct high-speed modulation. Thirdly, the high reflectivity of the output facet of the laser (33%) is broadband; any external reflections returning to the laser, other than at the lasing wavelength, are attenuated equally, reducing the possibility of destabilizing laser operation. On the other hand, the fiber Bragg grating external-cavity semiconductor laser has several major limitations. The output facet reflectivity of the laser chip has to be reduced to low levels. The addition of a fiber in the cavity requires good coupling and low residual subcavity reflections in order to reduce intracavity losses and poses negligible return loss to wavelengths other than within the bandwidth of the Bragg grating. The cavity length increases correspondingly by the incorporation of a reflective Bragg element at some point within the fiber, increasing the cavity round-trip time. What has the FGSL (or FGL) to offer to make it worth pursuing? First, the fiber Bragg grating has a temperature coefficient [6] of wavelength change 6–8 times lower than that of a semiconductor laser, potentially increasing the wavelength stability of the laser in a hybrid configuration. Second, fiber Bragg gratings can be routinely fabricated at precise wavelengths and are therefore suited to mass production; wavelength control makes them useful for wavelength division multiplexing (WDM) applications that require extremely tight tolerances [7]. Third, the longer laser formed by the incorporation of the external cavity reduces the linewidth in direct relation to the length [8]. Fourth, semiconductor lasers suffer from chirp as a result of cavity length change during direct modulation. The longer FGSL dilutes the carrier induced cavity length change in the semiconductor during modulation, by confining it to the much smaller fraction of the laser length. It is usual to quantify the operational sensitivity of a semiconductor laser by the linewidth enhancement factor a as [9,10]
Fiber Grating Semiconductor Lasers: The FGSL
a¼
4p dn=dN ; l dg=dN
349
ð8:1:1Þ
where l is the lasing wavelength, dn is the change in the refractive index of the laser, and dg/dN is the differential gain. In a semiconductor laser, the change in the gain and the refractive index is over the entire length of the laser Ls. With an extended cavity in which only part of the cavity is active, the influence of both these quantities is limited, with the result that the effect of the refractive index change (on the cavity length) is diluted. The change in the laser wavelength dl as a function of injected carriers is given by dl ¼ l
@lcarrier ; Llaser
ð8:1:2Þ
where @lcarrier is the change in the optical length of the semiconductor with injected current, and l is the lasing wavelength. The inverse dependence of the shift in the wavelength on the laser cavity length demonstrates the advantage of the longer cavity [11]. Further, the wide gain bandwidth of the semiconductor chip in conjunction with a fiber Bragg grating used to define the operating wavelength can increase the yield, potentially allowing the use of all chips on a wafer. This becomes especially important with the tight specification for WDM system wavelengths. Semiconductor production technology of DFB lasers requires that each laser be measured for wavelength prior to packaging and consequently has implications for the replacement of faulty lasers in WDM systems, where stocks may need to be maintained for each laser. The concept of retaining a fiber grating as part of the transmitter card, while replacing only the active gain medium when faulty, may have a cost advantage by reducing stocks. Finally, for telecommunications and sensor applications, lasers need to be pigtailed after fabrication. If the fiber grating is allowed to define the lasing wavelength, the two functions of operating wavelength and pigtailing may further reduce fabrication/packaging costs. These advantages are considered worthwhile to overcome the difficulties involved in optimizing laser chip-to-fiber coupling. The first semiconductor laser to use an external cavity of an etched in-fiber Bragg grating was reported by Brinkmeyer et al. [12]. This laser used a standard uncoated Fabry–Perot fiber pigtailed laser chip operating at 1.3 microns, which was fusion spliced to an etched fiber grating, forming a 2-meter-long cavity. With the twin coupled cavities (laser chip and laser chip with fiber grating), single-frequency operation of the laser was demonstrated with a linewidth of 50 kHz. The narrow bandwidth of the grating (26 GHz) was much less than the FP mode spacing (140 GHz), although the long cavity longitudinal mode spacing was only 50 MHz. The reflectivity of the front facet of the FP laser
Fiber Grating Lasers and Amplifiers
350
rFP 32%, while the reflectivity rfbg of the grating is very low, 0.01%; however, the combined coupled cavities results in a constructive interference with a contrast ratio of j rFP þ rfbg j2 =j rFP rfbg j2 1:07;
ð8:1:3Þ
so that when the cavity length is tuned for constructive interference, the modes of the long fiber cavity have a differential gain that is higher than the FP laser modes. This demonstration showed that the selectivity and additional reflectivity of the external cavity grating have a beneficial effect even in such a simple configuration. Etched gratings in fibers as external cavity mirrors have also been used with 1500-nm diodes [13]. Morey et al. [14] showed the use of a photoinduced fiber Bragg grating with a semiconductor laser to operate at the grating wavelength of 820 nm. A schematic of the FGSL device is shown in Fig. 8.1. An antireflection-coated (0.5% reflectivity) FP GaInAsP/InP buried-heterostructure 1.5-mm laser chip is coupled with a lensed fiber with a narrowband reflection grating spliced to it to form the external resonator [15]. The lensed fiber is aligned to maximize the output coupling and welded in place. This laser demonstrates a low chirp, which is restricted by the limited bandwidth of the grating and the reduced change in the cavity length according to Eq. (8.1.2). The linewidth of the laser for the 60-mm-long cavity was measured to be <50 kHz. The bias current varied between the threshold of 30 mA and 150 mA, changed the operating wavelength by <0.1 nm. The package temperature was controlled to 5 C. A potentially low-cost FGSL operating at 1.3 mm with 1 mW output power with an operating wavelength change of only 2 nm, over a temperature range of 100 C, has also been reported [16] for Access networks at 622 Mb/sec transmission rates. The intracavity interference effects cause the modes of the laser to hop. This has been observed in experiments with FGSLs [11]. It was shown that as the lensed end of the external cavity fiber is moved away from the AR coated facet of the laser, the laser output drops until it stops lasing in a cyclic manner. These AR coated front facet FP chip Diffusion splices
Llaser
Lensed fiber with intracore Bragg grating
Figure 8.1 The external fiber grating semiconductor laser with an AR-coated FP chip coupled to a lensed fiber [3]. The measured chirp was instrument limited to be <0.5 MHz when modulated with NRZ pulses at 1.2 Gb/sec [15].
Fiber Grating Semiconductor Lasers: The FGSL
351
experiments showed the importance of the phase of the reflected light entering the cavity. If the free spectral range of the subcavity of the semiconductor is much larger than that of the external grating reflector, then the laser wavelength mode will pull within the grating bandwidth. The differential gain between the different external fiber grating cavity modes determines which wavelength lases at any one time. Changing the length of the cavity is similar to changing the bias current, and therefore requires some mechanism to compensate this effect. In an effort to counter the detrimental effects of the variation of the phase as a function of bias current or modulation in the FGSL, a variation in the design using a chirped external grating reflector may be used, as is shown in Fig. 8.2 [17,18]. The principle of operation of chirped grating is to compensate for the current induced change in the optical cavity length of the semiconductor. As the length of the semiconductor cavity changes by dls to maintain an integral number of periods within the laser cavity, the wavelength shifts, altering the cavity length. In principle, the wavelength should shift smoothly as the bias current is altered. In this case, the grating had a deliberate chirp of 0.28 nm with decreasing wavelengths away from the gain medium, a reflectivity of 33%, and a grating length of 10 mm. The fiber lens had a hyperbolic shape to maximize coupling to the grating. The coupling efficiency of this type of lens can be as high as 98% [19]. With an HR coating on the rear and an AR coating on the output facet, the threshold of the device was measured to be 7 mA, operating singlefrequency at fiber output powers of >15 mW. A side-mode suppression ratio (SMSR) of 56 dB was also demonstrated at a bias current of 250 mA (25 mW output power). The linewidth was measured to be between 100 and 540 kHz (close to a mode hop), depending on the bias current. An important application of the fiber Bragg grating is for stabilization of erbium amplifier pump lasers operating at 980 nm [20,21]. Since the pump absorption band is narrow, it is useful to maintain the pump wavelength accurately. It was found empirically that placing a 4% reflector reflection grating AR coated front facet FP chip
Llaser
lL
Hyperbolic lensed fiber with intracore chirped Bragg grating
ls
Chirp = Δl / Lg
Figure 8.2 The chirped external fiber grating cavity laser [17]. The special AR-coated hyperbolic lens on the end of the fiber provides high-efficiency coupling of >98% [19], which in turn allows stable single-mode operation over a large output power (15 mW).
Fiber Grating Lasers and Amplifiers
352
approximately 50–100 mm away from the AR-coated facet of a high-power pump-laser chip caused it to lock to the grating. Temperature variations of 50 C pulled the wavelength by 0.2 nm, showing that the requirement for a thermoelectric cooler may not be necessary. Such a long cavity (as with the one described earlier [12]) works under the regime of “coherence collapse” [22], where the coherence length of the diode modes is much shorter than the laser cavity length. An alternative method of controlling the wavelength of the laser is to use the fiber grating at the rear facet of the chip [23]. The output coupler is a standard fiber pigtail. This device was fabricated in a package with all the in and out coupling with aspherical collimating lenses, as well as an isolator in the output path. The advantage of this scheme is the use of a highly reflective grating (>95%), potentially increasing the output power. The grating had a bandwidth of <0.4 nm, which was burnt-in at 350 C for 6 min. The fabrication technique is claimed to have a reproducibility of 0.05 nm, allowing devices to be fabricated to ITU-T standards for dense WDM. With temperature control of the package, both the grating and the laser chip are maintained at the same temperature. Consequently, the stability of the wavelength was reported to be better than 0.005 nm, with an output power stability within 0.02 dB. This laser, as is common with well-designed FEGSLs had a measured line-width of <30 kHz and a side-mode suppression ratio of >50 dB. A combination of a reflective amplifier and a large spot laser has resulted in a robust laser, which may be operated over a wide temperature range without temperature control [24]. This type of laser has good wavelength stability as a function of temperature and therefore is ideally suited to applications in Access to a local area network, with the potential of being a low-cost source. Figure 8.3 shows a schematic of this laser. The gain medium is a semiconductor chip, which has a passive tapered waveguide to allow the mode to expand so that it has a mode-field close to that of a standard telecommunications fiber. Further, the waveguide is arranged to terminate at an angle to the output facet. The large mode field size allows good coupling to a cleaved standard fiber [24,25], which can be covered in index matching gel to reduce end reflections to a very low level. The angled facet ensures low back reflection, virtually
Index matching gel
Angled front-facet large-spot semiconductor chip: No AR coating
Cleaved fiber end with intracore fiber Bragg grating
Figure 8.3 The large spot-angled facet, cleaved standard fiber FGSL [24].
Static and Dynamic Properties of FGLs
353
Figure 8.4 Photograph of packaged large spot laser FGL (photograph courtesy C. Ford, BT Laboratories).
eliminating the formation of a subcavity. No ripple due to intracavity reflections can be seen in the amplified spontaneous emission (ASE) spectrum. The positive effect of such a design is the lack of mode competition between subcavity modes as the current is ramped. A large-bandwidth (0.3-nm) fiber Bragg grating reflector allows the lasing mode to tune slightly in wavelength as the laser is modulated, with the virtual absence of mode hops. The alignment is simplified with the help of a mini silicon-optical bench. A key on the bench allows easy positioning and soldering of the laser chip. A silicon micro-machined vee-groove aligned to the laser completes the passive alignment of the assembly. A glass sliver is then used to bond the fiber in place with epoxy. The gap between the laser and the fiber-end is then filled with gel. The package may then be completed by injection molding [25] or by hermetic sealing. A photograph of the packaged laser is shown in Fig. 8.4.
8.2 STATIC AND DYNAMIC PROPERTIES OF FGLs The light–current characteristics are shown in Fig. 8.5. The kink-free curve shows mode-hop-free operation as a function of bias current. The extremely low hysteresis in the wavelength of the laser as the current is altered, shows the potential of such a device. Typical of most FGLs is the weak temperature dependence of the lasing wavelength when the grating is heated along with the laser. Mode hopping may be observed as the temperature is increased, as shown in Fig. 8.6. The cavity length determines the magnitude of the wavelength hop, and for this laser, it is 0.16 nm. Note that the overall change in the lasing wavelength is approximately 0.4 nm for a 25 C change in temperature. With an unheated grating, the drift in the wavelength is made almost insignificant. The increased stability of a short-cavity semiconductor laser in an external long passive cavity has also been demonstrated [11].
Fiber Grating Lasers and Amplifiers Output power mW
354 1.2 1 0.8 0.6 0.4 0.2 0
Power(up) mW Power(dn) mW
0
20
80
40 60 Current mA
100
Figure 8.5 L–I characteristics (up and down) of large-spot angled facet laser at 20 C with index-matched gel between laser facet and cleaved fiber end. (Courtesy M.C. Brierley, BT Laboratories.)
Wavelength nm
1552.55 1552.45 1552.35 1552.25 1552.15 15
20
25
30
35
40
Temperature deg C Figure 8.6 Wavelength versus temperature characteristics of the large mode field FGL with indexgel. The grating is heated along with the laser chip. (Courtesy M.C. Brierley, BT Laboratories.)
With an AR-coated front facet reflectivity of 5 10–5 and a grating reflectivity of 28%, it has been shown that this type of laser may be ramped from threshold to 100 mA without a mode hop, and with a wavelength change of only 0.1 nm [24]. The static and dynamic chirp measurements of 500-mm-long AR-coated (reflectivity 1 10–5) angled facet GaAsPInP buried-heterostructure amplifier chips [24] show that the static chirp of the dominant FGL mode is –0.06 GHz/mA, while the dynamic chirp is reduced to 0.016 GHz/mA. The fiber Bragg grating had a reflectivity of 30% with a FWHM bandwidth of 0.3 nm. This data is shown in Figs. 8.7a and 8.7b. This laser has also been shown to have extremely good thermal behavior, with the elimination of mode hops. The L–I characteristic at different temperatures shows kink-free operation over a wide temperature range, and the laser is therefore suitable for WDM applications. Figure 8.8 demonstrates the excellent coupling (high output power) and the smooth light output curves [30].
Static and Dynamic Properties of FGLs
355
40
–0.31 GHz / mA 14.2 GHz / mA
⎫ ⎬ ⎭
FP mode
1.68 GHz / mA –0.06 GHz / mA
⎫ ⎬ ⎭
External cavity mode
Wavelength chirp (A)
120 100 80 60 20 0
A
1 T = 20°C
140
0
20
40
60
80
700 600
0.5
400
0
300 200
–0.5
100 –1
100
0
2
4
B
Current (mA)
500
dn / dI = 0.016GHz/mA
Current (a.u)
Frequency shift (GHz)
160
6
8
0 10
Time (ns)
Figure 8.7 (a) Static chirp of external cavity and FP modes of the FGL; (b) shows the dynamic chirp under active modulation (from: Timofeev F.N., Bayvel P., Mikhailov V., Lavrova O.A., Wyatt R., Kashyap R., Robertson M., and Midwinter J.E., “2.5 Gbit/s directly modulated fibre grating laser for WDM networks,” Electron. Lett., 33(16), 1406– 1407, July, 1997. # IEE 1997, Ref. [30]).
Optical power (mW)
10 T = 10C T = 15C T = 20C T = 25C
8 6 4 2 0
0
20
40
60
80
100
Current (mV) Figure 8.8 The kink-free light output characteristics of an angled-facet FGL with a 7-mm external cavity, as a function of temperature. Measured side-mode suppression ratio is –45 dB and RIN is –145 dB/Hz (from: Timofeev F.N., Bayvel P., Mikhailov V., Lavrova O.A., Wyatt R., Kashyap R., Robertson M., and Midwinter J.E., “2.5 Gbit/s directly modulated fibre grating laser for WDM network,” Electron. Lett., 33(16), 1406–1407, July 1997. # IEE 1997, Ref. [30]).
As a wavelength selector for a laser, the fiber Bragg grating offers a unique solution. The broad-gain bandwidth of semiconductor lasers combined with the quality of single-mode fiber connectors resulted in a “wavelength-uncommitted” laser [26]. A schematic of this device is shown in Fig. 8.9. The basic device consists of an AR-coated laser chip with a short fiber pigtail terminated in a fiber connector. Fiber Bragg gratings at various wavelengths, also packaged in fiber-FC-PC connectors, are then used to define the operating wavelength of the laser. A set of gratings allows a large part of the gain band to be used.
Fiber Grating Lasers and Amplifiers
356
Output power (dBm)
Figure 8.9 Wavelength-uncommitted laser [26].
10 0 –10 –20 –30 –40 –50 –60 1.53
1.54
1.55 1.56 Wavelength, microns
1.57
Figure 8.10 SMSR of 50 dB for a fiber grating semiconductor laser with an output power of 2 mW.
A semiconductor laser was shown to lase from 1480 to 1573 nm. The SMSR was measured to be 50 dB for such a laser, at a power output of 1 mW in the fiber. Figure 8.10 shows the emission spectrum. This laser was shown to operate single-frequency at all wavelengths. However, the temperature has to be controlled carefully to prevent mode hopping and allow stable operation. A combination of the uncommitted laser concept and the large-spot laser should result in a cheap solution for wavelength selection. Apart from applications that require the use of a modulation technique external to the laser, direct modulation of FGLs is desirable for low-cost devices. A large number of factors, such as electrical capacitance and inductance, cavity roundtrip, bandwidth of the external grating, and the interplay between the sub- and supercavity reflections, influence the dynamic behavior of FGLs. It has been shown that an external cavity laser with strong feedback can operate with high stability [3]. Provided the laser is operated within mode hops, the modulated performance has been shown to be excellent at transmission rates in excess of 2.5 Gb/sec [27–30]. Further, application of a FGL for millimetric-wave generation has been investigated [31]. In the latter demonstration, the modulation frequency injection locks two otherwise unstable lasing modes within the grating reflection spectrum to produce a high spectral purity RF beat signal at 35 GHz.
Static and Dynamic Properties of FGLs
357
t10_50_3 h28n8_1 : Modulation bandwidth, I = 40,60,80,100mA 20 –3dB bandwidth: 15,5 GHz
15
Rel. amplitude (dB)
10 5 0 I = 100 mA
–5 –10 –15
I = 40 mA
–20 –25
Ith = 26 mA 0
2
4
6
8 10 12 14 Frequency (GHz)
16
18
20
Figure 8.11 The frequency response of a FGL comprising a high-speed strained MQW and 8-mm-long external cavity (from: Paoletti P., Meliga M., Oliveti G., Puleo M., Rossi G., and Senepa L., “10 Gbit/s ultra low chirp 1.55 mm directly modulated hybrid fiber grating semiconductor laser source,” in Tech. Digest of ECOC’97, pp. 107–110, 1997. # IEE 1997, Ref. [32]).
With a short external cavity multi-quantum-well (MQW) FGL, modulation at >10 Gb/sec is possible for optical fiber transmission [32]. This design relies on the high-differential gain and low intrinsic chirp of a strained MQW, ARcoated front facet FP laser and the high-speed chip design using semi-insulating blocking layers. The linewidth of this laser was measured to be 40 kHz (13 GHz roundtrip resonance; 8-mm external cavity), with –3 dB modulation bandwidth of 15 GHz. The frequency response of the laser for different bias currents is shown in Fig. 8.11. The external cavity resonance is clearly visible as the current is increased to 100 mA. Measurements have shown that the chirp of this laser remains <10 MHz/mA compared with 250 MHz/mA for a standard DFB laser. With a 2-mm-long external-cavity FGL, the frequency response is extended beyond 20 GHz [32].
8.2.1 Modeling of External Cavity Lasers Figure 8.12 shows a schematic of a semiconductor laser coupled to an external cavity. Also shown are the parameters that need to be taken into account for modeling the laser: the reflectivities of the front and rear facets, the reflectivity and length of the grating, and the phase offset between the laser and the grating
Fiber Grating Lasers and Amplifiers
358 Small-signal modulation R1
Bias tee
lreff (w)l 2
Phase offset j lrBraggl 2
Leakage current
Chip and package parasitics
DC bias
R2
Ld HR coating
Le AR coating
r 02
LB Fiber grating
Figure 8.12 Schematic diagram modeling of the FGL. (AR: antireflective coating; HR: high reflective coating) (from: Premaratne M., Lowery A.J., Ahmed Z., and Novak D., “Modelling noise and modulation performance of fiber Bragg grating external cavity lasers,” IEEE J. Selected Topics in Quantum Electron. 3(2), 290–303, 1997. # IEEE 1997, Ref. [33]).
(passive fiber cavity), as well as the electrical contribution due to leakage, parasitics in the packaging, bias, and modulation signals. Premaratne et al. [33] have presented a comprehensive rate-equation model for strong feedback taking account the many internal reflection of the composite cavity, based on the work of Park et al. [34], as well as the delayed fields from the reflections. Multiple reflections are accounted for by assuming that the field components are stationary [35]. Their analysis is therefore restricted to periodic and nonperiodic steady-state solutions. Transient response of FGLs has been investigated by Berger [36], while mode locking of the FGL to produce soliton like pulses has been modeled by Morton et al. [37,38]. The rate equations used to model the FGL are of the form [39,40] dEðtÞ 1 lnðrÞ ¼ io þ ðGðNÞ ascatt Þ þ EðtÞ; ð8:2:1Þ dt 2 tL where G(N) is the spatially averaged gain at the average carrier density, N, resonant cavity, o is the operating angular frequency with external feedback, ascatt is the scattering loss, or is the reference angular frequency of the field envelope, E ðtÞ, pffiffiffiffiffiffiffiffi E ðtÞ ¼ IðtÞei½or tþ’ðtÞ ; ð8:2:2Þ I(t) is the instantaneous intensity at time, t, and ’(t) the instantaneous phase. In Fig. 8.13 is shown the modeled steady-state L–I characteristics of a typical FGL with strong feedback and relatively large front-facet reflectivity of 0.04, and grating length of 8 mm. The choice of the initial phase condition at the interface determines the occurrence of the mode hop at around 30 mA, at either side of which the laser operates in a single longitudinal mode. The instability in this region is a result of the multimode nature of the long external cavity, predominantly due to the residual front-facet reflection.
Static and Dynamic Properties of FGLs
359
5 a H = 5.0 R1 = 0.9025 R2 = 0.04 kL = 1.0
Output power (mW)
4 3
Mode hopping
2 1
0
Threshold current = 22.28 mA
0
10
20 30 Bias current Iinj (mA)
40
50
Figure 8.13 The L–I characteristics for a FGL laser with 4 GHz external cavity with a relatively large front-facet reflectivity (from: Premaratne M., Lowery A.J., Ahmed Z., and Novak D., “Modelling noise and modulation performance of fiber Bragg grating external cavity lasers,” IEEE J. Selected Topics in Quantum Electron. 3(2), 290–303, 1997. # IEEE 1997, Ref. [33]).
Mode hopping can be eliminated by careful choice of operating temperature, as shown in Fig. 8.6, or by the reduction of the front-facet reflectivity. Interference effects due to delayed fields of the external cavity have been shown to be critical for predicting the stable performance of the FGL [33]. This effect of resonance-peak spectral splitting (RPSS) is shown in Fig. 8.14. As the modulation frequency (close to the roundtrip frequency) is varied as a function of bias current, the output of the laser undergoes a series of peaks and dips. At a bias current of 37 mA, a steady single resonance peak is established, leading to stable operation. This situation is similar to the instabilities in the lasing characteristics of FGLs observed by Timofeev et al. [11] under modulation. Modeling shows the importance of the intracavity interference [33] in the vicinity of the mode-hopping regime. The alpha parameter strongly influences the RPSS, with low values of a and front-facet reflectivity favoring stable operation. The relative intensity noise (RIN) spectra also show the appearance of RPSS as a is increased [33].
8.2.2 General Comments on FGLs It appears that reduced-length laser diode chips and short external cavities favor single-mode, mode-hop-free operation [11]. Further, low facet-reflectivity, high-speed MQW designs, and strong feedback from the fiber grating promote
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360
IM response (dB)
10 a H = 5.0 0
R2 = 0.042
–10
–20 Mo 3.5 du lat i
on
4.0 qu en c
fre
y ( 4.5 25 GH z)
30
45 40 35 mA) ( l e v Bias le
50
Figure 8.14 Change of RPSS versus bias current. Note the splitting position variation with bias current (4-GHz external cavity). The splitting may be reduced by a reduction in the front facet reflectivity (from: Premaratne M., Lowery A.J., Ahmed Z., and Novak D., “Modelling noise and modulation performance of fiber Bragg grating external cavity lasers,” IEEE J. Selected Topics in Quantum Electron. 3(2), 290–303, 1997. # IEEE 1997, Ref. [33]).
good-quality modulation characteristics [3,24,32]. Various designs for the FGL show the alternatives available in reducing front-facet reflectivity by the use of a reflective amplifier [24], increasing coupling by the use of hyperbolic lenses formed on the ends of the fiber [19], chirped gratings to promote smooth L–I characteristics and allow stable mode-locking as well as the generation of soliton pulses [17,18], high-speed MQW designs for wide-bandwidth operation [32], plug-in gratings to allow selection of lasing wavelengths [26], and the use of coherence collapse for the injection locking of pump lasers [21]. The benefits of FGLs remain in the ability to define the lasing wavelength [26] and the low chirp [15,28,32] in improving the utilization of laser chips from manufactured wafers.
8.3 THE FIBER BRAGG GRATING RARE-EARTH-DOPED FIBER LASER The concept of fiber lasers dates back to 1960. Snitzer [41] demonstrated a rod waveguide of smallish dimensions (0.5-mm diameter) doped with ions of neodymium and dielectric end mirrors, which produced pulsed radiation when pumped with a flash lamp. Although it may be argued that the device was not a true fiber laser, the difference is merely in the detail, since it was a waveguide. With the advent of low-loss single-mode fibers [42] and the subsequent rareearth-doped single-mode fibers [43], the early lasers still depended on external
The Fiber Bragg Grating Rare-Earth-Doped Fiber Laser
361
mirrors to choose the appropriate emission line. Although fibers enabled efficient end-pumping of these lasers, there were cases, as with Nd-doped fiber, in which the suppression of lasing of an adjacent competing line became difficult owing to the discrimination required of the resonator mirrors [44]. The problems were further compounded by the use of bulk reflectors or by the deposition of mirrors directly on the end faces of fibers [45]. In- and out-coupling had to be achieved through some form of mechanical alignment of mirrors, with high power densities at the mirror/fiber interface that usually needed a layer of index-matching oil for good optical contact. The first demonstration of the use of a narrowband in-line fiber reflector for a fiber laser used a 70% reflectivity grating etched into the core of a side-polished fiber [46]. While this scheme had the advantage of producing a narrow-bandwidth reflector (0.8 nm FWHM), it had the disadvantage of being a device that needed considerable mechanical processing. The principle of tunability had already been demonstrated using a bulk diffraction grating as a frequency-selective reflector [47]. Thus, the photosensitive fiber Bragg grating, which was narrowband, low loss, and of adjustable bandwidth, was wholly suited to application in fiber lasers. The availability of compact high-power pump lasers at 1480 and 980 nm has made the fiber laser simple to fabricate. The high gain available in rare-earth-doped fibers allowed a nominal 0.5% reflection grating and a 100% bulk mirror with 30 m of erbium-doped fiber laser to produce 300 mW of power at 1537.5 nm when pumped with 600 mW of 980-nm pump [48] in the first photoinduced grating-based fiber laser. The ability to write Bragg gratings directly into rare-earth-doped fibers compatible with photosensitive fibers has resulted in a variety of successful demonstrations using either single or multiple gratings to form the resonator [49]. In the following section, generic examples of some of these lasers along with their properties will be examined. Figure 8.15 shows a simple configuration of a single-cavity design of a fiber laser. The rare-earth-doped fiber has Bragg gratings as wavelength-selective mirrors. The laser is end-pumped via a WDM coupler normally through an optical isolator. The simple configuration allows a single-frequency laser [50,51] or twin laser configuration to be pumped in series with a single pump [52,53], or enables multiwavelength operation for sensor applications [54]. The transparency of the Intracore fiber gratings End pumped
Figure 8.15 A schematic of the fiber Bragg grating rare-earth-doped fiber laser. The simple configuration encloses a piece of rare-earth-doped fiber between two Bragg matched gratings and is end-pumped.
362
Fiber Grating Lasers and Amplifiers
gratings to all other wavelengths allows closely spaced wavelengths to be emitted from one laser source, or the cascading of the laser with an amplifier section in a master oscillator power amplifier (MOPA) configuration [55]. A review of these lasers covers some of the general aspects [56]. Some of these lasers are considered in the following sections. Spatial hole-burnt gratings may be used to stabilize semiconductor lasers. In this case, a piece of unpumped erbium fiber is used with a Bragg grating as the external cavity with an AR-coated semiconductor chip. The standing wave formed in the erbium doped fiber forms a population grating and line-narrows the laser. The 3-m-long cavity with a 1535-nm grating reflector was measured to have a linewidth of 1 kHz [57]. High-power, double-clad Yb: doped fiber lasers with integral gratings in the cladding have achieved output powers in excess of 6.8 W in a single longitudinal mode [58]. At 1090 nm, 9-W fiber lasers are available that can be used as pumps for amplifiers and Raman oscillators (Section 8.7) [59].
8.4 ERBIUM-DOPED FIBER LASERS The erbium ion in germanosilicate fiber has a three-level laser transition (4I13/2 ! 4I15/2), which may be modeled using the simplified three-level energy diagram of Digonnet [60]. A simple model for single-mode operation of an erbium fiber laser incorporating fiber gratings shows that the transient energy density per lasing mode is [61] dEn ðzÞ ¼ aIn ðzÞ gIn ðzÞ; dt
ð8:4:1Þ
where a and g are the gain and loss for the lasing mode n. With negligible propagation losses, the loss is given as g ¼ ln½Ro ðlÞRHR ðlÞ;
ð8:4:2Þ
where Ro(l) and RHR(l) are the reflectivity of the output coupling and high reflector gratings, respectively, at wavelength, l. The gain in terms of the steady-state ion population densities may be defined as a ¼ ðse N2 sa N1 Þ;
ð8:4:3Þ
where the emission and absorption cross-sections are se and sa, respectively. The gain for two-moded lasers may be simplified in terms of the saturation intensity in the strong pump regime to I1 ðzÞ I2 ðzÞ ; ð8:4:4Þ a a0 1 Isat Isat
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where the saturation intensity is a function of the emission and absorption cross-sections, and a0 is the small-signal unsaturated gain. The lasing photon intensities I1 and I2 are periodic within the cavity as a result of the formation of standing waves. Using Eq. (8.4.4) in Eq. (8.4.1), the intensity of each steady-state mode is g1;2 Isat 2 1 ð8:4:5Þ I1;2 : I1;2 ¼ 3 a0 3 The slope efficiency at single-mode operation is then lp sp g A1 ¼ ; 1 1 3l1 sa þ se a 0 Ap
ð8:4:6Þ
where sp is the ground-state absorption cross-section, A1 is the lasing mode area, and Ap is the pump mode area. Substituting Eq. (8.4.5) into Eq. (8.4.1) and solving for the loss for the second mode results in the condition for stable single-mode operation, g2 >
a0 2g1 þ : 3 3
ð8:4:7Þ
Equation (8.4.7) states that the loss of the second mode must be greater than a third of the single pass gain plus the cavity roundtrip loss for the first mode. The reflectivity of the grating may be computed by the methods presented in Chapter 4 and used to determine the reflectivity for a given length of doped fiber with an unsaturated gain of a0 per meter. With the gratings written in highly doped erbium-doped fiber, the additional gain within the gratings must also be taken into account, since the effective reflectivity is increased [62]. The best region for single-mode operation is when the free-spectral range is twice the expected shift in the lasing frequency induced by perturbations and cavity length changes. This is greatly assisted by making the cavity as short as possible.
8.4.1 Single-Frequency Erbium-Doped Fiber Lasers Erbium-doped-fiber grating lasers (EDFGLs) offer a simple and elegant solution for wavelength selection, providing a very narrow linewidth as well as a high degree of wavelength stability. They are also compatible with optical fiber systems and can be easily integrated with other fiber components, such as WDM couplers and fiber isolators. Tests performed on an externally modulated single-frequency EDFGL have confirmed its robust suitability for error-free high-speed application in transmission systems [50]. The laser is fabricated with 600 ppm GeO2:Al2O3 doped silica
Fiber Grating Lasers and Amplifiers
364
fiber with a refractive index difference of 0.023 (core diameter of 2.6 mm) and cutoff at 880 nm. The tight confinement ensures high efficiency, and aluminum reduces the concentration-quenching effects that reduce the lifetime of the upper laser level, also leading to instabilities [63]. The gain of this fiber was reported to be 10 dB/m, with an overall length of the laser of 4.4 cm, including gratings (98% O/P coupler and 100% broadband high reflector), using the linear cavity design shown in Fig. 8.15. Pumped at 1480 nm, the laser is operated close to threshold to ensure single-mode operation and amplified in a MOPA configuration through an optical isolator. The packaged laser is sensitive to vibration that drives relaxation oscillation at a frequency of 150 kHz. These oscillations can be actively controlled by using a feedback scheme to control the diode-pump laser and reduced to insignificant levels [50,64,65]. The error-free performance of several EDFGLs in transmission experiments at 2.5 and 5 Gb/sec been have reported [66,50,67], using external Mach–Zehnder modulators. Up to 60 mW of single-frequency power has been achieved using a MOPA configuration [68]. Other configurations for single-frequency operation use long gratings with a bandwidth less than the cavity mode-spacing [69,70], and operation at 1 micron using Nd:doped fiber has been shown with intracore gratings [71].
Composite Cavity Lasers There are several methods for achieving single- and multifrequency operation of EDFGLs. As outlined already, short lasers with narrowband reflectors are simple candidates; however, a composite cavity topology can enforce stable single-frequency operation by longitudinal mode control, adapted from semiconductor lasers [72–75]. The principle relies on a small additional feedback element in the form of a short Fabry–Perot, which modulates the gain spectrum of the main fiber laser cavity. Figure 8.16 shows the linear cavity configuration. The basic laser cavity gratings have reflectivities of 0.9 and 0.8 with a weaker reflection of 0.1 as the external reflector. The gain of the composite cavity is FS G1
FS G2 Er doped fiber 3.8 cm 4.8 cm
G3
1.5 cm
pump
output
ISO
WDM
Figure 8.16 A schematic of the composite cavity single-frequency EDFGL [53].
Erbium-Doped Fiber Lasers
365
Laser cavity feedback
0.8
A
FSR = 15GHz
0.6
0.4
0.2
0.0 –20
B
0 10 20 –10 Frequency shift, GHz
Figure 8.17 The reflectivity and mode selection spectra of the coupled-cavity EDFGL. Laser with feedback (solid line), and laser without feedback (dashed line) (from: Chernikov S.V., Taylor J.R., and Kashyap R., “Coupled-cavity erbium fiber lasers incorporating fiber grating reflectors,” Opt Lett 18(23), 2023 (1993) (after Ref. [53]).
modulated, increasing the discrimination between the modes. Since the lasing mode is influenced by the composite cavity, a single mode tunes with temperature changes but does not exhibit mode hops [53]. The 10-mm-long high erbium dopant concentration fiber (120 dB/m absorption at 1530 nm) is spliced to fiber gratings, forming a composite cavity 7 cm long; 980-nm pumping with a Ti: sapphire laser showed a threshold of 50 mW. The linewidth of this laser is 40 kHz using conventional heterodyne techniques. The gain bandwidth of such a laser with and without the extra etalon is shown in Fig. 8.17. With the external etalon, the gain is modulated at a frequency separation determined by the spatial separation and reflectivities of the gratings. The simulation in Fig. 8.17 shows the composite reflection spectrum of the two gratings without the additional feedback (dashed curve) and with feedback (solid curve). In this laser, the period of the frequency separation is 28 GHz without feedback and 7 GHz with the etalon approximately half the mode spacing of the composite laser. The use of a very short gain region with respect to the cavity length (a factor of 3–4) eliminates the relaxation oscillation observed in other cavity designs [50] using erbium fiber. Composite fiber gratings with ring or loop mirror cavities [76] are other possible configurations, and for single frequency operation with an intra-cavity frequency selector [77]. An example of this cavity is shown in Fig. 8.18 in a loop mirror arrangement. In this cavity, the loop mirror is a broadband mirror, while the external fiber Bragg grating is a band-selecting element. The isolators inside the loop-mirror ensure unidirectional operation, while the incorporation of an
Fiber Grating Lasers and Amplifiers
366 Amplifier Isolator
HR grating 50:50 coupler
10%
Termination
DFB Figure 8.18 The loop mirror cavity for single-mode operation, incorporating a DFB band-pass filter and an erbium-doped fiber amplifier (after Ref. [77]).
ultranarrow band-pass DFB fiber Bragg grating (see Chapter 6), also inside the loop, enforces single-frequency operation. The laser may be tuned by applying compressive or extensive strain on the intracavity DFB grating or, in the absence of the DFB grating, the external grating. Single-frequency operation of such a laser with a 0.075-nm band-pass DFB grating shows a linewidth of 2 kHz, with side-mode suppression of 50 dB. However, the long cavity with a mode spacing of 11.4 MHz mode hops and requires the use of stabilizing elements [77]. Replacing the DFB band-pass intra-cavity filter by an acousto-optic tunable filter (AOTF) may extend the principle of this type of a laser. Ramping the AOTF sweeps the output frequency of the laser with <0.1-nm linewidth [78]. To improve the strength of the grating, a specially deposited photosensitive ring cladding can be fabricated with a rare-earth-doped core. This scheme allows the core to have a high dopant concentration, while preserving the fiber’s photosensitivity and allowing short, high-strength grating to be inscribed for laser fabrication [79,80].
8.5 THE DISTRIBUTED FEEDBACK FIBER LASER The principle of the distributed-feedback (DFB) structure, generally applied to semiconductor lasers [81], is easily translated to doped fiber lasers. Figure 8.19 shows a schematic of the laser, which is identical to a Fabry–Perot laser,
End pumped
Phase shift
Figure 8.19 The EDFGL distributed feedback laser (EDFGL-DFB). The phase shift of p/2 forms a band pass in transmission.
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367
except that the two gratings are separated by only a quarter-wavelength (see Chapter 6). The phase shift generates a gap in the reflection spectrum, allowing narrow single-frequency operation. With the possibility of writing gratings directly into erbium-doped fibers (see Chapter 2), DFB lasers may be fabricated with ease, especially since long gratings (100 mm) require modest refractive index modulations to produce a high reflectivity. The first reported DFB laser operated at 1 micron and was fabricated in ytterbium-doped fiber [82]. It has been successfully demonstrated in erbium-doped fibers for 1500-nm operation [83,84] and extended by cascading five DFB lasers separated in wavelengths by 1 nm each, pumped by a single 1489-nm diode laser [85]. Simple modeling of a DFB laser may be achieved by including a gain (complex term) in kdc in [Eqs. (4.3.5) and (4.3.6)], so that Db is modified by ig, where g is the gain per meter. Numerical simulation of the transfer function in Chapter 4 [Eq. (4.8.22)] of the DFB is shown in Fig. 8.20 as a function of increasing gain. As the gain increases, the side modes of the DFB structure begin to lase. There are two possibilities for inducing a phase shift in the grating: one which is localized, or by distributing the phase shift along the length, as discussed in Chapter 6. The model takes account of the variation in the gain as a result of pump depletion. It has been shown for a 5-cm-long DFB laser that positioning the discrete p phase-shift away from the center of the laser, at 0.6 LDFB, increases the output power by as much as 60%. For a phase shift
Amplitude 40 6
20 0 –20
4 Gain
1.54996 1.54998
2
1.55 Wavelength, mm 1.55002 1.55004
0
Figure 8.20 The transmission spectrum of a 100-mm-long DFB laser as a function of the gain.
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368
distributed over 1 cm of the same laser, the optimum value acquires a slightly larger value of 1.3p radians [86]. The position of the phase shift changes as a direct result of spatial hole burning, since the intensity is highest at the center of the laser for a symmetrically positioned phase shift. The distributed phase-shift laser is as efficient as the discrete off-center phase-shifted laser.
8.5.1 Multifrequency Sources Dual-frequency and multifrequency sources can be built by combining techniques presented in the preceding sections. A novel and particularly simple arrangement is the four-grating coupled cavity arrangement based on the single-frequency laser shown in Fig. 8.16. By adding an extra grating matched to the external grating in Fig. 8.16, a dual-frequency laser is formed. In order to ensure the coupling between the cavities, the grating bandwidths are chosen to overlap slightly. This configuration produces a robust laser that performs as a single entity, and a schematic is shown in Fig. 8.21. The emission spectra consist of two single-frequency lasing modes at defined by the grating pairs, and tune without mode hops. However, the difference frequency tunes with temperature, strain, or pump power (thermally induced). The coupled cavity reported by Chernikov et al. [52] operated at a difference frequency of 59.1 GHz centered around 1545 nm, with a long-term average linewidth of 16 kHz and a stability of the dual frequency of 3 MHz. Multifrequency operation in ring lasers has been reported with the use of a grating with a multiple wavelength reflection spectrum. Up to 8 wavelengths have been shown to lase simultaneously. While single-frequency operation (10 kHz linewidth) has been demonstrated, in such long lasers, mode hopping is common, and special attention must be paid to stabilize the operation [87]. T 3 dB coupler at 1550 nm Pump at 980 nm
Output
ISO
1550nm M1
EDF
980nm
M2 PC1
WDM 980 / 1550 nm
PC2
M3
EDF 4.8 cm
M4 1.5 cm
Figure 8.21 The coupled-cavity dual-frequency source (from: Chernikov S.V., Kashyap R., McKee P.E., and Taylor J.R., “Dual frequency all fiber grating laser source,” Electron. Lett. 29(12), 1089, 1993. # IEEE 1993, Ref. [52]).
Bragg Grating-Based Pulsed Sources
369
8.5.2 Tunable Single-Frequency Sources The gain bandwidth of doped fibers is several tens of nanometers. Tunability is possible with fiber Bragg gratings by strain or temperature tuning (see Chapter 3). Ball and Morey [88] stretch-tuned a pair of Bragg gratings of a 100-mmlong erbium fiber laser and showed 9 GHz mode-hop-free operation. Since the gratings and the fiber length tune together, the laser remains stable. Compression tuning can extend the tuning range, and a tuning range of 32 nm has been reported for a short EDFGL (3 cm) in a MOPA configuration. Continuous single-frequency tuning was observed with the laser producing 3 mW of output. A feedback loop is necessary to stabilize the operation of the laser to reduce relaxation oscillations [89]. Compression tuning of a Yb/Er laser using a mirror and Bragg grating configuration has shown sub milliwatt thresholds and both strain and compression tuning ranges of up to 25 nm [90].
8.6 BRAGG GRATING-BASED PULSED SOURCES We have seen that the FGL can be used as a pulsed source for optical fiber transmission, either directly modulated or mode-locked [15,17,18,24,27,32]. We have also seen that the EDFGL is an excellent candidate as a source [50]. There are several alternative methods of generating short pulses in conjunction with gratings, for example, semiconductor laser pulse compression in a chirped grating [91,92], using the dual-frequency source and adiabatic soliton pulse compression [93] and linear actively mode-locked fiber laser [94]. Pulse compression of gain-switched DFB and FP lasers [95] is possible using a dispersive delay line, since the emitted pulse is chirped in time and frequency. If the dispersion of the delay line is opposite to that of the pulse, it may be compressed. Chirped gratings (see Chapter 7) are ideal candidates, since they are compact and the dispersion can be tailored for a particular application. The experimental setup is shown in Fig. 8.22. A pulse train at 500 MHz from a gain-switched DFB laser is reflected from a chirped grating and the recovered pulse is compressed. Figure 8.23 shows autocorrelation traces of the input and reflected pulse from the grating. The gain-switched pulse has chirped a bandwidth of 1.5 nm, and the 6-mm-long grating slightly filters the spectrum while recompressing the pulse. A small residual pedestal is due to the uncompensated part of the spectrum. The technique is simple and requires a minimum of control. The second scheme for compressing a sine wave into pulses is based on a combination of adiabatic perturbation and average soliton regimes of propagation. During an adiabatic perturbation, there is a balance between the dispersive and nonlinear contribution by a change in the soliton duration [96]. In the
Chirped FBG
Isolator DFB 50:50
Input pulse
Output
Wavelength, nm 0
1540
1542
1544
1546
1548
–5 –10 –15 –20 –25 –30
Transmission, dB
Figure 8.22 Experimental setup for pulse compression and transmission spectrum of the chirped fiber grating (from Gunning P., Kashyap R., Siddiqui A.S., and Smith K., “Picosecond pulse generation of <5 ps from gain-switched DFB semiconductor laser diode using a linearly step-chirped grating,” Electron. Lett. 31(13), 1066–1067, 1995. # IEEE 1995, Ref. [91]).
SHG intensity SHG intensity 1.0 1.0 0.8
0.8 0.6
0.6 27.8 ps (19.3 ps)
0.4 0.2
0.2
0.0
0.0
–100
A
7.3 ps (5.1 ps)
0.4
–50
0 Delay time, ps
50
100
–100
B
–50
0
50
100
Delay time, ps
Figure 8.23 Input pulse from the gain-switched DFB (a) and reflected pulse form the chirped fiber Bragg grating (from Gunning P., Kashyap R., Siddiqui A.S., and Smith K., “Picosecond pulse generation of <5 ps from gain-switched DFB semiconductor laser diode using a linearly step-chirped grating,” Electron. Lett. 31(13), 1066–1067, 1995. # IEEE 1995, Ref. [91]).
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average soliton regime, there is balance between the periodically varying dispersion and nonlinearity [97]. The use of this scheme allows the slow transformation of a modulated input signal into a soliton. An amplified optical sine wave is launched into a long length of fiber. It periodically undergoes self-phase modulation in a zero-dispersion section of a fiber, increasing the spectral content and linear dispersion in a high-dispersion part of the transmission line. By selecting the appropriate combination of dispersion and nonlinearity, the average dispersion of the link is reduced approximately exponentially. The reducing average dispersion continually compresses the optical sine wave into soliton pulses. A similar scheme for generating solitons was reported using dispersion decreasing fiber [98], based on the earlier predictions [99]. The source used for pulse generation is the dual-frequency EDFGL source shown in Fig. 8.21 [100]. The narrow line-widths of this source produce an optical beat signal. To eliminate stimulated Brillouin scattering, different germania concentration fibers are used in the 20-section comblike dispersion profiled fiber (CDPF), with a total length of 7.5 km. A beat signal at an amplified power of 190 mW and frequency of 59.1 GHz are converted into pedestal-free sech2-shaped pulses with a FWHM of 2.2 psec, at a wavelength of 1545 nm [93]. Several other configurations based on the beat-frequency generation may be found in Ref. [101]. Mode-locking of fiber lasers has been investigated in a variety of “figureeight” configurations [102] using fiber gratings to generate multi-wavelength pulses. Actively mode-locked dark-pulse generation from a praseodymiumdoped fiber laser with a chirped grating has also been reported [103].
8.7 FIBER GRATING RESONANT RAMAN AMPLIFIERS Raman scattering is a process in which a small fraction of the incident light is scattered by the vibrational modes within a material to generate a Stokes photon, downshifted in frequency. Stimulated Raman scattering (SRS) is a process by which the Stokes photon interacts with the pump photon to generate another Stokes photon and is described by the imaginary part of the nonlinear susceptibility, w(3)(–os, op, –op, os). The Stokes field grows exponentially with the length of the medium. Since its discovery [104], SRS has been a topic of considerable research [105]. SRS can be a very efficient process, strongly depleting the pump power. With the advent of optical fibers, the observation of SRS has become very easy because of the high power densities in the core, low optical loss, and long interaction lengths. The Raman gain, which depends on the scattering cross-section, has been measured in silica optical fibers [106]; the band-width, because of the amorphous nature of the glass, is extremely large,
Fiber Grating Lasers and Amplifiers
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extending over some 40 THz with a peak at 13 THz from the pump wavelength. Below the threshold for SRS, a signal photon, downshifted from the pump frequency, experiences gain if it lies within the gain band-width. This is the principle of Raman amplification [107]. If a fiber is placed between mirrors and pumped, the Stokes field sees a roundtrip gain of G ¼ e2gR I0 Leff ;
ð8:7:1Þ
where gR is the Raman gain coefficient, I0 is the intensity of the pump, Leff is the effective length of the fiber 1/apump (the fiber loss at the pump wavelength), and the factor of 2 is for the double pass. The pump intensity at which the Stokes field overcomes the cavity losses for a given set of mirror reflectivities is called the threshold for oscillation. For 10 m of fiber, a CW threshold can be reduced to 1 W [108,109]. As the pump power is increased, the Stokes field S1 increases until the threshold for the second-order Stokes S2 is reached, at which point energy is transferred to S2. At this point, a signal at the third Stokes frequency will experience gain, and so on. This is also the principle of the resonant Raman amplifier. This type of a multi-Stokes oscillator has been demonstrated by Stolen et al. [110,111], who generated five Stokes orders of independently tunable radiation. Perhaps one of the most elegant components that is the direct result of the high transparency of Bragg gratings outside the band stop is the Raman fiber grating laser, the RFGL. This laser has opened many opportunities in communications, by allowing amplification in any part of the communication spectrum by appropriate choice of pump lasers and fiber Bragg gratings. The general cavity configuration for a resonant Raman laser [112] is shown in Fig. 8.24. This laser has been shown to produce 1.5 W at 1485 nm when pumped by a diode-pumped Ybþ3 double-clad laser. The Ybþ3 pump at 1117 nm produces in excess of 6 W CW. The five stages of Stokes of conversion is remarkable, showing ultralow loss of <0.2 dB per grating.
Germanium-doped fiber Gratings 1480
1117-nm
Gratings
1315 1395
1175 1240
1175
1395 1240
1480 1395
Output coupler
Figure 8.24 Cascaded fiber grating resonant Raman 1480-nm pump laser for pumping erbium amplifiers [113].
Gain-Flattening and Clamping in Fiber Amplifiers
1175
WDM
1240
1.3/1.064mm
Blocking grating 1064
Gratings
Gratings
1300-nm input
373
1175 1117
1117
1240
Germanium-doped fiber
1064-nm pump
1300-nm amplified output
Figure 8.25 Schematic of a 1300-nm cascaded fiber grating resonant Raman amplifier [115].
For 1300-nm amplifiers, two configurations can be used. The linear cavity shown in Fig. 8.25 uses WDM couplers and a set of gratings to allow gain in the 1300 nm window when pumped by a 1.064 nm source. The gain is available at the fourth Stokes wavelength. These amplifiers have gains as high as 40 dB with saturated output powers of 24 dBm. With higher germania concentration (higher Raman gain) the pump power can be lowered to 300 mW while providing a gain of 25 dB [114]. The theoretical noise figure for Raman amplifiers is 3 dB [116], while the achieved figures are around 4.4 dB at 40 dB gain in a ring configuration [117,118]. The advantage of using bidirectional pumping as in a ring cavity is reduced cross-talk and polarization sensitivity. These amplifiers are increasing in importance as the requirement for the optical bandwidth increases. In regions of the communications window in which amplification is difficult, e.g., 1350–1500 nm, cascaded fiber grating resonant amplifiers are likely to provide unique solutions.
8.8 GAIN-FLATTENING AND CLAMPING IN FIBER AMPLIFIERS Rare-earth-doped optical fiber amplifiers are important components in transmission systems. The transmission bit-rate distance product is almost limitless in laboratory-based demonstrations [119]. This is primarily due to the “zerotransmission loss” through periodic amplification as well as the management of dispersion. The fiber amplifier has enabled undersea transmission over thousands of kilometers. With the increasing demand for bandwidth, wavelength division multiplexing (WDM) of optical channels is seen to be a viable solution for increasing transmission capacity at a given transmission rate, in point-to-point routes [120]. The important issues in amplified transmission systems for WDM applications are the available gain, the gain uniformity across the bandwidth of an
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amplifier, the robustness to transient switching of WDM channels in single and cascaded amplifier chains, and the overall noise figure. The gain spectrum and upper-state lifetimes of an erbium-doped fiber amplifier vary as a function of the core dopants. High-germanium erbium-doped fibers have a highly nonuniform gain with a bandwidth of 35 nm from 1530 to 1565 nm, peaking at approximately 1535 nm. The use of aluminum reduces the nonuniformity, making the gain spectrum flatter, while ytterbium as a codopant with erbium allows an efficient transfer of energy from the available high-power diode pumps at 980 nm (a wavelength at which erbium suffers from excited-state absorption, reducing efficiency) to the required transition in the 1550-nm wavelength region for amplification. The latter shifts the gain peak to >1540 nm, while narrowing the gain bandwidth. Along with the nonuniformity in the gain, the doped fiber amplifier is homogeneously broadened. While allowing amplification across the wide gain bandwidth, the gain available at any wavelength is dependent on the simultaneous presence or absence of other wavelength channels; thus, gain may be depleted from a saturating signal from an existing channel as another channel is switched on. As the gain in a single amplifier fluctuates, the problem is exacerbated with a chain of cascaded amplifiers, leading to severe cross-talk. Nonuniform gain across the bandwidth of the amplifier produces a wavelength-dependent low-frequency cross-talk penalty. Apart from the signal degradation, severe damage to components is a possibility because of spiking with intermittent interruption and resumption of transmission. The inversion and thus the gain is also pump-power dependent, leading to gain changes as the pump source ages. For analog transmission systems the problem is worse, because the local gain slope leads to harmonic distortion, degrading the received signal. Schemes have been developed to stabilize the gain of fiber amplifiers as well as flatten the gain variation over a wide bandwidth. Fiber gratings offer simple solutions to solve both these problems. In the following sections, we look at a specific example of gain flattening and gain stabilization and a combination of the two, using fiber gratings.
8.8.1 Amplifier Gain Equalization with Fiber Gratings In Chapter 4 the properties of side-tap gratings (STG) and long-period gratings (LPG) were described. Both types of gratings may be used as narrowband, wavelength-specific, loss-inducing components. In particular, the STG, which is a tilted Bragg grating, couples a narrowband at the short-wavelength side of the Bragg reflection wavelength to a continuum of the radiation field in the case of the unbounded cladding. This requires matching the cladding with an appropriate oil/polymer to destroy the well-defined boundary. For an appropriate blaze angle, which also minimizes Bragg reflection into the guided mode, the radiated
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375
light has an angular and spectral bandwidth. At this blaze angle, the period of the grating determines the wavelength of peak loss. Adjustment of the grating period at the same tilt angle enables the loss to be placed at any position within the gain spectrum of the amplifier. It should be noted that coupling to the radiation field for the first-order grating interaction is restricted to a local loss spectrum, close to the Bragg reflection wavelength, and hence a combination of several such spectra allows the fabrication of complex spectral loss features. By appropriate choice of filters, the filter loss may be matched to the inverse of the gain variation in the erbium amplifier to flatten the gain spectrum. Both single [121] and multiple [122] STGs have been used to tailor the gain spectrum of erbium amplifiers. In the first instance, a single, 4-dB peak-loss, 10-nm bandwidth (full-width) grating was placed at the 1533-nm peak of the erbium amplifier, eliminating the gain variation. By appropriate choice of the grating period (with the use of a phase mask) the peak loss is moved anywhere within the gain bandwidth. The inversion and therefore the gain shape is dependent on the pump power. The filter is thus appropriate for a given inversion (gain). Figure 8.26 shows eight such loss spectra with approximately identical shapes, but with different peak-loss wavelengths and insertion loss. Each filter is 8 mm long with an external blaze angle of 8 , chosen to minimize back reflection, written in a boron–germanium codoped fiber with a core diameter of 12 microns (see Chapter 3). The use of multiple STG filters allows tailoring complex gain spectra with a greater degree of accuracy. The peak deviation over 3 dB from a fixed gain of 13.7 dB across the bandwidth of the amplifier is conveniently reduced to 0.3 dB by the use of in-line STGs [122] and is shown in Fig. 8.27. The required loss spectrum is deconvolved into a number of contributions from individual gratings, prior to fabrication of the filter. With this information, the filter is
5
Loss, dB
4 3 2 1 0 1525
1535
1545
1555
1565
1575
Wavelength, nm Figure 8.26 Radiation loss spectrum of eight STG filters written at an external blaze angle of 8 in a fiber with a core diameter of 12 microns [122].
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376
With side tap filter
Gain, dB
18 16 14 12 10 1530
1540
1550
1560
1570
Wavelength, nm Figure 8.27 The ASE spectrum of a saturated erbium amplifier without and with a composite STG gain equalizing filter [122].
fabricated with the amplifier at its operating inversion level, allowing live gain tailoring. In this instance up to nine individual STGs were written to match the variation in the gain. Each grating can be 1 mm long, making the entire gain equalization filter to be less than 10 mm. Such a filter may be written using a single phase mask appropriately designed to give the desired loss at each wavelength, by scaling the length of each grating or with an appropriate amplitude shading [123]. A distinct advantage of the STG is that the uncompensated temperature sensitivity of the loss spectrum is similar to that for Bragg grating, making the filter intrinsically stable. With temperature compensation as with Bragg gratings, the variation in the loss spectrum may be eliminated altogether over the required operating temperature range. The tilt angle for the STG is chosen to minimize back-reflection into the guided mode (see Chapter 4). The transmission and reflection spectra of two STGs with peak-loss wavelengths separated by 10 nm are shown in Fig. 8.28.
Transmission loss/reflection, dB
1520 0
1540
1560
1580
–10 –20 –30
Transmission loss Reflection
–40 –50 –60 Wavelength, nm
Figure 8.28 The transmission and reflection spectra of two STGs written in the same fiber. The 6-dB peak-loss STG 1 at the shorter wavelength has a reflection of –35 dB. The second unoptimized STG at a larger tilt angle has a higher reflection of –21 dB.
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377
The combined transmission loss of the two gratings is 12 dB. The first grating is written at a blaze angle close to that required for a minimum reflection, while the second is written at a larger angle to increase the bandwidth of the composite filter. Each grating has a loss of approximately 6 dB. The first STG, which peaks at the shorter wavelength, has a reflectivity of –35 dB, while the unoptimized longer wavelength loss STG shows an increased reflection of 21 dB. Typically, the gain variation in the amplifier spectrum requires each STG to have a peak loss of less than 3 dB, resulting in a maximum back-reflection of <–40 dB. Thus, a concatenation of STGs may be used effectively to flatten the gain of an optical amplifier, with low back-reflection. The design of the fiber to alter the bandwidth of the filters has been discussed in Chapter 4. This approach allows finer structures in the gain spectra to be matched more closely [124]. The application of LPGs for tailoring the gain of optical amplifiers has also attracted interest. The main feature of the LPG is the coupling of the guided mode to a forward-propagating cladding mode, one of which is selected from a large number, to induce the desired loss at the appropriate wavelength within the gain spectrum. It should be remembered that several mode interactions, widely separated in wavelength, occur in tandem for a given grating period. As such, LPGs have been used to equalize the gain of erbium amplifiers [125,126] and as ASE-suppressing filters. The technique used for forming the gain-equalizing filter is identical to that for the STG and has been already described [122]. There are major differences between the STG and the LPG. The latter exhibits more than a single loss peak separated by 30–60 nm, depending on the type of fiber used for the filter. Written in standard telecommunications fiber, the temperature sensitivity of the LPG is roughly 4–5 times that of the STG. However, it has a low back-reflection into the guided mode of –80 dB. Unless the LPG is fabricated in special temperature-stabilized fiber, the amplifier gain equalization remains temperature sensitive, leading to gain tilt. Finally, the wavelength of the peak loss is a function of the refractive index of the material surrounding the cladding. With an appropriate low-index polymer coating, the cladding mode resonance is made insensitive to the surrounding material. The LPG has been used extensively in gain equalization of amplifiers and remains an important component. Another method of equalizing the gain of an erbium amplifier is by the use of two or more apodized reflection gratings in series with an optical isolator. The resultant spectrum is shown in Fig. 8.29. Gain equalization to approximately 0.5 dB may be achieved with this simple arrangement. Since gain flatness of the optical amplifier with filters is dependent on the level of inversion, it becomes particularly attractive to combine the filter with all-optical gain control, which is considered in the next section.
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Transmission, dB
1520 2
1530
1540
1550
1560
1570
0 –2 –4 –6
Gratings(1534/1559) Tx + ASE
–8 –10 Wavelength, nm
Figure 8.29 The simulated gain spectrum of an erbium amplifier equalized by two reflection gratings of lengths 0.2 and 0.061 mm centered at 1534 and 1559 nm. (The refractive index modulations for the two gratings are 3.5 10–3 and 1.06 10–3.)
8.8.2 Optical Gain Control by Gain Clamping There are a variety of ways of stabilizing the gain of erbium-doped fiber amplifiers. In particular, sampling the state of the amplifier and deriving some sort of feedback, for example, signal levels [127], ASE [128], or a dedicated probe [129], to adjust the amplifier’s pump power, or sacrificial injected signals have also been proposed [130,131]. These in turn limit the frequency response by introducing electrical delays in the feedback loop. A simpler and more elegant all-optical approach to gain stabilization of an erbium doped fiber amplifier was first reported by Zirngibl [132], using lasing action in a ring cavity configuration. In a homogeneously broadened system, the inversion and therefore the gain remains constant irrespective of input signal level or the number of input channels. The amplifier remains robust against transients and to the switching of channels, so long as the amplifier continues to lase at some wavelength within the amplification window. The gain is maintained at the expense of the flux in the lasing mode. This “gain-clamped” amplifier is especially useful for cascading and in dynamically changing optical networks. The amplifier design is greatly simplified by the use of a linear cavity made with narrowband, highly reflecting Bragg gratings [133]. Delevaque et al. demonstrated a gain-clamped erbium amplifier lasing at 1480 nm with the aid of narrowband grating reflectors, pumped at 980 nm. The properties of all-optical gain-controlled amplifiers, pumped at 1480 nm, and lasing at longer wavelengths, have been studied by Massicott et al. [134]. The cavity configuration used for gain control, pumped at 1480 nm, is shown in Fig. 8.30. The amplifier cavity contains a length of erbium-doped fiber with two narrowband Bragg-matched fiber grating reflectors. Along with these gratings, a STG
Gain-Flattening and Clamping in Fiber Amplifiers
Signal input ls
379
Er3+ doped Intracavity loss l l fiber
WDM Reflector ll
Laser cavity
Signal output ls
Reflector ll
Pump lp Figure 8.30 Amplifier with linear optical AGC.
filter with a specified intracavity, loss is included close to the output end to control the intracavity loss and reduce the effect of stray reflections. Above a certain threshold pump power at which the cavity gain equals the intracavity loss, lasing occurs at a wavelength l1, clamping the gain across the gain bandwidth due to the homogeneous nature of the transition. With an increase in the pump power, energy is stored in the lasing flux, while maintaining the inversion, and therefore the gain. Signal wavelengths experience a fixed gain up to a certain critical input level, at the expense of the lasing flux. Once the input signal is large enough to extract all the energy from the lasing mode, the amplifier ceases to lase. Thereafter, the amplifier inversion (and gain) is uncontrolled and is dependent on the pump power and signal levels as for a normal erbium-doped fiber amplifier. Figure 8.31 demonstrates the automatic optical gain controlled amplifier in operation. The amplifier consists of a 25-m length of erbium-doped fiber with a core diameter of 5.3 mm and refractive index difference of 0.013. The peak saturable absorption of the fiber is 6.1 dB/m with a background loss of 8 dB/km
Output / arb. log.
0 –10
Signal power
–20
–2dBm –3dBm
< Reduced laser power –30 –40 –50 –60 1510
1520
1530
1540
1550
1560
1570
Wavelength / nm Figure 8.31 Output spectra of all-optical AGC amplifier, lasing at 1520 nm [134].
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380
measured at a wavelength of 1.1 mm. The laser cavity is defined by two Bragg grating reflectors at 1520 nm, written in hydrogen-loaded GeO2-SiO2 fiber. The reflectivity of each grating is 94% with a 3-dB bandwidth of less than 0.4 nm. The splices dominate the cavity loss at 2.5 dB, single pass. The pump power in the fiber from a 1480-nm diode laser was approximately 80 mW. Figure 8.31 shows the compensatory effect of the control laser in the broadband output spectra. As the input signal is increased to –2 dBm from the small signal level (–30 dBm), there is more than 10 dB of reduction in the residual laser output power. The inversion (and therefore the gain) in both cases remains the same. The excess noise in the high signal case is an artifact due to the side modes of the signal DFB source. The evolution of the amplifier gain at 1550 nm, as a function of input signal level for four different pump powers is shown in Fig. 8.32. A gain of nearly 16 dB is maintained up to an input signal power level of about –5 dBm, at the maximum pump power level of 80 mW. The pump power no longer determines the amplifier’s gain in the gaincontrolled regime, only its maximum controlled output power. This is a desirable feature for a well-managed amplifier. The dynamic performance of amplifiers with and without gain control is compared in Figs. 8.33 and 8.34. To test the transient response of the amplifiers an input signal of –10 dBm is modulated at 54 Hz and the outputs monitored on an oscilloscope, as shown in Fig. 8.33.
Pump powers
Gain / dB
15
80mW 52mW 33mW 23mW
10
5
0 –40
–30
–20
–10
0
Input signal / dBm Figure 8.32 Gain characteristic of AGC amplifier. Lasing wavelength: 1520 nm; signal wavelength: 1550 nm; signal power: –30 dBm (from: Massicott J.F., Willson S.D., Wyatt R., Armitage J.R., Kashyap R., and Williams D., “1480 nm pumped erbium doped fibre amplifier with all optical automatic gain control,” Electron. Lett. 30(12), 962–963, 1994. # IEE 1994, Ref. [134]).
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381
Output / arb. lin.
Standard amplifier
AGC amplifier
Input signal 0
10
20
30
40
50
Time / ms Figure 8.33 1550-nm input signal modulated at 54 Hz (bottom trace), signal amplified using gain control (middle trace), and signal amplified without gain control (top trace) (from: Massicott J.F., Wilson S.D., Wyatt R., Armitage J.R., Kashyap R., and Williams D., “1480 nm pumped erbium doped fibre amplifier with all optical automatic gain control,” Electron. Lett. 30(12), 962–963, 1994. # IEE 1994, Ref. [134]).
2.5
Output / arb. lin.
2.0
No AGC
1.5 1.0
With AGC
0.5 0.0
0
10
20
30
40
50
Time / ms Figure 8.34 Cross-talk experienced by a contradirectionally propagating small signal 1560-nm probe to a modulated 1550-nm signal. Probe amplified using gain control (solid) and without gain control (dashed) [134].
Without the signal present, the population inversion builds up in the uncontrolled amplifier. When the signal is injected, the output overshoots, producing a spike before a new equilibrium is reached. In the optical gain-controlled amplifier, the spike is eliminated.
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Fiber Grating Lasers and Amplifiers
Additionally, the induced cross-talk is also eliminated, as shown in Fig. 8.34. A small counterdirectionally propagating probe at 1560 nm is strongly affected in the uncontrolled amplifier but remains unaffected with AGC. In the absence of AGC, the CW probe output power more than doubles when the saturating signal is blocked, whereas in the controlled case, a change of less than 0.5% in output is seen. To eliminate the residual laser power at 1520 nm, an additional STG [122] with a rejection of 30 dB is used. BER measurements performed at 2.5 Gb/sec show no penalty as a result of operating the amplifier in the optical gaincontrolled regime. A combination of both gain control and gain equalization forms a highly desirable amplifier. A GEQ filter composed of a concatenated set of STG filters, (as discussed in Section 8.8.1) added to the AGC amplifier output shows excellent GEQ-AGC. The flattened spectral shape is maintained for as long as the amplifier is operated within the gain-controlled range.
8.8.3 Analysis of Gain-Controlled Amplifiers For an amplifying fiber in which the Er3þ ion population inversion profile is approximated to be constant across the fiber core, the wavelength-dependent gain coefficient is given by gðlÞ ¼ GðlÞ N½ðseðlÞ þ saðlÞ Þn2 saðlÞ ;
ð8:8:1Þ
where se(l) and sa(l) are the emission and absorption cross-sections, respectively, N is the axial Er3þ ion density, n2 is the fraction of ions in the excited state, and G(l) is the confinement factor representing the overlap between the propagating mode and the radial ion density distribution. In an amplifier in which gain control is in operation, the population inversion, and hence n2, is set by the lasing condition, and amplifier gain calculations can be made without reference to the magnitudes of pump and signal power levels. The total linear loss at the laser cavity wavelength is L
Cavity loss ¼ e2gðlasÞ :
ð8:8:2Þ
Determination of the minimum required pump power to maintain gain control in the presence of signals of known magnitude is more involved. Approximate amplifier analyses (e.g., [135]) can be employed, but values so obtained substantially underestimate the actual power requirements. Principally, this is because no account is taken of pair induced quenching effects that degrade power conversion efficiencies even in low Er3þ ion concentration fibers [136].
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383
8.8.4 Cavity Stability The gain stability of the amplifier is determined by the stability of the control laser wavelength and the laser cavity loss. The laser wavelength is fixed by the narrow-linewidth grating reflectors that have a temperature sensitivity of 0.01 nm/ C. To avoid changes in cavity loss if drifting should occur, the use of one narrow- and one broader-band reflector is preferable. Reflections at the laser wavelength from other parts of the transmission system will alter the effective cavity loss, as will polarization dependence combined with birefringence in the fiber. The use of high-reflection gratings with an associated intracavity side-tap attenuator, as opposed to reflectors of lower reflectivity, offers greater resilience to stray light from other parts of the transmission system.
8.8.5 Noise Figure The signal-spontaneous beat-noise figure for an amplifier is often given as FðvÞ ¼
sp PðvÞ
hvdvðGðvÞ 1Þ
;
ð8:8:3Þ
where Psp ðvÞ is the spontaneous emission power at frequency v, measured in a bandwidth dv. Over a length of fiber in which the population inversion is constant, the noise figure for that inversion can be calculated using saðlÞ n1 1 FðlÞ ¼ 2 : ð8:8:4Þ seðlÞ n2 It can be seen that for a given wavelength, the best noise figure is obtained for the highest possible inversion, i.e., maximum n2/n1. In a full-length amplifier, whether or not it is gain controlled, the local population inversion varies along the fiber length. The overall noise figure of the amplifier is predominantly determined at the signal input end [137], as spontaneous emission generated at the input experiences the full amplifier gain before being detected within the bandwidth of the signal receiver. In an AGC amplifier, the evolution of the population inversion is, in part, determined by the laser power. For good overall noise performance, it is beneficial to minimize the laser power at the input in order to ensure that, as far as possible, the input inversion is determined by the shorter-wavelength pump source. This involves implementing as asymmetric a cavity configuration as possible. This is achieved by locating the bulk of the cavity loss at the signal output end of the amplifier and by choosing the wavelength of the control
Fiber Grating Lasers and Amplifiers
384 25
Gain F front loss
F and gain / dB
20
F back loss F modeleid
15 Laser: 1530nm 10
Pump: 1480nm 65mW
5 Length: 20m 0 1540
1545
1550
1555
1560
1565
1570
Wavelength / nm Figure 8.35 Gain and noise figure for an AGC amplifier in two cavity implementations (courtesy J. Massicott, BT Laboratories).
laser to be in a high-gain region of the spectrum requiring a correspondingly high cavity loss. Figure 8.35 shows the measured gain and noise figures of an AGC amplifier with a linear control laser cavity at 1530 nm [138]. The pump and signal powers are referenced to the input and output of the doped fiber and the measured noise figures are compared with data obtained using a full numerical amplifier model [139]. In one case the cavity loss grating is placed at the front end of the cavity, and in the other, at the back. The gain spectra for the two cases were identical to within experimental error, but the expected noise figure improvement is achieved when the cavity loss is located at the signal output end of the laser cavity.
8.9 HIGH-POWERED LASERS AND AMPLIFIERS As the previous sections have shown, one of the great successes of the FBG has been in applications in amplifiers and lasers. First, high-quality amplifiers would not have been possible without a pump-locking FBG. Second, gratings, both FBGs and LPGs, have been used to flatten and clamp the gain of amplifiers. Third, high-power Raman fiber lasers and amplifiers also would not have been possible without FBGs. This section reviews some of the advances in highpower lasers and amplifiers and introduces some of the concepts for the application of gratings in related areas of pulse shaping and higher-order mode amplification.
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385
8.9.1 Coupling of Laser Diodes to Optical Fiber with FBGs To have high-power optical fiber amplifiers and lasers, it is necessary to couple high-power light from a pump laser into a fiber. Although there are many ways of achieving this [140], here we consider examples of novel techniques using FBGs. The problem is coupling large stripe or light from a laser bar efficiently into a doped fiber. Normally, the doped core is pumped by light propagating in the cladding, using offset core geometries or secondary cladding with unusual shapes [141]. These techniques usually couple light into the cladding. It is a difficult problem to couple light directly into the core from a large stripe diode. However, it is relatively simple to end couple light into a large cladding. To achieve coupling into the core efficiently, it is necessary to couple the light propagating in the cladding into the core. This can be achieved in a couple of ways using FBGs or LPGs. Several schemes have been developed using LPGs to couple light into a core mode from cladding modes [142–144]. In these schemes, cladding modes phasematched via an LPG are converted into the core mode. The difficulty arises from the fact that a diode laser couples to several modes in the cladding and not all are phase-matched; consequently, the true coupling efficiency is not high, although the coupling to a specific cladding mode is excellent. It is therefore necessary to increase the “brightness” of the core by coupling more cladding modes into the core. Figure 8.37a shows a scheme for coupling high-power laser diode emission into the core of a single mode optical fiber. The high divergence light is focused into the cladding of the optical fiber by a lens formed at the end of the fiber. The shape of the lens determines the efficiency of coupling to a small extent; however, using hemispherical lenses it is possible to couple most of the light into a 125-mm diameter fiber. The light couples into several cladding modes, each with a bandwidth of the diode. If an LPG is imprinted into the core, a single cladding mode will couple into the core mode, over a bandwidth of the grating. Several LPGs with periods adjusted for each mode can be co-located in the fiber to allow each mode to couple into the core. If the fiber has a special parabolic refractive index profile in the cladding as shown in Fig. 8.37b, approximated by the stepped function, with a central single mode step-index core region, then it is possible to couple the cladding light efficiently into the core by using several LPGs or a single composite LPG. The graded index of the cladding ensures that the lengths and the refractive index modulation of the gratings remain roughly constant, but the periods change according to the core–cladding modes in question. It should be noted that the first five odd cladding modes (LP0n:2 to 6) carry >80% of the power. Thus, coupling simply the five modes results in a theoretical coupling efficiency of 70% into the core from a 100-mm wide stripe
Fiber Grating Lasers and Amplifiers
386 80 Coupling efficiency (%)
70 60 50 40 30 20 10 0 60
A
80
100
120
140
160
Working distance (microns)
Figure 8.36 The total coupling efficiency of the first five odd-cladding modes into the core for a graded/step index fiber (solid line) and for a standard step index fiber (dashed line) is shown as a function of working distance. Note the lower sensitivity for the graded index compared to the step index fiber. (Reproduced with permission from: Nemova G., and Kashyap R., “Highly efficient lens couplers for laser-diodes based on long period grating in special graded-index fiber,” Optics Commun. 261(2), 249–257, 2006. # Elsevier Press 2006.)
laser, as shown in Fig. 8.36. An additional benefit of this technique is that the working distance of coupling is relaxed enormously compared to a step index fiber, with a reduced variation in the coupling as a function of distance from the diode. The coupling to the new profile for a hemispherical lens is greatly improved to >70% with a depth of field (1 dB) of nearly 25 mm, compared to 60% and 5 mm for the step index fiber. Figure 8.37c shows the radial regions for the refractive index profile [145]. This technique holds the potential for improving the pumping of lasers and amplifiers through the core.
8.9.2 Hybrid Lasers: Dynamic Gratings Unpumped rare-earth doped fibers are useful for fabricating interesting devices such as tunable ultra-narrow band filters and lasers [146]. A standing wave in a doped fiber modulates the population of rare-earth ions into a population grating, leading to distributed feedback grating and thus narrow band reflectors, which can alter dynamically. These applications have been reviewed in an excellent article [147]. The use of doped fiber in an external cavity of a semiconductor laser was shown earlier [148,149] to produce a rather unique laser. This type of extended cavity works in the coherence-enhanced mode [150–152], although the doped fiber behaves as a two-level system and absorbs the radiation.
High-Powered Lasers and Amplifiers
387
nout Refractive index
1.47
rj Core nj
n1
r1 r2
1.465
1.46
n2
0
A
10
20 30 40 50 Radius (microns)
60
B High power large stripe LD
Multiple LPG for cladding mode coupling Step index core graded index clad
C
Lensed fiber
Figure 8.37 (a) Cross-section of the graded/step index fiber with a stepped index change across the cladding. (b) Refractive index profile of the step index core and graded index cladding. (c) Lens coupling schematic for coupling cladding light into the core. Each cladding mode is coupled to the core with a specific overlapped grating in the core.
A narrow-band FBG is used to reflect light back through a doped fiber and into an AR-coated semiconductor laser. The forward and backward propagating light establishes a standing wave in the cavity, which bleaches the absorption in the high-intensity regions [153] in the doped fiber. This creates a dynamic grating in which the modes compete for the gain. Within a few ns, the cavity stabilizes with the result that the coherence is enhanced by the formation of a stable distributed grating [11]. Dynamic gratings can be formed in any doped fiber in which the absorption can be bleached by self-pumping (i.e., two-level systems). Thus, the laser can operate anywhere within the absorption spectrum of the dopant. In the case of erbium-doped fiber, it should be possible to make lasers operating between 1480 and 1560 nm [153–155]. As the lifetime of the laser level is long lived, the dynamics of the laser are damped; any perturbations faster than the relaxation time of the laser do not affect its operation [156]. Thus, locking a semiconductor laser to an external reference requires low-frequency components, and therefore it is possible to lock two such lasers to within 2 Hz with relatively modest feedback loops [17], making this laser very “quiet” and highly stable. The laser has also shown potential for optical frequency domain modulation in
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Dynamic grating in doped fiber Diode laser With AR coated front facet
FBG
Figure 8.38 The single mode doped fiber external cavity laser (DFECL) with a 75% reflection FBG at one end of the fiber. The length of the external cavity is 30 cm. The unpumped absorption of the doped fiber is 30 dB/m at 1530 nm. (From reference [16].)
radio-over fiber applications using direct modulation [157]. This type of laser should find applications in sensors, phase-locked arrays, frequency standards, and RF generation, among other uses. Figure 8.38 shows the schematic of this laser with the dynamic grating. The refractive index change in a doped fiber is related to the change in the absorption via the Kramers–Kronig relations: c DnðoÞ ¼ P V p
o ð2
o1
Daðo0 Þ ðo0 Þ2 o2
do0 ;
ð8:9:1Þ
where the bandwidth o2 < o0 < o1 is the region in which the absorption change is significant, and P V is the principal value of the integral. The coupling constant of the grating, which is pump power dependent is kðPp Þ ¼
2Dn ; nl
ð8:9:2Þ
where n is the refractive index of the doped fiber. With the length, L of the doped fiber, the grating strength can be calculated as
R ¼ tanh2 k Pp L : ð8:9:3Þ The standing wave ratio gives a difference between the maximum and minimum index change, which determines the actual value of kac. Because a 15 dB/m change in the absorption is induced with 9 dBm, a refractive index modulation of 1.68 106 is induced in the fiber used, resulting in kacL 0.47 (R 17%) for a 28-cm-long grating with a bandwidth of 5.42 pm. With these parameters, the laser coherence increases dramatically, and the line width narrows to single frequency through mode suppression of all but the dominant oscillating mode, with a measured line width of 12.5 kHz at a wavelength of 1490 nm [13].
8.9.3 Fiber Lasers with Saturable Absorbers in the Cavity Using the principles outlined in the previous section, a 40-nm widely tunable, ultra-narrow-band laser has been demonstrated with an unpumped section of erbium-doped fiber. Figure 8.39 shows the ring cavity coupled to a
Toward Higher-Power Fiber Lasers and Amplifiers
389 Widely stretch tunable FBG
Erbium doped fiber 980nm pump laser
Dynamic grating R = 99% in saturable absorber Circulator
Polarization controller Isolator Output
Figure 8.39 Ring cavity with an unpumped external filter. The FBG has a bandwidth of 0.3 nm. The tunable grating is stretched by 4% without breaking. (From reference [19].)
section of unpumped erbium-doped fiber terminated with a narrow band 99% reflective FBG. The counter-propagating waves in the undoped fiber section, which are 4 m long, induce a dynamic grating to provide the additional feedback to narrow the line width of the laser to 750 Hz. The threshold for the 20-m-long erbium-doped fiber in the ring with a total cavity length of 35.7 m was measured to be 25 mW using a pump wavelength of 980 nm. An output power of between 20 and 32 mW was measured depending on the value of the output coupler between 5% and 95% [158]. The acrylic coating on the deuterium-loaded fiber is stripped in hot (>200 C) sulfuric acid and cleaned in acetone before inscription of the FBG. It is then recoated and annealed. This process maintains the strength to that of a pristine fiber. This type of laser holds the potential for high-power operation, although it would be interesting to see the dynamic effects of stimulated Brillouin scattering in this ultranarrow band laser at increasing powers. One would expect to see the laser descend into chaotic behavior.
8.10 TOWARD HIGHER-POWER FIBER LASERS AND AMPLIFIERS Perhaps the most dramatic adventure in the life of the FBG has been its application in high-power fiber lasers. One has seen how the FBG has enabled multi-Stokes oscillators and amplifiers to generate relatively high-power radiation for pumping erbium amplifiers at a wavelength of 1.48 mm, or even amplifiers at the difficult wavelength band of 1.3 mm with the use of ytterbium fiber lasers in Section 8.7. However, a most dramatic turn of events has occurred over the past few years in the area of ultra-high-power lasers and amplifiers. Highpower lasers are now commonplace in many applications such as welding and range finding. These have also been made possible by a simple FBG. Several
Fiber Grating Lasers and Amplifiers
390
companies have exploited the features of the FBG, low loss, in-fiber, highreflectivity, low-temperature sensitivity and high stability against laser radiation to generate ever-increasing powers, either in the single mode or in the multimode regimes. It will become apparent in this section that high-power fiber lasers themselves are used to pump other fiber lasers. A review of fiber lasers may be found in reference [159]. A particularly nice technique has enabled the pumping of short lasers to produce high optical powers. Phosphate-based silica glass has been shown to be an excellent host for a high concentration of ytterbium and erbium ions with negligible clustering [160]. The high concentrations allow the fabrication of ultrashort lasers with high efficiency. Using such a fiber with 5 dB/cm gain at 1530 nm, an end-pumped “short” Er:Yb doped fiber laser was shown to produce 200 mW when pumped with 800 mW. The cavity was made of the doped fiber with two narrowband FBGs fusion spliced on either end. One FBG had a bandwidth slightly wider than the other, but the laser operated in the single frequency regime. The lasing line width was measured to be 2 kHz. This system was extended to end pump a DFB fiber laser only 35 mm long, producing an output power of 160 mW and >50 dB side-mode suppression ratio [161]. A schematic of the device is shown in Fig. 8.40. To overcome spatial hole burning problems in high-power linear cavities, a high-power single frequency laser [162] based on a Fabry–Perot subcavity [163] in phosphate fiber combines the use of two quarter-wave plates in optical fibers [164], by splicing precisely cut pieces of polarization maintaining fibers to the ends of a short laser. Optical fiber wave plates have been used with polarization splitting couplers and chirped gratings to make polarization independent dispersion compensating band-pass filters [165], as shown in Figs. 6.38 through 6.40. The polarization maintaining fiber equivalent is a compact approach, which also has been used in wavelength conversion [166] to render it polarization insensitive, that leads to compact devices. Single-frequency laser emission at 1550 nm with an output power of 1.9 W was achieved in this erbium-doped phosphate fiber laser shown in Fig. 8.41a with 20 W of 980 nm pump power, without spatial hole burning, as the reflections from each end turns the linearly polarized light into counter propagating circularly polarized light in the cavity. DFB in grating in highly doped active phosphate single mode core
Laser emission
Phase-step
105 mm diameter multimode core pump fiber
Figure 8.40 Multimode pumped, 35-mm-long DFB laser in Er:Yb doped phosphate optical fiber [22].
Toward Higher-Power Fiber Lasers and Amplifiers 1% Er3+ + 8% Yb3+ doped 20%, 0.05nm fiber l/4 FBG 1 in PM fiber l/4
391
10 cm SMF sub-cavity 80% FBG 2 40% FBG 3 Core
Doped fiber OUTPUT
A
11 cm MMF Pump lasers
Teflon MM pump fiber
Doped fiber core
B Figure 8.41 (a) Single-frequency, 1.9-W output laser without spatial hole burning. (b) The generic details of the pumping scheme for such a fiber. (From reference [23].)
The F-P subcavity ensures that the fiber laser emits a single frequency [24]. Another novel feature of this short laser is the efficient pumping scheme applicable to short fiber lasers without the need for end pumping through the gain fiber. The generic technique uses one or several multimode optical fibers in physical contact with a gain fiber as shown in the cavity design of the laser described earlier and shown in Fig. 8.41b. A short fiber oscillator/amplifier source with a pulse energy of 20.4 nJ and peak power of 16.6 kW with 1.7 ps pulses at a repetition rate of 70 MHz, also based on a Yb:Er doped phosphate fiber and pumping scheme shown in Fig. 8.41b, has been reported [167]. A semiconductor saturable absorber mirror (SESAM) butt-coupled at the rear end of the laser was used with a 10% reflection, 8.2 nm bandwidth chirped FBG centered at 1558 nm to generate the pulses at 70 MHz in a 1.7-m long cavity. The amplifier used in this demonstration was 15 cm long and pumped by 20 W at a wavelength of 975 nm. The cavity was designed to have near zero dispersion in the anomalous dispersion regime (D ¼ 0.045 ps/nm). Androz et al. [168] reported 2.3 W of radiation at a wavelength of 1480 nm from a thulium doped ZBLAN fiber laser. As it is difficult to make highreflectivity FBGs using ultraviolet (UV) radiation in ZBLAN, the gratings for this laser were written using fs laser pulses. One of two Yb-doped fiber lasers operating at 1040 or 1064 nm, respectively, was used to pump a 2000-ppm thulium-doped ZBLAN fiber from Le Verre Floure´. The 2.9-mm diameter, 5.45 m long fiber with a numerical aperture (NA) of 0.235 with an LP11 cutoff at 870 nm was used with 90% reflectivity FBG as the high reflector with the 4% Fresnel output facet. It was reported that the FBG was annealed and
Fiber Grating Lasers and Amplifiers
392
Yb: Fiber laser
Residual pump
ISO
1480nm signal
OSA Tm+3:ZBLAN fiber
R = 90% FBG Figure 8.42 The Tm3þ:ZBLAN fiber laser configuration for 1480 nm wavelength emission with 2.3 W output power. (From reference [29].)
showed no degradation after months of operation in the laser. The optimum wavelength to pump this laser was found to be 1053 nm for a 3-m-long cavity with an output coupler of 10% reflectivity. However, pumping with 6 W at 1064 nm results in an output power of 2.25 W, with a laser line width of 140 pm and slope efficiency of 53%. It is speculated that a double clad fiber would allow the output power to scale to several watts. The laser is shown in Fig. 8.42. A prism is used to separate the desired 1480 nm output.
8.10.1 Fiber Raman Lasers The chirped FBG and narrow FBG combination has been used to generate Stokes radiation at high powers with near quantum-limited efficiency [169]. A 125-m-long nonpolarization preserving Corning PureMode HI980 optical fiber was used as the Raman laser cavity. The pump laser was a 15 W ytterbium-doped fiber laser. The fiber loss at the 1108-nm pump wavelength was 0.941 dB/km, and there was a loss of 0.811 dB/km at the emission wavelengths. Figure 8.43 shows a schematic of this device. A prism separates the output with polarization compensation being achieved by a tilted glass plate at the output. The high-reflector input FBG was a chirped grating centered at a wavelength of 1165 nm with a bandwidth of 5 nm and a flat reflectivity of 22 dB. Two output-coupler FBGs were used in the experiments: The first one was a narrowband FBG at a wavelength of 1165.4 nm (0.8 nm bandwidth) and with a reflectivity of 26%. The second one was a chirped FBG centered at 1163 nm with a FWHM bandwidth of 5 nm and a reflectivity of between 40% and 60% because of the chirp. These are spliced in turn at the
Toward Higher-Power Fiber Lasers and Amplifiers
15W Yb: Fiber laser @ 1108nm
393
Residual pump power
Stokes output
ISO
OSA
x20
x10 98:2% Tap
FBG1: I/C R =99.6 % 1162-1167nm
P1
FBG2: O/C
Pol. comp. glass plates
110m X = splice
Figure 8.43 Raman fiber laser with 110-m length of resonator fiber. (From reference [30].)
output of the resonator fiber with an insertion loss of 0.03 dB per splice for all gratings. Using this system, 7.2 W of output with a 9W-pump power at 1108 nm is achieved [30]. Fiber Raman lasers have been used to generate tunable high-power radiation. Emission over 60 nm has been achieved by using a pair of gratings tunable between 1075 and 1135 nm. The two FBGs imbedded in silicone resin cylinders were compression tuned. This tunable laser operated in single Stokes emission with a threshold pump power of 2.59 W and generated 5 W output with a pump power of 6.5 W, equivalent to a quantum efficiency of more than 93% [170]. Figure 8.44 shows the schematic of this laser. Replacing the output prism with a recirculating pump reflector grating would make the device fiber compatible.
Residual pump power
Yb: Fiber laser YLR-20-1064-LP
Stokes output
ISO
OSA x10
x16 99:1% Tap Tunable FBG1: I/C
P1
Tunable FBG2: O/C
X = splice
Figure 8.44 A widely tunable Raman fiber laser generating up to 5 W of radiation in the 1075–1135 nm window, pumped at a wavelength of 1108 nm. (From reference [31].)
394
Fiber Grating Lasers and Amplifiers
8.11 ULTRAHIGH-POWER LASERS AND AMPLIFIERS The damage threshold of silica fiber is 25 kW for a 35-mm mode-field diameter (MFD) ( 1.3GW-cm1). Therefore, high-power amplification in an optical fiber is possible, before damage limits the output power. However, stimulated Raman scattering (SRS) limits the output power to around 9 kW, if power extraction is at 1 kW-m1 for a 35-mm MFD fiber [20]. For a 12-m-long fiber amplifier, the limit reduces to 7 kW. Stimulated Brillouin scattering (SBS) is easily circumvented by a number of techniques, but by primarily increasing the bandwidth of the amplified signal. Although 3 kW CW output at in Yb: doped large area fiber has been demonstrated [171], great care must be taken to prevent damage of the output end. Large mode area multimode fibers can remain in single-mode operation without mode coupling, but it has to be a very high quality fiber. Coupling of the pump light of several kW also poses serious problems. For high-power operation, it would be good to find other solutions, and recently photonic crystal fibers, which are doped with erbium/ytterbium in the cladding and the core areas, have shown great promise for power scaling [172]. In this scheme, not only is the pumping of the fiber eased because the entire cladding is doped, but because of the large core mode area, high-power amplification is possible. The design of large mode area bend-resistant fibers has led to mode areas in the thousands of mm2 [173], and light propagation in ultra-large mode area higher-order mode multimode fibers has also been demonstrated [174]. Many schemes have been develop to produce high optical powers [175–178]. The use of FBGs at high powers requires heat management not only of the amplifier or laser fiber but also the FBG. Most of these schemes do not use FBGs. An alternative technique proposed for high-power amplification is based on the coupling of the fundamental mode in an input fiber to a cladding mode in a cladding doped fiber amplifier, where the amplification takes place [179]. This scheme offers a larger amplification volume while increasing the threshold for SRS. Additionally, the amplification may be chosen to be in a higher-order mode, which increases the mode area dramatically to over 7000 mm2. SRS is avoided by the larger MFD and at the end of 12 m of fiber; 2.2 kW output is predicted at a wavelength of 1.55 mm, before SRS occurs. The amplification could be improved greatly if Yb3þ/Er3þ:doped phosphate fibers are used [33]. These fibers also have a significant advantage over Ge:doped fibers: They do not show signs of photo-darkening at high powers [180]. At the output of this amplifier, a second LPG is used to convert the cladding mode back into a fundamental mode in a large mode area, amplifier cladding matched core optical fiber, shown in Fig. 8.45. This scheme may also be used with a photonic crystal fiber and could lead to the ultimate in high-power lasers,
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395 AMPLIFIER FIBER
INPUT FIBER SMF Input fiber
OUTPUT FIBER
Outer cladding LPG 2
LPG 1 Core
Large core fiber
Doped cladding
Figure 8.45 Er:doped cladding, higher-order mode (HOM) amplifier with two mode converting LPGs. The output fiber is short and simply provided to couple the HOM into the fundamental mode. (From reference [40].)
when combined with other schemes such as heavily doped phosphate fibers. However, heat management remains a serious issue despite the extremely high efficiencies of the fiber laser systems, and techniques will have to be developed to reduce the high temperatures if catastrophic failure is to be avoided in the long term. Finally, the use of fs pulsed lasers has allowed us to overcome the low photosensitivity in certain types of doped fibers, as well as resistance to thermal damage and stability at high-power operation. This scheme may also enable the writing of gratings in multimode optical fiber amplifiers [181] and help with the power scaling of lasers. High-power lasers and amplifiers of multikilowatt capacity may well find use in electrical power transmission and distribution systems, because the transmission loss of optical fibers is a fraction of electrical cables. It remains to be seen how the energy conversion from optical to electrical power will take place at these enormous power levels, as severe challenges exist in achieving this aim.
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Chapter 9
Measurement and Characterization of Gratings The transfer characteristics of a grating are of primary importance for a number of applications. For example, in high-bit-rate applications, it is necessary to know if the grating will impart additional dispersion and, if so, how much. Gratings can be used in a vast number of demanding applications, such as sensing in harsh environments, or in undersea optical fiber transmission that requires components to survive the 25-year design lifespan of the system. For long-term use, it is essential to know whether or not the grating will maintain its designed characteristics over the lifetime. It is also important to know, as it is for optical fibers, the integrity of its mechanical strength for the same reasons. Thus, reliability is a big issue. The transmission characteristics of certain gratings may be affected by the out-gassing and annealing processes more than others; the resonance wavelengths of all gratings drift because of the out-diffusion of molecular hydrogen in high-pressure sensitized fibers. Stress relaxation can complicate matters, by altering the induced refractive index modulation. Sensitivity of the Bragg wavelength with temperature and strain has to be taken into account for such applications as in band-pass filters. Gratings have to be annealed to stabilize their properties for long-term use. The bandwidth, reflection profile, and phase response of gratings require special measurement techniques for proper characterization. In this chapter we shall consider some of the parameters that are of importance and techniques that have been developed for characterization. These include reflectivity and transmission spectrum, bandwidth, average refractive index change and refractive index modulation coefficient, grating uniformity and quality of apodization, insertion loss, radiation loss, and group delay of chirped gratings, drift due to out-diffusion of hydrogen, temperature effects during measurements, PMD, and stress changes. Methods that have been reported for the measurement of thermal decay of gratings will be covered in the final section. The aim of this chapter is to provide an overview of the properties of optical fibers used for grating fabrication, including thermal annealing and characterization of fiber gratings and mechanical strength. 405
406
Measurement and Characterization of Gratings
9.1 MEASUREMENT OF REFLECTION AND TRANSMISSION SPECTRA OF BRAGG GRATINGS The nice thing about gratings is that their growth can be monitored during the inscription process. Since the fabrication is noninvasive, apart from stripping part of the coating, the input and output ends of the fiber are often accessible. Usually the source at the wavelength of interest is an edge-emitting diode, which provides sufficient output power for a variety of measurements. Alternatively, white light may be coupled into the fiber, although the dynamic range is limited. The amplified spontaneous emission from a fiber amplifier is a very good broadband source, and the choice is available to cover the 900–1700 nm wavelength band. It is normal to have either a circulator or a 50:50 fiber splitter between the source and the grating to be measured. The most sensitive method for detecting gratings is in reflection, and for this reason it is best to measure gratings in reflection for diagnostic purposes and display the signal on an optical spectrum analyzer. The basic apparatus for measuring Bragg grating reflection and transmission is shown in Fig. 9.1. The inset shows an alternative arrangement using the coupler. Reflections just above the noise floor of the spectrum analyzer are easily displayed. When a grating is written into the fiber, a reflection peak appears which may be checked for the wavelength. At the same time, the transmission spectrum shows no change until the grating reflectivity is a few percent. A useful guide is the 3.5% reflection from the cleaved far end of the fiber to calibrate the actual reflection from the grating (see Fig. 9.2). For a uniform grating with a reflectivity R, we can calculate the coupling coefficient, kacL, as pffiffiffiffi ð9:1:1Þ kac L ¼ tanh1 ð RÞ: With this information and the measured bandwidth, 2Dl, between the first zeroes (FWFZ), the grating length L is uniquely defined as per Eq. (4.6.14), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 ðkac LÞ2 þ p2 : ð9:1:2Þ 2Dl ¼ pneff L From the length, we may calculate the refractive index modulation vDn as in Eqs. (4.6.3) and (4.3.6), with a fringe visibility v, kac ¼
pvDn : l
ð9:1:3Þ
Eqs. (9.1.1) and (9.1.2) are plotted in Fig. 9.3 for three different values of grating length, 1, 2, and 8 mm, as a function of the coupling constant kac.
Reflectivity/ Transmission, dB
Measurement of Reflection and Transmission Spectra of Bragg Gratings
407
1550 1551 1552 1553 1554 1555 1556 0 –20 –40 –60 –80
–100 Wavelength, nm
Transmission Spectrum analyzer Grating under test
UV
Reflection
Coupler Source Isolator Broadband source
Reflection Circulator Figure 9.1 Apparatus to measure the transmission and reflection spectrum of Bragg gratings.
Reflectivity/ transmission, dB
1548 0 –20
1550
1552
1554
Lg = 4 mm
–60 –80 –100
1558
3.5% fiber end reflection
dn = 1e-7 (pk-pk) –40
1556
Rp Spectrum analyzer ~noise floor
Transmission Reflection
2Δl Wavelength, nm
Figure 9.2 The reflection spectrum of a 4-mm-long grating with a refractive index modulation amplitude of only 107. At this stage it is undetected in transmission. Also shown is the 3.5% reflection from the cleaved end of the fiber, assuming that there are no losses in the reflected light. The very weak Bragg reflected signal is easily detected. The noise floor for a spectrum analyzer resolution of 0.1 nm is shown only as an example.
408
Measurement and Characterization of Gratings 1 8
2 1
3
0.75 1
2
0.5
2
1
Reflectivity
FWFZ Bandwidth, nm
4
0.25
8 0 0
1000
2000
3000
0 5000
4000
k (per m) Figure 9.3 The reflectivity and bandwidth of three Bragg gratings as a function of the coupling constant kac at a wavelength of 1550 nm. The numbers refer to the lengths in millimeters. Note that for large values of the coupling constant, the grating bandwidth grows linearly. As a guide, the maximum refractive index modulation amplitude, Dn, for v ¼ 1, and overlap, ¼ 0.8, is 3 103 (at kac ¼ 5000 m1).
The data have been plotted for a Bragg wavelength of 1550 nm. The wavelength shift dl as the grating grows can be calculated from Eq. (4.6.4), dl ¼ 2l
Dn ; n
ð9:1:4Þ
where we remind ourselves that Dn is the ac index change and l is the Bragg wavelength at the start of the growth of the grating. As the grating grows, it shifts to longer wavelengths and this is shown in the transmission spectra in Fig. 9.4. Along with the shift is shown the effect of a nonuniform UV beam profile. This has been assumed to have a Gaussian
Transmission, dB
1551 0 –10
1552
1553
1554
1555
1556
B C
A
–20 –30
dn = 5e-4(A)
–40
dn = 1e-3(B) dn = 2e-3(C)
–50 Wavelength, nm
Figure 9.4 The shift in the Bragg wavelength and the appearance of the Fabry-Perot structure on the short-wavelength side of a Gaussian intensity profile grating as the UV-induced refractive index modulation amplitude increases.
Measurement of Reflection and Transmission Spectra of Bragg Gratings
Transmission, dB
1551 0
1552
1554
1553
409
1555
Gaussian
–10 –20 –30 –40
L = 4 mm dn = 1e-3(max)
–50
Uniform
–60 –70
Wavelength, nm Figure 9.5 A uniform amplitude profile grating compared with a Gaussian amplitude profile grating.
profile, as with many laser beams, and causes a chirp in the grating [1], since the Bragg wavelength is proportional to the effective mode index. There are two effects of the nonuniform UV beam profile: The grating acquires additional structure on the short-wavelength side (Fig. 9.4) as it grows, and the peak reflectivity drops for the same refractive index modulation amplitude, as is seen for the uniform profile grating in Fig. 9.5. Comparing the uniform and the Gaussian intensity profile grating, the effect on the bandwidth is only slight. The long-wavelength edge of the Gaussian profile grating is apodized. We now compare the Gaussian intensity profile with the Gaussian apodized grating, i.e., one in which the refractive index modulation changes with the length of the grating, but not the mode effective index (see Chapter 5), and find that the short-wavelength structure disappears and the peak reflectivity increases with apodization. The reason for this is that the Bragg wavelength of the apodized grating is constant and the reflection is not spread over a larger bandwidth, and so the effective length is longer. (See Fig. 9.6.) The maximum reflectivity can be calculated by measuring the transmission dip Td in dBs. The translation from the measured dip to the reflectivity is R ¼ 1 10Td=10 ;
ð9:1:5Þ
or from the peak of the reflected signal Rp below the transmitted signal it is (as shown in Fig. 9.2) R ¼ 10Rp =10 :
ð9:1:6Þ
The data are shown in Fig. 9.7. For example, in Fig. 9.2, the reflected signal is shown to be 70 dB below the transmission level. This translates to a reflectivity of 105%.
Measurement and Characterization of Gratings
Transmission, dB
410 1550 1551 1552 0.0 L = 4 mm –10.0 dn = 4e-3(max) –20.0 –30.0
1553
1554
Gaussian apodization
–40.0
1555
1556
Gaussian amplitude profile beam
–50.0 –60.0 Wavelength, nm
Figure 9.6 Comparison of chirp induced in a strong grating due to the amplitude profile of the writing beam and a Gaussian profile apodized grating with the same parameters. The FWFZ bandwidth is approximately the same, but the slope on the long wavelength side is different, as well as the structure on the short wavelength side.
1 Transmission dip
Reflectivity
0.8 0.6 0.4
Reflection peak below maximum transmission
0.2 0 0
5
10
20 15 T-dip or R-peak, dB
25
30
Figure 9.7 Reflectivity as a function of the dip in the transmission spectrum of a grating or as the reflection peak below transmitted signal.
Alternatively, a 10-dB transmission dip is equivalent to a reflectivity of 90%, 20 dB is 99%, and so on. It is assumed that there is no additional loss in the reflected signal as compared with the transmitted signal. If the loss is known, the transmitted level or the reflection peak must be adjusted accordingly. Special care needs to be taken when measuring transmission dips in excess of 30 dB because of the limited resolution of the spectrum analyzer. The slit width of the spectrum analyzer is not a delta function, and there is substantial leakage from the spectral region outside of the slit bandwidth. Integrated, it amounts to more signal being transmitted and affects the spectrum mostly at the dip in the grating transmission. There are several solutions to this problem. Obviously, a better spectrum analyzer is one, or a tunable laser source may be used in conjunction with a conventional spectrum analyzer, ensuring that the
Measurement of Reflection and Transmission Spectra of Bragg Gratings
Transmission, dB
1.540 0
1.542
1.544
1.546
1.548
–20
411
1.550
–Z/2
0
Z/2 L0
–40 –60 –80
–dL
Wavelength, mm
Figure 9.8 The measured grating is 4 mm long with an estimated index modulation of 4 103. The beam intensity profile had a Gaussian shape. The inset shows the change in the Bragg wavelength across the length of the grating.
scanning of the laser and the spectrum analyzers are synchronized [2] with an appropriate slit width. The combined side-mode suppression and the slit width reduces the captured noise. Such a measurement is shown in Fig. 9.8, in which the spectrum of a strong 4-mm-long grating spectrum with a transmission dip of >60 dB has been resolved. In addition to the very steep long-wavelength edge, it has structure on the short wavelength that is due to cladding mode coupling and Gaussian chirp. Clearly, chirp is not a feature that is desirable for simple transmission filters. We now consider the spectra of uniform period gratings and the effect of apodization. Figure 9.9 shows the reflection spectrum of unapodized and apodized (cos2 profile) gratings. The unapodized grating is nominally a 100% reflection grating, and the apodized one has identical length and refractive index modulation. The effect of apodization is to reduce the effective length to approximately L/2. As a result, the FWFZ bandwidth approximately matches the second 0 U
Reflectivity, dB
–10 –20 –30 –40
A
–50 –60 1541.0
1541.5
1542.0
1542.5
1543.0
1543.5
1544.0
Wavelength, nm Figure 9.9 Reflection spectra of 4-mm-long unapodized and cos2 apodized gratings with a refractive index modulation amplitude of 4 104.
412
Measurement and Characterization of Gratings
zeroes of the unapodized grating. Note that the reflectivity is also reduced (halved). To generate an apodized grating with the same bandwidth, the length has to be approximately doubled, and the coupling constant has to be adjusted, so that an 8-mm-long raised cosine apodized grating will have the same approximate bandwidth and reflectivity. In order to resolve the reduced side lobes for the apodized grating, the spectrum analyzer linewidth should be selected to remove artifacts and a false noise floor.
9.2 PERFECT BRAGG GRATINGS It is possible to make very high-quality uniform-period Bragg gratings. This is because optical fiber has very uniform properties. The theoretically calculated reflection, along with the measured spectrum, of a 30-mm-long grating is shown in Fig. 9.10. The grating was fabricated by scanning a phase mask with a UV beam [3]. The agreement between measurement and theory is very good, with the zeroes matching across almost the entire spectrum shown. Notice the slight deviations, especially at the first side-lobe zero (RHS), and the third side-lobe zero (LHS). These features are indicative of slight chirp and nonuniformity in the writing process. Nevertheless, this grating has 28,000 grating periods and shows a near-ideal response. One way to measure such a narrow bandwidth is to use a high-quality tunable laser source and a spectrum analyzer for reasons of resolution. It is difficult to measure such gratings accurately in transmission 1,556.65 0
1,556.75
1,556.85
1,556.95
Reflection, dB
–5 –10 –15 –20 –25 –30
Measured Reflection
–35
Calc. Reflection L = 29.5 mm Deltan = 3.8e-4
–40
Figure 9.10 Measured and computed reflection spectrum of a 29.5-mm-long fiber Bragg grating, produced by the scanned phase-mask technique [4]. The uniformity of the grating is indicated by the close agreement between the zeroes of the theoretical and measured response.
Phase and Temporal Response of Bragg Gratings
413
with a broadband source, since the bandwidth is almost the same as that of commercially available optical spectrum analyzers (0.07 nm FWHZ). Although this grating has a transmission dip of 14 dB, the spectrum remains unresolved in transmission with a spectrum analyzer. Gratings with such performance are particularly useful where the phase response is required along with the reflection characteristics in filtering applications, such as in pulse shaping and dark soliton generation [4].
9.3 PHASE AND TEMPORAL RESPONSE OF BRAGG GRATINGS
1.0
65
0.8
55
0.6
45
0.4
35
0.2
25
0.0 1541.5
1542.0
1542.5
1543.0
Total phase (radians)
Reflectivity
Figure 9.11 shows the computed reflection and accumulated phase-spectrum of a uniform-period unapodized Bragg grating. The measurement of phase of a grating can only be made by the measurement of the grating’s complex amplitude reflectivity. A technique has been proposed for the reconstruction of the phase of the grating using arguments based on causality and minimum phase performed on the measured reflection spectrum of a grating, with reasonable success [5]. This may be done by using interferometric techniques to characterize weak gratings (<20% reflectivity) [6–9]. These measurements have at best limited spatial resolution, or are difficult to implement, being interferometric. Other more direct methods include the use of a network analyzer for the measurement of dispersion [10,11]. The use of the network analyzer relies on the dispersion being constant over the frequency region of interest, and strictly it is better suited to measuring apodized chirped gratings. This technique has been applied to gratings to measure their dispersion [12,13]. Another method for testing of a grating or phase mask uses a probe transverse to the grating [14].
15 1543.5
Wavelength, nm Figure 9.11 The reflectivity and phase of a 4-mm-long unapodized grating with a refractive index modulation amplitude of 4 104.
414
Measurement and Characterization of Gratings 1541.5 0
1542.0
1542.5
1543.0
1543.5
U
Delay, ps
–10 –20
A
–30 –40 –50 Wavelength, nm
Figure 9.12 Dispersion of 4-mm-long unapodized (U) and apodized (A) gratings.
Typically, the accumulated phase for a 4-mm-long grating is a few tens of radians, as shown in Fig. 9.11. The group delays of the two gratings used for Fig. 9.9 are shown in Fig. 9.12. The strong dispersion at the edges of the band stop limits the useful bandwidth of a band-pass filter, especially in high-speed applications [15,16]. In Ref. [16] asymptotic expressions may be found for the dispersion on the long-wavelength side of the band stop of apodized gratings, useful for dense DWM systems. Note that the dispersion at the band edge of the grating reaches several tens of picoseconds for the unapodized grating and is much reduced in the apodized grating. Despite the apodization, there is a curvature in the center of the band stop, which may be useful to compensate for the chirp of a source [15]. The group delay shown in Fig. 9.12 is for gratings with uniform characteristics. Often there is a chirp involved, which has a significant effect on the dispersion. We note that the group delay characteristics of the uniform gratings remain symmetric about the center of the bandgap. With chirp in a grating, the group delay changes sign on one side of the band stop, and this is shown in the calculated response in Fig. 9.13. The unapodized grating acquires less pronounced zeroes and the group delay, a point of inflection. To characterize the group delay of gratings, a measurement setup based on the vector-voltmeter [17] is shown in Fig. 9.14. In this method, light from a tunable single-frequency source is modulated at a frequency f and is launched into a grating under test. The reflected (or transmitted) signal is compared via a circulator with the modulated input signal in a vector voltmeter. As the wavelength of the source is tuned, the delay in the reflected light from the grating changes. The vector-voltmeter compares the phase of the modulated light and translates it into a phase difference. Thus, the phase at the modulated frequency f is measured. With a judicious choice of the modulation frequency, e.g.,
Phase and Temporal Response of Bragg Gratings 1560.60
1560.65
1560.70
1560.75 200
–5
0
–10
–200
–15 –400
–20
Lg = 29.5 mm dn = 7.6e-5(pk-pk) chirp = 0.03 nm
–25
–600
Group delay, ps
Reflectivity, dBe
1560.55 0
415
–800
–30 Wavelength
Figure 9.13 Simulated response of a 29.5-mm-long grating with a 0.03-nm chirp and refractive index modulation of 7.6 105 (peak-to-peak).
Isolator
Circulator
LiNbO3 MZ
Tunable laser
modulator PC
Coupler 1 90%
2 Coupler 2
3
50 : 50
90 : 10 Oil
Fiber grating
1
277.7778 MHz 10%
Oil Reference
Wavemeter A B
Vector voltmeter
PD Reflection PD Transmission
Computer PD Figure 9.14 A schematic of the apparatus for the measurement of group delay, reflectivity, and transmission of gratings (after Ref. [17]).
277.77778 MHz, one degree of phase change is equivalent to a delay t of 10 psec, as [18] t¼
1 sec deg1 : 360 f
ð9:3:1Þ
Under computer control, these data may be acquired quickly with a minimum of processing. The resolution of the measurement may be increased by increasing the modulation frequency, but it is nominally 1 psec or better. With such a resolution, care needs to be taken, since temperature variations during the
Measurement and Characterization of Gratings
Reflectivity, dBe
1560.55 0
1560.6
1560.65
–5
–200
–15 –25
1560.75 200 0
–10 –20
1560.7
–400 L = 30 mm unapodized
–30
Delay, ps
416
–600 –800
Wavelength, nm Figure 9.15 The spectra of a 30-mm-long grating measured using the vector voltmeter.
course of the measurements can cause errors. Changes in the path lengths of the fiber used in the setup and also in the polarization affect the measurement. It is for these reasons that the paths are kept to a minimum and a reference frequency is generated optically after the modulator. Any amplitude and phase changes are then common to both the reference and the signal, minimizing errors. The reflection and group delay of a 30-mm-long grating of the type shown in the simulation of Fig. 9.13 are shown in Fig. 9.15. We note that the similarity is striking between the two grating spectra, in both reflectivity and group delay. It is therefore possible to characterize the measured spectra using simple simulation, since fiber-grating spectra can be so good. Note that the reflectivity spectra is in dB (electrical) ¼ 2 dB (optical). The grating reflectivity can be measured in transmission, to normalize the peak of the reflectivity spectra. The results for an identical but apodized grating are equally good. We now move on to the spectrum of a longer grating, of the type used in dispersion compensation. There are a few points that need to be remembered. In Chapter 7 the delay response of long gratings is considered in detail, and it is found that long gratings need to be characterized with high wavelength resolution if the GDR (group delay ripple) spectrum is to be resolved. In making measurements, a wavelength resolution of 1 pm is generally sufficient. The measured reflectivity and group delay of a raised cosine apodized, 100-mm-long grating is shown in Fig. 9.16. The data have been measured with 1 pm resolution, and the GDR is adequately resolved. We can measure the 3-dB optical (6-dBe) reflectivity bandwidth by comparing the transmitted signal level with Rp, remembering that the measurements are in dBe. This grating is designed to compensate the dispersion for 80 km of standard optical fiber, as has been presented in Chapter 7. With unapodized gratings the low resolution masks the detail and a comparison is shown in Fig. 9.17. Here two measurements of the same grating made with 10 pm and 1 pm are compared. The GDR is apparent with the higher resolution. With longer gratings this factor becomes even more critical.
Phase and Temporal Response of Bragg Gratings 1556.2
1556.4
900
–5 Reflectivity, dB
1556.6
700
0.35 nm
–10
500
–15
300
–20
100
–25
Delay, ps
1556.0 0
417
–100
Reflectivity, dB Delay, ps
–30
–300 –500
–35 Wavelength, nm
Figure 9.16 The reflectivity and delay of a 100-mm-long chirped grating with a chirped bandwidth of 0.75 nm [19], measured with a resolution of 1 pm.
800 1 pm-step 10 pm-step
700
Delay, ps
600 500 400 300 200 1553.8
1553.9
1553.9
1554.0
1554.0
1554.1
1554
Wavelength, nm Figure 9.17 Resolution dependence of the group delay of an unapodized chirped grating (after Ref. [17]).
As we have seen in Chapter 7, for 1-meter-long unapodized gratings, the GDR frequency has a period of 1.5 pm. To resolve the GDR, other means have to be adopted, such as the use of a multisection DFB laser [17], which can be electrically tuned. This is a very time-consuming task, since other factors such as the ambient temperature need to be controlled very accurately. The drift of the wavelength with temperature of a chirped grating is an issue. The grating
418
Measurement and Characterization of Gratings 0
Wavelength drift, nm
–0.1
10
20
30
40
50
Drift (nm) ~ –1.61 + 1.93 exp (–N/4.87)
–0.3 –0.5 –0.7 –0.9 –1.1 –1.3 –1.5 –1.7 Time (N ), days
Figure 9.18 The drift in the Bragg wavelength of a chirped grating with time due to deuterium out-diffusion [17].
should be temperature controlled in order to get an accurate measurement. With a change in the local Bragg wavelength of several pm/ C, a stable environment is essential for the measurement. Finally, we consider the drift of the Bragg wavelength with the out-diffusion of hydrogen for a fiber after the grating has been written. With a typical period of measurement of tens of minutes to an hour, the out-diffusion of hydrogen (or deuterium) must be taken into consideration, as has been shown in Chapter 2 [20,21]. In Fig. 9.18 we see the drift in the Bragg wavelength of a chirped grating monitored over a period of 45 days from the inscription of the grating, immediately after removal from the cold storage. The fiber is deuterium soaked at 200 bar at the start. By the end of 45 days, the total drift in the wavelength is approximately –1.65 nm, and it continues to shift very slowly. With technological improvements, it will be necessary to measure even longer gratings, perhaps longer than 10 meters. The measurement of one of these gratings (reflection spectrum of a 2-meter-long WDM channelized grating is shown in Fig. 9.19) can take several hours at picometer resolution. Here it becomes important that the grating be collectively maintained at the same temperature for the duration of the measurement. A fast technique has been reported by Ouellette et al. [22], which relies on the intrinsic birefringence B of the fiber. By alternatively measuring the orthogonal polarization reflected, S2, and launched, S1, signals from the spectrum analyzer, the group delay is shown to be nl S2 cos1 S1 þ S2 : ð9:3:2Þ t¼ 4pBc S1
Phase and Temporal Response of Bragg Gratings 1549.9 0
1553.9
1557.9
419 1561.9
1565.9
Reflectivity, dB
–5 –10 –15 –20 –25 –30 Wavelength, nm Figure 9.19 The reflection spectrum of a 2-meter-long grating fabricated at BT Laboratories. Both radiation and loss due to deuteration reduce the reflected signal from the shorter wavelength gratings. The combined loss of deuteration and radiation is 2 dB/m [26].
This method requires the calibration of the fiber birefringence. This technique may prove to be valuable, since it is simple, although there is no information available on the resolution. We remind ourselves that for the dispersive properties of gratings with a reflectivity of between 10 and 15 dB, the time delay t may be calculated as t¼
2neff Lg ; c
ð9:3:3Þ
where Lg is the length of the grating and c is the speed of light. For a signal pulse with a transform-limited bandwidth, Dl equal to the bandwidth of the grating chirp, the dispersion can be simply described as D¼
2neff Lg 2L ¼ ; cDl vg DL
ð9:3:4Þ
where L is the physical length of the grating, and vg is the group velocity of the pulse. The effects of dispersion can be considerable even in a short grating since the group delay in a fiber is 5 nsec m1, and in reflection it is doubled. We have seen in Chapter 7 how PMD can affect the GD of gratings. The measurement of PMD, or rather birefringence, can be made simply by launching light along the two orthogonal birefringent axes of the fiber grating and observing the wavelength shift. Meltz and Morey [23] reported a birefringence-induced Bragg wavelength shift of 0.1 nm, equivalent to an induced birefringence at 1550 nm of 2.3 105. This value is high and the Bragg wavelengths are therefore easily separated in a spectrum analyzer. By rotating the polarization of the UV writing beam so that it is along the fiber axis, the UV-induced birefringence can be reduced from 5% of the UV-induced refractive index change to around
420
Measurement and Characterization of Gratings
0.5% [24]. This area has received more attention recently, with the application of setup similar to that shown in Fig. 9.14 to measure the group delay difference of two polarization states. Also applied is the Jones matrix approach to map out all the polarization states in the fiber. The comparison between the two methods is good. Measured polarization-induced delay is reported to be 28 and 7 psec for two chirped gratings written in different fiber [25]. Clearly this is an area that will receive attention as the deployment of fiber gratings for dispersion compensation becomes widespread.
9.3.1 Measurement of the Grating Profile The use of simulation is an excellent method for characterizing the measured spectrum of a grating. By knowledge of the physical length, the reflectivity, and the shape of a grating, it is possible to identify a couple of parameters to choose in order to allow the simulation. This has been demonstrated for the spectra shown in Figs. 9.13 and 9.15. More complex structure can be analyzed theoretically, as has been amply demonstrated by Ouellette et al. [27] for stitching errors produced by phase masks. For example, random errors in the stitching cause the bandwidth of a uniform-period grating to broaden and acquire noise. The errors can be calculated from the simulation. Alternative techniques exist for assessing the refractive index profile of a grating. These include optical low-coherence reflectometry (OLCR), originally used to detect small flaws in optical waveguides [28,29], applied to fiber gratings by Lamblet et al. [30], and the method of side scatter from a grating, demonstrated by Krug et al. [31]. An interesting and very simple technique is the application of “heat scan” to probe the chirp in a grating [32,33]. The grating is probed with a fine hot wire while the reflection characteristics are being monitored. A detuning of the Bragg wavelength as a function of probe position measures the chirp. These are in addition to other methods already mentioned as being appropriate for assessing phase masks [9,14]. It is important to know the sources of “noise” on a grating, or correct for a flaw [34–36], since the out-of-band spectra may deteriorate. It should be mentioned that optical time-domain reflectometry (OTDR, or back-scatter) has also been used to locate gratings. In the next section we will consider the OLCR method and side-scatter as two techniques to assess Bragg gratings.
9.3.1.1 Optical Low-Coherence Reflectometry The scheme is based on a fast scanning Michelson interferometer, with the grating as the mirror in one arm, and a scanning broadband mirror (BBM) in the reference arm. Reflected light from the grating and the mirror interferes at
Phase and Temporal Response of Bragg Gratings
421 GUT
ELED
Coupler
PD
Phase modulator
Reference mirror
Figure 9.20 A schematic of the OLCR apparatus (after Ref. [37]).
a photodiode. The source has a large bandwidth so that the coherence length is short, and therefore interference is only visible over a short region of the grating when the path lengths are within the coherence length. The path difference between the arms is adjusted by moving the reference mirror so that different points within the grating are sampled. A schematic of the apparatus is shown in Fig. 9.20 [37]. The moving reference mirror is mounted on a motorized stage with a long scan length to allow easy adjustment of the paths. The phase modulator is provided to derive a lock-in signal. With reference to a 100% reflection, a cleaved end with approximately 4% end reflection registers a signal at –14 dB. The measurement has range with a noise floor at –140 dB with a bandwidth of 1 Hz. The source should have low spectral ripple to avoid artifacts. The coherence length, and therefore the resolution of the measurement in the fiber is lc ¼
4 ln 2 l2 ; p DlFWHM
ð9:3:5Þ
where DlFWHM is the source linewidth and l is the center wavelength, and the resolution R ¼ lc/2n where ng is the group index. For DlFWHM ¼ 55 nm at l ¼ 1300 nm, we get lc ¼ 18.3 mm and hence R ¼ 9.1 mm. The grating spectra are recorded by scanning the reference mirror. In a modified version of the setup, a rotating corner cube is used for increasing the speed of data acquisition. A typical OLCR spectrum is shown in Fig. 9.21. Here the front end reflection is followed by a decay in the signal with penetration depth; this is followed by another increase in the signal as the light exits the far end of the grating at a distance z ¼ 1.222 mm, equivalent to a physical length of z/neff ¼ 0.84 mm. The exponential decrease in the signal is proportional to the strength of the grating, while the initial fast rise at the entrance and soon after the first exponential decay is due to the abrupt starting and ending of the grating – a top-hat function. The further oscillations observed are due to the Fabry–Perot modes as the light rattles around within the grating. The measured spectra are the Fourier transform of the product of the amplitude reflectivity of the grating and the spectral distribution of the source. The free parameter is the refractive index modulation of the sinusoidal period, since the Bragg wavelength is
422
Measurement and Characterization of Gratings –20
OLCR signal [dB]
–40 –60 –80 –100 –120 –140 –1
0
1
2
3
4
5
6
7
Optical depth, z [mm] Figure 9.21 The OLCR spectrum of a 0.84-mm-long grating (courtesy Hans Limberger from: Lambelet P., Fonjallaz P.Y., Limberger H.G., Salathe´ R.P., Zimmer C., and Gilgen H.H., “Bragg Grating Characterization by Optical Low-Coherence Reflectometry,” IEEE Phot. Technol. Lett., 5, 565–567, 1993. # 1993 IEEE. [30]).
known, and the length of the grating is found from the length of the scan between the start of the spectrum and the reflection at the end of the grating. The zeroes of the OLCR spectrum are a very sensitive function of the refractive index modulation amplitude, and therefore provide an accurate value. The inverse Fourier transform of the OLCR data and deconvolution of the source spectrum give the grating spectrum. This is shown with the measured grating reflection spectrum in Fig. 9.22. The agreement is altogether excellent. This technique has been applied by Malo et al. [38] to measure the profile of an apodized grating. It is claimed that the relative precision with which the refractive index modulation may be measured is around 1% [30].
9.3.1.2 Optical Frequency Domain Reflectometry Optical low-coherence tomography or reflectometry (OLCT or OLCR) as described in this section is a powerful technique to characterize the reflection spectrum and spatially resolve the coupling constants of a grating. However, there are other schemes that allow characterization, also based on time or frequency domain spectroscopy. Techniques that allow the measurement of grating or other device parameters are optical time domain reflectometry (OTDR) and optical frequency domain reflectometry (OFDR), which is a coherent homodyne technique. These methods have their specific merits in terms of resolution,
Phase and Temporal Response of Bragg Gratings
423
100
80
60
40
20
measured calculated
0 1.283
1.284
1.285
1.286
1.287
1.288
1.289
Wavelength [mm] Figure 9.22 Measured and calculated reflection spectra (from the data obtained by OLCR measurement; see Fig. 9.21) as a function of wavelength for an in-fiber grating with a period of 0.443 mm, a length of 0.84 mm, and a modulation depth of Dn ¼ 1.16 103 (courtesy Hans Limberger from: Lambelet P., Fonjallaz P.Y., Limberger H.G., Salathe´ R.P., Zimmer C., and Gilgen H.H., “Bragg Grating Characterization by Optical Low-Coherence Reflectometry,” IEEE Phot. Technol. Lett., 5, 565–567, 1993. # 1993 IEEE. [30]).
speed of measurement, sensitivity and dynamic range, and accuracy. Both high- and low-coherence techniques of optical coherence tomography (OCT) have been applied as extremely powerful tools for probing scattering in turbid media (e.g., tissue) and have found applications in the medical diagnosis of ocular degeneration [39,40], dental investigations [41–43], cancer research [44], and in analyzing artwork [45,46], among others. OLCR can be used over a limited range of a few meters [47], whereas OTDR has been used to characterize the loss of optical fibers over 10 s of kilometers. The resolution in the latter system depends inversely on the pulse width, as the Rayleigh back-scattered signal is integrated over the time of the pulse. Thus, the resolution is of the order of a meter or two. The reach of the system also depends on the peak power launched into the fiber. Normally, the back-scattered signal is –30 dB below the input power level without including the fiber loss. OFDR has the advantage over the other techniques, as it is based on a high-power, swept-single-frequency laser. The larger signal and the fact that it is a coherent homodyne technique [48– 50] allow high spatial resolution of a millimeter over a distance of 1 km [51,52]. The fast sweep rate of the laser over a large wavelength range is useful for fast measurements with high sensitivity and resolution over hundreds of meters, and it is particularly suited to measuring the characteristics of gratings, such as reflectivity, phase response, dispersion, and for grating diagnostics
424
Measurement and Characterization of Gratings UT DUT PC ADC
3dB DUT Interferometer 3dB
3dB PC
v(t)
UDUT Uaux
3dB Tunable laser
Aux. Interferometer 3dB
~
Computer
Clock Figure 9.23 The OFDR measurement system for characterizing optical devices.
[43,53]. OFDR is also suited to Rayleigh back-scatter-based sensing using coherent homodyne detection for high spatial resolution [54]. A technique based on OFDR is described here and some of its attributes discussed for applications in the characterization and diagnosis of gratings, based on scanning the wavelength of a tunable laser. Referring to Fig. 9.23, a tunable single-frequency laser is coupled to two Mach-Zehnder interferometers (MZI) via a coupler. The split ratio of this coupler may be altered to suit the insertion loss of the device under test (DUT). The upper branch is the DUT interferometer, and light is routed to the DUT via a circulator (or coupler). Light transmitted through the coupler is detected by a photodiode (UT) connected to an analog-to-digital converter (ADC) and logged by a computer. The signal reflected by the DUT is routed back to the DUT-MZI to interfere with the signal in the reference arm and the fringes detected by another photodiode (UDUT) and similarly logged by the computer. The auxiliary interferometer is asymmetric and generates fringes at the output when the laser is scanned and is used as an absolute reference of wavelength change because the laser wavelength is not known accurately. This is also logged to normalize the data collected. A switch selects each input in sequence for sampling. The following analysis closely follows the description in reference 49. If the laser frequency is swept so that the instantaneous frequency is nðtÞ ¼ n0 þ g t;
ð9:3:6Þ
and the electric field of the light emitted by the laser is described as E0 ðtÞ ¼ jE0 jei2p½ðn0 gt=2Þt ;
ð9:3:7Þ
where g is the tuning rate, and assuming identical polarizations, the electric field amplitude of the signal (DUT and reference) at the output of the interferometer is the sum of the two component signals from the reference arm and the DUT,
Phase and Temporal Response of Bragg Gratings
Et ¼ E0 ðtÞ þ rE0 ðt tÞ;
425
ð9:3:8Þ
where t is the group delay chosen to be always greater than the reference arm, E0(t) is the electric field in the reference arm, and the second term is the delayed reflection from the DUT with a reflectivity of r. The current, it at the photodetector is proportional to the square of the quantity in Eq. (9.3.8), it ¼ kR E02 þ r2 E02 þ 2rE02 ðt tÞ ;
ð9:3:9Þ
where, k is a proportionality constant dependent on the polarization, loss, and other factors and the normalization of the electric field amplitudes, R, is the responsivity of the photodiode. Equation (9.3.9) describes a synchronous, theoretically quantum-limited coherent homodyne detection process. In practice, several factors influence the sensitivity, not least the spurious reflections and scattering within the DUT and elsewhere. The first two terms on the RHS of Eq. (9.3.9) are the dc terms from the DUT and the reference arm, and the third is the coherent mixing term. The dc terms add shot noise at the photodiode and therefore limit the sensitivity. However, the amplification afforded by the synchronous term gives excellent signal-tonoise ratio and is better than direct detection systems. Ignoring the dc terms, one can detect the synchronous frequency by using Eq. (9.3.6) in Eq. (9.3.9) and one arrives at the following expression: ð9:3:10Þ it ¼ i0 1 þ r2 þ 2rcosð2pg t t þ fd Þ where i0 ¼ kRE02 ;
ð9:3:11Þ
and fd is a constant phase term, the result of the phase difference between the reference arm and the point of insertion of the DUT. Here, f ¼ gt is the optical frequency shift in the time interval t´, from the group delay experienced by the light returning from the DUT and indicates its physical location, z. Describing the ac part of Eq. (9.3.10) in terms of instantaneous frequency, n(t) instead of time, iðuÞ ¼ 2i0 rcosð2pnt þ ’D Þ; ð9:3:12Þ where cc ¼ fd 2pu0t. Fourier transforming Eq. (9.3.12) provides the spectrum of the light reflected from the DUT in terms of time delay t, n0 ð þ Dn 2
F ðtÞ ¼ u0 Dn 2
iPD ðnÞei2pntdv:
ð9:3:13Þ
426
Measurement and Characterization of Gratings
The integration is over time, T, of the frequency scan, Dn ¼ gT of the tunable laser, and the spectrum is a function of the time delay. The spectrum from Eq. (9.3.12) therefore turns out to be F ðtÞ ¼ F þ ðtÞ þ F þ ðtÞ; and F þ ðtÞ ¼ i0 Dn j r j
ð9:3:14Þ
sin½pDnðt tz Þ ið2pn0 tyd Þ e ; pDnðt tz Þ
ð9:3:15Þ
where the * is the complex conjugate. The spectral content is the absolute value of jFþ(t)j, which is maximum at t ¼ tz, which is the point in the DUT being measured. The width of the sinc function in Eq. (9.3.15) determines the temporal resolution, Dt of the measurement, and therefore also the spatial resolution, Dz, vg vg ; ð9:3:16Þ Dz ¼ Dt ffi 2 2Dn where vg is the group velocity of in the DUT, and Dv ¼ gT. To avoid the peaks of the side lobes of the sinc function, which are 5% of the center introducing ghosts in the measurements, it is necessary to use an apodized window such as a Hamming function on Eq. (9.3.12) before the Fourier transform. This helps to suppress the side lobes significantly, but it reduces the spatial resolution depending on the window function used. Including the window function in the Fourier transform, the power spectrum is approximately jF ðtÞj2 A jrj2 h2 ðt td Þ;
jr 0 :
ð9:3:17Þ
When the reflection is continuous as in a FBG, the power spectrum represented by Eq. (9.3.17) becomes the spatial reflectivity of the DUT. Using such a simple system with a tunable laser with a bandwidth of 40 nm, a resolution of a few tens of microns is possible [50,55].
9.3.1.3 Side-Scatter Measurements Bragg gratings scatter radiation out of the fiber both within and outside of the bandgap. This is due to a number of reasons, not least radiation mode coupling of light into the cladding, which can be detected easily [56]. The writing process causes an asymmetry, which assists a directional coupling to the cladding modes (see Chapter 4). Apart from the coherent scattering mechanism, there is also incoherent scatter due to damage in the core as in a Type II grating [57]. This type of grating scatters light because of large surface irregularities at the core–cladding boundary. A third type of scattering mechanism is incoherent and is due to the inhomogeneity in refractive index modulation through the length of the grating. The latter has been investigated by Janos et al. [58].
Phase and Temporal Response of Bragg Gratings
427
There appears to be a pronounced scatter out of the fiber perpendicular to the direction of the writing beam. The observed anisotropy is consistent with the production of “scattering elements” within a few microns of the core. The scattering loss ranges from 0.2 dB/cm in highly doped Yb/Er phospho-silicate fiber with gratings inscribed with a pulsed laser 193-nm source, to 5 105 dB/cm for a boron-codoped fiber with a CW 244-nm source. A technique that measures the sideways diffraction from a fiber Bragg grating is used to characterize the grating refractive index profile. A schematic of this method is shown in Fig. 9.24. In this arrangement, light from a He–Ne laser is focused from the side of the fiber, incident at an angle yi to the normal. The schematic of the side-scatter measurement is shown in Fig. 9.24. As has been seen in Chapter 3 on fabrication of gratings with a phase mask, the incident light must have a wavelength less than the period of the grating in the fiber in order to have a first-order diffraction. Referring to Fig. 9.24, which defines the angles, and from phase matching conditions, we find [31] sin yr ¼ sin yi ¼ nclad sin yclad ¼ ncore sin ycore ¼ neff
lprobe ; lBragg
ð9:3:18Þ
where neff is the effective index of the fiber at the probe wavelength, so that the probe wavelength must be greater than the Bragg wavelength of the grating by a factor of the effective index of the fiber. The input light is reflected at the incident angle, yi. In the weak scattering limit, the cross-section si (as a fraction of the incident peak-power density) is given by the following expression,
Θi Θc
Θo
g z Θr
Detector
1st order Bragg reflected
Figure 9.24 The side-diffraction scheme for characterizing Bragg gratings (from: Krug P., Stolte R., and Ulrich R., “Measurement of index modulation along an optical fiber Bragg grating,” Opt. Lett., 20(17), 1767–1769, 1 September 1995. [31]).
428
Measurement and Characterization of Gratings
assuming that the grating has a pure sinusoidal period [31], and the focused spot size wi in the core is much greater than the core radius a: sin2 gcore : ð9:3:19Þ cos2 ycore Here Dn is the local refractive index modulation amplitude, k is the wave vector at the incident wavelength lprobe, and gcore is the angle between the reflected and incident beams and ignores reflection losses. Owing to the geometry of the scattering region, gcore is polarization sensitive and s-polarized light; the reflected power is maximum at gcore ¼ p/2. Typical parameters used for the experiment are a beam waist of 5 mm, using a 10-mm focal length focused 37 mm in front of the surface of the fiber to give a spot size of approximately 10 mm at the core (after focusing from the core– cladding surface) with an incident angle of 45.3 . The equivalent internal angle ycore ¼ 29.4 . The input power needs to be high for a good signal-to-noise ratio, and was reported in the experiments to be 5 mW. The resolution in this arrangement is limited to the spot size of 10 mm. The fiber grating is scanned in front of the fixed laser beam, so that the data may be recorded as a function of position along the grating. Good correlation between the measurement and the simulated transmission spectrum of the grating has been reported [31,35]. The side-diffraction profile of a Gaussian apodized grating is shown in Fig. 9.25. From this profile, it is simple to simulate the grating transfer function to establish a correlation. Normalized Bragg reflected power, P(z)/Pr(0)
si 1:66k2 a3 wi Dn2
1.0
0.5
0 0
1
2 3 Position, z [mm]
4
5
Figure 9.25 The profile of the light diffracted from a Bragg grating using the side-diffraction scheme (from: Krug P., Stolte R., and Ulrich R., “Measurement of index modulation along an optical fiber Bragg grating,” Opt. Lett., 20(17), 1767–1769, 1 September 1995. [31]).
Phase and Temporal Response of Bragg Gratings
uniform
Gaussian intensity
apodized
429
chirped apodized
Figure 9.26 Four commonly encountered grating refractive index modulation profiles.
The assumption of Eq. (9.3.19) restricts the application of this technique to gratings with slowly varying chirp by a variation of the grating period and to average refractive index (dl/L < 13), with no saturation in Dn. The chirp is limited by the numerical aperture of the focused beam. Figure 9.26 shows the refractive index modulation profiles of four gratings commonly encountered in Bragg grating technology.
9.3.2 Measurement of Internal Stress The refractive index changes induced by UV irradiation appear to affect the internal stress in the core [59]. There are conflicting observations, both of which are supported by experimental evidence [60]. In this section we consider the measurement of internal stress by optical means [61,62]. The technique is simple and requires the measurement of the state of the polarization of a focused spot of light (typically 3 mm diameter) transversely incident on the fiber, as it is scanned through the core region, implemented by Fonjallaz et al. [63]. It therefore is a polarimetric measurement, which requires the polarization of the incident beam to be at 45 to the orthogonal birefringent axes of the fiber. The fiber is immersed in index-matching fluid to minimize beam deviations. The transmitted light is analyzed as a function of the translation distance of the incident beam. The retardation d at the output provides the information on the stress distribution sz(z) from an Abel integral equation [61], 0 ð 1 R d ðyÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy; ð9:3:20Þ sz ðrÞ ¼ pC r y2 r2 where the prime indicates differentiation with respect to the transverse coordinate y, R is the radius of the fiber, and C is the stress-optic coefficient of silica. A change in the axial stress changes the refractive index by the stress-optic coefficient. The three components of the refractive index change, nr, ny, and nz,
430
Measurement and Characterization of Gratings
are related to the axial, circumferential, sy(r), and radial, sr(r) components of the stress-optic coefficients as [64] nr ¼ n0 B2 sr B1 ðsy þ sz Þ ny ¼ n0 B2 sy B1 ðsr þ sz Þ nz ¼ n0 B2 sz B1 ðsy þ sz Þ:
ð9:3:21Þ
In Eq. (9.3.9) the refractive index components are for light waves that have their electric field components in each of the three directions. The refractive index of the fiber is n0 without stress, and the stress-optic coefficients B1 and B2 are both positive, with B1 ¼ 4.12 105 mm2 kg1 and B2 ¼ 0.64 105 mm2 kg1 [65]. The radial component of the stress may be calculated from the axial component as [66] ð 1 R ð9:3:22Þ sr ðrÞ ¼ 2 sz ðsÞsds: r r
Axial stress, sz (kg mm–2)
The sign of the axial stress is found by measurements on silica fibers with and without strain, and through symmetry properties, sr(r) ¼ sy(r) ¼ sz(r)/2. The axial stress sy(r) indicates positive axial tension for a positive sign and compressive strain with a negative sign. By measuring the stress profile, the changes in the refractive index can be calculated as a function of the UV irradiation. Experiments performed by Fonjallaz et al. [63,67] have found that the axial stress increased with UV inscription of gratings, contrary to the stress-relief model [59]. The fibers are found to be either under slight axial compression or under tensile stress before UV irradiation (between –5 and þ1.6 kg mm2). The stress changes before and after UV irradiation of a fiber with a Ge concentration of 12% are shown in Fig. 9.27.
15
After UV 32 kJ cm–2
12mol%GeO2
10 5 0 –5 –70
–35
0
35
70
Radial position, r [mm] Figure 9.27 The measured radial stress profile of a fiber before and after the inscription of the grating (courtesy Hans Limberger [63]). (from: Fonjallaz P.Y., Limberger H.G., Salathe´, Cochet F., and Leuenberger B., “Tension increase corellated to refractive index change in fibers containing UV written Bragg gratings,” Opt. Lett., 20(11), 1346–1348, 1 June 1995.)
Strength, Annealing, and Lifetime of Gratings
431
The maximum stress on the axis of the fiber is found to be 14.2 kg mm2. An increase in the tension is equivalent to a reduction in the refractive index; the implication is that the stress changes counter the overall change in the UVinduced refractive index, since this is positive (the Bragg wavelength moves to longer wavelengths with increase in refractive index of the core). The net result of tension increase is that approximately 30% of the UV-induced refractive index change is negative and other factors, presumably compaction and changes in the UV absorption spectrum, increase the overall refractive index [68].
9.4 STRENGTH, ANNEALING, AND LIFETIME OF GRATINGS Reliability of fiber Bragg gratings is essential for long-term usage in telecommunications. There are two aspects that need to be taken into account: the mechanical strength and longevity of the grating. Mechanical strength of optical fiber is degraded through intense UV exposure, while the strength of the refractive index modulation of the grating begins to decay from the time of fabrication. The decay is slow but of concern unless treated. Careful handling and packaging of the UV-exposed grating may preserve the mechanical strength component. The issue of mechanical strength is common with the deployment of optical fibers; they need to be tested to assure 25-year survivability. The process of grating inscription generally requires the removal of the protective primary coating of the fiber prior to exposure. While a special polymer may be used for through coating inscription [69], this is not generally the case. The mechanical removal damages the surface unless chemical means are used [70]. UV exposure can further reduce the strength [71]. Some form of protective coating must be applied to conserve the strength, if the fiber is to be handled mechanically [72].
9.4.1 Mechanical Strength The issue of mechanical reliability of in-fiber Bragg gratings has been extensively studied [73–75]. The degradation in the strength of a fiber is due to the growth of cracks on the surface at tiny flaws. Stress concentration at these flaws propagates and causes the fiber to fail. The failure strength can be dramatically lowered over that of pristine fiber. A convenient method of comparing the strength of different fibers is by the measurement of dynamic fatigue. From the dynamic fatigue tests it is possible to predict the lifetime to failure by determining the distribution of flaws. The strength of optical fiber exposed to pulsed
432
Measurement and Characterization of Gratings
KrF radiation is dramatically reduced from a mean breaking strength of 4.8 GPa for pristine fiber to 1.2 GPa. Recoating the fiber immediately after exposure restores the breaking strength. With CW 244-nm radiation, the strength is almost unchanged [76], but the long-term survivability is compromised, while hydrogen loading has little influence on the breaking strength. For a failure probability of 1 103, the operational stress must be less than 1 GPa for a 20-year lifetime [76], using a test length of 100 mm and a grating of 8 mm.
9.4.2 Bragg Grating Lifetime and Thermal Annealing The thermal stability of Bragg gratings is of prime importance if fabricated components are to function properly over their required life. For example, a reduction in the reflectivity of a fiber Bragg grating used in an add–drop multiplexer from 40 to 30 dB could cause a degradation in cross-talk between channels. It is therefore essential to be able not only to predict the decay of the grating strength, but also to find ways to stabilize it. Experimental observations of the decay in the reflectivity of gratings have resulted in a model that predicts the lifetime of gratings. It is generally agreed that observations on all fibers other than hydrogen-loaded follow a power-law dependence, originally proposed by Erdogan et al. [77,78]. The essential differences are the exact values of the coefficients that are used in the model. The model proposes that the coupling constant of a grating kacL decays according to the following power law: ¼
kac L 1 ¼ : kac Ljt¼0 1 þ Ata
ð9:4:1Þ
Here the denominator on the LHS is the initial value of the coupling constant at the time of writing of the grating. The constants A and a are temperature dependent and are found by plotting the normalized coupling constant as a function of time. For convenience, the time parameter t may be normalized by unit time, e.g., 1 min to make Eq. (9.4.1) dimensionless. By measuring the decay of several gratings at different temperatures as a function of time, and fitting the data with Eq. (9.4.1), the values of A and can be evaluated for a particular fiber. According to the model, the exponent a ¼ T/T0 and A ¼ A0eaT, so that T0 is the fitted parameter when a is evaluated. Based on a theoretical approach, the model assumes that there is a distribution of trapped states after UV exposure (distribution of induced defects, DID). The thermal depopulation of these states into the conduction band has a release rate that is dependent on temperature and the energy E as vðEÞ ¼ v0 eE=kB T ;
ð9:4:2Þ
Strength, Annealing, and Lifetime of Gratings
433
where kB is the Boltzmann constant. States below the demarcation energy Ed may be easily depopulated at a given temperature T. The time taken to decay is related to the demarcation energy as t
1 : vðEd Þ
ð9:4:3Þ
In Fig. 9.28 is shown the decay of the normalized coupling constant of gratings elevated to three different temperatures; Equation (9.4.1) can be expressed in terms of the release rate as ¼
1 : 1 þ ½vðEÞta
ð9:4:4Þ
Using Eqs. (9.4.2)–(9.4.4), a Fermi–Dirac function describes the decay in the normalized coefficient, ¼
1 ; 1 þ eðEd E0 Þ=kB T0
ð9:4:5Þ
where E0 is the peak of the distribution. We note that A ¼ v(E)a and is dependent on the release rate at energy E and on a. The distribution of the defects (DID) is calculated from Eqs. (9.4.2) and (9.4.5) and is gðEÞ ¼
Nð0Þ eðEE0 Þ=kB T0 : kB T ½1 þ eðEE0 Þ=kB T0 2
ð9:4:6Þ
In Fig. 9.28 is plotted the decay of the normalized coupling constant as a function of time for three different values of temperature [79]. The trend in the decay of the strength of gratings is typical. However, the magnitudes vary
dn(120C) dn(225C) dn(300C)
Normalized coupling coeff.
1 0.8 0.6 0.4 0.2 0 0.0E+00
2.0E+05
4.0E+05 6.0E+05 Time, hrs
8.0E+05
1.0E+06
Figure 9.28 The thermal decay of boron–germanium codoped fibers, computed from the fitted data [79].
434
Measurement and Characterization of Gratings 200 1
400
600
0.1
0.01 Temperature, T (K) Figure 9.29 Relationship between A and the annealing temperature [79].
greatly. The computed A values from the fitted data to curves such as in Fig. 9.28 give the result shown in Fig. 9.29 in which a linear relationship may be seen when A is plotted on a logarithmic scale. This also ensures the validity of the data. Alpha is similarly plotted in Fig. 9.30. Again, the linearity of the data should be such that it should pass through zero. It turns out that the dependence of the decay of hydrogen-loaded boron– germanium fibers is much faster than that of unloaded fibers [79,80]. It has been speculated by the use of a more complex model that the DID in hydrogenloaded fibers is not a single Gaussian distribution but a flat top [81,82]. It has also been suggested that the distribution may be expressed as a stretched exponential [83]. However, there is no consensus as to a valid model. Riant et al. have pointed out the existence of at least two DIDs, and these are shown in Fig. 9.31. The contribution of each DID changes with temperature. It seems sensible that there may be several such DIDs, which play a role in determining the exact nature of the thermal decay of gratings.
0.2
Alpha
0.15 0.1 0.05 0 0
100
200
300
400
500
Temperature, T (K) Figure 9.30 The fitted a to data on boron–germanium codoped fiber [79].
600
Strength, Annealing, and Lifetime of Gratings
435
g(E)/N(0), (1/eV)
1 D1 D2
0.8 0.6 0.4 0.2 0 0
1
4
3
2
5
E(eV) Figure 9.31 The distribution of the energy states of two DIDs in hydrogen-loaded fibers [84]. One defect is centered at 1.35 eV and has a narrow distribution, whereas the second, at 2.67 eV, has a much broader distribution.
9.4.3 Accelerated Aging of Gratings Annealing a grating at an elevated temperature for a short time removes the fast decay, so that at lower temperatures, the decay rate slows down. This is the principle of accelerated aging [77,85]. The relationship between the annealing temperature T2 and the lifetime of a grating operated at temperature T1 is given by the simple relationship t2 ¼ eaT0 ½ðT1 =T2 Þ1 tT1 =T2 ;
ð9:4:7Þ
where the annealing time is t2 and the time over which the grating is to be used is t1. The parameter a has been defined in section 9.4.2. The data for boron–germanium codoped fiber [79] with a ¼ 1.31 102 K1 and T0 ¼ 2941 K is shown in Fig. 9.32, for an expected grating lifetime of
Aging time (minutes)
1.0E+08 1.0E+06
Aging time for 25 year lifetime (B:Ge fiber)
1.0E+04 1.0E+02 1.0E+00 1.0E−02 300
350
400
450
500
Accelerated aging temperature (K) Figure 9.32 Accelerated aging characteristics of boron–germanium codoped fiber [79], for a predicted lifetime of 25 years.
436
Measurement and Characterization of Gratings
25 years at 300 K. The relationship between the time of annealing and the anneal temperature shows that annealing at 480 K for 1 min is equivalent to 25 years at 300 K. In order to ensure that the grating meets the specification for the application, this initial “burning-in” phase must be taken into account, since there is a reduction in the refractive index modulation.
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[33] S. Sandgren, B. Sahlgren, A. Asseh, W. Margulis, F. Laurell, R. Stubbe, et al., Characterization of Bragg gratings in fibers with the heat-scan technique, Electron. Lett. 31 (1995) 665–666. [34] W.H. Loh, M.J. Cole, M.N. Zervas, R.I. Laming, Compensation of imperfect mask with moving fibre-scanning beam technique for production of fibre gratings, Electron. Lett. 31 (17) (1995) 1483–1485. [35] F. Ouellette, P.A. Krug, R. Pasman, Characterisation of long phase masks for writing fibre Bragg gratings, in: Photosensitivity and Quadratic Nonlinearity in Glass Waveguides: Fundamentals and Applications, vol. 22, OSA Technical Series, Optical Society of America, Washington, DC, 1995, pp. 116–119. [36] F. Ouellette, The effect of profile noise on the spectral response of fiber gratings, in: Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, vol. 17, OSA Technical Digest Series, paper BMG13, Optical Society of America, Washington, DC, 1997, pp. 222–224. [37] H.G. Limberger, P.Y. Fonjallaz, P. Lambelet, R.P. Salathe´, C. Zimmer, H.H. Gilgen, (1993) Fiber grating characterization by OLCR measurements, European Conference on Optical Fibre Communications, paper MoP2.1. [38] B. Malo, S. The´riault, D.C. Johnson, F. Bilodeau, J. Albert, K.O. Hill, Apodised in-fibre Bragg grating reflectors photoimprinted using a phase mask, Electron. Lett. 31 (3) (1995) 223–225. [39] J.G. Fujimoto, M.E. Brezinski, G.J. Tearney, S.A. Boppart, B.E. Bouma, M.R. Hee, et al., Optical biopsy and imaging using optical coherence tomography, Nature Med. 1 (1995) 970–972. [40] A.F. Fercher, Opthalmic interferometry, in: G. von Bally, S. Khanna (Eds.), Optics in Medicine, Biology and Environmental Research, Elsevier, Amsterdam, 1990, pp. 221–235. [41] D. Fried, J. Xie, S. Shafi, J.D.B. Featherstone, T.M. Breunig, C. Le, Imaging caries lesions and lesion progression with polarization sensitive optical coherence tomography, J. Biomed. Opt. 7 (2002) 618–627. [42] B.W. Colston, U.S. Sathyam, L.B. DaSilva, M.J. Everret, P. Stroeve, L.L. Otis, Dental OCT, Opt. Exp. 3 (1998) 230–238. [43] C.C.B.O. Mota, H.U.K.S. Kashyap, B.B.C. Kyotoku, A.S.L. Gomes, In vivo evaluation of enamel dental restoration interface by optical coherence tomography, VI Taller Interna´ ptica, Vida y Patrimonio, Capitolio De La cional Tecnola´ser II Reunio´n Internacional O Habana, La Habana, Cuba, Del 13 Al 16 De Abril De 2009, paper OPT 5, (2009). [44] A. Sergeev, V. Gelikonov, G. Gelikonov, F. Feldchtein, R. Kuranov, N. Gladkova, et al., In vivo endoscopic OCT imaging of precancer and cancer states of human mucosa, Opt. Exp. 1 (1997) 432–440. [45] H. Liang, M. Cid, R. Cucu, G. Dobre, A. Podoleanu, J. Pedro, et al., En-face optical coherence tomography – a novel application of non-invasive imaging to art conservation, Opt. Exp. 13 (2005) 6133–6144. http://www.opticsinfobase.org/abstract.cfm?URI¼oe-13 -16-6133 [46] H.U.K.S. Kashyap, C.C.B.O. Mota, B.B.C. Kyotoku, P.B. Santos-Filho, A.S.L. Gomes, Implementation of an Optical Coherence Tomography system for painting characteriza´ ptica, Vida y Patrimotion, VI Taller Internacional Tecnola´ser II Reunio´n Internacional O nio, Capitolio De La Habana, La Habana, Cuba, Del 13 Al 16 De Abril De 2009, paper OPO 1, 2009. [47] W. Sorin, D. Baney, Measurement of Rayleigh backscatter at 1.55 mm with 32 mm spatial resolution, IEEE Photon. Technol. Lett. 4 (1992) 374–376. [48] W. Eickhoff, R. Ulrich, Optical frequency domain reflectometry in single-mode fiber, Appl. Phys. Lett. 39 (1981) 693–695.
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[49] U. Glombitza, E. Brinkmeyer, Coherent frequency domain reflectometry for characterization of single-mode integrated optical waveguides, J. Lightwave Technol. 11 (1993) 1377–1384. [50] M. Froggatt, T. Erdogan, J. Moore, S. Shenk, Optical frequency domain characterization (OFDC) of dispersion in optical fiber Bragg gratings, in: Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, OSA Technical Digest Series, paper FF2, Optical Society of America, Washington, DC, 1999. [51] J.P. Von der Weid, R. Passy, G. Mussi, N. Gisin, On the characterization of optical fiber network components with optical frequency domain reflectometry, J. Lightwave Tech. 15 (1997) 1131–1141. [52] P. Oberson, B. Huttner, O. Guinnard, G. Ribordy, N. Gisin, Optical frequency domain reflectometry with a narrow linewidth fiber laser, IEEE Photon. Technol. Lett. 12 (2000) 867–869. [53] S. Kieckbusch, C. Knothe, E. Brinkmeyer, Fast and accurate characterization of fiber Bragg gratings with high spatial resolution and spectral resolution, in: Optical Fiber Communication, OSA Technical Digest Series, paper WL2, Optical Society of America, Washington, DC, 2003. [54] M. Froggatt, J. Moore, High resolution strain measurement in optical fiber with Rayleigh scatter, Appl. Opt. 37 (1998) 1735–1740. [55] B.J. Soller, D.K. Gifford, M.S. Wolfe, M.E. Froggatt, High resolution optical frequency domain reflectometry for characterization of components and assemblies, Opt. Exp. 13 (2) (2005) 666–674. [56] P.Y. Fonjallaz, H.G. Limberger, R.P. Salathe´, Bragg grating with directional, efficient and wavelength selective fiber out coupling, in: Tech. Digest of Opt. Fiber Commun., OFC ’95, paper WN3, 1995, pp. 160–161. [57] J.L. Archambault, L. Reekie, P.J. St. Russell, 100% reflectivity Bragg reflectors produced in optical fibres by single excimer pulses, Electron. Lett. 29 (5) (1993) 453. [58] M. Janos, J. Canning, M.G. Sceats, Incoherent scattering losses in optical fiber Bragg gratings, Opt. Lett. 21 (22) (1996) 1827–1829. [59] M.G. Sceats, P. Krug, Photoviscous annealing – dynamics and stability of photorefractivity in optical fibers, in SPIE 2044 (1993), 113–120. [60] See, for example, M. Douay, W.X. Xie, T. Taunay, P. Bernage, P. Niay, P. Cordier, et al., Densification involved in the UV based photosensitivity of silica glasses and optical fibers, J. Lightwave Technol. 15 (8) (1997) 1329–1342. [61] P.L. Chu, T. Whitbread, Measurement of stress in optical fiber and preform, Appl. Opt. 21 (1982) 4241–4245. [62] P.K. Bachmann, W. Hermann, H. Wher, D.U. Weichert, Stress in optical waveguides. 1: Preforms, Appl. Opt. 25 (7) (1986) 1093–1098. [63] P.Y. Fonjallaz, H.G. Limberger, R.P. Salathe´, F. Cochet, B. Leuenberger, Tension increase correlated to refractive index change in fibers containing UV written Bragg gratings, Opt. Lett. 20 (11) (1995) 1346–1348. [64] G.W. Scherer, Stress-induced index profile distribution in optical waveguides, Appl. Opt. 19 (1980) 2000. [65] W. Primark, D. Post, Photoelastic constants of vitreous silica and its elastic coefficient of refractive index, J. Appl. Phys. 30 (1959) 779. [66] H.W. Hermann, D.U. Wiechert, Stress in optical waveguides. 3: Stress induced index changes, Appl. Opt. 28 (1989) 1980–1983. [67] H.G. Limberger, P.Y. Fonjallaz, R.P. Salathe´, F. Cochet, UV induced stress changes in optical fibers, in: Photosensitivity and Quadratic Nonlinearity in Glass Waveguides: Fundamentals and Applications, vol. 22, OSA Technical Series, paper SAD4, Optical Society of America, Washington, DC, 1995, pp. 56–60.
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[68] H.G. Limberger, P.Y. Fonjallaz, R.P. Salathe´, F. Cochet, Compaction-and photoelasticinduced index changes in fiber Bragg gratings, Appl. Phys. Lett. 68 (1996) 3069–3071. [69] See, for example, D.S. Starodubov, V. Grubsky, J. Feinberg, D. Dianov, S.L. Semjonov, A.N. Guryanov, et al., Fiber Bragg gratings with reflectivity >97% fabricated through polymer jacket using near-UV radiation, in: Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, vol. 17, OSA Technical Digest Series, Optical Society of America, Washington, DC, post-deadline paper PD1, 1997. [70] H.G. Limberger, D. Valeras, R.P. Salathe´, G. Kotrotsios, Mechanical degradation of optical fibers induced by UV light, Proc. SPIE 2841 (1996) 84–93. [71] R. Feced, M.P. Roe-Edwards, S.E. Kanellopoulos, N.H. Taylor, V.A. Handerek, Mechanical strength degradation of UV exposed optical fibres, Electron. Lett. 33 (2) (1997) 157–159. [72] M.A. Putnam, C.G. Askins, G. Smith, E.J. Fribele, Method for recoating fiber Bragg gratings with polyimide, in: J.A. Slater (Ed.), Industrial and Commercial Applications of Smart Structures Technology, vol. 2044, SPIE, Bellingham, WA, 1997, pp. 359–362. [73] F.P. Kapron, H.H. Yuce, Theory and measurement for predicting stressed fiber lifetime, Opt. Eng. 30 (6) (1991) 700–708. [74] Y. Mitsunaga, Y. Katsuyama, H. Kobayashi, Y. Ishida, Failure prediction for long-length optical fibers based on proof testing, J. Appl. Phys. 53 (7) (1982) 700. [75] See, for example, in: Proc. of the First European COST Workshop 246 on Bragg grating Reliability, IOA-EPFL. 1995. [76] H.G. Limberger, D. Valeras, R.P. Salathe´, Reliability aspects of fiber Bragg gratings, in: Proc. of Optical. Fibre Meas. Conf., OFMC ’97, 1997, pp. 18–123. [77] T. Erdogan, V. Mizrahi, Decay of UV induced fiber Bragg gratings, in: Proc. Optical Fiber Conference, OFC ’94, 1994, p. 50. [78] T. Erdogan, V. Mizrahi, P.J. Lemaire, D. Monroe, Decay of ultraviolet-induced fiber Bragg gratings, J. Appl. Phys. 76 (1) (1994) 73–80. [79] S.R. Baker, H.N. Rourke, V. Baker, D. Godchild, Thermal decay of fiber Bragg gratings written in boron and germanium codoped silica fiber, J. Lightwave Technol. 15 (8) (1997) 1470–1477. [80] H. Patrick, S.L. Gilbert, A. Lidgard, M.D. Gallagher, Annealing of Bragg gratings in hydrogen loaded optical fibers, J. Appl. Phys. 78 (5) (1995) 2940–2945. [81] J.Z.Y. Guo, S. Kannan, P.J. Lemaire, Thermal stability of optical add/drop gratings for WDM systems, in: Tech. Digest of OFC ’97, paper ThJ6, 1997, p. 285. [82] S. Kannan, J.Z.Y. Gou, P.J. Lemaire, Thermal reliability of strong Bragg gratings written in hydrogen sensitized fibers, in: Tech. Digest of OFC ’97, paper TuO4, 1996, pp. 84–85. [83] R.J. Egan, H.G. Inglis, P. Hill, P.A. Krug, F. Ouellette, Effects of hydrogen loading and grating strength on the thermal stability of fiber Bragg grating, in: Tech. Digest of OFC ’96, paper TuO3, 1996, pp. 83–84. [84] G. Robert, I. Riant, Demonstration of two distributions of defect centers in hydrogen loaded high germanium content fibers, in: Tech. Digest of OFC ’97, paper WL18, 1997, pp. 180–181. [85] D.L. Williams, R.P. Smith, Accelerated lifetime tests on UV written intracore gratings in boron germanium co-doped silica fibre, Electron. Lett. 31 (24) (1995) 2120–2121.
Chapter 10
Principles of Optical Fiber Grating Sensors The brain and the five senses are the most sophisticated result of nature at its best. Emulating them remains an unreachable challenge for artificial intelligence.
In some sense, fiber Bragg gratings are not really appropriate for many applications requiring stability of the reflection wavelength. As discussed in Section 3.1.13, the parameters of the optical fiber waveguide are sensitive to its environment. This is a fact of life as the material properties of glass are a function of temperature, strain, vibration, and pressure, and therefore the waveguide itself is sensitive to the refractive index of a cladding as well. The belief that optical fibers are immune to electromagnetic radiation, an attribute that was often used in defense of applications in optical communications, was clearly misplaced; it is evident that fibers are indeed sensitive to radiation, as the very process of Bragg grating inscription is dependent on the exposure of the fiber to ultraviolet (UV) radiation. They are also sensitive to thermal radiation as it may be used to periodically modify the refractive index through heat, leading to long-period gratings. One can go further by observing that optical fibers are sensitive to nuclear radiation, as exposure increases transmission loss transiently at visible wavelengths. Thus, one could conclude that optical fibers devices that strongly depend on waveguide properties will just about sense anything and everything, but ultimately require a clever brain to discriminate between them. Fiber Bragg gratings (FBGs) fall into this category as the Bragg wavelength (using the terminology loosely to include long-period gratings) is coupled intimately to the guiding properties of the waveguide. Therefore, from being simple and highly discriminating components for a large number of applications, they also become highly sensitive devices for many sensing applications, spanning the more traditional to exotic, such as use in biomedicine, oil exploration, structural health monitoring, to name but a few. It is therefore not surprising to find that fiber Bragg gratings fabricated at visible wavelengths by the holographic interference method were researched 441
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Principles of Optical Fiber Grating Sensors
principally for sensor applications [1–3]. Meltz’s group demonstrated a number of interesting sensor applications of fiber Bragg gratings for strain, temperature, and vibration sensing using Bragg gratings that reflected in the visible as well as in smart structures such as in composites [3]. With the advent of fiber Bragg gratings at telecommunications wavelengths of 1550 nm [4], sensing technology became truly compatible with fiber optic communication systems. It was also recognized early on that Bragg gratings could offer extremely valuable solutions for distributed sensing, for example, in a network [5]. Others followed quickly with a variety of techniques for sensing with fiber Bragg gratings [6–8]. This chapter begins with a brief overview of the fiber Bragg grating as a potential sensor and then reviews the properties of the FBG used in sensing of the many quantities it is capable of measuring; the chapter concludes with the recent progress and future prospects of the FBG in sensing. A number of books and articles review the technology of sensing with Bragg gratings [9–14]. The intention of this chapter is not to introduce the topics covered widely elsewhere but to show sensing principles and how novelty in the design of FBGs can lead to enhanced performance, often allowing a number of parameters to be sensed simultaneously or with reduced complexity. Industry requirements for more cost-effective solutions do not automatically lead to the best technological solution winning the day; the decision is often based on a number of criteria including footprint, maintenance requirements, flexibility, robustness, and, of course, the initial investment. Traditionally, optical sensors have faced an uphill struggle with extremely conservative customers, especially in the gas and chemical industries, where well-tried, failsafe, and robust solutions override all other requirements, including sensitivity and cost, especially in harsh and dangerous environments. The innovation of fiber-based distributed sensing has provided unique solutions to the petrochemical industry, especially in the offshore oil-platform structural monitoring arena. These are harsh environments in which the stakes are very high in terms of the safety of life and security of the infrastructure (e.g., oil platform). There are many problems still to be solved in which FBGs may play a key role; however, widespread acceptance of the technology is first required before this will happen. FBGs are, however, on the brink of yet another revolution in sensing, driven by the requirements from the industry, which become increasingly demanding and more complex. First of all, it is important to remember that the FBG is a reflective device (i.e., it rejects light of a certain wavelength and must therefore be used in conjunction with a number of techniques to invert this function). These areas were reviewed in Chapter 6. Being a wavelength selective device, it is usually necessary to use the FBG sensor with a wavelength discriminating system such as a spectrum analyzer or other schemes that allow the quantity dependent
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443
on the measured parameter to be decoded. Other methods include interferometric techniques, such as homodyne detection, which can have extremely high sensitivity. The ultimate choice of the measurement technique depends on the requirements, as this has a large impact on complexity and cost.
10.1 SENSING WITH FIBER BRAGG GRATINGS
10.1.1 Principles of Sensing A sensor is a transducer that responds to the quantity being measured called the measurand. To remain unambiguous, the response of the sensor should be ideally a linear function of the measurand and strictly a single-valued parameter without hysteresis. Therefore, s ¼ kðoÞx;
ð10:1:1Þ
where s is the output of the sensor, x is the measurand, and k(o) is the sensor’s sensitivity or transfer function, which is a function of frequency, o. Any deviation from linearity causes errors and must be quantified for the sensor or the transfer function. However, in most instances, the transfer function is rarely perfectly linear and may wander as a function of time or deviate from an ideal linear response by some minimum and maximum value, called inferior and superior deviations [10]. These are lines parallel to the ideal curve and are determined at the time of calibration. The tolerance of the sensor’s response is the sum of the two deviations and is referred to as the error band. Another parameter of interest is the backlash, which is defined as the change, Dx, in the value of the measurand when it changes sign on the return cycle of a measurement over which the sensor registers no change (i.e., @s/@x ¼ 0). Hysteresis, Ds, is a step change in the sensor’s response when the sensor returns to its characteristic curve on the upward cycle of the measurand (i.e., @s/@x ¼ 1). Drift, which is generally permanent and cannot be compensated, may be due to several environmental conditions, sensor cycling, or storage. Sensitivity may be regarded as small or large signal and the range of the sensor, for example, the measurement of temperature defines the window over which measurements may be made with the required accuracy and repeatability or precision. Finally, the stability of the sensor is of primary importance in achieving an accurate measurement, and “burn-in” is a technique used to stabilize the sensor before it is used in the field. During burn-in, the sensor may be used above its operating range to ensure that it does not change when used within its specification. For FBGs, it is necessary to anneal the grating at elevated temperatures to guarantee its performance at the operating temperature; it also must be
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strained above the range of measurement to eliminate relaxation effects over the lifetime of the sensor. All of these parameters define the dynamic range, frequency response, sensitivity, and errors in sensing and ultimately determine the sensor performance. To understand how the FBG may be used for sensing, we consider the Bragg reflection wavelength as being dependent on an external parameter to be sensed, which is designated as X. Here X can be temperature, T, strain, e, hydrostatic pressure, P, or refractive index of the cladding, ncl. The functional dependence of the Bragg wavelength on these parameters can be calculated as lB ¼ 2neff L; dneff dlB d dL ¼2 þ 2neff neff L ¼ 2L dX dX dX dX
ð10:1:2Þ
¼ 2Ldneff þ 2neff La: 2Ldneff 2neff La dneff 1 dlB ¼ þ ¼ þ a; 2neff L neff lB dX 2neff L 0 1 dn dlB eff DX ¼ lB @ DlB ¼ þ aADX ; dX neff
ð10:1:3Þ
where, dneff/neff is the normalized sensitivity of the effective index of the mode, and a is the coefficient of physical length change dependent on the parameter X. For temperature change, the former turns out to be the thermally induced refractive index change, (1/neff)(dneff/dT), and a, the thermal expansion coefficient. Both of these parameters may be summed to lead to the thermo-optic x. From Table 10.1, simply using the measured values for a and coefficient, ~ the coefficient of refractive index change, a Bragg wavelength change of 13.6 pm- C1 may be calculated from Eq. (10.1.3). With strain, Eq. (10.1.3) transforms to DlB ¼ lB ð1 ra Þe;
ð10:1:4Þ
where, ra, the stress-optic coefficient is ra ¼
n2eff 2
½r12 sðr11 r12 Þ:
ð10:1:5Þ
r12 and r11 are coefficients of the stress-optic tensor, and s is Poisson’s ratio. For isotropic and homogeneous strain, DlB ¼ lB ð1 Pe Þe; where Pe is the strain-optic coefficient.
ð10:1:6Þ
Sensing with Fiber Bragg Gratings
445 Table 10.1
Physical parameters of fused silica (adapted from reference [46]) Parameter Density Melting point Softening point Annealing point Maximum continuous service temperature Maximum transient service temperature Specific heat capacity Coefficient of thermal expansion dn/dT Thermal conductivity Tensile strength Compressive strength Poisson’s ratio Modulus of elasticity (Young’s modulus, 25 C) Fracture toughness Permittivity (1 MHz, 25 C) Resistivity Dielectric strength (20 C)
Value
Units
2.2 10 1830 1600 1120 950 1200 703 5.5 107 1.19 106 1.38 110 690–1380 0.165 73 0.79 3.8 >1018 14-40 106
kg-m3 C C C C C J-kg1-K1 1 C 1 C W-m1-K1 MPa MPa
6
GPa MPa O-m ~ Vm
10.1.2 Fiber Designs for Sensing Sensing with an optical fiber Bragg grating is based on the variation of the Bragg wavelength as a function of the measurand. In many instances, the change in the Bragg wavelength is strongly dependent on the external “potential” such as temperature or pressure. Noting that the optical mode in the optical fiber has a spatial distribution not simply confined to the core alone, temperature has the effect of altering the refractive index of both the core and the cladding of the fiber, as well as altering its length. The former results in a change in the effective index of the mode in two ways: first, even if both the core and cladding refractive indexes change by exactly the same amount, the propagation constant of the mode also changes by exactly the same amount. Second, a differential change in the refractive index between the core and the cladding materials also influences the propagation constant. Recalling the effective mode index, neff is [15]: 0:996 2 Dn; ð10:1:7Þ neff ncl þ 1:1428 v where, v is the v-value of the fiber, and Dn is the core-cladding refractive index difference.
Principles of Optical Fiber Grating Sensors
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The change in the effective index occurs as a result of change in the v-value of the fiber for a given core-cladding refractive index difference. Differentiating Eq. (10.1.7), one arrives at the following condition for the change in the mode effective index, neff as a function of the v-value (1.5 < v < 2.5), dneff 1:23 1:98 Dn; ð10:1:8Þ dn v2 v3 from which one finds that value of neff is most sensitive at a v-valve of 1.3. Operating at this v-value means that the mode effective index will be the most sensitive to a change in the v-value. However, as the v-value is dependent on the core-cladding refractive index difference, any differential change in the refractive index of the core and the cladding will have the largest impact on the effective index of the mode. As the material in the core is usually different from that of the cladding (e.g., high germanium-doped core for large photosensitivity and pure silica cladding), a change in temperature will result in a change in the v-value. Operating at this point makes the fiber most sensitive to the measurement of temperature, based solely on the change in the refractive index. The differential refractive index effect is usually rather small and only significant when the temperature increase is large. However, it leads to a small nonlinearity depending on the exact guidance properties of the optical fiber. Figure 10.1 shows how the v-value changes for two values of the differential 2.8 2.7
Differential dn = 7e-6
2.6
V-Value
2.5 2.4 2.3 2.2 2.1
Differential dn = 5e-6
2 1.9 1.8 –100
0
100 200 300 Temperature, degrees C
400
500
Figure 10.1 The v-value as a function of the ambient temperature for two fibers for a coefficient of differential refractive index change of core and cladding of 7 106 (dashed line) and 5 106 (solid line). The significant change in the v-value and departure from linearity is evident over the temperature range shown. Note that single-mode operation ceases above 290 C and 400 C for the two fibers, respectively. This calculation ignores thermal expansion.
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447
Pcore / Ptotal , Pclad / Ptotal , dPcore / dv
change in the refractive index of the core and cladding. Note that even with a 5-ppm difference, the v-value of the fiber changes from single-mode to multimode operation with a temperature change of approximately 400 C. With 7 ppm, this changes to only 300 C! It should be noted that the length change associated with temperature in silica contributes to only around 10% of the overall change and is therefore dominated by the refractive index change (see Section 3.1.13). Another important aspect in FBG sensors is the variation in the optical mode field width and therefore the fraction of the power in the core as a function of the v-value. In other words, this may be considered as the mode overlap with grating in the core of the fiber. As the mode field width is a function of the v-value, the sensitivity of the grating as a sensor will change as a function of temperature depending on the exact v-value of operation. The power in the core may be calculated by solving the characteristic equation for the fundamental mode. Figure 10.2 shows that the power in the core changes nonlinearly depending on the exact operating point; for low v-values, the power alters rapidly if the refractive indexes of the core and cladding do not have the same temperature coefficients, but it has little effect at high v-values. Figure 10.2 also shows the change in the power in the core as a function of the change in the v-value. There is an optimum v-value of 1.1 at which the power reflected by the grating will be most affected by temperature or other external effects. This is not the same as the sensitivity of the grating to external influences, but it makes the
1.00
0.75
Pcore/Ptotal Pclad/Ptotal dPcore/dv
0.50
0.25
0.00 0.0
0.5
1.0
2.0 1.5 v-value
2.5
3.0
3.5
Figure 10.2 Power in the core as a fraction of the total power in the fiber for the fundamental LP01 mode as a function of v-value is shown (solid line) and the variation of the power in the core as a function of v-value (dashed-line). The power in the core has a maximum sensitivity at a v-value of 1.1. Also shown is the power in the cladding (dotted line). This approaches a maximum at low v-values. (Courtesy G. Nemova.)
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reflected power at the Bragg wavelength a function of the measurand. The Bragg wavelength will, of course, change as a function of the measurand. The data in Fig. 10.2 is for a grating that resides in the core of the fiber. However, there are other possibilities for the existence of the grating (e.g., in cladding-mode suppressed optical fibers) [16,17,18] in which the overlap of the mode with the grating is maximized for all v-values and the reflectivity is not affected by external influences, to first order. In this case, the reflectivity is also maximized for any given amplitude of the refractive index modulation of the grating and is therefore useful for high-sensitivity measurements. A third scheme, which is becoming more prominent for sensing, is the hollow core fiber [19] with a grating in the cladding region [20,21] or a hollow cladding for liquids or polymers with either LPGs or SPGs. The advantage of such systems with cavities is in the ability to engineer the sensor – either by using the hollow core to accommodate a liquid, liquid crystal [19,20], or polymer or by using liquids, polymers, or liquid crystals in the hollow cavity in the cladding as in photonic crystal structures [22–24] – to change the performance of the devices radically. An example of this type of structure is shown in Fig. 10.3. This figure shows a photosensitive cladding region, which is nearly matched to the refractive index of the silica cladding [20]. This can be done, for example, with codoping with fluorine and germanium to match the refractive index or with boron and germanium, which allows a much higher concentration of germanium and consequently a greater photosensitivity. This type of a fiber has a hollow core to allow the insertion of liquids and other materials such as liquid crystals [20]. With a liquid core the reflection from the grating is through an
Hollow core & photosensitive inner cladding
Cladding Grating Liquid core
A
B
Mode field
Figure 10.3 (a) A cleaved facet of a hollow cored glass fiber with a matched cladding photosensitive ring. The hollow core diameter and the inner cladding thickness are each 6 microns, (b) filled with a liquid of refractive index higher than that of the core. The mode field penetrates into the grating in the photosensitive cladding region. As the sensitivity to temperature of the refractive index of the liquid is significantly different to that of the glass and generally has a negative coefficient, the Bragg wavelength shift with temperature can also be negative. (From reference [20].)
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interaction with the evanescent field of the mode in the core. The change in the overlap of the core mode with the grating in this fiber as a function of the v-value is reversed compared to the grating in the core, shown in Fig. 10.2. Overlap of the mode with the grating is within the cladding for the liquid core. This increases with decreasing v-value, as the mode spreads farther out into the cladding. The sensitivity of the reflected power for such a grating as a function of v-value is the same as for the grating in the core, but with a negative sign. For such a fiber, the v-value can be made to vary strongly with temperature. The modulus of the liquid’s temperature coefficient of refractive index change is more than an order of magnitude larger than that of the doped silica cladding and negative ( –5 105 C1, compared with þ7 107 C1 for silica). Thus, the temperature sensitivity of the Bragg wavelength with temperature in this fiber can also be negative, depending on the v-value of operation. The dramatic change in the coefficients from negative to positive opens up new engineering possibilities for fiber grating sensors. Figure 10.4 shows the temperature dependence of a grating’s Bragg wavelength in liquid core fiber. The magnitude of the temperature coefficient of the Bragg wavelength for this fiber is more than 10 (0.262DT nm) that of the standard silica fiber (þ0.016DT nm). Note the strong shift in the Bragg wavelength, even though the temperature change is small. The second interesting aspect of the liquid cored fiber is that under strain, the refractive index of the core remains unchanged. However, the refractive index of the cladding does alter, and as the Bragg wavelength strain-optic coefficient of silica is positive, the wavelength shift is also positive. As subsequent sections will show, discrimination between these two quantities has been a big challenge and remains an active area for investigation. Figure 10.5 shows the shift in the Bragg 1547.0
Bragg wavelength, nm
1546.5 1546.0 1545.5
l = –0.262T + 1551.5nm
1545.0 1544.5 1544.0 18
19
20
21
22
23
24
25
26
27
28
Temperature, degrees C
Figure 10.4 Wavelength shift of a liquid cored fiber Bragg grating. (From reference [20].)
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ΔlB = 10.248e nm
lB, nm
1546.4 1546.0 1545.6 LC Core Bragg Wavelength Linear(LC Core Bragg Wavelength)
1545.2 1544.8 0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Strain, e (%) Figure 10.5 The strain dependence of an FBG in a liquid crystal cored fiber. The dependence is similar to a standard fiber, although the temperature dependence has a negative slope. (From reference [20].)
wavelength of a liquid cored fiber as a function of the applied strain. It is clear that the temperature and strain dependencies are in opposite directions. Thus, two sequential gratings in a liquid cored fiber and a standard fiber at different wavelengths can be used to discriminate between strain and temperature change, so long as both see the temperature change and only one sees the strain [25]. Capillary FBG sensors with a 6-mm diameter liquid core, for example, require less than 30 pL-m1 of liquid. For a typical 10-mm-long FBG sensor, a considerable potential advantage exists for measuring the refractive index of a sample, as only a small volume is required. FBG-based sensors have attractive properties for measuring refractive index as well as for biosensing. The reflection spectra of gratings in this type of a fiber are shown in Fig. 10.6. Strong gratings are possible by suitable design of the deposited cladding as well as the refractive index of the liquid core.
10.1.3 Point Temperature Sensing with Fiber Bragg Gratings Because they are small, fiber Bragg gratings are excellent point sensors. The small diameter of the optical fiber and the short grating length (typically a few mm) allows measurands to be sensed over a distance of only a few hundred wavelengths. The typical volume of a standard 125-mm diameter uncoated optical fiber is approximately 12.3 ml-m1, equivalent to 26.8 mg-m1. Therefore,
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0 Reflection (LC), dB Reflection (Cargile Oil), dB
Reflection, dB
–5 –10 –15 –20 –25 1544
1545
1546
1547
1548
1549
1550
Wavelength, nm Figure 10.6 Reflection spectrum of FBGs in a liquid crystal (LC) cored fiber (dashed line) and the reflection spectrum of an oil cored optical fiber. The spectra show that strong reflections are achievable with short gratings in cladding sensitive optical fibers. (From reference [20]).
a 1-mm-long FBG has a mass of 26.8 ng and has a heat capacity of 18.8 nJK1 – that is, a very small amount of energy is required to heat the grating by one degree Kelvin and is approximately half the amount required for a 1-mm long platinum thermocouple. Thus, for most applications it may be considered to be a noninvasive temperature sensor, not only because the fiber is electrically inert, but it also has a small sensor footprint. However, the temperature sensitivity of the FBG’s Bragg wavelength is 16 pm-K1 (see Section 3.1.13), requiring a sensitive instrument to measure such a change, especially when measuring small temperature excursions. Normally, an optical spectrum analyzer has a resolution of 10 pm, making it difficult to measure temperature changes of the order of one degree without increasing cost considerably. On the other hand, the fiber may have several FBGs written in sequence, each at a different wavelength. This sequential set of FBGs makes the optical fiber ideal for distributed point sensing of measurands. The parameter used for making measurements is wavelength. As an example, considering a measurement range of 50 C, the wavelength shift would be 0.8 nm. If the laser source used in sensing has a tunability of 100 nm, a maximum of 100/1.6 62 distributed point sensors can be accommodated on a single fiber. As the distributed FBG sensors may be used over several km without being limited by loss, it has a significant advantage over an equivalent distributed electrical sensor system. The reach of the electrical system is considerably shorter than 100 m; it not only requires 62 individual electrical sensors, but also twice as many lead wires to read the nodes, making the assembly heavy, cumbersome, expensive, potentially unreliable, and highly restricted in capability.
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10.1.4 Distributed Sensing with Fiber Bragg Gratings A typical distributed FBG point-sensing scheme is shown in Fig. 10.7. In this scheme, several FBGs of low reflectivity and distinct wavelength are concatenated in a single fiber. The measurement source for the system is either a continuously tunable single frequency laser or a broadband source such as an erbium-doped fiber amplifier. The detection system requires a spectrum analyzer, which has to be synchronized with the continuously tunable laser or simply be used directly with the broadband source. Generally, a baseline scan is recorded as a reference and subsequent measurements are compared to it. Any difference between the two is processed with a computer and displayed as the measurand. This system is simple to implement but restricted to sequential point sensing with a resolution limited by the spectrum analyzer, and is generally limited to a small number of sensors (50). To enhance the resolution of the system, an interferometric scheme may be used. This uses a simple phase-delayed Mach– Zehnder interferometer (MZI) de-modulator to detect dynamic signals. With this technique, the system shown in Fig. 10.7 may be used by replacing the spectrum analyzer with the interferometer followed by a wavelength de-multiplexer. The interferometer-based system is shown in Fig. 10.8. In this scheme, a set of distributed Fabry–Perot interferometer sensors made with a pair of identical FBGs separated by a fixed optical delay, DL, generates an interference signal only when the optical delay at the demodulating interferometer is also identical. A pulse applied to an electro-optic modulator provides temporal gating of sensors in the array. The detector array has individual photodiodes for each wavelength, and hence each sensor in the distributed chain [26] may be interrogated in time.
Circulator Broadband or Tunable source
Distributed FBG sensors l1
l2
l3....
Spectrum Analyzer
Computer Figure 10.7 A typical fiber-optic distributed sensor with FBG point sensors.
ln
Sensing with Fiber Bragg Gratings
Distributed FBG FP sensors
Sensor 1 ΔL
ΔL
l1
l1
l2
ΔL
l2
l3
l3
ΔL l1 WDM Filters ln
Detector Array
Demodulating Interferometer
ln
Pulse to EO modulator Circulator
Multiplexer
l1
453
Sensor Outputs
Figure 10.8 Interferometric distributed sensing scheme using FBG Fabry–Perot interferometers. (Adapted from reference [26].)
10.1.5 Fourier Transform Spectroscopy of Fiber Bragg Grating Sensors The use of Fourier analysis for monitoring parameters sensed by FBGS offers some key advantages. First, a single processing system can be used to detect a distributed set of sensed parameters (e.g., strain, temperature, or vibration). Second, the system is relatively straightforward; however, it does not over come the problems of polarization fluctuations and loss of the interference signal, and it requires the use of polarization control [27,28,29,30], polarization diversity techniques [31], or polarization maintaining optical fibers. The theory for the detection of the spectral position FBGs allows their tracking by continuous scanning. Figure 10.9 shows the scheme implemented by Davies and Kersey [32]. The detection system is in two parts: The first is simply the locking Michelson interferometer (MI) powered by a single frequency 1319-nm laser; the second is the same interferometer used in the 1550-nm window to measure the visibility function of the light reflected by the set of distributed gratings each at a different wavelength. The phase at the output of the Michelson interferometer is Df ¼
4pnleff DL l
¼
2px ; l
ð10:1:9Þ
where, l is the wavelength (1319 nm or nominally1550 nm), and x is the optical path length difference between the two arms of the MI. The intensity variation, which depends on the interference signal at the output of the interferometer, is aI0 2px 1 þ K l cos S¼ ; ð10:1:10Þ 2 l
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1.5 mm Broadband Source
WDM coupler
l1
Distributed FBG sensors l2 l3.... ln
Single Frequency Laser (1.319 mm) PZT Ramp FRM1 Fast Fourier Transform
1.5 mm
Output spectrum
FRM2
1.3 mm 1.319 mm control loop PD ∼ Reference Oscillator
Figure 10.9 Fast Fourier transform is performed on the signal at the receiver synchronized to the optical delay induced by the piezoelectric stretcher (PZT). The scanning of the 200-m-long fiber Michelson interferometer allows the visibility function to be recorded for each grating, as the delay is ramped over the free spectral widths of all the gratings. The single frequency laser is used to lock the interferometer. FRM1 and FRM2 are Faraday rotation mirrors. The dashedline box shows the extent of the remote sensing instrumentation, which is basically free of the sensing leg. (Adapted from reference [32].)
where a includes all the losses, including the reflectivity of the end mirrors of the MI, I0 is the source intensity, and Kl is the visibility of the interferometer at the superscripted wavelength, dependent on the splitting ratio of the coupler, polarization state, and the source power. If the optical path length difference is changed at a velocity, VOPL m-s1, then fringes will be seen at the output of the MI at a frequency of f ¼
VOPL : l
ð10:1:11Þ
For an extended wavelength or quasi-continuous source, Eq. (10.1.10) may be expanded to include all the reflections from the gratings as þ1 ð
SðxÞ ¼ 1
aI0 ðlÞ 2px 1 þ K l cos dl: 2 l
ð10:1:12Þ
S(x) describes the interferogram as the MI optical path length is varied. To describe the quasi-continuous reflected spectrum, a visibility function can be defined as F ðlÞ ¼ I0 ðlÞK l ; which leads to Eq. (10.1.10) being reformulated to
ð10:1:13Þ
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2 þ1 3 þ1 ð ð a 4 F ðlÞ 2px SðxÞ ¼ dl þ F ðlÞcos dl5: 2 Kl l 1
ð10:1:14Þ
1
The first term in Eq. (10.1.14) is the integrated power of the source, whereas the second term represents the interferogram of the individual spectra of the FBGs. The second term is related to the cosine Fourier transform of the source spectra, and a Fourier transform pair may be described resulting in the spectra of each grating as þ1 ð 1 2px F ðlÞexp i dl; ð10:1:15Þ GðxÞ ¼ pffiffiffiffiffiffi l 2p 1
and 1 F ðxÞ ¼ pffiffiffiffiffiffi 2p
þ1 ð
1
2px GðlÞexp i dl: l
ð10:1:16Þ
Comparison between Eq. (10.1.14) and the Fourier transform pair in Eqs. (10.1.15 and 10.1.16) leads to the other half for the real part of the interferogram: F ðlÞ ¼
2 a
þ1 ð
1
2px SðxÞcos dx: l
ð10:1:17Þ
Recording the interferogram as a function of the optical path difference induced by the PZT stretchers shown in Fig. 10.9 and taking a Fourier transform results directly the spectra of the FBGs. However, it is essential that the visibility be maintained during the measurement, which indicates the importance of polarization control. The PZT allows the fast scanning of the MI for both the 1319-nm light as well as the FBG wavelengths. The error signal generated from the interferometer with the 1319-nm light is used to drive the second PZT to maintain the interferometer at the optimum operating point and allows the normalization of the path length change. It should be noted that dispersion between the two wavelengths (the change in the effective index of the two wavelengths) needs to be taken into account to calculate the real path length change during the scanning of the PZT. The FBG sensor system is scanned with a synchronized ramp applied to a PZT to probe the spectral visibility function of the reflections from the FBGs. As a specified optical delay is equivalent to the measurement of the coherence length of the source (effectively the bandwidth of each individual FBG), the Fourier transform of the visibility function produces the spectrum of the FBGs. The system developed by Davies and Kersey [32] should see wider
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application with the recent advances and availability of inexpensive and compact fast Fourier transform (FFT) electronic hardware with large onboard memory and processing capability. Cheap wavelength division (WDM) components and telecommunication-grade single-frequency lasers may be used to replace the 1.319-mm source, making this system attractive for applications in distributed sensing of temperature, strain, and vibration.
10.1.6 Fiber Bragg Grating Fiber Laser Sensors Another type of FBG sensor is the active device in which a laser cavity forms part of the sensor. In this system, the tuning of the FBG changes the oscillation frequency of the laser, which can be detected by a scanning Fabry–Perot interferometer for ultrahigh resolution sensing of temperature, strain, or vibration. The FBG-based external cavity laser (see Chapter 8) can be an excellent vibration sensor, as the fiber external cavity is sensitive to any external perturbation. Hence, for “quiet” operation, the laser has to be packaged carefully to minimize the effects of external disturbances. On the other hand, this type of laser may be used for sensing as the frequency can be made to scan by stretching the fiber laser. Two types of sensors are based on lasers. The first one uses the entire laser as a composite sensor [33–35], whereas the second type uses a single FBG mirror to control the lasers operation [36]. Figure 10.10 and Fig. 10.11 show the schematic of these sensors. Figure 10.10 shows fiber distributed feedback (DFB) lasers, which are used remotely to measure strain or temperature. The narrowness of the laser emission [37] makes this sensor an extremely good device for sensing. The four lasers are independent and operate in the 1549-nm–1551-nm range. These cascaded lasers have a total forward output of 10 mW with 140 mW of pump power, with a back propagating signal of 26 mW. The measured line width was reported to be between 166 kHz and 345 kHz, with a relative intensity noise (RIN) of between 75 and 85 dB-Hz1/2. When used as polarimetric sensors, it is possible to discriminate between strain and temperature. The typical accuracy of this sensor is 3 me and 0.04 C. To extract the strain, the beat frequency generated by the difference between the emission wavelengths of the two polarizations is measured. Forward signal
WDM Coupler 1480nm pump
ISO
DFB Laser 1
2
3
ISO
4 EDFA
Backward Signal Figure 10.10 Remote sensing of strain and temperature using a fiber DFB laser sensor. (Adapted from reference [37].)
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1552.0
Wavelength (nm)
1551.2 1104 8.76kHz/me
1550.4
1101
1549.6
1098 1.136pm/me
1548.8
Beat frequency (MHz)
1107
1095
1548.0
1092 0
200
400
600
800 1000 1200 1400
Strain (me) Figure 10.11 Wavelength and beat frequency as a function of temperature applied to a DBR laser. (From reference [38].)
The sensitivity to strain and temperature was reported to be 150 MHz/me and 1 MHz/mK for the lasing frequency, and 8 kHz/me and 1.6 kHz/mK for the beat frequency. Hydrostatic pressure sensitivity in the configuration used was shown to be 900 Hz/Pa. In a slightly different approach, Shao et al. [38] used a short distributed Bragg reflective (DBR) laser construction with a phase-step only 2 mm from one end of an 8-mm-long laser Er:Yb-doped laser. To discriminate between strain and temperature, the following relationship may be used from the data presented in Figs. 10.11 and 10.12 for the variation in the beat frequency and central wavelength of the laser as a function of temperature and strain: dl de 1 1320 kHz= C 8:8pm= C ð10:1:18Þ ¼ 1414:5 dT 8:76 kHz=me 1:13pm=me dðDnÞ where de is the strain to be measured, and dT is the sensed temperature. Thus, by knowing both dl and d(Dn), one can measure the strain and temperatures independently. Sensitivity of the measurement system was shown to be 9.3 me and 0.05 C RMS over 0 – 1400 me and 10–52 C, respectively [38]. A schematic of the measurement system is shown in Fig. 10.13. In Fig. 10.14, a cantilevered arm to which strain is applied holds an FBG; reading of the strain is performed by making a comparison of the outputs of two photodiodes, one of which is spectrally filtered by an edge filter and the other which is not. This scheme is identical to an earlier one used for wavelength control of a laser proposed and implemented by Malyon [39]. As the strain increases, the output through the spectral filter increases as well. The ratio of
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1100
Wavelength (nm)
1548.6
1090 1548.5 1548.4
1080 8.88pm/°C
1070
–1.32MHz/°C
Beat frequency (MHz)
1110
1548.7
1060
1548.3
1050 1548.2 10
20
30
40
50
60
Temperature (°C) Figure 10.12 Wavelength and beat frequency of the DBR laser as a function of applied strain. (From reference [38].)
980nm pump laser
WDM
SMF28
Temperature Control Er:Yb DBR Laser
ISO
Translation stage PC
OSA
Polarizer RFSA
Photodiode
Figure 10.13 The independent strain and temperature measurement system using a short asymmetric DBR laser. RFSA ¼ radio frequency spectrum analyzer. (From reference [38].)
Spectral Filter Tx
Er3+ doped Fiber 980nm Pump
Mirror WDM
1550nm FBG on Cantilever
l Photodiode Signal Processor Photodiode
Figure 10.14 The schematic shows a cantilevered FBG strain sensor based on a fiber laser. The strain applied to the FBG changes the laser’s, emission wavelength and the spectral filter converts the output to an amplitude variation. The strain is directly proportional to the amplitude normalized to the emitted power using the second photodiode. (Adapted from reference [36].)
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l1l2 l3 l4 l pump
1550-nm tunable laser 50 % WDM l1 l2 l3 l4
96 cm 90 %
980-nm laser
1.07 m 90 %
1.13 m 90 %
90 %
70 cm S1
S2
S3
S4
l1
l2
l3
l4
Figure 10.15 A schematic of the implementation of an amplified sensor network. The FBGs are separated by couplers and lengths of erbium-doped fiber that act as a gain section, pumped by a common 980-nm laser. (After reference [41]. Reproduced with permission from: Abad S., Lo´pez-Amo M., Lo´pez-Higuera J.M., Benito D., Unanua A., and Achaerandio E., “Single and double distributed optical amplifier fiber bus networks with wavelength-division multiplexing for photonic sensors,” Opt. Lett. 24, 805–807, 1999. # OSA 1999.)
the outputs from the two photodiodes is directly proportional to the strain applied to the cantilevered arm. Of course, a pumped gain fiber may also be incorporated into the distributed sensor to provide additional sensitivity in the presence of high loss and is a topology used in theoretical [40] and practical implementation [41] of WDM type sensor networks, of which a schematic is shown in Fig. 10.15, and in a time division multiplexing scheme (TDM) [42]. The scheme shown in Fig. 10.15 allows a longer reach and many FBGs to be used; however, amplified spontaneous emission (ASE) is a limiting factor and the length and gain of each erbium-doped fiber needs to be adjusted, as does the pump power. The split ratios of the couplers and the gain of the entire system have to be balanced to achieve a well-behaved sensor system, and ASE noise accumulates as the number of sensors increase.
10.1.7 Measurement of Temperature with Fiber Bragg Gratings Section 10.1 describes how an FBG may be sensitive to temperature. There are two predominant causes for the change in the Bragg wavelength described by the thermo-optic coefficient, z; these are due to the change in the refractive index of the optical fiber and the change in the physical length of the fiber through thermal expansion: DlB ¼ lB xDT:
ð10:1:19Þ
The change in the normalized refractive index of silica is 5–10 106 C1, whereas the thermal expansion coefficient of silica is 5.2 10–7 C1 as discussed in Section 3.1.13. Thus, the ratio of the change in the refractive index and the expansion coefficient is 10, leading to x 10 – 5.52 10–6 C1
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depending on the type of fiber. This means that the contribution to the shift in the Bragg wavelength through thermal expansion of the fiber is only 10%, the Bragg wavelength shift being predominantly due to the change in the refractive index of the optical fiber. At a wavelength of 1550 nm, standard SMF28 has a coefficient of Bragg wavelength shift, DlB of 11 pm– C1. Although this is a small number, it is sufficiently sensitive for the measurement of large temperature excursions of >100 C. A typical change in the Bragg wavelength as function of temperature is shown in Fig. 10.16. As temperature sensing requires intimate contact of the FBGs with the sample, it is necessary to ensure that strain does not affect the measurement. An FBG must therefore either remain strain free or have some mechanism to compensate for strain over the range of measurement. To keep the FBG strain free, soft polymer glue (such as poly-dimethyl-siloxane, PDMS) may be used to make contact with the sample. However, the FBG must remain in an unstrained state despite the thermal expansion of the glue. The FBG may therefore be tethered on each end with a small bow on the fiber to compensate for the thermal expansion of the sample as well as the glue. Figure 10.17 shows a schematic of the FBG on a sample.
1540.4
Bragg wavelength, nm
1540.0
1539.6
1539.2
1538.8
–40
–20
0
20 40 Temperature, °C
60
80
Figure 10.16 The graph shows the shift in the Bragg wavelength of an FBG as a function of temperature in a boron codoped optical fiber (cladding mode suppressed). The slope is 9.1 pm- C–1.
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Soft glue FBG
Epoxy
Metal tube Epoxy in tube FBG housing
Optical Fiber Sample to be measured
Sample to be measured
A
B
Figure 10.17 The mounting of an optical fiber on a sample to avoid the effects of strain (a), and using a loose free-ended FBG in as a metal tube for measurement of temperature (b).
10.1.8 Strain Measurements with Fiber Bragg Gratings According to Eqs. (10.1.4) and (10.1.5), the shift in the Bragg wavelength is a function of strain. The two important factors that contribute to the shift in the Bragg wavelength are the stress-optic tensors and Poisson’s ratio, s. At room temperature, s ¼ 0.165. This means that there is a reduction in the diameter of a fiber equivalent to 17% of the applied isotropic axial tensile strain. The condition for thermal compensation of the wavelength of the FBG is given by equating Eq. (10.1.3) to zero, which results in the following condition: dneff ¼ neff a; dT
ð10:1:20Þ
or, a¼
1 dneff : neff dT
Note that here a is the effective thermal expansion coefficient of the FBG and its mount. The thermal expansion coefficient of silica from Table 10.1 is þ5.5 107 C1, which is an order of magnitude lower than is required for thermal compensation. Partial thermal compensation may be achieved by simply stopping the expansion of the FBG; this will reduce the thermal sensitivity of the FBG by around 10% (the contribution from thermal expansion of the fiber). By forcing the fiber into compression, complete compensation can be achieved. Therefore, a material with a magnitude of thermal expansion coefficient of the (1/neff)dneff/dT of the mode but with a negative sign, of –8.2 105 C1 would compensate the thermal sensitivity of the FBG. Other techniques using passive compensation of the temperature offer robust solutions. These may be achieved in several ways: As mentioned in Section 6.4 [43,44], an oriented liquid-crystal polymer may be used as a secondary cladding. This material has a negative thermal expansion coefficient in the axial direction so that a compressive strain is induced as the temperature
Principles of Optical Fiber Grating Sensors
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increases. When using composite thermal compensation (e.g., with the liquid crystal polymer coating), the relative tensile strengths of the fiber and polymer must be taken into account [43]. Based on this principle, a loose tube version of this scheme has been implemented in which the tube was made of the oriented liquid crystal polymer [45].
10.1.9 Fiber Bragg Grating Wavelength Temperature Compensation Techniques Of course, it is important to ensure that the dominant thermal expansion is from the contribution of the compensating scheme. This technique has been used to effectively compensate the sensitivity to the temperature in FBGs and Fig. 10.18 shows the compensation of the grating previously shown in Fig. 10.6. Residual thermal sensitivity is 0.8 pm- C1 over a 40 to þ70 C range, with complete compensation over a few degrees around þ40 C. What is apparent in Fig. 10.18 is the nonlinear compensation of the grating. This is because the temperature dependence of the FBG is not linear and may be described as lB ¼ l0 þ ADT þ BDT 2 þ . . . ;
ð10:1:21Þ
where A and B are constants, DT is the change in the temperature from a reference point, and l0 is the Bragg wavelength at the reference temperature. It is therefore always possible to achieve absolute temperature compensation at some temperature, depending on the setting point of the compensator. For the Bragg wavelength to be insensitive to temperature,
Wavelength, nm
dl ¼ A þ 2BDT ¼ 0: dDT
ð10:1:22Þ
1540.2
1540.0 –40
–20
0
20 40 Temperature, °C
60
80
Figure 10.18 A thermalized packaged FBG. Multiple graphs show the variation in the Bragg wavelength through temperature cycling. Note the nil variation in the wavelength as a function of temperature around þ40 C, which is the design wavelength for this compensator. The total variation in the wavelength is 45 pm over the 110 C temperature range.
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With the linear part compensated, the remaining factor, 2BDT ¼ 0. This is always possible when DT ¼ 0 (i.e., the set temperature for the FBG). In the example given, the thermal compensation was achieved with the help of a bimetallic strip. This compensator uses two strips of different metals, usually invar and steel welded back-to-back, with the steel below. Invar has a very low thermal expansion coefficient (a ¼ 1.2 106), whereas for steel it is much higher (a ¼ 11–13 106). The thickness of each metal is adjusted such that thermal expansion straightens a preformed bent composite strip. Clearly, the relative Young’s modulus and difference in thermal expansion coefficients determine the flexure of the bimetal. The bimetal previously in the shape of an arc relaxes the FBG fixed to it under strain at a low temperature. The compensation can be adjusted by the mechanical machining of the two components of the bimetal strip [47]. Before fixing, the FBG is strained and welded in place with a glass bead or epoxy. Figure 10.19 shows the scheme. Another, simpler technique uses a lever principle. The FBG is fixed on two supports, which move toward each other as a function of temperature [48]. By using a metal of the right length and thermal expansion coefficient, the Bragg wavelength shift with temperature can be compensated. A schematic of this temperature compensator is shown in Fig. 10.20. Here, the lever arms may be made of invar for low expansivity. The thermal compensation may be easily calculated as aInvar acompensator. From Fig. 10.17 and Eqs. (10.1.3) and (10.1.6), it is easy to show that b x : ¼ a lB ð1 Pe Þacompensator
ð10:1:23Þ
Equation (10.1.23) shows that temperature compensation of the FBG is by the simple choice of the ratio of the lever arms. A typical value for optical fibers, (1 – Pe) ¼ 0.78, which is equivalent to a Bragg wavelength shift of 12 nm/% strain at a wavelength of 1550 nm. Using a value for the thermally induced
Under Tension FBG Relaxed FBG
Epoxy Bimetallic Strip
Hot State
Cold State
A
Fiber
B
Figure 10.19 The fiber in the bimetallic temperature compensating system in the cold state is under high tension (a), whereas at high temperature, the bimetallic strip straightens and relaxes most of the strain (b). The strain relief compensates the temperature-induced shift of the Bragg wavelength by moving in the opposite direction (Adapted from reference [47].)
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Thermal relaxation Fiber under tension FBG
Weld b
Invar
Pivot Thermal expansion
a
Compensator Figure 10.20 An FBG temperature compensation scheme is shown using easily available materials. The thermal expansion coefficient of steel allows a ratio, r b/a ¼1, which results in a compact solution for a ¼ b ¼ 5 mm; however, the exact placement of the fiber becomes critical. This system may be used to enhance the temperature sensitivity in a controllable manner by tuning r ¼ b/a.
Bragg wavelength shift of 16 pm– C, the ratio of the lever arms is equal to unity when the thermal expansion coefficient of the compensator is acompensator ¼ 1:3 10 5 C 1 :
ð10:1:24Þ
This condition is easily satisfied, for example, with a certain quality of steel. The problem with such mechanical levers is the possibility of backlash and hysteresis because of ill-fitting pivots. To overcome this difficulty, another scheme may be devised, which replaces the pivots with welded joints. In this technique, a baseplate made of one metal of low expansion coefficient is welded to another of higher expansion coefficient, but with a shorter length as shown in Fig. 10.21. The FBG is fixed to the high-expansion coefficient material, the ends of which move toward each other, relaxing a prestrained FBG. For thermal compensation, the following condition has to hold: 2 bacompensator aainvar x ¼ : ð10:1:25Þ 2ða bÞ ð1 Pe Þ The factor of 2 has been retained, as the Lf ¼ 2(ab) is the length of the fiber. If aainvar bacompensator, then the required thermal expansion coefficient or the physical length of the device may be calculated from Epoxy b Fiber
2a FBG
Compensating material
invar
Figure 10.21 An FBG temperature compensating system. The thermal expansion of the compensating material is than that of invar (see text for details).
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acompensator ¼
Lf x : 2b ð1 Pe Þ
ð10:1:26Þ
Using Lf ¼ 30 mm, x ¼ 7 106 C1, and a steel compensator with acompensator ¼ 5 105 C1, the length of the compensator, b ¼ 5.4 mm. Finally, Fig. 10.22 shows an alternative arrangement of an FBG temperature compensator made of a liquid-crystalline polymer (LCP) tube [43,45]. In this arrangement, the prestrained FBG is held on each end of the LCP tube, and thermal compensation occurs using the principle of the strain relaxing with temperature. The advantage of a tight jacket made of LCP is that no prestraining of the FBG is required. Compensation occurs by compression of the fiber. This arrangement is superior to the prestrained method, as the fiber is far stronger in compression than it is under tensile strain (see Table 10.1). There are many different passive variants of compensation schemes, but all work on a similar principle of using materials with different thermal expansion coefficients [49–51]. Thermal compensation is necessary to measure strain. A simple method to actively compensate the thermal drift is by placing the FBG on a temperaturecontrolled Pelletier cooler while it undergoes strain. As this is an active method and requires power, it is seldom a reasonable proposition, and the passive methods described earlier are usually more desirable. In Fig. 10.22, any external strain applied to the LCP jacket will induce a Bragg wavelength shift, which will be, to first order, thermally compensated. With temperature compensation, FBGs may be used to measure a variety of parameters, such as strain, magnetic fields, pressure, acceleration, and displacement. A significant advantage of the tight jacket is the ability to withstand compressive stress. This method has been used to tune Bragg gratings [52], by placing the FBG in a carefully engineered mechanical system, ensuring that the off axis strain applied to the FBG remains zero; the later condition is extremely important, as any deformation under large compressive strain can shear the fiber. The system may be used to measure repetitive strain under constant temperature. A maximum tunability of 45 nm has been reported, with a maximum tuning speed of 21 nm-ms1, but this has a limited life as repeated cycling of the device results in failure of the FBG. Oriented Liquid Crystal Polymer Tube
FBG
Fiber
Epoxy on Ends Figure 10.22 The loose tube [45] or the tight-jacketed LCP [43] (which has a negative thermal expansion coefficient) FBG thermal compensator.
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In an elegant system, an FBG is placed in a large-diameter cylinder and aligned to its axis of a fiber under tension [53–55]. Injection molding the cylinder with a plastic results in a perfectly aligned composite structure, which is easily compressed axially to tune the grating over 80 nm without fracturing the fiber. As the FBG is in intimate contact with the polymer, this device may be cycled without damaging the FBG over long periods. Another issue is repeatability of the tuning curve. Figure 10.23 shows the schematic of this device, and Fig. 10.24 shows the tunability of the FBG over many cycles. This system offers a simple and reliable solution to the measurement of dynamic compressive strain and is a good candidate for commercialization, although a reference grating is FBG embedded in large diameter polymer cylinder Compression
Compression
Figure 10.23 A compressive strain-tunable FBG embedded in a large-diameter tight polymer jacket. (Adapted from reference [53].)
100 90 80
Efficiency [%]
70 60 50 40 30 20 10 0 1070
1080
1090
1100
1110
1120
1130
1140
Wavelength [nm] Figure 10.24 The tuning characteristics of the reflection spectrum of an FBG imbedded in a polymer cylinder. (After reference [53]. Reproduced with permission from: Be´langer E., Bernier M., Faucher D., Coˆte D., and Valle´e R., “High-power and widely tunable all-fiber Raman laser,” J. Lightwave Technol. 26, 1696, 2008. # IEEE 2008.)
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still needed to compensate for the effects of temperature. Under static compression, the polymer demonstrates slight drift and hysteresis and is therefore more suited to dynamic loading.
10.1.10 Pressure and Loading Reference [56] discusses the effect of secondary polymer coatings on the elasto-optic response of an optical fiber. The principle of operation is based on a lateral force being translated into a longitudinal one through Poisson’s ratio [see Eq. (10.1.5)]. The longitudinal strain on an FBG is given by [57] e¼
ð1 2sFBG Þ DP; EFBG
ð10:1:27Þ
where EFBG is Young’s modulus of the fiber. The wavelength shift, DlB, as a result of hydrostatic pressure, DP, may be described as [58,59] " # 2 lB ð1 2sFBG Þ neff ð2r12 r11 Þ 1 DP: ð10:1:28Þ DlB ¼ 2 EFBG As the pressure sensitivity of an optical fiber is 3 pm-MPa1 at 1550 nm, which is quite low, it is necessary to attach the grating to a transducer to increase its sensitivity. This requires a secondary coating to be applied to the FBG. The pressure sensitivity can be enhanced by immersing the FBG in a polymer. The extension of the polymer because of the lateral hydrostatic pressure is far greater than the optical fiber, as Young’s modulus of the polymer is much lower than that of the fiber. Thus, the axial strain of the fiber surrounded in polymer can be calculated from [57] e¼
sp AP DP: LFBG AFBG EFBG þ ðAP AFBG ÞEP LP
ð10:1:29Þ
Here, the subscripts p and FBG refer to the polymer and FBG, respectively. A and L are the respective cross-sectional areas and lengths of the subscripted regions. With a given lateral pressure, the shift in the wavelength of the FBG is modified from Eq. (10.1.28) to DlB ¼ lB ð1 Pe Þ
sp AP DP: LFBG AFBG EFBG þ ðAP AFBG ÞEP LP
ð10:1:30Þ
The differential enhancement, which is the ratio of the strain seen by the polymer-coated and bare fiber from Eqs. (10.1.26) and (10.1.29), is
Principles of Optical Fiber Grating Sensors
468
0
1
B C sp AP B C @ A LFBG ðAP AFBG ÞEP AFBG EFBG þ LP Ds ¼ : 1 2sFBG EFBG
ð10:1:31Þ
Assuming that LFBG LP and AFBG AP, we get Ds ¼
sP EFBG : ð1 2sFBG Þ EP
ð10:1:32Þ
Using typical values of sP ¼ 0.4 and sFBG ¼ 0.16, and EFBG ¼ 7 1010 N-m2, EP ¼ 1.8 106 N-m2 [60] gives a value of DS 2 104! Thus, a theoretical sensitivity of 60 nm-MPa1 is possible. In reality, the realized value is approximately 33.8 nm-MPa1 as reported [60]. Figure 10.25 shows the response of such a device with a silicone rubber coating for pressure sensing. The reflection of the FBG is monitored as a function of the applied pressure and may be displayed easily on a spectrum analyzer. The problem with this sensing scheme is the large wavelength shift, as it is difficult to monitor large dynamic strain because the scanning frequency of the spectrum analyzer is limited. ΔP = 0 MPa, Δl= 0 nm ΔP = 0.02 MPa, Δl= 0.391 nm
–30
ΔP = 0.05 MPa, Δl= 1.321 nm ΔP = 0.1 MPa, Δl= 3.013 nm
Reflection (dB)
ΔP = 0.2 MPa, Δl= 6.507 nm
–40
–50
–60
1540
1542
1544
1546
Wavelength (nm) Figure 10.25 The pressure sensitivity of the reflectivity spectrum of a polymer-embedded FBG sensor. (From reference [60]. Reproduced with permission from: Sheng H.-J., Fu M.-Y., Chen C.T.-C., Liu W.-F., and Bor S.-S., “A Lateral pressure sensor using a fiber Bragg grating,” Photon. Technol. Lett. 16(14), 1146, 2004. # IEEE 2004.)
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469
y = 1539.6 + 33.876x R = 0.99909
Wavelength (nm)
1546 1545 1544 1543 1542 1541 1540 1539 –0.05
0
0.05
0.1
0.15
0.2
0.25
Pressure (MPa) Figure 10.26 Linearity of the response of the polymer coated pressure sensor. (From reference [60]. Reproduced with permission from: Sheng H.-J., Fu M.-Y., Chen C.T.-C., Liu W.-F., and Bor S.-S., “A Lateral pressure sensor using a fiber Bragg grating,” Photon. Technol. Lett. 16(14), 1146, 2004. # IEEE 2004.)
Figure 10.26 shows the linearity of the sensor over a measurement range of 0.2 MPa. Although this sensor has a very linear response, its application is limited to high-resolution, low-pressure dynamic measurements. For larger dynamic range measurements, a smaller wavelength shift should be used, as it is easier to monitor the amplitude of the reflected signal on the slope of the grating reflection spectrum. By using a signal at the midpoint of the reflected power on one edge of the grating spectrum, a pressure change simply modulates the reflected power. Monitoring the amplitude of the reflected power allows measurement of dynamic pressure variations at high speeds. Cross-sensitivity may also be an issue for such sensors, as the FBG is inherently temperature sensitive with a sensitivity of 11.7 pm- C1, although the very large pressure sensitivity of the sensor dominates the measurement. Higher sensitivity than the bare fiber can be achieved by housing the FBG in a glass bubble; an enhancement of the sensitivity from 3.1 pm-MPa1 [61] to 32.5 pm-MPa1 is reported by Xu et al. [62]. Although this sensor is suitable for monitoring dynamic strain as the shift in the Bragg wavelength of the FBG is small enough to be tracked using a filter and power detector, drift as a result of temperature makes it less suitable for static pressure measurements. As originally predicted [63], imposing a superstructure on a uniform FBG causes coupling to forward-propagating cladding modes. This prediction has since been confirmed in a number of experimental demonstrations [64–68]. Using this principle, it is possible to have an FBG reflection simultaneous with coupling to cladding modes, both in the forward- and counter-propagating directions. This allows unique possibilities for simultaneous axial, transverse strain, and temperature sensing, as the cladding and core modes have different
Principles of Optical Fiber Grating Sensors
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sensitivities to temperature and strain. By ramping the temperature of a superstructured FBG free of axial strain, the slopes for the shift in the FBG wavelength and the LPG resonance for the LP01 ! LP04 cladding mode coupling are measured to be A ¼ 0.01 and C ¼ 0.04 nm C1, respectively, over a range 20–140 C. Then, at a fixed temperature of 20 C, the coefficients for pure axial strain are determined to be B ¼ 8.04 104 and D ¼ 1.98 103 nmme1, respectively. Finally, transverse load has the effect of splitting the LPG coupling into two resonances dependent on x- and y-polarizations. The latter set of data shows a sensitivity of E ¼ 57.73 and F ¼ 71.76 nm(kgmm)1 for the x- and y-polarizations, respectively. These data may be used with the following set of equations to determine the three parameters uniquely: axial strain, e, temperature, T, and transverse load, L, in Kg/mm: 0 1 2 30 1 0 1 Dl1 A B 0 T T @ Dl2 A ¼ 4 C D E 5@ e A ¼ ½K@ e A; ð10:1:33Þ Dl3 0 0 F L L where, Dl1 ¼ lFBG – lFBG,0, Dl2 ¼ lLPG,x – lLPG,0, and Dl3 ¼ lLPG,x – lLPG,y. Here the subscript 0 indicates the FBG’s unperturbed Bragg wavelength and the LPG’s unperturbed resonance wavelength, whereas the subscripts x and y indicate the polarization splitting induced by lateral strain. Once the system is calibrated, the constant [K] ¼ (A. . . F) is known, and hence all three measurands may be measured simultaneously. A schematic of the system used to make these measurements is shown in Fig. 10.27. In a similar system, but with a transient LPG, a strong uniform FBG may be subjected to a periodic pressure using a loaded etched metal grating in contact with the FBG for the measurement of lateral strain without the effects of birefringence. A V-groove allows a three-point contact across the fiber surface, thereby rendering the lateral strain uniform. The long-period grating generates
Glass SFBG Superstructure Epoxy FBG
Dummy Fiber
Glass Load
Epoxy
Fiber OSA
ELED
Temperature controller
Translation Stage
Figure 10.27 Schematic of temperature, axial, and lateral strain sensor system. (Adapted from reference [64].)
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higher orders in the reflection spectrum as the pressure grating induces a sampled FBG. As the loading on the metal grating increases, so does the reflectivity of the first and higher orders of the sampling period. Using this system, pressure may be measured; however, it is difficult to conclude from the results what is exactly the sensitivity of the sensor. It is not clear if the loading on the grating allows for the fact that two fibers support the weight, in which case the loading should be halved. However, with a suitable normalization, it would be conceivable that such a sensor could be used in pressure sensing, although its long-term mechanical survivability may be of concern.
10.1.11 Chirped Grating Sensors These sensors use the property of a chirped grating in several ways. The use of a chirped FBG (CFBG) in a loop mirror, as shown in Fig. 10.28, was first demonstrated by Margulis [69]. A chirped grating placed in a loop mirror has a periodic reflection and transmission spectra as a function of wavelength, similar to those shown in Figs. 10.28 and 10.29, because the Michelson interfermoter with oppositely chirped gratings in each arm is equivalent to the Sagnac loop mirror with a chirped grating in the middle of the loop. If the CFBG is subjected to strain or temperature, the spectrum shifts. Thus, this arrangement can be used as an effective sensor of both quantities. To measure either strain or temperature, both the reflected and the transmitted spectrum spectra are used simultaneously. The reflected spectrum is a complement of the transmitted spectrum (i.e., R ¼ 1 T). An FBG with Bragg reflection near the center of the bandwidth of the CFBG is used as the sensor. Figure 10.29 shows this scheme. Because the reflection and transmission spectrum overlap but are complementary, a wavelength shift in the FBG causes
Chirped FBG
BBS O
T Wavelength, nm Spectrum Analyser
Fiber Loop Mirror
Figure 10.28 Chirped FBG loop mirror sensor. The transmitted spectrum has a broad region in the center of the CFBG bandwidth when it is placed symmetrically within the loop mirror. BBS is a broadband source. At point O, the output is a linear function of wavelength. (After reference [69].)
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BBS
Sensor OC
lo
PT – PR Pout
PT + PR
Fiber Loop Mirror
R/T
Chirped FBG
OC
PD Wavelength, nm
Figure 10.29 Schematic of a dynamic or static temperature or strain sensor system using an FBG with a Bragg wavelength close to the center of the temperature-stabilized CFBG. The transmission and reflection spectra are shown for descriptive purposes, as the photodiodes see only the reflected power from the Bragg wavelength of the FBG. The operating point for this detection system is at point O shown in Fig. 10.28. OC is an optical circulator. (Adapted from reference [70].)
the transmitted signal to either increase or decrease, depending on the exact wavelength of operation, with a reversed response at the reflected port. Positioning the FBG’s wavelength, say to the left of the broadest spectral variation in Fig. 10.28 (point O), results in two outputs varying in opposite directions as a function of the temperature or strain applied to the FBG. The fringe spacing of the loop mirror when the CFBG is placed exactly in the center of the loop is dl ¼
l2 2neff Lg
;
ð10:1:34Þ
where l is the FBG’s Bragg wavelength, neff is the effective index of the mode, and Lg is the length of the Bragg grating. Surprisingly, this bandwidth is not dependent on the chirp of the grating but simply on its length. Using such a system, a strain resolution of 4.2 me and a sensitivity of 2.3 (ne)1 have been achieved. These parameters are dependent on the bandwidth of the CFBG and the power of the BBS. An interesting aspect of this system is that it allows the measurement of dynamic strain. An ac-signal is detected at the output of the detection system when a dynamic strain is applied to the FBG. With a fast power normalization system (as indicated in Fig. 10.29), high-frequency measurements in the kHz region should be possible. However, it is important to temperature stabilize the CFBG for high-resolution measurement. A dynamic strain resolution of 0.406 me-Hz½ has also been demonstrated [70].
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10.1.12 Acceleration Acceleration is a parameter of fundamental importance in a number of applications such as in the control of vehicles, as well as in the monitoring of civil infrastructure. It is usually necessary to limit the impact on structures and to keep within the safe limits of operating moving objects. Sensing acceleration requires an inertial mass to be attached to the FBG, which imparts a transient strain when undergoing acceleration [71]. The methods used to assemble the sensor head depend on the acceleration force to be measured as well as the required dynamic range. A schematic of the accelerometer is shown in Fig. 10.30. The fiber and FBG are embedded in a soft compliant material resting on a rigid baseplate. A mass is glued on the top of the compliant material (elastomer). When the mass undergoes acceleration, the inertial force squeezes the compliant material, imparting a compressive or tensile force on the FBG. This single degree of freedom allows the acceleration to be measured in the direction shown (Dy) in Fig. 10.30. The displacement, Dy of the mass, m is given by m ð10:1:35Þ Dy ¼ a; k where, k is the spring constant of the elastomer, and a is the acceleration. The mechanical coupling factor, G, of the elastomer to the FBG and Poisson’s ratio, r, leads to the following shift in the Bragg wavelength of the FBG: Dl ¼ x G r Dy;
ð10:1:36Þ
where x is the wavelength shift of the grating per unit strain. With the interferometer, the transfer function is then simply given by the phase change resulting from the wavelength shift in the MZI unbalanced by an optical path, D, as Sensor head Dy
Mass
FBG
BBS OC
Fiber
Compliant material
Base plate PZT Stretcher
Output
MZI PD
Detection System
Figure 10.30 Acceleration measurement system using an FBG embedded in an elastomer (compliant material). The interferometer is modulated at 20 kHz by applying an ac-voltage to the PZT stretcher from the detection system. The signal bandwidth is below 2 kHz, limited by the resonance of the sensor head. (Adapted from reference [71].)
Principles of Optical Fiber Grating Sensors
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m D S ¼ A 1 þ n cos x G r 2p 2 a ; k l
ð10:1:37Þ
where v is the visibility of the interferometer determined by the bandwidth of the FBG reflection, and A is power in the received signal, dependent on the total loss through the system. Finally, the dynamic response of the accelerometer is limited by the resonance frequency of the mechanical assembly as rffiffiffiffi 1 m f ¼ ; ð10:1:38Þ 2p k below which the response has been shown to be well behaved. Calibration of the sensor is carried out with the measurement of a static load to give a specified displacement. This graph of displacement versus load is used to calculate the force when sensing acceleration. During detection, a high-frequency signal (20 kHz) is applied to the PZT to allow measurements at the shifted frequency. The system is then used to detect a signal at the repetitive frequency of the applied acceleration (200 Hz). Figure 10.31 shows the dynamic response to an applied sinusoidal signal. The resulting sensitivity of the system with a bandwidth of 3 Hz has been shown to be 1 mg-Hz1/2 [71]. By reducing the mass of the sensor head, a higher frequency response may be possible. However, the signal processing frequency of 20 kHz should also be increased by using either an electro-optic phase modulator or a higherfrequency PZT.
Output Signal, dB relative to 1g RMS
measurement bandwidth = 3Hz 10 Signal: 1g RMS@200Hz
0 –10 –20 –30
⎯
Noise: –1mg/√Hz
–40 –50 –60
0
100
200
300
400
500
Hz Figure 10.31 Detected signal when a 1 g RMS 200 Hz sinusoidal perturbation is applied to the FBG-based accelerometer. The noise floor limit is also indicated on the graph. (Adapted from reference [71].)
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10.1.13 Vibration and Acoustic Sensing FBGs can also be used for vibration sensing. This sensor works on the principle of the change in the Bragg wavelength induced by the vibration, when the FBG is placed intimately in contact with the vibrating object. The stressinduced change in the Bragg wavelength can be extremely small, but there are several simple methods for translating the wavelength shift to an amplitude change, for example, by using one of the two edges of the FBGs spectral shape, as shown in Fig. 10.32. A narrow band FBG is glued to the vibrating object, for example, a wing of an aircraft, or embedded in a turbine blade. Light from a stable single-frequency laser is reflected from one of the cut-on or cut-off filter edges of the FBG. The amplitude of the reflected signal is modulated because the Bragg wavelength shifts approximately linearly as a function of vibration. By amplifying the received signal, small vibrations may be detected. Recently this was demonstrated by gluing an FBG to the vibrating belly of an acoustic guitar to act as a sensitive pickup [72]. FBGs may be used for acoustic sensing of ultrasonic waves both in bulk materials and in liquids. They have also been used to monitor ultrasound in vivo. The scheme of operation is simple. The FBG is used with a single-frequency laser tuned to the full width at half maximum (FWHM) point of the reflection spectrum. Any vibration is then detected as a change in the amplitude of the reflected signal at the acoustic frequency. An important aspect for ultrasound detection is that the ultrasound wavelength must be twice the length of the FBG, as otherwise the signal is significantly downgraded. Although grating lengths of 0.5 mm are
1.0
0.5
Amplitude modulated output
Transmission, AU
Transmission, AU
1.0
0.5
Vibration Induced Bragg wavelength shift
0.0 1549.85 1549.87 1549.89 1549.91 1549.93 1549.95
0.0 1549.8
Wavelength, nm
1550.2
1550.6
1551.0
Wavelength, nm Figure 10.32 The transmission spectrum of an FBG with an inset showing the effect of the vibration on the reflected power.
476
Principles of Optical Fiber Grating Sensors
possible, the bandwidth is large and the slope of the edge of the filter is low. On the other hand, a longer grating, typically 3 mm, has a roll-off approximately 6 as steep, increasing sensitivity, but with a lower frequency limit. Typically, in aluminum, the acoustic velocity is 6000 ms1, which limits the frequency response with a 3 mm grating to around 2 MHz, but it can be much higher in liquids. To increase the sensitivity dramatically, a DFB FBG structure, which has an ultra-narrow bandpass, may be used. An in-depth discussion of applications in acoustic sensing maybe found in reference [73]. Homodyne and heterodyne techniques may also be used to measure small amplitude vibrations, as discussed in Section 10.1.5. The noise characteristics of such a sensor are limited by the wavelength stability and characteristics of the source, as any rapid shift in the emission wavelength translates into an amplitude-modulated output signal, as does the vibration of the FBG.
10.1.14 Magnetic Field Sensing with Fiber Bragg Gratings Magnetic field may be sensed with a number of temperature-compensated grating systems, shown in Figs. 10.19, 10.20, and 10.21. In this device, a magnet placed close to one end of the compensating device results in an attractive force on the invar as F ¼ kB, where B is the magnetic field to be measured and k is a force constant, which depends on the stiffness and other material properties and the shape of the compensator. From Eq. (10.1.3) the change in the wavelength is then given by the measured characteristics of the device as dneff þ a mjHj; ð10:1:39Þ DlB ¼ lB neff where m is the permeability of the invar, and a is the sensitivity of the length of the fiber to the magnetic field, H. By calibrating the device against a known magnetic field, the shift in the Bragg wavelength then directly indicates the applied magnetic field independently of temperature [74]. Knowing the set temperature of the compensator, a total deflection is then limited by the initial strain used for the thermal compensation range, which is around 1.2 nm. By placing the source of the magnetic field at a suitable distance away from the package, the sensitivity of the device can be adjusted. Note that the device responds to the magnitude of the magnetic field strength and not the sign, unless a magnet is attached to the end of the thermal compensator. However, care must be taken with this measurement as the placement of the FBG package does modify the magnet field.
Evanescent-Field Refractive Index Sensors
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10.2 EVANESCENT-FIELD REFRACTIVE INDEX SENSORS
10.2.1 Fiber Bragg Grating–Based Refractive Index Sensors To measure refractive index change, the transmission or reflection characteristics of a Bragg grating have to be affected strongly. For this to happen, the FBG in the core of an optical fiber must be exposed to the medium to be tested. Thus, if a cladding diameter is reduced by etching, the evanescent field of the guided mode can be accessed. For example, the refractive index difference between the core and the cladding of an optical fiber is 0.5–1 102. The maximum change in the Bragg wavelength, which could be measured with reasonable ease by dipping the fiber into a liquid, would be of order of 1550 102/ncore nm, before the fiber stops guiding. This is a shift of 10 nm/102 refractive index change equivalent to 1000 nm/RIU, where RIU ¼ refractive index unit. This estimate ignores the effect of the overlap integral of the evanescent field and the surrounding liquid, and in reality the sensitivity is much lower when the mode is well guided, and reaches a maximum when v 1.0, as shown in Fig. 10.2. The data in Fig. 10.33 show the dependence of the fractional power in the core as a function of the core-cladding index difference. The effective index of the mode is proportional to the fractional power in the core and is plotted for a nominal core radius in microns equal to the wavelength in microns (1.55 microns). As the effective index of the mode and hence the
1.0 Pcore/Ptotal
0.8 0.6 0.4 0.2 0.0 0.00
0.02 0.04 0.06 0.08 Core-cladding refractive index difference
Figure 10.33 The fractional power in the core as a function of the refractive index difference between the core and the cladding. This calculation ignores the thin glass cladding around the core. Note that the change in the Bragg wavelength is ~80% of the maximum achievable (i.e., 8-nm shift for a refractive index change of 5 103, for the example in the text). However, the refractive index interval for a unit change in the Bragg wavelength gets larger as the refractive index difference increases.
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Bragg wavelength also varies nonlinearly as a function of the refractive index of the sensed medium, the sensitivity is thus dependent on the exact value of Dn. The Bragg wavelength shift is not a linear function of the change in the refractive index for any given Dn. Therefore, when large values of refractive index changes are to be measured, the nonlinearity of the curve must be taken into account for increased accuracy. As the power extends into the cladding, any absorption in the sensed medium attenuates the reflected signal, and the measurement is then susceptible to amplitude variation and a wavelength shift when detecting the change in the refractive index of the sensed medium. The fractional power in the core as a function of the refractive index difference between the core and the cladding is shown in Fig. 10.33. This calculation ignores the thin glass cladding around the core. As may be seen, the change in the Bragg wavelength is 80% of the maximum achievable (i.e., 8-nm shift for a refractive index change of 5 103, for the example, in the text). However, the refractive index interval for a unit change in the Bragg wavelength gets larger as the refractive index difference increases. Over a limited regime, the sensitivity of this type of an FBG refractive index sensor is 200 nm-RIU1 around a Dn ¼ 0.01. The practical difficulties in making a sensor with a diameter of a few microns renders this type of a sensor impractical. Figure 10.34 shows a schematic of the evanescent-field FBG refractive index sensor.
10.2.2 Long-Period Gratings–Based Refractive Index Sensors Another method for an increased sensitivity is based on long-period gratings (LPGs). In this sensor, the cladding supports the very mode that will be affected by the sensed medium. As the coupling to the cladding modes with the LPG works on the difference between the guided mode’s and the cladding mode’s effective indexes, this type of sensor can be inherently more sensitive than the FBG-based sensor. The difference in the core and cladding mode indexes is of the order of the core–cladding refractive index step. As this is usually around 102, any change in the cladding mode effective index has a large impact on mode coupling. As it was shown in Chapter 4, the LPG is 100–1000 more
Sensed medium Input
Transmitted light
Optical fiber
Reflected light FBG in etched clad fiber Figure 10.34 Schematic of FBG-based evanescent-field sensor. The transmitted light may also be used to measure the refractive index of the sensed medium.
Evanescent-Field Refractive Index Sensors
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sensitive to perturbations than an FBG, and the LPG makes a good temperature, strain, bend, or refractive index sensor. The nature of mode coupling to specific modes of the cladding allows some interesting dependences, such as the opposite sensitivities of two modes to the same perturbation [75,76].
10.2.3 Surface Plasmon-Polariton Sensors Interest in surface plasmon-polariton (SPP) sensors has been fueled by the need for highly sensitive refractometers, essential for biosensing applications. An SPP is an electromagnetic wave bound to a metal surface surrounded by two dielectric layers. It may be considered as a metal waveguide (as the real part of the permittivity of the metal is large, but negative) except that the electric field decays exponentially from the surface of the metal into the dielectric layers. Choosing an appropriate angle of incidence, as in high refractive index prism coupling of waveguides, an SPP may be excited. Commercially used for refractive index sensing in a bulk prism arrangement, the SPP excitation instrument couples a free space in-coming beam to an electromagnetic wave bound to a thin metal surface, usually gold with a thickness of 10 nm, using a Kretschmann technique [77,78] as shown in Fig. 10.35. As the magnitude of the real part permittivity of the metal layer is far higher (though negative) than that of glass, angular selectivity ensures phase matching. Thus, at specific angles of the incoming beam, light is preferentially coupled to the metal layer and therefore suffers attenuation instead of being reflected. By measuring the change in the coupling angle, the refractive index of a sensed medium in contact with the metal layer may be measured. To ease coupling to the metal layer, a metal grating has also been used for phase matching as well as for supporting the SPP. Figure 10.35 shows a schematic of this device with a metal grating.
Q
Sensed Medium
Metal Grating
Prism
Output Beam Reflection with Angle Q
Input Beam Rotation
With liquid Figure 10.35 The rotation of the prism or the tuning of the wavelength results in a coupling to the SPP at a specific angle. Measurement of the change in the coupling angle (or wavelength) with and without the sensed medium results in the calculation of its refractive index. (Adapted from reference [77].)
480
Principles of Optical Fiber Grating Sensors
The beam is coupled from the bottom of the metal layer, while the sensed liquid is placed on the top surface. By treating the surface of the prism with receptors, the presence of specific bacteria or biomolecules can also be detected. These sensors are capable of discriminating a change in refractive index of 105, but as with all such sensors, temperature stability is of prime importance during the measurement. One of the problems with such a system is the mechanical nature of the device, as angular changes have to be measured precisely. Although this system is commercially available, a more economic and compact system is desirable, with fewer moving parts and a smaller size. Alternative methods with waveguides have been proposed that try to address the mechanical and size issues [21,79,80]. There are, however, several differences between the waveguide SPP sensors that affect the sensitivity of the device. The first difference is that it can exist as a pure SPP, which is simply the SPP mode bound to the metal layer decaying exponentially away from the surface into the dielectric medium on either side [80]. The second is that the hybrid SPP, which is a combination of both an SPP part and an oscillatory part, can exist in the dielectric medium as part of a propagating mode [79]. The latter has a substantial part of its energy in the dielectric, whereas the former has all its energy in the SPP – hence the pure SPP label. Periodic structures, such as Bragg and long-period gratings have been proposed to enhance the sensitivity of the SPP sensor by decoupling the phasematching requirements from the waveguide and SPP supporting structure. All commercially available SPP instruments operate in the visible part of the spectrum, as the SPP resonance is around 500–600 nm in gold or silver metal layers. Moving to longer wavelengths poses certain challenges, not least of which is the process of the excitation of the SPP. The inherent advantage of using infrared (IR) wavelengths is the larger dimensions as well as the availability of reliable lasers in the 1500 nm region, a legacy of the developments in the telecoms sector. This was first proposed recently [80]. The coupling to the pure SPP has also been analyzed for polished fiber without grating assistance [81]. Owing to this sensor’s potential applications in biomedical sensing and to fully appreciate the functioning of this sensor, it is necessary to study its action in detail and is described in Sections 10.2.4 and 10.2.5.
10.2.4 Guided Wave Surface Plasmon-Polariton Sensors A surface plasmon-polariton (SPP) is a surface electromagnetic wave that propagates along the interface between two media with the real parts of permittivity of the opposite signs (e.g., dielectric and metal) [82]. The SPP field components have their maxima at the interface and decay exponentially in both media [82]. The small penetration depth of the SPP makes it an ideal tool for refractive
Evanescent-Field Refractive Index Sensors
n1
SPP
n3
481
metal L WL
WL n2
WFM
n2 n3
A
SPP
n1
FWM metal L
B
Figure 10.36 Illustration of the SPP fiber sensors. n1 is the refractive of the sample, n2 is the guiding layer’s refractive index, and n3 is the refractive index of the cladding. (a) The guiding layer is a fiber core (structure I). (b) The guiding layer is a dielectric cylinder (structure II).
index sensing. SPP fiber sensors are constructed by modifying a traditional SPP planar sensor system based on the scanning of the wavelength of light or the incident light angle [83–86], and it is usually referred to as a spectrally resolved sensing system. An alternative system is an integrated SPP fiber sensor (ISPP FS), which is a compact all-fiber sensor without moving parts. In this scheme, the SPP is excited by resonance coupling of the guided fiber mode with a properly designed short-period fiber Bragg grating (SPG) imprinted in the waveguiding layer of the fiber. This sensor requires a specially designed optical fiber. The waveguide is the dielectric layer, which has the highest refractive index in the fiber, such as a fiber core in the case of a standard fiber or any other finite fiber layer. The fiber mode oscillates in a guiding layer and exponentially decays in all other regions. For example, the fiber core mode of the classical fiber is a special case of a guided mode. The ISPP FS may be realized in two different forms. In the first case, the waveguiding layer is the fiber core and the metal layer is deposited on the cladding surrounding the core as shown in Fig. 10.36a. The sensed medium is placed on top of this metal layer. In the second scheme, the fiber’s guiding layer is a dielectric cylinder. A thin metal layer is deposited on the inner side of this cylinder, creating a metal capillary for the sensed sample and is shown in Fig. 10.36b. These alternative schemes are similar in the way they operate, but not in the introduction of the sensed medium into the sensor.
10.2.5 Theory of the Surface Plasmon-Polariton The waveguide mode and the SPP are solutions of Maxwell’s equations with standard boundary conditions set at the different layers of the fiber [87]. The permittivity of the metal layer is modeled by the Drude formula [88]
Principles of Optical Fiber Grating Sensors
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" eðoÞ ¼ e1
# o2p ; 1 oðo þ iGÞ
ð10:2:1Þ
where e1 is the high-frequency value of e(o), op is the plasma frequency, and G is its damping rate. Using cylindrical polar coordinates (r, y, z) for the structure, the dependence on z, y, and time t enters only by means of the second derivatives @ 2/@z2, @ 2/@y2, and @ 2/@t2, therefore solutions are sought in which all f;p field components contain a common factor expðbf;p gm z þ igy iotÞ, where bgm is the propagation constant superscripted f and p, corresponding to the fiber’s guided mode and the SPP, respectively. g is the azimuthal mode number. Subscript m is used to distinguish the different solutions of the dispersion relation for a given azimuth mode number g. For cylindrical geometry, except for the special case of g ¼ 0, the modes do not have pure transverse E or H character. The most important characteristic of each layer of the structure is its phase parameters 2 u2i ¼ w2i ¼ k20 n2i bf;p ; ð10:2:2Þ gm where k0 is the vacuum wave-number, and ni is the refractive index of the i-fiber layer. The fields of the modes of the cylindrical layered structure involve Bessel functions of the real argument with the arbitrary constants, which are oscillatory in character for ui2 > 0, and Bessel functions of imaginary argument with arbitrary constants, which are asymptotically exponential for w2i > 0. These arbitrary constants can be calculated from the continuity of the Ez, Hz, Ef, and Hf field components at the boundary layers and from the condition that the power carried by each mode is normalized to 1 W. The dispersion relations for the guided mode and SPP in the layered fiber structure are obtained from the continuity conditions for electric and magnetic components at the The normalized boundary layers. propagation constant of the guided mode nfgm ¼ bfgm =k0 has to be found in the range maxðn1 ; n3 Þ < nfgm < n2 , whereas for the SPP, the normalized propagation constant has to be in the range npgm > n2 . Fiber Bragg gratings have been used extensively for mode coupling in fibers. Being imprinted into the waveguide layer a short-period fiber grating (SPG) can couple a forward-propagation guided mode to a backward-propagating SPP, thus exciting the SPP without mechanical manipulation. The efficiency of SPP excitation is dependent on the grating’s reflectivity and is simulated by coupled-mode theory as described in detail in Chapter 4. To achieve efficient SPP excitation, the phase-matching condition between the propagation constants of the guided mode and the SPP must be satisfied; this can be done by choosing a period of the SPG for any predetermined wavelength. To achieve a good coupling constant, the fiber parameters such as refractive indexes and
Evanescent-Field Refractive Index Sensors
483
the radii of the fiber layers, and the grating parameters such as a grating strength (s) and the grating length, must be also be chosen. The spectral location for the resonance of the guided-mode–SPP reflection is given by dco-p þ kff =2 ¼ 0;
ð10:2:3Þ
where dco-p ¼ ðbf11 þ bp11 2p=LÞ=2, and L is the period of the grating, kff is the guided mode self-coupling constant, described by the relation k0 n22 s kff ¼ 2Z0
ðr2
2p ð
dF 0
Erf
rdr jErf j2 þ jEFf j2 ;
ð10:2:4Þ
rm
Eff
and are the component of the guided mode’s electric field, and where Z0 ¼ 377 O. Because an untilted fiber Bragg grating has been used here, the only nonzero coupling constants are those between the guided mode and the SPP having the same azimuthal numbers [89] (see Section 4.7.1). The guided mode-SPP coupling constant is kpf 1111
k0 n22 s ¼ 4Z0
ðr2
2ðp
dF 0
rdr Erp Erf þ EFp EFf ;
ð10:2:5Þ
rm
where sn2 is the amplitude of the SPG refractive index modulation [89].
10.2.6 Optimization of Surface Plasmon-Polariton Sensors To optimize the design of the sensor, the grating period should be as large as possible, and the grating strength and length should be as small as possible, while fixing the reflectivity at 70%. Choosing parameters of a specially designed fiber, such as refractive index and radii of the layers, two important issues must be taken into account: First, the fiber structure has to operate in the single-mode regime for a predetermined wavelength in order to eliminate modal interference; second, the fiber structure has to be large enough to be manipulated by hand (i.e., the fiber diameter has to be reasonably large). For the structure shown in the Fig. 10.36a, the core diameter is 26 mm with the refractive index n2 ¼ 1.44072; the cladding diameter is 30 mm with a refractive index n3 ¼ 1.44. These parameters ensure a single mode regime for wavelengths in the vicinity of l ¼ 1.55 mm. For the structure shown in Fig.10.36b, the single-mode regime can be realized when the thickness of the guiding layer is 3 mm with the refractive index n2 ¼ 1.442, the diameter of the sampling hole is 14 mm, and the diameter of the cladding with the refractive index
Principles of Optical Fiber Grating Sensors
484
n3 ¼ 1.44 is assumed to be infinite, and therefore the outer dimensions allow easy handling. The grating period is not dependent on the length of the grating but is a function the grating strength s according to Eqs. (10.2.2) and (10.2.3) through the self-coupling coefficient kff. The grating period will reduce with increasing grating strength for fixed fiber structure parameters in the weak perturbation regime. The efficiency of SPP excitation (i.e., the grating reflectivity) increases with increasing grating strength and grating length. Thus, to keep the grating reflectivity at 70%, we can manipulate two parameters: the length of the grating, which does not influence the grating period, and the grating strength, which influences the grating period. However, the grating period is dependent strongly on the thickness of the metal layer (D), as a consequence of the change in the SPP’s propagation constant. For both structures shown in Figs. 10.36a and b, the dependence on the metal layer thickness is shown in Figs. 10.37a and b, respectively. Figure 10.38 illustrates how the sensitivity and therefore the wavelength corresponding to the maximum (peak) of grating reflectivity of the two structures depend on the refractive index of the testing media (n1).
2.05 Structure I
1.95 n11p
1.85 1.75 1.65 1.55 1.45 55
A
105
155
205
255
5 Structure II
n11p
4 3 2 1 0
B
100
200
300
400
Δ (Å)
Figure 10.37 Plasmon-polariton propagation constant np versus the metal layer thickness D for structure I(A) and structure II(B).
Evanescent-Field Refractive Index Sensors
485
Wavelength (microns)
1.553 (a)
1.5525
Structure I
(b) 1.552
(c)
1.5515 1.551 1.5505 1.55 1.33
1.332
1.334 n1
1.336
1.338
Wavelength (microns)
1.5512 (a)
1.5510
Structure II
(b) 1.5508
(c) (d)
1.5506 1.5504 1.5502 1.5500 1.33
1.331
1.332 n1
1.333
1.334
Figure 10.38 Wavelength corresponding to the maximum of grating reflectivity (70%) versus the refractive index of surrounding media ns for both structures I(A) and II(B) shown in Figs. 10.36a and b, respectively.
The device sensitivity is for several metal layer thicknesses in the range ˚ –150A ˚ is shown in Figs. 10.23a and b. By reducing the thickness of the 80A metal layer, the device sensitivity can be increased, but at the same time in order to maintain the high-efficiency excitation of the SPP, the grating length and the strength of the grating must be increased; as a consequence, the grating period changes slightly. The parameters of the structures presented in Figs. 10.33a and b are given in the Table 10.2. Figures 10.38a and b show the dependence of the shifts in the grating’s reflectivity peak for a change in the refractive index of 105 of the sample media around a nominal refractive index n1 ¼ 1.33 (water) versus the metal layer thickness for structures I (a) and II (b), respectively. The sensitivity increases from approximately 1.5 to 3.0 pm per 105 change in n1, for structure I and 1.5 to 2.8 pm per 105 change in n1 for structure II, when the metal layer thickness
Principles of Optical Fiber Grating Sensors
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Table 10.2 Critical dimensions of the two SPP sensors for optimizing their response
Structure I
Structure II
(a) (b) (c) (a) (b) (c) (d)
L (cm)
L (mm)
s (104)
˚) D (A
9 5 4 7 6 5 4
420 469 496 477 497 509 519
4 2 1 5 3 2.3 2
100 130 150 80 100 120 150
˚ . The sensitivities of the structures are comparable decreases from 150 to 100 A for approximately equal metal layer thicknesses. The transmission spectrum of the grating can be used as an input signal for interrogating the sensor for both schemes. Because losses in the materials of the structures and coupling with the radiation modes have not been taken into account, the transmission of the grating is simply T ¼ 100% R. The bandwidth of the reflectivity spectrum at the first zeros is inversely proportional to the grating length as in the case of standard fiber gratings for counter-propagating resonance reflection (see Chapter 4). Assuming that a change in transmission of 0.01 dB can be measured results in a resolution of 1 ppm in refractive index change for both sensor structures. However, as with all SPP devices, temperature dependence of the refractive index is a factor that must also be taken into account. These two structures are different, as the locations of the testing region are inverted. In the first structure, the testing material surrounds the sensor on the outside, as it covers the open metal layer separated from the guided mode (in this structure, the guiding region is the large fiber core) by the thin buffer. This sample material can be infinite in volume. In the second structure, the testing material is located in the capillary tube formed by the metal layer deposited on the inner side of the waveguide along the fiber axis. The advantage of each of these structures depends on the applications. The metal layer thickness can be used for the control of the sensitivity of the sensor. The amplitude of refractive index modulation (grating strength) can be used for the optimization of the grating period. Increasing the grating strength reduces the grating period. However, both the grating length and the grating strength have an influence on the grating reflectivity. Increasing the grating length as well as the grating strength increases the grating reflectivity. Because the sensor monitoring process is based on examining the transmission spectrum of the grating, it is useful to keep the grating reflectivity at 70%. This may be realized by properly choosing the grating strength and the grating length;
Evanescent-Field Refractive Index Sensors
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however, increasing the grating strength reduces the grating period, which may be undesirable. Comparable resolutions can be achieved with these two fiber schemes. Experimental verification of surface plasmons excited by tilted FBGs has been reported [90]. In this sensor, a standard fiber with a tilted FBG inscribed in the core is coated with a thin layer of gold (10–20 nm). The tilt of the grating couples the guided mode to counter-propagating cladding modes, each of which has a distinct angle of propagation. At the appropriate angle for phase matching to the “pure” SPP in the gold layer, additional resonances are seen in the transmission spectra. These resonances are strongly affected by the refractive index of the surrounding medium. The wavelengths of the resonance peaks do not change, as these are defined mainly by the geometrical dimensions of the cladding and only very weakly by the refractive index of the surrounding medium [91]. The process of coupling to the SPP is twofold. First, the tilted FBG couples the guided core mode into the cladding modes, whose propagation constants are slightly modified by the gold metal layer. Because the formation of the cladding mode is governed by the reflection at the cladding-air interface and the gold layer simply enhances the reflection, the wavelengths at which the cladding mode resonances occur remain substantially unchanged. Thus, when a liquid is introduced on the surface of the fiber, the resonance condition for the cladding mode is not affected to any large degree; however, the phase-matching condition to the surface plasmon is strongly affected. Second, if there is no allowed cladding mode with the correct propagation constant to match the SPP’s phasematching condition, the SPP is not excited and the coupling shifts to the wavelength that closely matches it. Owing to the large number of cladding mode resonances, the distribution of energy between the modes makes it difficult to measure the refractive index of the surrounding medium accurately, as the peak wavelength for coupling to the SPP moves unpredictably with refractive index. The measurement of the refractive index of a surrounding liquid using a standard fiber and a tilted FBG is shown in Fig. 10.39. The weakly tilted grating results in a bandwidth of approximately 3.8 degrees, with a mean coupling angle of approximately 78 degrees (see Section 4.7 and Fig. 4.19). Comparing this scheme to the one in which the guided core mode has a substantial overlap with the SPP described earlier in Section 10.2.4, whether by tilted or untilted FBGs [80], the latter scheme should have a continuous and unambiguous measurement capability with high precision. This is because the coupling is directly between the core mode and the pure SPP and does not depend on an intermediate coupling to a cladding mode, which may or may not couple to the SPP owing to the mismatch in the phase-matching condition, resulting in an error of approximately 2.5 103. This is large when a required resolution in refractive index measurement is 105.
Principles of Optical Fiber Grating Sensors
Effective index of plasmon
1.435
lp = 454.49nD + 882.64nm
1542
1.430
1540
1.425
1538 1536
1.420 1534 1.415
1532
1.410
1530
Plasmon wavelength (nm)
488
1528 1.420 1.425 1.430 1.435 1.440 1.445 1.450 Refractive index of exterior medium, nD Figure 10.39 The refractive index of the surrounding medium measured by the standard fiber tilted grating SPP sensor. The graph shows an error in the measurement of 2.5 103 because of the two-step mechanism to couple to the SPP from the core-mode of the fiber. (Reproduced with permission from: Shevchenko Y.Y., and Albert J., “Plasmon resonances in gold coated tilted fiber Bragg gratings,” Optics Letts. 32(3), 211–213, 2007. # OSA 2007.)
An alternative system for exciting the SPP has also been implemented [92]. In this system, an optical fiber is butt in and outcoupled to and from a thin metal grating in a film of gold or copper. A schematic of this sensor is shown in Fig. 10.40. The grating in the metal film assists with phase matching to the SPP directly from the butt-coupled mode from the optical fiber. Efficient coupling is possible at a given wavelength, which is strongly affected by the refractive index of a surrounding medium. This simple scheme is easily fabricated on a disposable dielectric substrate for routine refractive index measurement of liquids placed in contact with the metal grating. However, in their
Thin metal grating sandwiched in dielectric Input optical fiber
Output optical fiber
Figure 10.40 A schematic of an SPP metal grating sensor fabricated on a dielectric substrate, excited by a butt-coupled optical fiber. The transmission spectrum shows a strong dip at the SPP phase-matching wavelength. (Adapted from reference [92].)
Long-Period Grating (LPG) Sensors
489 Liquid
Input optical fiber Thin metal grating on silica
Output optical fiber
Figure 10.41 Possible SPP refractive index sensor with thin metal film grating [93].
demonstration, the metal grating was sandwiched between two dielectric layers, making it impossible to access the SPP on the metal grating. By not depositing a polymer film on the surface of the grating, it would be possible to measure the refractive index by placing a drop of liquid on it, as shown in Fig. 10.41. Self-referencing in SPP sensors can be done using phase information. In this approach, one arm of a Mach–Zehnder interferometer is used as an SPR sensor, whereas the other arm in close proximity is used as reference. Any temperature change is therefore seen by both arms and to a large extent is canceled. The sensing is achieved by the interference of the unchanged signal in the reference and the phase change sensed by the SPP. Using a grating to excite the SPP enhances the sensitivity of the SPP sensors, as the coupling is strongly de-tuned by the presence of the measurand. The dispersion of the phase is linear close to resonance and is therefore given the refractive index directly. However, the sensitivity enhancement is restricted to a region close to the resonance condition, as no coupling is possible outside the Bragg wavelength using the hybrid SPP mode [94] and with the pure SPP [95].
10.3 LONG-PERIOD GRATING (LPG) SENSORS Another class of sensors that depends on the coupling of light propagating in the core of an optical fiber to forward propagating cladding modes is based on long-period gratings (LPGs; see Section 4.7.2). LPGs have specific properties that make it possible to differentiate between strain and temperature, twist and bending. Combining FBGs and LPGs allows the determination of several parameters in addition, such as vibration and acceleration. Because an LPG with a given period has the ability to couple the core mode to many cladding modes, each at a different wavelength (see Figs. 4.26 and 4.28), it is easy to understand that the sensitivities of each of the modes to temperature and strain will be different. The temperature sensitivity of the coupling wavelength of the core mode to a given cladding mode, m, is dlm dlm d@neff dlm 1 dL ¼ þL ; dT d@neff dT dL L dT
ð10:3:1Þ
Principles of Optical Fiber Grating Sensors
490
m core m where @neff ¼ ncore eff neff . Here, neff and neff are the effective mode indexes of the core and the mth cladding modes, respectively. Similarly, for sensitivity to strain,
dlm dlm d@neff dlm ¼ þL ; de d@neff de dL
ð10:3:2Þ
where, e is the applied strain, and L is the period of the grating. Clearly, either Eq. (10.3.1) or Eq. (10.3.2) may be equated to zero. As the two terms on the right-hand side (RHS) of both equations can have opposite signs, either positive or negative, it is possible to achieve either low sensitivity to either strain or temperature for one of the cladding mode resonances. For the modes on either side of this resonance, the sign of the sensitivity usually changes. Thus, it may be possible to attain a large positive or negative slope for the wavelength shift with strain. Figure 10.42 shows the shift in the wavelength of several cladding mode resonances in standard single mode fiber for a grating period of 280 mm as a function of temperature. Note, too, that the FBG sensitivity is also shown as curve “E.” The sensitivity for curve “A” is 93 pm- C1 [96], approximately an order of magnitude larger than for the FBG (11 pm- C1). Using an LPG with the mode “D,” it is possible to measure temperature independently of strain (Fig. 10.43); whereas an FBG in the same fiber allows the measurement of strain simultaneously (“E”), however, its temperature dependence (curve “E” in Fig. 10.42) has to be accounted for. 12 A
Wavelength shift (nm)
10 8 6
B C D
4 2
E
0 20
40
60
80
100
120
140
160
Temperature (°C) Figure 10.42 Wavelength shift as a function of temperature for four cladding modes (“A”– “D”) and an FBG (“E”). (From reference [96]. Reproduced with permission from: Bhatia V., “Applications of long-period gratings to single and multi-parameter sensing,” Optics Exp. 4(11), 457, 1999. # OSA 2007.)
Long-Period Grating (LPG) Sensors
491 A
6
Wavelength shift (nm)
5 4 E 3 2 1
B
0
C D 0
1000
2000 Micro-strain
3000
Figure 10.43 LPG resonance wavelength shift as a function of strain for the modes (“A”– “D”) shown in Fig. 10.39. Also shown is the strain dependence of an FBG (“E”). Note the near insensitive response for modes “B” and “D.” (From reference [96]. Reproduced with permission from: Bhatia V., “Applications of long-period gratings to single and multi-parameter sensing,” Optics Exp. 4(11), 457, 1999. # OSA 2007.)
Figure 10.44 shows an example of the temperature insensitivity of an LPG written into a Corning Flexcore fiber with a period of 40 mm with a resonance wavelength at 1141.5 nm. In this demonstration, the temperature is varied between 23.5 C and 127.2 C as the strain is applied to the fiber and remeasured. The total shift in the wavelength over a 100 C change in temperature is measured to be only 0.18 nm. The error in the strain measurement under these conditions is found to be 41me over the entire 127 C temperature change. The strain coefficient of this LPG sensor is measured to be 21.44 nm-(%e)1. For temperature sensing independently of strain, another LPG with a grating period of 340 mm, and a resonance at 1257.9 nm, is fabricated, also in Flexcore fiber. The near completely insensitive strain coefficient for this cladding mode is measured to be 0.4 nm-(%e)1, less than 1/50 of the sensitivity of the temperature insensitive case. As the strain is varied between 0 to 2100 me, the temperature measurement is accurate to 2.2 C over the 110 C measurement range. Figure 10.45 shows the measurement of temperature over a period of time-independent strain. Using other choices of LPGs, it is possible to differentiate between bendinduced strain, twist, and linear strain simultaneously with temperature. In one sensor [98], a superstructure FBG written into a fiber results in a reflection as well as coupling to cladding modes through the period of the superstructure. The sensitivities of the LPG’s and the FBG’s wavelength
Principles of Optical Fiber Grating Sensors
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Wavelength (nm)
1142
1140
1138 23.5 °C 76.7 °C 127.2 °C 1136 0
500
1000 1500 Micro-strain
2000
2500
Figure 10.44 Strain measurements independent of temperature using an LPG. (From reference [97]. Reproduced with permission from: Bhatia V., Campbell D.K., Sherr D., D’Alberto T.G., Zabaronick N.A., Ten Eyck G.A., Murphy K.A., and Claus R.O., “Temperature-insensitive and strain-insensitive long-period grating sensors for smart structures,” Opt. Eng. 36, 1872, 1997. # SPIE 1997.)
150
Temperature (°C)
125 100 75 50 25 60
20
40
60 Time (min)
80
100
120
Figure 10.45 Measurement of temperature using an LPG during the application of strain varying between 0 to 2100 me over the same period of time. The temperature was cycled to demonstrate the capability of the sensor. (From reference [97] . Reproduced with permission from: Bhatia V., Campbell D.K., Sherr D., D’Alberto T.G., Zabaronick N.A., Ten Eyck G.A., Murphy K.A., and Claus R.O., “Temperature-insensitive and strain-insensitive long-period grating sensors for smart structures,” Opt. Eng. 36, 1872, 1997. # SPIE 1997.)
Applications of FBG Sensors
493
sensitivity are such that they provide sufficient information for bending, temperature, and strain data to be determined uniquely. Further characteristics of LPGs for sensing may be found in reference [99].
10.4 APPLICATIONS OF FBG SENSORS From the preceding discussions, it is clear that the FBG and LPG are excellent devices for a variety of sensing applications. The small size and in-line nature make them ideal for a large number of applications. Here we consider some of the applications of FBGs.
10.4.1 Biomedical Sensing: Hydrostatic Pressure Sensing in Medicine Applications in biomedical sensing have steadily increased, from simple temperature measurement through in vitro [100], to in vivo distributed temperature sensing [101], to, more recently, the measurement of hydrostatic pressure in spinal fluids for the treatment of lumbar deformities and injuries [102]. In the latter application, an FBG is inserted into the viscoelastic shell between vertebrae through a thin hypodermic needle, as shown in Fig. 10.46. A standard strain gauge sensor is simultaneously inserted as a reference to measure the hydrostatic strain as the lumbar column is loaded. The pressure is measured by the FBG as described in Section 10.1.10 using Poisson’s ratio. The principle advantage of Load on Spine
Annulus Fibrosus (Viscoelastic Shell) Optical Fiber
Vertebrae Strain Gauge Transducer
FBG
Hypodermic Needle Hydrostatic Pressure
Nucleus Pulposus (Fluid Center)
Figure 10.46 Hydrostatic pressure sensor using an FBG. (Adapted from reference [103].)
Principles of Optical Fiber Grating Sensors
494
the FBG is its small form factor, as the diameter of the hypodermic needle is only slightly larger than the optical fiber (0.2 mm), minimizing tissue damage and discomfort to the patient. This technique benefits from the fact that the small form factor of the sensors do not interfere with other diagnostic tools such as ultrasonic imaging, although they are still visible to x-rays for manipulation and monitoring.
10.4.2 Respiration Monitoring Chest strain can be used for monitoring patients, as breathing is extremely difficult to monitor. In this application, an FBG is attached to an elastic belt, which is held in position just above the breast. As the patient breathes, the thorax cage distends and deflates rhythmically, providing a method for monitoring respiration. In addition, the frequency of the signal can be used to trigger corrective action should the patient be under stress [103]. Figure 10.47 shows a schematic of the respiratory sensor.
Optical Fiber FBG
Elastic Belt Figure 10.47 FBG respiratory belt to monitor breathing in patients. (Adapted from reference [103].)
10.4.3 Oil, Gas, and Mining More recent applications are in the oil, gas, and mining industries. For example, it is necessary to know when an oil or gas pipeline might fail. Thus, a quasicontinuous monitoring procedure is necessary to detect impending problems. A distributed FBG system is attached to the pipe and continuously monitored for vibration and or strain. Weakening of the pipe structure is detected through a sudden change in strain or vibration signature. Preventive action is thus possible before major oil or gas leaks occur. Temperature monitoring is also important for oil wells. Here, the temperature requirements for FBG sensors are quite demanding as the temperature
Applications of FBG Sensors
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range is large, up to 350 C. Although the FBG may survive this temperature, it is difficult for packaging and housing to survive such a high temperature over extended periods. The qualification process is far more difficult than for telecom applications and therefore remains an expensive solution. Higher temperature applications are envisaged in nuclear industries, for which continuous monitoring at close to 1000 C is necessary. Unfortunately, most FBGs cannot survive these temperatures over long periods, and special measures have to be taken. Possible solutions for this application may be achieved with micro-LPGs [104], femtosecond laser-written gratings [105], chemical composition gratings (CCGs) [106,107], thermal treatment [108] in phosphosilicate [109], and nitrogen-doped fibers, which should survive >1000 C, although the fabrication process may render them fragile [110].
10.4.4 Structural Health Monitoring Other applications of FBGs include monitoring the curing of concrete, as it is important to know when the chemical reaction is completed [111,112]. A system of distributed FBGs was used to monitor the traffic and the structural health of a bridge [113]. In this demonstration, a 346-m road bridge in Norway was monitored over a period of 18 months with 32 surface-bonded in-fiber Bragg gratings. Calibrated against two 0.15 nm temperature stabilized gratings to give an absolute wavelength reference and the rejection of common mode noise, a long-term stability of < 10 me over the test period with a precision of 5 me and a resolution of 1 me was demonstrated.
10.4.5 Tilt Sensors A simple temperature-independent technique for three-dimensional tilt sensing was proposed and demonstrated by Au et al. [114]. In this scheme, four FBGs are symmetrically arranged in a cross, attached to a mass in the center. Figure 10.48 shows a schematic of this device. The FBGs are stretched and tethered at each end, supporting the mass in the center. The FBGs have their distinct Bragg wavelengths and are inscribed in a single optical fiber for easy interrogation. As the plane of the sensor is tilted, gravitational force generates a proportional component in the direction of the tilt. This compresses one FBG while tensioning the FBG on the opposite side. Thus, the one FBG moves to a longer Bragg wavelength, while the opposite moves to a shorter wavelength. A simple geometric analysis allows the tilt of the platform to be directly calculated. Figure 10.49 shows the cross-sensitivity of the sensor to a directional tilt in the “main-tilt” and “off-tilt” planes. The off-tilt Bragg wavelengths remain substantially unaltered, although a strong effect is seen in the Bragg wavelengths of the main-tilt FBGs. The reported sensitivity for this sensor
Principles of Optical Fiber Grating Sensors
496 z
Attachment
y
FBG 4 FBG 2
FBG 1
FBG 3
Platform
mg
x
θ
Figure 10.48 A three-dimensional tilt sensor incorporating four FBGs. The mass pushes or pulls the relevant FBG uniquely, preserving tilt information in the FBG Bragg wavelength. (From reference [114].)
X-Z TILT
Wavelength shift, nm
0.12 Main tilt plane (Increasing tilt angle) Off-tilt plane (Increasing tilt angle) Main tilt plane (Decreasing tilt angle) Off-tilt plane (Decreasing tilt angle) Theoretical Prediction
0.08
l = 2.82e-3 q – 0.0015
0.04
0.0 0
5
10
15
20
25
30
35
40
Tilt angle, degrees Figure 10.49 A measurement of tilt in the x-z plane, showing the directional sensitivity. A similar result is seen for a y-z tilt. (From Reference [114].)
is 39.5 pm/ of tilt, with a resolution of 47 arc-seconds and an accuracy of 3 minutes of arc. Indeed, it should be noted that this sensor is capable of detecting not only the magnitude but also the direction of the tilt.
10.5 CONCLUSIONS AND FUTURE PROSPECTS Fiber Bragg gratings are appearing in a variety of sensing applications. In fact, the market for sensors is now either matching or exceeding the volume of applications in telecommunications. Certainly, the complete FBG sensor
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system remains expensive so long as it is based on spectrally resolved sensing; however, simpler techniques for sensing using FBGs are being proposed, which are based on amplitude and transmission measurements using discrimination. Coherent techniques add tremendous value to the capability of sensing with FBGs; these have potentially unprecedented sensitivity and resolution with key advantages of electromagnetic (EM) immunity with their nonelectrical, remote, and distributed sensing capabilities. Problems that need to be addressed are the high temperature survivability of FBGs, as glass undergoes plastic deformation over a period of time. Although CCGs may hold the promise for high-temperature sensing, the higher-temperature processing required to fabricate these gratings may compromise the intrinsic strength of the fiber, making them less attractive for strain measurements, unless special measures are taken to recover strength. On the other hand, fs laser written gratings, although less durable, may still remain a good solution, as the integrity of the fiber is not degraded by the writing process. It is yet to be seen how microstructured gratings [115] may fit this application, as no data are yet available. The field of sensing with FBGs is certainly secure, growing rapidly, and appears set for a long, steady, and healthy future in an increasing number of areas.
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[100] Y.J. Rao, D.J. Webb, D.A. Jackson, L. Zhang, I. Bennion, In-fiber Bragg grating temperature sensor system for medical applications, IEEE J. of Lightwave Technol. 15 (5) (1996) 779. [101] D.J. Webb, M. Hathaway, D.A. Jackson, S. Jones, L. Zhang, I. Bennion, First in-vivo trials of a fiber Bragg grating temperature profiling system, J. Biomed. Opt. 5 (1) (2000) 45. [102] C.R. Dennison, P.M. Wild, D.R. Wilson, P.A. Cripton, M. Dvorak, Pressure sensor for biological fluids and use thereof, Patent application: WO 2007/09/5752 A1, Pub. Date: Aug. 30, (2007). [103] G. Wherle, H.J. Kalinowski, P.I. Torres, L.C. Guedes Valente, Fiber optic Bragg grating strain sensor used to monitor the respiratory system, in: Proc. SPIE 14th International Conference on Optical Fiber Sensors, 4185 (2000) 310. [104] See: Helica™ Platform Sensors. www.chiralphotonics.com, 2009. [105] S.J. Mihailov, D. Grobnic, C.W. Smelser, P. Lu, R.W. Walker, H. Ding, Induced Bragg gratings in optical fibers and waveguides using ultrafast infrared laser and a phase mask, Laser Chemistry, vol. 2008, Article ID 416251. doi: 10.1155/2008/416251. [106] M. Fokine, Photosensitivity, Chemical Composition Gratings, and Optical Fiber Based Components, Doctoral Thesis, Royal Institute of Technology, Stockholm, Sweden, 2002. [107] M. Fokine, Section 2. Optical properties: Underlying mechanisms, applications, and limitations of chemical composition gratings in silica based fibers, J. Non-Cryst. Solids 349 (2004) 98–104. [108] D.D. Davis, T.K. Gaylord, E.N. Glytsis, S.G. Kosinski, S.C. Mettler, A.M. Vengsarkar, Long-period fibre grating fabrication with focused CO2 laser pulses, Electron. Lett. 34 (3) (1998) 302–303. [109] V.I. Karpov, M.V. Grekov, E.M. Dianov, K.M. Golant, S.A. Vasiliev, O.I. Medevedkov, R.R. Khrapko, Thermo-induced long-period fibre gratings, in: European Conference on Optical Communication (ECOC), 22–25 Sept. 1997, Edinburgh, published by IEEE, London, Conf. Publication No. 448, 1997, pp. 53–56. [110] E.M. Dianov, V.I. Karpov, A.S. Kurkov, M.V. Grekov, Long-period fiber gratings and mode-field converters fabricated by thermodiffusion in phosphosilicate fibers, 24th European Conference on Optical Communication (ECOC), 1998, Volume 1, 20–24 Sept. 1998 (1998) 395–396. doi: 10.1109/ECOC.1998.732615. [111] J. Echevarria, C. Jauregui, A. Quintela, M.A. Rodriguez, R. Garcia, G. Gutierrez, et al., Concrete beam curing process and flexural test with fiber-Bragg-grating based transducers, Proc. SPIE 4694, (2002) 271. doi:10.1117/12.472629. [112] P. Moyo, J.M.W. Brownjohn, R. Suresh, S.C. Tjin, Development of fiber Bragg grating sensors for monitoring civil infrastructure, Engineering Structures 27 (12) (2005) 1828–1834, SEMC 2004, Structural Health Monitoring, Damage Detection and LongTerm Performance. doi:10.1016/j.engstruct.2005.04.023. [113] Y.M. Gebremichael, W. Li, B.T. Meggitt, W.J.O. Boyle, K.T.V. Grattan, B. McKinley, et al., A field deployable, multiplexed Bragg grating sensor system used in an extensive highway bridge monitoring evaluation tests, IEEE Sens. J. 5 (3) (2005) 510. [114] S.K. Khijwania, H.T. Au, H.Y. Tam, Distributed Bragg reflector fiber laser based tilt sensor with large dynamic range, International Conference on Fiber Optics and Photonics, Delhi, India, 13–17 Dec. 2008. [115] See www.Chiralphotonics.com for super twisted submicron pitch glass fibers: Helica™ sensors, 2008.
Chapter 11
Femtosecond-Induced Refractive Index Changes in Glass But now I am six, I’m clever as clever. So I think I’ll be six now for ever and ever. —A.A. Milne, “The End,” The World of Christopher Robin
11.1 LIGHT PROPAGATION IN GLASS Femtosecond (fs) laser processing of glass was a discovery first reported by Hirao’s group [1,2]. They reported that exposing silica to high-peak power femtosecond laser pulses leads to the direct writing of complex refractive index structures, such as diffractive elements, waveguides, and local refractive index modification. However, it should be noted that the changes in the refractive index of pure silica through the exposure to ultraviolet (UV) lasers and the ensuing chemical changes had been reported much earlier [3] and studied in depth [4,5]. The refractive index engineering of glass using femtosecond pulses has led to interesting possibilities, the main one being the ability to alter the refractive index in virtually any material, including crystals. One should note that although optical fiber normally has unprecedented transparency to light, it becomes opaque at elevated temperatures and reaches a self-destructive regime relatively easily. The latter discovery [6–8] showed that if a glass sample were heated, the absorption coefficient would increase sufficiently to allow the light to be absorbed in a distance of microns, leading to a thermal runaway through further self-induced heating. This process is accompanied by plasma generation, which indicates the elevated temperatures reached. In an optical fiber, the process takes on a different form, as the energy is trapped in the core. The plasma leads to the rapid absorption of light, fueling the process and the damage, and the plasma propagates toward the source of 503
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Femtosecond-Induced Refractive Index Changes in Glass
the radiation. There is some debate as to the source of the generation of heat required to sustain the plasma, as it was conjectured that the absorbed light generates insufficient heat [9]. However, simulations have shown that, indeed soliton-like temperature profiles are generated [8,10] based on measured data, which show that sufficiently high absorption is present to fuel the generation of the plasma at elevated temperatures. This shock wave of the traveling temperature profile and through fast quenching of the glass, as most of the heat is lost by radiation at the extremely high temperatures, renders the optical fiber highly fragile after the fracture and the fiber can be easily broken by bending. Figure 11.1 shows the result of heating with a fusion splicer the end of an optical fiber carrying 1 W of laser light. A propagating damage process is initiated in the core of the optical fiber, in which the only possible direction of damage propagation is toward the laser. The process has been called the fiber “fuse.” Figure 11.2 shows how rapidly the temperature of the core and the cladding of an optical fiber increases as a result of absorption through plasma generation. In this instance, the glass is heated to above 1100 C with the result that the laser power in the core is absorbed within a small volume. The temperature rise of the core occurs in a few microseconds, before the heat is diffused into the surrounding glass. Thus, thermally driven processes in glass have a time frame of microseconds. At timescales shorter than this, heat is given as energy to the lattice, without being dissipated. In the example given earlier, heat is being generated continuously. However, laser pulses that arrive at a rate slower than the diffusion time cannot generate heat cumulatively. Silica is normally transparent to light over the visible and near infrared (IR) spectrum. However, as has been known for a long time, the complex dielectric constant is a function of the input intensity. The response of the glass to intense optical fields leads to dielectric breakdown and catastrophic damage [12,13], at which point cavitations occur and voids can form within the bulk material. This
Figure 11.1 Damage sustained to an end of an optical fiber through the self-propelled mechanism. The damage is accompanied by a strong plasma emission. (Adapted from reference [11].)
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104 Core
˚K
103
102 Surface 101 10–5
10–4
10–3
10–2 10–1 Time, seconds
10–0
10
100
Figure 11.2 The self-induced heat generation in silica optical fiber. The two vertical lines indicate the time for the temperature to increase to 104 K from the initiation of the process (adapted from reference [7]). These calculations are for a mode diameter of approximately 8 microns. A spot size of 1 micron will reduce the time by a factor of 64 to achieve the same temperature. Silica will thus reach its melting point of 2000 C in a few microseconds.
type of breakdown is related to the third-order nonlinear susceptibility of glass, resulting in self-focusing, being the outcome of the transverse intensity profile of a focused beam. The center of the beam has higher intensity than its wings, leading to a weak refractive index profile, which closely resembles the intensity profile. The fast collapse of the propagating beam increases the intensity parametrically, leading to optical damage. The high intensities can lead to breakage of molecular bonds through intense ionization and the generation of heat, with enormous pressures building up within the bulk of the glass, resulting in crack formation. Self-trapping can occur via self-focusing when the input power in the beam reaches a certain critical threshold, Pcr. This may be understood as an interplay between diffraction and self-focusing, an equivalent of a transverse soliton. The balance between these two leads to the formation of a stable propagating beam, which is self-trapped, unable to diffract.
11.1.1 Theoretical Background To understand the phenomenon of nonlinear photo-induced refractive index change, let us recall that the spot size of a weakly diffracting Gaussian beam in a medium is o0 ¼
0:61l ; NA
ð11:1:1Þ
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Femtosecond-Induced Refractive Index Changes in Glass
where NA is the numerical aperture of the beam, and l is the wavelength, and NA ¼
0:61l : o0
ð11:1:2Þ
If we now consider the case in which the incoming beam induces a refractive index change in the medium, in order to overcome diffraction of this beam, we require a waveguide with the equivalent NA to trap it. The NA for a medium with a refractive index of n1 in the center surrounded by a refractive index, n2, is defined as 1=2 NA ¼ n21 n22 ; ð11:1:3Þ which may be further simplified as NA ¼ ð2nDnÞ1=2 :
ð11:1:4Þ
Here we have simplified the difference in the refractive index, Dn ¼ n1 – n2, and n is the average refractive index when the difference is very small. This NA is indicative of the strength of the trapping of the light in a material. We further recall that the change in the refractive index, Dn, as a result of the nonlinear intensity, I dependent refractive index term, n2, of a material is Dn ¼ n2 I:
ð11:1:5Þ
Introducing Eq. (11.1.5) into Eq. (11.1.4), we get NA ¼ ð2n n2 IÞ1=2 :
ð11:1:6Þ
Equating Eqs. (11.1.6) and (11.1.2), we get ð2n n2 IÞ1=2 ¼
0:61l ; o0
ð11:1:7Þ
from which I¼
ð0:61lÞ2 : 2n n2 o20
ð11:1:8Þ
The power is dependent on the profile of the beam, and in the case of the Gaussian trapped beam, p Pcr ¼ ð11:1:9Þ o2 I; 4 0 where Pcr is the critical power required to reach self-trapping. Combining Eqs. (11.1.8) and (11.1.9) leads to Pcr ¼
pð0:61lÞ2 : 8n n2
ð11:1:10Þ
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Power above this value eventually leads to self-collapse at a distance zsf from the input plane of the glass. This is described by the following approximate equation: zsf ¼
2no20 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi; l0 P=Pcr 1
ð11:1:11Þ
where P is the power in the beam when it is above the critical power. Having set out the scene for self-trapping and optical damage induced by the power in a laser beam, we can now appreciate the requirements for processing glass to fundamentally alter its properties. Using the coefficient of nonlinear refractive index of silica, n2 ¼ 3.2 1016 cm2-W1, the self-focusing threshold can be calculated to be 3 MW at a wavelength of 1 micron, equivalent to around 1014 W cm2 for a focal spot of 1 micron. This means that a single pulse with a peak power of this value will propagate spatially unaltered in silica, forming a spatial soliton. Power in excess of this value will lead to catastrophic damage through self-collapse of the beam depending on the NA, as will be seen later. It is the regime between these two values that is of interest in the laser processing for the refractive index modification of glass. For a 100 fs pulse, the required energy to process glass has to be in excess of a few mJ per pulse. Figure 11.3 shows the average power required to process silica for different pulse widths and repetition rates. Ignored in this graph is the effect of heat generation and accumulation, which is an increasingly important effect as the repetition rate increases beyond a few kilohertz. For a single pulse of a few femtoseconds, the nonlinear effect of self-focusing and near catastrophic dielectric
Average power, W
1.0E+04
10ps
1.0E+03
1ps
1.0E+02
100fs
1.0E+01
Power, W (10fs)
1.0E+00 1.0E−01 1.0E−02 1.0E−03 1.0E−04 1.0E−05 1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+07
1.0E+08
Repetition rate, Hz Figure 11.3 Average power required for a given repetition rate for different pulse widths. To the left of the vertical line, avalanche ionization is more likely, whereas to the right, thermal effects begin to have a significant effect.
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Femtosecond-Induced Refractive Index Changes in Glass
breakdown occurs almost instantaneously without the heat having the time to dissipate. Thus, the modification to the glass is accompanied by minimal heat generation. However, with increasing repetition rates, multiple pulses arriving in the same region cause the temperature to rise, leading to thermal stress and possibly cracking and serious damage. By moving the sample at higher speeds, the average exposure can be reduced. The speed of translation of the glass sample can be traded off against the repetition rate and energy of the pulse to use heat as another parameter in glass processing. For self-propagating damage using CW light as seen in Fig. 11.2, Dianov et al. [14] reported that this destructive effect in optical fibers leads to an increase in the refractive index of the glass adjacent to the damaged region [7]. Therefore, what are the processes that lead to the modification of the refractive index of glass at fs or ps pulses? In our picture presented earlier, we have only considered the linear absorption in glass as a function of temperature. When the energy of the photon is smaller than the band gap, no absorption is possible for CW light. For self-trapping and catastrophic damage, the nonlinear refractive index of glass leads to self-collapse by the local perturbation of the electron clouds. However, as the intensity increases through the action of focusing, several processes come into play. First, a nonlinear absorption term, dependent on multiphoton absorption, can become significant by which several photons can be absorbed simultaneously to excite valence electrons into the conduction band. When this happens, the absorption increases further and causes more photons to be absorbed. Under these circumstances, the electron accelerates with the electric field of the laser pulse and leads to avalanche ionization on a time frame of the femtosecond pulse. At the same time, there is also a probability of electrons tunneling from the valence band to the conduction band. However, these two processes compete, and avalanche ionization dominates for femtosecond pulses. With longer pulses, the seeding of the nonlinear absorption is through defects in glass and thus the process becomes unpredictable and nonuniform [15–17]. However, in a certain regime, the deposition of energy within the small volume of glass can be highly controlled with the use of femtosecond pulses and leads to repeatable avalanche ionization with multiphoton absorption. The effect of self-focusing is reduced by the generation of the plasma, as it has a negative nonlinear coefficient [4]. This can also lead to a very well defined micromachining of glass with the plasma. As seen before, the balance between self-focusing of the focused beam and de-focusing by plasma leads to a formation of a soliton, which can propagate without altering its transverse shape and is manifest by the appearance of a filament. When the absorbed energy becomes large, voids form [18] as seen in damage propagation [7], indicating that these effects have a similar source. The interplay between filamentation and self-focusing and their role in the induced refractive index change is discussed in detail in reference [19]. Figure 11.4 is a useful graph reported by
Light Propagation in Glass Structural damage Filament Breakdown Damage Multifilament
10 Threshold energy (mJ)
509
1
Smooth refractive index change Self-focusing
Pure FL Pure OB
No photoinduced change 0.1 10
20
30
40
50
60
70
Focal length (mm) Figure 11.4 Energy density diagram for processing glass with laser pulses. The shaded regimes indicate thresholds for various effects. The region bounded by pure optical breakdown (OB), physical damage, self-focusing, filamentation (FL), and multiple filamentation provides the sweet spot for modifying the refractive index in a controlled fashion. Below the self-focusing limit, there is no induced refractive index change. The diagram is for a wavelength of 1550 nm. (Courtesy of Dr. Re´al Valle´e.)
Salimina et al. [19] indicating the different regimes influencing material processing of glass. Between the two diagrams of Fig. 11.3 and Fig. 11.4, a choice between pulse width and energy may be made. Thermal effects can lower the threshold for micromachining of glass. Thus, the region of high-repetition rate must be used with care, as ablation may take place rather than refractive index modification, as a result of lattice heating during the time of the pulse or by the arrival of subsequent pulses before heat has had a chance to dissipate. The characteristic time for heat dissipation from the volume of the focused region is of the order of a microsecond; thus, one is generally limited to modifying refractive index at repetition rates of less than a few megahertz, unless alternative techniques are used, such as burst mode engineering [20]. In this technique, a high-repetition rate laser is used in a burst mode (i.e., a pulse picker is used to gate several femtosecond pulses from a train of continuous mode-locked high-repetition rate pulses. These gated pulses are focused on the glass periodically as it is being translated. The lowered threshold through controlled heating is used to modify the refractive index of the glass for short bursts so that heat does not accumulate sufficiently to lead to damage. The technique is therefore also suitable for inscribing Bragg gratings during the inscription of a waveguide, as the burst-mode pulses may be synchronized with the translation of the glass to inscribe a Bragg grating [21].
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Another consequence of the dependence of the refractive index change on the intensity of the fs pulses is that the shape of the grating is no longer sinusoidal. The fringe pattern is a cosine squared function; however, the refractive index change in the initial stages of inscription follows cosn, where n is a value that depends on the ratio the band gap of Germania and of the energy of the photon. Thus, for 800-nm radiation (1.55 eV), n ¼ 5, and for 1300 nm, n ¼ 9 photons. It is therefore clear that the use of fs 1.3 and 1.5 mm radiation for refractive index engineering is difficult, unless it is a cascaded process with a minimum of 6 photon absorption into the 240-nm band. However, at high intensities, self-focusing dramatically increases the intensity at the focus, allowing multiphoton processes to be triggered. Smooth refractive index change is possible by the use of low NA lenses at energies above the filamentation threshold (see Fig. 11.4). Shorter wavelength light appears to generate unstable refractive index change compared to lower-energy photons, presumably because it is much harder to reach optical breakdown through direct multiphoton absorption into the band gap with lower energy photons. The use of 1.3-mm radiation for refractive index modification has been reported with 36 fs pulses [22]. Figure 11.5 shows the number of photons required for three different energy gaps in glass: 5.6, 7.6, and 9 eV for five different photon energies. It is clear that with longer wavelengths, damaging the glass through multiphoton ionization becomes quite difficult. However, nonlinear processes such as self-focusing are not avoidable, leading to more stable conditions for inducing controllable refractive index modification, compared to pulse breakup and multifilamentation formation 1.5 1.3 1.0 0.8
0.5
Wavelength, microns
0.24
12
Number of photons
10 9eV process 7.6eV process 5.6eV process
8 6 4 2 0 0
1
2
3 Energy of photon, eV
4
5
6
Figure 11.5 Energy of the laser photon versus the number of photons required for multiphoton ionization for three band gaps commonly considered in Germania-doped silica glass. Very tight focusing and high peak powers are necessary to initiate optical breakdown at long wavelengths.
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with higher-energy photons. It should be noted that weak plasma, an indication of multiphoton absorption, is generated when the electric fields at the focus (or close to the focus) are large enough to strip some of the electrons from the atoms. When this happens, the intensity of the beam is clamped at ITH, since those photons with a high-enough intensity are absorbed and removed from the pulse. As the focusing beam propagates further, the beam readjusts, with more photons attaining the threshold for nonlinear absorption, which are also removed. Thus, the beam may propagate without ever self-focusing. When the intensity increases further, depletion of the peak part of the pulse continues to self-stabilize the intensity as a result of nonlinear absorption. Just above threshold, insufficient plasma is generated to significantly defocus the beam [23]. A schematic of the depletion model is shown in Fig. 11.6. This type of self-limiting behavior is independent of whatever the mechanism for the depletion, whether it is multiphoton or avalanche ionization [23]. This then contributes to the blue fluorescence indicative of plasma, remains below the onset of supercontinuum generation, and is close to the threshold of refractive index modification above this intensity (see Fig. 11.4). The interplay among optical breakdown, supercontinuum generation, fluorescence, and refractive index change has been discussed in several publications [19,24–26], and Hirao
Direction of propagation & focus
Lens ITH
Figure 11.6 Focusing of an fs laser beam, which is clamped due to the intensity at the peak attaining the threshold for multiphoton absorption. (Adapted from reference [23].)
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Femtosecond-Induced Refractive Index Changes in Glass
et al. [27] offer an interesting review of possibilities with fs laser processing of materials. The model by Rayner et al. [23] is based on self-limiting absorption. In their experiments, they showed clearly that the intensity of fs pulses decreases with increasing intensity when the beam is focused within the glass (i.e., as the intensity increases), and the transmission reduces, being clamped at some value given by the onset of nonlinear absorption, independent of the point of focus within the glass. They also note that the absorption begins to saturate for very high intensities 2 1014W-cm2. To ascertain the onset of nonlinear absorption instead of self-phase modulation, they observed that continuum generation was noticeably absent at these intensities; rather a blue shift was seen, an indication of a weak plasma generation.
11.1.2 Point-by-Point Writing of Fiber Bragg Gratings with Femtosecond Lasers First, the fs laser is by definition a broadband source and therefore is at the extreme end of spectral incoherence for lasers. Thus, all the limitations that prevail for sources of low coherence are that much more important when the fs laser is used for writing FBGs. For IR lasers, point-by-point writing with a diffractionlimited spot is not sufficient to allow a first-order grating to be inscribed unless a UV femtosecond source is used. Assuming that the spatial beam is TEM00, one is limited to a spot size of the order of the central wavelength of the laser. For Ti: sapphire lasers, this is around 0.8 mm, and 1.55 mm for the fs erbium fiber laser. Given that the period for the grating for use at 1.55 mm is 1 micron, it is necessary to write higher-order gratings to overcome the focusing limitation. Thus, a second or higher-order grating may be written with a period of 2 mm, comfortably attainable by the lasers. Second, the broadband nature of the pulse requires that sufficient energy is available despite pulse broadening through dispersion in optical components. Third, the tight focusing limits the refractive index change to the focal area, which is generally much smaller than the core diameter of the fiber [30,31]. Thus, some kind of a lateral scanning mechanism has to be employed to give a uniform exposure for each period, also requiring multiple pulse exposure for each period. Figure 11.7 shows the experimental setup to write gratings with the tightly focused fs laser pulses. At a repetition rate of 1 kHz energy per pulse of 0.2–2 mJ, and with a spot size of a few microns, a lateral scanning rate of 0.05 Hz is used with a piezoactivated translation of 10 mm to create a uniform refractive index change across the core and the cladding to reduce the cladding mode loss. A faster scanning rate of 8 Hz with a NanoAutomationW stage (P-752) and a 50% chopper
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Lens
Cladding fs laser beam
Core
Piezo oscillator Figure 11.7 Schematic of a point-by-point writing scheme, showing the translation of the focusing lens to expose the entire core with fs laser pulses. The scanning piezoelectric transducer moves the lens and hence the beam across the core of the fiber to induce a uniform refractive index change.
operating at 950 Hz to reduce the average power from an 800 nm, 115 fs, 1 kHz repetition rate Ti: sapphire laser has also been used for grating fabrication in ZBLAN fiber [28].
11.1.3 Femtosecond Laser Writing with a Phase Mask The small coherence length of the fs laser poses some constraints on the writing of FBGs with a phase mask. First, it is not possible to write FBGs with a standard tunable Talbot interferometer by scanning (see Figs. 3.5, 3.7– 3.8), because the coherence length is too short to give a uniform visibility over the length of the grating more than a few microns long. The different wavelength components of the fs pulse diffract at different angles and therefore do not overlap at the same point at the fiber. Second, tuning the interferometer increases the walkoff between the wavelength components and thus cannot produce good visibility fringes. However, the phase mask may be used to scan a grating into a fiber by placing it directly in front of the fiber (see Fig. 3.12 and Fig. 11.8). In this method, the zero and first diffraction orders of the fs beam de-phase in a very short distance (coherence length) and cannot interfere if the fiber is placed just a millimeter behind the phase mask. With a higher-order phase mask, this lack of multiple interference is ensured and the first orders overlap, being temporally coherent, and interfere at the fiber. Simulations show that immediately behind the phase mask, multiple interferences take place between all diffraction orders [see Fig. 11.9a] but disappear
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Input beam
Piezo-movement
Phase mask
Optical fiber
Distance L Cylindrical lens Figure 11.8 Focusing arrangement for using fs laser pulses with a phase mask. The distance between the lens and the mask should be sufficient to avoid damage. The cylindrical lens is scanned slowly when tight focusing is used.
in less than 1 mm to produce clean fringes from the interference between the 1 orders. The advantage of using a higher-order phase mask [e.g., N ¼ 2 in Eq. (3.1.4)] is that the angles for the 1 diffraction orders are small [see Eq. (3.1.2)] in the UV ( 7 ) and therefore the beams continue to overlap much farther away from the phase mask, easing the placement of the fiber [29]. However, at a wavelength of 800 nm, the angle increases to 23 . This requires the use of wider beams to maintain the overlap between the orders. Different diffraction orders emerge behind the phase mask at different angles and thus have different effective velocity in the direction of the incident laser beam. Thus, the orders only fully overlap immediately behind the phase mask. Further away from the mask, the pulses of different orders cannot overlap, as they are temporally too short. These result in the overlap of only the 1 orders at the correct distance away from the mask, which have high enough intensity to write the grating, giving a clean fringe pattern, shown in Fig. 11.9b. The direct result of this type of inscription is shown in Figs. 11.10a and b for the two orthogonal axes relative to the laser beam [when viewed at right angles (a) or along the direction (b) of to the inscribing beams]. One notes the clean fringe pattern extending across the core, with the grating penetrating deeper in the direction of the beam [Fig. 11.10a], compared to the orthogonal direction [Fig. 11.10b]. The width of the fringe pattern can, of course, be changed, if the beam is scanned across the fiber core, because the beam is focused into a small spot. Thus, by placing the fiber a certain distance away from the mask, only single diffraction orders can interfere, leading to clean uniform fringes. In Fig. 11.9b, one can see that after a distance of approximately 1 mm away from the phase mask, interference between the 1 orders is maintained, so long as the beams are wide enough to overlap at the fiber. There is always a loss of fringes at the edges of the beam, because the angular dispersion of the orders confines the interference to the half-diamond-shaped region, as is shown in Fig. 11.11.
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Normalized amplitude
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 Dis 1 tan 0 ce a (m long –1 ma m) sk
150
–2
100 50 mask m e fro c n a t Dis (mm)
0
Normalized amplitude
A
0.5 0.4 0.3 0.2 0.1 0 +2 Dis +1 tan 0 ce alo (m ng –1 m) ma sk
B
1150
–2
900
950
1000
1050
1200
1100 ask
mm e fro c n a ) Dist (mm
Figure 11.9 Interference fringes as a function of distance away from a 2 micron pitch phase mask, using 150 fs pulses. (a) The fringes formed by the interference of multiple orders as a function of distance from immediately behind the phase mask. (b) The multiple orders walk off and only the 1 orders interfere as the distance from the phase mask approaches 1 mm. (From Reference [29]. Reproduced with permission from: Smelser C.W., Mihailov S.J., Grobnic D., Lu P., Walker R.B., Ding H., and Dai X., “Multiple-beam interference patterns in optical fiber generated with ultrafast pulses and a phase mask,” Opt. Lett. 29(13), 1458–1460, 2004. # OSA 2004.)
Femtosecond-Induced Refractive Index Changes in Glass
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8.2 mm core 1.6 mm pitch
A
Laser beam
∼ 50 mm 8.2 mm core 1.6 mm pitch ∼ 28 mm
B
Figure 11.10 Microscope images of photo-induced index modulation in SMF-28 fiber produced with 125 fs, 800-nm laser pulses, with the fiber 3 mm from a 3.213 mm phase mask. (a) Fiber is imaged near the core and is viewed normal to the beam axis. (b) Fiber is rotated 90 and is viewed along the beam axis. (From Reference [30]. Reproduced with permission from: Grobnic D., Smelser C.W., Mihailov S.J., Walker R.B., and Lu P., “Fiber Bragg gratings with suppressed cladding modes made in SMF-28 with a femtosecond IR laser and a phase mask,” IEEE Photonics Technol. Lett. 16(8), 2004. # IEEE 2004.)
–1st order –2nd order
+1st order
Dead zone Dead zone
+2nd order
Fiber Single fringe region Phase mask Multiple fringe region Fs beam Figure 11.11 Grating inscription directly behind the phase mask, showing the regions of multiple and single fringe formation. Also shown are the dead zones on the either side of the diffracting beams. Depending on the coherence length of the source, additional fringes may form if the pulses overlap within the coherence length. The zero order and the diffraction orders >2 have been neglected.
It should also be noted that a small region exists on each end where the two orders do not overlap, but so long as the intensity of each order is below the threshold for inducing a refractive index change, it does not affect the inscription process.
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The situation for fringe formation changes dramatically if the pulses are long (e.g., a nanosecond), as the “length” of the pulse is long enough to allow overlap with sufficient coherence over a much deeper distance behind the phase mask [31]. In air, a 50-fs pulse is only 15 mm long; therefore, slight misalignment or displacement of the fiber behind the phase mask is all that is necessary to select the first orders for fringe formation. However, it is important to note that the transverse dimension (along the length of the fiber) of the beam is small (1 mm); it restricts considerably the size of the triangular region behind the phase mask, reducing the depth of field. Therefore, a wide beam should be used for writing gratings to maximize the overlap and the depth of field. This is possible with regenerative amplifier Ti: sapphire laser systems, which have sufficient energy to write long gratings (>35 mm) in a single exposure with cylindrical focusing. Nevertheless, multiple interferences produce period doubled and halved fringe patterns in the overlapping multiorder zone immediately behind the mask. A schematic for the inscription of FBGs with a phase mask and fs laser pulses, shown in Fig. 11.10, highlights the regions of interest. First, the singleorder interference region begins some distance away from the phase mask [29]. Second, the regions adjacent to the interference fringes remain dead zones, in which only a single beam exists. Third, if the pulses are long enough, the first and second orders can also interfere, but slightly displaced from the main singleorder fringe region at the center. Higher-order period phase masks reduce the angular dispersion and increase the overlap of orders, providing a larger depth of field.
11.1.4 Infrared Femtosecond Laser Inscription of Fiber Bragg Gratings The attraction of using IR femtosecond lasers, such as at a wavelength of 0.8, 1.3, and 1.5 mm, is that they could reduce the possibility of optical damage of the coating and therefore ease the fabrication of FBGs. In practice, the coating should be removed as it can affect grating production even with fs laser pulses at 800 nm [32]. High-quality gratings have been difficult to obtain through the coating, and Kondo et al. demonstrated this when writing long period gratings [33]. These gratings showed large background losses because of the formation of scattering centers and an incomplete overlap of the mode and the induced refractive index change. Nevertheless, with care, it is possible to write through the coating and achieve good results [32], and gratings 26 mm long with a reflectivity of 50 dB with a 2 dB out-of-band loss have been demonstrated using a point-by-point writing technique. The strength of the fiber is also preserved, as also demonstrated by straining the fibers, which resulted in a maximum
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wavelength shift of the coated FBG of 21 nm compared with 13 nm for a stripped fiber before breakage. Curiously, it was noted that writing through the coating reduced the cladding mode loss, although the authors offered no comment. It is possible that the coating allowed a larger cross-section grating to be written in the core and the cladding by slightly defocusing the beams. The focusing arrangement for through-the-coating writing was 100 objective with an NA of 0.55 and a pulse energy of 1 mJ, twice that required for bare fiber inscription. The grating writing speed was 1.07 mm/s at a repetition rate of 1 kHz, giving a second-order grating at 1550 nm. In this demonstration the beam was not scanned, so that cladding mode loss was present. Longer wavelength inscription excludes the possibility of fabrication of firstorder gratings as the light can only be focused to the diffraction limit. Thus, these wavelengths are good for writing long period gratings or higher-order period FBG. As Fig. 11.10 showed, with longer wavelengths the nonlinear process responsible for the change in refractive index change increases upward of fifth order and thus requires higher peak powers [19,31,32]. An important factor governing the efficiency and cladding mode loss of an FBG is the poor overlap between the grating and the mode because of the narrow beam that is created using cylindrical lens focusing with fs laser pulses. This is shown in Figs. 11.12a and b. The thin pencil beam has an oval shape and therefore results in a refractive index change in the core, which is asymmetric. It is to avoid such an effect that scanning across the core is used to make the refractive index change uniform. Using the scanning technique results in cladding mode loss suppressed FBGs, and a transmission and reflection spectrum is shown in Fig. 11.13. This high-quality grating has very low insertion and cladding mode loss [34]. Perhaps the most significant study of the application of fs lasers has been in the direct writing of waveguides in several different materials, which was nicely presented in a review article by Della et al. [35]. A number of demonstrations of FBGs in standard SMF28 optical fibers [36], pure silica [37], hydrogen-loaded standard fibers [38], ZBLAN [28], rare-earth doped [39] and sapphire fibers [40], lithium niobate splitter at 633 nm [41], and Mach–Zehnder interferometer at 1.55 mm [42]. Gratings in these materials have led to the fabrication of fiber lasers of high quality and power [43]. For example, a 7.8 W Raman laser has been demonstrated using a broadband chirped FBG at 1163 nm, with a conversion efficiency into the first Stokes of 94% in a 110-m long fiber [44]. More recently, outstanding FBG results have been achieved with fs laser writing. The understanding of the interplay between filamentation, optical breakdown, refractive index change, single-order interference, as well as from the scanning of the interfering beams across the core to create a uniform refractive index change has truly given way to a new dimension in optical components. High
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reflectivity (98.5%) chirped gratings, with a bandwidth of 310 nm, have been demonstrated in hydrogen-loaded standard fibers [36] with an exposure time of 30 seconds. Figure 11.14 shows the reflection spectrum of the amazing grating. It should be noted that such FBGs are a direct challenge to traditional thin-film technology, with broadband reflectors being their main strength. Thus, for the first time, the FBG is able to do as much as traditional thin film
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technology. Higher reflectivity with such a broad bandwidth in FBGs should be possible in the near future. The refractive index change, Dn(z), induced by the fs writing was reported to be 3.5–4.5 103 in hydrogen-loaded and 2–2.5 103 in nonhydrogenloaded fibers. It was also shown that the gratings survive temperatures >500 C over extended periods of time [19].
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11.1.5 Strength of Grating One of the important parameters of FBGs is their resistance to thermal decay. UV-induced gratings decay when processed at several hundred C. For attaining higher resistance, damage grating (Type IIA) may be used. However, more recently, IR fs and ps written gratings have been shown to have better properties with respect to temperature. These gratings fall into the category of Type II IR gratings, which are damage gratings with fs pulses, but have an element of color center as well as damage for the ps IR gratings, both fabricated with an energy of 1 mJ/pulse. Refractive index changes of approximately 1.5 103 and 1.7 103 are induced for an exposure to only a few dozen 125 fs and 1.6 ps pulses, respectively. Smelser et al. [45] presented an excellent comparison for SMF28 fiber in which a comparison is made between Type I-UV and IR and Type II-IR gratings. The decay of these gratings as a function of temperature is shown in Fig. 11.15. It may be seen that Type II-IR fs gratings survive exposure to 1000 C, for extended periods, whereas Type II-IR ps gratings decay initially on annealing and then stabilize thereafter at roughly half the initial reflectivity. The reflection and transmission spectra of strong gratings written in 2000 ppm Tm: ZBLAN fiber is shown in Fig. 11.16. The unresolved transmission loss is in excess of 30 dB and was written with 806 nm wavelength IR fs pulses. Unfortunately, these gratings cannot survive high temperatures and readily decay around 260 C because of the low glass transition temperature [27]. It should be noted that the refractive index change does not always have a positive sign on exposure to fs laser pulses. Typically, in pure silica, the refractive index change is negative, and thus it is hard to form waveguides by direct laser writing. In ZBLAN, too, the refractive index change is predominantly negative, being the result of an expansion in the core and compression at the edges of the laser affected zone, reported in reference [28]. The expansion reduces the refractive index, whereas the edges gain higher refractive index. Overall the refractive index change is negative. A curious phenomenon of nanograting formation, also called laser-induced surface structures (LIPSSs), was observed soon after the laser was invented. This type of nanostructure is formed by the physical separation of the glass into a grating of air and glass with a period of 150–350 nm depending on the physical spacing of the pulses impinging on a glass sample [46]. The direction of the nanograting is perpendicular to the polarization of the laser. Again, the period of the grating strongly depends on whether the intensity is above or below the self-focusing threshold. Above the threshold with low NA focusing results in a period of a grating l/3n, which is 170 nm, and it was suggested that the third harmonic of the laser played a role in the grating formation [46]. Others have noted a period of l/2n with a period of 270 nm [47]. This grating can be used as a polarizer or a diffraction grating and is relatively easy to fabricate.
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Figure 11.15 The decay of Type I UV (open circles), Type I-IR fs (black squares), Type II-IR fs (open squares), and Type II-IR ps (black circles). (a) The short time ramps against temperature. (b) The extended exposure to 1000 C. Reproduced with permission from: Smelser C.W., Mihailov S.J., and Grobnic D., “Formation of Type I-IR and Type II-IR gratings with an ultrafast IR laser and phase mask,” Opt. Express 13, 5377–5386, 2005. # OSA 2005.
11.2 CONCLUSION It is clear that fs laser technology offers an excellent route to writing gratings into many different materials. Care must be taken as identical gratings are more difficult to fabricate than UV-induced ones. Fs lasers, for example, based on the regenerative amplifier Ti: sapphire, are not entirely turnkey systems, although major strides have been made in making these more user-friendly. An area gaining in prominence is the direct writing of complex refractive structures in the bulk of glass, for which the reader is directed to a recent review [35]. For example, it is possible to write waveguides inside glass and fabricate three-dimensional splitters and interferometers. This technology has opened new areas of research by converging refractive components with micromachined devices,
Conclusion
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Wavelength (nm) Figure 11.16 Transmission and reflection spectra of a grating written into 2000 ppm Tm: ZBLAN fiber. The reflection is >99.9% at the Bragg wavelength. (From reference [28]. Reproduced with permission from: Bernier M., Faucher D., Valle´e R., Saliminia A., Androz G., Sheng Y., and Chin S.L., “Bragg gratings photoinduced in ZBLAN fibers by femtosecond pulses at 800 nm,” opt. Lett. 32(5), 2007.)
such as microfluidic channels and waveguides, to create a set of powerful tools and sensors for biomedical applications. This rich area is fascinating as it allows microscopic structures to be created by laser processing, a consequence of multiphoton absorption in polymers. The technology of grating writing with fs lasers is still in its infancy, despite major improvements in understanding of the physics of the phenomenon of induced refractive index change. In the coming years, one should expect to see a similar explosion in the activity of fs grating writing, as has been the case with UV-written lasers. However, the two approaches certainly complement each other. The highly controllable UVwritten techniques are likely to remain in the limelight despite the increasing competition from fs pulse writing. Advances in high-power, fiber-based fs lasers will no doubt have an enormous impact on the technology of fs written gratings and micromachining by reducing the operating costs and enhancing the flexibility of the systems. However, to date, the costs remain high in comparison to UV sources, which are easily accessible and have revolutionized FBG devices. The truly broadband FBG possible with fs lasers is a major advance toward challenging the capabilities of thin film technology. Although FBG devices are unlikely to replace thin films because of the mass production capability of the latter technology, the waveguide nature of the FBG is an enormous benefit. Compatibility with optical fibers and waveguides, flexibility, and the possibility of splicing with negligible insertion loss are attributes of the FBG that thin films may not be able to match, despite lower costs.
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The strong fs-written gratings will indeed help improve the power-handling capability of optical fibers. A word of caution though: The optical fiber is made of glass and has captured the light genie in its core. As the optical powers continue to increase, the relationship between the optical fiber and light will become increasingly fragile and delicate. The light used to process glass may well turn out to be the devil rather than the genie.
REFERENCES [1] K.M. Davis, K. Miura, N. Sugimoto, K. Hirao, Writing waveguides in glass with a femtosecond laser, Opt. Lett. 21 (1996) 1729. [2] K. Miura, J. Qiu, H. Inouye, T. Mitsuyu, K. Hirao, Photowritten optical waveguides in various glasses with ultrashort pulse laser, Appl. Phys. Lett. 71 (1997) 3329–3331. [3] M. Rothschild, D.J. Ehrlich, D.C. Shaver, Effects of excimer laser irradiation on the transmission, index of refraction, Appl. Phys. Lett. 55 (1989) 1276. [4] J.H. Kyung, N.M. Lawandy, UV light induced selective etching in borosilicate glasses for micro patterning, Electronics Letters 32 (5) (1996) 451–452. [5] J. Zhang, P.R. Herman, C. Lauer, K.P. Chen, M. Wei, 157-nm laser-induced modification of fused-silica glasses, Proceedings of SPIE 4274 (2001) 125–132. [6] R. Kashyap, K.J. Blow, Observation of catastrophic self-propelled self-focusing in optical fibres, Electron. Lett. 29 (1) (1988) 47–49, 7. [7] R. Kashyap, High average power effects in optical fibres and devices, Proceedings of SPIE, in: Limberger H.G., John Matthewson M. (Eds.), Reliability of Optical Fiber Components, Devices, Systems, and Networks, vol. 4940, Bellingham, WA, SPIE, 2003, pp. 108–117. [8] R. Kashyap, A. Sayles, G.F. Cornwell, Heat flow modeling and visualisation of catastrophic self-propelled damage in single mode optical fibres, in: Special Mini-Symposium at the Optical Fibres Measurement Symposium, vol. 2966, Boulder, CO, 1996, pp. 586–591. [9] T.J. Driscoll, J.M. Calo, N.M. Lawandy, Explaining the optical fuse, Opt. Lett. 16 (13) (1991) 1046–1048. [10] D.P. Hand, P.J.St. Russell, Solitary thermal shock waves and optical damage in optical fibers: the fiber fuse in optical fibers, Opt. Lett. 13 (9) (1998) 767–769. [11] R. Kashyap, Self-propelled self-focusing damage in optical fibers, in: Duarte F.J. (Ed.), The Proc. Of the Xth. International Conf. on Lasers, Stateline, Nevada, Lake Tahoe, STS Press, 1987, pp. 859–866. [12] R.R. Alfano, S.L. Shapiro, Observation of self-phase-modulation and small-scale filaments induced by light pulses in transparent media, Phys, Rev. Lett. 24 (11) (1970) 592–594. [13] E. Yablonovitch, N. Bloembergen, Avalanche ionization and limiting diameter of filaments induced by light pulses in transparent media, Phys. Rev. Lett. 29 (14) (1972) 907–910. [14] E.M. Dianov, V.M. Mashinskii, V.A. Myzina, Y.S. Sidorin, A.M. Streltsov, A.V. Chickolini, Change of refractive index profile in the process of laser-induced fiber damage, Sov. Lightwave Commun. 2 (1992) 293–299. [15] D. Du, X. Liu, G. Korn, J. Squier, G. Mourou, Laser-induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs, Appl. Phys. Lett. 64 (1994) 3071–3073. [16] A.P. Joglekar, H. Liu, E. Meyhofer, G. Mourou, A.J. Hunt, Optics at critical intensity: Applications to nanomorphing, Proc. Natl. Acad. Sci. 101 (16) (2004) 5856–5861.
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[17] B.C. Stuart, M.D. Feit, A.M. Rubenchik, B.W. Shore, M.D. Perry, Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses, Phys. Rev. Lett. 74 (1995) 2248–2251. [18] E.N. Glezer, M. Milosavljevic, L. Huang, R.J. Finlay, T.H. Her, J.P. Callan, E. Mazur, Threedimensional optical storage inside transparent materials, Opt. Lett. 21 (1996) 2023–2025. [19] A. Saliminia, N.T. Nguyen, S.L. Chin, R. Valle´e, The influence of self-focusing and filamentation on refractive index modifications in fused silica using intense femtosecond pulses, Opt. Commun. 241 (2004) 529–538. [20] P.R. Herman, H. Zhang, Ultrashort-pulsed laser direct writing of strong Bragg grating waveguides in bulk glasses, in Conference on Optical Communication/National Fiber Optic Engineers Conference, 2008. OFC/NFOEC 2008, San Diego, Paper OThV4, 1–3. [21] H. Zhang, P.R. Herman, Femtosecond laser direct writing of chirped Bragg grating waveguides inside fused silica glass, in Proc. Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides, BGPP 2007. Optical Society of America, Washington, D.C., Paper BTuD4T, Quebec City, Quebec, Canada. [22] R. Valle´e, Private communication. [23] D.M. Rayner, A. Naumov, P.B. Corkum, Ultrashort pulse non-linear optical absorption in transparent media, Opt. Exp. 13 (9) (2005) 3208–3217. [24] S.L. Chin, F. The´berge, W. Liu, Filamentation in nonlinear optics, Appl. Phys. B. 86 (2007) 477–483. doi: 10.1007/s00340-006-2455-z. [25] J.B. Ashcom, R.R. Gattass, C.B. Schaffer, E. Mazur, Numerical aperture dependence of damage and supercontinuum generation from femtosecond laser pulses in bulk fused silica. J. Opt. Soc. Am. B 23 (11), 2317–2322. doi:10.1364/JOSAB.23.002317. [26] S.M. Eaton, H. Zhang, M.L. Ng, J. Li, W.J. Chen, S. Ho, et al., Transition from thermal diffusion to heat accumulation in high repetition rate femtosecond laser writing of buried optical waveguides, Opt. Exp. 16 (13) (2008) 9443–9458. [27] K. Hirao, Y. Shimotsuma, J. Qiu, K. Miura, Femtosecond laser induced phenomena in glasses and photonic device applications. Mater. Res. Soc. Symp. Proc. 850 Materials Research Society, paper MM2.1.1. [28] M. Bernier, D. Faucher, R. Valle´e, A. Saliminia, G. Androz, Y. Sheng, S.L. Chin, Bragg gratings photoinduced in ZBLAN fibers by femtosecond pulses at 800 nm, Opt. Lett. 32 (2007) 454–456. [29] C.W. Smelser, S.J. Mihailov, D. Grobnic, P. Lu, R.B. Walker, H. Ding, et al., Multiplebeam interference patterns in optical fiber generated with ultrafast pulses and a phase mask, Opt. Lett. 29 (13) (2004) 1458–1460. [30] D. Grobnic, C.W. Smelser, S.J. Mihailov, R.B. Walker, P. Lu, Fiber Bragg gratings with suppressed cladding modes made in SMF-28 with a femtosecond IR laser and a phase mask, IEEE Photonics Technol. Lett. 16 (8) (2004) 1864–1866. [31] C.W. Smelser, D. Grobnic, S.J. Mihailov, Generation of pure two-beam interference grating structures in an optical fiber with a femtosecond infrared source and a phase mask, Opt. Lett. 29 (15) (2004) 1730–1732. [32] A. Martinez, I.Y. Khrushchev, I. Bennion, Direct inscription of Bragg gratings in coated fibers by an infrared femtosecond laser, Opt. Lett. 31 (11) (2006) 1603–1605. [33] Y. Kondo, K. Nouchi, T. Mitsuyu, M. Watanabe, P.G. Kazansky, K. Hirao, Fabrication of long period fiber gratings by focused irradiation of infrared femtosecond laser pulses, Opt. Lett. 24 (10) (1999) 646–648. [34] Courtesy of Dr. Real Valle´e. [35] G. Della Valle, R. Osellame, P. Laporta, Micromachining of photonic devices by femtosecond laser pulses, J. Opt. A: Pure Appl. Opt. 11 (2009) 013001–013019. doi:10.1088/ 1464-4258/11/1/013001.
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[36] M. Bernier, Y. Sheng, R. Valle´e, Ultrabroad fiber Bragg grating using femtosecond pulses, Opt. Exp. 17 (5) (2009) 3285–3290. [37] S.J. Mihailov, C.W. Smelser, D. Grobnic, R.B. Walker, P. Lu, H. Ding, et al., Bragg gratings written in all-SiO2 and Ge-doped core fibers with 800 nm femtosecond radiation and a phase mask, J. Lightwave Technol. 22 (1) (2004) 94–100. [38] C.W. Smelser, S.J. Mihailov, D. Grobnic, Hydrogen loading for fiber grating writing with a femtosecond laser and a phase mask, Opt. Lett. 29 (18) (2004) 2127–2129. [39] D. Grobnic, S.J. Mihailov, R.B. Walker, C.W. Smelser, Characteristics of strong Bragg gratings made with femtosecond IR radiation in heavily doped Er3þ and Yb3þ silica fibers, in: Bragg Gratings, Photosensitivity and Poling in Glass Waveguides (BGPP ’07), Quebec City, Canada, paper BTuC4, 2007. [40] D. Grobnic, S.J. Mihailov, C.W. Smelser, H. Ding, Sapphire fiber Bragg grating sensor made using femtosecond laser radiation for ultra high temperature applications, Photon. Technol. Lett. 16 (11) (2004) 2505–2507. [41] L. Gui, B. Xu, T.C. Chong, Microstructure in lithium niobate by use of focused femtosecond laser pulses, Photon. Technol. Lett. 16 (2004) 1337. [42] C. Mendez, G.A. Torchia, D. Delgado, I. Arias, L. Roso, Fabrication and characterization of Mach-Zehnder devices in LiNbO3 written with femtosecond laser pulses, Proc. of IEEE/LEOS Workshop on Fibers and Optical Passive Components (2005) 131. [43] E. Wikszak, J. Thomas, J. Burghoff, B. Ortac¸, J. Limpert, S. Nolte, et al., Erbium fiber laser based on intracore femto-second-written fiber Bragg grating, Opt. Lett. 31 (2006) 2390–2392. [44] R. Valle´e, E. Be´langer, B. De´ry, M. Bernier, D. Faucher, Highly efficient and high power Raman fiber laser based on broadband chirped fiber Bragg gratings, J. Lightwave Technol. 24 (12) (2006) 5039–5043. [45] C.W. Smelser, S.J. Mihailov, D. Grobnic, Formation of Type I-IR and Type II-IR gratings with an ultrafast IR laser and phase mask, Opt. Exp. 13 (2005) 5377–5386. [46] Q. Sun, F. Liang, R. Valle´e, S.L. Chin, Nanograting formation on the surface of silica glass by scanning focused femtosecond laser pulses, Opt. Lett. 33 (220) (2008) 2713–2715, and reference therein. [47] V.R. Bhardwaj, E. Simova, P.P. Rajeev, C. Hnatovsky, R.S. Taylor, D.M. Rayner, et al., Optically produced arrays of Planar nanostructures inside fused silica, Phys. Rev. Lett. 96 (2006) 057404.
Chapter 12
Poling of Glasses and Optical Fibers As seen in Chapter 2, a large number of effects take place in glass fibers exposed to ultraviolet (UV) radiation. Electronic bonds break, defect centers are formed, and charges are redistributed. Because glass is an excellent insulator, charge displacement is limited at room temperature. However, one can ask the question, what should one expect if together with exposure to UV, the glass were subjected to a strong electric field to organize and promote drift of charges in a preferential direction? Likewise, could charge movement be enhanced by heating the fiber under the application of an external field? This chapter deals with these questions. Not surprisingly, it is found that a nearly permanent strong electric field can be recorded in glass systems, much as a nearly permanent refractive index change can be recorded under UV exposure. As a consequence of this recorded field, the glass gains a second-order optical nonlinearity, which can be exploited in important applications such as frequency doubling and electro-optic phase control. This chapter discusses the physical processes that take place, the methods used for characterization, the models, and the applications of this effect.
12.1 OPTICAL POLING For more than a century it has been known that some dielectrics develop a permanent charge distribution when subjected to heat and a to dc voltage bias. This effect is long lived at room temperature and similar to a magnet in which the sample creates a magnetic field; here the sample develops an internal electric field and is named an electret (after O. Heaviside in 1885) or thermoelectret. Electrets nowadays find diverse applications, for example, in foil microphones that do not require a separate power supply for bias. In 1937, Nadjakov discovered that some dielectrics acquire permanent electric polarization when exposed to light under an electric field. The process is one type of photosensitivity, where charge separation takes place. In photoelectrets, the polarization persists in the dark but can be erased by illumination, and 527
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this is the principle used in a modern photocopying machine. In early studies, photoelectrets were created under an electric field 105 V/m. Later, it was shown that the photoelectrets could also be created in high-resistivity semiconductors by an internal photovoltaic field [1] or as a result of the different mobilities of electrons and holes [2]. As discussed in Section 2.1, the lowest optical nonlinearity expected in amorphous or symmetrical materials is the third order, governed by the w(3) tensor. However, when transparent materials with a strong recorded internal electric field [represented by Edc in Eq. (2.1.8)] are subjected to an additional applied field, they exhibit a nonlinear behavior described by the linear electro-optic effect. Similarly, it is possible to expect second harmonic generation in the presence of an internal field (or external electric field, by electric field-induced second-harmonic generation, or EFISH), arising from the third-order mixing of two electric field components of the optical fundamental wave and one electric field component of the recorded field in the material. It can be said that the recorded field breaks the symmetry and allows for the appearance of secondorder nonlinear effects, which are prohibited otherwise. The process of creation of a second-order optical nonlinearity in glasses was first described in the intriguing report that silicate fibers could be prepared for second harmonic generation [3,4], as mentioned in Chapter 1. The process consisted of simply launching high-power radiation at a wavelength of 1.06 mm and waiting (for hours): The second harmonic light grew from a hardly detectable level to some milliwatts of average power, at which point the process saturated (Fig. 12.1). Once prepared, the fiber could frequency double light efficiently upon reillumination at the same infrared wavelength. The green second harmonic light power generated in the fiber was sufficient to pump a synchronously pumped dye laser [4]. Optical poling, as the preparation process is now called, attracted much interest. The slow increase of the conversion efficiency through this preparation process exceeds by many orders of magnitude the minute signals that could be expected from higher-order nonlinear contributions such as magnetic dipole and electric quadrupole [5,6]. Although it had been reported previously that weak frequency doubling and sum frequency mixing could take place in fibers [7,8], this preparation process was different, as it could be observed over time, similarly to the early Bragg gratings described by Hill et al. exploiting photosensitivity. In contrast to early reports of Bragg gratings, however (see Section 1.1), the preparation results could be readily reproduced in various laboratories around the world. Efficient frequency doubling in glass fibers was unexpected not only because silica glass exhibits inversion symmetry but also because the phase velocities of the fundamental (generating) and second harmonic (generated) waves are so different (i.e., lack of phase matching). Questions on the basic mechanism
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TIME (HOURS) Figure 12.1 Growth of second harmonic generation in an optical fiber as a function of time. ¨ sterberg U., and Margulis The inset shows the signal on a linear scale (same units). From [4]: O W., “Dye laser pumped by Nd YAG laser pulses frequency doubled in a glass optical fiber,” Opt. Lett. 11(8), 516–518, 1986. # (1986) Optical Soc. of America.
of conversion, the slow growth, the memory effect in the fiber, the high efficiency, and phase matching were thus some of the puzzling features of optical poling. The growth of the second harmonic light during preparation for second harmonic generation (SHG) was exponential with time. This suggested that the increase in the frequency-doubled light was parametrically dependent on the amount of SH light already in the fiber (i.e., dI2o/dt I2o). When intense SH light was injected as a seed into the fiber together with the fundamental infrared (IR), the preparation was greatly accelerated [9]. Thus, standard telecommunication fibers could be used to generate second harmonic light as well, rather than being restricted to some obscure types of fibers used in experiments.
12.1.1 A Grating for Quasi-Phase Matching When the phase velocity of the pump and signal waves are not equal, the waves move in and out of phase periodically over a distance referred to as the coherence length given by
Poling of Glasses and Optical Fibers
530
Lc ¼
p ; Dk
ð12:1:1Þ
where the phase mismatch Dk ¼ k2o 2ko ¼ 2oðn2o no Þ=c:
ð12:1:2Þ
For a pump wavelength of 1.06 mm, the period observed in silicate fibers is 2Lc 40 mm, after which the net 2o signal returns to zero. Therefore, the phase mismatch must be overcome for frequency doubling to be an efficient process. This was recognized in the early days of nonlinear optics in 1962 by Bloembergen and coworkers [10], who suggested that if the phases were reversed at every half coherence length, the second harmonic signal could grow as it propagated in the nonlinear medium. For maximum efficiency, this phase reversal occurs at exactly every half coherence length. Here, the group velocities of second harmonic and pump waves are not equal and the waves still slide relative to each other, although the phases are matched. The process is called quasi-phase matching (QPM).
12.1.2 Recording a Grating for SHG A self-organized grating satisfying the QPM condition is created in optical poling. Such a grating has been unambiguously visualized by etching a frequency doubling fiber and inspecting it under a phase contrast microscope [11]. This periodic structure that is typically up to 10 centimeters in length [12] explains the relatively high conversion efficiency obtained in optical poling [4,13]. As for the physical process behind the creation of a w(2), because both o and 2o waves contribute to the preparation of the fiber for frequency doubling, it is natural to look for ways in which these two waves can create a QPM grating with the required periodicity. One such process occurs when two photons of the fundamental beat with one photon of the second harmonic through thirdorder nonlinear mixing, leading to optical rectification, exp½ið2ko k2o Þz; E0 ¼ Eo Eo E2o
ð12:1:3Þ
and an optically rectified term is created, which is temporally invariant and oscillates spatially with the exact period required for QPM in the fiber [9]. If this optically produced dc term could orient defects, then the grating would account for phase matching. It is now known, however, that the magnitude of the dc electric field created by optical rectification is some orders of magnitude smaller than that required for alignment of the defects, but it is accepted that the mixing of two infrared pump photons with one second harmonic photon as proposed in reference [9] is involved in the creation of the second-order
UV Poling
531
nonlinearity by optical poling and that the resultant dc field is proportional to the internal field. The dominant mechanism behind optical poling was described by the coherent photovoltaic effect introduced in reference [14]. Spatially periodic charge ejection results from the photoionization of defects caused by both two photons of fundamental and one photon of second harmonic light, which create an asymmetric photocurrent. The charges released drift in the gradient of the optical field until they are trapped. In fibers, trapping takes place preferentially at the core–cladding interface. The dark conductivity of the glass limits charge accumulation, and the preparation process saturates. A large number of subsequent studies confirmed the original description of the coherent photovoltaic mechanism. One of the issues resolved was whether the required periodic field was created by aligned dipoles or by periodic charge accumulation. The latter was found to be dominant [15]. A frequency doubling conversion efficiency of a few percentage points has been achieved in Ge-doped silica fibers from 1.064 mm to 0.532 mm [4,13], but is only possible at high pump intensities. Phase mismatch at lower powers and optical erasure at high intensities set a limit to the efficiency of the process. The recorded electric field amounts to 105 V/m, and the nonlinear coefficient, w(2) induced is limited to 103 pm/V. Although optical poling has been extended to other centrosymmetric materials (e.g., plastics), few practical applications of the process have been exploited until present.
12.2 UV POLING UV exposure of the core of the fiber is the basis for FBG fabrication. From color center formation and erasure measurements of optical poling, it was known that high-energy UV photons are capable of releasing charges from silica glass. Optical poling in the presence of cladding pumping with UV radiation was found to greatly accelerate the preparation process for frequency doubling [16]. In 1995, the technique of UV poling was introduced [17]. The Ge-doped core of a twin-hole fiber was exposed to ArF laser radiation at 193 nm in the presence of a strong electric field (8 107 V/m). After exposure, the fiber was reported to exhibit an electro-optic coefficient r 5.8 pm/V. Because r ¼ 2 w(2)/n4, for n ¼ 1.47 (at 633 nm), the reported induced effective w(2) corresponded to w(2)eff 13.4 pm/V. For a device 1.8 cm long, the Vp reported was only 18 V at 633 nm. It is interesting to note that the same fiber could be poled at room temperature with only 800 V applied for 60 minutes [17]. The electro-optic nonlinear coefficient induced under these conditions was only six times lower than after UV exposure –
532
Poling of Glasses and Optical Fibers
that is, w(2)eff 2.2 pm/V [17]. The fiber was also poled while an FBG was recorded, and both electro-optic amplitude modulation and wavelength tuning were also reported [18]. In extensive subsequent work carried out by Fujiwara and various coworkers, very high nonlinear coefficients (r 12 pm/V or w(2)eff 27 pm/V in Ge-doped films [19]) and w(2)eff 11 pm/V in bulk glass preforms [20] were reported through UV poling. The extremely high values of the second-order nonlinear coefficient were attributed to a combination of two factors [21]: crystallization of the glass under electric fields above 5 106 V-m1 that led to a large increase in the w(3) [22,23] and an internal space charge field due to the formation of Ge E0 centers. The starting value of the w(3) of the Ge-doped silica glass used was reported to be 7 1021 m2-V2, growing after crystallization to 1 1019 m2-V2 (500 larger than the value commonly used for fused silica w(3)silica 2 1022 m2-V2 even with a limited fill-factor of the induced crystals [24]). Various studies discuss increasing the stability of the effect [21]. After many failed attempts to repeat the measurements of UV poling in various laboratories, interest has waned. Observations, such as room temperature poling even at a relatively low bias voltage, significant polarization dependence in poled fibers (even without UV exposure), and the huge values of the secondorder nonlinearity induced are in contrast to results from other groups. For example, in reference [25] the effective w(2) induced was at least 100 smaller than reported in reference [17] under similar poling conditions. It is possible that the fibers and base materials used in the large w(2) experiments with UV poling had special features that are to date unknown and without these parameters it may be impossible to repeat the described results. The high nonlinearity claimed in UV poled silicate glasses remains to this day an enigma but nevertheless a source of continuing inspiration.
12.3 THERMAL POLING OF GLASS
12.3.1 Glass Electrets Electrets recorded in glass by heat and high voltage have been studied for a long time [26,27]. The electronic industry often makes use of a thin glass layer on semiconductor structures for insulation, and the behavior of the glass films under strong electric fields and high temperature is of great interest [28,29]. Likewise, glass is a cheap material used for insulation in high-voltage cables, transformers, and capacitor banks. In this environment, the temperature can easily exceed room temperature, and over time an electret can be created starting from the insulator material. Catastrophic breakdown is costly, and the properties of glass under high voltages and at high temperatures need to be known so
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that breakdown can be anticipated [30]. Another application of glass electrets for a large number of industries is in anodic bonding, when a high voltage bias is applied to the heated glass sample to create a strong air-tight binding between a metal or semiconductor electrode and the glass [31,32]. Charge displacement in most glasses, including silicates, is dominated by cationic transport. Two main processes can occur in heated glass resulting from the application of high voltages. The first one is the drift of the positive ions rendered mobile by the high temperature from the anode to the cathode side of the sample. The cations move collectively and can accumulate near the negative electrode or recombine partially or completely. In the absence of cation replenishment from the electrode on the anode side, the movement of ions in the glass results in the creation of a region depleted of mobile species (i.e., negatively charged and of higher resistivity). Most of the potential difference applied to the heated glass sample can end up falling across this thin space charge region because of the high resistivity. The electric field that was originally across the whole sample thus increases by a large factor, comparable to the ratio of the thicknesses of the sample and the depletion region. The effect is long lived when the sample is cooled to room temperature with the high voltage bias still on, as the charge distribution freezes because of the Arrhenius increase in the resistivity. The second process that takes place in heated glass subjected to high voltages comes from a preferential displacement of ions from their equilibrium position on a microscopic scale [33]. Every cation has a number of equivalent positions in the glass matrix around the defect to which it is bound, usually an oxygen deficiency center. The frequency of jumps between these sites of equal energy increases when the sample is heated. Upon the application of an electric field, the occupation probability of the sites closer to the cathode becomes higher than for the anode site. This bias in one direction creates microscopic dipoles and also results in the macroscopic polarization of the entire sample. Here, too, the process becomes long lived when the sample is cooled with the high voltage still on, but in contrast to the first case, the polarization of the sample is throughout its entire thickness. It should be added that the creation of a space charge in the glass is inevitably associated with the creation of new dipoles in the material, because most negative sites produced from cation displacement are subjected to a strong electric field that shifts the center of mass of the electronic cloud. These dipoles are concentrated near the anodic surface of the sample. The creation of a space-charge region and the polarization through dipoles take place simultaneously, and in principle the sample can acquire a second-order optical nonlinearity, which is the sum of the two contributions. The dipolar contribution is usually the largest in polymers, but in glasses the space charge effect generally dominates. Electret formation in heated glasses is usually referred to as thermal poling.
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12.3.2 Creating a Second-Order Nonlinearity In 1991, Myers and coworkers [34] discovered that a significant second-order optical nonlinearity was induced in silica glass through thermal poling, leading to the formation of an electret. The procedure consisted of heating a 1-mmthick silica disk to a temperature in the vicinity of 280 C, applying high voltage (e.g., 3–4 kV) for a few minutes, cooling the sample to room temperature with the high-voltage bias still on, and finally switching off the poling voltage, as schematically illustrated in Fig. 12.2. The nonlinearity was probed by shining intense infrared radiation from an Nd:YAG laser into the sample and monitoring the green second harmonic signal generated in transmission. The second-order nonlinear coefficient induced was long lived and had a value of w(2) 1 pm/V. Although several times smaller than in LiNbO3, it was nevertheless three orders of magnitude larger than achieved with optical poling. Numerous research groups easily reproduced the procedure. In this pioneering work, a number of striking features were identified. The nonlinear region was much thinner than the 1.6-mm-thick silica disks used, and it was always located on the surface adjacent to the anode electrode. By etching the sample and monitoring the residual second harmonic signal, the authors determined that its depth typically measured a few microns. Various types of commercial-grade fused-silica glass were tested including Infrasil, Herasil, and Homosil, all of which showed roughly the same order of induced nonlinearity, with a trailing Suprasil (synthetic silica), in which the SHG level was only 10% of the other glasses examined. This result indicated that the impurity level is highly relevant, because the contamination in Suprasil is typically one order of magnitude lower than in other types of fused silica. Myers et al. also reported that the necessary voltage did not scale with the sample thickness, because thinner samples subjected to proportionally lower voltages did not pole at all. However, a large nonlinearity was induced when thin and thick samples were poled in series under the high-voltage bias. More recent and detailed studies show that thinner samples subjected to the same poling voltage and thus to a higher poling field develop a second-order nonlinearity more rapidly [35]. When dissociation, recombination, and mobility are taken HV 280 °C
HV 20 °C
20 °C Silic
Silic
(1)
(2)
Silic
(3)
Figure 12.2 Schematic diagram of thermal poling procedure described in [34]: Myers R.A., Mukherjee N., and Brueck S.R.J., “Large second-order nonlinearity in poled fused silica,” Opt. Lett. 16(22), 1732–1734, 1991. # (1991) Optical Soc. of America.
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into account, a minimum threshold voltage required for poling (900 V) was predicted and observed [36]. Earlier, the question was raised as to the mechanism behind the creation of a second-order optical nonlinearity in a silica glass electret. Both charge transport followed by trapping and bond reorientation could in principle account for an effective w(2) through [34] ð2Þ
weff ¼ 3Edc wð3Þ þ ðNmbEdc Þ=ð5kTÞ;
ð12:3:1Þ
where Edc is the recorded field, b is the molecular second-order hyperpolarizability, m is the dipole moment, and k is Boltzmann’s constant. This issue was studied in reference [37] but as discussed in Section 12.6, the body of experimental evidence gathered over the years indicates that the space charge effect is dominant in the great majority of cases.
12.3.3 Other Poling Techniques An electret with a second-order optical nonlinearity is formed whenever glass is permanently charged – no matter what mechanism was employed to create the charge distribution. High-energy electrons [38–40] and protons [41] can be directed into the glass substrate to cause a charge imbalance, which results in the appearance of an effective w(2). Electrostatic charging of optical fibers with a single positive pole can be also be used to induce second-order optical nonlinearity. To learn more about charging fibers, please refer to the additional section on this book’s companion Web site: www.elsevierdirect.com/companions/9780123725790. Likewise, corona poling has also been used to induce a second-order optical nonlinearity in silica and other glasses (discussed later) and also in planar structures, such as films and waveguides [42–44]. Corona poling is the most widespread technique used to pole polymers such as polyvinylidene fluoride (PVDF) [45]. Limiting the current supplied during poling is advantageous when dealing with glasses of high conductivity as compared with silica, because thermal runaway through resistive heating is avoided. The stability of corona poled thin films of Corning 7059 glass on Pyrex was found to be relatively poor [46], with the long-lived component (t 103 hours) attributed to relaxation of the structure at or near the interface [47]. High-energy photons such as g- [48,49] and x-rays [50–52] can also be employed to break chemical bonds, and the simultaneous presence of a driving field (and often high temperature) causes charge separation and poling. The development of femtosecond writing of Bragg gratings (see Chapter 3) opened the possibility of bringing together the techniques of poling with the creation of plasma with submicron spatial resolution in silica glass. Multiphoton processes in the glass change the structural composition and locally enhance (by a factor 2) the induced SHG [53]. Likewise, UV femtosecond pulses have also been used as the excitation source (rather than heat) to release charges [54].
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A very local source of heating – a CO2 laser – has been used by two groups to thermally pole internal electrode fibers [55,56]. The technique has a potentially good spatial resolution that may be exploited in the future for QPM structures. As heat can be deposited rapidly, this may enable poling of the fiber during the drawing process, although it has so far not been demonstrated. Efforts have been made to optimize poling parameters [57–59]; however, at present, poling with a CO2 laser is not widely used.
12.4 CHARACTERIZATION TECHNIQUES Characterization techniques required to elucidate the physics involved in poling and for optimization need to address two main challenges: the space-charge region created in poling, which is often thin (10 mm), and the relatively weak nonlinear coefficient induced. Other difficulties encountered are sample-tosample variation, the high, dark resistivity of silica glass, and the existence of several mobile ionic species. Characterization techniques address the optical nonlinearity induced by measuring SHG, the electro-optic effect, or provide complementary information, such as the electric field distribution or the ionic content of the sample near the space-charge region after poling.
12.4.1 Measurement of the Nonlinear Optical Coefficient The phase change induced by the electro-optic effect across a relatively thinpoled sample is generally insufficient for monitoring the induced optical secondorder nonlinearity. Thus, the characterization of the electro-optic effect by the propagation of probe light along a poled waveguide, where the phase change accumulates over longer lengths is highly advantageous, as will be discussed.
Maker Fringes The most straightforward characterization of w(2) involves shining an infrared laser beam into a poled sample and measuring the SHG signal for various incidence angles, as proposed by Maker et al. in 1962 [60] and used by Myers et al. in 1991 [34]. A schematic diagram of such a measurement setup is shown in Fig. 12.3 [36]. One detects the integral of the frequency-doubled light generated through the entire thickness of the sample. This integration can result in constructive and destructive interference depending on the apparent thickness of nonlinear material traversed by the probe beam. The so-called Maker fringes result from the interference between the second-harmonic wave “bound” to the
Characterization Techniques
537 Harmonic beam-splitter
Silica sample lw
qe
Pulsed source lP = 1.064 mm
lw
lw
PMT l2w
l2w GLAN polarizer
Rotation stage
Band pass filter 532 nm Data acquisition
Figure 12.3 Example of Maker fringe measurement set-up. From [36]: Quiquempois Y., Godbout N., and Lacroix S., “Model of charge migration during thermal poling in silica glasses: Evidence of a voltage threshold for the onset of a second-order nonlinearity”, Phys. Rev. A (Atom Mol. Opt. Phys.) 65(4), 043816 1–14, 2002. # (2002) American Physical Society.
incident fundamental field and “free” second-harmonic field as the length changes when the incidence angle is varied. It is possible to calibrate the frequency doubling efficiency against the signal generated using a reference sample (of quartz, for instance) and determine the absolute value of the w(2) recorded. However, as a consequence of the integration, one loses information as to the location of the second-order nonlinearity. Solving the inverse problem for determining the profile and sign of the nonlinearity as a function of depth from the measured Maker fringe pattern therefore becomes difficult. The relation between the SH signal P2o and the pump peak power Po for incidence angle yi is [61] l ð ð2Þ 2 2 2 2 2o Po tan yi w ipz exp Tðyi Þ ð12:4:1Þ dz : P2o ¼ 3 2 e0 c n2o pw0 Lc cos yi 2 0
Here, T(yi) is a correction factor that takes into account the losses resulting from Fresnel reflection at the surfaces, w0 is the beam radius, and the coherence length Lc is defined in Eq. (12.1.1). The main advantage of the Maker fringe technique is that the measurement is nondestructive, and one can relatively quickly estimate if the nonlinearity is restricted to a thin layer near the surface (when a broad single fringe is detected as the angle is changed) or distributed across the bulk of the sample or on both surfaces (when many fringes are detected), as shown in Fig. 12.4. Most measurements with fused silica result in a nonlinearity restricted to the anodic surface of the sample, as first reported by Myers [34]. In some cases – for example, when the sign of the poling voltage is reversed – one can induce a nonlinear layer located on each side of the sample [62], and occasionally bulk and surface contributions have been identified at the same time [63]. High-purity glass such as
Poling of Glasses and Optical Fibers
538
SH power (nW)
30
1.5
1.0
20
0.5 10 –50
0 Propagation angle
50
–50
0 Propagation angle
50
Figure 12.4 Example of Maker fringes for a 1-mm-thick Infrasil sample (left) when the nonlinearity is induced only near the surface and for a 1-mm-thick Suprasil sample (right) when the nonlinearity is deep and interference is seen when the sample is rotated. Adapted from reference [61]: Quiquempois Y., Martinelli G., Dutherage P., Bernage P., Niay P., and Douay M., “Localisation of the induced second-order non-linearity within Infrasil and Suprasil thermally poled glasses,” Opt. Commun. 176(4–6), 479–487, 2000. # (2000) Elsevier Science B. V.
Suprasil tends to result in a deeper depletion region than fused silica [64]. The Maker fringe technique has also been widely employed to estimate the maximum value of w(2) induced in poled glasses other than silica, as described in Section 12.7.5. The uncertainty in the fringe measurement translates into a large uncertainty in the inferred nonlinear profile. As a consequence, the literature has a number of conflicting reports on shapes of the distribution of the second-order nonlinear layers, based on Maker fringe measurements. Some improvements have been introduced to the straightforward Maker fringe technique since 1962. It was realized that refraction from air to glass limits the maximum probe angle, and the introduction of prisms [65] and of semihemispherical lenses [61,66,67] eliminates this problem – the maximum probing angle becoming close to 90 . A noncollinear, background-free setup has also been described with shorter coherence length than a conventional setup and thus provides better spatial resolution [68]. It was also realized that the information obtained from a pair of identical samples (two halves of the same specimen, for instance) stacked against each other or separately gave much more information about the distribution of the nonlinearity than a single sample [69,70]. An inverse Fourier transform technique was further developed to obtain the nonlinear profile from the measured fringes [71], the technique culminating in a powerful interactive procedure to determine the amplitude and the sign of the nonlinear profile [72].
SHG Microscopy Another simple nondestructive technique for measuring the second-order nonlinear coefficient in poled glass involves probing the sample from the side
Characterization Techniques
539
with a focused laser beam propagating orthogonal to the direction of the poling field. The spatial distribution of w(2) is studied by scanning the beam relative to the sample. The technique can be generally referred to as a SHG microscopy and is compatible with optical fibers, which can be cleaved without the need for polishing. The first such measurement in poled bulk glasses was carried out in 1993 [38]. A 50-mm-diameter probe IR beam impinged on the (polished) side of a sample and the near-field pattern analyzed with a microscope and TV camera showed that the SHG signal peaked at a depth 12 mm under the anodic surface. The width of the SHG spot was 7 mm. The technique was also successfully used with poled waveguides [73,74] and fibers [75,76]. When scanning the cross-section of a poled planar waveguide or fiber with an intense probe beam, SH is generated at the charge accumulation layers. A map of the charge distribution can thus be obtained. Most subsequent SHG microscopy studies in fibers [76–78] also show a narrow (3-mm-wide) SHG layer buried by several microns inside the glass, as illustrated in the example of Fig. 12.5. The information obtained by SHG microscopy is valuable, but the interpretation of results requires care. Frequency doubling is not usually detected along the entire depletion region where the field is supposed to be strong or even at the anodic surface where the electric field peaks just after poling. Rather, SHG takes place nearly exclusively in the negatively charged layer that marks the end of the depletion region. The most likely explanation for this observation is that cleaving or side polishing must change the original field distribution of the sample, because positive surface charges are attracted to the negatively charged layer when suddenly exposed to the air. SHG is likely to be influenced by fringing fields on the surface from which the probe beam enters the sample, complicating matters further.
Figure 12.5 SHG microscopy of a poled D-shaped fiber. The (incomplete) light ring around the hole accommodating the anode electrode indicates the charge accumulation layer. From [77]: Honglin A., and Fleming S., “Investigation of the spatial distribution of second-order nonlinearity in thermally poled optical fibers,” Opt. Exp. 13(9), 3500–3505, 2005. # (2005) Optical Soc. of America.
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Poling of Glasses and Optical Fibers
12.4.2 Etching Etching has been used for decades as a tool for providing information on the effects of poling in glasses [79,80]. In optically poled fibers, etching allowed the imaging of the frequency doubling grating [11]. Carlsson and coworkers showed that a depleted region formed in cation-rich glasses had several properties modified strongly, such as the refractive index, the IR reflection spectrum, the mechanical breaking strength, various structural properties, chemical durability, and electrical resistance [79]. In particular, the depleted surfaces of silicate glasses are more resistant to attack by hydrofluoric acid (HF) than untreated glass, the etching rate dropping by as much as 25% in the example reported in reference [79]. By removing successive layers of the poled sample, one can determine where the SHG takes place [34]. One can also determine the electric field distribution, if the latter can be related to the rate of chemical attack. Etching as a characterization tool is destructive and the information obtained ambiguous, because various simultaneous processes can cause a change in the etching rate. Among the parameters known to affect etching of glasses are the fictive temperature and atomic bond angle [81], densification and crystallization [82], the level and type of doping [83], optical exposure [84,85], and the electric field and the charge distribution in the material [86]. However, as long as the results are interpreted with care, etching can assist in revealing some of the processes taking place in poling with submicron resolution. Myers et al. [34] used etching to study the extent of the layer with a second-order nonlinearity induced in poled silica. Etching from the anode side reduced exponentially the SHG signal, and a sample poled for 2 hours had a characteristic (1/e2) depth of 6.8 mm, whereas one poled for 15 minutes had a typical depth of 3 mm. The space-charge region of poled silica glass was also studied by etching in a configuration where the recorded electric field is parallel to the etched surface [87]. After poling, the samples were cleaved and etched, and the surface analyzed with an Atomic force microscope (AFM). Ridges and valleys were observed and correlated to the profile of the depletion region. Care is also needed here as in SHG microscopy, because cleaving the sample is likely to modify the electric field distribution obtained just after poling, as new positive charges are attracted to the negatively charged layer now exposed to air. These experiments have been used to validate a two-ion model developed to describe poling of silica glass [88]. Subsequently, the same technique has been used with an optical microscope rather than an AFM and the edge of the depletion region visualized [89]. Etching a cleaved poled twin-hole fiber surface has also been shown to reveal the extent of the depletion region [90,91]. This is particularly valuable because it is otherwise difficult to determine how the depletion region overlaps with the fiber core. An intriguing feature noticed is that the imaged depletion region is a ring around the entire hole accommodating the electrode, even on the side opposite to the cathode (see, for example, Fig. 12.6), possibly because the creation of the depletion region is ultimately dictated by the voltage and not
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541
Figure 12.6 Image of the cross-section of the fiber of Fig. 12.5 after etching in HF for a few seconds. The etching rate is different at the charge accumulation layer and a similar (incomplete) ring is observed. From [77]: Honglin A., and Fleming S., “Investigation of the spatial distribution of second-order nonlinearity in thermally poled optical fibers,” Opt. Exp. 13(9), 3500–3505, 2005. # (2005) Optical Soc. of America.
by the field applied. The etching technique has proved useful in the study of the time evolution of the depletion region in poled fibers [92,93]. An in situ interferometric technique was reported to optically monitor the thickness removed by chemical attack [94] with submicron resolution. This enables the etching rate in the space charge region to be monitored while the depletion region is gradually eroded. If the etching rate change is related to the strength of the electric field normal to the surface of the sample [86], a map of the electric field distribution is obtained. A systematic abrupt variation in etching rate after a few microns etching is related to the end of the depletion region, and a narrow (<1 mm) buried negatively charged layer is inferred from these measurements [94]. The interferometer used to study the time evolution of the depletion region shows a recorded field [95,96], which peaks at 4 108 V/m. The interferometric technique to measure the etching rate has been made much more powerful [97] with the in situ measurement of the SH generated by the sample during the etching process, as illustrated schematically in Fig. 12.7. As implemented by the group at Lille University, this allows the determination of the contribution to the total SH signal and the value of the w(2) of every layer of material removed. The distributions of the second-order optical nonlinearity and of the electric field in the poled glass can thus be evaluated. The technique makes use of the experimental procedure of always studying samples originating from the same silica mother rod, and therefore greatly increases the reproducibility of the data. The powerful technique in reference [97] leads to a much better understanding of poling and the experimental observation of various effects not previously reported. Using this technique allows a measurement of the time
Poling of Glasses and Optical Fibers
542
Nd: YAG
w
w
Amplifier
Photodiode
HF
Amplifier
Photodiode 2w
Poled sample
Filter
Figure 12.7 Schematic of interferometer measuring simultaneously the etching rate and SHG adapted from the text in [97]: Kudlinski A., Quiquempois Y., Lelek M., Zeghlache H., and Martinelli G., “Complete characterization of the nonlinear spatial distribution induced in poled silica glass with a submicron resolution,” Appl. Phys. Lett. 83(17), 3623–3625, 2003. Reprinted with permission. # (2003) American Institute of Physics.
evolution of the true w(2) profile during poling [98] and the observation of a sign reversal of the second-order nonlinearity in some samples [99]. It also confirms that the short circuit condition [100] (i.e., the integral of the electric field when moving from the anodic to the cathodic surface is zero) is satisfied in poled glasses [101]. Fine details, such as a spatially varying alteration of the w(3) near the surface of poled silica samples, can also be identified [102].
12.4.3 Elemental Analysis of the Surface and Other Techniques The nondestructive laser-induced pressure pulse technique (LIPP) has been used to characterize the charge distribution in poled glass [100,103,104]. In this technique, a short duration (3 ns) laser pulse hits a target attached to the poled sample and creates a pressure wave that displaces the stored charge, creating a current signal. Information is thus obtained about the sign and distribution of the charge in the sample. Studies with poled soda-lime glass [100] with pressed-on electrodes allowing for injection of hydrogen gave similar results to those obtained by simultaneous etching and SHG of silica [97]. They reveal a buried negative charge layer causing a strong electric field near the anodic surface and a weak reverse field in the bulk. A multilayered and reverse charge distribution is also measured in the absence of hydrogen injection, and the buried positive charge is assigned to a pileup of Ca ions [100]. The LIPP technique with poled silica [104] also shows that different samples have different charge distributions recorded and that these evolve on a timescale of a few days.
Characterization Techniques
543
One limitation of LIPP is the few microns (depth) spatial resolution achieved. It is valuable then to complement LIPP studies with an elemental analysis technique that allows mapping of ions to extract information about the electric field and the recorded charge distribution. Various techniques can be employed [100], including nuclear reactions between impinging protons and sodium atoms in the glass. Likewise, hydrogen depth profiles can be evaluated with the use of the elastic recoil detection technique (ERD) [105]. Rutherford backscattering spectrometry (RBS) [106] can determine the presence and profile of heavier elements such as calcium. The main drawback of such measurements is the shallow depth that can be studied, but the resolution is excellent (tens of nm). In poled soda-lime glass, removal of sodium, replacement of Na by H-ions, and the piling-up of calcium have been identified [100]. Likewise, energy dispersive x-ray spectroscopy (EDS) has been shown by a number of groups (e.g., [107]) to map the missing ions in the depletion region. The lack of sensitivity for light ions (e.g., Naþ) is a limitation, but the sample can be cleaved and analyzed from the side over the entire region of interest. A highly relevant study of the elemental content of fused silica under poling conditions for inducing a second-order optical nonlinearity describes the use of secondary ion mass spectroscopy as a tool for near-surface elemental analysis [108]. The experiments show that sodium and lithium migrate from the anodic surface. After poling for 20 minutes at 275 C, the depletion region width for sodium reaches 25 mm, whereas that for lithium becomes 10 mm thick. Excellent correlation is found between the layer depleted of lithium and the ridge in etching experiments, which is related to the edge of the nonlinear layer. Accumulation of Liþ, indiffusion of hydrogen, and other cations are also detected. Secondary ion mass spectometry (SIMS) studies of planar waveguides with a silver anode electrode show that during poling, the metal enters the glass in high concentrations [109]. Various other characterization studies report the profile of the second-order nonlinearity and the distribution of the charges and the electric field in thermally poled fused silica samples. The nondestructive techniques employed include the noncollinear frequency conversion of a pair of femtosecond pulses colliding at an angle [110], in a setup similar to that of reference [68], EFISH used as a means of examining the electric field distribution during and after poling [111], and the measurement of the surface potential with a field meter [112]. When a sample studied is an optical fiber or a planar waveguide, Bragg gratings can provide valuable information about the value of the refractive index under an external electric field. Using this principle, a poled section of an internal electrode fiber was spliced inside an FBG Fabry–Perot interferometer [55]. Under the application of voltage, it is possible to measure the index change, the secondorder nonlinearity, and the recorded electric field in the fiber for various poled lengths. It is also observed that UV poling and thermal poling (with a CO2 laser as a heating source) result in opposite signs of the nonlinearity. In waveguides with a Bragg grating, the w(3) is largely unaffected by poling and the polarization dependence is weak [113].
Poling of Glasses and Optical Fibers
544
12.5 FUNDAMENTAL AND PRACTICAL ISSUES
12.5.1 Cation Mobility Time-resolved studies are valuable for a better understanding of the dynamics of poling, as shown by Mukherjee et al. [114]. The rise times and the (openand short-circuit) fall times of the SHG signal investigated as a function of temperature, over the relatively limited interval 207 C–267 C, show that the activation energy is in the range 1.1 eV to 1.3 eV, consistent with the activation energy for Naþ in silica. Interestingly, in the studies at 267 C, the rise time for SHG is a few seconds, in contrast with the 5- to 15-minute poling time normally required to reach the maximum SHG efficiency. The high mobility of cations in glasses at elevated temperatures is essential for the creation of a space charge. It is found experimentally that the secondorder nonlinearity can only be induced in silica above 200 C [115]. This is attributed to the movement of the cation of highest mobility, which is Naþ, followed by Liþ and Kþ. Hydrogen is believed to play a major role in poling, but rather than being uniformly distributed in the silica glass, mobile hydrogen species are injected into the glass from the surface layer or from the environment. At higher temperatures, diffusion rates increase and the recording field must also be increased to counter erasure. The temperature dependence for a fixed applied voltage and poling time is illustrated in Fig. 12.8. The maximum efficiency is measured to be at 280 C. 3.0
SHG (a.u.)
2.5 2.0 1.5 1.0 0.5 0.0 200
250
300
350
Temperature (⬚C) Figure 12.8 SHG as a function of temperature in poled silica. Applied voltage: 4 kV, poling time: 15 minutes. From [115]: Myers R.A., Mukherjee N., and Brueck S.R.J., “Temporal and spectral studies of large w(2) in fused silica, in: Bragg gratings, Photosensitivity and Poling Conference BGPP 2003,” Proc. of the SPIE 2044, The International Society of Optical Engineering, Quebec City, Canada, pp. 2–10, 1993. # (1993) SPIE.
Fundamental and Practical Issues
545
Sodium-ion contaminants in fused silica are located mainly at negatively charged AlO4 trapping sites and jump across the silica landscape to the next equivalent defect [116,117,118]. The dependence of the mobility is strong with temperature. For example, whereas the mobility of the sodium ion, mNa in silica films extrapolated to 280 C is 2 1015 m2V1s1, this value drops to 1017 m2V1s1 at 250 C [119]. The value 2 1015 m2V1s1 is in excellent agreement with the one used to fit the experimental data to an ion-exchange model [120] described in detail in Section 12.6.3. Another value for the mobility of sodium ions in silica given in the literature is 7.5 1015 m2V1s1 [87]. The values of mNa inferred from [29] and [121] are very different from the ones used by researchers working in the field of poling (by orders of magnitude). They are based on an activation energy 0.7 eV, much lower than the values 1.1–1.3 eV estimated from poling dynamics experiments [114]. There is also evidence that the mobility of sodium is strongly reduced in the hydrated surface layer of silica glass [117], and this could create a nonuniform distribution of values for the mobility, which is generally taken to be uniform everywhere in the glass sample. The relative mobility of Liþ in Type II glass (GE204) was estimated to be mLi 0.15 mNa at 380 C, whereas in Type I glass (GE214) the estimate was mLi 0.075 mNa at 350 C [116]. The relative mobility for Kþ was projected to be 40 and 200 times lower than of Naþ in those references. Protons are less mobile in silica glass, but attached to water molecules as hydrons [122] they can diffuse through the glass matrix with a relatively high diffusion constant. The ratio of the mobilities of sodium and H-related species that best fit the data in reference [120] is mNa/mH 2200, leading to a value mH 1 1018 m2V1s1. It is worthwhile to note that the value of the mobility of hydrons is apparently still lower than that of Kþ, although the latter is believed to be less important than the former in the poling process. There is ample evidence that various types of cation move from the anodic surface during poling, not only Naþ (see, for example, Fig. 12.13). The values of the mobilities given earlier are measured for neutral glass. In poling, however, after the migration of sodium toward the cathode the glass gains a negatively charged depletion region. Cationic transport there is greatly facilitated, because a cation can easily find an unoccupied negative defect site in its neighborhood, in contrast to pristine glass. It is therefore possible that some of the mobility values used for modeling poling need to be revised, incorporating the fact that the glass becomes negatively charged during the poling process. Cationic transport in a charged material is largely unexplored in poling literature. Engineering the charge distribution through the selective displacement of different cations may in the future lead to larger second-order nonlinear coefficients.
546
Poling of Glasses and Optical Fibers
12.5.2 Defects and Water Poling of silica cannot be attributed to the breakage of the Si-O bond (>7 eV), and thus it is defect related. Early papers correlated the amount of OH in the glass with the efficiency of the poling process in fused silica and in sol-gel glass [123,124], but this correlation was not reproduced in other studies (e.g., reference [125]). Subsequent investigations found that the second-order nonlinearity increased with the introduction of Na and water into the anodic surface of the glass [126]. Sodium was introduced from a polymethyl methacrylate (PMMA) film and a maximum SHG efficiency was obtained for a 4 ppm Na concentration in the film, with the nonlinear layer located near the anode. In contrast, molecular water introduced in an autoclave at 150–220 C for 6–96 hours caused a large increase of the nonlinearity in the entire sample thickness. The authors suggested that poling depends on a conduction process where water molecules release protons that drift through the glass [126], but it is likely that mobile protons attach to water molecules and drift as hydronium ions H3Oþ. The effect of preannealing showed that bulk SH could indeed be induced, but the results here depended on the batch of silica glass used [127]. Even Ge defects in undoped silica have been correlated to SHG in poled glass [128]. An interesting study was carried out in reference [129] where a drop of a saturated solution of NaF was deposited on the anodic and also on the cathodic surface of the high purity silica sample before poling. It was found by SIMS analysis that fluorine could be injected into the glass and the SHG signal generated on the cathodic surface increased by two orders of magnitude, becoming comparable to that on the anodic surface. Investigations about role of defects in thermal poling are still lacking. It is difficult and expensive to perform systematic studies of contamination without varying at the same time many other parameters. Although the anodic nonlinearity can be attributed to the depletion of cations such as Naþ and Liþ, in highpurity glass such as Suprasil, the nonlinear layer is much thicker, and the presence of water and OH seems to play a more important role. The contribution of OH bonds could, for instance, be investigated in more detail in H2-loading experiments [130]. An interesting and puzzling experiment has been performed, where rapidly quenched Ge-silicate glass exhibits SHG on both surfaces without ever being subjected to an external electric field [131]. The effect is attributed to a deformation of the glass structure near the surfaces caused by a gradient of the stress. Because no crystallization is observed, the sample must gain a spontaneous polarization.
Fundamental and Practical Issues
547
12.5.3 Charge Movement When a poling voltage (e.g., 4 kV) is applied to a silica disk (1 mm thick), a nearly uniform field is established (4 MV-m1). Cations dominate current transport in silica, and mobile species start drifting from the anode to the cathode. High-mobility ions move away from the anode first. If the electrodes are blocking and ions that drift toward the cathode are not replenished, a region forms at the anode, which is depleted of mobile species. The region is negatively charged, as the movement of positive ions results in a charge imbalance. The negative charge is immobile, and the excess electrons are trapped mainly at aluminum defects and oxygen deficiency centers. Because the mobile species have drifted away, the resistivity of this region is orders of magnitude higher than that of pristine glass, and most of the applied voltage falls across the depletion region. To a first approximation, in the limit all the voltage bias falls across a region that is a few microns thick, giving rise to an electric field E 4 108 V-m1. The creation of a completely depleted region benefits from a positive feedback effect that is typical in poling. If the depletion is incomplete, any mobile ions remaining in the region are subjected to an even larger field that extracts them easily, so that they quickly drift away from the depletion region toward the cathode. The total charge moved by poling is comparable to that expected from a charging capacitor. After the w(2) has been recorded, in the limiting case when all the applied voltage Vapp falls across the depletion region and d L, where d is the thickness of the depletion region and L is the sample width, from Gauss’s law f 2EA ¼ 2
Vapp A Q ¼ ; e d
ð12:5:1Þ
where A is the area and the dielectric constant e ¼ er eo (the relative dielectric constant er ¼ 3.8 for silica). The approximation that the absolute value of the recorded field is equal to the applied voltage divided by the width of the depletion region is based on the assumption that the displaced charges create a voltage equivalent to the poling bias. This works well for a voltage divider, because the resistance of the depletion region becomes much larger than the resistance of the bulk. Note, however, that just like the applied field can be smaller than the recorded field, at least in principle, so can the poling voltage be smaller than the equivalent recorded voltage (i.e., the external bias required to compensate the recorded field). The total charge displaced is Q ¼ erAd;
ð12:5:2Þ
where e is the electronic charge, and r is the density of mobile cations. Therefore, one finds [132] 2er e0 V 1=2 : ð12:5:3Þ d¼ er
Poling of Glasses and Optical Fibers
548
Qualitatively, the depletion region width increases with the poling voltage and decreases with the concentration of mobile ions. If w(2) is given by the space charge term of Eq. (12.3.1), that is, ð2Þ
weff ¼ 3Edc wð3Þ ;
ð12:5:4Þ
then the relation between the recorded second-order nonlinearity and the applied voltage is determined to be V erV 1=2 ð2Þ weff ¼ 3wð3Þ ¼ 3wð3Þ : ð12:5:5Þ d 2er e0 One then sees that it is advantageous to pole at higher voltages to increase the induced second-order nonlinearity [133]. From the density of silica (2.2 103 kg-m3), assuming a content of one cation for every 106 SiO2 molecules, the density of ions available is rcat 2.2 1022 cations-m3. For a poling voltage of 4 kV, one finds from Eq. (12.5.3) a depletion region width of d 8 mm, in qualitative agreement with measurements. For an electrode area 1 cm2, this gives a total charge Q 3 mC (i.e., 2 1013 singlecharge ions, or 2.4 104 charges per cubic micron). It should be noted that poling of other silicate glasses such as soda lime has been shown experimentally to result in a space charge region of comparable thickness, although the cation concentration is four to five orders of magnitude larger than in silica. This is in clear disagreement with Eq. (12.5.3) and implies that most of the charge in soda lime is neutralized during poling. If a depletion region in soda lime could store all excess negative charge left behind by the cations removed during poling, it would create a field >1 MV-mm1, far exceeding the breakdown limit of any known material. Whereas the heated bulk glass with the original cation contamination behaves as a resistor, the depleted glass is an excellent insulator and can be well described as a capacitor. A nonlinear behavior is expected [134], because the capacitance value decreases as the depleted width expands. From an RC description, it is possible to estimate the characteristic RC time of the circuit, t1, at the limiting value of d [134]: L er e0 1=2 : ð12:5:6Þ t1 ¼ m 2erV For a mobility m ¼ 2 1015 m2V1s1 and L ¼ 1 mm, one finds t1 ¼ 5 102 seconds. The measurements of the poling current is useful for revealing the evolution of formation of the electret [135], but extra care needs to be taken when measuring currents in the nA range created by a high-voltage bias, because field emission at the edge of the electrodes and air ionization are typical occurrences at high temperatures. Current measurements that lack guard rings often lead to
Fundamental and Practical Issues
549
errors (i.e., are not directly related to the ionic current flow in the glass). For short poling times, the current is seen to decrease rapidly in fused silica. The fast current decay component is associated with the movement of sodium and possibly lithium. Longer time constants of the current are associated with the movement of less mobile species. The drift of ions toward the cathode can result in charge accumulation or cancellation. The mechanisms taking place on the cathode side have been studied to a much lesser extent than for the anode. From experiments with soda-lime glass (sodium content 16%) in which a white deposit is seen to appear on the cathodic surface of the sample, it is believed that cations reaching the cathodic interface are neutralized with the possible creation of oxide of the type Na2O [107]. Although the amount of such oxide expected in fused silica is small and difficult to observe, experiments indicate that all charges reaching the cathode are effectively neutralized.
12.5.4 Electrodes Thermal poling of glasses has been made with various low-resistivity electrodes. Thus, press-on stainless steel, silicon, and gold electrodes, as well as aluminum and gold evaporated films, have been successfully employed [34]. Contrary to evaporated metal films, in which the entire electrode area is in contact, press-on electrodes can have a layer of air several microns thick between the electrode and the sample surface. Nevertheless, the resistance of such an air layer is usually small compared with the resistance of the glass, and the poling of glass is successful despite the imperfect physical contact. A careful comparison made between the use of n-type doped silicon, p-doped Si, intrinsic Si, or Au or Al evaporated films [136] shows that the thickness of the depletion region is significantly thinner in the former case, suggesting that the n-doped material leads to the injection of fewer positive charges. An aging effect has also been reported [137], in which the nonlinear spatial distribution changes at room temperature over several months. Twin-hole fibers have been widely used for poling, as described in Section 12.9. The internal electrodes are surrounded by glass and electrical breakdown is prevented even for high (108 V-m1) poling fields. In such fibers, plain- or Au-coated microwires of aluminum, tungsten, and stainless steel have been employed by inserting them into fiber holes on both sides of the core. Here, too, the contact between the wire and the hole wall is imperfect and varies along the device being poled. When high voltage is applied at high temperature, it is possible that the air barrier can be overcome by air ionization. However, when the temperature and the bias voltage are reduced after poling, it is expected that the exact position of the wire in the hole makes a difference for device performance. In some applications, such as frequency doubling or electric field sensing
550
Poling of Glasses and Optical Fibers
(see, for example, Section 12.9.2 and “Voltage Sensing” under Section 12.9.3), the electrodes should be removed after poling. In this case, the wire electrodes may be advantageous because they are more easily removed. Fibers drawn with an internal electrode and a conductive coating based on carbon have been developed for poling long fiber lengths for sensor applications [138–140]. To minimize the uncertainty caused by the position of the wire and to simplify fabrication, various alloys such as AuSn and AgSn [92] have been inserted as liquids into the fiber holes at high pressure, filling the entire available crosssection even after solidification [141]. Poling works effectively and reproducibly under these conditions, even with a molten AgSn alloy. However, attempts to pole fibers with molten BiSn alloy have failed. This was attributed to the injection of cations from the liquid metal into the depletion region being formed (the metal used being too “dirty”). Anodic bonding is sometimes observed when soft glass is poled, causing the electrode to remain permanently bonded to the glass.
12.5.5 Spatial Resolution Although air ionization in the neighborhood of the electrode guarantees that the glass surface is at nearly the same potential as the metal itself, it also implies that fine details of the electrode shape are lost during poling. The ensuing reduced spatial resolution is a severe problem when periodic structures with micron-size dimensions are required, for example in quasi-phase matching. Two approaches have been used to deal with air ionization. The first one is based on the ability to erase the effect of poling by means of a well-focused, high-energy UV [142,143] or e-beams [144]. Uniform poling is first carried out, and then the detailed structure is erased at a later stage. UV erasure is a useful tool for the manufacture of QPM fiber components, as discussed in Section 12.9.2. Vacuum poling [145,146] provides a spatial resolution 1 mm by removing any possibly mobile ions from the surroundings of the anode. Good spatial resolution is also obtained by poling in silicone oil [147]. It should be possible to achieve a similar effect by covering a lithographically prepared electrode with a high-resistivity plastic coating that prevents the spread of the high voltages to regions of the device without the metal electrode.
12.6 THE POLING PROCESS IN DETAIL The physics of poling naturally depends on the glass system under consideration and on the entire treatment to which the material is subjected (e.g., aging in humid atmosphere and polishing, etc.). However, common features of poling
The Poling Process in Detail
551
are known. In this section, the formation of an electret in silica glass and the various mechanisms that contribute to the recorded second-order optical nonlinearity are studied in more detail. An early experiment designed to evaluate the role of dipole orientation versus charge displacement and trapping, with the measurement of the ratio between the nonlinearity induced parallel and orthogonal to the poling field, could not differentiate between the two mechanisms [37]. Even without taking into account experimental problems such as charge redistribution as a result of cutting and polishing the surfaces, the ratio of 3:1 found is consistent both for space charge and dipole orientation models. Another experiment addressed the same issue [148] and once again a ratio of 3:1 was measured. Although the evidence for the dominance of space charge is circumstantial, it is also discussed abundantly in the literature. Despite a large number of studies in poled silica glass, it has not been possible to identify a significant role for dipole orientation in the creation of a second-order nonlinearity. It is possible that dipole orientation occurs at very low temperatures, as claimed in early thermally stimulated depolarization current studies of silicate glass [149]. The theories devised to describe thermal poling have consequently converged more and more to the space charge model.
12.6.1 Poling for Short Time Intervals In the simplest ideal case, after poling, a glass electret is formed by a space charge region a few microns wide, where the density of negative charge is uniform. On the other hand, the bulk of the glass remains unaffected, assuming that the mobile ions do not pile up within the glass and that they are neutralized at the cathode surface. As stated in Section 12.5.3, from experimental evidence, the currently accepted description of poling generally assumes that the cations reaching the cathode are neutralized, rather than accumulated. If the poling process is interrupted by cooling the sample to room temperature, the charge distribution in the glass is frozen. Once the bias voltage is removed, the negative charges trapped in the depletion region attract free positive charges from the environment to accumulate at the anodic surface of the sample. A charged parallel plate capacitor is then formed. Some positive charges are also expected to accumulate on the cathodic surface of the sample. For samples much thicker than the depletion region (e.g., 1 mm versus 10 mm), the number of charges attracted to the cathodic surface is insignificant [98]. In air, the short-circuit condition is satisfied, in which the potential difference between the anodic and cathodic surfaces of the sample is zero. In order words, no work is done by the poled sample when external charges are displaced, from one surface to the other, and
Poling of Glasses and Optical Fibers
552
ð E dl ¼ 0:
ð12:6:1Þ
This implies that besides the thin region of strong electric field that exists pointing from the anodic surface into the glass (i.e., into the depletion region), a weak reverse electric field is also present in the bulk glass pointing from the cathodic to the anodic surface [101]. In this regime, the uniform negative charge distribution in the depletion region and Gauss’s law imply that the electric field is strongest on the anodic surface of the sample and decreases linearly, approaching a value very close to zero at the end of the depletion region. If the third-order nonlinear susceptibility w(3) can be assumed to be uniform, a triangular electric field profile implies a second-order nonlinearity with a triangular shape, with the apex at the anodic surface and diminishing to zero at the end of the depletion region. This condition has been verified experimentally in silica for poling times 5 minutes (see Fig. 12.11, presented later in the chapter). A schematic representation of the charges in the sample, the net density of immobile charge recorded in the glass, and the electrical field distribution are depicted in Fig. 12.9. The shortcircuit condition means that the two areas indicated by (a) and (b) in Fig. 12.9 are equal. Depletion region + -- -- -+ + -- -- -+ + -- -- -+ + -- -- -+ + -- -- -+ + - - +
+ +
Electric field
Net charge density
+
A
B
Figure 12.9 Illustration of the charge- and electric-field distribution in poled silica glass for short poling times after the poling bias is removed. The areas above and below the horizontal axis are equal.
The Poling Process in Detail
553
The initial single carrier model in which the depletion of sodium is considered as the sole contributor responsible for the induced w(2) is inconsistent with the charge dynamics [114] and with repoling studies with inverted polarity [88], which show hysteresis and incubation periods. The model does explain, however, the evolution of the nonlinear layer and the electric field distribution recorded for short poling times, as evidenced in Fig. 12.11, presented later.
12.6.2 Poling for Long Time Intervals Charge migration continues if poling is not interrupted and the sample is subjected to high voltage and heat for longer time intervals. In this longer time frame, neutralization of the negative sites allows the depletion region to expand [34,35,88,95,150]. Neutralization can take place in various ways: (1) A new positive charge may be injected from the atmosphere or from the hydrated glass surface and trapped at negative sites [88], (2) negative ions (such as O) may drift toward the anode [151], and (3) electrons may be ejected from the sample [152,153]. All these effects are seen in silicate glasses, but the dominant neutralization mechanism occurring in poled silica has not been decisively identified in experiments. However, recent literature assumes that the injection of a lower mobility species dominates neutralization, hydronium (H3Oþ) being the main ion exchanging places with the removed high-mobility cations [88]. This ionexchange process is well documented in soda-lime glass, even in the absence of a driving field [154]. Hydrogen ions can either be injected from the atmosphere or be present already in the glass sample, but water molecules are likely to be injected from the environment. Not surprisingly, poling in vacuum [145] or when the electrodes block the entrance of hydrogen and water on the surface [100] results in different charge dynamics. The contamination level of hydrogen atoms at the surface of the sample is expected to be 1016 atoms-cm2 [155]. It is interesting to observe that this corresponds to a number of hydrogen atoms two orders of magnitude larger than the total number of cations displaced from the depletion region in fused silica. For long poling times (e.g., >1 hour at 280 C), most of the negative sites left behind by the cations removed from the depletion region are neutralized. This neutralization is efficient deep into the depletion region, but it is less complete near the surface [120]. However, the deepest layer of negative charge – the one that marks the extremity of the depletion region – is never neutralized. This thin layer has a width <1 mm [94] and separates the high resistivity depletion region from the bulk glass. Therefore, for long poling times, the depletion region is almost entirely neutral and ends in a thin layer rich in negative charge. The remaining field in the depletion region becomes quite uniform (Fig. 12.10), and to a large extent so does the second-order nonlinearity, the profile of which
Poling of Glasses and Optical Fibers
554 Depletion region − − − − − − − − − − − −
+ +
Electric field
Net charge density
+ − + +− + + + − + + + − +
A1 B1
Figure 12.10 Illustration of the charge- and electric-field distribution in poled silica glass for long poling times after the poling voltage is removed. The areas above and below the horizontal axis are equal.
flattens out (w(3) variation accounts for the nontrivial proportionality between field and the induced w(2) [102]). For a typical cation concentration in fused silica, the electric potential varies a few hundred volts across this thin negatively charged region from the bulk to the depletion region. Here again, a positive layer of charges is attracted to the anodic surface of the glass from the environment to cancel the field outside the glass sample after poling is completed and the voltage bias switched off. Figure 12.10 illustrates this by the two areas A1 and B1 being once again equal. As before, a parallelplate capacitor is formed, and the second-order nonlinearity induced is only significant in the depleted region of the sample. The structural modification in the depletion region that follows from poling has been reported in several publications [156–158]. Loss of contrast during etching, optical scattering, x-ray microscopy, and plain microscopy also indicate that a crystalline phase can be created in the glass, mainly within the thin negative layer at the edge of the depletion region [159]. Although the edge of the depletion region moves deeper into the sample as poling progresses, the layer with a phase transition does not seem to become thicker with increased poling time. Similar crystallization and phase separation has been identified in poled fibers [160]. The significance of the structural change at the edge of the depletion region has so far not been discussed in detail in the literature.
The Poling Process in Detail
555
12.6.3 Models The excellent agreement between experiments and a two-ion model [88,120] obtained when a low-mobility species (e.g., H3Oþ) replaces a highly mobile one (e.g., Naþ) is illustrated by the example given in Fig. 12.11 (from reference [120]). The measured profile of the induced w(2) in samples poled for different time intervals (Fig. 12.11a) is compared with the theoretical predictions of the nonlinear profile expected according to the model (Fig. 12.11b). The technique for the experimental determination of w(2) was that of measuring SHG while etching with the arrangement shown in Fig. 12.7 and the application of “layer peeling” [97]. Figure 12.12a shows the theoretical and experimental values of the peak w(2), and part (b) shows the width of the depletion region for various poling times, both determined with the same techniques as in Fig. 12.11. The ion-exchange model is based on solving Poisson’s equation and the equation of continuity for the electric field E and the carrier concentration, which in one dimension can be written as [120] @pNa @ðpNa EÞ @ 2 pNa ¼ mNa þ DNa @t @x2 @x @pH @ðpH EÞ @ 2 pH ¼ mH þ DH @t @x2 @x @E e ¼ ð½pNa pNa ð0Þ þ ½pH pH ð0ÞÞ: @t e 1.2 0.04 0.03 0.02 0.01 0.00
0.8
0
1
2
1min 5min 10min 30min 100min
0.4
0.0
c (2) (pm/V)
1.2
c (2) (pm/V)
ð12:6:2Þ
0.8
0.4
0.0 0
2 4 6 8 Depth under anode (mm)
10
0
2 4 6 8 Depth under anode (mm)
10
Figure 12.11 Data for 500-mm-thick samples. (a) w(2) profiles experimentally obtained with the “layer peeling” method, for samples poled for 1 (see insert), 5, 10, 30, and 100 minutes. (b) The corresponding w(2) spatial distributions obtained with the two-charge carrier model. From [120]: Kudlinski A., Quiquempois Y., and Martinelli G., “Modeling of the w(2) susceptibility time evolution in thermally poled fused silica,” Opt. Exp. 13(20), 8015–8024, 2005. # (2005) Optical Soc. of America.
Poling of Glasses and Optical Fibers
556 1.4
10 Nonlinear width (mm)
1.2 c (2) (pm/V)
1.0 0.8 0.6 0.4 0.2 0.0
A
0
20
40
60
80
Poling duration (min)
8 6 4 2 0
100
B
0
20
40
60
80
100
Poling duration (min)
Figure 12.12 Time evolution of the maximum value of the w(2) induced (a) and of the width of the nonlinear layer (b). Squares correspond to experimental data, and solid lines represent numerical simulations performed with the two-carrier model from [120]: Kudlinski A., Quiquempois Y., and Martinelli G., “Modeling of the w(2) susceptibility time evolution in thermally poled fused silica,” Opt. Exp. 13(20), 8015–8024, 2005. # (2005) Optical Soc. of America.
Here, pNa and pH are the carrier concentrations of sodium and of the hydrogenrelated species, and D and m are the respective diffusion constants and mobilities. The boundary conditions are important to define the Ð dynamics of the problem. One can relate the applied voltage to the field: Edx ¼ Vapp, where the integration limits are zero and L (sample thickness). The initial sodium concentration in the sample is assumed to be uniform and zero for the hydrogen-related species. The actual value of pNa (t ¼ 0) determined from the slope of the triangular profile of w(2) measured for short poling times is 9.5 1022 m3, which corresponds to 1.6 ppm. Finally, the model of reference [120] assumes an injection rate into the anodic surface to allow for ion exchange proportional to the electric field at that surface. Thus, it is assumed that @pH ¼ sH E; ð12:6:3Þ @t x¼0 where sH is a constant determined from the best fit of the model. By setting the mobilities to mNa ¼ 1.5 1015 m2-V1.s1, mH ¼ 2 1018 m2-V1.s1 and sH ¼ 5 1012 m2-V1.s1, one finds excellent agreement of the shape, behavior in time, and absolute values between theory and experiments. The agreement is equally good for the studies carried out with a 200-mm-thick sample [120]. One may wonder how it is possible that a relatively simple model as the one of references [88,120] can describe poling so well. For example, it is known from various studies of elemental analyses of poled glass that more than two species move under the high temperature and high fields created by poling. SIMS measurements correlate well the widths of the second-order nonlinearity with the
The Poling Process in Detail
557
region depleted of lithium, not sodium (which is much wider). Injection of metallic ions such as aluminum, lead, and zinc has been reported in various experiments. Likewise, a pileup of cations of relatively low mobility, such as calcium and potassium, at the edge of the depletion region has been also identified in various studies. A complete model of poling should take into account these and other factors, such as the presence of various defects in the pristine bulk glass and surface imperfections (e.g., the presence of water molecules). To explain all features identified, such as the spatially dependent increase in w(3), it may be required to include even the orientation of polar entities. Various other models have been proposed to describe the physics of poling. For example, the concept of ionic waves was introduced in reference [161] to explain deep and shallow poling. The nonlinearity induced is deep in synthetic silica, and when the depleted layer width exceeds the coherence length, it gives rise to short period oscillations, which are identified as a threshold for bulk nonlinearity in the Maker fringe measurements. The recording of second-order optical nonlinearity through poling invoking three distinct fields was described in a series of articles and summarized in references [162] and [163]. In the initial phase of the process, a shielding field is established that cancels the poling field. If poling is interrupted, a residual shielding field remains and gives rise to a w(2). For longer poling times, ionization sets in through negative charge emission toward the anode or by the injection of positive charge from the anode into the glass. The ionization field thus created reduces the shielding field. Eventually, the ionization dominates and the sign of the nonlinearity reverses. The residual field recorded is then dictated by the ionization field. Although this model can explain the observations of various experiments [162], so can the ion-exchange model described previously [88,120]; ionization simply reduces the number of available negative carriers trapped in the space charge region. The dissociation and recombination of charge during thermal poling were considered in reference [36]. The model predicts that a minimum poling voltage is required to establish a second-order nonlinearity, and its value was experimentally determined to be 900 V: For voltages below this value, no SHG could be observed [36]. Charge separation and trapping under steady-state conditions and UV excitation was investigated theoretically in reference [164].
12.6.4 Erasure and Stability The second-order nonlinearity induced in glasses needs to be stable over very long periods – decades for poling to be useful in commercial applications. This long-term requirement has to be met even if storage and operation take place at, say, 70 C, with or without a voltage bias. Because the second-order nonlinearity is first induced at moderate temperatures (typically below 300 C) and because it
Poling of Glasses and Optical Fibers
558
can be erased by heat alone [34], studies are required to demonstrate that the effect is long lived, not only for researchers but also for potential users of the technology. Although one can find a number of papers in the literature showing decay and erasure of the nonlinearity, few articles demonstrate long-term stability. The electric field recorded in poled silica glass can be canceled by neutralizing the immobile negative charge in the depletion region and at its boundary with the bulk glass. This neutralization only requires the drift of high-mobility cations over a few microns, from bulk toward the anodic surface of the sample. Besides diffusing, the mobile cations in the bulk of the heated sample are driven by the small but finite reverse field pointing toward the negative charges from the cathode surface. If the sample is poled for a short time, so that the single carrier regime applies, the process is expected to be entirely reversible. Erasure by heat should return the sample to its pristine state. However, if the sample is poled for a longer time interval so that a low mobility ion replaces the highly mobile species, neither heat alone nor heat with a reverse bias will remove the slow species from the sample. Nevertheless, erasure of the second-order nonlinearity can still take place, because the recorded field can be canceled even if a region of the glass is still depleted of a particular type of ion that was exchanged with another cation. One such example is shown in Fig. 12.13, in which SIMS analysis shows that even after removal of the previously induced second-order optical nonlinearity by heat, the region near the anodic surface can still be
H count ratio (to 30Si)
1H
10–3 0.002 7Li
10–4
39K 23Na
(x 0,2) 0.001
10–5
Li, Na, & K count ratio (to 30Si)
10–2
0.003
10–6 0.000
0
10
20
30
40
Approximate depth under anode (mm) Figure 12.13 Anodic region of a depoled sample. Original poling was for 155 min at 275 C, depoling (V ¼ 0) for 1 min followed by cooling. No second-harmonic signal is observed after depoling, but depletion regions of sodium (34 mm wide) and lithium (16 mm wide) are still clearly evident through SIMS analysis. Figure 6 from [108]: Alley T.G., Brueck S.R.J., and Wiedenbeok M.,“ Secondary ion mass spectrometry study of space-charge formation in thermally poled fused silica,” J. Appl. Phys. 86(12), 6634–6640, 1999. Reprinted with permission. # (1999) American Institute of Physics.
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559
depleted of sodium and lithium [108]. It should be remarked that when more than one type of cation moves in the glass to form a depletion region (e.g., Naþ and Liþ), erasure may acquire different time constants, each one associated with one carrier. Experiments have been carried out to examine the w(2) distribution after various annealing times to shed light on the erasure mechanisms [165]. A relatively low poling temperature (250 C) was used to slow down the process. Two groups of samples were poled, one for a short time (5 minutes) and the other one for a long time (100 minutes). After poling, the high voltage was switched off with the samples still at high temperature. This allowed annealing of the samples for different time intervals without an uncertain warming up time. Samples poled for 5 minutes had a triangular nonlinear profile that erased from the cathode to the anode side as a function of the annealing time, as illustrated in Fig. 12.14a. This result is consistent with a depletion region in which the charge distribution is uniform and the recorded electric field has a triangular profile, as schematically shown in Fig. 12.9. The samples poled for 100 minutes annealed with more complicated dynamics, as seen in Fig. 12.14b, consistent with the charge distribution schematically shown in Fig. 12.10. Fitting the simulations to the experimental data requires the modification of the mobility value used previously, perhaps indicating that the charge movement is more complex than assumed in reference [165]. It has been reported that the stability of thermally poled silica glass is critically dependent on the Type of fused silica used as well as the storage conditions [166]. Although type I (electrically fused) silica samples showed no decay in time, Type II (flame fused) silica exhibited a marked reduction in the SHG signal over a period of 400 hours, except for a sample poled in air and stored in dry atmosphere. This study has important bearings on the stability of poled silica devices.
c (2) susceptibility [pm/V]
c (2) susceptibility [pm/V]
0.5 0.5
Poling time: 5 min Not annealed Annealed for 5 min Annealed for 15 min Annealed for 30 min
0.4 0.3 0.2 0.1 0.0
0.3
Annealed for 100 min
0.2 0.1 0.0
0
A
Poling time: 100 min Not annealed Annealed for 15 min Annealed for 30 min
0.4
1 2 3 Depth under anode [mm]
0
4
B
2 4 6 8 Depth under anode [mm]
10
Figure 12.14 Spatial distribution of the second-order nonlinearity remaining in annealed samples poled at 250 C for 5 minutes (a) and 100 minutes (b). From [165]: Quiquempois Y., Kudlinski A., Martinelli G., Quintero G.A., Gouvea P.M.P., Carvalho I.C.S., and Margulis W., “Time evolution of the second-order nonlinear distribution of poled Infrasil samples during annealing experiments,” Opt. Express 14(26), 12984–12993, 2006. # (2006) Optical Soc. of America.
Poling of Glasses and Optical Fibers
560
Another highly relevant study was carried out with a comparison of the thermal stability of different types of glasses under isothermal annealing [167]. The decay of the second-order nonlinearity was fitted with two time constants. Aluminosilicate and aluminoborosilicate glasses exhibited characteristic decay time constants at least five orders of magnitude longer than a silica reference sample. The enhanced stability was attributed to a strong decrease in the mobility of the alkali ions because of the glass composition. Here, poling of the silica sample was carried out at 280 C, whereas all other samples were poled at higher temperatures. The most stable sample was poled at the highest temperature (420 C). The stability of poled fibers and devices is a topic that deserves more attention in the literature. Short w(2) decay times (some days) have been observed in normal [168] and special fiber [169]. This led to the development of a fiber with a borosilicate glass ring around the anode to trap negative charges and improve stability [168]. However, long-term stability was reported in reference [170], in which Vp was found to be unaffected at room temperature over a period exceeding 2 years. The discrepancy may arise from the difference in materials, poling conditions, or the presence of air in the internal electrodes of references [169] and [168] compared to a solid metal electrode filling the entire cross-section of the holes in reference [170].
12.7 ROUTES FOR INCREASING THE SECOND-ORDER OPTICAL NONLINEARITY The low value of the induced w(2) in practical devices is the main reason for the lack of widespread use of poled glasses. In poled silica glass fibers, for instance, the typical value of the induced second-order optical nonlinearity is only 0.2–0.3 pm/V [171]. Poled fiber devices, therefore, need to be long to compensate for the low nonlinearity. This limits the electrical bandwidth of the devices and increases electrode resistance as well as electrode-induced loss. Long QPM structures are also required for frequency doubling applications, imposing severe fabrication tolerances and limiting bandwidth. Therefore, the field of poling glasses, planar waveguides, and fibers would gain enormous momentum if the recorded nonlinearity could be improved, because the nonlinear conversion depends on the square of the nonlinear coefficient. Much effort has been directed at trying to achieve such a goal since the pioneering work by Myers et al. in 1991 [35]. Progress in this area is summarized in the present section. As noted previously, in the vast majority of cases, poling is well described by the nonlinear term ð2Þ
weff ¼ 3wð3Þ Erec :
ð12:7:1Þ
Routes for Increasing the Second-Order Optical Nonlinearity
561
Increasing the second-order nonlinear coefficient then involves increasing the recorded field, for example, by optimizing the poling procedure and improving the breakdown voltage, or increasing the third-order susceptibility, through special doping, exploiting resonances, or simply replacing silica by materials with higher intrinsic w(3).
12.7.1 Poling Methods (Optimization and Novel Techniques) Several studies address the question of optimization of the poling conditions of silica glass. The temperature required for maximizing w(2) in silica is 280 C, as shown in Fig. 12.8 [115]. However, this optimum temperature is related to the applied voltage. By increasing the poling voltage and the poling temperature, a higher w(2) can be obtained [172]. Otherwise, in accordance to Eq. (12.5.5), it is found that the recorded w(2) increases with the voltage applied during poling, because the recorded field is proportional to the charge displaced. In Suprasil, alternating the poling polarity at a particular frequency has been found to increase the induced second-order nonlinearity by one order of magnitude [62,64]. The value of the induced w(2) then is w(2) 0.3 pm/V, approaching the nonlinearity achieved in fused silica of much higher cation impurity content (e.g., Infrasil). The depletion region was found to be as wide as 80 mm, suggesting that the technique may be useful in poling optical fibers, where the induced nonlinearity is high whereas reducing the loss caused by a near-lying electrode [64].
12.7.2 Increasing E-Field Breakdown Little effort has been placed on trying to increase the breakdown limit of silica in studies related to poling. One valuable experimental procedure is to insert low-resistivity glass (e.g., soda-lime) slices between the metal electrodes and the silica sample to be poled [34]. This limits the maximum current, and higher poling voltage can be applied without causing electrical breakdown. It is also possible and advantageous to increase the poling voltage at a time close to the end of the poling process when the temperature is being reduced [173,174], such as a two-stage poling [175].
12.7.3 Increase x(3) through Poling Various papers report an increase in the third-order nonlinear susceptibility as a consequence of poling. Phase transformation via UV exposure and crystallization in the presence of a poling field were described to lead to w(3)final 15 w(3)initial [176] in Ge-doped silica. However, an increased w(3) was also reported
562
Poling of Glasses and Optical Fibers
in thermally poled waveguides (increase 2 times) [177], fibers (2–3 times) [178,179,180] and glasses (1–2 times) [102] and not only correlated to material phase transformation. Explanations for the increase in the w(3) were attempted [181], invoking local field corrections. In contrast, various reports also exist reporting that the w(3) does not change significantly in waveguides [113] and fibers [170] after poling. The w(3) is a measure of the polarizability and therefore an intrinsic property of the material. However, charging, ion exchange, and crystallization are reported consequences of poling. It is expected that when the amorphous material undergoes a phase change and crystallizes, the w(3) is also modified. Even in the absence of a structural change, it is known that charges and embedded ions modify the polarizability of the medium [182]. Changes of w(3) in the vicinity of the space charge layer are perhaps not surprising. On the other hand, the w(3) modification is likely to be restricted to the neighborhood of the space charge and therefore remain local. Variations of w(3) over a few microns may explain the discrepancy of results from various authors, which probe the effect with a light guiding core situated at various distances from the anode electrode [102].
12.7.4 Increasing x(3) through Resonance and Doping Exciting results of a 25 pm/V nonlinearity induced in silica glass poled with silver-painted electrodes have been reported [109,183]. Frequency doubling measured in reflection used to characterize the poled samples showed that surface plasmon resonance (lp 0.40.5 mm) was responsible for the large enhancement of w(3) (2o, o, o). The induced electro-optic effect exploiting w(3) (0, o, o) unfortunately remains low. It has been impossible to achieve comparable w(2) values in transmission measurements through glass samples as in reflection, and it is likely that the high value of measured w(2) is associated with a large and unavoidable absorption, which hinders the exploitation of the large w(2) in frequency doubling applications. The enhanced w(3) expected from doping fibers and planar waveguides with rare earths [184] has surprisingly not been exploited in poling.
12.7.5 Glasses Other Than Silica There are a large number of articles describing poling of glass systems other than silica with the objective of achieving a large w(2). Some of these studies have been motivated by plain scientific curiosity and others by the concrete goal of increasing the induced nonlinearity, as the w(3) of most glasses is higher than in silica. Also, soft glasses such as soda-lime and borosilicate glass are considerably cheaper than pure silica and facilitate the combination of poling with the
Routes for Increasing the Second-Order Optical Nonlinearity
563
manufacture of planar waveguides by ion exchange. The summary of the work listed here is not exhaustive but gives examples of the glass systems investigated in conjunction with thermal poling. A relevant question in this context is, is a high refractive index glass with a large nonlinear coefficient induced by poling but smaller than that of lithium niobate, useful for commercial applications? It is likely that only those glasses that can be pulled as fibers will be considered useful in the future because of the developmental work required. Short device length crystals may continue to dominate the market.
Nanocrystals The search for glasses with high third-order nonlinearity, in many cases, has focused on the presence of nanocrystals of various types. Crystallization often leads to a large increase in the nonlinear coefficient [185–188]. However, it also brings about increased scattering and propagation losses in waveguides/fibers [189], which may prevent the use of the material in most applications. Relatively high w(2) has been reported – for example, 1.2 pm/V in CdS-nanocrystal alkaliborosilicate glass and 5 pm/V in borophosphate glass with NaNbO3 nanocrystals (see the section titled “Phosphates”) [190]. Crystallization of Ge-SiO2 glass under UV excitation in the presence of an electric field has been also reported [22], and the microcrystals themselves were found to possess an intrinsic second-order nonlinearity [191].
Heavy Metal Oxides Glass compositions incorporating heavy ions (heavy metal oxides) of large polarizability have also been investigated. The motivation for using such glasses despite the mismatch with standard optical fibers and problems with splicing is that the w(3) increases rapidly with the refractive index according to Miller’s rule [192] (e.g., the nonlinearity expected when the refractive index exceeds 1.74 is one order of magnitude larger than in silica). Among others, glasses containing Bi and Pb have been poled thermally. Poling of silica glass with Bi2O3 incorporated at various concentrations as a network modifier or former results in a second-order nonlinear coefficient as high as 0.64 pm/V [193]. This investigation is particularly relevant because high-quality planar waveguides and optical fibers can be fabricated with highcontent Bi glass. The best poling temperature is found to be close to glass transition temperatures and the poling process involves proton migration, glass ionization, and ion injection [193]. Unfortunately, the maximum recorded field in bismuth borate decreases with increasing Bi2O3 content, preventing the creation of a very large w(2). As early as in 1993, lead silicate glass was poled by electron implantation, and a second-order nonlinear coefficient w(2) 0.7 pm/V was measured. PbO/
564
Poling of Glasses and Optical Fibers
B2O3 glass was poled thermally, resulting in a deep (bulk) distribution of the recorded nonlinearity [194]. This is attributed to the depletion of Hþ near the anodic surface and of Pb2þ in the bulk [195]. Irradiation of commercial ZF7 glass (71% PbO) with electrons (total charge 0.29 C/m2) results in an efficient w(2) 4 pm/V [196,197]. The second-order nonlinearity is found to increase (linearly) with the Pb content of the glass. The same glass thermally poled gives a w(2) 7 pm/V [197,198]. Other work on PbO containing glass includes poling of silicates [199] and PbO-Bi2O3-Ga2O3 [200]. A large (15 pm/V) second-order nonlinearity was induced in thermally poled Pb-glass waveguides [201]. The sign of the nonlinearity was opposite to that induced in poled silica. By polishing the side of a sample at a shallow angle, the nonlinear region could be studied with high spatial resolution. An average electro-optic coefficient four times larger than in fused silica was reported.
Tellurites Since the early 1990s, tellurium-based glasses have been discussed in the context of poling [202]. A linear relation was found between the optimum poling temperature and the glass transition temperature [203]. Heat treating NaO2ZnO-TeO2 glasses (at 300 C) and then poling them at the (lower) optimum temperature (280 C) results in a much weaker nonlinear coefficient than just poling without the initial heat treatment, suggesting that the glass undergoes a transformation near the anodic surface before heating. This is in contrast with the findings in reference [175], in which a two-stage poling procedure gave good results for the TeO2-Bi2O3-ZnO system: The first stage consisted of poling at low voltage but at a temperature exceeding Tg, and the second stage consisted of a conventional poling treatment. The stability of Na-rich telluride glass was found to be much worse than for Zn rich in reference [203]. The value of w(2) induced is 0.45 pm/V. WO2 – TeO2 glass has also been poled, and a nonlinear coefficient w(2) 2.1 pm/V is induced at 250 C [204]. The decay time of the second-order nonlinearity is relatively short (15 days). Tellurite glass samples containing lead (70%TeO2 – 25%Pb(PO3)2–5%Sb2O3) have been poled in vacuum and studied after poling with infrared reflectometry, Maker fringe SHG, and dielectric susceptibility spectrometry at various temperatures. The results show migration of cations (Naþ) from the anode side of the sample with a structural transformation in the space charge region [205], leading to a second-order nonlinearity one order of magnitude larger than in poled silica. An interesting result is obtained with ZnCl2-rich TeO2-B2O3 glass, in which the induced second-order optical nonlinearity is found to depend on the amount
Routes for Increasing the Second-Order Optical Nonlinearity
565
of ZnCl2 in the composition. The recorded w(2) is attributed to the movement of anions from the cathode to the anode side of the thermally poled sample [206].
Chalcogenides Like tellurides, chalcogenide glasses have been drawn to good quality optical fibers. Several such glass systems have been poled. For example, a comparative study of thermal poling was carried out for GeSx glasses, where x was varied from 3 to 6 at various poling temperatures [207]. The SHG efficiency peaked for x ¼ 5 and near the glass transition temperature. A second-order nonlinear coefficient as large as w(2) 8 pm/V has been reported in Ge25Sb10S65 [208]. The high value is assigned to a large w(3) 1.7 1020 m2 V2 and a relatively strong recorded field Erec 1.5 108 V m1. The refractive index of the glasses investigated is n 2.1. One common issue for all such nonsilica systems is the stability of the induced nonlinearity. It is often the case that the nonlinear coefficient decays rapidly after poling. For example, in reference [207], the second-order nonlinearity decayed with a typical time constant of 10–15 hours. This is of great concern to any attempts at making devices with poled fibers. In chalcogenides, whereas Gallium-containing glasses experience relaxation by recombination a few days after poling, Ge25Sb10S65 glasses exhibited SHG up to 6 months after the poling treatment. A useful technique was described in reference [209], where the stability is increased from hours to months by repeating the poling procedure a few (4) times. It would be of interest to determine if this procedure only applies to chalcogenide glasses, or the stability of poling can be improved in other glasses with a similar technique.
Phosphates Poling of commercial phosphate glass (Schott IOG-1) has also been reported [210]. The authors observe through Maker fringe measurements that a secondorder nonlinearity is recorded both in the bulk and near the anodic surface of the sample, but with opposite signs. The bulk component is more stable with temperature, and a single carrier model with a nonblocking cathode accounts for most of the observations [210]. Borophosphate glass containing Nb and Na oxides [211] shows the highest SH signal obtained from these glasses when poled: only 13% of that in poled silica, although w(3) (for 20% Nb2O5) is 26 times larger than for Herasil. Furthermore, the SH signal is found to decay in minutes [211]. Likewise, borate glasses of various compositions also show much less SHG than poled fused silica [212]. Although recent studies indicate a high nonlinearity (w(2) 5 pm/V) in such glass systems [190], it is associated with a crystallization of the sodium niobate phase [213] and is discussed in reference [185].
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Poling of Glasses and Optical Fibers
Soft Silicates Soda lime is an attractive material to process by poling. It is inexpensive, highly compatible with the ion-exchange techniques for waveguide fabrication, and the base material for other glass systems, such as with semiconductor doping (colored glass). Poling of soda lime and other soft silicate glasses (e.g., borosilicate) is hindered by the massive currents induced with the high-voltage bias at high temperature, as the sodium content is typically 15%. Thermal runaway can be prevented by raising the voltage in sufficiently small steps or by using a current limited high-voltage supply [214,215]. As the current becomes gradually limited by the formation of a depletion region, the bias can be increased further to the few kilovolts level. It is found in poled soda lime and borosilicate that the initial second-order nonlinearity induced is comparable to that of poled fused silica, but the SHG level decays rapidly both spontaneously and stimulated by the probe beam [214,215]. Thermal poling of soda lime leads to a massive atomic rearrangement [158]. Because the refractive index changes considerably, waveguides can be fabricated by poling [216,217]. Bulk and near-surface second-order nonlinearity were induced in BK7 borosilicate glass [218] by poling at a high temperature (400 C) for long times (e.g., 4 hours). The decay of the second-order nonlinearity was fast; the one associated with the bulk was slower than for the near-surface one, which dropped to half the original value at room temperature after 2 hours [218]. Various characterization techniques (SHG microscopy, etching, energy dispersive spectroscopy (EDS), x-ray diffraction (XRD) analysis) were used in poled Schott D263 borosilicate glass [219] and Pyrex glass [220]. Depletion of Kþ is identified and of Naþ inferred. As in soda-lime glass, accumulation of sodium oxides on the cathodic surface is observed, confirming the neutralization of the mobile Naþ migrating ions on the cathode side. A useful way of improving the stability of the induced second-order nonlinearity in soft glasses such as soda lime is reported in reference [221], in which the temperature and voltage are raised in a current controlled way. Perhaps a similar technique can be employed for stabilizing the nonlinear coefficients induced in other glasses, such as chalcogenides.
Comparison of Different Poled Glass Materials Table 12.1 illustrates some of the highest values expected when various glass systems undergo thermal poling. A common trend found for most materials is that although the third-order nonlinearity is much larger than the w(3) in silica, the lower field recorded results in similar (but often lower) w(2) than in silica (1 pm/V). In the few cases where the w(2) induced is significantly larger than in
Glass system
Typical poling conditions
Silica
280 C/4kV/15 min
NanoXtal Na12Nb15P8O65 Bi2O3–ZnO–B2O3 ZF7 (PbO)
310 C/1 kV/0.5 hr 210 C/2.5 kV/0.5 hr 300 C/4 kV/3 min
0.1 mm thick PbO film WO2-TeO2 Ge25Sb10S65 Ga5Ge20Sb10S65 IOG1 (phosphate) Soda lime Soda lime BK7 Borosilicate Ge-silicate Ge-silicate
Eff. w(2)
Decay time (RT)
(pm/V)
Type of system
Ref.
1 5
Years ?
Bulk Bulk
[34] [190]
0.64 7
? ?
Bulk Bulk
[193] [196]
15
?
[201]
250 C/3 kV/20 min 170 C/4 kV/5 min 230 C/4 kV/30 min once; repeated four times 150 C/1-2 kV/40–120 min 280 C/slow increase to kV/1 hour up to 250 C, two stages/1 kV/hours 350 C/slow increase to 3.5 kV/30 min 300 C/4 kV/2 hours
2.1 8 4 4 0.04 1
15 days >6 months
Planar waveguide Bulk Bulk Bulk
3 days > months >10 months minutes
Bulk Bulk
[210] [214,215]
0.4
months
Bulk
[221]
1.6–2
2 hours
Bulk
[213]
0.04
?
[222]
280 C/4 kV/15 min –>1 hour
0.2
Days [168]; Years [170]
Channel waveguide Fiber component
350 C/3.5 kV/15 min 4 slabs in series 275 C/3 kV/15 min
[204] [208] [209]
Routes for Increasing the Second-Order Optical Nonlinearity
Table 12.1 Typical x(2) values induced in different glass systems
[222,223,168,170]
567
568
Poling of Glasses and Optical Fibers
silica glass, the question must be asked as to whether or not it is possible to make a fiber or planar waveguide device with such a material competitive with crystalline devices.
12.8 POLED FILMS AND WAVEGUIDES
12.8.1 Materials and Systems Planar waveguides in glasses allow for the integration of several functions such as high port count splitting and interferometry in diverse configurations. Active control of such devices is of great interest, particularly if associated with the low propagation, and fiber coupling losses are possible in glass waveguides (see, for example, reference [224]). The combination of poling with planar waveguides makes natural use of the guided propagation in the thin induced nonlinear layer. Possible applications include electro-optic control of waveguides and interferometers as well as frequency conversion (e.g., SHG) in lithographically fabricated periodic QPM structures. Some of the techniques used for the manufacture of planar waveguides, such as plasma-enhanced chemical vapor deposition (PECVD), make use of synthetic materials of extremely high purity. In such systems, the contamination levels are low, and the creation of a second-order nonlinearity through poling involves mainly the interfaces, where stress-related defects exist and charge trapping occurs [205]. In other planar systems, such as those deposited by sol-gel films on a substrate, materials are not so pure and a space charge region can be created by the displacement of cations or the polarization of anions (such as OH [225]), although the interfaces in the material still play a role [226]. Any granular structure of the material with surface states can also contribute to the recording of an electric field [227].
12.8.2 Physics and Characterization Soon after the report by Myers et al. [34], a sputtered film of Corning 7059 glass on a borosilicate substrate was corona poled for frequency doubling [42,43]. A w(2) < 1 pm/V was reported. Phase matching was accomplished by tapering the waveguide so that the material dispersion canceled the waveguide dispersion at a particular thickness, a scheme not reproduced since in inorganic glasses. A channel waveguide written using a 30-keV electron beam was thermally poled at 270 C and a modulator with an active length of 4.8 mm operating at 633 nm showed 32 mrad phase shift for an applied field of 7.3 106 V-m1 [228].
Poled Films and Waveguides
569
A 2 2 silica-on-silicon Mach–Zehnder interferometer (MZI), poled at 300 C with a 4-kV bias, resulted in switching with a 20-dB extinction ratio, 4-dB insertion loss, and a rise time <100 ns at a voltage Vp 1.7 kV [229]. The phase modulating section was 36 cm long, and the (nearest) distance between the anode electrode and the core of the waveguide was 9 mm. Poled waveguides are also used in the investigation of fundamental properties of poled glass. A second-order nonlinearity that was polarization independent is observed in waveguides [230]. Although the initial value of w(3) differed by 25% for the two orthogonal polarization eigenstates, no change was observed after poling [113], in contrast to observations made by other authors using waveguides [177] and fibers (see Section 12.7.3). An increase in the value of w(3) by a factor 1.9 was measured after poling [177] and found to be independent of the poling voltage.
12.8.3 Quasi-Phase Matching Thermal poling of a fused silica QPM waveguide with periodic electrodes made by photolithography has been demonstrated [231]. Phase matching over 2 mm at 1064 nm was accomplished, and a nonlinearity equivalent to d11/ 200 of that of quartz was estimated from the measurements. An alternative technique reported to create a QPM structure makes use of a thermally poled planar fused silica that is exposed to spatially periodic illumination with UV, erasing the second-order nonlinearity (i.e., the recorded field) at the correct period (48 mm) to guarantee phase-matched SHG [232]. A similar technique has been exploited for frequency doubling in fibers, as described in Section 12.9.2. A considerable amount of work was carried out trying to exploit the surface plasmon resonance from in-diffused silver for frequency doubling, in which a nonlinear coefficient 25 pm/V was reported [109,183,233]. Other noble metals were also investigated [234]. The main difficulty encountered in exploiting the resonance was avoiding the loss incurred by the silver clusters [234,235]. Quasi-phase matched structures were fabricated [236], but no efficient conversion has been demonstrated. It is possible that the presence of silver reduces the recorded electric field in the glass, thus reducing the benefit of increasing the third-order nonlinearity.
12.8.4 Bleaching The development of nanocrystals in glasses for poling has not yet led to a breakthrough in the exploitable w(2) for electro-optic or frequency-doubling applications. The expected resonance enhancement in w(3) is to a large extent
570
Poling of Glasses and Optical Fibers
compensated for by an increase in absorption, scattering losses, and a reduction of the field recorded in the space charge region. On the other hand, noble metal nanocrystals open the doors to a large number of other interesting applications of poling. Soda-lime glass can have a surface layer of metal nanocrystals (e.g., Ag a few microns deep) fabricated through an ion exchange process. When the sample is poled, the nanoclusters are imbedded in the depletion region under extremely high electric fields and ionized. Positive silver ions then drift away from the anode toward the negatively charged layer at the end of the depletion region. The negative charge left behind is balanced by field emission of electrons from the metal nanocrystals first into the matrix and ultimately into the anode electrode. These two combined effects lead to the gradual dissolution of the silver nanoclusters. The sample becomes discolored (bleached) if the metal layer originally had color [237–240]. A similar effect is also observed with gold [241,242]. Bleaching of the glasses leads to greatly increased transparency or coloring. Bleaching of Ag-nanoparticles under simultaneous electric field and illumination has also been reported [243], which may have interesting applications in electric-field-assisted and laser-induced dissolution experiments. Another interesting application of poling is the possibility of implanting Naþ ions in a process described as poling-assisted ion implantation [244]. The authors used a sodium-containing glass slide on the anodic surface of a Tebased glass (2Ag2O 3Na2O 25ZnO 70TeO2 in mol %) and poled it at 300 C with a 3-kV bias. They found that the Te-glass became colored through the injection of Naþ ions, and that Agþ ions were reduced to Ag metal, creating silver nanoparticles that precipitate in the vicinity of the anodic surface [244].
12.9 POLED FIBERS Silicate fibers are by far the best waveguides one can fabricate, and the confined and uniform propagation allows for the accumulation of nonlinear contributions over lengths exceeding L > 106 l. Therefore, even weak nonlinear coefficients can result in efficient frequency doubling or >p-phase shifts in electro-optic fibers. The recognition that in order to subject the core of an optical fiber to a large electric field, one needs to bring the electrode near the core, led to the use of etched micro-fibers, D-shaped fibers, and fibers with internal electrodes (i.e., twin-hole fibers). The latter configuration is particularly convenient as voltage breakdown can be avoided. Following the developments in optical poling, but still before the report by Myers et al. in 1991 [34], a study was carried out in which a twin-hole fiber with internal electrodes was exposed to high-power blue light illumination in the presence of an applied DC field [245]. The purpose was to promote the
Poled Fibers
571
alignment of defects excited optically with the field. This scheme allowed for optical poling, much in the same way as with UV poling reported in 1996. A twin-hole fiber was also used with liquid indium/gallium electrodes in 1986 for electro-optic (Kerr) phase modulation [246]. The superposition of a 400-V bias to a sinusoidal modulation voltage led to a phase shift of 1.7 radians/m for a signal of 67 Vrms. This Kerr device was 30 m long [246]. Soon after thermal poling of silica was reported in 1991 [34], two groups described thermal poling of fibers [222,223]. In early thermal poling experiments with fibers, acoustic resonances were observed, possibly enhanced by a lack of proper anchoring of the fiber to a rigid substrate [222,247]. Various applications of poled fibers followed and are summarized in Section 12.9.3. Recently, poling of photonic crystal fibers (holey fibers) has been reported [248,249]. The design freedom can help in extending the period for quasi-phase matching [249], but the greatest interest is in the future possibility of poling photonic crystal fibers fabricated with high-nonlinearity glasses. A competing technique to the one discussed in this chapter is that of polishing a standard telecommunications fiber to expose its core and deposit on it a film of polymer with a high electro-optic coefficient. Excellent performance is achieved at room temperature after poling the film, such as low loss and 40 Gb/sec capability.
12.9.1 Physics and Characterization Although the underlying mechanisms of poled fibers and waveguides are the same as for bulk silica, some differences arise from the special geometry and material composition of core and cladding. The second-order nonlinearity induced in fibers is studied in a number of ways. Interferometers are used for the characterization of the linear electrooptic effect. They become a powerful tool when used in situ [222,250], so that the dynamics of the linear electro-optic effect is tracked as a function of poling time [178]. The second-order nonlinearity is evaluated while poling (or depoling), and simultaneous temperature and current measurements can be performed [58]. Various studies of the time evolution of the depletion region in poled fibers were carried out with SHG microscopy [76,77]. They showed that similar to waveguides [251], the core of the fiber behaves as a barrier [252] that can be overcome [253]. In one experiment, the depletion region extended all the way to the cathode electrode [78], which potentially eliminates the need for using fibers with a core near one of the holes (anode) [254]. Generally, however, the core of the twin-hole fiber is fabricated a few microns from the anode electrode, and for short poling time the negative
572
Poling of Glasses and Optical Fibers
charges are still mostly located between the anode and the core. For longer poling times, on the other hand, the depletion region can become sufficiently wide to encompass the core. When this happens, neutralization results in most of the remaining negative charges to be located in a thin layer between the core and the cathode (see Fig. 12.10). These negative charges are the main source of the electric field experienced by the core. The effective displacement of the negatively charged layer from the anodic side to the cathodic side of the core leads to a sign change during poling [92] and explains, with the model discussed in Section 12.6.3, some of the effects originally attributed to the growth of an ionization field [162].
12.9.2 Quasi-Phase Matching Ever since the report that thermally poled silica could lead to second-order nonlinearity 103 times larger than achieved with optical poling, periodically poled fibers and waveguides became a goal pursued by various groups. QPM electrode structures were fabricated in waveguides [231] and in fibers as early as 1991 [255], using photolithography [256] and, more recently, through ablation of a Ag-film inside a twin-hole fiber [257]. Although the coherence length in glasses (20 mm) is two to three times longer than in ferroelectric crystals and this simplifies fabrication, the nonlinear coefficient available is much lower in silica and thus longer devices are required. Although the conversion efficiency scales with the square of the length, the bandwidth reduces with the inverse of the length of the QPM grating. SHG in fibers induced by a periodic electric field and exploiting an external electric field–induced second harmonic generation was demonstrated in 1989 [258]. A periodic electrode pressed against a fiber that had its core exposed through polishing led to phase matching, the angle between the fiber and the electrodes defining the phase-matched mode. In a series of reports, a D-fiber with one internal electrode and one periodic electrode fabricated on the flat by photolithography has been successfully used for frequency doubling. To maintain the required spatial resolution, the fibers were poled in vacuum. A tunable picosecond Ti:sapphire laser was converted into the blue [259,260], and 76 mW average power was obtained at 422 nm (for a 35-mW input power). The period for phase matching grating was 25 mm, and the device was 1.8 cm long [260]. In a similar setup, nanosecond pulses from a 1.532-mm high-power Er-doped amplifier were frequency doubled into the red by a QPM poled fiber of length up to 7.5 cm and periodicity of 56.5 mm. A top conversion efficiency of 30% and average power 6.8 mW were demonstrated [261].
Poled Fibers
573
Uniform thermal poling followed by periodic UV erasure has also been used as a means to create a QPM structure, not only in bulk silica [142,232] but also in fibers [262]. The setup exploited the high-resolution provided by side exposure with 244 nm UV beams, and the precision required for periodically erasing long structures is available from FBG fabrication facilities. Continuous manufacture of a meter-long QPM w(2) grating is possible without the need for photolithography. Frequency doubling from a fiber laser to 777 nm was demonstrated in a twin-hole fiber that was thermally poled uniformly and then had the wire electrodes removed before periodic UV erasure [262]. A 2.4% average conversion efficiency was achieved in an 11.5-cm-long device pumped with 108 W peak power, corresponding to a normalized conversion efficiency ¼ 2.2 102 %-W1. Mechanical compression of the QPM grating mounted on a flexible slab previously used for extreme tuning of FBG led to a broad tuning range (45 nm) of the frequency doubling grating [263]. In the experiment, it is observed that the spectral profile suffers little alteration over the entire wavelength range. The combination of thermal poling and UV erasure is potentially a powerful technique to fabricate truly competitive frequency doubling fibers. The top trace in Fig. 12.15 illustrates the schematic diagram of the setup used in this experiment and the bottom trace illustrates the SH power detected. The requirement for extreme uniformity on the fibers used for this type of application can be somewhat relaxed with the use of chirped w(2) gratings [264]. Another reported application of thermally poled fiber that undergoes periodic UV erasure is for the creation of pairs of photons for quantum communication experiments [264].
12.9.3 Applications of Electro-Optic Fibers Periodic structures in poled fibers are suited for wavelength conversion and in particular for frequency doubling. In this type of application, the electrodes have little or no use after poling is completed, because electrical control is not needed and the electrodes only introduce a loss. This is in contrast with applications of poled fibers where an external voltage is used to control the refractive index through the electro-optic effect. Here, the same electrodes used to provide for the poling voltage and displace ions at high temperature can also be used for the control signal causing phase modulation once poling is completed. However, although even imperfect contact between the electrodes and the glass surface is sufficient for the creation of an electret because of air ionization and the low current levels involved (Section 12.3.1), after poling the requirements placed on the electrodes are more stringent. Room temperature operation is usually
Poling of Glasses and Optical Fibers
574 Idle position
Elastic beam
Movable block Δz
Flexible slab
Screw
d
Power (a.u)
Second harmonic power (dB)
PPSF Lb
–36 –38 –40 –42 –44 –46 –48 –50 –52 –54 –56 –58 –60 –62 –64
44.84nm fundamental wavelength tuning Idle point
20 Source spectrum
10 0 1530
1540
1550
1560
1570
1580
1590
1600
1610
1620
Fundemental wavelength (nm) Figure 12.15 (top) Schematic diagram of the compressor used for tuning the phase-matched wavelength in a frequency doubling fiber fabricated by periodic UV erasure [263]. (bottom) Relative power spectra of the SH for various IR pump wavelengths. From [263]: Canagasabey A., Corbari C., Zhaowei Z., Kazansky P.G., and Ibsen M., “Broadly tunable second-harmonic generation in periodically poled silica fibers,” Opt. Lett. 32(13), 1863–1865, 2007. # (2007) Optical Soc. of America.
preferred, and the voltage levels used in applications are hopefully much lower than the kilovolt levels used to displace ions during poling, so that ionization of the air in the neighborhood of the electrodes is more restricted. Therefore, most applications of poled fiber devices using the electro-optic effect involve D-shaped fibers with film electrodes or twin-hole fiber devices with alloy electrodes filling the entire cross-section of the holes, in which the contact between electrode and the glass is intimate and the components fabricated are reproducible.
Poled Fibers
575
Phase Modulation The simplest application for an electro-optic fiber is in phase modulation. A manufacturing technique was developed for D-fibers, which allows the distance between the electrode and the core of the fiber to be reduced by polishing, while keeping the ends of the fiber free for splicing, as schematically shown in Fig. 12.16 [266]. Further development and characterization carried out in reference [267] shows that devices have an induced w(2) 0.3 pm/V, and a 12-cm-long component has a Vp 75 V at a wavelength of 0.63 mm. The product Vp. L ¼ 900 V.cm exhibited at red wavelengths is perhaps not impressive, considering that a 2.5-cm long LiNbO3 modulator often has Vp < 5 V at 1.5 mm (i.e., a product Vp. L ¼ 12.5 V.cm in the infrared). The poled fiber device may, on the other hand, have advantages over crystalline modulators under high-power illumination. The fiber phase modulator probed with red light has low loss and shows good room temperature stability over 4 months, the w(2) decaying only 10% after 1000 hours at 90 C. Phase modulators have also been developed with twin-hole fibers filled with AuSn alloy electrodes. The devices are side polished for contacting, poled, and packaged as shown in Fig. 12.17. Fabrication here also involves processing a Polyamide
600
“D” fiber Optional electrode
Polished poled fiber segment (12 cm long, 13mm thick, r~0.3 pm/V poled by –5kV @255⬚C for 10 min.)
500
Thick polyamide layer
Polished fiber & upper electrode
Interferometer signal (a.u.)
Si wafer
400 300 LiNbO3
200 100 0 –100
Vt
Fiber ends for splicing
–200
aJi(pV/Vk)
–300 0
25 50 75 100 125 150 175 200 225 250 Driving voltage (V)
Figure 12.16 (left) Scheme of fabrication procedure of an all-fiber phase modulator. From [266]: Long X.C., Brueck S.R.J., “ Large-signal phase retardation with a poled electrooptic fiber,” IEEE Photon. Technol. Lett. 9(6), 767–769, 1997. # (1997) Institute of Electrical and Electronics Engineers. (right) Interferometric measurement of Vp. From [267]: Interferometric measurement of Vp (Long X.C., Myers R.A., and Brueck S.R.J., “A poled electrooptic fiber,” IEEE Photon. Tech. Lett. 8(2), 227–229, 1996). # (1996) Institute of Electrical and Electronics Engineers.
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Poling of Glasses and Optical Fibers
Figure 12.17 Packaged electro-optic fiber phase modulators. Dimensions (left) are 5 5 0.2 cm. (Courtesy of O. Tarasenko, Acreo.)
number of fibers in parallel. Typical performance parameters at a wavelength of 1.55 mm show single-mode operation, a Vp 110 V, 3–4 dB insertion loss, effective w(2) ¼ 0.27 pm/V, a poled fiber length ¼ 80 cm, an electrical bandwidth >10 MHz, PDL 1.5 dB, and a differential group delay 0.12 ps. Electro-optic fiber phase modulators are processed under high voltage and high temperature and can thus withstand a high-voltage bias (e.g., 5 kV) at room temperature after poling. Typically, a total phase shift of several tens of p-radians results from such a bias. Such a large phase adjustment is seldom necessary in applications involving modulation or interferometry. On the other hand, the resulting group delay corresponds to only tens of microns additional optical path, less than achievable by stretching the fiber with a piezoelectric actuator. An expansion of the group delay can be accomplished exploiting the dispersion in a chirped fiber Bragg grating [268]. Applications of an all-fiber phase modulator are diverse, ranging from quantum communication (entangled states can be constructed with phase rather than polarization control [269]) to fiber optic gyroscopes (where the quadrature point is reached by adjusting the bias of a phase modulator [270]).
Amplitude Modulation Phase modulation is converted to amplitude modulation using interference. Various types of fiber interferometers have been built incorporating a poled fiber for the characterization of the electro-optic (Pockel’s) coefficient through the switching voltage Vp [178,222,267,271,272]. Besides providing information on the voltage required for a p-phase shift, the interferometer configuration allows for amplitude modulation of the optical signal and switching when the voltage bias is close to Vp.
Poled Fibers
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Amplitude modulation in a poled fiber Mach–Zehnder interferometer (MZI) has been used to transmit the video information recorded on a DVD with a bandwidth 6 MHz on an optical carrier. The arm lengths of the MZI were 1 m long and the active fiber measured 37 cm. Full switching required Vp ¼ 217 V at 1.55 mm, but for analogue modulation, a much lower contrast (10%) was acceptable for good quality image transmission over a 5-km fiber spool. Active stabilization (for temperature fluctuations) was not necessary but could be implemented by varying the bias slowly on top of the video signal. The scheme employed for this demonstration is illustrated in Fig. 12.18 [272]. Amplitude modulation has also been exploited for mode locking and for Q-switching of fiber lasers. In mode-locking experiments, a ring cavity is constructed incorporating an electro-optic fiber in a MZI, an isolator, an erbium-doped fiber amplifier (EDFA), a 1-nm tunable Fabry–Perot filter, and a polarization controller. A 1.4-V sinusoidal drive signal at 12 MHz provided by a radio frequency (RF) generator and resonant with the 79 ns cavity roundtrip is used. With precise adjustment of the frequency (df/f 106), the 2% amplitude modulation of the fiber modulator results in subnanosecond pulses [273]. Q-switching of fiber lasers is a useful application area for electro-optic fiber modulators in a MZI, which combine the potential of high optical power handling, high-speed operation, high repetition rates, and good extinction ratio. The gain is accumulated, and the laser Q-switches when the MZI is activated. Preliminary experiments were carried out with a 40-m-long commercial fiber amplifier as gain medium, much longer than ideal, in a linear or ring DVD 20 Vpp
Fiber link PD
3 dB
Active arm 3 dB
out
Fiber modulator CW laser Reference arm in
Figure 12.18 Amplitude modulator for video transmission with a poled fiber. From [272]: Margulis W., and Myre´n N., Progress on fiber poling and devices, in: Optical Fiber Conference OFC05, Optical Fiber Communications Conference Technical Digest, Optical Soc. of America (USA), Anaheim, Ca, Vol. 4, p. 3 paper OThQ1. # (2005) Optical Soc. of America.
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578
Signal [a.u.]
3
2
1
0 –100 –80
–60
–40
–20
0
20
40
60
80
100
Time [ms] Figure 12.19 Q-switched pulse using an electro-optic fiber modulator. From [170]: Margulis W., Tarasenko O., and Myre´n N., Electrooptical fibers, in: Proceedings of the SPIE 6343, The International Society of Optical Engineering, Quebec City, Canada (2006), pp. 634319–634328. # (2006) SPIE.
configuration [170]. Single 4 ms-long Q-switch pulses are generated for 200-V drive pulses at a repetition rate <1 kHz, as illustrated in Fig. 12.19.
Switching and Tuning The MZI may be used in a 2 2 switching configuration (i.e., depending on the relative phase of the two arms), the light can exit either one of two output ports. Protection switching can thus be implemented with a pair of 2 2 switches. Protection switching at 1.55 mm comprising an electro-optic fiber was demonstrated in error-free transmission at 10 Gb/s over 820 km of installed fiber. The protection link consists of an 80-km fiber path with a dispersion compensation module. Error-free transmission at 40 Gb/s was also achieved, despite the measurable distortion of the 2-ps-long pulses used in the experiment [274]. Mismatch of the arm lengths in a fiber MZI results in a periodic signal as a function of wavelength, as illustrated in Fig. 12.20. The peak transmission wavelengths sweep under the application of voltage, with the transmission function repeating itself at a voltage 2Vp. In the example shown in Fig. 12.20, the MZI can be tuned by 3 nm with the application of 38 V. This is remarkable, considering that the refractive index changes by only 1.2 106 in each electro-optic fiber in the interferometer. In other words, while an FBG would be tuned by only 1.8 pm at Vp, the interferometer increases the tuning range by a factor close to 2000. Direct tuning of FBG using a thermally poled fiber has been reported, with a modest wavelength excursion (e.g., 40 pm at 3 kV bias) [275–277]. Although the tuning range of a MZI is large, the wavelength selectivity is poor. To improve this, the interferometer can be used inside a laser cavity.
Poled Fibers
579 U=0V
0
U = 38 V
Transmission (dB)
–2 –4 –6 –8 –10 –12 –14 1540
1545
1550
1555
1560
Wavelength (nm) Figure 12.20 Push-pull electro-optic fiber Mach–Zehnder interferometer.
Step-wise tuned lasing has been demonstrated with an electro-optic fiber modulator by locking the laser wavelength to one of several peaks of a sampled FBG [278]. Wavelength selection with electro-optic fibers is achieved in microseconds.
Polarization Control Electro-optic materials are birefringent, and the refractive index change induced with voltage is generally polarization dependent. In poled glass waveguides [113,230,279] and in fibers [280–283], however, it has been found that the dependence on polarization is weak, typically <15%. One illustration of such a phenomenon is shown in Fig. 12.21, in which the phase shift measured for light polarized parallel and perpendicular to the internal electrodes is plotted as a function of applied voltage. The difference in Vp for both polarizations is small. This is in contrast with frequency doubling in poled fibers, which does show strong polarization dependence (the intensity ratio is nearly 9:1 [284]). The lack of polarization dependence of the electro-optic effect in silica glass has been attributed to a consequence of electrostriction [113,279,280]. The small difference in the refractive index shift induced for orthogonal polarizations in poled fibers is advantageous for making polarization-independent devices for optical communications. On the other hand, it becomes difficult to exploit the electro-optic effect in silica fibers to achieve efficient polarization rotation [285], although even a small rotation is sufficient in some cases [282]. A fiber MZI has been exploited for polarization rotation, in which each electro-optic fiber arm of the interferometer carried light in one polarization, and full rotation was achieved with 76 V [283].
Poling of Glasses and Optical Fibers
580
Number of pi phase shifts [rad]
16 P polarization (30.5 mW)
14
S polarization (24.7 mW)
12 10 8 6 4 2 0 0
500
1000
1500
2000
Volts [V] Figure 12.21 Number of p-radian phase shifts measured for both polarization eigenstates of a poled fiber as a function of applied voltage. From [283]: Tarasenko O., Myre´n N., Margulis W., and Carvalho I.C.S., All-fiber electrooptical polarization control, in: Optical Fiber Communication Conference – OFC 2006, Optical Society of America, Washington DC, (2006) paper OWE3. # (2006) Optical Soc. of America.
Voltage Sensing Silica fibers are compatible with high voltage sensing because of the excellent electrical isolation and the lightweight of a fiber-based sensor head. Intrinsic all-fiber voltage sensing using a poled twin-hole silica fiber has been reported in a number of publications [282,286–289]. After poling, the electrodes are removed from the holes. The fiber is coiled helically around the high voltage insulator, allowing the electric field to be integrated between ground and the high-voltage conductor to obtain the potential difference. A low coherence interferometer is used for data acquisition and processing at 5000 samples/sec. The weak birefringence in the poled fiber induced by the electric field (see Fig. 12.21) causes fringes to appear because of the beating of the two polarization modes, modulating the broad but otherwise smooth amplified spontaneous emission (ASE) spectrum from an EDFA. The phase depends linearly on the voltage, and the spectral information can be translated into the potential difference to be determined. In this application, small electric field–induced birefringence can be read by a sensitive interrogation system, and the application is not critically dependent on the performance parameters of the poled fiber (such as loss and effective w(2) induced). Research is ongoing toward the realization of a full-scale voltage sensor, improving signal-to-noise performance, using longer poled fiber lengths and solving packaging issues [289].
References
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12.10 CONCLUSIONS Unlike FBGs, which found immediate applications and were introduced into the market soon after the pioneering work by Meltz and coworkers [290], poled glass devices have not been commercialized to date. Many practical applications have been reported, but a clear market need that cannot be satisfied by existing technologies has not been identified. High-voltage sensor systems based on poled fibers are being developed and may be introduced in the medium-term future. Perhaps the devices closest to commercialization at present are phase modulators fabricated with twin-hole Ge-doped silica fibers. Besides the natural parameters associated with poling, such as Vp and loss, various other considerations are of paramount importance, such as short and long-term stability, reproducibility, manufacturability, packaging, and ultimately price. For electro-optic devices, additional technical criteria need to be considered, including optical power handling capacity, differential group delay (DGD), polarization dependent loss (PDL), and electrical and optical bandwidth. In QPM components for frequency conversion, new effects may play an important role, such as refractive index change through color center formation and optical erasure. The dream of achieving the tantalizing goal of 10 pm-V1 in glass has not been realized after more than a decade of research. However, the long, lowloss poled glass fibers offer a distinct advantage over other technologies and may, for some time, remain a solution looking for a problem. It is conceivable that the first applications of poled fibers may be in nonlinear parametric effects such as in photon-pair generation for quantum-cryptography, as these require fiber-compatible devices and for which the poled fiber has many of the required parameters. In the near future, if the nonlinear coefficient may be demonstrated to be sufficiently large and stable, their commercial success would be guaranteed. Until then, glass poling may remain a fascinating research area.
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Appendix I A.1 CALCULATING GRATING PARAMETERS This section lists useful formulae to allow the quick calculation of grating parameters.
A.1.1 FBGs 1. The bandwidth, Dl of an FBG between the peak and the adjacent first zero is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l2 Dl ¼ kac Lg þ p2 ; ðA1:1Þ 2pneff Lg where Lg is the length of the grating and the coupling constant, kac is kac ¼
2p n Dn ; lB
ðA1:2Þ
lB is the Bragg wavelength, and Dn is the amplitude of the refractive index modulation, 2pNz dnðzÞ ¼ 2nDn þ nDncos ; ðA1:3Þ Lg where N is the order of the grating with a period, Lg, and is the overlap factor of the modes with the refractive index modulation, nDn ¼ 2n nDn;
ðA1:4Þ
where n is the visibility of the fringe pattern, nDn is the average refractive index change seen by the mode, and n is the nominal refractive index of the core. For strong gratings, kac Lg >> p, and the bandwidth is Dl ¼
l2 Dn : lB
ðA1:5Þ
For weak gratings, the condition when kac Lg << p, the bandwidth is Dl ¼
l2 2neff Lg
:
ðA1:6Þ
597
598
Appendix I
2. The reflectivity of the grating at the Bragg wavelength, lB, is R ¼ tanh2 kac Lg :
ðA1:7Þ
The reflectivity, r, in % is: rð%Þ ¼ 1 10
Tdip ðdBÞ 10
ðA1:8Þ
where Tdip is the transmission dip in dBs at the Bragg wavelength. 3. The group delay in a chirped grating is t¼
2ng Lg ; cDlg
ðA1:9Þ
where, Dlg is the chirp bandwidth of the grating, c is the speed of light, and ng is the group index of the mode.
A.1.2 LPGs 4. The bandwidth of a LPG between the peak and the first zero is DlLPG ¼
l2resonance Dneff Lg
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4kn!01 Lg ; p
ðA1:10Þ
where the ln cladding mode has an effective refractive index difference to the core LP01 mode of Dneff. k is the coupling constant for the selected mode interaction.
A.2 MATERIAL PROPERTIES–BASED PARAMETERS The dependence of the Bragg wavelength of an FBG in SMF28 on the temperature is typically 16 pm- C, and the strain dependence is 10 nm/%. In high germania-boron co-doped fiber, the thermal sensitivity of an FBG is 10 pm-K1, and the strain coefficient is 0.95 pm/me. The thermal dependence of the stress-optic coefficient is 0.21 0.03 fm me1 K1 [1]. The thermal expansion coefficient of silica is 106 K1. The thermo-optic coefficient of SMF28 is 9.2 106 – K1.
References
599
A.3 USEFUL PHYSICAL CONSTANTS Boltzmann’s constant kB ¼ 1.3806503 1023 m2 kg s2 K1 Planck’s constant, h ¼ 6.626 1023 J-s Charge of an electron, e ¼ 1.626 1019 C Energy of photon at 1 micron, 1.24 eV
REFERENCES [1] M.J. O’Dwyer, C.C. Ye, S.W. James, and R.P. Tatam, Thermal dependence of the strain response of optical fibre Bragg gratings, Meas. Sci. Technol. 15 (2004) 1607–1613.
Index A Abel integral equation, 429–430 Absorption, 22–23 effect of hydroxyl and OD, 26, 29 ground state, 363 OH/OD, 39t two-photon, 3, 19, 21, 22–23 Access, 352–353 Acousto-optic tunable filter (AOTF), 366 Activation energy, 544, 545 Active stabilization, 577 Adiabatic perturbation, 369–371 soliton pulse compression, 369 Aging, accelerated, 435–436 Aging effect, 549 Air ionization, 548–550, 573–574 Alpha parameter, 359 Alternate polarity, 561 Amplified spontaneous emission (ASE), 376f, 377, 378, 378f Amplifier erbium, 351–352 fiber, 373–384 gain clamping (see Gain clamping) gain flattening (see Gain flattening) gain saturation, 362–363 noise-figure, 383–384 reflective, 352–353, 359–360 semiconductor chip, 357–358 semiconductor saturable absorber mirror (SESAM), 391 Amplitude modulation, 576–578 Angled facet, 352–353, 352f, 354, 354f, 355f Anisotropy, UV induced, 422–423 Annealing, 24f, 405, 431–436, 434f isothermal, 560 time, 559 thermal, 405, 432–434 Anodic bonding, 532–533, 550 Antenna, 158 Anti-reflection (AR), 56–57, 72, 350, 350f, 351–352, 354, 355–356, 358f, 362, 386–387, 391f
Apodization, 189–190 bit-error rate penalty, 325 Blackman, 191 Cauchy, 192 cosine, 192, 192f, 328–329 effect on group delay, 192, 192f Hamming, 190–191 Hanning, 190–191 long-wavelengthedge, 211, 312–313, 315, 315f, 316–317, 320, 321, 328–329 moire´ effect, 195, 198–199, 207f, 208, 210, 211–212, 211f raised cosine, 192, 192f requirements for fabrication, 199–201, 212–213 self, 193–196, 195f shading functions, 190–192, 203 short wavelength, 211 sinc, 85, 189–190, 192, 201f, 202–203, 204, 205–206 super-step-chirped gratings, 211 tanh, 191, 192, 192f, 316–317, 327, 328–329, 329f Apodization methods position weighted fabrication, 201–203 self-apodization, 193–196 symmetric stretch, 208–212 strain vs. length of grating, 210, 210f top hat reflection gratings, 201–203 variable diffraction efficiency phasemask, 198–199 Apozidation, asymmetric, 313–315, 316–317 Apozidation methods amplitude mask, 196–197 moving fiber/phase mask, 203–207 multiple printing, 199–201 two exposure/double exposure, 196, 197f, 198–199 Arm length mismatch, 578 Athermalization, 248. See also Temperature compensation Attenuation, 15
601
602 Autocorrelation, 369 Average soliton. See Soliton
B Bandpass filter, 217, 365–366, 366f, 390–391 Bragg reflection coupler (BRC), 259–269 concatenated chirped gratings, 226–227 DFB, distributed feedback, 218–229, 219f, 220f, 222f, 285, 287f DFB stop bandwidth, 229, 231 Fabry-Perot (FP), 229–233 full width at first zeroes (FWFZ), 238–239, 276, 408f, 411–412 Gires-Tournois, 282–284 Interferometer (GTI), 284 chirped, 283–284 grating frustrated coupler (GFC), 259–269 grating Mach-Zehnder interferometer (GMZI-BPF), 245–250, 277f grating Michelson interferometer, 233–245 asymmetric, 240–245 Michelson interferometer transmittance, 236, 265 moire, 218, 228f, 229–233, 233f, 238–239 MZI, long term stability, 247–248 polarization independent, 274–275 polarization rocking, 218, 274–277, 279 polarization splitting (PBS-BPF), 255–258 superstructure, 218–229, 228f Winstom filter, 286f Bandwidth co-directional coupling, 145 electrical, limiting, 560 FBG, 148, 150, 151 rocking, 147 Beam forming, 337–339, 338f Bessel function, 31–32, 123–124, 157, 235 Beta-eucryptite, 247–248 Beta-quartz, 247–248 Birefringence, 75–76, 145, 146, 147, 221, 274, 276–277, 322–325, 418–420, 579, 580–581 bend induced, 75 fiber, 75–76, 147, 418–419 polarization coupler, 74, 75–76 polarization maintaining, 97–99 stress-induced, 29–30, 32–33, 274, 430–431 UV induced, 419–420 Bit error rate (BER), 252, 305–306, 325
Index Bit rate, 301, 305–306, 322–323, 325–326, 328–329, 329f, 331f Blazed grating, 152–157, 165f angle, 61–62, 65–66, 137 tilt, 56f, 60–61, 61f, 87, 96, 136, 138, 146, 152–153, 162, 164, 165–166 Boltzmann constant, 432–433 Bond angle, 540 orientation, 535 Bound field, 141 Boundwave, 129–130 Bragg grating fabrication Bragg wavelength dependence on bend, 86, 87 chirp by non-uniform strain, 87–88 continuously chirped gratings, 86–91 elevated temperature, 82–83 etched tapers for chirped gratings, 88–89 higher spatial order masks, 72–74 phase mask, 55–59, 56f tunable, 101–103 fabrication, 57–59 interferometer, 59–64 point-by-point writing, 74, 75f, 78, 85–86, 99–100 scanned phase mask, 66–69 single pulse, 77–78 tilted gratings, 65–66 Bragg gratings bandwidth between first zeroes, 147, 150 CCG (see chemical composition grating) chemical composition grating, 41–43 continuous writing, 79–80, 93, 94 electro-optical push-pull method, 94–96, 97–99 fabrication of, 53–101 M-shaped, 180 step-chirped, 91–92 fabrication bulk interferometer, 53–55 fabrication by projection, 73 fabrication of ultra-long, 79–81 lifetime, 431–436 phase-only sampling, 181–182 reflection spectrum, 406–412, 407f, 413, 418 reflectivity, 406 reliability, 405, 431 temperature dependence of, 81–83, 444 functional dependence, 462 liquidcore, 449, 450f decay, 432
Index temperature compensation, 462–467 transmission spectrum, 405 measurement of, 406–412 uniform group delay, 151 group delay measurement, 414–415, 416, 418–420 phase delay, 151 measurement of phase delay, 414–416 properties of, 148–151 uniform period gratings, 301, 306–307, 307f, 311–312, 320 properties of, 148–151 wavelength of reflection peak, 406, 409, 410, 410f Bragg reflection coupler (BRC) OADM, 260–266 theory, 261–266 Bragg wavelength, 20f, 26, 30, 32–33, 36, 56 sensitivity to strain and temperature, 155 tuning, 60–61, 81–86, 97, 98f, 456, 466–467 UV induced shift, 26 Breakdown, limit, 548, 561 Burn-in, 435–436
C Carrier, 348, 349 density, 358 Cation neutralization at electrode, 553 mobility, 544–545 Cavity loss, 347–348, 372, 379–380, 382, 383, 384 Cavity round trip, 347–348, 356, 363 Charge accumulation, 531, 538–539, 539f, 541f, 549 cancellation, 549 displacement, 527, 533, 551 dissociation, 557 electron ejection, 553 frozen distribution, 551 injection positive, 553 movement, 527, 547–549, 559 negative, uniform distribution, 552 neutralization cations, 551 neutralization of negative, 558–559 recombination, 557 separation, 557 transport, 535 trapping, 557
603 Charging fiber, 547, 561–562 Charging capacitor, 547 Chemical durability, 540 Chirp induced by moving fiber/phase mask, 203–207 static and dynamic, 354 Chirped grating approximate delay, 308 asymmetric apodization (see Apodization) chirp rate, 176 compression ratio, 305 continuously chirped, 91 effect of apodization, 325 figure of merit (FOM), 305 high-power, 347–348 linear chirp, 91–92 pulse broadening, 322–323 quadratic chirp, 91–92 random refractive index modulation amplitude, 327 Bragg wavelength, 331, 333f group delay difference, 318–319 transmission spectrum, 311–312 symmetric and asymmetric apodization comparison, 316–317, 317f Chirped moire´ grating, 207f Chirped pulse amplification, 301 Codirectional coupling of guided modes, 175 power in crossed state, 144 power in uncoupled state, 144 radiation modes, 152–170 Codoping aluminum, 34, 34t boron, 6, 26–28 germanium, 22–28 nitrogen, 26 phosphorus, 5–6 Coherence collapse, 359–360 length, 105 Coherent photovoltaic effect, 531 CO2 laser, 536, 543 Color center formation, 531, 581 Comblike dispersion profiled fiber (CDPF), 371 Compression, axial, 430–431 Conduction band, 19 Conductive coating, 549–550 Copropagating modes, 130
604 Core dopant Al/Tb, 34t Al/Yb/Er, 34t cerium, 37 erbium, 34 germanium, 21 neodymium, 360–361 phosphorous, 5–6 rare earth doped fiber photosensitivity, 366 rare earth doping, 34 tin, 23, 28, 37 ytterbium, 5–6 Coupled-mode equation, 125, 126, 127 modes, 122–123 theory, 124–139 Counterpropagating modes, 139–142 Coupler coupling constant, 142–145, 306–307 ac, 140, 142, 150 dc, 140, 143–144 contra-directional, 145 cross, 144 self, 143 coupling length, 145 grating frustrated (GFC) (see Bandpass filter) polarization splitting, 145–147 transfer matrix, 235 Coupling, 142–145 codirectional, 142–145 co-propagating modes, 146 Coupling constant dc (see Visibility) dc self-coupling, contra-directional, 145 modulation depth (see Visibility) radiation modes, 125, 138f Critical angle, 133, 134 Cross-section, emission and absorption, 362–363 Crystallization, 532, 540, 546, 554, 561–562, 563, 565 Current, limited (current), 535 Cut-off angle, radiation mode, 134 CW lasers, 121–122
D Dark soliton, 332–334 Decibel, electrical, 325 Defects, 4–5, 8–9 absorption at 240nm, 26, 37
Index detection of, 20–21 electron spin resonance, 20 GeE[EQUATION], 18–19, 20f, 36–37 GeO, 19, 20f, 22 germanium, 19, 20f in glass, 18–20 higher order ring structure, 36 non-bridging oxygen center (NBHC), 20 oxygen deficiency, 18–19 oxygen deficient center (GeODC), 29 paramagnetic, 18–19, 21 peroxy radical, 20 phosphorous, 20 role of (Defects), 546 sub-oxides, 18 Delay line, 335, 369 Dense WDM, 352. See also WDM Densification, 36, 38, 540 Depletion region formation, 547 neutralization of, 553 resistivity of, 547 width, 547, 555 Detuning, 190–191, 248 Deuteration. See Hydrogen loading DFB bandpass filter, 229–233 apodization of, 232–233 bandwidth, 230–232 chirped grating, 238 flat-top bandpass, 223–224 multiple phase-shift, 227 quality factor, 223 Dielectric constant, 17 Diffraction, 198–199 Dioxide, germanium and silicon, 18 Dipole orientation, 551 Dirac-delta function, 122–123 Dispersion diagram, 134 effect of chirp, 303 measurement, 419 polarization mode dispersion, 322–325 in chirped gratings, 322–325 Dispersion compensation, 325–326, 420 chromatic dispersion, 303 dispersion compensation, 420 dispersion compensation grating (DCG), 301–302 systems measurement, 325–330 Distribution of induced defects (DID), 433, 434
Index Dopants, fluorine, 4–5 Doped fiber, 362–366, 384, 385, 386–387 amplifier, 381f, 389–393 erbium, 362–366, 378–380, 386–387 cladding doped, 394 doped fibre laser, 362–366 gain clamping, 378–382 gain control, 378–382, 381f analysis, 382 gain equalization, 374–377, 382 ytterbium (Yb), 34, 362, 367, 390, 391–392, 393f, 394 Drift, negative ion, 553
E Eigenmode, 120 Eigenvalue, 124, 262 equation, 124 Elastic limit, 8 Elastic recoil detection, 542–543 Electret glass, 532–533, 535, 550–551 thermoelectret, 527 Electrical isolation, 580–581 Electrical resistance, 540 Electric field, periodic, 572 Electric field induced second harmonic generation, 528, 543 Electric polarization, 527–528 Electric quadrupole, 528–529 Electrodes absence of, 533 alloy, 550 cathode, 571 effect, 549 gold, 549 liquid gallium, 571 liquid indium, 571 loss, electrode-induced, 560 one internal, 572 periodic (Electrodes), 569, 572 press-on, 549 QPM, 572 removal of, 549–550 resistance, 560 silicon, 549 silver-painted, 562 Electrons, high energy, 535 Electro-optic coefficient, 531, 564, 571, 576 effect, 15
605 fiber, 575, 579 interferometer control of, 568 materials, 8–9 phase modulator (EOPM), 94–96, 95f phase-modulator, fiber, 576 push-method, 94–96 waveguide control, 568 waveguide modulator, 568–569 Electron hopping, 19 tunneling, 19 Electron beam, e-beam, 66–67 Electrostriction, 579–580 End-pumping, 360–361 Energy dispersive x-ray spectroscopy (EDS), 542–543 Energy density, 22–23 Error free transmission, 578 Etching rate, 540, 541–542, 541f, 542f ring around the electrode, 540–541 External cavity, 357–359 laser modeling, 357–359 External grating reflector, chirped, 351 Extinction, 233 Extinction ratio, 568–569, 577–578 Eye-closure penalty, 328, 329f
F Fabrication of long period grating (LPG), 78 mode converting grating, 278–280 polarization converting grating, 75–76 step-chirped grating, 91–92 super-step-chirped gratings, 93–94 superstructure grating, 85 Fabry-Perot (FP) effect of absorption, 231 free spectral range (FSR), 231–232 reflectivity, 231–232 Femtosecond laser, 503, 512–513 cavitation, 504–505 damage, 503–505, 507–509, 514f, 517–518 grating, 521 filamentation, 509f, 510–511, 518–520 fluorescence, 511–512 grating in ZBLAN fiber, 512–513, 518–520, 521, 523f ionization, 504–505, 507f, 508–509, 510–512
606 Femtosecond laser (Continued) melting, 505f multi-photon absorption, 508–509, 510–511 nonlinear/multi-photon absorption, 508–509, 510–511, 512 optical/dielectric breakdown, 504–505, 508, 509f, 510–512, 518–520 order walk off, 513, 515f plasma, 503–504, 508–509, 511, 512 refractive index change, 503, 505–506, 508–509, 509f, 510–512, 513f, 514–516, 518, 519f, 520, 521, 522–523 decay of Type I-IR and II-IR grating, 522f Self-focusing, 504–505, 507–509, 509f, 510–511 threshold, 521 Soliton, 504–505, 507–509 titanium sapphire laser (Ti:sapphire), 512–513, 517, 520f, 522–523 ultra-broadband reflection grating, 520f Femtosecond laser writing burst-mode, 509 fs erbium fiber laser, 512 with higher-order phase-mask, 513–514, 517 IR laser, 517–518 with phase-mask, 513–517 point-by-point, 512, 512 UV, 512 Fermi-Dirac function, 433 Fiber design, 445–450 dispersion, 328 D-shaped, 539f, 570, 573–574 fuse, 504 grating laser (FGL), 353–360 grating semiconductor laser (FGSL), 347–353 internal electrode, 536, 543, 570 micro, 570 photonic crystal, 571 pigtail, 352, 355–356 preform collapse, 18 rear facet grating laser, 357–358 twin-hole, 531, 540–541, 549–550, 570–572, 573–574, 575–576 Fiber grating laser, modeling, 357–359 Fiber laser, 347–353 rare earth, 360–362
Index Fictive temperature, 540 Field increasing the recorded, 561 ionization, 557, 571–572 poling, 534–535, 549–550, 551, 557, 561–562 recorded, reduction of, 569–570 shielding, 557 space charge, 532 Filter band-pass, 366f band-stop, 226–227 blocking, 226–227 Bragg reflection coupler (BRC), 260–266 gain flattening, 373–384 in-coupler Bragg grating, 259–269 mode converters, 278–280 narrow band, 229 polarization rocking, 145–147 rocking, bandwidth, 275 sidetap, 270–274 sidetap design diagram, 164–165 Forward radiating modes, 138 Fourier transform, 190 Fraunhofer diffraction, 155–157 Free spectral range (FSR), 231–232 Frequency doubling, efficient, 528–529, 570 Front facet reflectivity, 347–348, 354, 359 Fused silica, 4, 8, 26–27, 75 physical parameters, 23–25, 34t
G Gain control, all-optical, 378–382 control, automatic (AGC), 379–380 analysis, 382 equalization, 374–377 flattening, 373–384 spectrum, 364–365, 373–374 stability, 383 tilt, 376–377 Gain clamping, 378–382 Gamma rays, 535 Gaussian, 69–70, 105–106, 385 GDR, cascading of non-ideal shapes, 327 Glass bismuth oxide, 563 borosilicate, 560, 562–563, 566 as a capacitor, 548 chalcogenide, 565 deformation of, 546
Index heavy metal oxide, 563–564 lead silicate, 563–564 Pb, 564 phosphate, 565 quenched, 546 as a resistor, 548 soda lime, 542–543, 549, 553, 566, 570 tellurium, 564 Glass sliver, 353 Gordon-Haus jitter, 230–231 Grating dispersion tunable, 325–326 fabrication, 103 lifetime, model, 431–436 period, 330 profile, 426 reliability, 431 schemes for lumped, 340 strength, 431–436 strength, hydrogen loading, 431–432 thermal stability, 432 Grating-assisted coupler (GAC), 259–260 Grating-frustrated coupler (GFC), 266–269 Grating lifetime, decay, 432 Grating longevity, 431 Grating type Bragg grating, 1–2 long period grating (LPG). See Long period grating (LPG) sidetap, 270–274 bandwidth, 271 step-chirped grating, 91–92 tilted, 65–66 Type I, 22–23, 104 Type II, 23, 104 Type II, damage, 23 Type IIA, 22–23, 28, 29–30, 104 Group delay, 419–420, 576 ripple (GDR), 416 Guided mode, 134–135, 139–142, 377 cutoff angle, 134–135
H Hamming function. See Apodization Hanning function. See Apodization Heat generation, UV induced, 29 Helical coil, 580–581 Heterodyne, 41 High repetition rates, 577–578 High-speed operation, 577–578 High voltage sensing, 580–581
607 Hydrogen loading deuterium, 31 diffusion time, 33 in-diffusion of hydrogen, 31–32 out-diffusion, 31–32, 32f, 33f temperature dependence of outdiffusion, 31–32 Hydrogen, surface, 553 Hydronium, 546, 553 Hysteresis, 353
I Image transmission, 577 Index matching gel, 352–353 Injection rate, 556 Insertion loss, 375 Intimate contact, 573–574 Inter-modal coupler, guided mode, 146 Interferometer Fabry-Perot (FP), 221 Lloyd, 69–72 Mach-Zehnder, 245–250 Michelson, 233–245, 258 phase mask, 59–64 prism, 69–72 silica block, 64 International Telecommunication Union, 352 Inverse Fourier transform, 538 Inversion symmetry, 528–529 Ionic waves, 557 Ions injection of, fluorine, 546 pile up of, 542 sodium, neutralization of, 566
K Kerr effect dc, 17 dc Kerr constant, 17 intensity dependent refractive index, 17 optical, 17 Kramers-Kronig relationship, 7–8, 36 Kronecker’s delta function, 122–123
L Laser composite cavity, 364–366 distributed feedback (DFB), 366–369 modeling, 367 dual frequency, 368 fiber Raman, 392–393
608 Laser (Continued) four grating coupled cavity, 368 gain-switched DFB, 369 large spot, 352–353, 355–356 long external cavity, 347–348 longitudinal mode control, 364–365 multifrequency, 368 Raman fiber grating laser (RFGL), 372 ring, 368 short external cavity, 357 single-mode operation, condition, 394 tunable single frequency, 426–427 wavelength uncommitted, 355–356 Laser induced pressure pulse, 542 Layer peeling, 555 Lensed fiber, 350 Linear electrooptic effect, 571 Linewidth, 348–349 enhancement factor, 348–349 LIPP, 542 Local area network, 352–353 Long period grating (LPG), 32–33, 169, 287–288 angular distribution of radiation, 271 bandwidth, 169 effect of overlay refractive index, 169 loss at boundary, 169 sensitivity to UV induced refractive index change, 167 transmission spectrum, 167–168, 290f Loop mirror, 234 Low coherence interferometer, 580–581 Luminescence, UV induced, 30
M Mach-Zehnder interferometer, 245–250 athermalization, 247–248 cascaded, 246 poled, 577 Macroscopic polarization, 533 Maker fringes, 536–538 Manufacturability, 581 Master oscillator power amplifier (MOPA), 361–362 Maxwell’s equations, 121 Michelson interferometer, 233–245 bandpass filter apodized gratings, 240–241 chirped gratings, 242 counter-chirp, dissimilar length gratings, 244–245 identical chirped gratings, 282f
Index path imbalance, 243, 258 reverse chirped gratings, 255 scanning, 420–421 Mirror, broad band, 235 Microscopic dipoles, 533 Miller’s rule, 563 Mobility cation, 544–545 in charged material, 545 hydrogen, 544 lithium, 545 potassium, 545 protons, 545 sodium, 545 Mode, 122 hop, 351, 352–353, 354, 355–356 hop-free, 359–360 Mode field width, 158 Mode-locked laser, 369 Mode-locking, 577 Modes coupling, types of, 132–139 Modulation amplitude, 127 refractiveindex, 119, 126–129, 155–157, 169, 177–178, 181–182 nonsinusoidal, 132 large, 183 Moire´, 83 Multi-quantum well (MQW), 357 Multiple bandpass filter, 233 MZI. See Mach-Zehnder interferometer
N Nanocrystals, 563 Nanoparticles dissolution, 570 metal, 570 silver, 570 Noise figure Raman, 373 signal, spontaneous beat noise, 383 Noise floor, 406 Nonlinear coefficient, stabilization of, 566 Nonlinearity, 15 Nonlinear layer, edge of, 543 Normalized waveguide parameters, 123–124
O Optical add-drop multiplexer (OADM), 248–250 programmability, 252–253 reconfigurable dispersion compensating (RDC-ADM), 255 reconfigurable (ROADM), 255
Index Optical bandwidth, 581 Optical circulator ADM (OC-ADM), 252–253 coherent power penalty, 249–250 incoherent power penalty, 252 short wavelength loss, 251 Optical circulator (OC), 250–255 Optical erasure, 581 Optical exposure, 540 Optical fiber fabrication, 3–4 fabrication (MCVD), 18, 27–28 preform, 18 Rayleigh scatter, 4–5 Optical fiber amplifier, 347 Optical low-coherence reflectometry (OLCR), 420–429 Optical-time domain reflectometry (OTDR), 420 Optical coherence tomography, 335 Optically induced changes, 38–41 Optically induced effects photorefractive, 16–18 photosensitive, 16–18 Overlap integral, 130–132 Optical nonlinearity, 528 profile, 536–537 Optical rectification, 530–531
P Packaging, 581 Parallel plate capacitor, 551, 554 charged, 551 Parallel processing, 575–576 Penalty, 382 Period, pure sinusoidal, 423–424 Permittivity, 16 Phase mask, 55–59 chirp correction, 67–69 coherence length, 529–530 diffraction angle from phase mask, 56 diffraction efficiency, 56–57 diffraction order, 55–56 etch depth, 57 fabrication with electron-beam, 57–59 holographic fabrication, 57–59 non-normal incidence, 56–57 normal incidence, 56–57 period, 55–56 phase mismatch, 530 phase reversal, 530
609 relationship to Bragg grating period, 56 relationship to Bragg reflection wavelength, 53–54 stitching, 67–69 tunability, 62 zero-order, avoidance, 61–62, 61f zero-order minimization, 60–61 Phase matching, 129–130 condition, 130, 132 copropagation, 130 detuning, 130 diagram, 135f, 138f types of, 139 Phase modulation, 575–576 Phase separation, 554 Phase shift, 176, 576 Photochromic, 16 Photolithography, 569, 572, 573 Photosensitivity cladding, 26–27 of high pressure hydrogen loaded fiber, 29–30 summary of mechanisms, 35–37 Photosensitization boron-germanium co-doping, 26–27, 37 cold soaking in hydrogen, 26, 29–33, 37–38 flame brushing, 21, 37 hot-hydrogenation, 21, 22, 26, 37 phosphorus-fluorine, 27–28 summary of route to, 37–41 tin-germanium, 28 Piezoelectric stretcher, 230–231, 254 Pockels’ coefficient, 576 Point reflector, 263 Poisson’s equation, 555–556 Polarizability, charge-modified, 561–562 Polarization independence, 569 induced, 16 modes, 580–581 rotation, 579–580 Polarization beam splitter, 257–258 Polarization dependence, 326 no, 579–580 strong, 579 weak, 579 Polarization mode dispersion (PMD), 322–325 induced delay, 324–325 shift in reflection wavelength, 325
610 Polarization rocking filter. See Bandpass filter bandwidth, 276 fabrication, 276 filter rocking angle, 276 Mach-Zehnder, transmission function, 276–277 Polarization splitting coupler, 256–257 Polarization splitting (PBS-BPF), dispersion compensation, 257–258 Polarization wave, 133 Poled fundamental properties of, 569 glass, 566–568 polymers, 535 Poled channel waveguide, 568–569 Poling, 11 assisted, 570 Bragg gratings, 8–9 corona, 535 current measurement, 548–549 dynamics, 545 glasses, 8–9 increasing the voltage, 561 ion-implantation, 570 long, 553–554 model, 555–557 optical, 527–531 optimum temperature, 561 periodic illumination with UV, 569 reversed, 537–538 risetime, 544 room temperature, 531–532 short, 551–553 in silicone oil, 550 single carrier, 565 stability, 557–560 temperature dependence of, 544 thermal, 532–536 time (Poling), 553 UV, 531–532 vacuum, 550 voltage, 8, 534 Polymer, 78 Polyvinylidene fluoride (PVDF), 535 Population, inversion, 383–384 Power handling, 577–578, 581 Poynting’s vector, 122–123, 157 Preform collapse, 18, 22 Preform fabrication in reducing atmosphere, 22 Price, 581
Index Prism, 538 Prism interferometer, dependence of grating length, 70–71 Protection switch, 578 Protons, high energy, 535 Pulse compression, 305 re-, 301–302, 305, 308, 313, 335f shaping, 331–334, 336, 413 spectral, 334 multiplication, 336–337 Pulsed laser, semiconductor DFB, 366–369 Pump bidirectional, 373 laser stabilization, 351–352
Q QPM. See Quasi-phase-matching Q-switching, 577–578 Quantum cryptography, 581 Quartz alpha, 19 beta, 247–248 Quasi-phase matching, 569 self organized grating, 530
R Radiated field, 153–154 Radiation even-azimuthal order modes, 164 field, 136–137 loss, 87 modes, 76 odd-azimuthal order modes, 164 scattered power, 160 start wavelength, 137 untilted grating, maximum angle of, 136–137 wavelength vs. angle in STG and LPG, 138–139 Radiation modes coupling, 152–170 Radiation spectrum of STG, 164–165 with photosensitive cladding, 163–164 Raised cosine, 192 Raman, 371–373 amplifier, 372 amplification, 371–372 cross-section, 371–372 gain coefficient, 372 scattering, 371–372 stimulated Brilluoin, 371, 389, 394
Index stimulated Raman, 371–372, 394 scattering (SRS), 371–372 Raman amplifier, 372, 373 Rate equation, 358 Ray propagation, 133, 133f diagram, 133 dispersion diagram, 134, 137–138 RBS. See Rutherford backscattering Reflection, 352–353, 406, 407f, 411–412 back, 375, 376–377 Bragg, 406, 412 first zeroes, 148f, 150 peak, 408–409 reflectivity, 409, 410, 411–412, 413, 416 fiber end, 406 STG with photosensitive cladding, 163–164 without photosensitive cladding, 89 Refractive index, 425, 426f ac modulation index, 127 concentration dependence of refractive index change, 23–25, 25f dc index, 127 dependence on energy density, 22–23 effective mode index, 123–124 group index, 7 modulation, 23, 350f, 359–360 origin of, 6–8 profile, 4–5 Sellmeier expression, 7 thermally induced change in hydrogen loaded fiber, 30 UV induced change, 22–25 internal stress, 429–431 UV induced growth rate of, 22–23, 41, 42f UV radiation induced growth of, 29 Refractive index modulation, 23, 230–231 non-sinusoidal, 132 non-uniform, 317–319 uniform, 127, 128f Relative group delay, 192 Relaxation, 405 Reproducibility, 581 Resolution spatial, 550 submicron, 535, 540, 541 Resonance peak spectral splitting (RPSS), 359 Resonance, vibrational, 94–96 Resonator, 230–231 Ring cavity, 577
611 Ring configuration, 577–578 Rocking filter, 147 coupling length, 147 rocking period, 146 rotation angle, 146–147 Rutherford backscattering, 542–543
S Scanning electron microscope (SEM), 57–59 Secondary ion mass spectroscopy, 543, 546, 556–557, 558–559, 558f Second harmonic generation, 8–9, 528 Second-order nonlinearity, 528 decay of, 560 erasure of the, 557–560 measurement of, 571 non-destructive, 537–538 in situ, 541–542 stability of, 557–560 Second-order susceptibility (w2) erasure of, by UV, 569 chirped grating, 573 increasing, 561 profile of, evolution of, 541–542, 556f sign reversal, 541–542 value of the induced, 560, 561 Seed light, 529 Self-coupling, 143 Semiconductor, 362 Semihemispherical lens, 538 Sensor, 74, 443–476, 477–492, 493–497 acceleration, 465, 473–474, 489–490 accelerometer, 473–474 acoustic, 475–476 amplified, 459f biomedical, 493–494 chirped FBG, 471 de-modulating, 452 detection, 452, 472, 473f Fourier transform, 453–456, 454f homodyne, 442–443, 476 heterodyne, 476 differential refractive index change, 445, 446–447 discrimination between quantities, 449–450, 496–497 distributed, 441–442, 452, 455–456, 459, 496–497 temperature, 493 erbium-doped fiber, 452, 459 fiber design, 445–450
612 Sensor (Continued) hollow core fiber, 448 loop mirror, 471 long period grating, 441, 470–471, 478–479, 489–492, 494–495 magnetic field sensing, 476 negative temperature sensitivity, 448f, 449, 450f, 461–462, 464f, 489–490 network, 441–442 number of sensors, 452, 459 WDM, 459 oil, 441, 442, 451f, 494–495 point, 450–451 pressure, 441, 444, 445, 457, 465, 467–471, 493–494 respiration, 494 Sagnac, 471 strain/stress-optic, 444, 449–450, 461 surface Plasmon-Polariton, 479–480 bio sensing, 449–450, 479 Bragg grating assisted, 480–481, 488f tilted grating, 487, 488f hybrid, 480, 489 Kretschman technique, 479 long period grating, 480 metal grating, 479, 488–489 optimisation of, 483–489 prism technique, 479, 480 pure, 480, 489 refractive index sensing, 479, 480 thermo-optic, 444, 459 temperature compensation, 462–467 invar, 463, 464–465, 476 thermal expansion coefficient, 444, 459, 461, 463, 465 tilt, 495–496 vibration, 441–442, 453, 475, 476, 489–490, 494 Serrodyne, 79 SHG microscopy, 538–539 Short circuit condition, 541–542, 551–552 Short wavelength loss, 266 Side lobe, 151 Side-mode suppression ratio (SMSR), 351, 352 Side-polished fiber, 360–361 Side-scatter, 422–429 Sidetap, 270–274 Sidetap filter bandwidth, 279–280 design diagram, 278 zero back reflection, 164–165
Index Side tap grating (STG), 136 angular distribution of radiation, 138–139 sensitivity to UV induced refractive index change, 154–155 spectrometer, 153 spectrum analyzer, 153 Signal wave, 141 backward propagating, 126 Sign change, 571–572 Silica Herasil, 534 Homosil, 534 impurity level, 534 Infrasil, 534 Suprasil, 534 temperature dependence of grating length, 83 tetrahedra, 18 thermal expansion coefficient, 83 Silica block interferometer, dependence of grating length, 64 Silicon optical bench (SLOB), 353 Silver nanoclusters, 570 SIMS. See Secondary ion mass spectroscopy Simulation methods for gratings, 171 Simulation of gratings Bloch theory, 120 effective index, 409, 423–424 layer-peeling, 183 T-matrix, (see transfer matrix method) Gel’Fand-Levitan-Marchenko coupled integral, 120, 182–183 Gel’Fand-Levitan-Marchenko inverse scattering, 171, 182–183 phase only sampling, 181–182 Rouard’s method for thin films, 119–120, 171, 177–178 transfer matrix method, 171, 172–177 restrictions of, 176–177 Single-frequency laser, 361–362, 363–366, 388, 390–391 high power, 390–391, 391f Slowly varying envelope approximation (SVEA), 125 Small signal gain, 362–363 Sodium concentration of, 556 of different glasses, 560 long term, 548–549 of poled silica, 542 stability, 564 Sodium oxide, accumulation of, 566
Index Soliton, 358, 371 Space charge region, 533, 536, 540, 541, 548, 551, 557, 564, 568, 569–570 Spatial hole burning, 368, 390–391 Spectroscopy secondary ion mass, 543 Spectrum analyzer, 153, 406, 407f, 410–411, 412, 419–420 effect of linewidth, 412 Spontaneous polarization, 546 Step-chirped grating (STG), 306–319 design diagram, 311, 312 sections vs. chirp, 93–94, 322f Stepwise tuned laser, 578–579 Stokes field, 372 Strain, 38 Bragg wavelength dependence, 87–89 Stress internal, 429–431 UV induced, 430–431 Stress gradient, 546 Stress-optic coefficient, 38 Stress-relief model, 430–431 Structural modification, 554 Super-step-chirped grating (SSCG), 93–94, 319–322 high resolution reflectivity and group delay, 322 join, 320 join and group delay, 321 join and reflection spectrum, 320 long, 320, 321, 322 Supermodes, 266 Superstructure, 85 Superstructure grating, 227–229 Bragg wavelengths, 84 transmission spectrum, 232–233 wavelength spacing, 84–85 Surface potential, 543 Susceptibility, 16 2 x 2 Switch, 578 Switching voltage, 576 Symmetry, 130–132 Systems measurements, 325–330 Systems simulation, 327–330
T Talbot effect, temporal, 336, 337f interferometer, 59f, 94–96
613 Temperature compensation, 462–467, 476 effect of, 405, 443–444 tuning, 325–326 Tension, axial, 430, 461, 463f, 466–467, 495 Tetra-chloride germanium, 18, 27–28 silicon, 18 Sn, 28 Thermal decay, hydrogenated B-Ge fibers, 28 Thermally stimulated depolarization current, 551 Thermal runaway, 535, 566 Third harmonic generation EFISH (see electric field induced second harmonic generation) electric field induced second harmonic generation, 528 Third-order nonlinearity, 17, 528 increasing the, 569 third-order nonlinear mixing, 530–531 Third-order polarizability, increasing the, 561 Third-order susceptibility (w3) increased value of, 569 resonance enhancement of, 569–570 spatial variation of, 541–542 Three-level laser, 362 Threshold voltage, 534–535 Tilted gratings, shortening due to fringe depth, 65–66 Transfer characteristics, 405 Transfer matrix elements codirectional, 176 counterpropagating, 167 Rouard’s method, 177–178 Transmission bit error rate (BER), 252, 325 dip, 412–413 error floor, 305–306 eye closure, 328–330 pulse broadening, 322–323 Transverse momentum of STG, 158 Trapping sites, 545 Tuning, compression, 393 Type IIA, B-Ge, 28
U UV erasure, 550 UV femtosecond pulses, 535
614 UV lasers high coherence, 106–110 low coherence, 105–106 types of, 108t UV trimming, 67–69, 238, 245–247
V Vector-voltmeter, 414–415, 415f, 416f Vee-groove, silicon micro-machined, 353 Video information, 577 Visibility, 106, 127 Voltage divider, 547
W Water molecule, 545, 546, 553, 556–557 Wave equation, 122 Waveguides fabrication, 566 normalization constant, 124 optical fiber, 122 orthogonality relationship, 122 perturbation, 126 planar, 20
Index Wavelength-division-multiplexing (WDM), 301–302, 349, 354, 361, 364f, 368f, 373–374, 379f coupler, 363, 372f, 373 dense, 352 Wavelength selection, 578–579 Wire contact, 549–550 micro, 549–550 position in hole, 549–550
X XeCl, 108t X-rays, 535
Y Ytterbium (Yb), 5–6, 362, 369, 372, 390, 391–392, 393f, 394 DBR laser, 458f doped laser, 457
Z Zeroes (FWFZ), 147, 148f, 150, 406, 412–413, 412f, 414, 421–422 ZBLAN, 391–392, 512–513, 518–520 Tm, 521, 523f
