Photonic Microresonator Research and Applications (Springer Series in Optical Sciences, 156)

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OPTICAL SCIENCES founded by H.K.V. Lotsch Editor-in-Chief: W. T. Rhodes, Atlanta Editorial Board: A. Adibi, Atlanta T. Asakura, Sapporo T. W. Hänsch, Garching T. Kamiya, Tokyo F. Krausz, Garching B. Monemar, Linköping H. Venghaus, Berlin H. Weber, Berlin H. Weinfurter, München

156

Springer Series in

OPTICAL SCIENCES The Springer Series in Optical Sciences, under the leadership of Editor-in-Chief William T. Rhodes, Georgia Institute of Technology, USA, provides an expanding selection of research monographs in all major areas of optics: lasers and quantum optics, ultrafast phenomena, optical spectroscopy techniques, optoelectronics, quantum information, information optics, applied laser technology, industrial applications, and other topics of contemporary interest. With this broad coverage of topics, the series is of use to all research scientists and engineers who need up-to-date reference books. The editors encourage prospective authors to correspond with them in advance of submitting a manuscript. Submission of manuscripts should be made to the Editor-in-Chief or one of the Editors. See also www.springer.com/series/624 Editor-in-Chief William T. Rhodes Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [email protected] Editorial Board Ali Adibi Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [email protected] Toshimitsu Asakura Hokkai-Gakuen University Faculty of Engineering 1-1, Minami-26, Nishi 11, Chuo-ku Sapporo, Hokkaido 064-0926, Japan E-mail: [email protected] Theodor W. Hänsch Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Straße 1 85748 Garching, Germany E-mail: [email protected] Takeshi Kamiya Ministry of Education, Culture, Sports Science and Technology National Institution for Academic Degrees 3-29-1 Otsuka, Bunkyo-ku Tokyo 112-0012, Japan E-mail: [email protected] Ferenc Krausz Ludwig-Maximilians-Universität München Lehrstuhl für Experimentelle Physik Am Coulombwall 1 85748 Garching, Germany and Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Straße 1 85748 Garching, Germany E-mail: [email protected]

Bo Monemar Department of Physics and Measurement Technology Materials Science Division Linköping University 58183 Linköping, Sweden E-mail: [email protected] Herbert Venghaus Fraunhofer Institut für Nachrichtentechnik Heinrich-Hertz-Institut Einsteinufer 37 10587 Berlin, Germany E-mail: [email protected] Horst Weber Technische Universität Berlin Optisches Institut Straße des 17. Juni 135 10623 Berlin, Germany E-mail: [email protected] Harald Weinfurter Ludwig-Maximilians-Universität München Sektion Physik Schellingstraße 4/III 80799 München, Germany E-mail: [email protected]

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Ioannis Chremmos · Otto Schwelb · Nikolaos Uzunoglu Editors

Photonic Microresonator Research and Applications

123

Editors Ioannis Chremmos School of Electrical and Computer Engineering National Technical University 157 80 Athens Zografou Greece [email protected]

Nikolaos Uzunoglu School of Electrical and Computer Engineering National Technical University 157 80 Athens Zografou Greece [email protected]

Otto Schwelb Department of Electrical & Computer Engineering Concordia University 1455 Maisonneuve Blvd. West, EV005.126 Montreal QC H3G 1M8 Canada [email protected]

Springer Series in Optical Sciences

ISSN 0342-4111 e-ISSN 1556-1534 ISBN 978-1-4419-1743-0 e-ISBN 978-1-4419-1744-7 DOI 10.1007/978-1-4419-1744-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010922071 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

It has been observed that in science and technology, as in other areas of human development, there is a remarkable coincidence in the appearance of ideas related to a relatively narrow discipline. Although one can argue that with the ascent of global communication, increasing access to information and with the facility of modern travel, the rapid maturing of closely connected ideas is a natural consequence, yet, it is still with immense fascination that we record the emergence and growth, in a mere two decades, of an area in optical sciences: photonics. Documented by innumerous scientific papers and an entire industry devoted to exploit its economic potential, photonics has developed into a well-established discipline. To satisfy the need for information on the theory and practice of photonic devices design, a number of books have been published recently, aiming to assist both students and practitioners. Somewhat distinct from these books, the present volume presents a broad panorama of all aspects of the state of art in photonic microresonator research and applications, including established design principles as well as creative novel approaches which will, no doubt, engender new ideas in the reader. The book has been written for those who have an interest in recent progress and trends in photonic devices and device design, and for those who would enquire about their potential, so it is aimed at technologists, engineers, and physicists who have at least a rudimentary background in electromagnetics, optical electronics, and wave propagation. However, the applications of photonics include not only telecommunications and signal processing, but also instrumentation as well as sensing, in particular displacement, chemical, and biological sensing. For this reason we expect a significant audience also from automotive and aerospace designers, chemists, pharmaceutical researchers, and life scientists. The subject is both well defined and broad. This statement appears to be contradictory only until the viewer’s standpoint is defined. For a non-specialist photonic microresonator research and applications appear to be a narrow segment of electrical engineering or high-speed telecommunications. On the other hand, in the past two decades the field has blossomed to become so diverse that we encountered considerable difficulty confining it to 19 chapters. These chapters present a compact overview of optical microresonators in communications, signal processing, basic and advanced filter design, active tuning, and band-limited mirror construction. They also deal with applications related to high-Q photonic crystal (PC) v

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Preface

microcavities and PC circuits, photonic molecules, microcoil and microsphere resonators, and microresonators tuned by micro-electro-mechanical systems (MEMS). Two chapters are devoted to sensing, one of which focuses on biodetection methods. Not to be left out, we included in our coverage photonic switching schemes, polarization rotation phenomena in ring resonators, the impressive slow light applications in coupled microring and PC cavities, the novel concept of spectral engineering, and radial Bragg resonators and nanolasers. Even though various microresonator design techniques are met and used throughout the book, a chapter is dedicated to the sophisticated spatial coupled-mode analysis of integrated microresonator filters. The last chapter examines various benchmarks to assess the perfomance of microresonators in real-word telecommunications. Thus, through the chapters one can appreciate the fractal nature of science: the closer one approaches the more details unfold. We often use the prefix ‘micro’ when referring to ring shaped or other resonators. This prefix is but loosely related to their actual size, except for the fact that these devices are all so small that they are indiscernible to the naked eye. As of today, the characteristic dimensions of photonic devices, the fabrication feature dimensions, and the distances between waveguides are measured in micrometers and nanometers. The trend is to reduce these dimensions without sacrificing performance caused by increased radiation loss, surface roughness scattering, or undesirable signal interference. How much signal processing power can be crammed into a few cubic wavelengths – that is the question. We must not close this Preface without a very important note on notation. As much as one would like to see a uniform set of symbols used throughout the book, it soon becomes clear that this is a hopeless goal when confronted by so many contributors and such multifarious topics. There is also the fact to be considered that there are only so many symbols in the alphabets we use. Conflicts are therefore here to stay; in world affairs as in notation used in scientific manuscripts. However, we have made a dedicated effort to clarify the terminology for easy reading of the text as well as to double check all equations so that they can be used with confidence. We would like to acknowledge the authors for their contributions and for their diligence in providing us with clarifications, corrections, and updates to their manuscript during the long months of our ruminations and gestation. Our thanks also go to the Springer staff for the high-quality professional work and continuous assistance during this project. Athens, Greece Montreal, QC, Canada Athens, Greece

Ioannis Chremmos Otto Schwelb Nikolaos Uzunoglu

Contents

1 Fundamental Principles of Operation and Notes on Fabrication of Photonic Microresonators . . . . . . . . . . . . . Landobasa Y.M. Tobing and Pieter Dumon

1

2 Circular Integrated Optical Microresonators: Analytical Methods and Computational Aspects . . . . . . . . . . . . . . . . . Kirankumar Hiremath and Manfred Hammer

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3 Polarization Rotation in Ring Resonators . . . . . . . . . . . . . . Francesco Morichetti 4 Series-Coupled and Parallel-Coupled Add/Drop Filters and FSR Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . Yasuo Kokubun and Tomoyuki Kato 5 Advanced Microring Photonic Filter Design . . . . . . . . . . . . . Vien Van 6 Band-Limited Microresonator Reflectors and Mirror Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Otto Schwelb and Ioannis Chremmos 7 Slow and Stopped Light in Coupled Resonator Systems . . . . . . Shanhui Fan, Sunil Sandhu, Clayton R. Otey, and Michelle L. Povinelli 8 Processing Light in Reconfigurable Directly Coupled Ring Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrea Melloni 9 Microresonators with Active Tuning . . . . . . . . . . . . . . . . . Qianfan Xu 10

Performance of Single and Coupled Microresonators in Photonic Switching Schemes . . . . . . . . . . . . . . . . . . . . Jacob B. Khurgin

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87 115

139 165

181 205

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Contents

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Single Molecule Detection Using Optical Microcavities . . . . . . . Andrea M. Armani

253

12

Microfiber and Microcoil Resonators and Resonant Sensors . . . . Fei Xu and Gilberto Brambilla

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13

Photonic Crystal Ring Resonators and Ring Resonator Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weidong Zhou, Zexuan Qiang, and Richard A. Soref

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High-Q Photonic Crystal Microcavities . . . . . . . . . . . . . . . . Qiang Li and Min Qiu

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15

Radial Bragg Resonators . . . . . . . . . . . . . . . . . . . . . . . . Jacob Scheuer and Xiankai Sun

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Photonic Molecules and Spectral Engineering . . . . . . . . . . . . Svetlana V. Boriskina

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Fundamentals and Applications of Microsphere Resonator Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vasily N. Astratov

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MEMS-Tuned Microresonators . . . . . . . . . . . . . . . . . . . . Ming-Chang M. Lee, Ming C. Wu, and David Leuenberger

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Microresonators for Communication and Signal Processing Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lin Zhang and Alan E. Willner

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contributors

Andrea M. Armani Mork Family Department of Chemical Engineering and Materials Science and Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, CA, USA, [email protected] Vasily N. Astratov Department of Physics and Optical Science, University of North Carolina at Charlotte, Charlotte, NC, USA, [email protected] Svetlana V. Boriskina Department of Electrical and Computer Engineering, Boston University, Boston, MA, USA, [email protected] Gilberto Brambilla Optoelectronics Research Centre, University of Southampton, Southampton, Hampshire, UK, [email protected] Ioannis Chremmos School of Electrical and Computer Engineering, National Technical University, 157 80 Athens, Zografou, Greece, [email protected] Pieter Dumon Universiteit Gent, Sint-Pieterniewstraat, Ghent, Belgium, [email protected] Shanhui Fan Ginzton Laboratory, Stanford University, Stanford, CA, USA, [email protected] Manfred Hammer Department of Applied Mathematics, MESA+ Institute for Nanotechnology, University of Twente, Enschede, The Netherlands, [email protected] Kirankumar Hiremath Department of Mathematics, Institute for Scientific Computing and Mathematical Modeling, University of Karlsruhe, Karlsruhe, Baden-Württemberg Germany, [email protected] Tomoyuki Kato Photonics Laboratory, Network Systems Laboratories, Fujitsu Laboratories Ltd., Kawasaki, Japan, [email protected] Jacob Khurgin Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD, USA, [email protected] Yasuo Kokubun Graduate School of Engineering,Yokohama National University, Hodogaya-ku, Yokohama, Japan, [email protected]

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Contributors

Ming-Chang M. Lee Department of Electrical Engineering and Institute of Photonics Technologies, National Tsing Hua University, HsinChu, Taiwan, [email protected] David Leuenberger CSEM Centre Suisse d’Electronique et de Microtechnique SA, Zürich, Switzerland, [email protected] Qiang Li Photonics and Microwave Engineering, Royal Institute of Technology (KTH), Kista, Sweden, [email protected] Andrea Melloni Dipartimento di Elettronica e Informazione, Politecnico di Milano, Via Ponzio, Milano, Italy, [email protected] Francesco Morichetti POLICOM – Dipartimento di Elettronica e Informazione, Politecnico di Milano, Fondazione Politecnico di Milano, Milano, Italy, [email protected] Clayton R. Otey Department of Applied Physics, Ginzton Laboratory, Stanford University, Stanford, CA 94305, [email protected] Michelle L. Povinelli Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, CA, USA, [email protected] Zexuan Qiang Department of Electrical Engineering, NanoFAB Center, University of Texas at Arlington, Arlington, TX, USA, [email protected] Min Qiu Photonics and Microwave Engineering, Royal Institute of Technology (KTH), Kista, Sweden, [email protected] Sunil Sandhu Department of Electrical Engineering, Ginzton Laboratory, Stanford University, Stanford, CA 94305, [email protected] Jacob Scheuer School of Electrical Engineering, Department of Physical Electronics, Tel-Aviv University, Ramat-Aviv, Israel, [email protected] Otto Schwelb Department of Electrical and Computer Engineering, Concordia University, Montréal, QC, Canada, [email protected] Richard A. Soref Sensors Directorate, Air Force Research Laboratory, AFRL/RYNC, Hanscom Air Force Base, MA, USA, [email protected] Xiankai Sun Department of Applied Physics, California Institute of Technology, Pasadena, CA, USA, [email protected] Landobasa Y. M. Tobing Nanyang Technological University, 50 Nanyang Avenue Singapore, [email protected] Vien Van Department of Electrical & Computer Engineering, University of Alberta, Edmonton, AB, Canada, [email protected] Alan E. Willner Optical Communication Laboratory, Department of Electrical Engineering, University of Southern California, Los Angeles, CA, USA, [email protected]

Contributors

Ming C. Wu Department of Electrical Engineering & Computer Sciences, University of California, Berkeley,CA, USA, [email protected] Fei Xu Department of Materials Science and Engineering and National Laboratory of Solid State Microstructures, Nanjing University, Nanjing, Jiangsu, China, [email protected] Qianfan Xu Department of Electrical and Computer Engineering, Rice University, Houston, TX, USA, [email protected] Lin Zhang Optical Communication Laboratory, Department of Electrical Engineering, University of Southern California, Los Angeles, CA, USA, [email protected] Weidong Zhou Department of Electrical Engineering, NanoFAB Center, University of Texas, Arlington, TX, USA, [email protected]

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Chapter 1

Fundamental Principles of Operation and Notes on Fabrication of Photonic Microresonators Landobasa Y.M. Tobing and Pieter Dumon

Abstract Light confining microresonators based on evanescent wave propagation and whispering gallery (WG) modes have received much attention in the past decades, due to their conceptual similarity with their standing wave counterparts, improvements in fabrication technology, and their versatility in realizing various functions in telecommunications, sensing, measurement, and instrumentation. In this chapter the general concepts, design principles, and practical realizations of optical microresonators are briefly introduced. Using a simple but generic model, important design parameters such as the Q-factor, the finesse, the free spectral range, the intensity buildup, and the effects of loss are derived in general terms from basic principles. The discussion on cavity design is completed by reviewing several intrinsic properties of available material systems, such as the refractive index contrast, which is essential for field confinement, which limits the resonator geometrical size, contributes to material loss, and influences the nonlinear response. Finally, fabrication techniques of microring and WG resonators are also outlined, from the surface tension-mediated processes in silica microspheres and microtoroids to the wafer-based technologies such as deep ultraviolet (DUV), electron beam, and nanoimprint lithography. Some notable examples of fabricated resonators are discussed and compared.

1.1 Introduction Optical microcavities have generated tremendous research progresses in many aspects of optical science, such as all-optical nonlinear switching [1–3], modulators [4–6], all-optical signal processing [7–9], biochemical sensing [10–12], slow-light structures or optical buffers [13–19], wavelength division-multiplexed (WDM) optical filters for optical networks and on-chip optical interconnects [20–27], L.Y.M. Tobing (B) Nanyang Technological University, Nanyang Avenue, Singapore, 639798, Singapore e-mail: [email protected] I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_1,  C Springer Science+Business Media, LLC 2010

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L.Y.M. Tobing and P. Dumon

ultralow-threshold microlasers [28–29], enhancement of Raman processes [30] as well as Raman lasing [31–33], and in fundamental experiments in quantum physics, e.g., cavity quantum electrodynamics (QED) [34]. All the above applications are made possible by the strong light confinement in a small modal volume. By adjusting the shape, size, and material composition, the microresonator can be designed to support a spectrum of optical modes with required polarization, frequency, and field patterns. The most fundamental form of optical resonator is the Fabry–Pérot (FP) etalon [35], which consists of a slab-shaped optical medium with highly parallel sides surrounded by a medium with a different refractive index. The buildup of the intracavity intensity caused by reflections from the interfaces in this standing wave resonator, aided by constructive interference between the incoming and the intracavity fields, depends on the time spent by the light bouncing back and forth inside the cavity and is determined by the reflectivity of the interfaces. The resonances are periodically spaced in frequency with a free spectral range (FSR) that is inversely proportional to the optical cavity length (inter-mirror distance). Highly reflecting mirrors in place of the interfaces result in resonances with high finesse, i.e., in high FSR to bandwidth (BW) ratio. The simple yet elegant concept of the FP resonator is the fundamental principle of operation found in all optical microresonators, both standing wave and traveling wave. In traveling wave resonators, the closed loop offers spatial separation between the waves impinging on and emerging from the couplers, preventing the formation of standing waves. In principle, there are so far two well-established mechanisms of light confinement and guidance inside the volume of an optical microresonator. The first is the conventional mechanism of total internal reflection (TIR) and the existence of evanescent waves, where the guiding medium must be optically denser, i.e., have a higher refractive index, than the surrounding one in order to achieve light confinement. The second is the photonic bandgap (PBG) found in artificial optical media having a spatial periodicity in one, two, or three dimensions, termed photonic crystal (PC) [37], which is a result of the phenomenon of Bragg reflection causing the formation of frequency bands where propagation of light is prohibited by the destructive interference of field harmonics inside the crystal. Highly confined optical modes can be achieved in these bands when certain defects are introduced in the otherwise perfectly periodic crystal. With PC defect modes [38–40], the light can be confined in a size comparable to its wavelength (λ/n), where λ is the vacuum wavelength and n is the medium refractive index. Currently, the most highly established traveling wave-integrated optical resonator is the microring (MR) [41] and the whispering gallery (WG) microresonator, based on the phenomenon of continuous light TIR along the boundary of a rotationally symmetric optical medium, such as a microdisk [42], microsphere [43], or microtorus [36]. Note that MRs are not necessarily circular (racetrack) and that polygonal resonators [44] are also based on TIR. MR resonators are usually designed to have a narrow width so that only a single guided mode is supported which is a bent version of the conventional straight waveguide mode. Generally speaking, MRs supporting low radial order have very small width to bend radius ratio, while for WG modes

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Principles of Operation and Notes on Fabrication

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this ratio is close to unity. Similar concepts are found in rotationally symmetric ringor disk-shaped defects in a PC environment, as already demonstrated in annular Bragg gratings [45] and PC ring resonators [46], where light is confined by the PBG created by the surrounding configuration. Some of these types of microresonators are illustrated in Fig. 1.1. Microcavities based on WG and ring modes, generically called recirculating cavity modes, have received considerable attention due to their relative ease of fabrication (in comparison with their PC counterparts), design flexibility, and suitability for planar integration (microspheres excepted). In this chapter, we attempt to give a brief introduction to the operating principle of recirculating wave cavities and their experimental aspects, such as fabrication principles and material systems.

Microdisk

Microsphere

Microring

Polygon

Annular Bragg grating (ABG)

Microtoroid

Micro-pillar (micro-post)

Fig. 1.1 The various forms of photonic microresonators. The schematic of the microtoroid is courtesy of Vahala from [34], reprinted by permission from Macmillan Publishers Ltd

1.2 General Characteristics of MR and WG Resonators The most basic configuration of the MR resonator is shown in Fig. 1.2, which consists of a ring-shaped waveguide coupled to either one or two optical waveguides. The cavity mode is excited by evanescent coupling to closely spaced optical waveguides, which function as semi-reflecting mirrors. Direct analogy can be drawn between recirculating resonators and standing wave resonators. In the case of a ring resonator coupled to one waveguide, one relates to a Gires–Tournois interferometer [47] with one partially and another totally reflecting mirror. In the case of a ring resonator coupled to two bus waveguides, one has a Fabry–Pérot interferometer with two partially reflecting mirrors. The resonance condition is satisfied when the (effective) circumference of the ring, or generally the round-trip length, is equal to an integer multiple of the optical wavelength inside the medium. This translates to a series of Lorentzian-shaped transmission curves evenly spaced in frequency by the FSR, with the resonance

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Single coupling

Gires-Tournois interferometer

Double coupling

Fabry-Pérot interferometer

Fig. 1.2 The schematics of traveling wave resonators and their standing wave analogs

linewidth characterizing the storage time of photons inside the cavity. The photon lifetime can be normalized to one optical cycle, known as the quality factor (Q), or the cavity round-trip time, known as the cavity finesse (F). The maximum achievable Q-factor is defined as Qint , which depends only on the intrinsic cavity loss. When the resonator is coupled to the external world, the Q-factor further decreases due to the loss imported by the coupler (Qext ). As a result the final quality factor −1 −1 Qload is made up of these two components: Q−1 load = Qint + Qext . The MR cavity consists mainly of two key elements, namely the optical feedback and the coupler. These two elements affect each other in the design of high-Q optical resonances, and their relationship to each other determines the cavity intensity buildup. Under the condition called over-coupling, the power coupling dominates over the round-trip loss, and the circulating light acquires enhancement in intensity (intensity buildup) while experiencing delay (phase buildup). The intensity buildup increases with decreasing coupling strength to the point where the coupled power is equal to the cavity round-trip loss, which is known as the critical coupling in the microwave and optical terminology [48–50]. The intensity buildup at critical coupling is the highest and the phase becomes discontinuous. In the under-coupled situation, on the other hand, the coupled power is less than the cavity round-trip loss and the slope of the phase spectrum changes sign, which resembles the spectrum of the atomic susceptibility in the presence of absorption. Ensuring the right coupling condition is crucial in applications that require enhancement of nonlinear interaction or light–matter interaction in general. In the sections that follow, the mechanisms of coupling and cavity loss will be briefly discussed. Briefly revisiting basic concepts of an MR cavity from a designer’s point of view, one recognizes that the major defining quantities of an isolated (uncoupled) MR are the average (effective) perimeter length L and the complex propagation constant β–jα of the circulating mode, β being the phase constant and α being the amplitude (or field) attenuation coefficient due to the intrinsic loss mechanisms of the cavity: radiation, material absorption, and loss due to imperfections such as surface roughness. On resonance, the phase acquired by the wave after a complete round-trip is an integer multiple of 2π , i.e., βL = 2π N, with N the mode number. Equivalently, the perimeter is an integer multiple of the effective wavelength on resonance, or L = Nλ0 /neff , where the effective refractive index is defined as neff = cβ/ω and λ0 = c/f0 is the free-space wavelength at the resonance frequency f0 (usually the design or center frequency of the device). The resonances are separated by the FSR,

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given in frequency by the inverse of the round-trip time delay FSR = c/(ng L), or in wavelength by FSR = λ20 /(ng L), where ng = c(dβ/dω) is the group index. Notice that ng characterizes the group velocity of a wave packet with nonzero bandwidth (BW) around frequency ω and should be distinguished from neff which characterizes the phase velocity of a mode with harmonic time dependence, i.e., a purely monochromatic wave. Furthermore, neff is generally frequency dependent which implies that n g = neff +ω(∂neff /∂ω), and therefore only a slowly varying function neff (ω) justifies the frequently used approximation ng ∼ = neff . This is often true across the very narrow bandwidth of high-Q resonators. The uncoupled MR is characterized by a Q-factor defined by

Qint = ω0

π neff Stored energy 2π N ω0 τR = = = , Intrinsic power loss 2αL αλ0 2αL

(1.1)

where τ R = Lneff /c is the cavity round-trip time. An equivalent definition of the Q-factor is 2π times the ratio of the cavity lifetime (τ ) over the period of the optical wave (2π /ω0 ): Q = ω0 τ . This is consistent with (1.1) if one recognizes that the cavity lifetime, i.e., the time required for the field intensity to decay by a factor of e, is given by τ = (2γ )−1 , with γ being the amplitude decay rate in time. The latter is connected with the amplitude attenuation coefficient by equating the expressions for the round-trip loss in time and in space, namely γτ R = αL. The familiar definition of the Q-factor, Q = ω0 / ω, also stems from the previous relations where the resonator linewidth is apparently given by ω = 2γ . In the case of an MR coupled to a single bus waveguide (see Fig. 1.2) with power coupling coefficient K, the power transmittance through the waveguide is calculated to be

T=

(t − σ )2 + 4tσ sin2 (θ/2) ∼ (αL − K/2)2 + (ω − ω0 )2 τR2 , = (αL + K/2)2 + (ω − ω0 )2 τR2 (1 − tσ )2 + 4tσ sin2 (θ/2)

(1.2)

where the round-trip phase θ = βL acts as a normalized frequency parameter, σ = exp(–αL) is the field round-trip loss (σ includes coupler loss, if any), and t = (1–K)1/2 is the field transmission coefficient through the coupler. Expression (1.2) exhibits distinct transmission minima at the MR resonances at θ = 2π with an approximately inverse Lorentzian lineshape which is revealed by the second approximate expression of (1.2) being valid for αL, K1, and (ω–ω0 )τ R 1. Critical coupling is defined as the condition of zero transmittance, when all the power is absorbed by the cavity, a condition which is obtained only on resonance and when t = σ , i.e., when the cavity round-trip loss equals the coupling coefficient (1–σ 2 = K). It is easy to show that the maxima of 1–T are characterized by a FWHM, given in terms of the round-trip phase by θ FWHM = 4 sin−1 [(1–tσ )/2(tσ )1/2 ]. The loaded

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Q-factor of the system is defined by Qload = f0 /FWHM, which can be approximated by Qload = 2π N/ θ FWHM . For sufficiently small K and α, 1–tσ 1 and we obtain Qload ∼ =

2π N ∼ 2π N . = 2(1 − tσ ) K + 2αL

(1.3)

Using (1.1), and defining the external Q-factor Q ext = 2π N/K (1.3), reveals −1 −1 the fundamental result Q−1 load = Qint + Qext , indicating the reduction of the Qfactor due to the external power loss caused by coupling to the waveguide. This is a general rule of resonator theory and it also applies to the case of MRs coupled to more than one waveguides and/or cavities in which case the loaded Q is given by  −1 −1 Q−1 = Q + Q where the summation is over all contributors of external ext,i load int loss coupled to the resonator. Another measure of resonator performance is the so-called finesse (F), which is defined as the ratio of the FSR over the BW or, equivalently, as 2π times the ratio of the loaded cavity lifetime (τ load = Qload /ω0 ) over the round-trip group delay (τ 0 = ng L/c); thus F=

FSR τload neff Qload = 2π . = FWHM τ0 ng N

(1.4)

From (1.3) and (1.4) it follows that the physical interpretation of finesse is proportional to the number of round-trips before the light intensity in the MR decays by a factor of e. The Q-factor is simply this number multiplied by the mode number, i.e., the number of field oscillations. A common feature of resonances, critical in nonlinear applications, is the buildup of the MR intensity due to the constructive interference of the circulating wave within the cavity. The ratio of the MR field intensity at resonance and the incident intensity is called the build-up factor. In our example this is given by B = K/(1–tσ )2 ∼ =4K/(2αL+K)2 . Intuitively, the build-up factor can be derived from the coherent summation of the 2τ /τ 0 = F/π intra-cavity field components with amplitude K1/2 that interfere over the field cavity lifetime (2τ = γ −1 ), i.e., EMR = (F/π )K1/2 Einc , from which the previous expression for B is obtained as B=|EMR / Einc |2 . Under negligible intrinsic cavity loss (α→0), B is inversely proportional to the coupling strength (B∼ =4/K) and also proportional to the finesse (B∼ =2F/π ), if ng ∼ =neff . Note that for a symmetric add-drop filter the latter expression becomes B∼ =F/π . The circulation of light inside an MR, or generally any optical resonator, gives rise to the group delay of the optical signal, one of the most important enabling properties of a photonic microresonator. This delay is the result of the enhanced phase sensitivity of the device and is measured by the derivative of the phase delay with respect to frequency: τ =−

dϕ dϕ = − τ0 , dω dθ

(1.5)

1

Principles of Operation and Notes on Fabrication

7

where ϕ is the phase of the complex field transmission coefficient and τ 0 = dθ /dω = ng L/c is the round-trip group delay. For an MR coupled to a single waveguide, the group delay at resonance is τ = τ 0 σ K/[(1–σ t)(σ –t)]. For α→0 this becomes τ = 4τ 0 /K = Bτ 0 , i.e., the normalized group delay equals the build-up factor and is directly related to the finesse, as expected from the physical meaning of the latter. The above basic concepts are present in any system incorporating optical microresonators-coupled mutually and/or to external waveguides.

1.2.1 Evanescent Coupling Power transfer between two optical modes occurs when there is a significant field overlap between the modes. Rigorous studies of optical couplers based on various analytical methods are available [51, 52], and it has been concluded that the coupling strength of an ideal coupler is dependent on the interaction length between the two optical modes and the extent of the overlap, i.e., the proximity of the waveguides. Efficient power transfer occurs only when the two optical modes are phase-matched, and there are many ways to achieve this. For example, prism-coupling can be used to couple light into a microsphere (Fig. 1.3, left panel). This is in fact the earliest concept of coupling light into a microresonator. By TIR at the prism– microsphere interface, the light experiences a Goos–Hanchen shift in the evanescent field enabling a phase match and efficient coupling [53, 54]. In the toroidal cavity coupler (middle panel), a tapered fiber is used to achieve the phase-matching condition between the fiber and the cavity mode [54–59]. Recently, tapered fibers have also been used for coupling to high-Q PC defect modes [60, 61]. Another approach to evanescent coupling of MR resonators, now used almost exclusively, is the lateral and the vertical coupling shown in the right panel of Fig. 1.3 [21, 62, 63]. Both approaches have their own advantages and disadvantages. In lateral coupling, the fabrication can be done in one step because both the resonator and the waveguides are fabricated on the same layer. However, the gap separation is limited by the fabrication tolerances of the lithography system, and the index contrast range is limited by the core/cladding materials restriction.

Microsphere Tapered fiber

Lateral coupling

Microtoroid Microprism

Fig. 1.3 Different evanescent field coupling schemes

Vertical coupling

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L.Y.M. Tobing and P. Dumon

A racetrack geometry can be introduced to increase the interaction length while keeping the gap separation fixed. In the vertical coupling scenario, the gap separation is not adjusted lithographically, but rather epitaxially. This offers the advantage of a smooth surface which reduces coupling loss due to surface roughness. Since the adjustment of the gap separation is epitaxial, a much smaller gap separation can be defined and the control over the gap separation is much better. However, the disadvantage is that multiple steps of growth are required and sometimes flipchip technique is needed to vertically integrate the two structures [63]. On the other hand, vertical coupling removes the index contrast restriction of lateral coupling, enabling a greater flexibility in the choice of the buffer material and an enhanced index contrast between waveguide and buffer. As a result a more effective energy confinement can be achieved with significantly reduced radiation loss and bending radii. In terms of practicality, lateral coupling offers simplicity in the fabrication while vertical coupling is an enabling technology for ring radii less than ∼5 μm. Ideally, a coupler simply performs a power transfer by means of evanescent field coupling. Realistically, however, there are also higher order effects such as the additional phase shift due to self-coupling, which has been shown to give undesirable effects in filter design [64]. Furthermore, the coupling is inherently phase-mismatched because the optical modes of the circular cavity and the waveguide are inherently different. This imparts additional loss known as coupling loss, which becomes important when ultralow loss is required. If the cavity supports several radial modes with the same azimuthal order, it is also possible to have coupling to all these modes, thereby reducing the coupling efficiency. This can be prevented with a single-mode ring geometry in which only the lowest radial mode is supported.

1.2.2 Sources of Loss in MR and WG Resonators In an ideal situation where the optical medium is lossless the light can be confined indefinitely and the Q-factor can be arbitrarily high. However, in reality the cavity mode suffers from radiation loss and material absorption. Furthermore, there is additional loss originating from phase-mismatch coupling, and fabrication induced scattering loss due to sidewall roughness. All these losses contribute to limit the achievable Q-factor of the optical microcavity. Many applications require high-Q resonances to be realized in small modal volumes (V), i.e., high Q/V ratios. However, the Q-factor scales exponentially with cavity dimensions; thus achieving both high Q and small modal volume is rather contradictory unless the overall loss is reduced. It is therefore important to discuss the origin of the various loss mechanisms in WG and MR resonators. 1.2.2.1 Radiation Loss Azimuthal wave propagation of WG and MR modes have an intrinsic radiation loss, due to the fact that the field needs to maintain its modal shape as it bends along

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Principles of Operation and Notes on Fabrication

9

the curved waveguide. As the phase velocity must be proportionally higher at larger radius, there exists a situation where the speed of light cannot increase any further because it has already reached the maximum allowable speed in a given medium (c/n). As a consequence, a fraction of the mode not satisfying the condition of retaining the modal shape will radiate away. The nature of radiation loss can be better understood from the perspective of conformal mapping [51, 62, 65], by which the curved waveguide is transformed into an equivalent straight waveguide. Figure 1.4 compares the propagation characteristics of straight and curved waveguides. In the straight waveguide (left panel), the mode field is symmetric and completely guided by TIR and the normalized propagation constant (effective index, neff ) lies between the cladding and core index (ncl
Waveguide mode ncl

ncore

Bend mode

ncl

ncl

ncore ncl

Δu n(x)

ncore

neff

neff ncl Straight waveguide

Leaky mode Curved waveguide

Conformal mapped index n(u)

Fig. 1.4 Comparison between straight and curved waveguides. The ncore , ncl , and neff signify core, cladding, and effective index (propagation mode), respectively. The inset shows the radiated field from a curved silicon waveguide

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L.Y.M. Tobing and P. Dumon

1.2.2.2 Mode-Conversion Loss In the case of coupling between two identical waveguides, the power transfers from one waveguide to the other with unity conversion efficiency. This is because the field profiles are identical. However, in the case of different field profiles, as in the coupling between an MR resonator and a straight waveguide, it is clear that the efficiency must be less than unity. Such mode-conversion loss often dominates the losses, particularly when the radiation loss and the waveguide loss are practically negligible [66, 67]. Thus it is important to understand the nature of this loss and how to minimize it. The mode-conversion loss in MR and WG cavities may be present in two ways, as illustrated in Fig. 1.5 In a racetrack cavity (left panel), distributed coupling takes place in the straight section, while the curved section is used as an optical feedback. The difference in the effective indices of the straight and curved waveguide mode imparts a Fresnel reflection, and a conversion efficiency less than unity results from the non-perfect mode overlap. It should be noted that there is no coupling loss in the straight section, because the modes are identical. Coupling loss of this type may be reduced if the materials used to construct the cavity provide high-index contrast, i.e., strong mode confinement, and merely impart a slight field asymmetry in the curved section. The other way of reducing the loss is by lateral offset of the straight section relative to the curved ones, so as to match the mode “center of mass” to increase the mode overlap [67]. The second type of coupling loss is present in lumped element or point-coupling schemes in ring geometry. When the intra-cavity field is reasonably strong, as illustrated by the dashed lines (Fig. 1.5, right panel), there is a tendency to extend the azimuthal propagation of cavity mode to the waveguide. This excites the higher order modes of the bus waveguide, which have larger propagation angle than what is required for single-mode guidance. Thus, the light propagates in zigzag manner while giving off radiation at every reflection at the cladding. One method to reduce this loss is by decreasing the width of the straight waveguide, so as to limit the propagation angle of the coupled light inside the waveguide. This method was demonstrated in the fabrication of a high-Q ring resonator with 1.5 μm radius [68].

Radiation

Fig. 1.5 The illustration of mode-conversion loss in racetrack and ring resonators. The gray arrows represent the mode loss as a result from Fresnel reflection and field overlap (for racetrack resonator) and phase-mismatch (ring resonator). The dashed circle denotes the azimuthal propagation of the coupled mode around the microring

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Principles of Operation and Notes on Fabrication

11

1.2.2.3 Sidewall Scattering Loss The propagation loss due to sidewall roughness is the major contributor of the cavity round-trip loss. It originates from imperfect fabrication processes as, for example, in dry etching. The sidewall scattering loss depends on the correlation length, roughness standard deviation, index contrast, and the field magnitude at the sidewall [67, 69, 70]. There are two ways to reduce sidewall scattering loss. In the first way, the waveguide cross section is configured in horizontal geometry so that it supports only the quasi TE mode [71]. This geometry is advantageous from the fabrication point of view, because typically roughness is induced when deep etching is involved, and in the horizontal geometry the waveguide thickness is smaller than the width. The second way is sidewall roughness reduction, which has been proposed in different material systems such as silicon-on-insulator (SOI) [72], polymers [73], and silica [36]. In SOI material systems, by taking advantage of the existence of native oxide in silicon, sidewall smoothening can be done by thermal oxidation of the wave guide/resonator, which is usually silicon. Because such a process is CMOS compatible, the thickness of silica can be controlled down to the nanometer scale. Based on this technique, propagation losses of between 0.8 dB/cm and 3 dB/cm have been experimentally demonstrated [72, 74], some of the lowest reported for on-chip fabricated resonators. In a polymer material system, the smoothening is implemented by thermal reflow. Here, by annealing the polymer optical structure 10–20◦ below the glass transition temperature, the viscosity is reduced and the fluidity of the polymer is enhanced: the smoothening then occurs due to the surface tension. Using this technique, improvement of ∼74 dB/cm in propagation loss has been reported [73]. Although the reflow technique for silicon has been implemented, its control remains difficult. Another surface tension-assisted smoothening can be found in silica microspheres and microtoroids, where both the atomically smooth surface and the resonator itself are formed at the same time. The exhibited Q-factors based on this technique are some of the highest among all microresonators, in the order of 108 –109 .

1.2.2.4 Nonlinear Absorption Although optical resonators provide intensity buildup that is useful for the enhancement of nonlinear processes, certain nonlinear effects such as two-photon absorption and free-carrier effects can cause additional nonlinear loss which degrades the resonator performance. Two-photon absorption (TPA) occurs in semiconductor materials, where an electron in the valence band is excited by two coincident photons in the conduction band. This effect is especially dominant when the photon energy is half the energy bandgap or larger. The generation of these free carriers (due to TPA) creates additional loss known as free-carrier absorption (FCA), and a slow thermal effect that changes the resonance wavelength. These nonlinear effects can be readily seen at moderate on-chip power. It should be noted that the FCA in the above case is a higher order effect generated from TPA (i.e., second-order

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L.Y.M. Tobing and P. Dumon

effect), while the thermal effect is the higher order effect from the nonlinear FCA (tertiary effect). The converse, however, is not true. Introducing free-carrier effects does not give higher order TPA. In some applications, the free-carrier effect is used as a means of modulation or switching in silicon resonators [1, 71, 75]. 1.2.2.5 Bulk Material Loss Some of the optical losses originate from the material itself. Phenomenologically, the optical loss for a defect-free material is approximated as [76] α = αUV exp (λUV /λ) + αR λ−4 + αIR exp ( − λIR /λ),

(1.6)

where αUV , αR , and αIR describe the ultraviolet (primarily electronic), Rayleigh, and infrared (multiphonon) losses, respectively. The wavelengths λUV and λIR are at the edges of the material transparency window. In an amorphous material there are also extrinsic losses (e.g., chemisorption of OH ions in fused silica) and losses due to material imperfection, inclusions, and inhomogeneities. In semiconductor materials, there is also a loss associated with intrinsic doping levels (e.g., silicon). However, compared to other loss mechanisms, this type of loss is very small.

1.3 Resonator Material Systems Fabrication of MR and WG resonators has been carried out for a variety of material systems, such as silica (SiO2 ), silica-on-silicon, silicon (Si), silicon-on insulator (SOI), silicon nitride (SiN) and oxynitride (SiON), polymers, semiconductors such as GaAs, InP, GaInAsP, GaN, and crystalline materials such as lithium niobate (LiNbO3 ) and calcium fluoride (CaF2 ). All these key materials have their own advantages and disadvantages in terms of their linear and nonlinear functionalities, their ease of fabrication processing, and the range of refractive index contrasts that determine the allowable sizes of the resonator features. General properties of these materials are outlined in Table. 1.1 Silica glass has a wide transparency window (from visible to infrared), low intrinsic loss, and is also compatible with fiber-optic technology. The material loss is limited to Rayleigh scattering, thanks to the mature fiber-optic technology that has minimized intrinsic loss around the infrared region due to water and OH contaminants [77]. The fabrication of silica microspheres (or microtoruses) is generally based on the surface tension mechanism, which forms atomically smooth surfaces leading to ultrahigh-Q-factors. The index contrast between silica and air is small in comparison to semiconductor materials (InP, GaAs, Silicon, etc), thus the cavity size scales inversely. For the above reasons, fabrication of silica microspheres (or microtoruses) is easier compared to the planar MR based on semiconductor materials. Typical microspheres have a radius around 100 μm. In terms of nonlinear functionalities, foreign atoms can be introduced (e.g., implantation of Er and Yb atoms into silica) and nonlinear layers can be deposited

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Principles of Operation and Notes on Fabrication

13

Table 1.1 General properties of optical microresonator material systems Demonstrated loss (dB/cm)

Material system

Refractive index

Index contrast (ncore –nclad )/nclad

Fused silica (SiO2 ); silica-on-silicon. Silicon-on-insulator (SOI) Silicon nitride (SiN) Silicon oxynitride (SiON) R Hydex  Polymer III–V compounds

1.44–1.47

∼0.47

3.47

∼1.39–2.47

1.44–1.99 1.44–1.99

∼0.37–1 <0.045

∼1.5–4 ∼0.15

1.44–1.7 1.3–1.6 3.17–3.37

∼0.17 ∼0.2–0.6∗ ∼0.06 (vertical), <2.37(lateral)

∼0.06 ∼3–17 ∼10–20

∗ The

0.8–3.4

∼0.6 index contrast is obtained by vertical confinement through isotropic etching

on top of silica [78, 79]. Due to the exceptionally high-Q-factor generated by this material, several parametric processes have been demonstrated at ultralow power, such as Raman lasing [28], erbium-doped lasing around 1.5 μm [29], and green up-conversion lasing [80]. Microspheres fabricated by melting the tip of an optical fiber are generally difficult to integrate on planar chips. A better integrability is offered by silica-on-silicon fabrication technology where microtoroid and microdisk resonators are formed by heating the top silica part of short silicon posts acting as supports [81]. In silicon-on-insulator (SOI) technology one can have a much larger index contrast (Si/SiO2 ∼ 3.45/1.45) compared to that in silica technology, which enables the fabrication of resonators with reduced dimensions [1, 2, 25, 74, 82]. In terms of fabrication the SOI material system is perfectly compatible with CMOS-based processing, which has been well established for deep-submicron electronic chips [13, 66, 74]. The fact that silicon is transparent for λ > 1.1 μm makes SOI-based resonators suitable for telecommunication wavelengths and propagation losses of 0.8–3 dB/cm [66, 72, 74] have been demonstrated using CMOS-compatible processes. Indeed, superb fabrication control is one of the advantages of using the SOI material system. In terms of nonlinear functionalities, the nonlinear effect in silicon remains limited in comparison to other material systems because of its inherent crystal structure and the fact that it is an indirect-bandgap material, although free-carrier and thermo-optic effects have been shown to be useful for active photonic functions [1, 2, 75]. Although SOI technology offers a high-index contrast (∼130%), the refractive index is not tunable. The relatively recent material systems, on the other hand, namely the silicon nitride (SiN) [23, 83], the silicon oxynitride (SiON) R [22], have tunable refractive indices of ∼30% (for SiN [84, 85], and the Hydex R and SiON) and 17% (for Hydex ), and a demonstrated propagation loss in the range of 1.5–4 dB/cm (SiN) [23, 83], ∼0.15 dB/cm (SiON) [85], and a remarkably R ). low for ∼0.06 dB/cm [86] (Hydex

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The III–V semiconductor compounds are very suitable for photonic active devices due to their intrinsically direct material bandgap transition, as well as their high intrinsic nonlinear coefficients. The refractive index contrast is almost as high as in SOI material systems. Intrinsic nonlinear-optics (Kerr effect) and electro-optics coefficients (Pockels effect) as well as mechanisms of electro-absorption (EA) and the quantum confined stark effect (QCSE) can be readily used to realize optical modulation and switching functions. Furthermore, the so-called bandgap engineering which confines electrons (and holes) in potential wells can greatly enhance nonlinear interaction between photons and atoms, for example, in 2-D quantum wells, 1-D quantum wire, and 0-D quantum dots [87]. Despite the active-friendly feature of III–V compounds, realizing passive functions becomes a challenge for the same operating wavelength because of the light–matter interaction that enhances waveguide loss, e.g., the existence of TPA and FCA. The demonstrated propagation loss based on submicron III–V waveguides reported to date is between 10 and 20 dB/cm [67, 88, 89], which is still moderately high. In terms of availability, III–V compounds are expensive to fabricate. Polymer is best used for active functions, similar to III–V semiconductors. The successes in organic light emitting devices (OLED) has spurred interest in polymeric-integrated optical devices. Active media, such as chromophores, organic laser dyes, and rare-earth light-amplifying complexes, can be locally introduced via doping [90]. The softness of a polymer permits a non-semiconductor approach in fabricating optical devices. A nanoimprinting method [91] has been suggested as a path to mass production of polymeric optical devices. The reported propagation losses of different polymeric waveguides range from 3 to 17 dB/cm [73, 91, 92]. The refractive index of available polymer materials is in the range of 1.3–1.6 [4, 92, 93], corresponding to a possible index contrast of ∼23%, and consequently large minimum bending radii which limits the integration density. To date, polymer ring resonators only with radii larger than 25 μm have been fabricated [4, 91, 93]. It should be emphasized that low material cost and flexibility for active and passive functions are the main features of polymer systems.

1.4 Principles of Fabrication The fabrication technologies used to realize high-Q and small modal volume photonic microresonators have evolved significantly over the last two decades. The earliest demonstrated optical resonator had the form of a liquid droplet and was followed by its solid-state counterparts, fused silica microspheres, a few years later (see the review of [94]). The effort then continued to on-chip microresonators based on planar wafer technologies in the last decade. Using recent advances of nanofabrication, a very high-Q nanocavity with effective modal volume approaching one photon, and therefore suitable for quantum optics experiments, has been experimentally realized [87]. In this section, various fabrication techniques from surface tension induced to wafer-based approaches will be briefly outlined, together with the recent examples of demonstrated microresonators.

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Principles of Operation and Notes on Fabrication

15

1.4.1 Surface Tension-Induced Microresonators (STIM) 1.4.1.1 Microspheres The microsphere resonators are formed by means of surface tension and have been demonstrated in liquid, amorphous, and crystalline forms. In liquid form, the microresonator is generally manifested in the form of a micron-sized droplet with a perfectly spherical surface. The use of liquid resonators is limited because of their evaporative nature and their difficulty to be manipulated, unlike their solid-state counterparts. However, numerous studies have shown that liquid resonators are useful in spectroscopy enhancement, fluorescence, and lasing in dyes. Stimulated Raman scattering was also studied in CS2 , CCl4 , water, glycerol, and other droplets [94]. The first solid-state microsphere was demonstrated in fused silica [43]. By melting the tip of a silica optical fiber by hydrogen flame or electric arc heating [81], the fiber tip evolves into a perfect spherical surface as a result of minimizing the surface energy. The microsphere immediately solidifies as the flame or electric arc is removed from the fiber tip. Both the electric arc heating and the accurate fiber positioning can be achieved using standard fiber splicing equipment [81], while the microsphere diameter is controlled by adjusting the initial size of the fiber tip. Reproducible microsphere size and shape, using electric arc heater, has been demonstrated. The diameter of the microsphere was in the order of 50–100 μm with demonstrated Q-factors in the order of ∼109 . It should be noted that silica microspheres are very sensitive to external contaminants which lead to additional loss, such as OH and water absorption. Thus, cavity experimentation requires an inert environment. The most common approach to excite WG modes in a silica microsphere is by a tapered silica fiber (Fig. 1.3). The taper width is controlled by slowly pulling the fiber which is locally heated by a hydrogen flame or an electric arc. Typically, taper thicknesses of less than 5 μm are often required for efficient coupling, suggesting the delicate nature of the process. The practical concern in STIM is both coupling to the WG modes efficiently and resting on-chip at the same time. Although taper–fiber coupling has been proved to be a powerful coupling tool, it is clearly not suitable for on-chip integration. The design of the optical coupler becomes complicated in microsphere configurations due to the fact that the mode profile of the WG mode is different from that of the optical waveguide. Numerical analysis [56, 81] has shown that WG mode forms a zigzag propagation along the equatorial line, forming a belt-like mode profile composed of azimuthal modes with multiple polar lobes. Efficient coupling requires matched modal propagation constants as well as mode profiles, and thus an optical waveguide that confines light in silica is needed. However, realizing such a wave guide is difficult because there is no available material of lower refractive index than silica for providing light confinement. An unconventional waveguide geometry was designed to solve this problem and termed the stripline pedestal anti-resonant reflecting optical waveguide (SPARROW) [81]. The key principle of SPARROW can be seen as a cavity, formed

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by a silica layer sandwiched between two thin silicon slabs (Fig. 1.6a). By careful adjustment of the thickness of the silica that is sandwiched between the two silicon layers, the light of a particular wavelength is brought into anti-resonance condition rejecting the light to enter the cavity. As a result the light is fully reflected and localized at the upper silica layer (Fig. 1.6b). The microsphere then is positioned to have significant mode overlap with the SPARROW (Fig. 1.6c). Alternatively, one can have two SPARROWs to realize a wavelength-drop application, with the straight waveguide serving to couple the light into the microsphere, and the oblique waveguide functioning exclusively as the drop port (Fig. 1.6e). Each of these wave guides is carefully aligned to the polar lobes of the WG mode (in this case there are two lobes), and the evanescent coupling occurs between each waveguide mode and one of the lobes. Experimentally, power transfer of 50% was demonstrated in the range of 20–50 MHz linewidth, using this technique [81]. (a)

Low index (Silica)

Substrate

Drop

Polar lobes

High index (Silicon) (b)

Through

Fiber stem

(c)

(d) Top view (e)

Input

Fig. 1.6 (a) The schematic of SPARROW geometry. (b) The numerically simulated mode profile of SPARROW. (c) SPARROW coupler configuration. (d) The fabricated microsphere. (e) The coupling scheme for wavelength-drop device. Figures (b) and (d) are courtesy of Laine [81]

1.4.1.2 Microtoroid It is apparent that although microsphere resonators exhibit ultrahigh-Q-factors, they are not clearly suitable for on-chip integration. The microtoroid [36], on the other hand, offers a more promising approach because it combines the ultrahigh-Q-factor derived from surface tension, with the wafer processing technique that gives more control over resonator dimensions. As outlined in Fig. 1.7, first a circular silica disk is lithographically defined by dry etching, followed by isotropic etching (using XeF2 gas) to remove the silicon underneath the microdisk in order to provide 3-D mode confinement in the resonator. At this stage, the remaining silicon functions as a post that supports the silica microdisk. Finally, the structure is irradiated with a CO2 laser

1

Principles of Operation and Notes on Fabrication

17 4. CO2 laser illumination

1. Deposition Silica

D

Silicon

d

2. Dry etch of Silica

Fabricated microtoroid 3. Wet etch of Silicon (XeF2)

Fig. 1.7 Process steps in fabrication of a microtoroid. The SEM micrograph of the microtoroid is courtesy of Armani et al. [36], reprinted by permission from Macmillan Publishers Ltd

(∼10.6 μm wavelength), which melts the silica and through surface tension forms a toroidal surface finish at the periphery. It should be noted that surface formation only occurs at the disk periphery. This is because the heat conductivity of silicon is much higher than that of silica, which rapidly transfers the generated heat to the silicon post. Using this technique, microtoroids with principal diameters (D) of 80–120 μm and torus thickness (d) of 5–10 μm have been reported, with exhibited Q-factor in the order of ∼108 . The optical mode of a toroid circulates at the periphery of the toroidally shaped silica cavity, rendering the modal volume to be many times smaller than that of microspheres. Numerical calculations have shown that the microtoroid can localize WG modes in about three times smaller effective area than the microsphere [95]. The WG mode is evanescently excited by a tapered fiber. The fiber taper provides ideal coupling scheme due to the fact that both the resonator and the fiber are made of silica material, and the taper width is readily adjusted to match the toroid thickness.

1.4.2 MR Resonators Based on Wafer-Based Processing Techniques The wafer-based fabrication of MR resonators is typically divided into three main stages, namely the wafer preparation, lithography, and etching. In the first stage, the template for the MR is designed by depositing vertical-layered materials with different refractive indices in order to support vertical light confinement. The methods of material deposition may vary depending on the material systems, the required precision, and the resonator structure. The available methods include molecular beam epitaxy, chemical vapor deposition, physical deposition techniques (sputtering, electron beam evaporation), and wafer bonding (e.g., SOI material system). For

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L.Y.M. Tobing and P. Dumon

example, molecular beam epitaxy (MBE) and metalo-organic chemical vapor deposition (MOCVD) are normally used for growing multilayered III–V semiconductor materials, while the plasma-enhanced chemical vapor deposition (PECVD) technique is more suitable for growing composite materials like silicon nitride (SiN), silicon oxynitride (SiON), as well as amorphous silicon (aSi). In the case of polymers, however, typically spin-coating technique is employed to sequentially spin different layers of polymer onto the substrate. In the second stage, the resonator structure is imaged onto a wafer through a multi-step lithography process. The photosensitive photoresist is first spread on the wafer by the spin-coating technique. Then a photomask with the resonator structure is imaged on the resist by radiation exposure. There are various types of lithographies ranging from optical to electron (ion) beam, and the resolution of a particular lithography system is limited by the imaging property of the corresponding operating wavelength. Some relevant data for various lithography types are optical (λ = 160–430 nm, feature size ∼130 nm), extreme ultraviolet (λ = 13 nm, feature size ∼45 nm), X-ray (λ = 0.4–4 nm, feature size ∼25 nm), or electron (ion) beam (λ = 3 pm, feature size ∼10–20 nm) [96]. Typically, polymer is used as a photoresist for its remarkable sensitivity to light, which, after exposure of the resist, can be either hardened (polymerized) or softened (un-polymerized), depending on whether it is a negative or a positive photoresist. The resist is then developed in a solution that preserves the imaged resonator pattern while removing the excess resist. In the third stage, the resonator pattern is transferred to the wafer by removing the material not protected by the resist using specially designed etchants. The etching can be all-directional with the same etch-rate (isotropic) or directional (anisotropic), depending on the desired resonator geometry as well as the resonator material system. Wet etching is a purely isotropic chemical process that uses acid solutions to selectively dissolve the exposed layer. An example is given in the fabrication of the microtoroid (see Fig. 1.7), where wet etching is used to remove the high-index silicon layer underneath the silica micordisk, thereby providing vertical light confinement for the WG resonator. Anisotropic etching, on the other hand, is a guided etching process that is used for producing vertical sidewalls in wafer-based microresonators. The most common form of anisotropic etching is the dry etching technique based on plasma generated from RF discharge. The removal of materials in dry etching can be either purely physical (i.e., by heavy bombardment of energetic nonreactive ions, e.g., argon), or a combination of physical and chemical, as, for example, in reactive ion etching (RIE). In some situations deep vertical etching (e.g., in III–V semiconductor-based microring resonator) or increased etching rate (e.g., in etching hard materials like GaN) are required. This can be overcome by increasing the plasma density and ion energy. In the system called inductively coupled plasma (ICP), two coupled RF units are used to acquire independent control of ion energy and plasma density. Furthermore, the plasma density can be two orders of magnitude higher compared to that in the conventional RIE [97], thereby making ICP-RIE a suitable tool for deep anisotropic etching.

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19

Some examples of fabricated MR resonators based on the above wafer-based processing techniques are briefly described below. 1.4.2.1 Deep Ultraviolet (DUV) Lithography There have been dedicated efforts in developing a DUV lithography system as a high-throughput and high-resolution fabrication tool for photonic components [13, 74, 98, 99]. The main reason is the much shorter wavelengths used in the system (λ = 248 nm, 193 nm) as compared to that in the conventional lithography system (λ = 365 nm), and the fact that it is also compatible with the CMOS processing used for fabricating electronic IC chips. The processing steps are outlined in Fig. 1.8. The 200 mm silicon-on-insulator (SOI) wafer is prepared with specified top silicon and buried oxide (BOX) thicknesses. The resonator structures are created in the top silicon layer, and the whole top silicon layer is etched to provide lateral light confinement. The BOX layer functions as the lower cladding and also to prevent light leakage to the silicon substrate. Typically a BOX thickness of 1 μm is sufficient to give negligible substrate leakage loss. The dry etching used in patterning the resonator structures introduces surface roughness, and usually short thermal oxidation is used to smoothen the sidewalls. At this stage, resonator structures with air as upper and lateral claddings have been fabricated. Alternatively, additional oxide deposition can be done to define uniform waveguide cladding. Structures with air cladding have stronger confinement, thus resonators can be fabricated in smaller dimension compared to those with oxide cladding. However, the resonator easily reacts with external contaminants which can significantly degrade the Q-factor. On the other hand, the structure with oxide cladding is completely sealed from the external world, which means there is no contaminant

SiO2 (BOX)

SiO2 (BOX)

SiO2 (BOX)

Si-substrate

Si-substrate

Si-substrate

1. Wafer preparation

2. Dry Etching (RIE)

3. Thermal Oxidation

SiO2 (BOX) Si-substrate 4. Oxide deposition

Fig. 1.8 The processing steps in DUV lithography. The two fabricated ring resonators are courtesy of Interuniversity Micro Electronic Center (IMEC) from Bogaerts et al. [74], reprinted with permission from IEEE

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issue in the second structure. Furthermore, the roughness scattering loss for the oxide cladding is lower than that for air cladding, due to the lower index contrast. Based on these reasons, oxide-cladding structures offer more advantages than the air-cladding ones. Based on the DUV feature size that is in the order of 100 nm, the optical waveguide can be evanescently coupled to the MR resonator with gap separation as close as 150–200 nm without significant distortion due to proximity effects. The inset of Fig. 1.8 shows examples of fabricated racetrack (distributed) and pointcoupled (lumped element) MR resonators. The process is optimized for 220 nm waveguide thickness and 450 nm waveguide width. Based on this process, microring resonators with radius down to 1.5 μm and loaded Q-factor up to 62,000 (corresponds to a finesse of ∼600) have been fabricated [100]. The waveguide loss measured in this work is as low as 2.4 dB/cm, with negligible radiation loss for bending radii larger than 5 μm. This suggests an intrinsic Q-factor and corresponding intrinsic finesse of ∼320,000 (R = 5 μm) and ∼3,600 (comparable to the Q/V) ratio), respectively. The problem of DUV lithography, however, is that the process is optimized only for a specific waveguide thickness which limits fabrication flexibility. 1.4.2.2 Electron Beam Lithography (EBL) Electron beam lithography provides the ultimate resolution because the electron, in principle, can be accelerated to arbitrarily high voltage which translates to arbitrarily short electron wavelength. The resolution of EBL, however, is limited to 10 nm, which is caused by the fact that secondary electrons are emitted around the exposed area. The fabrication steps of EBL are essentially similar to optical lithography, with the distinction that the resonator structures are serially written on the mask, and therefore unsuitable for mass production, because the writing time is very long. Thus, EBL is a suitable tool for research where high-throughput is not required, but rapid prototyping of smaller structures is necessary. A specific issue with EBL is the stitching of writing fields that is necessary for larger devices. This leads to unwanted loss and reflections, but may be partially solved by multiple patterning techniques. Some examples of fabricated SOI MR resonators based on EBL are shown in Fig. 1.9. The obtained waveguide propagation loss is 1.9–4 dB/cm [25, 26, 82, 101], corresponding to intrinsic Q-factors of 200,000–300,000 in telecommunication wavelengths with R = 5 μm [101]. The prominent feature of EBL is that a very narrow gap between waveguides can be defined without significant optical proximity effects, which can be detrimental in higher order filter design. The left panel of Fig. 1.9a shows a third-order optical filter based on three coupled rings (R = 2.5 μm) with different coupling strengths. As can be seen from the middle and right panels, the gap separation can be tuned from 100 nm to 350 nm without distortion, unlike in the case of DUV lithography where the resolution is limited to ∼150 nm. The choice of mask is also crucial for fabricating resonators with smooth sidewalls. In Fig. 1.9b, it is shown that the use of negative resist (HSQ) greatly decreases the sidewall roughness compared to the

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5μm (a)

3rd order filter

~100nm

~350nm

(b)

Fig. 1.9 (a) Example of fabricated ring resonators using EBL. (b) Different sidewall surface roughness for different masks. These SEM images are courtesy of Xiao [25, 26], reprinted with permission from IEEE and OSA

conventional positive tone resist (PMMA). Apart from the SOI material system, fabrication of MR resonators using EBL have also been demonstrated in silicon nitride (SiN) [23] and III–V semiconductor materials (GaAs, InP, etc) [7, 8, 9, 67, 102]. 1.4.2.3 Nanoimprinting Lithography (NIL) Nanoimprinting lithography is a high-throughput and high-resolution technique that combines the resolution of EBL with the mass processing capability of a conventional lithography system [103]. Typical process steps of NIL are schematically shown in Fig. 1.10. First, the master pattern is fabricated using a combination of EBL and dry etching. Then the UV-curable polymer is spun onto the master pattern in order to replicate the inverse pattern of the resonator structures. The polymer mold is solidified by a specific thermal treatment which depends on the material choice. Such a polymer mold can function directly as the replica of the master structures or as the second template to make the harder replica, for example, to make hard SiO2 mold in the hot-embossing technique [93]. Next, a polymer solution is dropped onto the substrate and is depressed with appropriate pressure by the replica made earlier. The polymer is heated to a temperature above glass transition temperature to soften the polymer and increase its plasticity. The depression force is carefully adjusted to ensure that the entire pattern inside the replica is completely transferred to the polymer layers. Afterwards, the polymer is again solidified and the replica is removed. It should be noted that it is important to deposit a non-adhesive layer to the replica before patterning, because the patterned structure is easily broken during polymer–replica separation [93]. Depending on the resonator material system, the patterned polymer can be used as the final material for the resonator structure [91], or as a mask to fabricate the final structure [104]. The other crucial factor is the polymer residual layer after the replica–polymer separation, which can harmfully affect the desired light guiding properties. Typically, the residual layer is mitigated

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(1)

(2)

(3)

(5c) (5a) (4) (6) (5b)

Fig. 1.10 The schematic of process flow for NIL. (1) The master pattern is fabricated and the replica is molded. (2) The mold is pressed on the polymer deposited on the substrate. (3) The depression force is adjusted until the patterns in the mold are completely transferred to the polymer. (4) Mold and polymer are separated. The final resonator structure can be fabricated in three ways. In (5a), the waveguide structure is deposited on lower index material. In (5b), the light confinement is enhanced by isotropic etching of the lower cladding. In (5c) the polymer functions as the mask required for the final structure shown in (6). The SEM images are examples of fabricated resonators based on (5b) Courtesy of Guo [93], reprinted with permission from American Institute of Physics

either by dry etching after replica separation, or by adjusting the thickness of the polymer prior to replica depression. MR resonators have been fabricated by different types of nanoimprint technologies, soft lithography replica molding [91], hot-embossing technique [93], and ultraviolet nanoimprint lithography (UV-NIL) [104]. Polymer MR resonators generally have intrinsic Q-factors on the order of 10,000 (for R = 100 μm), fabricated by soft lithography [105]. This is due to the fact that polymer is generally an amorphous material and therefore has higher material loss than crystalline materials such as silicon. Furthermore, compared to semiconductor materials, the index contrast is much lower and the resonator needs to be fabricated with a much larger bending radius. Although the surface roughness can be significantly reduced by thermal reflow of the polymer to yield a loss of ∼3.4 dB/cm [73], the per-turn loss is still

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quite high because of the large resonator dimensions. Resonators fabricated using UV-NIL have recently been demonstrated using SOI material systems [104], with measured propagation loss of ∼3.5 dB/cm and loaded Q-factor of Qloaded ∼ 47,000 (for R = 10 μm). The intrinsic Q-factor of these resonators is believed to be as high as those fabricated using E-beam and deep-UV lithography, since they have a comparable waveguide propagation loss.

1.5 Concluding Remarks The general aspects of MR and WG resonators as well as their fabrication principles were briefly introduced in this chapter. The MR and WG resonators are in many ways analogous to the most conventional Gires–Tournois and Fabry–Pérot resonators, where the evanescent couplers are functionally identical to the mirrors and the resonator circumference determines the cavity resonance frequency. The concept of evanescent coupling and the sources of cavity loss were treated. The design of photonic microresonators is in many aspects related to the fundamental trade-off between cavity dimensions, which determine the modal volume and the FSR, and the quality factor, which is related to the frequency selectivity (finesse) and overall device sensitivity. The advances of nanofabrication technology have enabled the realization of such cavities through various lithographic techniques. Fabricated resonators based on planar technology have loaded Q-factors in the order of ∼105 with bending radii down to 5 μm. On the other hand, resonators based on surface tension (microspheres and microtoroids) have Q-factors three orders of magnitude higher, but at the expense of more difficult planar integration. Acknowledgments The authors acknowledge the partial support from Academic Research Fund (ARC16/07-T2061204RS) and National Research Foundation (NRF-G-CRP 2007-01). LYMT wishes to dedicate this chapter to the late Prof. Chin Mee Koy who had been a good mentor, friend, and advisor

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Chapter 2

Circular Integrated Optical Microresonators: Analytical Methods and Computational Aspects Kirankumar Hiremath and Manfred Hammer

Abstract This chapter discusses an ab initio frequency domain model of circular microresonators, built on the physical notions that commonly enter the description of the resonator functioning in terms of interaction between fields in the circular cavity with the modes supported by the straight bus waveguides. Quantitative evaluation of this abstract model requires propagation constants associated with the cavity/bend segments, and scattering matrices, that represent the wave interaction in the coupler regions. These quantities are obtained by an analytical (2-D) or numerical (3-D) treatment of bent waveguides, along with spatial coupled mode theory (CMT) for the couplers. The required CMT formulation is described in detail. Also, quasi-analytical approximations for fast and accurate computation of the resonator spectra are discussed. The formalism discussed in this chapter provides valuable insight into the functioning of the resonators, and it is suitable for practical device design.

2.1 Introduction Resonances in optical microcavities are explored for a variety of applications [1–3]. Single or cascaded microresonators not only in the form of rings, disks, and spheres but also in other forms like squares, rectangles, or flower-like microgears (disks with angularly periodically varying radii), and arranged in various configurations, led to a multitude of interesting phenomena [4, 5]. In this chapter we focus on the most common microresonator configuration applied in integrated optics, consisting of a ring- or disk-shaped cavity which is evanescently coupled to two parallel bus waveguides. K. Hiremath (B) Department of Mathematics, Institute for Scientific Computing and Mathematical Modeling, University of Karlsruhe, Karlsruhe, Germany e-mail: [email protected]

I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_2,  C Springer Science+Business Media, LLC 2010

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Quite frequently the functioning of these resonators is discussed on the basis of a frequency domain model [6–8], where the interaction between the cavity and the straight waveguides is represented in terms of scattering matrices for the coupler regions. Interferometric resonances are established by segments of bent waveguides that connect these couplers. By treating the coupler scattering matrices and the bend mode propagation constants associated with the cavity as free parameters, one can estimate the response of the microresonators [9–11]. As a step beyond, one might try to solve the parametric model from first principles, i.e., calculate all parameter values for given geometry and material properties. Although this is obviously essential for realistic device design, rather few ab initio studies of that kind exist so far. Initial attempts can be found in [12–14], which differ with respect to the methods and approximations that are employed to obtain the modal basis solutions for the curved cavity segments and to predict the interaction between the cavity and the bus waveguides. For the latter task, approaches based on coupled mode theory (CMT) [15, Sections 1.4, 4.2.4], [16] are applied. The CMT arguments in the former studies are basically derived for the interaction of parallel straight waveguides and can be extended to low loss cavities (typically very large radius and/or high-index contrast). But the situation is characteristically different for cavity modes with non-negligible losses. It is possible to reformulate the CMT approach to overcome the above shortcomings [17, 18]. The simulations rely on frequency domain modal solutions for bent waveguides and curved interfaces on radially unbounded domains, which can be computed analytically for the 2-D setting [19]. In 3-D, numerical means have to be employed, like the film mode matching method [20]. For given real frequency, these modal solutions have complex angular propagation constants with a suitably fast decay in the radial direction [19], such that they can be conveniently used as basis fields in the frequency domain coupled mode description. The coupled mode equations can be derived from a variational principle [17] or by means of reciprocity techniques [21]. This leads to an ab initio frequency domain spatial CMT model of circular microresonators which is a straightforward implementation of the conventional traveling wave microresonator viewpoint [8, 9]. Once facilities for determining bend mode propagation constants and coupler scattering matrices are at hand, adaptation of the model to different configurations should be relatively simple, requiring merely modifications in the initial analytical reasoning. This concerns, e.g., cavity shapes with piecewise straight segments (“racetrack” resonators [22]) and resonators with only a single bus waveguide for resonant phase shifting [23] or with perpendicular bus waveguide cores [24]. For cases like the coupled optical resonator waveguides [25] where the intercavity coupling needs to be taken into account, a CMT formalism as in Section 2.4 with bend modes of the two cavities as basis fields would be required. Extension to larger composites with parallel coupled cavities [26, 27], serial configurations [26, 28], or even mesh-like filters [24, 29, 30] should be straightforward by means of scattering matrix operations, given the input–output characteristics of the single cavity resonator elements.

2

Analytical and Computational Aspects

31

The present approach differs from models based on time domain modes for the entire circular cavities [26, 31]. These are solutions with integer angular mode number and complex eigenfrequency [12, 32]. Due to their radially growing fields, they are not directly suitable for the frequency domain CMT framework. Therefore we use the frequency domain model of the bent waveguides as discussed in Section 2.3. As these modal solutions have complex valued angular mode numbers, they do not constitute valid solutions for the full rotationally symmetric cavities and hence are not useful to access directly the (complex) resonance frequencies of the isolated cavities. Still, by taking into account their interaction with the straight waveguides, one can construct approximate solutions for the system “cavity + bus waveguides” at given wavelengths and thereafter estimate resonance frequencies. It is possible to translate between both viewpoints [12, 32]. Alternatively, rigorous numerical tools can be employed to model the resonators. The most prominent among these are the finite difference time domain (FDTD) method [33–35] and its (discontinuous Galerkin) finite element variants [36–38]. However, already in the 2-D setting these simulations turn out to be timeconsuming; the computational effort required in 3-D is expected, at least at present, to be prohibitive for practical design work. Integral equation methods are also applied for efficient analytical solutions of eigenvalue and scattering problems for specific 2-D configurations of microring and disk cavities [39–41]. Unfortunately, the extension to 3-D appears to be far from obvious. In the subsequent sections, we discuss the coupled mode theory approach in detail. Section 2.2 describes the “standard” resonator model formulated directly for multimodal cavities. Evaluation of the abstract equations requires propagation constants of the relevant cavity modes and the coupler scattering matrices. These quantities are relatively easy to obtain in a 2-D setting, where also rigorous numerical data suitable for reliable benchmarking is conveniently available. The 2-D configurations might also be of interest as effective index projections of actual 3-D structures [13]. Therefore we first discuss in detail 2-D configurations of bent waveguides in Section 2.3 and couplers in Section 2.4. Numerical approaches for the efficient evaluation of the resonator spectrum are presented in Section 2.5. In Section 2.6 these ingredients are combined to simulate full 2-D microresonators; the extension to three spatial dimensions follows in Section 2.7. In both cases the CMT results are compared with data from other independent numerical methods. Conclusions of the present analysis are given in Section 2.8.

2.2 Analytical Framework Consider a microresonator consisting of a ring- or disk-shaped dielectric cavity, evanescently coupled to two parallel straight waveguides. In the laterally coupled configuration (Fig. 2.1, left; a top view of a real 3-D device, also the 2-D setting of Sections 2.3, 2.4, 2.5 and 2.6), these waveguides are placed in the x–z-plane just as the cavity is, whereas in the vertically coupled configuration (Fig. 2.1, right),

32

K. Hiremath and M. Hammer x

In

PI

a wc

0

Drop

L

b˜ ws PD

˜ B

PT

B g

PD

(b)

b

PI

y

Through

(I) A

˜ L

R

x z

nc nb

g˜

a˜

˜ A ( II )

ns PA

Add

(a)

z

PT

Fig. 2.1 The “standard” microresonator model : A laterally (a) or a vertically (b) coupled resonator is functionally decomposed into two bent–straight waveguide couplers (I) and (II), which are interconnected by cavity segments of lengths L and L˜ outside the couplers. Schematics for the laterally coupled resonator (a): A cavity of radius R, core refractive index nc and width wc is placed between two straight waveguides with core refractive index ns and width ws , with gaps of width g and g˜ between the cavity and the waveguides. The cladding refractive index is nb (Reprinted with Permission Copyright 2005 Elsevier)

they are positioned at different y levels. Each of these configurations has its own advantages and disadvantages [10]. We chose a frequency domain description, where a time harmonic optical signal ∼ exp (iωt) of given real frequency ω corresponding to vacuum wavelength λ is present everywhere. In line with the most common view on circular microcavities [8, 9], the resonators are functionally divided into two bent–straight waveguide couplers, which are connected to each other by segments of the cavity. Semi-infinite pieces of straight waveguides constitute the external connections, where the letters ˜ B˜ (external) and a, b, ã, b˜ (internal) denote the coupler ports. A, B, A, Assume that the interaction between the optical waves in the cavity and in the bus waveguides is negligible outside the coupler regions. Also assume that all transitions inside the couplers are sufficiently adiabatic, such that back-reflections do not play a significant role for the resonator functioning. We further restrict the model to unidirectional wave propagation, as indicated by the arrows in Fig. 2.1. Depending on the specific configuration, these assumptions can be justified or not; for the examples in Sections 2.6, 2.7 they appear to be adequate. Suppose that the straight waveguides support Ns guided modes with propagation constants βsq , q = 1, . . . , Ns . For the cavity, Nb bend modes are taken into account. Due to the curvature, their propagation constants γbp = βbp − iαbp , p = 1, . . . , Nb , are complex valued [19]. Here βbp and αbp are positive real valued quantities representing phase constants and attenuation constants of the cavity modes. All these modes are power normalized. Let the variables Aq , Bq , and ap , bp denote the directional amplitudes of the properly normalized “forward” propagating (clockwise direction, cf. Fig. 2.1) basis modes in the respective coupler port planes, combined into amplitude (column) vectors A, B, and a, b.

2

Analytical and Computational Aspects

33

Then the response of coupler (I) can be represented in terms of its scattering matrix, which relates the amplitudes of the outgoing waves to the amplitudes of the corresponding incoming modes as ⎞ ⎛ 0 a− ⎜ A− ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎝ b ⎠ = ⎝ Sbb B Ssb ⎛

0 0 Sbs Sss

S− bb S− sb 0 0

⎞⎛ ⎞ a S− bs − ⎟⎜ Sss ⎟ ⎜ A ⎟ ⎟. 0 ⎠ ⎝ b− ⎠ B− 0

(2.1)

Here the superscripts − indicate the amplitudes of backward (anticlockwise) propagating waves and the zeroes implement the assumption of negligible backreflections. The entries of the submatrices Svw with v, w = b, s represent the “coupling” from a particular mode of the waveguide w to a particular mode of the waveguide v. A fundamental property of any linear circuit made of nonmagnetic materials is that the transmission between any two “ports” does not depend upon the propagation direction, i.e., the full scattering matrix of the reciprocal circuit is symmetric [15, Section 1.3.2]. This argument even holds for circuits with attenuating materials, in the presence of radiative losses, and irrespective of the particular shape of the connecting cores. It relies crucially on the precise definition of the “ports” of the circuit, where independent ports can be realized either by mode orthogonality or by spatially well separated outlets. Assuming that the above reciprocity requirements are satisfied for the bent– straight waveguide couplers, one expects that the bidirectional coupler scattering matrix is symmetric (as we shall see in Sections 2.4.4, 2.4.5, Figs. 2.6, 2.7, 2.8 and 2.9, the numerical results give evidence that this is indeed the case). For the submatrices this implies that the following equalities hold (T denotes the transpose): − T − T − T T Sbb = (S− bb ) , Ssb = (Sbs ) , Sbs = (Ssb ) , Sss = (Sss ) .

(2.2)

If coupler (I) is defined symmetrical with respect to the central plane z = 0, and if identical mode profiles are used for the incoming and outgoing fields, then one can further expect the transmissions from A to b to be equal to the transmission from B− to a− [15, Section 1.3.2.] Similarly, the transmissions from a to B and from b− to A− are equal: Sbs = S− bs ,

Ssb = S− sb .

(2.3)

As a result of (2.2) and (2.3), also the unidirectional scattering matrix S=

Sbb Sbs Ssb Sss



associated with the clockwise propagation through coupler (I) is symmetric:

(2.4)

34

K. Hiremath and M. Hammer − T S− bs = (Ssb ) .

Sbs = (Ssb )T ,

(2.5)

The physical interpretation of the above statements is that “the coupling from the straight waveguide to the cavity is equal to the coupling from the cavity bend to the bus waveguide”. A completely analogous reasoning applies to the second coupler, where the tilde ˜ B, ˜ and a˜ , b˜ at the port planes, and the unidi(∼) identifies the mode amplitudes A, ˜ rectional scattering matrix S related to coupler (II), such that the coupler operation is represented as



b˜ ˜ a˜ . = S ˜ ˜ A B





b a =S , B A

(2.6)

Outside the coupler regions the bend modes are used for the description of the field propagating in the cavity, with the angular/arc-length dependence given by their propagation constants (cf. (2.20)). Hence the amplitudes at the entry and exit ports of the connecting cavity segments are related to each other as a = G b˜

and

˜ b, a˜ = G

(2.7)

˜ are Nb × Nb diagonal matrices with entries Gp, p = exp ( − iγbp L) where G and G ˜ p, p = exp ( − iγbp L), ˜ respectively, for p = 1, . . . , Nb . and G We are interested in the case where modal powers PIq = |Aq |2 and PAq = |A˜ q |2 are given at the input port A and at the add port A˜ of the resonator, and we need to calculate the transmitted power PTq = |Bq |2 at port B and the backward dropped ˜ This is equivalent to solving the linear system estabpower PDq = |B˜ q |2 at port B. ˜ Due to the linearity of the lished by (2.6) and (2.7) for B and B˜ in terms of A and A. device the restriction to an excitation in only one port, here port A, with no incoming ˜ = 0, is sufficient. Then one obtains add-signal A ˜ sb G ˜ −1 Sbs + Sss )A, B˜ = (S ˜ −1 Sbs )A B = (Ssb GS˜ bb G

(2.8)

for amplitudes of the outgoing guided modes in the through- and drop-ports, and b = −1 Sbs A,

˜ −1 Sbs A b˜ = S˜ bb G

(2.9)

˜ for the internal mode amplitudes in the cavity, where  = I − Sbb GS˜ bb G. The spectral response of the resonator is found by computing the transmitted (PTq ) and dropped (PDq ) power for a series of wavelengths with the help of (2.8). Resonances manifest as peaks in PDq along with simultaneous dips in PTq . The pronounced wavelength dependence at the resonance is caused by the “resonance denominator” . For further insight, consider (2.8) and (2.9) when  is singular, ˜ has a unit eigeni.e., it has an eigenvalue zero, or, equivalently, when Sbb GS˜ bb G value. This corresponds to a cavity field which reproduces itself after propagating

2

Analytical and Computational Aspects

35

consecutively along the right cavity segment, through coupler (II), along the left cavity segment, and finally through coupler (I). The self-consistent reproduction of the cavity field represents the resonance condition in the present abstract multimode cavity model. Practical evaluation of — the so far only parameterized equations — (2.8) and (2.9) requires values for the phase and attenuation constants γbp = βbp − iαbp of the ˜ and the scattering matrices S, S˜ of couplers (I) and cavity modes, contained in G, G (II). Corresponding procedures will be the subject of the following sections, where the concepts are first discussed in 2-D, then extended to the full 3-D setting.

2.2.1 Monomodal Resonators For fully symmetric microresonators (i.e., identical couplers (I), (II), connected by equal cavity segments ˜· = ·) with single mode cavity and bus cores, further evaluation of expressions (2.8) and (2.9) is fairly standard [8, 9]. Let Sbb = |Sbb |eiϕ , Sss − Sbs Ssb /Sbb = ρeiψ with ϕ, ρ, and ψ real. Then the dropped power is given by PD = P I

|Ssb |2 |Sbs |2 e−αLext , 1 + |Sbb |4 e−2αLext − 2|Sbb |2 e−αLext cos (βLext − 2ϕ)

(2.10)

and the expression for the through power reads PT = PI

|Sss |2 (1 + |Sbb |2 ρ 2 e−2αLext − 2|Sbb |ρe−αLext cos (βLext − ϕ − ψ)) . 1 + |Sbb |4 e−2αLext − 2|Sbb |2 e−αLext cos (βLext − 2ϕ) (2.11)

Here Lext = L + L˜ is the total length of those parts of the cavity which are not already included in the couplers. β and α are the phase and attenuation constants of the single relevant bend mode of the cavity. In principle, all quantities in (2.10) and (2.11) are wavelength dependent. Hence the rigorous way to determine the resonator spectrum would be to evaluate all relevant quantities for a series of wavelengths. Some further insight, however, can be obtained with the approximation that significant changes in the drop and through power, on a limited wavelength interval, originate exclusively from the cosine terms in (2.10) and (2.11) that include the phase information (this term corresponds to the resonance denominator  in (2.8) and (2.9)). To take into account some non-negligible length l of the cavity segments in the coupler regions, write the phase term as βLext −2ϕ = βLcav −φ, where Lcav = 2π R is the complete cavity length, and φ = 2βl + 2ϕ (a corresponding procedure is also applied to the phase term in the numerator of (2.11)). Further consider only the wavelength dependence of the phase constant β as it appears explicitly in the term βLcav − φ. In this way, one incorporates the wavelength dependence of the phase change βLcav for the entire cavity, but disregards the wavelength dependence of the phase change φ that is introduced by the interaction with the port waveguides.

36

K. Hiremath and M. Hammer

Resonances (i.e., maxima of the dropped power) are now characterized by singularities in the denominators of (2.10) and (2.11), which occur when cos (βLcav − φ) = 1, i.e., when the total phase gain after one cavity round trip is a multiple of 2π . This leads to the resonance condition β=

2mπ + φ  : βm , Lcav

for integer m.

(2.12)

For a resonant configuration, the dropped power is given by PD |β=βm = PI

|Ssb |2 |Sbs |2 e−αLext . (1 − |Sbb |2 e−αLext )2

(2.13)

Note that properly computed values of Ssb , Sbs , and Sbb already include the losses along the parts of the cavity inside the couplers. Therefore Lext in (2.13) (and in those places of (2.10) and (2.11) where attenuation is concerned) must not be replaced by Lcav . For the present case of monomodal resonators, one can now derive the following familiar expressions that characterize the resonances [8]. Here we only summarize the results. • The free spectral range (FSR) λ is defined as the wavelength difference between two successive maxima of the dropped power (or minima of the through power) and is given by  λ2 

λ ≈ , neff Lcav m

(2.14)

where λ is the resonance wavelength of the m-th order resonance and neff = λβm /2π is the effective mode index of the cavity mode. Here we assumed that the effective group index is equal to the effective mode index. • The full width at half maximum (FWHM) 2δλ measures the width of the resonance peak at a level of half the resonance value, i.e., the sharpness of the peak. For the resonance of order m one obtains 

 1 αL/2 λ2 −αL/2  . (2.15) − |Sbb |e 2δλ = e π Lcav neff m |Sbb | • The ratio of the FSR and the FWHM at a specific resonance is called the finesse F: F=

λ |Sbb |e−αL/2 . =π 2δλ 1 − |Sbb |2 e−αL

(2.16)

• The quality factor Q can be viewed as the ability of the cavity to confine the field in space or time, which determines the sharpness of the resonance peaks. It is

2

Analytical and Computational Aspects

37

defined here1 as the ratio of the resonance wavelength to the FWHM Q=

neff Lcav neff Lcav |Sbb |e−αL/2 λ = =π F. 2 −αL 2δλ λ 1 − |Sbb | e λ

(2.17)

Hence, for a circular resonator with radius R and cavity length Lcav = 2π R, one obtains Q = kRneff F for the relationship between Q and finesse F. Again, one can improve the accuracy of the above expressions by using the effective group index of the cavity mode in place of the effective mode index neff [8].

2.3 Waveguiding Along Bent Cores An analytical description of bent slab waveguides is apparently well known [15, Section 5.2.4], [43, 44], but seemingly hardly ever evaluated rigorously. Bessel and Hankel functions of large complex order and large argument are involved. Probably this constitutes a major obstacle, such that most authors resorted to approximations. However, by using uniform asymptotic expansions of the Bessel/Hankel functions, we found that a quite rigorous analytic treatment of the problem is possible [19]. Consider a bent slab waveguide with the y-axis as the axis of symmetry as shown in Fig. 2.2. For the present 2-D treatment, all material properties and the electromagnetic fields are assumed to be invariant in the y direction. Being specified by the radially dependent piecewise constant refractive index n(r), the waveguide can be seen as a structure that is homogeneous along the angular coordinate θ . Hence we choose an ansatz for the bend modes with pure exponential dependence on the azimuthal angle, where the (dimensionless) angular mode number is commonly written as a product γ R with a reasonably defined bend radius R, such that γ can

z r ns

nf

θ 0

d

nc x

R

Fig. 2.2 2-D bent slab waveguide. The core of thickness d and refractive index nf is embedded between an interior medium (“substrate”) with refractive index ns and an exterior medium (“cladding”) with refractive index nc . The distance between the origin and the outer rim of the bend defines the bend radius R (Reprinted with Permission Copyright 2005 Springer)

1 In the time domain, the Q factor Q = ω/(2δω) is defined as the ratio of the optical power stored in the cavity to the cycle averaged power radiated out of the cavity [42]. The larger the Q factor, the longer the optical field is trapped inside the cavity.

38

K. Hiremath and M. Hammer

be interpreted as a propagation constant. Note that the definition of the bend radius (here the outer rim, as in Fig. 2.2), and consequently the definition of the cavity size, is to some degree arbitrary. The consequences of different choices of R are explained in [19]. In the cylindrical coordinate system (r, y, θ ), solutions for the optical electric field E and magnetic field H are sought in the functional form (in the usual complex notation) of bend modes



E (E˜ r , E˜ y , E˜ θ ) i(ωt−γ R θ) . (r, θ , t) = ˜ y, H ˜ θ ) (r) e ˜ r, H H (H

(2.18)

The tilde indicates the mode profile, and γ is the propagation constant of the bend mode. Since the electromagnetic field propagating through the bent waveguide is lossy, γ = β − iα is complex valued, where β and α are the real valued phase and attenuation constants. Inserting the ansatz (2.18) into the Maxwell curl equations, one obtains two decoupled sets of equations: one set for transverse electric (TE) waves with nonzero ˜ r , and H ˜ θ and a second set for transverse magnetic (TM) waves components E˜ y , H ˜ y , E˜ r and E˜ θ . Within radial intervals with constant with nonzero components H ˜ y (TM) satisfy a refractive index n, the principal components φ = E˜ y (TE) or φ = H Bessel equation

1 ∂φ γ 2 R2 ∂ 2φ 2 2 + k − )φ = 0 + (n ∂r2 r ∂r r2

(2.19)

of complex order γ R, where k = 2π/λ is the given real valued vacuum wave number. By solving (2.19) piecewise in radial intervals with constant refractive index, together with polarization dependent material interface conditions and suitable boundary conditions at r → 0 (bounded solutions) and at r → ∞ (outgoing fields), one obtains a dispersion equation (transverse resonance condition) for the bent waveguide. For a given frequency ω, discrete values of γ are to be identified as roots of this equation in the complex plane, where we employed numerical means in the form of a secant method. Bessel/Hankel functions with complex order are evaluated on the basis of uniform asymptotic expansions [45]. The outcome is a set of complex bend mode propagation constants with accompanying complex mode profiles. The modes are labeled by the number of radial minima in the absolute value of the principal component. Despite their oscillatory behavior in the outer region, the bend modes can be rigorously power normalized and they satisfy certain orthogonality conditions. We refer to [19] for further details on the solver and on bend mode properties.

2

Analytical and Computational Aspects

39

2.3.1 Bend Modes

|Ey|

Figure 2.3 illustrates results of the former procedures. For large bend radius, the bend mode resembles closely the familiar, well confined mode of a symmetric straight slab waveguide. Closer inspection reveals that the tail of the bent mode is protruding more outward into the cladding region than inward into the substrate region. For smaller bend radii, the confinement decreases and the relative level of the field in the cladding grows, along with a shift of the absolute field maximum toward the outer rim of the core. With increasing bend radius one observes an almost exponential decrease of the attenuation constant, whereas the phase constant of the bent mode approaches the phase constant of the corresponding straight waveguide mode [19]. 30

30

30

20

20

20

10

10

10

0

0

x [μm]

−3

−2

−1

0 (r−R) [μm]

1

2

3

−3

0 −2

−1

0 (r−R) [μm]

1

2

3

−3

11

6

4

10

5

3

9

4

2

8 −4

3 −4

−2

0 z [μm]

2

4

−2

0 z [μm]

2

4

1 −4

−2

−1

−2

0 (r−R) [μm]

0 z [μm]

1

2

2

3

4

Fig. 2.3 TE0 mode profiles for bent waveguides with (ns , nf , nc ) = (1.0,1.5,1.0), d = 0.5 μm, λ = 1.05 μm for bend radii R = 10, 5, 3 μm. First row: radial dependence of the absolute value of the basic electric field component E˜y . The profiles are power normalized, with the global phase adjusted such that E˜ y (R) is real and positive. Second row: snapshots of the propagating bend modes according to (2.18). The gray scales correspond to the levels of the real, physical field Ey . neff = γ /k are 1.3232−i 6.4517·10−11 , 1.2930−i 7.5205·10−6 and 1.2576−i 6.8765·10−4 , respectively

The effect of “bending” and the lossy nature of the bend modes are illustrated best by the snapshots of the physical fields in the second row of Fig. 2.3. For the bent waveguide with R = 3 μm, the radiative tails of the field in the cladding are clearly visible. Just as for straight configurations, the confinement of bend modes depends upon the refractive index contrast; for sufficiently wide waveguides, higher order modes can be supported [19].

2.3.2 Whispering Gallery Modes If the core width of a bent waveguide with sufficient curvature is increased, a regime can be reached where the modes are guided by just the outer dielectric interface, while the precise location of the interior interface becomes irrelevant. The model of Section 2.3 also covers such configurations with the formal choice ns = nf in

40

K. Hiremath and M. Hammer

Fig. 2.2. The solutions are called whispering gallery modes (WGMs). Figure 2.4 illustrates the first four lowest order WGMs that are supported by a structure with the parameters of the previous bent waveguide segments, where the interior has been filled with the core material. While the fundamental field TE0 is well confined to the waveguide, the higher order modes spread far beyond the core. Although these modes attenuate fast, we shall see in Section 2.6.2 that they can influence the response of a resonator with a cavity made up of the present bend segments. 30

TE

TE1

0

TE2

TE3

y

~

|E |

20

10

0 −5

−3

−1 0 1 (r−R) [μm]

3

5 5

x [μm]

6 TE0

−3

−1 0 1 (r−R) [μm]

3

5 −5

TE

−3

−1 0 1 (r−R) [μm]

3

5 −5

1

−3

TE

TE

2

−1 0 1 (r−R) [μm]

3

5

3

4 2 0

−5

0 z [μm]

5

−5

0 z [μm]

−5

5

0 z [μm]

5

−5

0 z [μm]

5

Fig. 2.4 TE polarized whispering gallery modes; the plots show the absolute value |E˜ y | of the radial mode profile (top) and snapshots of the propagating physical field Ey (bottom). The effective mode indices γj /k for bend radius R = 5 μm are 1.32793 − i 9.531 · 10−7 (TE0 ), 1.16931 − i 4.032 · 10−4 (TE1 ), 1.04222 − i 5.741 · 10−3 (TE2 ), and 0.92474 − i 1.313 · 10−2 (TE3 ), for λ = 1.05 μm. All modes are power normalized

2.4 Bent–Straight Waveguide Couplers Capitalizing on the availability of analytical bend modes, we now proceed to the coupler regions. One of the many variants of coupled mode theory (CMT) [15 Sections 1.4, 4.2.4], [16] will be applied here to model the interaction of the optical waves. The formulation takes into account that multiple modes in each of the cores may turn out to be relevant for the functioning of the resonators [17]. Consider the coupler configuration shown in Fig. 2.5a. The basis fields for the CMT description are the time harmonic modal solutions associated with the isolated bent (b) and straight cores (c). In line with the assumptions of Section 2.2 only forward propagating modes are considered, where for convenience we choose the z-axis of the Cartesian system as introduced in Fig. 2.5 as the common propagation coordinate for all fields. Let Ebp , Hbp , and εb represent the modal electric fields, magnetic fields, and the spatial distribution of the relative permittivity of the bent waveguide. For the CMT formalism, the bend mode field ansatz (2.18) in the polar coordinates is expressed in the Cartesian x–z-system, such that the basis fields for the cavity read

Ebp Hbp



(x, z) =

E˜ bp ˜ bp H



(r(x, z)) e−iγbp Rθ(x, z) .

(2.20)

2

Analytical and Computational Aspects

41

(a))

((b)

z

b nb 0

R (xl z i)

θo θi

nc

nb

B ns

wc

g

ws

(c)

z

(xr zo) nb

z r θ

a

x

0

x

x

0

A (E H )

(E bp H bp

b)

(E sq H sq

s)

Fig. 2.5 The bent–straight waveguide coupler configuration (a), coupler (I) of Fig. 2.1. The interaction between the modal fields supported by the bent and straight cores is restricted to the computational window [xl , xr ] × [zi , zo ]. Inside this region the optical field is represented as a linear combination of the modal fields of the bent waveguide (b) and of the straight waveguide (c) (Reprinted with Permission Copyright 2005 Elsevier)

˜ bp are the radial dependent electric and magnetic parts of the mode Here E˜ bp and H profiles; γbp are complex valued propagation constants. Consistent with the definition in Section 2.3, the bend radius R indicates the position of the outer curved interface (Fig. 2.5). Similarly, Esq , Hsq , and εs denote the modal fields and the relative permittivity associated with the straight waveguide, which are given by

Esq Hsq



(x, z) =

˜ sq E ˜ sq H



(x) e−iβsq z .

(2.21)

Here positive propagation constants βsq characterize the phase changes with the propagation along the z-direction. Now the total optical field E, H inside the coupler region is approximated by a linear combination of the modal basis fields (2.20) and (2.21) as

E H

(x, z) =

Nv v=b, s i=1

Cvi (z)

Evi Hvi

(x, z),

(2.22)

with a priori unknown amplitudes Cvi that are allowed to vary with the propagation coordinate z. For ease of notation, these amplitudes are combined into amplitude vectors C = (Cb , Cs ) = ((Cbi ), (Csi )). The governing equation for C is derived using a variational principle. Unlike, e.g., in [12], here no “phase matching” arguments appear.2 Via the transformation (r, θ ) → (x, z), the tilt of the wave front of the bend modes (2.20) is explicitly taken into account.

2 The arbitrariness in the choice of R renders the actual value of γ virtually meaningless [19]. Only the product γ R is relevant.

42

K. Hiremath and M. Hammer

2.4.1 Coupled Mode Equations Consider the functional

  (∇ × E) · H∗ − (∇ × H) · E∗ + iωμH · H∗ + iωε0 εE · E∗ dx dz, F(E, H) = (2.23) a 2-D restriction of the 3-D functional given in [15 Section 1.5.6, Equation (1.98)]. For the present 2-D setting, the above curl operators are interpreted with the convention of vanishing derivatives ∂y = 0. The Maxwell equations ∇ × E = −iωμH and ∇ × H = iωε0 εE form a necessary condition for stationarity of F with respect to variations of (E, H). By inserting the trial field (2.22) into the functional (2.23), we restrict F to the fields allowed by the coupled mode ansatz. For the “optimal” solution of the curl equations in the form of the field (2.22), the variation of F(C) is required to vanish for arbitrary variations δC. Disregarding boundary terms, the first variations of F at C in the directions δCwj , for j = 1, . . . , Nw and w = b, s, are

δF =



Nv  dCvi ∗ Mvi,wj dz − c.c, − Fvi,wj Cvi δCwj dz

(2.24)

v=b,s i=1

where c.c. indicates the complex conjugate of the preceding integrated term,

Mvi,wj =

  az · Evi × H∗wj + E∗wj × Hvi dx,

Fvi,wj = −iωε0

(ε − εv )Evi · E∗wj dx,

(2.25) (2.26)

and where az is the unit vector in the z-direction. Consequently, one arrives at the coupled mode equations Nv v=b,s i=1

Mvi,wj

Nv dCvi − Fvi,wj Cvi = 0, for j = 1, . . . , Nw , and w = b, s, dz v=b,s i=1

(2.27) as a necessary condition for F to become stationary for arbitrary variations δCwj . The same expression is obtained from the complex conjugate part of (2.24). One can alternatively derive these coupled mode equations by means of a reciprocity identity [15 Section. 4.3]. In matrix notation, (2.27) reads M(z)

dC (z) = F(z) C(z). dz

(2.28)

2

Analytical and Computational Aspects

43

Due to the functional form of the bend modes and the varying distance between the bent and straight cores, the coefficients M, F are z-dependent. For explicit representations of these equations in the single mode case Nb = Ns = 1, see [8].

2.4.2 Coupler Scattering Matrices In practice we solve the coupled mode equations numerically. The solution can be represented in terms of a transfer matrix T (cf. Section 2.4.3) that relates the CMT amplitudes at the output plane z = zo to the amplitudes at the input plane z = zi of the coupler as C(zo ) = T C(zi ).

(2.29)

We still need to relate the transfer matrix, obtained directly as the solution of the coupled mode equations on the limited computational window, to the coupler scattering matrix as required for the abstract model of Section 2.2. Outside the coupler region [xl , xr ] × [zi , zo ], it is assumed that the interaction between the fields associated with the bent waveguide and the straight waveguide is negligible. In this region, the individual modes propagating undisturbed with the harmonic dependences on the respective propagation coordinates are given by

E˜ bp −iγbp R(θ−θi ) for θ ≤ θ , i ˜ bp e H

E˜ bp ˜ bp e−iγbp R(θ−θo ) for θ ≥ θo , Hbp

ap



E˜ Aq ˜ sq e−iβsq (z−zi ) for z ≤ zi , H sq

E˜ sq e−iβsq (z−zo ) for z ≥ zo . Bq ˜ Hsq (2.30)

Here ap , Aq and bp , Bq are the constant external mode amplitudes at the input and output ports of the coupler, as introduced in Section 2.2. The coordinate offsets zi , θi and zo , θo are as defined in Fig. 2.5. Because of the assumption of vanishing external interaction, it is expected that outside the coupler the modes of straight waveguide propagate with constant amplitudes (with suitable phase changes). While the modes of the straight waveguide are typically well confined, the leaky bend modes may extend far beyond the bend waveguide, even reaching the region of the straight waveguide core. This affects intrinsically the way the CMT adjusts the amplitudes Cs (zo ) in (2.29). The physical field around the exit planes of the CMT window consists of a superposition of the outgoing guided modes of the straight waveguide and a non-guided part due to the leaky bent modes. To extract the required external amplitudes Aq , Bq , we therefore project the total coupled field on the modes of the straight waveguide. Exploiting their orthogonality properties, the projection at the coupler output plane z = zo yields

44

K. Hiremath and M. Hammer

Bq exp (iβsq z) = Csq +

Nb p=1

Cbp

Mbp, sq , Msq, sq

(2.31)

where the mode overlaps Mmi,nj occur already in the coupled mode equations (2.28). A similar procedure is applied to relate the coefficients Aq to the amplitudes Csq at z = zi . There is no need of such a projection for the external bend mode amplitudes, since the field strengths of the straight waveguide modes is usually negligible in the respective angular planes, where the major part of the bend mode profiles is located. Here merely factors are introduced to adjust for the offsets of the angular coordinates in (2.30). Thus, given the solution (2.29) of the coupled mode equations in the form of the transfer matrix T, the scattering matrix S that relates the amplitudes ap , bp , Aq , Bq of the external fields as required in (2.6) is S = Q T P−1 ,

(2.32)

where P and Q are (Nb + Ns ) × (Nb + Ns ) matrices with diagonal entries Pp, p = exp ( − iγbp Rθi ) and Qp, p = exp ( − iγbp Rθo ), for p = 1, . . . , Nb , followed by the entries Pq+Nb , q+Nb = exp (− iβsq zi ) and Qq+Nb ,q+Nb = exp ( − iβsq zo ), for q = 1, . . . , Ns . A lower left block is filled with elements Pq+Nb , p = Mbp, sq /Msq, sq z=z i  and Qq+Nb ,p = Mbp, sq /Msq, sq z=z , for q = 1, . . . , Ns and p = 1, . . . , Nb , o respectively, that incorporate the projections. All other entries of P and Q are zero. Indeed, as observed in Sections 2.4.4 and 2.4.5, the projected amplitudes |Bq |2 (or the related scattering matrix elements |Ssq, wj |2 ) become stationary, when viewed as a function of the exit port position zo , while at the same time the associated CMT solution |Csq (z)|2 (or the elements |Tsq, wj |2 of the transfer matrix) exhibits an oscillatory behavior. Still, in the sense of the projections one can speak of “noninteracting, decoupled” fields. This justifies the restriction of the computational window to z-intervals where the elements of S (not necessarily of T) attain constant absolute values around the input and output planes.

2.4.3 Remarks on the Numerical Procedures Equation (2.28) is solved numerically on a rectangular computational window [xl , xr ] × [zi , zo ] as shown in Fig. 2.5. For given z, the integrals (2.25) and (2.26) are numerically computed by the trapezoidal rule [46] using a uniform discretization of [xl , xr ] into intervals of length hx . Subsequently, a standard fourth order Runge–Kutta scheme [46] is applied to solve the coupled mode equations over the computational domain [zi , zo ], which is split into intervals of equal length hz . Exploiting the linearity of (2.28), the procedure is formulated directly for the transfer matrix T, which gives

2

Analytical and Computational Aspects

45

dT (z) = M(z)−1 F(z) T(z) dz

(2.33)

with initial condition T(zi ) = I, where I is the identity matrix, such that C(z) = T(z) C(zi ). While the evaluation of the resonator properties via (2.32) and (2.8), (2.9) requires only the solution T = T(zo ) at the coupler output plane z = zo , examination of the evolutions of T(z) and S(z) turns out to be instructive.

2.4.4 Couplers with Monomodal Bent Waveguide Consider bent–straight waveguide couplers formed by straight and circularly bent cores of widths ws = 0.4 μm and wc = 0.5 μm with refractive index nc = ns = 1.5, embedded in a background with refractive index nb = 1. The bend radius R and the distance g between the cores are varied. The CMT simulations are carried out on a computational window of [xl , xr ] × [zi , zo ] = [0, R + 10] μm × [ − R + 1, R − 1] μm, if R ≤ 5 μm, otherwise on a window of [xl , xr ] × [zi , zo ] = [R − 5, R + 10] μm × [ − 8,8] μm, discretized with step sizes of hx = 0.005 μm and hz = 0.1 μm. First we look at the interaction of waves for vacuum wavelength λ = 1.05 μm and bend radius R = 5 μm. For this setting both constituent waveguides are single modal, with propagation constants γ /k = 1.29297−i 7.5205·10−6 for the TE0 bend mode and β/k = 1.3137 for the straight waveguide. The CMT analysis generates 2 × 2 transfer matrices T and scattering matrices S that can be viewed as being z-dependent in the sense as discussed for 2.33. Figure 2.6 shows the evolution of the matrix elements with the position z = zo of the coupler output plane. TE

1

|T 0.75

|

TE

2

|T

b0,b0

|Sb0,b0|

2

2

|

s0,s0

|Ss0,s0|

2

0.5 0.25

−4

|Tb0,s0|2

|Ts0,b0|2

|Sb0,s0|2

|Ss0,b0|2

−2

0

zo [μm]

2

4 −4

−2

0

2

4

zo [μm]

Fig. 2.6 Elements of the TE transfer matrix T and scattering matrix S as a function of the output plane position zo , for couplers as introduced in Section 2.4.4 with R = 5 μm and g = 0.2 μm, at λ = 1.05 μm (Reprinted with Permission Copyright 2005 Elsevier)

The matrix elements To, i and So, i relate the amplitudes of an input mode i to an output mode o. Thus for the present normalized modes the absolute squares can be viewed as the relative fractions of optical power transferred from mode i at the input plane z = zi to mode o at the output plane z = zo . After an initial interval,

46

K. Hiremath and M. Hammer

where these quantities remain stationary, one observes variations around the central plane z = 0, which correspond to the interaction of the waves. Here the mutually nonorthogonal basis fields are strongly overlapping; it is therefore not surprising that the levels of specific components of |To,i |2 and |So,i |2 exceed 1 in this interval. After the region of strongest interaction, near the end of the z-computational interval, one finds that the elements |Tb0, i |2 that map to the bend mode amplitude become stationary again, while the elements |Ts0, i |2 related to the output to the straight mode still show an oscillatory behavior. This is due to the interference effects as explained in Section 2.4.2. The proper amplitudes of the modes of the bus channel can be extracted by applying the projection operation (2.31); the corresponding matrix elements |Ss0, i |2 attain stationary values, such that the “coupling strength” predicted for the involved modes does not depend on the position of the coupler output plane. The final scattering matrix S that enters the relations (2.8) and (2.9) should be considered a static quantity, computed for the fixed computational interval [zi , zo ]. From the design point of view, one is interested in the elements of this matrix (the “coupling coefficients”) as a function of the resonator/coupler design parameters. Figure 2.7 summarizes the variation of S with the width of the coupler gap, for a series of different bend radii. TE

TE

1

2

R

2 2 |Sb0,s0| |Ss0,b0|

3 μm

2

|Sb0,b0| |Ss0,s0|

0.8

R

0.6

3 μm

5 μm

0.4

5 μm

10 μm

10 μm

15 μm

0.2 0

15 μm 0

0.2

0.4

0.6

g [μm]

0.8

1 0

0.2

0.4

0.6

0.8

1

g [μm]

Fig. 2.7 Scattering matrix elements |So,i |2 versus the gap width g, for couplers as considered in Section 2.4.4 with cavity radii R = 3, 5, 10, 15 μm, for TE polarized waves at λ = 1.05 μm (Reprinted with Permission Copyright 2005 Elsevier)

For all radii, one observes that for large gap widths, the non-interacting fields lead to curves that are constant at levels of unity for |Ss0, s0 |2 (full transmission along the straight waveguide), moderately below unity for |Sb0, b0 |2 (due to the attenuation of the isolated bend mode), and zero for |Sb0, s0 |2 and |Ss0, b0 |2 (decoupled fields). As the gap width decreases, the growing interaction between the modes in the two cores increases the cross coupling |Sb0, s0 |2 , |Ss0, b0 |2 and decreases the selfcoupling |Ss0, s0 |2 , |Sb0, b0 |2 . This continues until a maximum level of power transfer is attained (where the level should depend on the “phase mismatch” between the basis fields, though a highly questionable notion in case of the bend modes [19]).

2

Analytical and Computational Aspects

47

If the gap is further reduced, the cross-coupling coefficients decrease, even if the strength of the interaction is increased. This is due to “forth and back coupling”, where along the propagation axis a major part of the optical power is first transferred completely from the input channel to the adjacent waveguide, then back to the input core [47]. Therefore one should distinguish clearly between the magnitudes of the coefficients (2.26) in the differential equations that govern the coupling process, and the solution of these equations for a finite interval, the net effect of the coupler, represented by the scattering matrix S. For the symmetric computational setting used for the simulations in Figs. 2.6 and 2.7, the reciprocity requirement, i.e., the symmetry of S (see Section 2.2), is satisfied appropriately. In Fig. 2.6, the curves related to |Ss0, b0 |2 and |Sb0, s0 |2 end in nearly the same level at z = zo . One observes some deviations for configurations with very small bend radii and gaps close to zero. For such extreme cases, the underlying ansatz (2.22) of CMT may not be valid. Otherwise the symmetry of the scattering matrices provides a useful means to assess the accuracy of the CMT simulations, beyond merely the power balance constraint.

2.4.5 Couplers with Multimodal Cavity Segments Next we consider couplers that consist of a straight waveguide close to a single bent interface that supports a range of WGMs. A parameter set similar to Section 2.4.4 is adopted, with nc = ns = 1.5, nb = 1.0, R = 5 μm, ws = 0.4 μm, and g = 0.2 μm, for a reference wavelength λ = 1.05 μm. A few WGMs supported by the curved interface are illustrated in Fig. 2.4. The CMT analysis is carried out on a computational window [xl , xr ] = [0,15] μm, [zi , zo ] = [ − 4,4] μm with large extent in the (radial) x-direction, in order to capture the radiative parts of the lossy higher order bend fields. Step sizes for the numerical integrations are hx = 0.005 μm, hz = 0.1 μm, as before. It is not a priori evident how many basis fields are required for a particular simulation. Figure 2.8 shows the effect of the inclusion of the higher order WGMs on the evolution of the primary coefficients of the matrix S. The self-coupling coefficient |Sb0,b0 |2 of the fundamental bend field is hardly influenced at all, and there is

TE

1 |S 0.9

|

b0,b0

|S

|

0 z μm o

2

4

−4

|S

s0,b0

0.2

0.1

−2

TE

0.3 2

b0,s0

0.2

0.8

−4

TE

0.3 2

0.1

−2

0 z μm o

2

4

−4

TE

1 2

|S

|

s0,s0

0.9

|

2

0.8

−2

0 z μm o

2

4 −4

−2

0 z μm

2

4

o

Fig. 2.8 Effect of the inclusion of higher order bend modes on the evolution of the scattering matrix for the multimode coupler of Section 2.4.5. Results for TE waves with one (dashed line), two (dash-dotted line), three (solid line), and four cavity modes (dotted line) taken into account (Reprinted with Permission Copyright 2005 Elsevier)

48

K. Hiremath and M. Hammer

only a minor effect on the cross-coupling coefficients |Ss0, b0 |2 and |Sb0, s0 |2 . But the self-coupling coefficient |Ss0, s0 |2 of the straight mode is reduced by a substantial amount with the inclusion of the first order bend mode, due to the additional coupling to that basis field. Apparently, for the present structure it is sufficient to take just the two or three lowest order bend modes into account. This is one of the advantages of the CMT approach, where one can precisely analyze the significance of the individual basis modes (cf. the comments in Section 2.6.2). With three cavity fields and the mode of the straight waveguide, the CMT simulations lead to 4 × 4 coupler transfer and scattering matrices. The evolution of the 16 matrix elements follows similar qualitative trends as in Fig. 2.6, albeit with additional intricacy due to the multimodal cavity [17]. It turns out that, for the present case, the coupling between the bend modes themselves is practically negligible. According to Fig. 2.9, the elements of the scattering matrix exhibit a similar variation with the gap width as found for the former monomode couplers (cf. Fig. 2.7). With growing separation the cross-coupling coefficients tend to zero. The constant levels attained by the self-coupling coefficients of the bend modes are determined by the power the respective mode loses in traversing the computational window. Also here, with the exception of configurations with almost closed gap, we see in the central and right plots that the cross-coupling coefficients with reversed indices coincide, i.e., the simulations obey reciprocity.

TE

1

TE

TE |S

0.8

|2

|Ss0,b0|2

2

2

b0,s0

|Sb1,s0|

0.6

2

|Sb2,s0|

0.4

|S

2

b0,b0

0.2

|

|Sb2,b2|2 0

0.2

0.4 0.6 g [μm]

|S

|Ss0,b1|

|2

|Sb0,b1|2

2

|Sb0,b2|2

2

|S

b1,b0

|Sb2,b0|

2

|S

|Ss0,b2|

|

b2,b1

|2

b1,b2

2

|Sb1,b1|

|Ss0,s0|2 0.8

10

0.2

0.4 0.6 g [μm]

0.8

1 0

0.2

0.4 0.6 g [μm]

0.8

1

Fig. 2.9 Scattering matrix elements |So,i |2 versus the gap width g for the couplers of Section 2.4.5. The CMT simulations are based on three WGMs (indices b0, b1, b2) and on the field of the straight waveguide (index s0) (Reprinted with Permission Copyright 2005 Elsevier)

2.5 Spectrum Evaluation Having access to the bend mode propagation constants and the coupler scattering matrices, (2.8) permits to compute the resonator response. This can be done in several ways with adequate efficiency. • Direct method: In principle the spectral response can be obtained by repeating all calculations for a series of wavelengths. This requires recalculating the bend mode propagation constants and scattering matrices.

2

Analytical and Computational Aspects

49

• Interpolation of reduced scattering matrices: A substantial computational overhead can be avoided if one calculates the relevant quantities merely for a few distant wavelengths and then interpolates between these values. The interpolation procedure, however, should be applied to quantities that vary slowly with the wavelength. In line with the reasoning at the end of Section 2.2.1, one expects that the wavelength dependence of the transmission is determined mainly by the phase gain along the cavity, which is caused by a comparably slow wavelength dependence of the bend mode propagation constants γbp , but multiplied by the cavity lengths ˜ If a substantial part of the cavity is already covered by the couplers, then L, L. the matrices S exhibit fast phase oscillations with the wavelength, such that S is not directly suitable for interpolation [47]. The coupled wave interaction might introduce additional slow wavelength dependence. To separate the two scales in S, divide by the exponentials that correspond to the undisturbed wave propagation. This gives the reduced scattering matrix S = Q0 S (P0 )

−1

.

(2.34)

Here P0 and Q0 are diagonal matrices with entries P0j, j and Q0j, j as defined for P and Q in (2.32). Formally, one can view S as the scattering matrix of a coupler with zero length, where the interaction takes place instantaneously at z = 0. ˜ is compensated by redefining This modification of S, applied analogously to S, the lengths of the external cavity segments as L = L˜ = π R, by changing the ˜ accordingly, and, where necessary, by taking into account the matrices G and G altered phase relations on the external straight segments. After these modifications, the new matrices G and G˜ capture the phase gains of the cavity fields along the full circumference. These show only slow wavelength dependence, just as S and S˜ , such that they can be successfully interpolated [47]. • Assumption of a constant scattering matrix: As an extreme variant of the former approximation, (2.8) are evaluated with the scattering matrix for a central reference wavelength, together with rigorously computed or interpolated cavity mode propagation constants.

2.6 Circular Microresonators in Two Spatial Dimensions The ingredients discussed so far are now combined into a simulation tool for entire resonator structures. We compare the results of the CMT approach with finitedifference time domain (FDTD) simulations based on a second order Yee mesh with total field/scattered field formulation and artificial transparent (perfectly matched layer, PML) boundary conditions [17, 48]. While the present examples consider exclusively TE polarized fields, the abstract reasoning in Section 2.2 and the CMT formalism in Section 2.4 are just as well applicable for TM polarization [17].

50

K. Hiremath and M. Hammer

2.6.1 Microring We consider the symmetric ring resonator with monomode cavity made up of the couplers of Section 2.4.4. In line with the assumptions leading to (2.8) and (2.9), the fundamental mode of the bus waveguides is launched at the input port with unit power, with no incoming field at the add port. Figure 2.10 shows the spectral response for parameters nc = ns = 1.5, nb = 1.0, wc = 0.5 μm ws = 0.4 μm, R = 5 μm, g = g˜ = 0.2 μm in a wavelength interval around the former arbitrarily chosen design wavelength λ = 1.05 μm. The further computational setting is as given for Fig. 2.6. 1.2 TE

CMT

FDTD

Direct

Quadratic

Linear

Direct

∼

With S ′, S ′ at λ=1.05 μm

PT/PI

0.8

T

I

D

P /P ,P /P

I

1

0.6 0.4 TE

TE

0.2

P /P D

0

1.02

I

1.04

1.06 λ [μm]

1.08 1.02

1.04

1.06 λ [μm]

1.08 1.02

1.04

1.06

1.08

λ [μm]

Fig. 2.10 Relative transmitted power PT and dropped power PD versus the wavelength for a ring resonator as discussed in Section 2.6.1; Left: CMT and FDTD results. Center: CMT results with spectrum evaluation by the direct and by the interpolation method with nodal wavelengths 1.015 μm and 1.085 μm (linear), or 1.015 μm, 1.05 μm, and 1.085 μm (quadratic interpolation). Right: Spectrum evaluation by the direct and the constant scattering matrix method (Reprinted with Permission Copyright 2005 Elsevier)

One observes the familiar ring resonator resonance pattern with dips in the transmitted power and peaks in the dropped intensity. According to Fig. 2.7, the present parameter set specifies configurations with rather strong interaction in the coupler regions (|Sb0,s0 |2 = 30%), such that the resonances are relatively wide, with a substantial amount of optical power being directly transferred to the drop port also in off-resonant states. The CMT results are compared with FDTD simulations (for numerical details, see [17]). As seen in the left plot of Fig. 2.10, we find an excellent agreement between the CMT and the FDTD results for TE polarization (though one observes minor deviations for the TM case, where the fields are discontinuous [17]). Even in the present 2-D setting, these FDTD calculations typically require a computation time of several hours, while the CMT analysis (with interpolation) delivers the entire spectrum in just a few minutes. The central plot of Fig. 2.10 shows the resonator spectrum as obtained by interpolating bend mode propagation constants and CMT scattering matrices for only two (linear interpolation) or three (quadratic interpolation) distinct wavelengths, according to Section 2.5. While small deviations remain for the linear approximation, on the scale of the figure the curves related to quadratic interpolation are hardly distinguishable from the direct CMT results. The right most plot of Fig. 2.10 shows

2

Analytical and Computational Aspects

51

that the assumption of a constant scattering matrix is perfectly reasonable for the current setting. Minor deviations appear only far from the reference wavelength. Thus the interpolation approach provides a very effective means to predict accurately the resonator spectrum, in particular if narrow dips/peaks in the responses of high quality resonators have to be resolved, such that the direct evaluation would be computationally expensive. We shall exploit this later on for the 3-D simulations. The principal field components for off-resonance and resonant configurations are illustrated in Fig. 2.11. In the off-resonance state one observes the large through transmission and small amplitudes of the waves in the drop-port along with minor wave amplitudes in the cavity. At the resonances, the straight transmission is almost suppressed. A major part of the input power arrives at the drop-port, and the leaky nature of the ring mode can be clearly observed.

Fig. 2.11 CMT results for the microring structure of Fig. 2.10. Snapshots of the principal components of the physical TE field, off-resonance (first plot) and at resonance (second plot). For visualization purposes the coupler computational window has been extended to [zi , zo ] = [ − 4, 8] μm (Reprinted with Permission Copyright 2005 Elsevier)

2.6.2 Microdisk Here we look at the symmetrical microdisk resonator that is constituted by two of the multimode couplers of Section 2.4.5. The computational setting and all parameters are identical to the data given in Section 2.4.5, for gap widths g = g˜ = 0.2 μm. First consider the spectral response obtained by CMT computations, where besides the mode of the straight waveguide, different sets of WGMs are used as basis fields. The curves in the left plot of Fig. 2.12 exhibit only specific extrema from the full spectrum with similar extremum levels. Hence these resonances can clearly be assigned to the respective TE0 or TE1 WGM. As these modes circulate along the cavity with different propagation constants, individual resonance conditions are satisfied in general at different wavelengths. TE2 plays obviously only an inconsequential role.

52

K. Hiremath and M. Hammer TE

PT /PI, PD/PI

TE

0

1

1

TE

CMT: 0,1

2

CMT: 0,1,2

0.8 0.6

TE 0.4 0.2 0

1.02

1.04

λ [μm]

1.06

1.08

1.02

1.04

λ [μm]

1.06

1.08

Fig. 2.12 TE power spectrum of the microdisk resonator of Section 2.6.2. CMT analysis with different sets of basis modes. Besides the mode of the straight waveguide, only one WGM (TE0 , TE1 , or TE2 ; left), or the two and three lowest order WGMs (right) is/are taken into account (Reprinted with Permission Copyright 2005 Elsevier)

The effect of the inclusion of higher order WGMs on the resonator response is shown in the right plot of Fig. 2.12. While the fundamental and first order WGMs are essential for the present resonator, inclusion of the second order WGM into the CMT analysis leads only to minor changes. Thus for this microdisk configuration, it is sufficient to take into account the two lowest order cavity modes as basis fields to predict reliably the spectral response. This was already evident in the coupler analysis of Fig. 2.8. Due to negligible interaction among the cavity modes themselves (which might be caused by the presence of the couplers, i.e., the perturbation through the bus waveguides), the resonance locations in the combined CMT analysis (right plot) coincide well with those predicted by the single mode calculation (left plot). Similar conclusions can be drawn by inspection of the local mode amplitudes b = (bq ), as functions of the wavelength, that are predicted by the CMT model [17]. The comparison of CMT and FDTD spectra in Fig. 2.13 shows a quite satisfactory agreement. The right plot validates the interpolation approach of Section 2.5. As before, we see that the quadratic interpolation of the scattering matrix entries and propagation constants leads to curves that are almost indistinguishable from the directly computed results. Figure 2.14 shows examples for the corresponding field distributions. At offresonance, most of the input power is directly transferred to the through-port. At the wavelength corresponding to one of the minor resonances, the field pattern in the cavity exhibits a nearly circular nodal line corresponding to the radial minimum in the profile of the first order WGM (cf. Fig. 2.4). As seen in Fig. 2.13, here the first order mode carries most of the power inside the cavity. The deviation form the circular pattern is caused by the interference with the fundamental WGM, which is also excited at this wavelength with a small power fraction. The major resonance related to the fundamental mode is of higher quality, with much larger intensity in the cavity, almost full suppression of the waves in the through-port and nearly complete drop of the input power.

2

Analytical and Computational Aspects

53

1.2 TE

1

CMT

FDTD

Direct

Quadratic

Linear

PT/PI, PD/PI

P /P T

0.8

I

0.6 0.4 TE

0.2 0

PD/PI 1.02

1.04

1.06

1.08

1.02

1.04

λ [μm]

1.06

1.08

λ [μm]

Fig. 2.13 Power transmission through the microdisk resonator of Section 2.6.2. Left: CMT and FDTD spectra for TE modes. Right: CMT spectra (four basis modes) computed directly, and by interpolation of data evaluated at the nodal wavelengths 1.015 μm, 1.085 μm (linear), and 1.015 μm, 1.05 μm, 1.085 μm (quadratic interpolation) (Reprinted with Permission Copyright 2005 Elsevier) 6

x [μm]

4 2 0 −2 −4 Re(E ) y

Re(E )

Re(E )

−6 λ = 1.055 μm −6 −4 −2 0 2 z [μm]

λ = 1.0483 μm −6 −4 −2 0 2 z [μm]

λ = 1.043 μm −6 −4 −2 0 2 z [μm]

y

4

6

y

4

6

4

6

Fig. 2.14 Snapshots of the real physical electric field for the microdisk resonator of Section 2.6.2; CMT simulations with four basis modes. The wavelengths correspond to an off-resonance state (left) and to minor (center) and major resonances (right). The gray scale levels of the plots are comparable (Reprinted with Permission Copyright 2005 Elsevier)

2.7 Circular Optical Microresonators in 3-D So far we restricted ourselves to two spatial dimensions, in order to explain concepts and phenomena behind the CMT model, and for purposes of rigorous numerical assessment. There are practical circumstances, however, where the 2-D setting is definitely inadequate, e.g., when an effective index approximation seems not reasonable, when the assumption of decoupled polarizations appears to be inappropriate, when the vectorial nature of the fields might be important (as in the case of cavity or bus cores with pronouncedly hybrid modes), or the obvious case of vertically coupled microresonators. One then has to resort to fully 3-D simulations. The abstract resonator model in Section 2.2 remains applicable, irrespectively of the number of spatial dimensions. With the exception of an additional integration along the third,

54

K. Hiremath and M. Hammer

vertical y-axis, the CMT formalism for the couplers is essentially identical to what has been discussed in Section 2.4. Thus the extension of the present CMT resonator model to 3-D [18] should be straightforward, in principle. The real additional complexity is the task of generating the required basis fields. Analytic solutions, as in 2-D, for modes of straight and — especially — bent waveguides in 3-D, i.e., with 2-D cross sections, do not exist; numerical mode solvers have to be applied. For the simulations discussed in this section we could rely on a semi-analytical technique based on film mode matching (FMM) [20, 49, 50]. The modal eigenvalue problem is addressed by dividing the waveguide cross section plane into vertical slices such that the permittivity profile is constant along the horizontal/radial axis. On each slice the modal field is expanded rigorously into eigenfunctions (modes of 1-D multilayer slab waveguides) associated with the local refractive index profile, where the sets of eigenfunctions are discretized by Dirichlet boundary conditions sufficiently far above and below the interesting region around the waveguide core. The 3-D modes are obtained by connecting the expansions on the individual slices such that the full field satisfies all continuity requirements at the vertical interfaces and shows the appropriate behavior in the outermost regions. The rigorous mode profile approximations are represented quasi-analytically, which proves to be advantageous for the subsequent use as basis fields within the CMT formalism (integrations). For the 3-D coupler introduced in Fig. 2.15a, the coupled field ansatz (2.22) applies, with an additional nontrivial dependence on the vertical y-axis, introduced by the—now truly vectorial—mode profiles (2.21) and (2.20) of the straight and bent cores. The functional (2.23) and consequently the matrix elements (2.25) and (2.26) receive an additional y-integration. Otherwise the reasoning of Sections 2.4.1, 2.4.2, and 2.4.3 remains valid.

y

(a)

(b)

yt R nd s ns

yb xi

hd g 0

nc x nf w

hs

(c)

xo

Fig. 2.15 (a): Coupler setting in 3-D, a cross section perpendicular to the direction of propagation at z = 0. A disk cavity of radius R, core refractive index nd , and height hd is coupled to bus waveguides of core refractive index nf , width w, and height hs . Here the disk and the bus waveguide are placed at different levels, at a vertical distance s and at a horizontal position g. Negative values for g represent overlapping components. ns and nc are the refractive indices of the substrate and cladding regions. (b) and (c): Choices for constituting structures for the CMT analysis. In (b) the substrate is included into the cavity mode analysis, whereas it is excluded in (c) (Reprinted with Permission Copyright 2005 Elsevier)

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55

One should be aware that the choice of the constituting structures, i.e., of the refractive index profiles for which the basis modes are calculated, is not at all unique. For the structure of Fig. 2.15a, for example, the substrate could be included into the computation of the cavity mode (b) or omitted (c). In the first case the permittivity profile εb of the cavity is closer to the true permittivity ε of the full coupler. Hence their difference, i.e., the perturbation in (2.26), is small, and one can expect a better overall approximation. Option (b) also allows to take the influence of the substrate on the cavity modes into account. As an added benefit, the integrals (2.26) in the coupled mode equations extend only over the disk and the straight waveguide cores, not over the substrate domain. As a disadvantage, one has to recalculate the cavity modes for different vertical separations s. This can be avoided in the setting (c) of Fig. 2.15. In general, the choice of the constituting structures and the selection of the basis fields is a matter of physical intuition and of convenience for the subsequent numerics. For the present configuration with low to moderate index contrast between substrate and cladding, we observed hardly any difference [18]. The performance of the coupler of Fig. 2.15 is affected by both the vertical separation s and the relative horizontal core position g. Their influence on the elements of the coupler scattering matrix is shown in Fig. 2.16. While a rigorous explanation of these variations on the basis of modal interaction strengths, as in the 2-D case, turns out to be difficult, the qualitative behavior can still be understood. For large horizontal separations g, due to less mode interaction, all self-coupling coefficients (a) and (d) tend to 1, and the cross-coupling coefficients (b), (c), (e), and (f) vanish. Also, the cross-coupling coefficients satisfy the self-consistency requirement of reciprocity. A 1

(a)

(b)

s = 1.0 μm

(c)

s = 1.0 μm

|Ss b0|2

0.8

|S

|Sb1 b0|2

|2

|S

|So i|2

b0 s

0.6 |Sb0 b0|2

0.4

|S

b1 b1

|2

|2

|Sb2 b0|2

|Sb1 s|2

|Sb0 b2|2

|S

|2

|S

s b2

|Sb2 b2|2

0.2

b0 b1

|Ss b1|2

b2 b1

|Sb2 s|2

|2

|Sb1 b2|2

s = 1.0 μm 0 1

(d)

s = 0.5 μm

(e)

s = 0.5 μm

(f)

s = 0.5 μm

|Ss b0|2

0.8

|S

|Sb1 b0|2

|2

|S

|So i|2

b0 s

0.6 |Sb0 b0|2

0.4

|S

b1 b1

−2

−1 g [μm]

0

|Sb1 s|2

|Sb0 b2|2

|2

|S

s b2

b2 b1

|Sb2 s|2

1 −3

−2

−1 g [μm]

|2

|Sb2 b0|2

|S

|Sb2 b2|2

0.2 0 −3

|2

b0 b1

|Ss b1|2

0

|2

|Sb1 b2|2

1 −3

−2

−1 g [μm]

0

1

Fig. 2.16 Scattering matrix elements of the 3-D couplers of Fig. 2.15 versus the relative horizontal core position g for different vertical separations s. The CMT computation is based on the single mode of the straight core (index s), together with the first three lowest order modes (b0, b1, b2) of the disk cavity. (a, d): self-coupling coefficients; (b, e): cross-coupling straight/bent core; (c, f): cross coupling between bend modes. The coupler consists of a straight waveguide with w = 2.0 μm, hs = 0.140 μm, nf = 1.98, ns = 1.45, and nc = 1.4017 and a disk cavity with nd = 1.6062, hd = 1.0 μm, R = 100 μm (Reprinted with Permission Copyright 2005 Elsevier)

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K. Hiremath and M. Hammer 1

(a) (b)

PT/Pin, PD/Pin

0.8 0.6 P 0.4

P

D

D

s = 1.0 μm g=0

s = 1.0 μm g = −1.0 μm

0.2 0 1

(d)

(c) PT/Pin, PD/Pin

0.8 0.6 P

P

D

0.4

D

s = 0.5 μm g = −1.0 μm

s = 0.5 μm g=0

0.2 0 1.548

1.549

1.55 λ [μm]

1.551

1.552 1.548

1.549

1.55 λ [μm]

1.551

1.552

Fig. 2.17 Spectral response of the vertically coupled microdisk resonator consisting of two identical couplers as in Fig. 2.16 with different vertical separations s and horizontal positioning g (Reprinted with Permission Copyright 2005 Elsevier)

reduction of the vertical separation ((a), (b), (c) vs. (d), (e), (f)) increases the strength of the interaction. For the small vertical distance, also the interaction (f) between the cavity modes, affected by the presence of the straight core, is clearly no longer negligible. Comparisons, as far as possible, of these CMT results with simulations by a beam propagation method show a reasonable agreement [18]. The resultant effect of the relative vertical and horizontal core positions on the spectral response of the full 3-D vertically coupled microdisk resonator is depicted in Fig. 2.17. The data is computed with a constant scattering matrix at the reference wavelength λ = 1.55 μm [18] (cf. Section 2.5). For configuration (a) with moderate interaction strength, one observes a set of three resonances that appear periodically, each corresponding to one of the three low-loss cavity modes involved. Reducing either the horizontal separation (b) or the vertical distance (c) leads to much stronger coupling with deteriorated resonances. The resonance characteristics disappear altogether in the somewhat extreme situation (d).

2.8 Concluding Remarks The ab initio frequency domain model, as discussed in this chapter, originates from the physical notions that are commonly used to describe the functioning of circular microresonators. Bend modes supported by the segments of the ring or disk-shaped cavities, together with modal fields of the straight bus cores, constitute the basis for the quantitative coupled mode theory of the evanescent wave interaction in the coupler regions.

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57

For the present frequency domain description, it is straightforward to take into account material dispersion. Since the spectral response is evaluated as a scan over vacuum wavelengths, the quantities that enter the CMT equations can be determined directly for the material properties at the respective wavelengths. The remarks from Section 2.5 on interpolation schemes for efficient spectrum evaluation apply as well. Unlike with other common, purely numerical methods, here we have convenient access to all local modal amplitudes, which allows characterizing and analyzing the resonances. One observes that, for most reasonable configurations, only one or a few cavity modes play a significant role. It is then possible to accurately predict the spectral response of the device in question by very efficient, quasi-analytical calculations. Our thorough study of the 2-D version of the model serves to explain all concepts; the examples permit a thorough benchmarking versus rigorous finite difference time domain calculations. Extension of the formalism to realistic resonators in 3-D is straightforward. Here in particular the computational advantages of the CMT approach are revealed, since hardly any other, even moderately efficient simulation tool is available for practical design work. Beyond the vertically coupled microdisk resonator of Section 2.7, the approach has shown to be sufficiently flexible to handle also quite exotic 3-D configurations, like the device with hybrid ring cavity and pedestal waveguides of Ref. [18]. Finally, the CMT model is ideally suited to incorporate small changes in the configuration, e.g., for purposes of the evaluation of fabrication tolerances, or for predicting the effects of tuning mechanisms [8, 51]. Acknowledgments This work was carried out as a part of the project “NAIS” (IST-200028018), funded by the European Commission. K. R. Hiremath also acknowledges support by the DFG (German Research Council) Research Training Group “Analysis, Simulation and Design of Nanotechnological Processes,” University of Karlsruhe. The authors thank R. Stoffer for his hard work on the 3-D simulations. They are grateful to H. J. W. M. Hoekstra, E. van Groesen, and their colleagues in the NAIS project for many fruitful discussions.

References 1. Vahala, K. Optical microcavities. World Scientific Singapore (2004) 2. Michelotti, F., Driessen, A., et al. (eds.) Microresonators as building blocks for VLSI photonics, volume 709 of AIP conference proceedings (2004) 3. Heebner, J., Grover, R., et al. Optical micro-resonators: Theory, fabrication, and applications. Springer (2007) 4. Landobasa, Y. M., Darmawan, S., et al. Matrix analysis of 2-D microresonator lattice optical filters. IEEE J. Quantum Electron. 41, 1410–1418 (2005) 5. Popovi´c, M. A., Manolatou, C., et al. Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters. Opt. Express 14, 1208–1222 (2006) 6. Stokes, L. F., Chodorow, M., et al. All single mode fiber resonator. Opt. Lett. 7, 288–290 (1982) 7. Yariv, A. Universal relations for coupling of optical power between microresonators and dielectric waveguides. IEE Electron. Lett. 36, 321–322 (2000) 8. Hammer, M., Hiremath, K.R., et al. Analytical approaches to the description of optical microresonator devices. In Michelotti, F., Driessen, A., et al. (eds.), Microresonators as building blocks for VLSI photonics, volume 709 of AIP conference proceedings, 48–71 (2004)

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9. Okamoto, K. Fundamentals of Optical Waveguides. Academic Press, USA (2000) 10. Klunder, D.J.W., Krioukov, E., et al. Vertically and laterally waveguide-coupled cylindrical microresonators in Si3 N4 on SiO2 technology. Appl. Phys. B. 73, 603–608 (2001) 11. Klunder, D.J.W., Balistreri, M.L.M., et al. Detailed analysis of the intracavity phenomena inside a cylindrical microresonator. IEEE J. Lightwave Technol. 20, 519–529 (2002) 12. Rowland, D.R., Love, J.D. Evanescent wave coupling of whispering gallery modes of a dielectric cylinder. IEE Proc.: Optoelectron. 140, 177–188 (1993) 13. Chin, M.K., Ho, S.T. Design and modeling of waveguide coupled single mode microring resonator. IEEE J. Lightwave Technol. 16, 1433–1446 (1998) 14. Cusmai, G., Morichetti, F., et al. Circuit-oriented modelling of ring-resonators. Opt. Quantum Electron. 37, 343–358 (2005) 15. Vassallo, C. Optical waveguide concepts. Elsevier, Amsterdam (1991) 16. Hall, D.G., Thompson, B.J., (eds.) Selected papers on coupled-mode theory in guided-wave optics, volume MS 84 of SPIE milestone series. SPIE Optical Engineering Press, Bellingham, Washington USA (1993) 17. Hiremath, K.R., Stoffer, R., et al. Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory. Opt. Commun. 257, 277–297 (2006) 18. Stoffer, R., Hiremath, K.R., et al. Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory. Opt. Commun. 256, 46–67 (2005) 19. Hiremath, K.R., Hammer, M., et al. Analytic approach to dielectric optical bent slab waveguides. Opt. Quantum Electron. 37, 37–61 (2005) 20. Prkna, L., Hubálek, M., et al. Field modeling of circular microresonators by film mode matching. IEEE J. Sel. Top. Quantum Electron. 11, 217–223 (2005) 21. Stoffer, R., Hiremath, K.R., et al. Comparison of coupled mode theory and FDTD simulations of coupling between bent and straight optical waveguides. In Michelotti, F., Driessen, A., et al. (eds.), Microresonators as building blocks for VLSI photonics, volume 709 of AIP conference proceedings, 366–377 (2004) 22. Van, V., Absil, P., et al. Propagation loss in single-mode GaAs-AlGaAs microring resonators: measurement and model. IEEE J. Lightwave Technol. 19, 1734–1739 (2001) 23. Absil, P.P., Hryniewicz, J.V., et al. Compact microring notch filters. IEEE Photonics Technol. Lett. 14, 398–400 (2000) 24. Chu, S.T., Little, B.E., et al. Cascaded microring resonators for crosstalk reduction and spectrum cleanup in add-drop filters. IEEE Photonics Technol. Lett. 11, 1423–1425 (1999) 25. Yariv, A., Xu, Y., et al. Coupled-resonator optical waveguide: a proposal and analysis. Opt. Lett. 24, 711–713 (1999) 26. Little, B.E., Chu, S.T., et al. Microring resonator channel dropping filters. IEEE J. Lightwave Technol. 15, 998–1005 (1997) 27. Grover, R., Van, V., et al. Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters. IEEE J. Lightwave Technol. 20, 900–905 (2002) 28. Hryniewicz, J.V., Absil, P.P., et al. Higher order filter response in coupled microring resonators. IEEE Photonics Technol. Lett. 12, 320–322 (2000) 29. Chu, S.T., Little, B.E., et al. An eight-channel add-drop filter using vertically coupled microring resonators over a cross grid. IEEE Photonics Technol. Lett. 11, 691–693 (1999) 30. Little, B. E., Chu, S. T., et al. Microring resonator arrays for VLSI photonics. IEEE Photonics Technol. Lett. 12, 323–325 (2000) 31. Manolatou, C., Khan, M. J., et al. Coupling of modes analysis of resonant channel add drop filters. IEEE J. Quantum Electron. 35, 1322–1331 (1999) ˇ 32. Prkna, L., Ctyroký, J., et al. Ring microresonator as a photonic structure with complex eigenfrequency. Opt. Quantum Electron. 36, 259–269 (2004) 33. Taflove, A., Hagness, S.C. Computational electrodynamis: The finite difference time domain method. Artech House, Norwood, MA, USA, 2nd edition (2000) 34. Hagness, S.C., Rafizadeh, D., et al. FDTD microcavity simulations: Design and experimental realization of waveguide coupled single mode ring and whispering gallery mode disk resonator. IEEE J. Lightwave Technol. 15, 2154–2165 (1997)

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35. Koos, C., Fujii, M., et al. FDTD-modelling of dispersive nonlinear ring resonators: Accuracy studies and experiments. IEEE J. Quantum Electron. 42, 1215–1223 (2006) 36. Sacks, Z.S., Lee, J.-F. A finite-element time-domain method using prism elements for microwave cavities. IEEE Trans. Electromagn. Compat. 37, 519–527 (1995) 37. Carpes Jr., W.P., Pichon, L., et al. Efficient analysis of resonant cavities by finite element method in the time domain. IEE Proc. Microw. Antennas Propag. 147, 53–57 (2000) 38. Ji, X., Lu, T., et al. Discontinuous galerkin time domain (DGTD) methods for the study of 2-D waveguide-coupled microring resonators. IEEE J. Lightwave Technol. 23, 3864 – 3874 (2005) 39. Boriskina, S.V., Benson, T.M., et al. Effect of a layered environment on the complex natural frequencies of two-dimensional WGM dielectric-ring resonators. IEEE J. Lightwave Technol. 20, 1563–1572 (2002) 40. Boriskina, S.V., Benson, T.M., et al. Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators. Opt. Quantum Electron. 35, 545–559 (2003) 41. Boriskina, S.V., Nosich, A.I. Radiation and absorption losses of the whispering-gallery-mode dielectric resonators excited by a dielectric waveguide. IEEE Trans. Microw. Theory Tech. 47, 224–231 (1999) 42. Jackson, J.D. Classical electrodynamics. John Wiley and Sons, Inc., 3rd edition (1998) 43. Lewin, L., Chang, D.C., et al. Electromagnetic waves and curved structures. Peter Peregrinus Ltd. (On behalf of IEE), Stevenage, England (1977) 44. Pennings, E.C.M. Bends in optical ridge waveguides, modelling and experiment. Ph.D. thesis, Delft University, The Netherlands (1990) 45. Abramowitz, M., Stegun, I.A. Handbook of mathematical functions (applied mathematics Series 55). National Bureau of Standards, Washington, DC (1964) 46. Press, W.H., Teukolsky, S.A., et al. Numerical recipes in C. Cambridge University Press, 2nd edition (1992) 47. Hiremath, K.R. Coupled mode theory based modeling and analysis of circular optical microresonators. Ph.D. thesis, University of Twente, The Netherlands (2005) 48. Stoffer, R. Uni- and Omnidirectional simulation tools for integrated optics. Ph.D. thesis, University of Twente, Enschede, The Netherlands (2001) 49. Sudbø, A.S. Film mode matching: a versatile numerical method for vector mode fields calculations in dielectric waveguides. Pure Appl. Opt. 2, 211–233 (1993) 50. Prkna, L., Hubálek, M., et al. Vectorial eigenmode solver for bent waveguides based on mode matching. IEEE Photonics Technol. Lett. 16, 2057–2059 (2004) 51. Hiremath, K.R., Hammer, M. Modeling of tuning of microresonator filters by perturbational evaluation of cavity mode phase shifts. IEEE J. Lightwave Technol. 25, 3760–3765 (2007)

Chapter 3

Polarization Rotation in Ring Resonators Francesco Morichetti

Abstract Densely integrated photonic devices require high-index contrast waveguides supporting small bending radii with negligible loss. However, when optical waveguides are tightly curved, polarization rotation (PR) effects can arise. In this chapter, PR in bent waveguides is theoretically and numerically investigated and its dependence on the main waveguide parameters is derived. It is shown that the efficiency of PR linearly increases with the refractive index contrast, is strongly inhibited by birefringence, and vanishes in the case of waveguides with a refractive index profile symmetrical in the direction orthogonal to the plane of the bend. In resonant structures PR is strongly enhanced and can severely distort the frequency response of ring resonators (RRs). Cavity-enhanced PR in RRs is discussed, and design rules to mitigate or enhance PR are provided. The detrimental effects due to undesired PR on the behavior of common architectures including RRs are shown. Examples of devices exploiting cavity-enhanced polarization rotation to implement highly efficient polarization rotators are presented.

3.1 Introduction High-index contrast optical waveguides capable of supporting small bending radii with negligible loss are required to drive photonics toward ultra-dense integration scale [1–5]. However, in a tightly curved waveguide, guided modes can become hybrid and polarization rotation (PR) effects can arise [6, 7]. On the one hand, PR in bent waveguides can be exploited for the realization of integrated polarization rotators and several strategies have been proposed to increase the rotation efficiency, such as the use of waveguides with highly slanted sidewalls [8]; on F. Morichetti (B) POLICOM - Dipartimento di Elettronica e Informazione, Politecnico di Milano, Milano, Italy; Fondazione Politecnico di Milano, Milano, Italy e-mail: [email protected]

I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_3,  C Springer Science+Business Media, LLC 2010

61

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F. Morichetti

the other hand, when PR is undesired, it can severely impair the expected behavior of optical devices. These effects are magnified when a bent waveguide is closed in a ring-resonator (RR) configuration, because at the resonance frequencies PR is enhanced by light trapping and becomes strongly frequency dependent [9, 10]. This chapter starts from the modeling of PR in bent waveguides, then focuses on its effects on RR-based optical devices. In Section 3.2 the relations between the PR efficiency and the main parameters of the waveguide (bending radius, waveguide shape, and refractive index profile) are pointed out. Particular attention is devoted to the role of the waveguide birefringence, which is responsible for the phase-matching condition necessary for the building-up of the process. The effects of PR in RRs are discussed in Section 3.3, where conditions for full PR are derived, by means of a simple approach based on the theory of poles-zeros loci. The role of round-trip losses and birefringence is also discussed. The impact of undesired PR in optical devices including RRs is discussed in Section 3.4. Examples of efficient polarization rotators exploiting cavity-enhanced PR in RRs are presented in Section 3.5.

3.2 Polarization Rotation in Bent Waveguides In this section PR in bent waveguides is modeled by means of the coupled mode theory (CMT) and its dependence on the main waveguide parameters is analytically derived and discussed with the help of a numerical case analysis.

3.2.1 Modeling Polarization Rotation in Bent Waveguides Figure 3.1 shows a schematic of a bent waveguide with the cross section lying in the x–y plane and the curvature plane coincident with the x–z plane. A bent waveguide supports proper modes that are orthogonal to each other, but differs from the modes of the straight waveguide as much as the curvature of the bend is increased [11]. Typically the bend modes are characterized by a higher degree of hybridness with respect to the straight waveguide modes, where the hybridness is defined as the ratio between the minor and the major transverse components of the electric field,

y z

ρ

TEin

Fig. 3.1 Schematic of a bent optical waveguide

x

TEout TMout

3

Polarization Rotation in Ring Resonators

63

that is, Ey /Ex for the transverse electric (TE) mode and Ex /Ey for the transverse magnetic (TM) mode [12]. The different hybridness degree of the bend’s modes with respect to the straight waveguide modes is responsible for the power exchange between the two orthogonal TE and TM modes of the straight waveguide [13]. To model PR in bends it is first convenient to choose suitable reference fields, which are defined in Section 3.2.1.1 and then employed in the CMT model discussed in Section 3.2.1.2. 3.2.1.1 The Choice of the Reference Fields Let us define the bend modes as b,n,u (u = x,y), where the major transverse component of the electric field is Ex in b,n,x and Ey in b,n,y . The subscript n (n= 0,1, . . .) indicates the mode order and is omitted throughout the analysis when not strictly required. The propagation of the overall electric field (x,y,z) along the bend can be expressed as (x, y, z) =



  bn,u b,n,u (x, y) exp −jβb,n,u z ,

(3.1)

u=x,y n

where βb,n,u are the propagation constants of the bend modes and the complex amplitude bn,u do not depend on the propagation length z because of mode orthogonality. Alternatively, it is possible to describe propagation in the bend referring to the TE modes x and TM modes y of the straight waveguide, provided that their mutual coupling is taken into account. For the sake of simplicity, the modes of the straight waveguide are assumed with no hybridness, i.e., the electric field of the TE (TM) mode is considered to be fully x-polarized (y-polarized). However, all the presented results hold even if the straight waveguide modes have some initial degree of hybridness, as typically happens in the case of high-index contrast waveguides. Referring to the straight waveguide modes, field propagation can be described as   (3.2) b˜ n,u (z)n,u (x, y) exp −jβn,u z , (x, y, z) = u=x,y n

where βn,u are the propagation constants of the straight waveguide modes and complex amplitudes b˜ n,u (z) take into account mode coupling within the bend. More generally, any generic set of functions providing a complete function space can be used. Here we conveniently define the optical fields ˆ b,u (x, y) = 



gn,u n,u (x, y)

(3.3)

n

as a linear combination of all the modes of the straight waveguide with the same ˆ b,u describes polarization state. The coefficients gn,u are chosen in such a way that  the overall field that, within the bend, has the same polarization state as the mode

64

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ˆ b,x and  ˆ b,y is u of the straight waveguide. The need for introducing the fields  due to the fact that the field in the bend can have a transversal distribution significantly different from that of the straight waveguide fundamental modes 0,x and 0,y , even when the curvature is not particularly tight. Therefore, it is not generally possible to express the field in the bend as a linear combination of 0,x and 0,y only and higher order modes are also required. The number of required higher order modes depends on the waveguide parameters, namely the index contrast and geometry, and on the curvature. 3.2.1.2 Coupled Mode Theory in Bent Waveguides Inside the bend, the x- and y-polarized fields can be written as [14] ˆ b,x (x, y) exp ( − jβx z), Ex (x, y, z) = ax (z)

(3.4)

ˆ b,y (x, y) exp ( − jβy z), Ey (x, y, z) = ay (z)

(3.5)

where au (z) (with u = x, y; v = x, y) is the slowly varying complex amplitude. The propagation constants βx and βy of the TE and TM fundamental modes of the straight waveguide are conveniently assumed as reference wavenumbers for the propagation of Ex (x, y, z) and Ey (x, y, z), respectively. The power exchange between the two polarization states can be efficiently modeled by means of the CMT [14, 15] which leads to the following set of coupled equations [16]: dax = −jκxx ax − jκxy exp (j β0 z) ay , dz day = −jκyx exp (−j β0 z) ax − jκyy ay , dz

(3.6) (3.7)

with the coefficients kuv (with u = x, y; v = x, y) defined as in [14] κuv

1 = 2βu



+∞

−∞

dx

+∞

−∞

dy ln

n2t

  ˆ∗ ∂ ∂ b,u ˆ b,v  , ∂v ∂u

(3.8)

where

x nt = n(x,y) 1 + ρ

(3.9)

is the cross-sectional refractive index profile of the straight waveguide equivalent to the bend of radius ρ. The term β0 = βx − βy = 2π B0 /λ takes into account the phase mismatch due to the birefringence B0 of the waveguide. ˆ b,x and  ˆ b,y , The coefficients κuv , that in (3.8) are calculated from the fields  can be expressed as a function of the straight waveguide modes by substituting

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Polarization Rotation in Ring Resonators

65

ˆ b,u can be written (3.3) into (3.8). In the hypothesis of small curvature (x/ρ  1),  as [17] ˆ b,u (x, y) = 0,u (x, y) + 

Au 1,u (x, y), ρ

(3.10)

where 0,u and 1,u are the fundamental mode and the first higher order mode of the straight waveguide, respectively. Equation (3.10) coincides with the (normalized) truncation to the first higher order mode of (3.3), with g0,u = 1 and g1,u = Au /ρ. The parameter Au does not depend on the radius ρ and the higher order mode 1,u is leaky in single-mode waveguides. For tight curvatures and multimode waveguides this approximation could fail and either higher order guided modes or radiation modes could become necessary for an accurate modal expansion [18]. By substituting (3.10) in (3.8), the dependence of κuv on the bending radius ρ can be derived analytically, leading to the following expressions [10]: κ0,xy κ0,yx , κyx = , ρ ρ κ0,yy κ0,xx = 2 , κyy = 2 , ρ ρ

κxy =

(3.11)

κxx

(3.12)

where the constants κ0,uv only depend on the straight waveguide index profile. Equation (3.11) states that the cross-coupling coefficients κuv , representing the strength of the coupling between the two polarization states in the bend, scale with the inverse of the bending radius. In most practical cases κxy and κyx can be assumed equal, i.e., κxy  κyx = κp . The self-coupling coefficients κuu , representing the perturbation induced by the curvature on the phase-shift of the u-polarized field, scale with the inverse of the squared bending radius. As a result, the difference between the phase constants of the TE and the TM mode in the bend is given by

β = β0 +

β1 , ρ2

(3.13)

where β1 = κ0, xx − κ0,yy . A complete power transfer between the two coupled modes can occur only if

β = 0. According to (3.13), at large bending radii β approaches β0 and complete PR requires a zero-birefringence waveguide (B0 = 0). On the other hand, for tight curvatures, the effect of the curvature on the phase mismatch cannot be neglected and total conversion is achieved only in non-zero-birefringence waveguides ( β0 = 0) and only for a particular value of the radius ρ compensating for the straight waveguide birefringence. The phase constants of the proper modes of the bend, b,x (x, y) and b,y (x, y), are given by βb,u = β¯ ± δ,

(3.14)

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F. Morichetti

where β¯ =

κxx + κyy βx + βy + 2 2

(3.15)

is the average phase constant of the two modes and  δ=

β 2 + κp2 . 4

(3.16)

In (3.14) the subscript u in the phase constants of the hybrid bend mode refers to the direction of the major transversal component of the electric field. For waveguides with symmetry also in the y-direction, κp vanishes [10] and the phase constants of the bend modes reduce to βb,x = βx + κxx and βb,y = βy + κyy [17]. From (3.6) and (3.7) it is possible to derive the transmission matrix Tb that relates the output to the input amplitudes of the TE and TM modes of a bend of length L [9, 19] ¯

Tb = e−jβL



 cos φ − jR sin φ −jS sin φ . −jS sin φ cos φ + jR sin φ

(3.17)

The power exchange, governed by the parameter φ = δL, is spatially periodic with period given by the beat length π . 2δ

(3.18)

|Tb,21 |2 = S2 sin2 φ, |Tb,11 |2 + |Tb,21 |2

(3.19)

LB = The PR efficiency, which is defined as [6] K=

reaches maxima equal to S2 = 1 − R2 = κp2 /δ 2 . When β = 0 (S = 1), complete PR takes place after a length LB /2. If | β| increases, LB decreases and the power exchange occurs over a shorter length, but a complete PR never occurs.

3.2.2 A Numerical Case Analysis A numerical case analysis is discussed in this section with the aim of pointing out the role of the main geometric and optical waveguide parameters in the PR process along a bend. Simulations were performed by means of a full-vectorial beam propagation method tool making use of the conformal transformation method [20], that is, the bent waveguide is transformed into an equivalent straight waveguide with an index profile described by (3.9). The field at the input of the bend is assumed to have TE polarization.

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Polarization Rotation in Ring Resonators

67

Figure 3.2a shows the cross section of a rib-shaped waveguide with a height h = 1.8 μm and a refractive index ng = 1.537 and ns = nc = 1.445 in the core and cladding regions, respectively. The slab thickness is t = 0.4 μm and the sidewalls are slanted by an angle θ . The birefringence B0 of the waveguide versus the waveguide width w is reported in Fig. 3.2b for three values of θ . In the width range shown in the figure, B0 increases almost linearly versus w and zero-birefringence condition ( β0 = 0) can be obtained by several combinations of θ and w.

(a)

1

(c)

θ

0.8

600 μm

B0

K

0.6

2 1 0 –1

x 10-4

0.4

(b)

0.2

200 μm

θ = 90° θ = 80° θ = 70°

2

400 μm

2.1

2.2

2.3

2.4

0 0

h = 0.4 μm t = 1.8 μm θ = 70°

ρ 5

10

15

L [mm]

w [μm] Fig. 3.2 (a) Cross section of a rib-shaped waveguide with slanted sidewalls; (b) simulated birefringence of the straight waveguide shown in (a) for different values of the angle θ; (c) PR efficiency for w = 2.2 μm (solid lines) and w = 2.11 μm (dashed lines) for three different values of the bending radius and θ = 70◦ [10]. Reprinted with permission. Copyright 2006 IEEE

3.2.2.1 Bending Radius Figure 3.2c shows in solid lines the PR efficiency in the zero-birefringence waveguide (w =2.2 μm and θ = 70◦ ) of Fig. 3.2b for several values of the bending radius. Bending loss is negligible down to a radius of 200 μm. At large bending radii (ρ > 600 μm), total PR occurs because the term β1 /ρ 2  1 of (3.13) is negligible and the conversion process is perfectly phase matched. At smaller bending radii, the rate of PR becomes higher, because of the 1/ρ dependence of κp [see (3.11)], but the efficiency drops down under the effect of the phase mismatch induced by the curvature. In such conditions total conversion can be achieved only for a suitable non-zero-birefringence waveguide, as shown by dashed lines of Fig. 3.2c, where a waveguide of reduced width w =2.11 μm and with B0 = −5.4 · 10−4 is considered. In this waveguide total conversion occurs only when ρ  200 μm, while at larger radii conversion decreases because the straight waveguide birefringence is not compensated by the curvature. The dependence of parameters β and κp on the bending radius is summarized in Fig. 3.3. Marks represent results derived from electromagnetic simulations, while

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F. Morichetti

dashed lines show the numerical fitting with relations (3.11) and (3.12). For these waveguides, the effect of the curvature on β cannot be neglected for bending radii smaller than 600 μm.

κp , Δβ [m–1]

600

400

κp

200

0

–200

Δβ 200

400

600

ρ [μm]

800

1000

Fig. 3.3 Values of κp and β evaluated from the curves of Fig. 3.2c for w = 2.2 μm (asterisks) and w = 2.11 μm (circles). Dashed lines indicate the numerical fitting with the expressions given by (3.11). In both cases κ0,xy  8.35·10−2 , while the parameter β1 = 8.57 · 10−6 m for w = 2.2 μm and β1 = 6.78 · 10−6 m for w = 2.11 μm [10]. Reprinted with permission. Copyright 2006 IEEE

3.2.2.2 Waveguide Index Contrast PR in bends depends also on the refractive index contrast n = (ng − nc )/nc of the waveguide. In Fig. 3.4 the coefficient κ0,xy of (3.11) is numerically evaluated for several rib-shaped waveguides with different n. In order to make the comparison

Fig. 3.4 Coupling coefficient κ0,xy versus the refractive index contrast n. The height of all waveguides is fixed to h = 0.68w [10]. Reprinted with permission. Copyright 2006 IEEE

3

Polarization Rotation in Ring Resonators

69

at equivalent propagation regimes, all the waveguides are single mode (just below the first higher order mode cutoff) and have the same dimensional ratios, as sketched in the inset of the figure. Results show that κ0,xy scales almost linearly with n, increasing by nearly three times as the index contrast moves from n = 5% to

n = 20%. Therefore, the higher the n, the higher the waveguide sensitivity to PR effects. It is easy to show that the magnitude of PR per radiant of curvature increases with the square of n. In fact, referring for simplicity to the synchronous coupling case [ β = 0 and δ = κp in (3.16)], from (3.19) PR efficiency can be written as     K = sin2 κp L = sin2 κ0,xy α , (3.20) with α the angle described by the bend. 3.2.2.3 Cross-Sectional Geometry Besides refractive index implications, polarization conversion in bent waveguides strongly depends on the shape of the waveguide. If the optical modes of the straight waveguide exhibit an even symmetry in the direction orthogonal to plane of the bend (y-direction), the coupling coefficients κxy and κyx of (3.6) and (3.7) vanish and no polarization conversion takes place [10]. In Fig. 3.5 the PR efficiency K of several types of single-mode waveguides is shown. All the considered waveguides have the same birefringence B0 < 10−4 , with the exception of waveguide (b) with B0 = 1.9 · 10−3 . Waveguides (a), (e), and (f) have all even y-symmetry and their PR efficiency is more than 40 dB below the efficiency of other waveguides. Waveguides with different upper and lower claddings, such as (b) and (c), and rib waveguides (d) do not strictly need slanted sidewalls to give rise to polarization conversion. Therefore, such index profiles 0

(d)

–10

(c)

K [dB]

–20

(b)

–30 –40

(e)

–50 –60 –70 –80

(a) 0

1000

2000

3000

(f) 4000

L [μm] Fig. 3.5 PR efficiency in bent waveguides with different cross section for ρ = 400 μm. Core index is 1.513, cladding index 1.4456 (in the case of waveguides (b) and (c) the upper cladding index is 1.2) [10]. Reprinted with permission. Copyright 2006 IEEE

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F. Morichetti

should be avoided if PR is not desired. Waveguide (b), obtained by substituting the upper cladding of the zero-birefringence channel square waveguide (a), has a reduced beat length LB as a consequence of its high birefringence B0 = 1.9 · 10−3 . It is worthwhile to note that waveguides (e) and (f), in spite of their complicated and curious cross sections, do not lead to polarization conversion.

3.3 Polarization Conversion in Ring Resonators In this section the effects of PR due to waveguide bending on the behavior of RRs are discussed. We start from the simple case of the RR phase-shifter, which represents the basic building block for many architectures, then several examples are reported to show the impact of the PR in RR-based devices. In all the devices considered in this section, PR is assumed to occur only in the RR’s bent waveguide, while no PR is considered in the coupling section between a RR and a bus waveguide, or between two RRs. Actually, coupling sections between bent waveguides or coupling sections made of waveguides with a different sidewall tilt could provide themselves an additional contribution to PR. These effects, not considered in the following for the sake of simplicity, can be modeled by using the CMT [21] and straightforwardly added to the PR given by the curvature of the RR’s waveguide.

3.3.1 Ring-Resonator Phase-Shifters As sketched in Fig. 3.6a, a RR phase-shifter consists of a RR coupled with a bus waveguide by means of a directional coupler. The fields in the RR are related to the fields in the bus waveguide by the transmission matrix Tc of the directional coupler

(a)

(b) OTE

OTE

E1 ITE ITM

OTM E1, TM

E1, TE

OTM

Tc, TM

Tc,TE

E2

Tb E2, TE ITE

E2, TM ITM

Fig. 3.6 The RR phase-shifter (a) and its double-ring equivalent circuit in presence of polarization coupling in the bent waveguide (b)

3

Polarization Rotation in Ring Resonators



I O



j TE = t TM



1 r

−r −1

71



E1 E2



TE = Tc TM

E1 E2

TE , TM

(3.21)

√ with -jt being the field coupling ratio of the coupler and r = 1 − t2 . In case of PR in the bent waveguide, the behavior of the RR phase-shifter can be modeled according to the equivalent circuit of Fig. 3.6b. The equivalent circuit consists of a double-ring structure, where the first coupler (Tc,TE ) and the third coupler (Tc,TM ) model the coupling between the RR and the bus waveguide. The central coupler of the equivalent circuit models the optical bend of the RR and it is associated with the transmission matrix Tb given by (3.17). Indeed, from a circuit point of view, the TE–TM coupling in the bend acts similar to the coupling occurring between the two guides of a directional coupler: In this analogy, the bend modes play the role of the even and odd proper modes of the directional coupler, whereas the TE and TM modes, which are coupled in the bend, can be associated with the coupled modes of the directional coupler. The transmission matrix of the double-ring structure of Fig. 3.6b can be simply derived by cascading the transmission matrices of the three couplers of the equivalent circuit. The transfer function, written in Z-transform notation, between the TE (TM) input field and the TE (TM) output field, defined as Hee = OTE /ITE (Hmm = OTM /ITM ), is thus given by Hee = mm

r˜z−2 − (1 + r2 )˜z−1 cos (φ) ± jRt2 z˜−1 sin (φ) + r , 1 + r2 z˜−2 − 2r˜z−1 cos (φ)

(3.22)

where the ± sign, respectively, refers to Hee and Hmm . Similarly, for the crosspolarization TE–TM (Hem = OTE /ITM ) and TM–TE (Hme = OTM /ITE ) transfer function, one obtains jSt2 z˜−1 sin (φ) Hem = , 2 1 + r z˜−2 − 2r˜z−1 cos (φ) me

(3.23)

¯ r ), γ being the RR round-trip loss. Equations where z˜−1 = γ z−1 = γ exp ( − jβL (3.22) and (3.23) hold in the simplest case of polarization-independent directional coupler, Tc,TE = Tc,TM , and state that Hem always coincides with Hme , whereas Hee = Hmm only if β = 0 [i.e., R = 0 in (3.17)]. In this section, the parameters φ, ¯ and β have the same meaning as defined in Section 3.2. β, Owing to polarization coupling, the RR phase-shifter exhibits a second-order spectral response, with two poles in both transfer functions of (3.22) and (3.23). The two poles, common to Hee (Hmm ) and Hem (Hme ), split in complex conjugated positions zp = r exp ( ± jφ) and two zeros appear in Hee (Hmm ). The phase of the poles is always associated with the phase constant of the bend modes b,x (x, y) and b,y (x,y), given by (3.14), that are the real resonating modes of the cavity.

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F. Morichetti

In the absence of polarization coupling (κp = 0), the transfer functions Hme and Hem vanish, whereas Hee and Hmm reduce to the well-known expression for a RR phase-shifter [22], r − z˜ −1 e∓j

βLr 2

, H ee =

βLr 1 − r˜z−1 e∓j 2 mm

(3.24)

with only one pole zp = rγ exp ( ∓ j βLr /2) internal to the unitary circle and one external zero zz = r−1 γ exp ( ∓ j βLr /2). For lossless rings, the two singularities lie in reciprocally conjugated positions. 3.3.1.1 Phase-Shifters with β = 0 Figure 3.7a shows the loci of the zeros and the poles of a RR phase-shifter with neither birefringence (R = 0) nor loss (γ = 1) as a function of φ = κp Lr . As κp increases from κp = 0, where a pole-zero cancellation takes place at z = r, the two zeros of Hee (Hmm ) approach moving along the real axis, overlap at z = 1, then become complex conjugates on the unit circle. The critical coupling φcr,0 that brings the two zeros onto the unit circle and gives total conversion is 2r . 1 + r2

cos φcr,0 =

(3.25)

The spectral response over a free spectral range (FSR) of a phase-shifter with

β = 0, t2 = 0.5, and γ = 1 is shown in Fig. 3.8, for values of φ equal to 0, 0.34 (φcr,0 ), and π/2, corresponding to a PR efficiency K per round trip equal to 0%, 11.1%, and 100%, respectively. For φ = φcr,0 a wide transmission zero appears in 1

0

2

φ = 0

le s po

0

φ cr

2

0

2 po

– 0.5 ro ze

–1

2

5 Imaginary Part

φ cr,0

– 0.5

(b) φ = π/ 2

0. 5 Imaginary Part

1

(a)

φ = π/ 2

les r ze

os

φ = 0 φ cr,0

s

0 .5 1 Real Pa rt

–1 1 .5

0

0 .5 1 Real Pa rt

1

.5

Fig. 3.7 Loci of the zeros and poles of (a) a lossless and (b) a lossy (γ = 0.8) RR phase-shifter with t2 = 0.5 and β = 0 as a function of φ [10]. Reprinted with permission. Copyright 2006 IEEE

3

Polarization Rotation in Ring Resonators

73

Hee (Hmm ), due to the two overlapped zeros, while in Hem (Hme ) the two poles give rise to a second-order maximally flat response, centered at the resonant frequency of the TE (TM) mode (marked onto the frequency axis). For higher values of φ total conversion always occurs, but at two distinct frequencies within the FSR. When φ = π/2 (K = 1), all zeros and poles lie on the imaginary axis and the transfer functions become symmetrical with half the FSR of the original structure. In this condition, the equivalent double-ring structure of Fig. 3.6b degenerates to a single RR with a length 2Lr for unity coupling ratio of the central coupler. In a lossy RR phase-shifter the poles and the zeros move according to Fig. 3.7b, where parameters t2 = 0.5, γ = 0.8, and β = 0 are considered. Total conversion can still take place, provided that the condition r < γ is satisfied, and the critical coupling condition can be generalized as cos φcr =

r(1 + γ 2 ) . γ (1 + r2 )

(3.26)

However, in the presence of loss, critical coupling is generated by only a single zero located on the unit circle. As a consequence, the notch in Hee (Hmm ) response is narrower than in the lossless ring of Fig. 3.8 and Hem (Hme ) does not exhibit a maximally flat response at unit conversion. If r > γ , both zeros always lie inside the unit circle and complete conversion is never achieved. 5

φ=0

0 |Hee|2, |Hem|2 [dB]

Fig. 3.8 |Hee |2 = |Hmm |2 (dashed curve) and |Hem |2 = |Hem |2 (solid curve) of lossless phase-shifters with t2 = 0.5 and β = 0 for three values of φ (φcr,0 = 0.34) [10]. Reprinted with permission. Copyright 2006 IEEE

–5

φ = π/2

–10 –15

φ = φcr,0

–20 –25 –30 –0.5

TE (TM)

– 0.25

0 0.25 Frequency / FSR

0.5

3.3.1.2 Phase-Shifters with β = 0 When β = 0, complete PR can take place only in the presence of loss (γ < 1). Let us consider, for example, a RR with βLr = π/2. As shown in Fig. 3.9, for

74

F. Morichetti

κp = 0 the two poles lie in z = γ r exp ( ± jπ/4). The Hee spectral response has one zero external to the unitary circle and the other zero, in reciprocally conjugated position, cancels with the pole in the negative imaginary plane. As κp increases, the two zeros move describing a spiral in the complex plane (the external one anticlockwise, the internal one clockwise) and the external zero crosses several times the unit circle, this giving unitary conversion for several values of κp . In Fig. 3.9 the positions of the singularities for the two smaller values of φ providing critical coupling (φcr,1 and φcr,2 ) are marked. φ = π/2

1

φ cr,1 φ =0

Imaginary Part

0.5

φ cr,2 0

–0.5

–1 –1

– 0.5

0

0.5

1

Real Part

Fig. 3.9 Loci of the zeros (dotted lines) and poles (solid lines) of a lossy (γ = 0.8) RR phaseshifter with t2 = 0.5 and βLr = π/2 as a function of φ

In the lossless case (γ = 1), as κp increases, the two zeros of Hee move similar to the lossy case of Fig. 3.9, describing a spiral in the complex plane, asymptotically approaching the unit circle, but never crossing it. Therefore, critical coupling and total conversion never occur at any frequency. Figure 3.10 shows the measured frequency-domain response of a RR phaseshifter with β = 450 m−1 , γ = 0.84, and κp = 380 m−1 [9]. The device, with t2 = 0.38, FSR = 100 GHz, and ρ = 274 μm, was fabricated by using silicon oxynitride technology with an index contrast n = 6% [23]. The input mode is quasi-TE polarized. While far from resonances a weak conversion, lower than 22.5 dB, occurs, at resonance PR is strongly enhanced and a considerable amount of TM polarization is produced. Marks indicate the position of the TE and TM resonances in the absence of polarization conversion. The deeper notch occurs at the resonance of the hybrid mode closer to the resonant frequency of the input mode, so that, in the case of a TM-polarized input field, the deeper notch would be the left one.

3.4 Impact of Polarization Conversion in RR-Based Devices Undesired polarization conversion in bent waveguides can cause severe detrimental effects in optical devices based on RRs. In this section, two architectures commonly

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Polarization Rotation in Ring Resonators

75

Fig. 3.10 Cavity-enhanced PR in a RR phase-shifter: measured (solid curve) and simulated (dashed curve) intensity TE–TE transmission (Hee ) and TE–TM transmission Hem in case of TE polarized input field [9]. Reprinted with permission. Copyright 2004 The Optical Society of America

employed to implement filtering operations in wavelength division multiplexing (WDM) optical transmission systems are discussed: an optical interleaver consisting of a ring-loaded Mach–Zehnder Interferometer (MZI) and a channel selector made of a chain of direct coupled RRs. The input polarization state is assumed TE in all the examples discussed in this section. Moreover, the discussion is limited to the case of lossless RRs (γ = 1), even though the extension to lossy rings is straightforward.

3.4.1 Ring-Loaded Mach–Zehnder Interferometers Figure 3.11a shows a balanced MZI with both arms loaded with a RR phase-shifter, in the following referred to as 2R-MZI. The RRs are employed to introduce a phase-shift with a nonlinear characteristic versus frequency, resulting in a spectral response that outperforms the sinusoidal spectral response of a conventional unbalanced MZI. Figure 3.11b shows in solid line the spectral response at the cross port of the 2R-MZI in the absence of polarization conversion in the RR (the spectral response at the bar port is identical but shifted by FSR/2). A maximum flatness second-order spectral response is obtained when the power coupling ratio of the two-directional couplers of the MZI is 0.5 and that of the RR’s couplers is 0.824 [22]. The FSR of the device coincides with the FSR of the RRs. The normalized bandwidth B at 20 dB power attenuation is 0.19 FSR and the passband, defined at 1 dB power attenuation, is 0.39 FSR wide. In the presence of PR in the two RRs, the symmetry at the output ports is maintained, so that only one of the two ports needs to be investigated. Figure 3.11b shows in dashed lines the effect of an increasing PR on the overall output power, given by the sum of TE and TM output power, in the simplest case β = 0. For

76

F. Morichetti

(a)

(b)

0

κp

Pin

Lr

TE

Pbar TE+TM

Lr

Pcross

Transmission [dB]

–10 –20

Δβ L Δβ L

π

0.25

0.5

–30 –40

Pcross TE+TM

–50 –0.5

–0.25

0

Frequency / FSR Fig. 3.11 (a) Schematic of a balanced MZI loaded with two RR phase-shifters (2R-MZI); (b) spectral response of the 2R-MZI when βLr = 0 and βLr = 0.6π for increasingly higher values of κp Lr (0, 0.12π , and 0.25π )

κp Lr = 0.12π , corresponding to a round-trip conversion efficiency K = 13.5%, B reduces to 0.1 FSR. The extinction ratio, defined as the ratio between the minimum (in-band) and the maximum (off-band) attenuation of the device, drops to less than 15 dB for κp Lr = 0.25π (K = 50%). When PR in the bend is not phase matched [ β = 0 in (3.16)], the spectral response of the 2R-MZI is asymmetrically distorted and the bandwidth B narrows more significantly, as shown by dotted lines of Fig 3.11b for βLr = 0.6π . This result is apparently counter-intuitive, because the magnitude of PR decreases with

β, as discussed in Section 3.2.1. Actually, the spectral distortion mainly originates from the relative shift of the TE and TM resonant frequencies of the two RRs, whose frequency spacing increases linearly with β. When δLr = φ = Mπ , the effective round-trip conversion efficiency is zero and the only effect on the 2RMZI spectrum is a frequency shift by FSR/2, causing the two ports of the device interchange each other. Figure 3.12 shows in a contour map the normalized bandwidth B/FSR at 20 dB power attenuation of the 2R-MZI versus κp Lr and βLr . The 2R-MZI bandwidth is maximum, B/FSR = 0.19, in the lighter gray points, where φ = δLr = Mπ and the round-trip conversion efficiency K = 0. As much as δLr differs from Mπ , the bandwidth is increasingly limited by the PR. Since the conversion efficiency decreases with | β|, the impairment reduces from the bottom to the top of the map. Asterisks and squares mark the points corresponding to the spectral responses of Fig. 3.11b for | β|Lr = 0 and | β|Lr = 0.6π . To give a practical example, let us consider a 2R-MZI realized with the waveguide of Fig. 3.2 with θ = 70◦ . The RR bending radius is assumed ρ = 1200 μm (corresponding to a FSR of about 25 GHz) and κp = 70 m−1 . Since κp Lr = 0.17π , from the map of Fig. 3.12 the lowest optimal value of | β|Lr /2π resulting in a zero round-trip PR [φLr = π in (3.19)] is 0.98 (marked in the map as P1 ).

Polarization Rotation in Ring Resonators

Fig. 3.12 Contour map of the normalized bandwidth at 20 dB power attenuation B/FSR of the ring-loaded 2R-MZI interferometer as a function of the round-trip coupling ratio κp Lr and the round-trip phase mismatch | β|Lr . The points corresponding to the spectral responses shown in Fig. 3.11b are marked with asterisks (| β|Lr = 0) and squares (| β|Lr = 0.6π ). P1 and P2 mark two possible working points when the waveguide of Fig. 3.2 is employed to realize the 2R-MZI

77

3

B / FSR

0.19

2.5

0.1 |Δβ | L r / 2π

3

0.05

2

P2 1.5

P1

1 0.5 0

0

0.2

0.4

0.6

κ p Lr /π

0.8

1

The required birefringence B0 = ±1.9 · 10−4 can be achieved, for instance, by either widening the waveguide of Fig. 3.2 up to w = 2.5 μm or by narrowing it down to w = 1.9 μm. In order to keep the bandwidth B larger than 0.19 FSR, B0 must be controlled better than δB0 = ±3 · 10−5 , that is, the waveguide width tolerance is δw = ±35 nm. However, the sensitivity to birefringence fluctuations decreases from the bottom to the top of the map, so that for any value of | β|Lr /2π exceeding 1.75 (point P2 ) the device bandwidth is preserved. Since this condition requires a high birefringence waveguide (|B0 | > 3.4 · 10−4 ), a large waveguide widening could result in the occurrence of higher order modes, while a large waveguide narrowing could cause higher bending loss in the RR. This implies that an accurate design of the waveguide cross section is required.

3.4.2 Direct Coupled Ring-Resonator Filters As a second RR-based architecture, let us consider the structure of Fig. 3.13a, consisting of N direct coupled RRs. The spectral characteristics of direct coupled ring-resonator filters are suitable for managing channel selection operation on WDM channel grids [24, 25]. Figure 3.13c shows the frequency-domain response at the output ports of a coupled RR filter made of three cascaded rings (N = 3). The coupling ratio of the directional couplers are t12 = t42 = 0.61 and t22 = t32 = 0.13. In the absence of PR (solid line), at the through port the bandwidth at 20 dB attenuation is 0.1 FSR. Dashed lines show the effects of PR in the case of zero-birefringence waveguides ( β = 0). The return loss of the filter, given by the power level at the through port, rapidly degrades with κp , and B < 0.01 FSR for κp Lr = 0.1π . At the drop port, the

78

F. Morichetti (a)

(b)

through

t

drop in

t t

t

t

Transmission [dB]

0

drop port –10

–20

through port

–30

–40 –0.3

–0.2

–0.1

0

0.1

0.2

0.3

Frequency / FSR

Fig. 3.13 (a) Schematic of a direct coupled RR filter; (b) spectral response of a filter with N = 3 in the case κp = 0 and β = 0 (solid lines), κp Lr = 0.1π and β = 0 (dashed lines), and κp Lr = 0.4π and βLr = 4π (dotted lines) [10]. Reprinted with permission. Copyright 2006 IEEE

bandwidth at 1 dB attenuation reduces from nearly 0.15 FSR to 0.07 FSR. For the same value of κp Lr = 0.1π , a waveguide with birefringence of about βLr = 4π would not introduce any degradation in the spectral response of the filter. Dotted curves of Fig. 3.13c show that, for the case βLr = 4π , the spectral response at both the through and drop ports is almost undistorted up to κp Lr = 0.4π .

3.5 Polarization Converters Based on Ring Resonators As discussed in Section 3.4, in most applications PR in bent waveguides is detrimental and methods to mitigate its effects are required. On the other hand, PR in optical resonators can also be exploited to realize efficient polarization rotators [26, 27]. In this section basic design rules for the realization of polarization converters based on RRs are provided. For the sake of simplicity, with respect to the model reported in Section 3.2.1, here we assume that the bend eigenmodes b,u can be expressed as a superposition of the fundamental modes u of the straight waveguide only. This simplified approach does not lack generality and can be extended to the model described in Section 3.2.1. Furthermore, in order to maximize the PR efficiency, all the presented devices are assumed to fulfill the phase-matching condition ( β = 0). Without loss of generality, we assume as an input field the x-polarized mode x of the straight waveguide.

3.5.1 Model Based on the Bend’s Eigenmodes In Section 3.2.1 PR in a bend is modeled by using the transmission matrix Tb given by (3.17), describing the coupling of the modes u (u = x,y) of the straight waveguide along the bend. Alternatively, propagation in the bend can be described

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by referring directly to the eigenmodes b,u of the bend. The propagation constant βb,u of the bend modes is related to the eigenvalues λu = exp ( − jβb,u L) of matrix Tb , where βb,u = β ± δ (u = x,y), β being the average phase constant of the TE (x ) and TM (y ) fundamental modes of the straight waveguide (see Section 3.2.1). The field distribution of the bend eigenmodes can be expressed as a superposition of the straight waveguide modes, where the weights to be used in the linear combination are the eigenvectors of the matrix Tb . By following conventional rules for matrix diagonalization, matrix Tb can be written as a cascaded product Tb = QQT of a diagonal matrix  = diag[λx ,λy ] and an orthogonal matrix Q = [Qx Qy ], where the superscript T indicates matrix transposition. The columns of Q are the eigenvectors Qx =

√1 2

  √ √1 + R , Qy = − 1−R

√1 2

 √ √1 − R 1+R

(3.27)

of Tb . The straight waveguide modes (L) = [x (L) y (L)]T at the output of a bend with a length L can be related to the input modes (0) = [x (0) y (0)]T as (L) = Tb(0) =  = Q QT Tb Q QT (0) = = QQT (0).

(3.28)

Left multiplying both terms of (3.28) by QT , QT (L) = QT (0),

(3.29)

one can observe that the field QT (L) at the output of the bend is related to the input field QT (0) through the diagonal matrix Λ. The absence of off-diagonal terms in Λ implies that the fields QT  are decoupled inside the system described by Tb , i.e., they coincide with the proper modes of the bend b = QT . The hybridness Hu of the bend modes [12], defined as the ratio between the minor and the major transverse components of the electric field, that is, Ey /Ex (Ex /Ey ) for the mode b,x (b,y ), is directly derived from the expression of the eigenvectors (3.27) as √ 1−R . Hu = ± √ 1+R

(3.30)

The phase matching condition β = 0, leading to R = 0 and δ = κp in (3.16)– (3.19), corresponds to the case |Hu | = 1 of maximally hybrid bend modes, thus confirming that the higher the hybridness of the bend modes with respect to the straight waveguide modes, the more effective the PR [13]. Conversely, Hu vanishes in the absence of polarization coupling, that is, when κp = 0 and R = 1: In this case, the bend modes and the modes of the straight waveguide are strictly coincident.

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F. Morichetti

3.5.2 Parallel-Coupled Resonators In Section 3.3, it was demonstrated that PR alters the spectral response of a RR coupled with a bus waveguide from that of a first-order phase shifter to that of a second-order polarization filter. Such an equivalence is here generalized for an arbitrary number N of RRs side-coupled to a common bus waveguide, as in the structure shown in Fig. 3.14a [27]. For the sake of simplicity, all the N rings are assumed identical and lossless. PR occurring in the bent waveguide of each RR gives a round-trip conversion efficiency K = S2 sin2 (φ), according to (3.19), while the bus line is assumed to be free of PR. Fig. 3.14 (a) Polarization converter consisting of N identical RRs coupled to a common bus waveguide. Polarization coupling is assumed to take place only in the bent waveguides of the rings. (b) The equivalent circuit of the structure (a) with PR occurring only between RRi ,TE and RRi ,TM

K

in RR1

(a)

K RR2 RRN

K out

OTM

ITM RR1,TM

K

(b) ITE

RR2,TM

RRN,TM

K

K RR1,TE

RR2,TE

RRN,TE

OTE

The device of 3.14(a) is equivalent to the structure shown in Fig. 3.14b, where the polarization coupling K between the x and y modes is replaced by a directional coupler placed between resonators RRi ,TE and RRi ,TM . The portion of field u entering each RR can be decomposed in the superposition of the two bend modes N b,u . Due to their mutual orthogonality, the overall transfer function Hb,u of each mode b,u is simply given by the Nth power of the lossless RR phase shifter transfer function, that is, 

N Hb,u

r − exp ( − jβb,u Lr ) = 1 − r exp ( − jβb,u Lr )

N ,

(3.31)

where βb,x = β +κ, βb,y = β −κ [see (3.14)], and t2 = 1−r2 is the power-coupling coefficient between the RRs and the bus waveguide. Strictly speaking (3.31) holds only if the birefringence of the bus waveguide is zero. Total conversion from x (TE) to y (TM) is obtained when the device of Fig. 3.14 introduces a π phase-shift between the proper modes of the bend, that is, N N − ∠Hb,y = (2M + 1)π , ∠Hb,x

(3.32)

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with M an integer. At the resonant frequency of the straight waveguide mode x (y ), βLr is an integer number of 2π and relation (3.32) becomes

(1 − r2 ) sin (κp Lr ) Mπ π + . = tan 2N N (1 + r2 ) cos (κp Lr ) − 2r

(3.33)

Equation (3.33) provides a useful relation to evaluate κp required to achieve total N PR, as a function of the number N of RRs. Once Hb,u is known, the TE–TE (Hee ) and TE–TM (Hem ) transfer functions of the device can be directly obtained as QTu Hb Qu , N N where Hb = diag(Hb,x , Hb,y ). The spectral response of the device of Fig. 3.14 is shown in Fig. 3.15 for t2 = 0.4 and M = 0. When N increases, no significant differences are visible in the transfer function and the second-order spectral response of the RR phase-shifter, shown for instance in Fig. 3.8, is almost preserved. The main benefit carried by the increasing number of resonators is that the value of κp , and hence the required round-trip PR K, necessary to obtain total rotation at the RR resonance, significantly reduces. Such property can be exploited in all circumstances where it is more practical to increase N than κp . In fact, to increase κp , either small bending radii are needed, for which radiation losses could become critical, or waveguides with strongly y-asymmetrical refractive index profile (see Section 3.2), typically requiring non-conventional technological processes. On the other hand, the main drawback of using several RRs is the need for making all the RRs resonate at the same frequency; this can be accomplished by locally heating the bent waveguide of the RR by means of a microheater placed on the top of the waveguide [28]. 0 –5

Powerr [dB]

–10 –15 –20

Hem

–25 –30

N=1 N =1 N=2 N =2

–35

t2 – 40 –1

Hee

= 0.4 – 0.5

=3 N N=3

0

0.5

1

frequency / FSR Fig. 3.15 Frequency-domain response of the polarization converter of Fig. 3.14 for N = {1, 2, 3}. The corresponding values of φ = κp Lr and round-trip conversion efficiency K are respectively, φ = {0.08π , 0.033π , 0.021π } and K = {6.2%, 1.1%, 0.4%} [27]. Reprinted with permission. Copyright 2006 IEEE

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F. Morichetti

3.5.3 Direct Coupled Resonators The spectral characteristics of periodic polarization rotators can be improved by using coupled resonator optical waveguides (CROWs) made of N direct coupled RRs [29, 30]. As an example of a CROW polarization rotator, let us consider the device of Fig. 3.16a, where only the last RR of the chain induces PR. In practice, to inhibit polarization coupling in the first N-1 rings, a suitable birefringence can be introduced by slightly narrowing or widening the waveguides, as explained in Section 3.2.1. Alternatively, PR in the Nth ring can be reduced or strengthened by acting on the bending radius ρ of every RR, according to (3.11). The equivalent circuit of the CROW of Fig. 3.16a is the four-port architecture shown in Fig. 3.16b, which has the structure of a filter with 2N direct coupled resonators. The coupling coefficient of the central directional coupler of the equivalent filter, between rings RRN,TE and RRN,TM , is given by the PR efficiency K induced by the last RR of the CROW. The TE and the TM modes, which under the hypothesis β = 0 have the same propagation constant, resonate in all the RRs of the chain at the same frequencies. Yet, in the equivalent circuit, the TE mode is only in the first N rings, while the TM mode is only in the last N rings. If all the N directional couplers are polarization independent, the equivalent filter has a symmetrical topology and its spectral response can be tailored by means of well-known synthesis techniques [31].

Fig. 3.16 Polarization converter consisting of N direct coupled RRs (a) and its equivalent circuit when polarization coupling occurs only in the last ring RRN (b)

out

(a ) RR1

in

RRN-1

K RRN

OTM

OTE

(b)

RR1,TE

ITE

RRN,TE

RRN,TM

RR1,TM

ITM

Figure 3.17 shows the TE–TM spectral response Hem of the CROW polarization rotator of Fig. 3.16 for an increasing number N of resonators. The PR induced by the last RR is fixed to κp Lr = 0.08π (K = 6.2%). The coupling coefficients ti2 between RRi-1 and RRi , which are reported in Table 3.1, define a maximally flat response with B/FSR  0.15, where B is the 3 dB bandwidth of the PR. When N increases the conversion band becomes extremely flat with sharp transitions at the edges. The architecture considered in the example can reach extinction ratios of nearly 50 dB

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0 N=1

Hem

–10

N=2

Power [dB]

N=3

–20 –30 0

–40

-40

–50 –60 –1

Hee

-20

-60 -0.1

–0.5

0

0

0.5

0.1

1

frequency / FSR Fig. 3.17 Spectral response Hem of the coupled-RR polarization converter of Fig. 3.16 for an increasing number of resonators N = {1, 2, 3}. For clarity the TE–TE spectral response Hee is shown only in the inset, around the central frequency where total polarization conversion occurs. The coupling coefficients ti2 between RRi−1 and RRi are reported in Table 3.1 [27]. Reprinted with permission. Copyright 2006 IEEE

Table 3.1 Power-coupling coefficients between the RRs of the CROWs considered in Fig. 3.17 [27]. Reprinted with permission. Copyright 2006 IEEE N

t12

t22

t32

1 2 3

0.4 0.72 0.88

− 0.14 0.29

− − 0.09

by cascading only two resonators. The high extinction ratio comes from the fact that propagation in a CROW is allowed only at frequencies nearby ring resonances [30], whereas, away from resonance, an input field cannot propagate through the structure and does not experience any PR. To give an idea of the feasibility of the proposed polarization rotators, assuming FSR = 100 GHz (bending radius nearly 300 μm for a glass waveguide), the highest value of κp required by the structures presented in this section is about 130 m−1 (K = 6.2%) and can be significantly reduced by using a parallel-coupled RR architecture. As discussed in detail in Section 3.2, such values of κp can be easily obtained by suitably designing the cross-sectional geometry of conventional integrated optic waveguides, without the need for extremely small bending radii.

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3.6 Concluding Remarks The analysis reported in this chapter reveals that PR in a bent waveguide is strictly related to the characteristics of the straight waveguide. Waveguides with an asymmetric cross section in the direction orthogonal to the plane of the bend, such as rib waveguides or uncovered waveguides, are expected to suffer from PR effects much more than symmetrical buried waveguides. The strength of PR, expressed by the coupling coefficient κp , linearly increases with the waveguide refractive index contrast: this implies that PR has to be taken into strict consideration when high-index contrast technologies are employed. The efficiency of PR is mainly limited by the birefringence of the straight waveguide, but, in the case of tightly bent waveguides, the curvature itself contributes significantly to the phase matching of the process and cannot be neglected. In the case of RRs, polarization conversion is strongly enhanced and can severely affect the spectral response of optical devices utilizing RRs, even at low values of κp . Its impact can be mitigated by either reducing the parameter κp or by increasing waveguide birefringence. The first solution, achieved by using waveguides with properly symmetric cross sections, has the advantage of canceling PR effects in zero-birefringence waveguides. In the second approach, the exploitation of waveguide birefringence to reduce PR has to be combined, in many applications, with the need for polarization-insensitive devices. For instance, it is possible to use birefringent waveguides to realize RRs that are nearly polarization insensitive only within a spectral range of interest: This happens, for instance, when the hybrid modes of the bend resonate at the same frequency, but at different orders. Cavity-enhanced PR in RRs can also be exploited for the realization of spectrally periodic polarization rotators capable of managing the polarization state of optical channels in WDM optical systems. For instance, by adjusting the resonances and FSR of these devices with respect to the WDM channel grid, it could be possible to rotate the polarization state of subsets of selected channels, in order to realize polarization (de)interleavers and polarization splitters/combiners to be used in systems based on polarization diversity transmission.

References 1 Fukazawa, T., Ohno, F., et al. Very compact arrayed-waveguide-grating demultiplexer using Si photonic wire waveguides. Jpn. J. Appl. Phys. 43, 673-675 (2004) 2 Bogaerts, W., Baets, et al. Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology. J. Lightwave Technol. 23, 401–412 (2005) 3 Tsuchizawa, T., Yamada, K., et al. Microphotonics devices based on silicon microfabrication technology. IEEE J. Sel. Top. Quantum Electron. 11, 232–240 (2005) 4 Xia, F., Sekaric, L., et al. Ultracompact optical buffers on a silicon chip. Nature Photon. 1, 65–71 (2006) 5 Xu, Q., Fattal, D., et al. Silicon microring resonators with 1.5-μm radius. Opt. Express 16, 4309–4315 (2008) 6 van Dam, C., Spiekman, L.H., et al. Novel compact polarization converters based on ultra short bends. IEEE Photon. Technol. Lett. 8, 1346–1348 (1996).

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7 Sakai, A., Fukazawa, T., et al. Estimation of polarization crosstalk at a micro-bend in Si-photonic wire waveguide. J. Lightwave Technol. 22, 520–525 (2004) 8 Obayya, S.S.A., Rahman, B.M.A., et al. Beam propagation modeling of polarization rotation in deeply etched semiconductor bent waveguides. IEEE Photon. Technol. Lett. 13, 681–683 (2001) 9 Melloni, A, Morichetti, F., et al. Polarization conversion in ring resonator phase shifters. Opt. Lett. 29, 2785–2787 (2004) 10 Morichetti, F., Melloni, A., et al. Effects of polarization rotation in optical ring-resonator-based devices. J. Lightwave Technol. 24, 573–585 (2006) 11 Marcatili, E.A.J. Bends in optical dielectric guides. Bell Syst. Tech. J. 48, 2103–2132 (1969) 12 Obayya, S.S.A., Somasiri, N., et al. Full vectorial finite element modeling of novel polarization rotators. Opt. Quantum Electron. 35, 297–312 (2003) 13 Little B.E., Chu, S.T. Theory of polarization rotation and conversion in vertically coupled microresonators. IEEE Photon. Technol. Lett. 12, 401–403 (2000) 14 Lui, W.W., Hirono, T., et al. Polarization rotation in semiconductor bending waveguides: a coupled mode theory formulation. J. Lightwave Technol. 16, 929–936 (1998) 15 Yeung, C., Rozzi, T., et al. Crosspolarization coupling in curved dielectric rib waveguides. Inst. Elect. Eng. Proc. - Pt. J 135, 281–284 (1988) 16 Kwon, M.-S. Comments on ‘Effects of polarization rotation in optical ring-resonator-based devices’. J. Lightwave Technol. 24, 5119–5119 (2006) 17 Melloni, A., Carniel, F., et al. Determination of bend mode characteristic in dielectric waveguides. IEEE J. Lightwave Technol. 19, 571–577 (2001) 18 Lenz, D., Erni, D., et al. Quasi-analytic formalism for mode characteristics in highly overmoded rectangular dielectric waveguide bends. J. Opt. Soc. Am. A 22, 1968–1975 (2005) 19 Cusmai, G., Morichetti, F., et al. Circuit-oriented modelling of ring-resonators. Opt. Quantum Electron. 37, 343–358 (2005) 20 Heiblum M., Harris, J.H. Analysis of curved optical waveguides by conformal transformation. IEEE J. Quantum Electron. 11, 75–83 (1975) 21 Stoffer, R., Hiremath, K.R., et al. Cylindrical integrated optical microresonators: modeling by 3-D vectorial coupled mode theory. Opt. Comm. 256, 46–67 (2005) 22 Madsen, C.K., Zhao, J.H. Optical filter design and analysis: a signal processing approach. John Wiley, New York (1999) 23 Melloni, A., Costa, R., et al. Ring-resonator filters in silicon oxynitride technology for dense wavelength-division multiplexing systems. Opt. Lett. 28, 1567–1569 (2003) 24 Little, B.E., Chu, S.T., et al. Microring resonator channel dropping filters. J. Lightw. Technol. 15, 998–1005 (1997) 25 Little, B.E., Chu, S.T., et al. Very high-order microring resonator filters for WDM applications. IEEE Photon. Technol. Lett. 16, 2263–2265, (2004) 26 Bianucci, P., Fietz, C.R., et al. Whispering gallery mode microresonators as polarization converters. Opt. Lett. 32, 2224–2226 (2007) 27 Morichetti, F., Melloni, A. Polarization converters based on ring-resonator phase-shifters. IEEE Photon. Technol. Lett. 18, 923–925 (2006) 28 Morichetti, F., Melloni, A., et al. Error-free continuously-tunable delay at 10 Gbit/s in a reconfigurable on-chip delay-line. Opt. Expr. 16, 8395–8405 (2008) 29 Yariv, A., Xu, Y., et al. Coupled-resonator optical waveguide: a proposal and analysis. Opt. Lett. 24, 711–713 (1999) 30 Melloni, A., Morichetti, et al. Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures. Opt. Quantum Electron. 35, 365–379 (2003) 31 Melloni A., Martinelli, M. Synthesis of direct-coupled-resonators bandpass filters for WDM systems. J. Lightwave Technol. 20, 296–303 (2002)

Chapter 4

Series-Coupled and Parallel-Coupled Add/Drop Filters and FSR Extension Yasuo Kokubun and Tomoyuki Kato

Abstract The shape of the filter response can be controlled by a combination of ring resonators, such as series coupling, parallel coupling, and cascade topology. In particular series coupling is effective to realize box-like spectrum responses, which are required for wavelength filtering in WDM systems for photonic networks. In this chapter, the transfer function of series-coupled ring resonators is derived using the matrix method, and the optimum conditions for coupling coefficients to realize the box-like filter response are described. In addition, the free spectral range (FSR) extension method using the Vernier effect in series-coupled ring resonators with dissimilar radii will be discussed. Last, the transfer function of parallel-coupled ring resonators is derived and an application to interleaving is introduced.

4.1 Filter Synthesis by Coupled and Cascaded Topologies For wavelength filters used in dense wavelength division multiplexing (DWDM) systems, the following requirements must be satisfied: (a) the width of the passband must be wide enough to cover the signal spectrum (typically 50–80 GHz), (b) the passband spectrum response should have a flat top in order to prevent the deformation and loss of the signal spectrum, (c) the transition from the passband to the rejection band should be narrow, i.e., the roll-off should be steep, (d) the cross talk in the rejection band must be sufficiently small, (e) the spectrum response must be temperature and polarization independent, and (f) the dispersion in the passband must be sufficiently small. However, it is difficult for a single microring resonator to satisfy all these requirements because its spectral response, expressed by a Lorentzian function, has a sharp peak and involves a large, wing-like cross talk in the rejection band. Thus Y. Kokubun (B) Graduate School of Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan e-mail: [email protected] I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_4,  C Springer Science+Business Media, LLC 2010

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Y. Kokubun and T. Kato

a filter response synthesis is needed. Fortunately, the desired filter response can be synthesized by the combination of coupled and cascaded topologies. There are three basic topologies for combining ring resonators, i.e., series coupling (also called direct coupling), parallel coupling, and cascade topology, as shown in Fig. 4.1. The advantages and drawbacks of these topologies are summarized in Table 4.1 [1, 2].

Fig. 4.1 Basic topologies of ring resonator combinations

Table 4.1 Topologies of ring filter circuit Topology

Advantages

Drawbacks

Cascade

Expansion of FSR Loss increase due to center wavelength Reduction of cross talk mismatch Series coupling Expansion of FSR Small fabrication tolerance Flat-top passband Parallel coupling Flat-top passband Small fabrication tolerance

4.2 Series-Coupled Ring Resonators Since the detailed analytical methods to derive the transfer matrix of series and parallel-coupled ring resonators are given in Chapter 2 , only a brief outline of the derivation process is given in this section.

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4.2.1 Transfer Function of Series-Coupled Ring Resonators When ring resonators are cascaded as shown in Fig. 4.1a, the overall spectrum response is expressed by the multiple product of the responses of individual resonators, either for the through port response or the drop port response depending on which port is connected to the input port of the following resonator. In the cascaded connection, each ring resonator has only one input port. Then the resonant condition of individual ring resonators does not depend on the connection topology. On the other hand, when resonators are directly coupled as shown in Fig. 4.1c and d, each resonator is coupled to two waveguides, i.e., the bus waveguide and/or the other adjacent resonator(s). When the resonator is coupled to more than one waveguide, the input fields interfere with each other and the resonant condition is changed. This complicates the calculation of the transfer characteristics of seriescoupled microresonator filters [3–6]. The transfer function of series-coupled ring resonators can be derived using the transfer matrix method [7–13] as shown below. The series-coupled ring resonator configuration consists of coupling regions between waveguides and recursive delay lines as shown in Fig. 4.2. If no loss occurs at the coupling regions and the waveguides are matched, i.e., the propagation constants of the coupled waveguides are equal, the field amplitudes at the output ports of the coupling region, shown in Fig. 4.3a, are expressed by 

E c1 E c2

 =

 √  √   Ec1 t −j κ √ √ · , Ec2 t −j κ

Fig. 4.2 Double (second-order) series-coupled ring resonator

Fig. 4.3 Components of ring resonator: (a) coupling region, (b) recursive delay line

(4.1)

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Y. Kokubun and T. Kato

where t is the power transmittance and κ is the power coupling efficiency of the coupler. If the coupler is lossless, κ = 1–t and the matrix is unitary. Equation (4.1) can be rewritten as a relation between (Ec2 ,E c2 ) and (Ec1 ,E c1 ) as 

Ec2 E c2



 =C·

 Ec1 , E c1

(4.2)

where  √  − t√ 1 · . C= √ −1 t 1−t j

(4.3)

The propagation in the ring resonator shown in Fig. 4.3b is expressed by 

Er2 E r1



 =R·

 E r2 , Er1

(4.4)

where   √ −1 jβl2 a2 ·e 0 R= √ −jβl1 , 0 a1 e

(4.5)

where β is the propagation constant and ai is the power transmittance through length li (i = 1 or 2) in the resonator. Note that R corresponds to clockwise propagation while R −1 corresponds to counter-clockwise propagation. Cascading (4.2), (4.3), (4.4), and (4.5) at each stage, the transfer matrix of a second-order series-coupled ring resonator is given by 

ED EA

 =

Cb2 · R−1 r2



 ET · Cr1 · Rr1 · Cb1 · . EI

(4.6)

From (4.6), the transfer function from input to through port ET /EI and the transfer function from input to drop port ED /EI are obtained as follows [14]: √ √ √ √ a tb e−2jβL − a (1 + tb ) tr e−jβL + tb ET , = √ EI atb e−2jβL − 2 atb tr e−jβL + 1

(4.7)

√ √ ED a(1 − tb ) 1 − tr e−jβL , = √ EI atb e−2jβL − 2 atb tr e−jβL + 1

(4.8)

where tr is the power transmittance of coupling region between ring resonators and tb is that between bus and ring waveguides. We assumed that Cb1 = Cb2 , i.e., tb = tb1 = tb2 , l11 = l12 = l21 = l22 = L/2, and a11 = a12 = a21 = a22 = a1/2 .

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Series-Coupled and Parallel-Coupled Add/Drop Filters and FSR Extension

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The power transmittance to the through port PT = |ET /EI |2 is given by 2 √ d+ kn − cos (βL) + d− (1 − d+ kn ) , PT = A √ 2 1 + A kd − cos (βL) 

(4.9)

where A=

4atb (1 − atb )2 (1 − tr ) 2

kn =

,

a (1 + tb )2 tr

(1 + a)2 tb (1 + atb )2 tr . kd = 4atb

(1 ± a) , d± = 4a

, (4.10)

On the other hand, the power transmittance to the drop port PD = |ED /EI |2 is given by PD =

1+A

√

p kd − cos (βL)

2 ,

(4.11)

where p=

a (1 − tb )2 (1 − atb )2

.

(4.12)

4.2.2 Optimum Condition When the coupling efficiencies in a series-coupled ring resonator are larger than the critical coupling [15], the peak in the drop port response and the notch in the through port response split into the number of ring resonators in the series-coupled ring resonator, regardless of whether the ring resonators are identical or not. In a double series-coupled microring resonator, when the ratio of coupling efficiencies κ r /κ b is large, the passband has two peaks (overcoupling), i.e., the passband involves a ripple. When the ratio of coupling efficiencies κ r /κ b is small, the passband has a single peak (undercoupling), although the peak transmittance decreases. Thus an optimization of coupling efficiencies is required for a box-like spectrum response. 4.2.2.1 Through Port Response From (4.9), the shape of the through port response at the resonant wavelength varies depending on the coupling efficiencies. At critical coupling presented below, (4.14), the transmittance at the resonant wavelength is reduced to almost zero (−∞ dB). We define this condition therefore as the optimum condition for the through port response. This condition is equivalent to that in which the numerator of (4.9) is equal to zero at the resonant wavelength. Let us define trt as the optimum tr for the through port under this condition. Then the relation between tb and tr is given by

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trt =

(1 + a)2 tb a (1 + tb )2

.

(4.13)

Then the optimum condition of coupling efficiency is expressed as κrt =

[a − (1 − κb )] · [1 − a (1 − κb )] a (2 − κb )2

,

(4.14)

where κ rt is the optimum κ r and κ = 1 − t. This condition is called critical coupling [15]. Condition (4.17) is plotted in Fig. 4.4. When the resonator involves loss, κ b has a threshold value for the optimum condition. Since 0 < κ rt < 1, κ b must be in the range 1 − a < κb < 1.

(4.15)

Fig. 4.4 Optimum coupling efficiencies for through port response

The above condition is equivalent to tb < a. Thus κ b > 1 − a, i.e., the power coupling efficiency between the bus and the resonator should be stronger than the loss in the resonator [14]. 4.2.2.2 Drop Port Response The optimum condition to flatten the drop port response is obtained by making the second-order derivative of (4.11) with respect to cos(βL) equal to zero at the resonant wavelength. Let us define trd as the optimum tr for the drop port under this condition. The analytical relationship between tb and trd is given by trd =

4atb (1 + atb )2

.

(4.16)

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From the latter, the optimum condition for the coupling efficiency is expressed as 

1 − a (1 − κb ) κrd = 1 + a (1 − κb )

2 ,

(4.17)

where κ rd is the optimum κ r for this condition. Condition (4.17) is plotted in Fig. 4.5. In the lossless case, this condition is identical to the optimum condition for the through port response. Fig. 4.5 Optimum coupling efficiencies for drop port response

When the coupling efficiency is optimized and there is no loss in the resonator, a deep notch in the through port response and a box-like drop port response are simultaneously realized. However, a real resonator involves propagation loss, and so optimum conditions of coupling efficiencies for the through port and drop port responses are different. Thus a lossy series-coupled microring resonator cannot simultaneously realize both a deep notch in the through port response and a box-like drop port response [14].

4.2.3 High-Order Series-Coupled Ring Resonators The transfer function of high-order series-coupled ring resonators can be derived in the same manner as (4.6) by cascading the coupling matrix C of (4.3) and the half-ring propagation matrix R of (4.5). Defining the matrix product by M, the total transfer matrix is expressed by 

ED EA



 =M·

 ET . EI

(4.18)

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Using the elements mij of M, the transfer functions from input to through port ET /EI and to drop port ED /EI are obtained as follows: ET m01 =− , EI m00 m00 m11 − m01 m10 ED = . EI m00

(4.19) (4.20)

On the other hand, when the coupling efficiencies are assumed to be weak, the through port transfer function TN (ω) is expressed by the following continued fraction [16]: TN (ω) = 1 −

κb jϕ + + 2 jϕ +

κb

κr1 κr2

jϕ + · · · +

,

(4.21)

κr(N - 1) κb jϕ + 2

where κ b is the coupling efficiency between the bus line and the resonator, κ ri is the coupling efficiency between the ith and (i + 1)th resonator, numbered from the input, φ = (ω − ωc )nL/c, ωc is the center angular frequency (resonant angular frequency), n is the equivalent (effective) index of waveguide, L is the circumference of the resonator, and c is the speed of light. On the other hand, by assuming lossless resonators, the drop port transfer function DN (ω) is expressed as |DN (ω)|2 = 1 − |TN (ω)|2 , κb DN (ω) =

 N−1 √ i=1

(4.22)

κri

Denominator of second term of TN (ω)

.

(4.23)

The transfer function expressed as a continued fraction is the same as that derived with the matrix formulation when the coupling efficiencies are weak and the resonators are loss-free. Explicitly, the transfer functions in the drop port of second- to fourth-order seriescoupled ring resonators are given by D2 (ω) =

D3 (ω) =

(jϕ)3

√ κb κr1 (jϕ)2 + κb jϕ + κb2 /4 + κr1

+ κb

(jϕ)2

,

κb κr1  2  , + κb /4 + 2κr1 · jϕ + κb κr1

(4.24)

(4.25)

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Series-Coupled and Parallel-Coupled Add/Drop Filters and FSR Extension

√ κb κr1 κr2   D4 (ω) =  . (jφ)4 + κb (jϕ)3 + κb2 /4 + 2κr1 + κr2 (jϕ)2 2 +κb (κr1 + κr2 ) jϕ + κb2 κr2 /4 + κr1

95

(4.26)

4.2.3.1 Butterworth Filter Response The spectral response of a Butterworth filter has no ripple; the passband is maximally flat. The transfer function of a nth order Butterworth band-pass filter is

PBn (ω) =



1

ω − ωr 1+A δω

2n ,

(4.27)

where A is the resonant intensity and ωr is the resonant angular frequency. 4.2.3.2 Chebyshev Filter Response The spectrum response of a Chebyshev filter involves ripples with equal peaks and valleys, and the roll-off from the passband to the stopband is steeper than that of the Butterworth filter. The transfer function of an nth order Chebyshev filter is obtained by PCn (ω) =

1 1 + ACn (ω − ωr )2n

,

(4.28)

where Cn (x) is the nth order Chebyshev polynomial, A is the resonant intensity, and ωr is the resonant angular frequency. Some low-order Chebyshev polynomials are given below: C2 (x) = 2x2 − 1 C3 (x) = 4x3 − 3x . C4 (x) = 8x4 − 8x2 + 1 C5 (x) = 16x5 − 20x3 + 5x C6 (x) = 32x6 − 48x4 + 18x2 − 1

(4.29)

4.2.3.3 Box-Like Filter Response by Butterworth Condition The coupling efficiencies should be controlled to optimize the spectrum shape. The optimum coupling efficiency conditions were derived for the all-pass configuration [15, 17, 18] and for the lossless series-coupled ring resonator [16]. To realize the Butterworth filter response, the spectrum response must obey (4.27). Thus, the required conditions for the coupling coefficients can be obtained

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by demanding that the denominators of (4.24), (4.25), and (4.26) reduce to the denominator of (4.27) and are found to be κb2 = 0.25κb2 , 4√ ( 2 − 1)κb2 = = 0.1036κb2 , 4

κb2 = 0.125κb2 8 √ . (4.30) (3 − 2 2)κb2 = = 0.0429κb2 4

κr1−2 =

κr1−3 =

κr1−4

κr2−4

Here κ ri-j is the coupling efficiency κ ri between (i − 1)th and ith ring for jth-order series coupling. Using the Butterworth condition, a box-like filter response can be designed. In Fig. 4.6 the theoretical spectrum responses of Butterworth filters with different orders can be compared. It can be seen that higher order Butterworth filters can realize a more box-like filter response, exhibiting flatter passband and steeper roll-off from passband to stopband. The advantage of the series-coupled microring filter is that the box-like response can be realized without increasing the insertion loss, in contrast to the Gaussian-like response of arrayed waveguide grating (AWG) filters. Fig. 4.6 Spectrum responses of Butterworth filters of different orders

Let us now define the shape factor (SF) as a measure of the box-like shape by [5]

SF =

−1 dB bandwidth . −10 dB bandwidth

(4.31)

Then the SF of the nth-order series-coupled microring resonator filter with optimum coupling condition is given by [14]  SFn =

 10−1 (1 − 10−0.1 ) 10−0.1 (1 − 10−1 )

1 2n

.

(4.32)

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The SFs of Butterworth filters from first to fourth order are summarized in Table 4.2.

Table 4.2 Comparison of SF of Butterworth filters of orders 1–4 Order of Butterworth filter

Shape factor (SF)

Single ring Double series coupling Triple series coupling Quadruple series coupling Gaussian (AWG)

0.17 0.41 0.55 0.68 0.32

Since the filter response of a single ring resonator is expressed by a Lorentzian function, the shape factor is as small as 0.17, and due to the wide wing of the Lorentzian function, the cross talk is not sufficiently small. On the other hand, the filter response of the quadruple series-coupled ring resonator filter is much more box-like than the other three, exhibiting an SF of 0.68, a fourfold improvement compared to the single ring. Figure 4.7 shows an example of a box-like spectrum response realized by the quadruple series-coupled microring resonator [1, 19]. In this 3D structure, shown in Fig. 4.7a, the bus waveguides are vertically separated at the crossing point of the cross-grid, and as a result the scattering loss at the crossing point of the bus line is eliminated [20]. A large shape factor of 0.66, close to the theoretical value of 0.68, was obtained.

Fig. 4.7 Box-like spectrum response realized with a stacked quadruple series-coupled microring resonator: (a) 3D structure, (b) measured drop port response

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4.3 Vernier Effect As mentioned above, the extended roll-off and the insufficient flatness of the passband of a Lorentzian characteristic is unsatisfactory for high-bit-rate transmission. In addition, the FSR is limited to approximately 20 nm, and the cross talk is insufficiently low. Higher order series-coupled microring filters consisting of rings with different radii can solve these problems by utilizing the Vernier effect [5, 16, 21] as shown in Fig. 4.8.

0

R2

R1

–10

Transmittance [dB]

Fig. 4.8 Principle of FSR extension by the Vernier effect in series-coupled ring resonators with dissimilar radii

–20 –30 –40

R1=40μm

–50

R2=30μm R1-R2 series coupling

1540

1550

1560

R 1 R2 1570

Wavelength λ[nm]

4.3.1 Design of FSR Extension Factor The center wavelength (resonant wavelength) and the FSR (in wavelength units) of a single ring resonator with radius R are, respectively, expressed as λ0 =

2π R · neff , N

FSR (wavelength units) =

λ20 , 2π R · bneff

(4.33) (4.34)

where neff is the effective index of the ring waveguide and N is the angular mode number. The total FSR of high-order series-coupled ring resonators consisting of two ring radii R1 and R2 (to be specific we choose R2 < R1 ) is expressed by [21] m1 FSR1 = m2 FSR2 = FSRtotal ,

(4.35)

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where m1 and m2 are relatively prime integers or coprimes. To maximize the extension of the FSR, the following relation must be satisfied: m2 = m1 − 1.

(4.36)

Then, by redefining m1 = M as the FSR extension factor [22], (4.35) can be rewritten as M · FSR1 = (M − 1) · FSR2 = FSRtotal .

(4.37)

Substituting (4.34) into (4.37), the following relation is obtained: R1 =

M R2 . M−1

(4.38)

In the actual design, the minimum of R2 is determined by the bending loss. Then, FSR2 is determined from (4.34). If the desired FSR is greater than FSR2 , then the FSR extension factor M should be designed as  M=

FSRdesired FSR2

 .

(4.39)

int

where [ ]int denotes the integer part of the argument. Once R2 and M are computed, the adjacent cross talk is uniquely determined for a given order of a series-coupled filter as described in Section 4.3.4.

4.3.2 Optimum Arrangement of High-Order Series-Coupled Microring Resonators for Cross Talk Reduction Although the Vernier effect is effective to extend the FSR of series-coupled ring resonators, the adjacent channel cross talk may increase when the FSR expansion factor M is large. In addition, the geometrical arrangement of the ring resonators with different radii is nontrivial and must be considered to reduce the cross talk when the order of series coupling is greater than three [21]. In this section, the optimum arrangement of series-coupled microrings with different radii is described, from the cross talk point of view, and an optimum combination of ring radii is obtained for quadruple series coupling. Additionally, a physical insight is given to the optimum arrangement [12]. In the following analysis, the combinations of two different ring radii, i.e., R1 and R2 , are assumed to be 100 μm and 95 μm, respectively. Furthermore, the coupling efficiencies are assumed to be optimized so as to obtain the Butterworth filter response. First, the spectrum response of a double series-coupled ring filter consisting of different radii R1 and R2 is shown in Fig. 4.9b. The cross talk at the center of the rejection band is −19.25 dB, and the adjacent channel cross talk is −4.79 dB. The cross talk in the rejection band is not decreased sufficiently. On the other hand, the

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Fig. 4.9 Drop port spectrum response of double series-coupled microrings with different radii: (a) definition of parameters, (b) drop port spectrum response

FSR is extended by a factor of 20, which can be calculated from (4.38). In the double series-coupled rings, there are only two possible geometric arrangements of two different ring radii R1 and R2 , i.e., R1 -R2 and R2 -R1 . The spectrum responses of these arrangements are identical, because only the common resonant wavelengths of rings R1 and R2 are transmitted while the other resonant wavelengths of the individual rings are suppressed. Second, the responses of a triple series-coupled ring filter consisting of ring radii R1 and R2 are shown in Fig. 4.10a, b. Figure 4.10a shows the filter response when the rings are symmetrically arranged as R1 -R2 -R1 . The cross talk at the center of the rejection band is −25.16 dB. Although the adjacent channel cross talk is not observed clearly because of the poor resolution of this figure, it actually reaches

Fig. 4.10 Drop port spectrum responses of triple series-coupled microrings for different arrangements of two radii R1 and R2 : (a) R1 -R2 -R1 , (b) R1 -R1 -R2

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0 dB, as shown by the dashed line in Fig. 4.12a. It is deduced from this figure that the Vernier effect is not effective for the symmetric triple series-coupled ring filter. Figure 4.10b shows the filter response of triple series-coupled rings arranged asymmetrically as R1 -R1 -R2 . The coupling efficiencies between bus and ring and between ring waveguides are the same with those in the former symmetric case. The cross talk at the center of the rejection band is −19.08 dB and the adjacent channel cross talk is −4.67 dB. Output is obtained only at the wavelength where the resonant wavelengths of individual ring resonators are equal and other resonant peaks are suppressed. However, the adjacent channel cross talk cannot be improved markedly compared to the double series-coupled rings; it is deduced therefore that the Vernier effect is not very effective in the triple series-coupled ring resonator even if the ring resonators are arranged asymmetrically. Finally, the spectrum responses of quadruple series-coupled rings consisting of different radii R1 and R2 are shown in Fig. 4.11a–c. Figure 4.11a shows the filter response when the rings are symmetrically arranged as R1 -R2 -R2 -R1 . The rejection

Fig. 4.11 Drop port spectrum responses of quadruple series-coupled microring for different arrangements of two radii R1 and R2 : (a) R1 -R2 -R2 -R1 , (b) R1 -R2 -R1 -R2 , (c) R1 -R1 -R2 -R2

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Fig. 4.12 Magnified filter response of adjacent channel for different arrangements: (a) R1 -R2 -R2 R1 , (b) R1 -R2 -R1 -R2 , (c) R1 -R1 -R2 -R2

band cross talk is −35.36 dB. Although the adjacent channel cross talk is not observed clearly because of poor resolution, it actually reaches 0 dB, as shown by the solid line in Fig. 4.12a, similar to the triple series-coupling case with the symmetric arrangement R1 -R2 -R1 . Figure 4.11b shows the filter response for another arrangement, R1 -R2 -R1 -R2 , which is antisymmetric against the center, as will be shown in Fig. 4.13. The coupling efficiencies were assumed to be the same as those for the former arrangement. The rejection band cross talk is −39.88 dB. In this case, the adjacent channel cross talk is also 0 dB, as shown by the solid line in 4.12b. Figure 4.11c shows the filter response of the asymmetric arrangement R1 -R1 -R2 -R2 . The coupling efficiencies between waveguides were again assumed to be the same with those in the previous case. The rejection band cross talk is markedly reduced to −52.11 dB, and the adjacent channel cross talk is also reduced to −19.52 dB. Thus, the filter response can be greatly improved by quadruple series coupling in an asymmetric arrangement of two different ring resonators.

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Fig. 4.13 Definition of subsets in the antisymmetric arrangement R1 -R2 -R1 -R2 of quadruple series coupling

4.3.3 Physical Insight into the Quadruple Arrangement Let us consider the filter response of the arrangement R1 -R2 -R1 of the triple seriescoupled ring resonator filter, shown in the inset of Fig. 4.10a. When the input wavelength is λ2 , which is the resonant wavelength of ring R2 and is slightly different from the resonant wavelength of ring R1 , ring R1 is off-resonance, and so the transmittance from the input bus waveguide to ring R2 is very small. This situation is very similar to a single ring resonator with ring radius R2 and with very small coupling efficiency between bus and ring. In a single ring with very small coupling efficiency to the bus, the filter response at the resonant wavelength reaches 0 dB and the bandwidth is very narrow, i.e., the Q factor is quite high. This is the case for the adjacent channel in Fig. 4.10a and can be observed in the magnified filter response of the adjacent wavelength channel, shown with a dashed line in Fig. 4.12a. The above situation occurs also in the quadruple series-coupled rings with the symmetric arrangement of R1 -R2 -R2 -R1 as shown in the inset of Fig. 4.11a. The resonator pair (R2 -R2 ) at the center of the arrangement R1 -R2 -R2 -R1 can be regarded as a single resonator unit. Therefore, the filter response of the adjacent channel is similar to that of the R1 -R2 -R1 triple series-coupled ring, as shown with the solid line in Fig. 4.12a, except for the double peak at λ2 due to the series coupling of the two R2 rings. Next, let us consider the antisymmetric arrangement of R1 -R2 -R1 -R2 of quadruple series-coupled rings, shown in Fig. 4.13. The subset arrangement R1 -R2 -R1 enclosed by the dashed line in Fig. 4.13 has a similar filter response to the triple series-coupling arrangement R1 -R2 -R1 , shown with a dashed line in Fig. 4.12a around the wavelength λ2 . The difference is the existence of ring R2 between the subset R1 -R2 -R1 and the output bus waveguide. However, at the resonant wavelength λ2 , the transmittance from this subset to the bus is strong (close to unity) and so the situation is very close to the direct coupling from the subset to the output bus waveguide, i.e., the case of the arrangement R1 -R2 -R1 of triple series coupling. On the other hand, at the resonant wavelength λ1 , the situation is very close to the subset R2 -R1 -R2 enclosed by the dash–dotted line in Fig. 4.13. Therefore, the filter response of the adjacent channel of the arrangement R1 -R2 -R1 -R2 has a double-peaked shape, as shown by the solid line in Fig. 4.12b.

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Finally, let us consider the asymmetric arrangement R1 -R1 -R2 -R2 . This can be regarded as two coupled pairs of double series-coupled rings, i.e., R1 -R1 and R2 -R2 . The basic filter response is that of the double series-coupled ring, as shown by the dashed line in Fig. 4.12c. Because each single ring R1 or R2 is replaced by the double series-coupled pair R1 -R1 or R2 -R2 , the roll-off of the double series coupling is steeper than that of single-ring coupling. Therefore, the cross talk of quadruple series coupling is reduced much more than that of double series coupling, as shown by the solid line in Fig. 4.12c. In conclusion, the asymmetric arrangement is the optimum arrangement for quadruple series coupling.

4.3.4 Adjacent Channel Cross Talk Let us consider a quadruple series-coupled ring filter, the coupling efficiencies κ b , κ r1 and κ r2 of which are chosen to give the maximally flat passband (see two last equations of (4.30)). Because the adjacent channel cross talk is optimally reduced by the asymmetric arrangement R1 -R1 -R2 -R2 , let us consider the subset double series couplings R1 -R1 and R2 -R2 . When the coupling efficiencies κ b , κ r1 , and κ r2 are fixed, the following equation must be satisfied (since the finesse, in the lossless case, is determined by the coupling efficiency): FSR2 FSR1 = = Finesse. FWHM1 FWHM2

(4.40)

From (4.34), (4.37), and (4.40), the following equation is derived: M · FWHM1 = (M − 1) · FWHM2 .

(4.41)

Now let us define the difference between FSR1 and FSR2 in wavelength units as δλ. Then adjacent cross talk is uniquely determined by δλ/FWHM2 . From (4.40) and (4.41), this ratio is linearly proportional to the ratio of the finesse to M as follows [12]:

FSR2 − FSR1 M−1 FSR1 δλ = = 1− · FWHM2 FWHM1 M FWHM1 . (4.42) 1 = · Finesse [κb , κr1 , κr2 = const.] M This equation reveals that the adjacent channel cross talk is simply determined by 1/M, because the finesse is uniquely determined by the flat passband condition, namely from the coupling efficiencies. Figure 4.14 shows the relation between the adjacent channel cross talk and the FSR expansion factor for a quadruple series-coupled microring filter with asymmetric arrangement R1 -R1 -R2 -R2 . In the range of the FSR expansion factor M from 5 to 40, the following linear relation can be used to approximate the curve [12]:

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Fig. 4.14 Relation between adjacent channel cross talk and FSR extension factor for quadruple series-coupled microrings

Ct (dB) = 4.02 · (10 log10 M) − 71.8.

(4.43)

4.4 Dispersion The influence of dispersion is small in a single microring resonator [22], because the second-order dispersion (group delay dispersion) is zero at the center of the resonant frequency [23]. However, in high-speed transmission systems over 10 Gbps, the spectrum bandwidth of the signal is broad and so the third-order dispersion should be taken into account.

4.4.1 Definition of Group Delay and Dispersion Let us define the transfer function to the output port (through port or drop port) by E0 (ω) = A(ω)e−j(ω) ,

(4.44)

where E0 (ω) is the complex transfer function, A(ω) is the amplitude, and (ω) is the phase of the transfer function. The phase (ω) is expressed as (ω) = tan

−1

  Im (E0 (ω)) . − Re (E0 (ω))

(4.45)

The group delay τ g is given by the derivative of (ω) with respect to the angular frequency:

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τg (ω) =

∂ (ω) . ∂ω

(4.46)

The group delay dispersion (GDD) is the derivative of group delay with respect to the angular frequency [24]: GDD (ω) =

∂τg (ω) ∂ 2  (ω) . = ∂ω ∂ω2

(4.47)

The chromatic dispersion (CD) is given by the derivative of τ g with respect to the wavelength λ: CD (λ) =

∂ ∂τ (λ) = ∂λ ∂λ



∂ (ω) . ∂ω

(4.48)

4.4.2 Group Delay Dispersion in High-Order Series Coupling The drop port transfer functions of double to quadruple series-coupled rings under the Butterworth condition are given by (4.24), (4.25), and (4.26). From the transfer functions, the group delay and the GDD can be calculated using (4.44), (4.45), and (4.46). Figure 4.15 shows the group delay of single- to sixth-order Butterworth seriescoupled ring resonators with the same 3 dB bandwidth (FWHM). The horizontal axis is normalized by the frequency of the group delay peak of a single ring resonator. It is seen that the Butterworth ring resonator filters have two peaks within the 3 dB bandwidth. However, the slope of the curve is zero at the resonant wavelength. This means that the GDD is zero at the resonant wavelength and that the filter does not induce excessive pulse broadening.

Normalized group delay τ

2

Fig. 4.15 Group delay of Butterworth series-coupled ring resonator filters

1.5 1 0.5 0 -2

1st 2nd 3rd 4th 5th 6th -1 0 1 Frequency (ω-ωc)/FWHM

2

Series-Coupled and Parallel-Coupled Add/Drop Filters and FSR Extension

Fig. 4.16 Group delay dispersion of Butterworth series-coupled ring resonator filters

Normalized group delay dispersion

4

15 10 5

107

1st 2nd 3rd 4th 5th 6th

0 -5 -10 -15 -2

-1 0 1 Frequency (ω-ωc)/FWHM

2

In contrast, the GDD of single- to sixth-order Butterworth series-coupled ring resonators with the same 3 dB bandwidth (FWHM) is shown in Fig. 4.16. It is seen that the GDD takes its local maximum around the edges of the 3 dB bandwidth and is antisymmetric with respect to the center resonant wavelength. Therefore, the average of GDD is zero and the pulse broadening induced by the ring resonator is small. However, the third-order dispersion of a Butterworth filter is not zero, because the slope of the curve at the center is not zero. This means that ringing will appear at the tail of a pulse. The filter with zero third-order dispersion is called a Bessel–Thompson filter and the condition to realize it is given in [24]. However, the shape of the spectrum response of the Bessel–Thompson filter is less box-like than the Butterworth filter and it has a single peak. The pulse broadening due to the deformation of the signal spectrum should be taken into account when the 3 dB bandwidth of the signal is close to that of the filter.

4.5 Parallel-Coupled Ring Resonator The high-order parallel-coupled ring resonator has a box-like passband and periodic spectrum. By taking advantage of these characteristics, an interleaver can be realized. An interleaver is a wavelength demultiplexer that divides multiplexed wavelengths into two alternating sets of multiplexed signals with twice the initial wavelength spacing, as shown in Fig. 4.17. By placing an interleaver at a stage prior to the wavelength demultiplexer, it is possible to multiplex or demultiplex wavelength-multiplexed signals with narrower wavelength spacing, even if a demultiplexer with the same channel spacing is used. This configuration offers the advantage of easy expansion of the number of multiplexed wavelengths, while using the existing system.

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Fig. 4.17 Function of an interleaver

4.5.1 Transfer Function of Parallel-Coupled Rings The fundamental topology of a parallel-coupled ring resonator filter is shown in Fig. 4.18. Note that each single ring can be replaced by an odd number of seriescoupled rings. When single ring resonators are used in the parallel-coupling scheme, the relation between the input, drop, through, and add port field amplitudes, EI , ED , ET , EA , respectively, can be expressed by the matrix equation 

EI ED





 ET = T1 · L1 · T2 · L2 · · · Ln−1 · Tn · , EA

(4.49)

where n is the order of parallel coupling and Ti is the transfer matrix of the ith ring. The latter is obtained from the single-ring matrix M of (4.18) and is expressed in terms of its elements as ⎤ 1 m10 ⎢ m11 m11 ⎥ ⎥ Ti = ⎢ ⎣ m00 m11 − m01 m10 m01 ⎦ . m11 m11 ⎡

−

(4.50)

Li is the transfer matrix of the straight arms between rings i and (i +1) and is given by

Fig. 4.18 Fundamental topology of quadruple parallel-coupled ring resonator filter

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 0 ejβLi,i+1 . Li = 0 ejβLi,i+1

109



(4.51)

The transfer functions in the drop and through ports can then be obtained from (4.49).

4.5.2 Interleaving Using Parallel-Coupled Microrings Wavelength filters for an interleaver are required to have box-like filter responses with a flat passband occurring periodically in the spectrum. In the design of an interleaver using a parallel-coupled ring resonator, the ring radii of individual ring resonators should be equal, i.e., Ri = R for all i, in order to ensure the periodic spectrum response. Additionally, there are four issues to be considered: the order of parallel coupling, the length of the straight waveguide between rings, the coupling efficiency between ring and bus waveguides, and the apodization of coupling efficiencies [25]. Figure 4.19 shows the dependence of the drop port spectrum response on the order of parallel coupling. The length of the straight waveguide between successive rings is assumed to be π R, the length of half a ring perimeter, for purposes of phase matching. It is seen that when the order of parallel coupling is increased beyond three, side lobes appear in the stopband. This is because the periodic wavelength characteristics of the parallel-coupled ring filter are subject to the effect of multipath interference through the bus waveguide connections, similar to that of a

Fig. 4.19 Drop port spectral response of parallel-coupled ring resonators

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Mach–Zehnder interferometer, in addition to the individual resonant characteristics of the ring resonators. To apply parallel coupling for interleaving it is therefore necessary to suppress these side lobes. The length of the bus waveguide between rings should be an integral multiple of π R, so that Mach–Zehnder-like interference is utilized to enhance the box-like spectrum response in addition to the ring resonances. Figure 4.20 shows the dependence of the drop port spectrum response of double parallel-coupled ring resonators on the length of the bus waveguide between resonators. It is observed that the period of the side lobes becomes smaller as the length of the bus waveguide becomes longer. However, their magnitude does not change significantly.

Fig. 4.20 Dependence of drop port spectral response of double parallel-coupled ring resonators on the length of the bus waveguide between rings

Several methods have been reported for the reduction of the side lobes [26, 27]. Apodization is an effective technique to reduce the side lobes by gradually adjusting the coupling efficiency. Specifically, the coupling efficiency of the ring resonators to the bus, in the higher order parallel coupling case, is gradually reduced in the outward direction. The sidelobe variations of the drop and through ports with varying coupling efficiency for the quadruple parallel-coupled rings of Fig. 4.18 are shown in Fig. 4.21a, b. With apodization (κ 1 and κ 4 are made smaller), the sidelobes are reduced and the passband of the through port is flattened. In addition, the passband width is expanded. Finally, the measured spectral characteristics for the drop and through ports of quadruple parallel-coupled microrings are shown in Fig. 4.22 [25]. It is noted that interleaved responses with box-like shape were obtained for both drop and through ports. In addition, the polarization dependence was eliminated and the resonant wavelengths were adjusted to the ITU Grid, which was standardized to 100 GHz spaced channels centered at 193.10 THz.

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111

Fig. 4.21 Spectral responses of quadruple parallel-coupled ring resonators with apodization of the coupling efficiencies: (a) drop port response, (b) through port response

Fig. 4.22 Measured spectral response of quadruple parallel-coupled ring resonator with apodization of the coupling efficiency

4.6 Conclusion Coupled microring resonator filters can satisfy most of the optical filter requirements through a filter response synthesis using a combination of ring resonators, such as series coupling, parallel coupling, and cascade topology. In the cascade filter, the overall filter response is expressed by the product of responses of individual filters and the FSR is expanded. In series coupling, the passband is flattened and the FSR is expanded. Parallel coupling can function as a multiport interleaver. To synthesize the filter response using these topologies, optimum design conditions should be considered.

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References 1. Kokubun, Y. Vertically coupled micro-ring resonator filter for integrated add/drop node. IEICE Trans. Electron. E88C, 349–362 (2005) 2. Kokubun, Y. High index contrast optical waveguides and their applications to microring filter circuit and wavelength selective switch. IEICE Trans. Electron. E90-C, 1037–1045 (2007) 3. Hryniewicz, J.V., Absil, P.P., et al. Higher order filter response in coupled microring resonators. IEEE Photon. Technol. Lett. 12, 320–322 (2000) 4. Little, B.E., Chu, S.T., et al. Very high-order microring resonator filters for WDM applications. IEEE Photon. Technol. Lett. 16, 2263–2265 (2004) 5. Yanagase, Y., Suzuki, S., et al. Box-like filter response and expansion of FSR by vertically triple coupled microring resonator filter. J. Lightwave Technol. 20, 1525–1529 (2002) 6. Barwicz, T., Popovic, M.A., et al. Fabrication of add-drop filter based on frequency-matched microring resonators. J. Lightwave Technol. 24, 2207–2218 (2006) 7. Barbarossa, G., Matteo, A.M., et al. Theoretical analysis of triple-coupler ring-based optical guided-wave resonator. J. Lightwave Technol. 13, 148–156, (1995) 8. Orta, R., Savi, P., et al. Synthesis of multiple-ring-resonator filters for optical systems. IEEE Photon. Technol. Lett. 7, 1447–1449 (1995) 9. Madsen, C.K., Zhao, J.H. A general planar waveguide autoregressive optical filter. IEEE J. Lightwave Technol. 14, 437–447 (1996) 10. Lee, H.S., Choi, C.H., et al. A nonunitary transfer matrix method for practical analysis of racetrack microresonator waveguide. IEEE Photon. Technol. Lett. 16, 1086–1088 (2004) 11. Poon, J.K.S., Sheuer, J., et al. Matrix analysis of microring coupled-resonator optical waveguides. Opt. Express 12, 90–103 (2004) 12. Goebuchi, Y., Kato, T., et al. Optimum arrangement of high-order series coupled microring resonator for crosstalk reduction. Jpn. J. Appl. Phys. 45, 5769–5574 (2006) 13. Chen, W., Wang, Z., et al. General ring resonator analysis and characterization by characteristic matrix. IEEE J. Lightwave Technol. 23, 915–922 (2005) 14. Kato, T., Kokubun, Y. Optimum coupling coefficients in second-order series-coupled ring resonator for non-blocking wavelength channel switch. IEEE J. Lightwave Technol. 24, 991–999 (2006) 15. Yariv, A. Critical coupling and its control in optical waveguide-ring resonator systems. IEEE Photon. Technol. Lett. 14, 483–485 (2002) 16. Little, B.E., Chu, S.T., et al. Microring resonator channel dropping filters. IEEE J. Lightwave Technol. 15, 998–1005 (1997) 17. Yariv, A. Universal relations for coupling of optical power between microresonators and dielectric waveguides. Electron. Lett. 36, 321–322 (2000) 18. Menon, V.M., Tong, W., et al. Control of quality factor and critical coupling in microring resonators through integration of a semiconductor optical amplifier. IEEE Photon. Technol. Lett. 16, 1343–1345 (2004) 19. Kokubun, Y., Hatakeyama, Y., et al. Fabrication technologies for vertically coupled microring resonator with multilevel crossing busline and ultra-compact ring radius. IEEE J. Sel. Top. Quant. Electron. 11, 4–10 (2005) 20. Hatakeyama, Y., Hanai, T., et al. Loss-less multilevel crossing of busline waveguide in vertically coupled microring resonator filter. IEEE Photon. Technol. Lett. 16, 473–475 (2004) 21. Schwelb, O., Frigyes, I. Vernier operation of series coupled optical microring resonator filters. Microwave Opt. Technol. 39, 257–261 (2003) 22. Suzuki, S., Kokubun, Y., et al. Ultra-short optical pulse transmission characteristics of vertically coupled microring resonator add/drop filter. IEEE J. Lightwave Technol. 19, 266–271 (2001) 23. Schwelb, O. Transmission, group delay and dispersion characteristics of single-ring optical resonators and add/drop filters – A tutorial overview. IEEE J. Lightwave Technol. 22, 1380–1394 (2004)

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24. Kato, T., Kokubun, Y. Bessel-Thompson filter using double-series-coupled microring resonator. IEEE J. Lightwave Technol. 26, 3694–3698 (2008) 25. Ito, T., Kokubun, Y. Fabrication of 1 × 2 interleaver by parallel-coupled microring resonator. Electron. Commun. Japan (John Wiley) 89, 56–64 (2006) 26. Little, B.E., Chu, S.T., et al. Filter synthesis for periodically coupled microring resonators. Opt. Lett. 25, 344–346 (2000) 27. Griffel, G. Vernier effect in asymmetrical ring resonator arrays. IEEE Photon. Technol. Lett. 12, 1642–1644 (2000)

Chapter 5

Advanced Microring Photonic Filter Design Vien Van

Abstract This chapter reviews the development of high-order photonic filters based on coupled microring resonators for optical spectral engineering applications. Advanced microring filter architectures capable of realizing general infinite impulse response optical transfer functions with finite transmission zeros are presented. Techniques for the analysis and exact synthesis of these devices to achieve a prescribed filter response are also described in detail. Examples of advanced optical filter designs such as high-order pseudo-elliptic filters and filters with maximally flat group delay are given to illustrate the potential applications of these general microring architectures in optical spectral engineering.

5.1 Introduction Over the past decade there has been a growing interest in the application of microring resonators in advanced optical spectral and dispersion engineering. Progress in microphotonic fabrication techniques has enabled microring resonators with extremely high quality (Q) factors to be realized in various material systems, with typical values ranging from 104 to 106 [1–4]. Improved fabrication control has also allowed increasingly complex device structures consisting of multiple coupled microring resonators to be demonstrated, with notable examples such as high-order filters consisting of up to 11 serially coupled microring resonators fabricated in glass [5] and coupled resonator optical waveguide (CROW) structures consisting of 100 microracetracks in silicon-on-insulator [6]. The compact sizes, high Q factors, and dispersive nature of microring resonators make them attractive for a wide range of integrated photonics applications, such as add/drop filters for wavelength division

V. Van (B) Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada e-mail: [email protected] I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_5,  C Springer Science+Business Media, LLC 2010

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multiplexing (WDM) systems [7], optical cross-connects [8], dispersion compensators [9], optical delay lines [10], and microwave photonic filters [11]. Since many of these applications have stringent spectral requirements such as flat-top passband, sharp band transition, high isolation level, constant group delay, or linear dispersion slope, the realization of these devices often requires the use of multiple microring resonators arranged in complex coupling topologies. The design of microring devices that can meet these spectral specifications involves the exact and selective placement of the poles and zeros of the device transfer function, much like in the design of electrical filters. Indeed, by leveraging design and synthesis techniques for analog and digital circuits, advanced microring photonic device architectures can be realized with the complexity and functionality approaching that of state-of-the-art microwave filters. Although various microring architectures have been proposed for filtering applications, only three general architectures have emerged to date that can exactly realize optical transfer functions with independent poles and zeros and for which direct synthesis procedures are available. These architectures are the sum–difference all-pass microring filters [12–14], the direct-coupled microring filters with twodimensional (2D) coupling topology [15, 16], and most recently the microring ladder filters [17, 18]. In this chapter we present a detailed description of each of these general microring filter architectures, along with techniques for analyzing and synthesizing the devices to achieve a prescribed filter response. The organization of the chapter is as follows. The general infinite impulse response (IIR) optical transfer function of microring filters is described in Section 5.2. Sections 5.3, 5.4, and 5.5 are devoted to the analysis and synthesis of, respectively, the sum–difference all-pass microring filters, the direct-coupled microring filters, and the microring ladder filters. Concluding remarks summarizing the chapter are given in Section 5.6.

5.2 General IIR Optical Transfer Functions The design of microring photonic filters begins with the specification of the desired optical transfer function in either the complex frequency domain (s-domain) or the unit delay variable domain (z-domain). Due to the feedback nature of microring resonators, microring filters exhibit IIR with the transfer function having the general form M 

H(s) =

P(s) k=1 =K N Q(s)  k=1

in the s-domain or

(s − zk ) (5.1) (s − pk )

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117 M 

P(z) k=1 H(z) = =K N Q(z) 

(z − zk ) (5.2) (z − pk )

k=1

in the z-domain. In the above, K is a constant and pk and zk are the poles and zeros, respectively, of the filter. The number of poles N corresponds to the order of the filter, which determines the number of resonators in the network, while the number of transmission zeros, M ≤ N, determines the coupling topology or the number of coupling elements for certain types of microring architectures. For passive microring filters, the poles are restricted to the left half of the s-plane, or to the region inside the unit circle in the z-plane. The locations of the poles and zeros are chosen to achieve specific amplitude, phase, or group delay characteristics of the filter. Given a set of prescribed spectral characteristics, the problem of determining a suitable transfer function in the s- or z-domain is known as the filter approximation problem, for which a large body of literature exists in the field of analog and digital filter design (e.g., [19, 20]). Given the transmission response H(s) or H(z) of a filter, the complementary reflection response F(s) or F(z) can be determined from the Feldtkeller relation for a lossless network, H(s)H(− s) + F(s)F( − s) = 1

(5.3)

˜ + F(z)F(z) ˜ =1 H(z)H(z)

(5.4)

in the s-domain or

˜ ˜ = F ∗ (1/z∗ ) are the parain the z-domain. In (5.4), H(z) = H ∗ (1/z∗ ) and F(z) Hermitian conjugates of H(z) and F(z). Letting H = P/Q and F = R/Q, we obtain from (5.3) and (5.4) R(s)R(− s) = Q(s)Q(− s) − P(s)P(− s),

(5.5)

˜ − P(z)P(z). ˜ = Q(z)Q(z) ˜ R(z)R(z)

(5.6)

By factoring the polynomial on the right-hand side of (5.5), we obtain two sets of roots, one lying in the left-half s-plane and the other in the right-half s-plane. Any combination of one root from each set can be used to form the polynomial R(s), although it is common to choose the zeros in the left-half plane, yielding a minimum phase reflection response. In the z-domain, the polynomial R(z) can be obtained from (5.6) in a similar manner, with the minimum phase roots being those that lie inside the unit circle. In the design of microring filters it is generally more convenient to synthesize s-domain transfer functions using techniques based on the energy coupling formalism of microring resonators, whereas synthesis of z-domain transfer functions

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can be more naturally performed using the power coupling formalism. The energy coupling formalism provides a description of the energy transfer between the resonators based on the time-domain coupled mode theory and is valid in the limit of weak coupling between the microrings. Energy coupling analysis of microring networks naturally yields device transfer functions in the s-domain. The advantage of designing microring filters in the s-domain is that the transfer functions of a large class of well-known filters such as inverse Chebyshev, elliptic function, and linearphase filters are readily available. However, since the energy coupling formalism assumes weak coupling between the microrings, s-domain synthesis methods are strictly accurate only for narrowband filters whose passbands are much smaller than the free spectral range (FSR) of the resonators. For broadband microring filters, a more accurate design must be sought in the z-domain based on the power coupling formalism. Power coupling analysis of microring resonators leads to periodic transfer functions which are analogous to those of digital filters in the discrete time domain, where the inverse of the unit delay variable, z−1 , represents the round-trip delay of the microring. Filter transfer functions in the z-domain may be obtained from s-domain transfer functions using the bilinear transformation, or by direct approximation of the target filter response using optimization techniques such as the Remez exchange algorithm [20].

5.3 Sum–Difference All-Pass Microring Filters The first microring filter architecture capable of realizing general IIR optical transfer functions was proposed by Jinguji [21] based on a cascaded lattice of Mach– Zehnder interferometers (MZIs), each loaded with a single all-pass (AP) microring resonator. Madsen subsequently proposed a more simplified, although less general, architecture consisting of a single MZI with each arm loaded with a high-order all-pass microring network [12–14]. The device can be regarded as the optical realization of the sum–difference AP digital circuit, which is well known in the field of digital filter design [22, 23]. Figure 5.1 shows a schematic of the sum–difference AP digital filter and its optical realization using a MZI structure. In both devices an input signal is split into two halves, which then pass through two separate AP networks with transfer functions A1 (z) and A2 (z), and subsequently recombine as sum and difference at the outputs. In the optical circuit, the AP transfer functions are realized using AP microring networks. The lower arm of the MZI has a differential phase shift θ to compensate for the phase difference between the two AP networks. At the cross port and bar port of the MZI, the transfer functions H(z) and F(z), respectively, are given by H(z) =

−j P(z) [A1 (z) + A2 (z)e−jθ ] = , 2 Q(z)

(5.7)

F(z) =

R(z) 1 [A1 (z) − A2 (z)e−jθ ] = . 2 Q(z)

(5.8)

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Advanced Microring Photonic Filter Design (a)

119

allpass

1/2

+

H (z) =

1 [ A1 ( z ) + A2 ( z )] 2

+

F ( z) =

1 [ A1 ( z ) − A2 ( z )] 2

A1(z) input

A2(z)

−1

allpass

(b)

1/2

bar

input

A1(z) 3dB

θ

A2(z)

F(z) 3dB

H(z) cross

Fig. 5.1 (a) Schematic of a sum–difference AP digital filter and (b) its optical realization using a MZI structure

For an Nth-order filter, where N is assumed to be even, each of the AP networks A1 (z) and A2 (z) is of order N/2. Their transfer functions can be expressed in general form as A1 (z) =

˜ 1 (z) −N/2 −jφ D z e 1, D1 (z)

(5.9)

A2 (z) =

˜ 2 (z) −N/2 −jφ D e 2, z D2 (z)

(5.10)

where D1 (z) and D2 (z) are polynomials of degree N/2. Substituting the above into (5.7) and (5.8) gives −j ˜ 2 (z)D1 (z)e−j(θ+φ2 ) ]z−N/2 , ˜ 1 (z)D2 (z)e−jφ1 + D [D 2 1 ˜ 2 (z)D1 (z)e−j(θ+φ2 ) ]z−N/2 , ˜ 1 (z)D2 (z)e−jφ1 − D R(z) = [D 2

P(z) =

Q(z) = D1 (z)D2 (z).

(5.11) (5.12) (5.13)

It is evident from the above expressions that P, R, and Q are polynomials of degree N. A sum–difference AP circuit consisting of two N/2-order AP microring networks can thus realize a transfer function with N poles and a maximum of N transmission zeros. The poles of the filter circuit are comprised of the poles of the two AP transfer functions, as evident from (5.13). In addition, one can show from (5.11) and ˜ and R(z) = z−N R(z), ˜ which (5.12) that, apart from a phase factor, P(z) = z−N P(z) implies that the zeros of P(z) and R(z) must appear in reciprocal conjugate pairs (zk , 1/z∗k ) [13].

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(a)

N/2

(b)

θ

φ1

φ2

φN/2

2

1

2

N/2

1

κ1

κ2

κN/2

κN/2+1

κk

κN

N/2+1

k

N

φN/2+1

φk

φN

θ

N/2+1

N/2+2

N

Fig. 5.2 Realization of the AP transfer functions A1 (z) and A2 (z) using microring resonators in (a) the parallel-cascaded configuration and (b) the serial coupling configuration

In the optical circuit, the transfer functions A1 (z) and A2 (z) are realized using AP microring resonators in either the parallel-cascaded configuration as illustrated in Fig. 5.2a or the serial coupling configuration as in Fig. 5.2b. The parallel-cascaded configuration is more attractive since it is simpler to design and also easier to fabricate. Each AP microring resonator in the array with coupling coefficient κ k and phase shift φ k has a transfer function of the form

Tk (z) =

τk − e−jφk z−1 , 1 − τk e−jφk z−1

(5.14)

$ where τk = 1 − κk2 is the transmission coefficient and z−1 = σ rt exp(−jωTrt ) for a microring with round-trip time Trt = 1/FSR and round-trip amplitude attenuation σ rt = exp(−αL), L being the ring circumference, and α the attenuation constant. The transfer function of a cascaded array of N/2 AP microring resonators is simply the product of the transfer functions of the individual microrings,

A(z) =

N/2 % k=1

N/2 % p∗ z − 1 τk − e−jφk z−1 k = e−jφk , −jφ −1 1 − τk e k z z − pk

(5.15)

k=1

where pk = τk e−jφk is the pole associated with the kth AP microring. The above equation shows that the kth pole of the AP transfer function A(z) has amplitude given by the transmission coefficient τ k and phase given by the phase shift φ k of microring k.

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5.3.1 Filter Synthesis In the design of the sum–difference AP microring filter, the AP transfer functions A1 (z) and A2 (z) are first determined from the prescribed transfer functions of the filter. Given the transfer functions H(z) = P(z)/Q(z) and F(z) = R(z)/Q(z), we obtain from (5.11) and (5.12) ˜ 1 (z)D2 (z)z−N/2 e−jφ1 = jP(z) + R(z), D

(5.16)

˜ 2 (z)z−N/2 e−j(θ+φ2 ) = jP(z) − R(z). D1 (z)D

(5.17)

The roots of the polynomial on the right-hand side of (5.16) are then computed and those roots which lie inside the unit circle are chosen to be the roots of D2 (z). Similarly, the roots of D1 (z) are chosen from those of the polynomial in (5.17) that lie inside the unit circle. Knowledge of D1 (z) and D2 (z) allows the AP transfer functions A1 (z) and A2 (z) to be constructed, which can then be realized using cascaded arrays of AP microring resonators as shown in Fig. 5.2a. Specifically, from (5.15) it is seen that each pole pk = τk e−jφk in the AP transfer function requires a microring resonator with transmission coefficient τ k and phase shift φ k . The phase shift θ in the lower arm of the MZI is chosen by evaluating (5.7) or (5.8) at a particular frequency. For example, by evaluating (5.7) at z = 1, we obtain θ = ∠A2 (1) − ∠{2jH(1) − A1 (1)}.

(5.18)

In general the sum–difference AP microring filter can realize an Nth-order filter using N + 2 coupling parameters (including the two 3 dB couplers) and N + 1 phase shifters. The architecture can also realize up to a maximum of N transmission zeros. A disadvantage of the sum–difference AP architecture is that it requires asynchronously tuned microring resonators whose relative phase shifts have to be individually and precisely controlled.

5.3.2 Design Example We consider as an example the design of a linear-phase microring filter with maximally flat responses in both the amplitude and group delay. This type of filters has important applications in WDM communication systems such as low-dispersion filters and optical delay lines. The general s-domain transfer function of a linearphase flat-top filter has been derived for microwave filters in [24]. Using the bilinear transformation, we obtain the z-domain transfer functions of a sixth-order linearphase flat-top filter with a 10 GHz bandwidth and a 2 THz FSR. The coefficients of the polynomials P(z), R(z), and Q(z) are shown in Table 5.1. The above transfer functions can be realized using a sum–difference AP microring architecture with a third-order AP microring network in each arm of the MZI. The polynomial D(z) of each AP network is also shown in Table 5.1. From the roots of D1 (z) and D2 (z) we

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Table 5.1 Coefficients of the polynomials of the transfer functions of a sixth-order linear-phase flat-top filter k (z−k )

R(z)

Q(z)

D1 (z) = D∗2 (z)

0.9300 −5.5801 13.9503 −18.6005 13.9503 −5.5801 0.9300

1 −5.8550 14.2852 −18.5907 13.6105 −5.3150 0.8649

−1.0750 − 0.0253j 3.1476 + 0.0490j −3.0726 − 0.0237j 1

P(z)

0 1 2 3 4 5 6

0 0.3427E−3 −1.3719E−3 2.0584E−3 −1.3719E−3 0.3427E−3 0

obtain the microring coupling coefficients κ k and phase shifts φ k , which are shown in Fig. 5.3a along with a schematic of the filter. In Fig. 5.3b and c, we plot the amplitude and group delay responses, respectively, of the synthesized filter for the ideal case of no loss in the microrings and the case where the microring waveguides have a moderate propagation loss of 5 dB/cm. In both cases the device responses are seen to exhibit a flat-top passband and a constant group delay of 65 ps over a 10 GHz bandwidth.

1

π

(b)

φ2

φ1

(a)

κ1

2

κ1

φ3 κ2

3

κ3

κ2

F

κ1 = 0.1656 κ 2 = 0.2297 κ 3 = 0.2473

κ3

4

5

6

−φ1

−φ2

−φ3

H (c)

0

70

transmission (dB)

–10

60

group delay (ps)

|H(z)|2

–20 –30 2

|F(z)|

–40 –50 –60 –40

φ1 = 0.03681 φ2 = −0.02036 φ3 = 0.007107

50 40 30 20 10

–20

0

20

frequency detune, Δf (GHz)

40

0

–30

–20

–10

0

10

20

30

frequency detune, Δ f (GHz)

Fig. 5.3 (a) Schematic of a sixth-order sum–difference AP microring filter with flat-top amplitude and group delay responses. (b) Amplitude and (c) cross port group delay responses of the synthesized microring filter for the lossless case (solid lines) and the case of 5 dB/cm propagation loss in the microrings (dashed lines)

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5.4 Direct-Coupled Microring Filters Direct-coupled microring filters of 2D coupling topology are generalizations of the more well-known serially coupled microring filters, which can realize only all-pole transfer functions. In microwave coupled-cavity filter design, it is well known that transfer functions with transmission zeros require 2D coupling topologies with nonadjacent resonator coupling. Similar coupling topologies can also be implemented in the optical domain using microring resonators to realize more general IIR optical transfer functions. Figure 5.4 shows a schematic of a network of N microring resonators in the most general 2D coupling topology, in which every microring i is coupled to every other microring j via energy coupling coefficient μi,j . Microrings 1 and N are also coupled to the input and output bus waveguides, respectively, via coupling coefficients μi and μ0 . For generality, we assume the microring resonators to have non-identical resonant frequencies, with the amount of detuning in the resonant frequency ωk of microring k from the center frequency ω0 of the filter passband denoted by ωk = ωk – ω0 . Loss in each microring is represented by an intrinsic cavity lifetime τ , which is related to the amplitude attenuation constant α in the microring waveguide via 1/τ = αvg , where vg is the group velocity. μ1,N through

μ13

st

si input

μi

1

μ12

μ1,N−1

2

μ3,N drop

μ3,N−1

sd N−1

3 μ23 μ2,N−1 μ2,N

N μo

Fig. 5.4 Schematic of a direct-coupled microring filter consisting of N resonators in the most general 2D coupling topology [33]. Reprinted with permission. Copyright 2008 SPIE

Let sˆi , sˆt , and sˆd denote the amplitudes of the instantaneous energy signals at the input, through, and drop ports, respectively. The amplitude of the instantaneous energy stored in microring k is denoted by âk (t). Energy coupling analysis of the network of N mutually coupled resonators in Fig. 5.4 yields the system of timedomain coupled mode equations [15, 25] ⎡

⎤ ⎡ aˆ 1 (jω1 − 1/τi ) −jμ1,2 −jμ1,3 ⎢ aˆ 2 ⎥ ⎢ −jμ1,2 (jω2 − 1/τ ) −jμ2,3 ⎥ ⎢ d ⎢ ⎢ aˆ 3 ⎥ ⎢ −jμ1,3 −jμ2,3 (jω3 − 1/τ ) ⎢ ⎥=⎢ dt ⎢ .. ⎥ ⎢ .. .. .. ⎣ . ⎦ ⎣ . . . aˆ N −jμ1,N −jμ2,N −jμ3,N

⎤ ⎤⎡ ⎤ ⎡ −jμi sˆi ··· −jμ1,N aˆ 1 ··· −jμ2,N ⎥ ⎢ aˆ 2 ⎥ ⎢ 0 ⎥ ⎥ ⎥⎢ ⎥ ⎢ ··· −jμ3,N ⎥ ⎢ aˆ 3 ⎥+⎢ 0 ⎥ . ⎥ ⎥⎢ ⎥ ⎢ ⎥⎢ . ⎥ ⎢ . ⎥ .. .. . . ⎣ ⎦ ⎦ ⎣ . . . . ⎦ aˆ N 0 · · · (jωN − 1/τo ) (5.19)

In the above 1/τ i and 1/τ o represent the total rates of energy extracted from resonators 1 and N due to both intrinsic loss and energy coupling to the input and output bus waveguides

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V. Van

1 τ(i,o)

=

μ2(i,o) 2

+

1 . τ

(5.20)

Assuming harmonic time signals sˆi = si ejωt and aˆ k = ak ejωt , the system of equations in (5.19) can be written in matrix form as [(s + 1/τ )I + L + jM]a = b,

(5.21)

where s = j(ω – ω0 ), a = [a1 ,a2 , · · · ,aN ] t , b = [ − jμi si ,0, · · · ,0] t , I is the N × N identity matrix, and M is an N × N symmetric coupling matrix having the general form ⎡

ω1 ⎢ μ1,2 ⎢ ⎢ M = ⎢ μ1,3 ⎢ . ⎣ .. μ1,N

μ1,2

ω2 μ2,3 .. . μ2,N

μ1,3 μ2,3

ω3 .. . μ3,N

··· ··· ··· .. . ···

⎤ μ1,N μ2,N ⎥ ⎥ μ3,N ⎥ ⎥ .. ⎥ . ⎦

ωN

(5.22)

and L is a diagonal matrix representing the rates of energy being extracted from the system via the input and output waveguides L = diag[μ2i /2 , 0, . . . 0, μ2o /2 ].

(5.23)

Note that for the special case of serially coupled microring filters, the coupling matrix M has a simple tridiagonal form. Performing eigenvalue decomposition of the matrix −(L + jM) as − (L + jM) = Q · D · Q−1 ,

(5.24)

a = Q[(s + 1/τ )I − D]−1 Q−1 b

(5.25)

we obtain from (5.21)

or more explicitly an = −jμi si

N

Qn,k Q−1 k,1

k=1

s + 1/τ − pk

, n = 1 to N,

(5.26)

where pk is the kth eigenvalue in D. Using the relationships st = si − jμi a1 and sd = −jμo aN , the transfer functions of the microring filter at the drop port, H(s), and through port, F(s), can be explicitly expressed as H(s) = −μi μo

N

QN,k Q−1 k,1

k=1

s + 1/τ − pk

=

P(s) , Q(s)

(5.27)

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Advanced Microring Photonic Filter Design

F(s) =

1 − μ2i

125

N

Q1,k Q−1 k,1

k=1

s + 1/τ − pk

=

R(s) . Q(s)

(5.28)

The above transfer functions have N poles which, in the absence of loss (1/τ = 0), are given by the eigenvalues of the matrix −(L + jM). Also, the coefficient of the N  QN,k Q−1 highest power term of the polynomial P(s) in (5.27), aN−1 = k,1 , is zero k=1

since it is the product of row N of Q and column 1 of Q−1 . Thus P(s) has a maximum degree M = N − 2, yielding a maximum of N − 2 transmission zeros for the drop port transfer function H(s) of the microring filter.

5.4.1 Filter Synthesis Synthesis of the direct-coupled microring filter is facilitated by first transforming the microring network into the electrical domain and then applying well-known electrical filter design techniques to determine the required microring parameters. In this section we first derive an equivalent electrical network model for the directcoupled microring filter of the most general coupling topology. The coupling matrix technique for synthesizing coupled-cavity microwave filters is then applied to determine the microring parameters. Finally optimization of the coupling topology and reduction to the canonical form is implemented.

5.4.1.1 Equivalent Electrical Network Model If the coupling strengths between the microring resonators in the network are weak, each microring k can be represented to first order by an LC oscillator with resonant frequency ωk = (Lk Ck )−1/2 [26]. The system of N-coupled microring resonators in Fig. 5.4 can then be modeled by an equivalent electrical network of coupled LC oscillators as shown in Fig. 5.5a. The mutual coupling between oscillators i and j is represented by admittance inverter Xi,j , whose equivalent capacitive circuit is shown in Fig. 5.5b. The susceptance elements Xi,j as well as the bus-to-ring coupling susceptances Xi and Xo correspond directly to the energy coupling coefficients: Xi,j √ √ = μi,j , Xi = μi / 2, and Xo = μ0 / 2 [26]. Analysis of the electrical network in Fig. 5.5a yields the following equation for the nodal voltages vk : ⎡

−Y1 ⎢ jX1,2 ⎢ ⎢ jX1,3 ⎢ ⎢ . ⎣ .. jX1,N

jX1,2 −Y2 jX2,3 .. . jX2,N

jX1,3 jX2,3 −Y3 .. . jX3,N

··· ··· ··· .. . ···

⎤⎡ ⎤ ⎡ ⎤ jX1,N v1 −jXi V1 ⎢ ⎥ ⎢ ⎥ jX2,N ⎥ ⎥ ⎢ v2 ⎥ ⎢ 0 ⎥ ⎥ ⎢ ⎥ ⎢ jX3,N ⎥ ⎢ v3 ⎥ = ⎢ 0 ⎥ ⎥, ⎢ . ⎥ ⎢ ⎥ .. ⎥ .. . ⎦ ⎣ ⎦ ⎦ ⎣ . . . vN −YN −jXo V2

(5.29)

126

V. Van X2,N

(a)

X2,k X1,N X1,k

I1 V1

Xk,N

v1

LC1

Xk–1,k

X23

X12

Xi

Xk,k+1

XN–1,N

LCk

LC2

I2

vN

vk

V2

X V2 o LCN

jXi,j (b)

− jXi, j

Xi,i,jj

− jXi, j

Fig. 5.5 (a) Equivalent coupled LC oscillator circuit model of the 2D direct-coupled microring filter topology. (b) Capacitive circuit of the admittance inverter Xi,j [15]. Reprinted with permission. Copyright 2007 IEEE

where Yk = s − j ωk is the admittance of oscillator k. Using the relations v1 = −I1 /jXi and vN = −I2 /jXo at the input and output ports, we can express (5.29) as (sI − j X)V = I,

(5.30)

where I is the N × N identity matrix V and I are the voltage and current arrays t  V = −I1 / j Xi , v2 , v3 , · · · vN−1 , − I2 / j Xo ,

(5.31)

 t I = j Xi V1 , 0, 0, · · · 0, j Xo V2 ,

(5.32)

and X is an N × N symmetric coupling matrix ⎡

ω1 ⎢ X1,2 ⎢ ⎢ X = ⎢ X1,3 ⎢ . ⎣ .. X1,N

X1,2

ω2 X2,3 .. . X2,N

X1,3 X2,3

ω3 .. . X3,N

... ... ... .. . ...

⎤ X1,N X2,N ⎥ ⎥ X3,N ⎥ ⎥. .. ⎥ . ⎦

ωN

(5.33)

Since Xi,j = μi,j , it is seen that the coupling matrix X of the electrical network is identical to the energy coupling matrix M in (5.22) of the microring network.

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5.4.1.2 Coupling Matrix Synthesis The above equivalent electrical network allows synthesis techniques developed for coupled-cavity microwave filters to be directly applied to the design of microring filters. In a well-known approach [27], the real and symmetric coupling matrix X is diagonalized in the form X = T ·  · Tt ,

(5.34)

where T is a real orthonormal matrix and  is a diagonal matrix containing the eigenvalues λk of X. The voltage array V in (5.30) can be solved to give V = (sI − jX)−1 I = T (sI − j)−1 Tt I = ZI,

(5.35)

where Z = T(sI − j)−1 Tt is the N × N impedance matrix of the circuit network. From (5.35) the two-port short-circuit admittance matrix of the network can be obtained as     N 2 1 Xi Xo T1,k TN,k Xi2 T1,k y11 y12 . (5.36) = Ysc = 2 y21 y22 Xo2 TN,k s − jλk Xi Xo TN,k T1,k k=1

The above expression shows that the eigenvalues λk of X can be obtained from the poles of the admittance parameters. Given an Nth-order filter circuit with shortcircuit admittance matrix expressed in the form Ysc =

N k=1

1 s − jλk



(k)

(k) 

(k)

(k)

ξ11 ξ12 ξ21 ξ22

,

(5.37)

(k)

where ξi,j is the residue of admittance yi,j at the pole jλk , we equate (5.36) and (5.37) to obtain the following expressions for the input and output coupling elements Xi2 = Xo2 =

N k=1 N

(k)

ξ11 = μ2i /2,

(5.38)

& (k) ξ22 = μ2o 2.

(5.39)

k=1

The elements in the first and last rows of the matrix T are next computed from $

(k)

2ξ11

T1,k = '

μi

, (

(5.40) (k)

TN,k = T1,k sgn imag ξ12

) * .

(5.41)

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V. Van

The remaining rows of the orthonormal matrix T can be obtained by Gram–Schmidt orthogonalization. Finally, the coupling matrix X = M is determined from T and  using (5.34). The short-circuit admittance matrix Ysc of the equivalent electrical network can be determined from the prescribed optical transfer functions of the microring filter. Identifying the through-port transfer function F(s) = R(s)/Q(s) as the S11 parameter of the equivalent electrical network, we first obtain the input impedance Zin of the circuit as Zin (s) =

Q(s) + R(s) 1 + F(s) = . 1 − F(s) Q(s) − R(s)

(5.42)

By decomposing the numerator term Q(s) + R(s) into a sum of an even polynomial, M(s), and an odd polynomial, N(s), it can be shown that the elements of Ysc can be determined from [28]: + N(s) +M(s), if N is even, M(s) N(s), if N is odd, +  P(s) +M(s), if N is even, y12 (s) = y21 (s) = P(s) N(s), if N is odd. 

y11 (s) = y22 (s) =

(5.43) (5.44)

(The polynomial N(s) is not to be confused with N, the number of resonators). By performing partial fraction expansion on y11 (s) and y12 (s) in the form of (5.37), the (k) (k) poles λk and residues ξ11 and ξ12 can be obtained, from which the coupling matrix X = M along with the input and output coupling coefficients, μi and μo , can be determined. Usually the above network synthesis is performed for a prototype filter with normalized center frequency ω0 = 1 rad/s and cutoff frequency ωc = 1 rad/s. For a bandpass filter of bandwidth B (rad/s), √ the energy coupling parameters are scaled as μ˜ i,j = μi,j (B/2) and μ˜ (i,o) = μ(i,o) B/2, where the tilde denotes parameters of the new bandpass filter with bandwidth B. Finally, in the physical implementation of the microring filter, the energy coupling coefficients are converted to the power coupling coefficients for microring resonators with a specified FSR using the relations √ κi,j = μ˜ i,j /FSR and κ(i,o) = μ˜ (i,o) / FSR. 5.4.1.3 Optimization of the Coupling Matrix The above synthesis procedure may initially yield a matrix M which corresponds to a coupling topology that is not realizable due to physical layout constraints, such as one which requires a microring to be coupled to too many other microrings, or one which contains microring quadruplets with cross couplings, as shown in Fig. 5.6a. Another undesirable coupling topology is one which leads to coupling between counter-propagating modes in the microrings, such as in the triplet configuration shown in Fig. 5.6b. These non-realizable or undesirable coupling topologies can be converted to simpler and realizable ones by applying similarity transformations such

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Advanced Microring Photonic Filter Design

129

(b)

(a) 1

2

4

3

3

1

2

Fig. 5.6 Examples of undesirable microring coupling topologies: (a) a quadruplet with cross couplings; (b) a triplet causing coupling between counter-propagating modes in the microrings

as Jacobi rotations to the matrix M without disturbing its eigenvalues. Each Jacobi rotation yields a new coupling matrix M according to M = R(θr ) · M · Rt (θr ),

(5.45)

where R(θ r ) is an N × N rotation matrix and θ r is the rotation angle chosen to annihilate an undesirable coupling element in the original coupling matrix M [28, 29]. By applying a series of such rotations, a new coupling matrix can be obtained which yields the optimum coupling topology with the fewest number of coupling elements and, for some filters, the fewest number of negative coupling elements. In general the optimum filter architecture is given by the canonical coupling topology, which has the form of a folded array. A procedure for converting an arbitrary coupling topology to the canonical form is given in [28]. The canonical topology requires 3N/2 coupling elements to realize an Nth-order transfer function with a maximum of N − 2 transmission zeros. The direct-coupling topology is thus more efficient than the sum–difference AP filter in terms of the number of design parameters required. Also, the device layout is very compact and the design is much simplified since the microring resonators are all synchronous. One major disadvantage of the direct-coupled microring architecture, however, is that it requires negative coupling coefficients for transfer functions with transmission zeros located on the imaginary jω-axis, as illustrated in the example below. Negative coupling elements are generally difficult to realize with microring resonators, requiring either microracetracks with long coupling lengths such that the coupling angles are between 3π /2 and 2π [15] or precise layout geometries [30].

5.4.2 Design Example We consider in this section the design of a sixth-order pseudo-elliptic optical filter with a 25 GHz bandwidth, 0.05 dB ripple in the passband, and −40 dB stopband rejection. Choosing a design with four transmission zeros, the prototype transfer functions meeting the above specifications are given by the polynomials [15] P(s) = 0.1128(s4 + 4.5861s2 + 4.9410), R(s) = s6 + 1.6716s4 + 0.7452s2 + 0.0589, Q(s) = s6 + 1.9528s5 + 3.5783s4 + 3.8036s3 + 3.1742s2 + 1.7013s + 0.5606. (5.46)

130

V. Van

The filter transfer function has six poles at pk = {−0.0744 ± j1.0609, −0.2961 ± j0.9154, −0.6059 ± j0.4103} and four transmission zeros at zk = {±j1.6900, ±j1.3153}. The complementary transfer function, R(s)/Q(s), also has six zeros on the imaginary axis at zk = {± j0.3166, ± j0.7834, ± j0.9786}. The pole-zero diagram of the filter is shown in Fig. 5.7a and its spectral responses are plotted in Fig. 5.7b. 2

–10 transmission (dB)

imag (jω )

1 0.5 0

–0.5 –1 –1.5 –2

0

poles zeros of P(s) zeros of R(s)

1.5

|F(s)|2

|H(s)|2

–20 –30 –40 –50 –60

–1.5 –1 –0.5 0 0.5 real ( σ )

1

1.5

–70 –40

–30

–20 –10 0 10 20 frequency detune, Δ f (GHz)

30

40

Fig. 5.7 (a) Pole-zero diagram of a sixth-order pseudo-elliptic microring filter. (b) Target spectral responses at the drop port, |H(s)|2 , and through port, |F(s)|2 , of the filter

From the complementary transfer function R(s)/Q(s) of the filter we first obtain the input impedance Zin of the equivalent network using (5.42). The short-circuit admittance elements y11 = y22 and y21 = y12 are then determined in accordance with (5.43) and (5.44), yielding the poles and residues as shown in Table 5.2. Using (5.38) and (5.39) we compute the input and output coupling coefficients to be μi = μo = 1.3974. The first and last rows of the orthonormal matrix T are next calculated using (5.40) and (5.41), and the remaining rows are obtained by Gram–Schmidt orthogonalization. Using T and the eigenvalue matrix , the coupling matrix M = X can now be determined from (5.34) as (only the upper half is shown since the matrix is symmetric): ⎡

⎤ 0 −0.3228 0.4824 −0.5568 −0.1639 −0.0578 ⎢ −0.9508 −0.2870 0 0 −0.3228 ⎥ ⎢ ⎥ ⎢ 0.4592 0 0 0.4824 ⎥ ⎢ ⎥. M=⎢ −0.4912 0.1646 0.5568 ⎥ ⎢ ⎥ ⎣ 0.9892 0.1639 ⎦

(5.47)

0 The above matrix corresponds to a coupling topology that is not physically realizable since it requires microring 1 to be coupled to all the other five microrings. By applying a series of matrix rotations as described in [28], the above coupling matrix is reduced to the canonical form given by the simpler matrix

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131

Table 5.2 Poles and residues of the short-circuit admittances of the equivalent electrical filter network Poles of y11 and y12

ξ 11 = ξ 22

ξ 12 = ξ 21

j1.1429 −j1.1429 j1.0507 −j1.0507 j0.4635 −j0.4635

0.0734 0.0734 0.1638 0.1638 0.2510 0.2510

−j0.0734 j0.0734 j0.1638 −j0.1638 −j0.2510 j0.2510

⎤ 0 0.8209 0 0 0 0.0578 ⎢ 0 0.5393 0 0.2840 0⎥ ⎥ ⎢ ⎢ 0 0.7820 0 0⎥ ⎥ ⎢ M1 = ⎢ ⎥. 0 −0.5393 0⎥ ⎢ ⎥ ⎢ ⎣ 0 −0.8209 ⎦ 0 ⎡

(5.48)

A schematic of the canonical microring filter is shown in Fig. 5.8a. In the coupling matrix M1 all the diagonal elements are zero, implying that the microring resonators are synchronously tuned to the center frequency ω0 of the filter passband. Note also that the coupling elements μ45 and μ56 are negative. Assuming microrings with FSR of 1 THz, the power coupling coefficients are computed and shown in the table in Fig. 5.8a. In Fig. 5.8b the spectral responses of the synthesized filter are plotted for both the case of no loss in the microrings and the case when there is a 5 dB/cm propagation loss. In the absence of microring loss, the synthesized filter

(a)

κ56

κo

κ45

(b) 0

κ16

sd st si

5

1 κi

4 κ25

2 κ12

κ34 3

κ23

-10

transmission (dB)

6

-20

|H(s)|2

-30 -40 -50

κi = κo = 0.3916

κ16 = 0.004538

κ12 = −κ56 = 0.06447

κ25 = 0.02231

-60

κ23 = −κ45 = 0.04236

κ34 = 0.06142

-70 -40

(FSR = 1THz)

|F(s)|2

-20

0

20

40

frequency detune, Δ f (GHz)

Fig. 5.8 (a) Schematic of the sixth-order canonical microring filter. (b) Spectral responses at the drop port (|H(s)|2 ) and through port (|F(s)|2 ) of the microring filter for the lossless case (solid lines) and the case of 5 dB/cm loss in the microrings (dashed lines)

132

V. Van

response is seen to be in good agreement with the target response. We also note that predistortion techniques can be applied to the design of coupled microring filters to recover the desired filter spectral shape in the presence of loss in the microrings [31].

5.5 Microring Ladder Filters Microring ladder filters are parallel-cascaded arrays of symmetric microring networks with phase shift elements between adjacent stages [18]. These filters can realize transfer functions with zeros that are quadrantally symmetric in the s-plane. In its simplest form, the microring ladder architecture consists of an array of symmetric microring doublets, as shown in Fig. 5.9 [17]. Each doublet in stage k is composed of two serially coupled microring resonators with inter-ring energy coupling coefficient μk and symmetric bus-to-ring coupling coefficients μbk . The microrings are assumed to be lossless and synchronously tuned to the same resonant frequency ω0 . Connecting adjacent microring stages are two parallel bus waveguides of identical lengths L, with a possible differential π -phase shift in the upper waveguide denoted by the factor γk = e−jπ = −1. Note that the output signals at the drop port and through port of each microring stage propagate in the forward direction to the next stage. Such a microring network is referred to as a feed-forward network. By contrast, in a parallel-cascaded array of odd-order microring stages, the drop and through port signals propagate in opposite directions, forming feedback loops which give rise to additional poles in the structure. These poles cannot be independently controlled since there are not enough free design parameters in the structure. As a result it is not possible to exactly synthesize a given target transfer function using arrays of feedback microring networks; instead only approximate solutions can be found using numerical optimization, yielding filter designs that are generally less efficient than those based on arrays of feed-forward networks.

a0 input

a1

ak−1

ak

aN

μb1

γ2 μb2

γk μbk

μbN

μ1

μ2

μk

μN

μbk

μbN

μb1

L b1

μb2

L bk−1

bk stage k

bar

F(s)

H(s) bN

cross

Fig. 5.9 Schematic of a microring ladder architecture consisting of N cascaded microring doublet stages [17]. Reprinted with permission. Copyright 2008 IEEE

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133

The transfer function of the microring ladder filter in Fig. 5.9 can be obtained by transfer matrix analysis. In the s-domain the transfer matrix Mk of the symmetric microring doublet in stage k has the symmetric form Mk =

  1 Fk (s) jKk , Gk (s) jKk Fk (s)

(5.49)

where s = j(ω − ω0 ) and Kk = μ2bk μk , Fk (s) = s2 + (μ2k − μ4bk /4), Gk (s) = s

2

+ μ2bk s + (μ2k

(5.50)

+ μ4bk /4).

The transfer matrix of the two bus waveguides connecting microring stages k–1 and k with differential phase shift factor γ k is given by k = e

−jβL



 γk 0 , 0 1

(5.51)

where γ k = ±1 and β is the propagation constant of the waveguides. The total transfer matrix of the N cascaded stages is given by T = (MN N )(MN−1 N−1 ) · · · (M2 2 )M1 .

(5.52)

The transfer functions of the ladder filter can be obtained from (5.52). Neglecting a common phase factor e−jβ(N−1)L , the transfer functions at the cross port and bar port, respectively, of the device can be expressed as H(s) = jPN (s)/QN (s) and F(s) = RN (s)/QN (s), where PN (s), RN (s), and QN (s) are polynomials satisfying the recursive relations Pk (s) = γk Kk Rk−1 (s) + Fk (s)Pk−1 (s), Rk (s) = γk Fk (s)Rk−1 (s) − Kk Pk−1 (s), Qk (s) = Gk (s)Qk−1 (s) =

k %

(5.53) Gn (s).

n=1

Starting with P1 (s) = K1 , R1 (s) = F1 (s), and Q1 (s) = G1 (s), where K1 , F1 , and G1 are given by (5.50), it can be deduced from the above recursive relations that PN (s) is a polynomial of degree 2(N – 1) while RN (s) and QN (s) are polynomials of degree 2N. Furthermore, since Kk and Fk (s) are even-order polynomials, both PN (s) and RN (s) are also even-order polynomials, implying that their roots are either located on or symmetrically distributed about the jω-axis in the s-plane. Thus the microring ladder filter with N stages can realize a transfer function with 2N poles and up to 2(N – 1) quadrantally symmetric transmission zeros. Each microring doublet in the

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V. Van

array is responsible for generating a pair of complex conjugate poles in the transfer function of the ladder. Since the transfer matrices Mk in (5.49) are symmetric with identical diagonal elements, their products are commutative, Mk Mk−1 = Mk−1 Mk . Thus two microring doublets connected by waveguides with no differential phase shift can be interchanged without affecting the response of the ladder filter. On the other hand, if two microring doublets are connected by waveguides with a differential π -phase shift, then it can be shown that Mk k Mk−1 = k (Mk−1 k Mk )k , where k = diag[−1, 1]. In this case the order of the two stages can be exchanged if a π -phase shift is also added before and after the two stages. These commutative properties of the Mk matrices can be used to reduce the number of phase shift elements in the array by permuting the order of the stages. In general it is always possible to reduce a given cascaded array to the canonical form which contains at most one π -phase shift element. A procedure for reduction to the canonical form is given in [18].

5.5.1 Filter Synthesis Given the prescribed filter transfer functions H(s) and F(s), the coupling parameters {μk , μbk } and phase shift γ k in each stage of the microring ladder can be determined using a procedure based on the order reduction technique for digital two-port lattice networks [32]. The synthesis procedure begins with the parameter extraction of the last microring stage N and proceeds backward until the first stage is reached. Each microring doublet stage k is completely characterized by a pair of conjugate poles {pk , p∗k } in the filter transfer function. Given a pole pk lying in the upper left-hand side of the s-plane, the coupling coefficients of the corresponding microring stage k are determined from μ2k = −2Re{pk }, (5.54) μbk = Im{pk }.

(5.55)

Note that the above expressions always yield positive values for the coupling coefficients. Knowledge of μk and μbk then allows us to construct the transfer matrix Mk of stage k as in (5.49), which can then be de-embedded from the cascaded array. The transfer functions of the array after stage k is removed are obtained by solving (5.53) for Pk−1 (s) and Rk−1 (s). The results are Pk−1 (s) =

Fk (s)Pk (s) − Kk Rk (s) , Gk (s)Gk ( − s)

Rk−1 (s) = γk−1

Fk (s)Rk (s) + Kk Pk (s) . Gk (s)Gk ( − s)

(5.56) (5.57)

The phase shift factor γ k of stage k appearing in (5.57) is chosen to be either 1 or −1 such that the ratio Rk−1 (pk−1 )/Pk−1 (pk−1 ) = j. The procedure is then repeated with the parameter extraction of the next stage, k−1, until the first stage is reached.

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135

Note that due to the commutative property of the transfer matrices Mk , the order in which the poles are used to extract the coupling parameters of the microring stages does not affect the final filter design, except for a possible permutation of the stage order and a difference in the phase shifts between adjacent stages. The ladder structure obtained in the above synthesis procedure usually contains more than one π -phase shift element in the array. The commutative properties of the transfer matrices are then used to reduce the design to the canonical form containing at most a single π -phase shift element. The canonical form of the double-microring ladder architecture requires 3N/2 coupling parameters and one π -phase shift element to realize an Nth-order transfer function with N − 2 quadrantally symmetric transmission zeros. An important advantage of the ladder architecture over the 2D direct-coupling topology is that all the coupling coefficients in the microring stages are always positive. Also, the ladder architecture is potentially more amenable to the realization of very high-order microring filters since it allows complex optical transfer functions to be decomposed into multiple stages of simple microring doublets, which can be independently designed, optimized, and fine-tuned.

5.5.2 Design Example We consider the microring ladder design of the sixth-order pseudo-elliptic filter whose transfer functions are given in Section 5.4.2. Two designs are presented: design {4, 2} consisting of an array of a fourth-order microring network cascaded to a microring doublet stage and design {2, 2, 2} consisting of an array of three microring doublet stages. The schematic and coupling parameters for each design are shown in Fig. 5.10. The designs have identical spectral responses, which are plotted in Fig. 5.11 for the case of no loss in the microrings and the case where there is a 5 dB/cm propagation loss. Note that the microring networks in both designs all have positive coupling coefficients, in contrast to the direct-coupled microring filter

(a)

(b)

κ12 κ23

κb1

2

1

3

4

κb2 5

κ14 κ12

π

κb1

κb1 = 0.3269 κ23 = 0.0777 κ b2 = 0.2157

κ56 κb2

π

κb1

κb2

1

3

2

4

κ1

κ2

6

κ12 = 0.0231 κ14 = 0.0378 κ56 = 0.0719

κb3 5

κb1

κ b1 = 0.1081 κb2 = 0.3085 κb3 = 0.2157

κb2

κ3 6

κb3

κ1 = 0.0833 κ2 = 0.0322 κ3 = 0.0719

Fig. 5.10 Alternative microring ladder filter designs: (a) design {4, 2} consisting of a fourthorder microring network cascaded to a microring doublet stage and (b) design {2, 2, 2} consisting of three microring doublet stages

136

0 –10

transmission (dB)

Fig. 5.11 Spectral responses at the drop port (|H(s)|2 ) and through port (|F(s)|2 ) of the microring ladder filter designs {4, 2} and {2, 2, 2} for the lossless case (solid lines) and the case of 5 dB/cm loss in the microrings (dashed lines)

V. Van

|F(s))|2

–20

2

|H(s)|

–30 –40 –50 –60 –70 –40

–20

0

20

40

frequency detune, Δ f (GHz)

in Section 5.4.2, which requires two negative coupling elements to realize the same transfer function.

5.6 Summary Advanced microring filter designs were presented for realizing general IIR optical transfer functions with transmission zeros. Of the three general filter architectures considered, the direct-coupled microring filter is the most compact and efficient in terms of the minimum number of design parameters required. However, for certain types of filter transfer functions the direct-coupling topology suffers from the disadvantage that it requires negative coupling coefficients, which are generally difficult to implement. The microring ladder architecture eliminates the need for negative coupling elements while retaining many attractive features of the direct-coupling topology such as synchronicity and compactness. The architecture is especially attractive for constructing very high-order optical filters since it allows complex transfer functions to be constructed in a modular stage-by-stage fashion. With continuing improvement in the fabrication technology and post-fabrication tuning capability, it is expected that microring filter technology will play an increasingly important role in advanced optical spectral and dispersion engineering applications.

References 1. Vahala, K.J. Optical microcavities. Nature 424, 839–846 (2003) 2. Grover, R., Absil, P.P., et al. Vertically coupled GaInAsP-InP microring resonators. Opt. Lett. 26, 506–508 (2001) 3. Barwicz, T., Popovic, M., et al. Microring-resonator-based add-drop filters in SiN: Fabrication and analysis. Opt. Express 12, 1437–1442 (2004)

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4. Xiao, S., Khan, M.H., et al. Compact silicon microring resonators with ultra-low propagation loss in the C band. Opt. Express 15, 14467–14475 (2007) 5. Little, B.E., Chu, S.T., et al. Very high order microring resonator filters for WDM application. IEEE Photon. Technol. Lett. 16, 2263–2265 (2004) 6. Xia, F., Sekaric, L., et al. Ultracompact optical buffers on a silicon chip. Nature Photon. 1, 65–71 (2007) 7. Chu, S.T., Little, B.E., et al. Compact full C-band tunable filters for 50 GHz channel spacing based on high-order micro-ring resonators. Optical Fiber Communication Conference, PDP9 (2004) 8. Little, B.E., Chu, S.T., et al. Microring resonator arrays for VLSI photonics. IEEE Photon. Technol. Lett. 12, 323–325 (2000) 9. Madsen, C.K., Chandrasekhar, S., et al. An integrated tunable chromatic dispersion compensator for 40 Gb/s NRZ and CSRZ. Optical Fiber Communication Conference, FD9-1 (2002). 10. Yang., J., Fontaine, N.K., et al. Continuously tunable, wavelength-selective buffering in optical packet switching networks. IEEE Photon. Technol. Lett. 20, 1030–1032 (2008) 11. Coppinger, F., Madsen, C.K., et al. Photonic microwave filtering using coherently coupled integrated ring resonators. Microwave and Opt. Technol. Lett. 21, 90–93 (1999) 12. Madsen, C.K. Efficient architectures for exactly realizing optical filters with optimum bandpass designs. IEEE Photon. Technol. Lett. 10, 1136–1138 (1998) 13. Madsen, C.K. General IIR optical filter design for WDM applications using all-pass filters. J. Lightwave Technol. 18, 860–868 (2000) 14. Madsen, C.K Zhao, J.H. Optical filter design and analysis: A signal processing approach. John Wiley & Sons, New York (1999) 15. Van, V. Synthesis of elliptic optical filters using mutually-coupled microring resonators. J. Lightwave Technol. 25, 584–590 (2007) 16. Prabhu, A.M., Van, V. Realization of asymmetric optical filters using asynchronous coupledmicroring resonators. Opt. Express 15, 9645–9658 (2007) 17. Liew, H.L., Van, V. Exact realization of optical transfer functions with symmetric transmission zeros using the double-microring ladder architecture. J. Lightwave Technol. 26, 2323–2331 (2008) 18. Prabhu, A.M., Liew, H.L., et. al. Generalized parallel-cascaded microring networks for spectral engineering applications. J. Opt. Soc. Am. B 25, 1505–1514 (2008) 19. Weinberg, L. Network analysis and synthesis. McGraw-Hill, New York (1975) 20. Cappellini, V., Constantinides, A.G., et. al.: Digital Filters and Their Applications. Academic Press, London (1978) 21. Jinguji, K. Synthesis of coherent two-port optical delay-line circuit with ring waveguides. J. Lightwave Technol. 14, 1882–1898 (1996) 22. Ansari, R., Liu, B. A class of low-noise computationally efficient recursive digital filters with applications to sampling rate alterations. IEEE Trans. Acoust. Speech Sig. Proc. 33, 90–97 (1985) 23. Regalia, P.A., Mitra, S.K., et al. The digital all-pass filter: A versatile signal processing building block. Proc. IEEE 76, 19–37 (1988) 24. Rhodes, J.D. A low-pass prototype network for microwave linear phase filters. IEEE Trans. Microwave Theory Tech. 18, 290–301 (1970) 25. Little, B.E., Chu, S.T., et al. Microring resonator channel dropping filters. J. Lightwave Technol. 15, 998–1005 (1997) 26. Van, V. Circuit-based method for synthesizing serially-coupled microring filters. J. Lightwave Technol. 24, 2912–2919 (2006) 27. Atia, A.E., Williams, A.E., et al. Narrow-band multiple-coupled cavity synthesis. IEEE Trans. Circuits Syst. 21, 649–655 (1974) 28. Cameron, R.J. General coupling matrix synthesis methods for Chebyshev filtering functions. IEEE Trans. Microwave Theory Tech. 47, 433–442 (1999)

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29. Cameron, R.J. Advanced coupling matrix synthesis techniques for microwave filters. IEEE Trans. Microwave Theory Tech. 51, 1–10 (2003) 30. Popovic, M.A. Sharply-defined optical filters and dispersionless delay lines based on loopcoupled resonators and ‘negative’ coupling. IEEE Conf. Lasers Electro-Optics CThP6 (2007) 31. Prabhu, A.M., Van, V. Predistortion techniques for synthesizing coupled microring filters with loss. Opt. Comm. 281, 2760–2767 (2008) 32. Jinguji, K., Kawachi, M. Synthesis of coherent two-port lattice-form optical delay-line circuit. J. Lightwave Technol. 13, 73–82 (1995) 33. Van, V., Prabhu M.A. Multiple coupled microresonator devices for advanced spectral shaping applications. Proc. SPIE 6872, 68720S (2008)

Chapter 6

Band-Limited Microresonator Reflectors and Mirror Structures Otto Schwelb and Ioannis Chremmos

Abstract An overview of microring resonator architectures, fabricated using both fiber and integrated technologies, whose function is to reflect nearly the entire incident signal within a specified band of frequencies, is presented. The signal components that lie outside this band, called the stopband, are either transmitted through the device to exit through another port, or dissipated. The stopband reflectivity is ideally close to unity, while outside the stopband the reflectivity is very small; the device performs the function of a band-limited optical mirror. A primary application for such a mirror is in semiconductor laser devices, where its frequency selective properties are exploited to eliminate unwanted resonances of an active medium with a broad gain characteristics. Another application involves the mirror group delay characteristics in reflection, which can, in some architectures, be shaped to compensate for the group delay distortion of an incident signal, or control the Q factor of an oscillator. Comparisons of the reflection coefficient provided by various architectures in terms of resonator loss and as a function of the coupling strength to the input waveguide are included.

6.1 The Optical Fiber Loop Mirror Early proposals to use optical fiber-based loop mirrors in conjunction with fiber lasers were presented by Mortimore [1] and Urquhart [2]. The fiber-based loop mirror, shown in Fig. 6.1, consists of a nominally 3 dB directional coupler and a length of fiber waveguide serving as a feedback line connecting the output ports of the coupler. There are two signal paths from the input port back to the input port. These travel through the feedback line in opposite direction and arrive in phase back at the input port. On the other hand, the signals that feed the other free port of the coupler O. Schwelb (B) Department of Electrical and Computer Engineering, Concordia University, Montréal, QC, Canada e-mail: [email protected] I. Chremmos et al. (eds), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_6,  C Springer Science+Business Media, LLC 2010

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1o

a1 b1

2o

K L α

b2

Fig. 6.1 Fiber loop mirror. Due to the destructive interference between counterpropagating waves there is no output at port 2 when the power coupling coefficient K = 0.5 (3 dB). The loss√ per turn is σ = exp(–αL) ≈ 1. The direct connections of the coupler, indicated by solid lines, are (1–K), √ the coupled connections, indicated by dashed lines, are j K. The amplitude loss of the coupler, if √ any, is denoted by (1–γ ) ≈ 1–γ /2, where γ is the fractional power loss

cancel each other, because they acquire a 180◦ phase difference on account of the 90◦ phase shift each time the signal crosses the coupler. Note that counterpropagating waves travel the same distance in the fiber, suffering the same loss. Assuming the coupler to be a lossless fused or lumped element (as opposed to a distributed parameter) device, or its losses to be absorbed into those of the fiber, the scattering matrix elements of the mirror can be obtained by inspection: , + S11 = S22 = b1 a1 =j2σ K (1 − K) exp (−jβL)

(6.1)

+ S21 = S12 = b2 a1 =σ (1 − 2 K) exp (−jβL)

(6.2)

and

where K is the power coupling coefficient of the coupler, L the length of the loop, β the phase constant, σ = exp ( − αL) the per turn loss (σ ≤ 1), and α the amplitude attenuation coefficient of the fiber. Letting K = 1/2–ε, where ε is a small deviation from the nominal value, we find that |S11 |2 ∼ = σ 2 and |S21 |2 = 4σ 2 ε2 . In practice σ is very close to unity and ε  1, therefore mirror functionality is well approximated. In the analysis above an ideal, symmetric and lossless coupler has been assumed. Corrections to the phase relationship between direct and coupled waves in a coupler with measurable differential phase shift between even and odd modes have been reported in [3]. Such a distortion would result in a polarization dependence of the mirror. While the length of the loop is arbitrary a number of issues must be considered. First, to achieve high reflectivity σ should remain close to unity, therefore the αL product should be held to a minimum. Second, if the mirror is used in conjunction with a laser, L must be commensurate with its coherence length, and the traveling time along the fiber with its coherence time. Finally, the birefringence of the fiber, a cumulative effect along its length, should not cause any serious interference with the

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operation of the mirror. Remember that the effective refractive indices, and thus the propagation constants, are different for the orthogonal polarizations of a birefringent fiber, giving rise to different optical lengths for the loop. This last caveat, the most critical of the three, has been addressed in detail, invoking Jones calculus, in [1].

6.2 Discontinuity- And Grating-Assisted Reflectors The most serious shortcoming of the fiber loop mirror is its lack of band limitation, which is observed only as a result of the frequency characteristics of the coupler, and to some extent caused by the birefringence of the fiber. However, these properties do not allow bandwidth (BW) control. Perhaps one of the simplest reflectors, offering a relatively well-defined BW, is the discontinuity-assisted ring resonator (DARR), which consists of a fiber or integrated microring resonator into which a discontinuity is deliberately inserted. The resonator with the discontinuity is then coupled to a bus waveguide as shown in Fig. 6.2. The discontinuity couples clockwise and counterclockwise traveling waves, effectively coupling the two degenerate resonances of the ring, converting a first-order resonator into a second-order one, with the attendant sharpening of the BW and the splitting of the resonance frequencies [4]. To obtain a relatively narrowband reflector the resonant ring must have a large Q factor which, in turn, mandates a minimal external load, i.e., a small power coupling coefficient K. As shown in the analysis below, there is a direct relationship between K and the optimum reflection coefficient r d to produce a large reflection at the input port. Although the circuit represented in Fig. 6.2 is simple and can easily be analyzed by inspection, we outline here a more general scattering matrix treatment that offers significant advantages in cases of more complex architectures, where

a1 b1

b2

K

1o 3

o 2 4

L rd Fig. 6.2 The discontinuity-assisted ring resonator (DARR) mirror. The discontinuity, whose reflection coefficient is rd and whose location within the ring is arbitrary, couples counterpropagating waves and under proper conditions gives rise to a large reflected signal at port 1. The discontinuity can be a notch, a gap, or a short length of waveguide with altered refractive index

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analysis by inspection or a solution of the often used node equations becomes far too cumbersome. At the outset observe that Fig. 6.2 represents a feedback-assisted four-port network where the four-port is the coupler, and the feedback circuit consists of the waveguide of length L, incorporating the discontinuity, connecting ports 3 and 4. The scattering matrix of the coupler [5] relates the four exiting waves bi , i = 1 to 4, to the four incident waves a i through the linear relation b = Sa, where b = [b1 b2 b3 b4 ]T , a = [a1 a2 a3 a4 ] T (the T superscript stands for transposition) and ⎡

√ 0 , ⎢ 1−K S= 1−γ ⎢ ⎣ √0 j K

√ √ ⎤ 1 − K √0 j K   ⎥ j K √ 0 ⎥ = SA S B , √0 SC SD j K √ 0 1−K⎦ 0 1−K 0

(6.3)

where γ is the fractional power loss. The subdivision of S into 2 × 2$submatrices + appears on the right. One often associates a coupling angle ϕ = tan−1 K (1 − K) √ √ to K, thus: cos ϕ = 1 − K and sin ϕ = K, quantities which, in what follows, will often be abbreviated by c (not to be confused with the speed of light) and s, respectively. Note that this description of the coupler assumes symmetry, i.e., zero differential phase between even and odd modes, and exactly π /2 phase shift across the coupled (dashed line) connections. When ports 3 and 4 are connected through a feedback line, as in Fig. 6.2, characterized by a 2 × 2 embedding matrix SE , it can be shown [5] that the scattering matrix of the resultant two-port is  −1 SC . SR = SA + SB S−1 E − SD

(6.4)

In the present case the feedback line is the resonator with length L and propagation constant k = β − jα into which the discontinuity is inserted. The location of the discontinuity turns out to be immaterial. While its reflectivity must be carefully controlled, the fabrication of the discontinuity is technology dependent. In a fiberbased DARR it can be a narrow gap, i.e., a very short Fabry–Pérot cavity as in [6]. In integrated optics technology it can be a single step-index dielectric or a shallow (h  λ) notch, as suggested in [7]. We assume a bilaterally symmetric discontinuity, most conveniently represented by [8]  rd jtd , Sd = jtd rd 

1

(6.5)

where td = (1 − rd2 ) 2 is the transmission coefficient of the lossless discontinuity. In practice the reflection coefficient is small, in the order of 10−1 to 10−4 therefore, to

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obtain a high Q resonance, K must be small as well, usually between –10 and –40 dB. The refractive index nd and length ld of a defect layer are determined from    1 − exp ( − j2kd ld )   , rd = ρ (6.6) 1 − ρ 2 exp ( − j2kd ld )  where kd is the wavenumber of the defect layer, neff the effective index of the ring waveguide and   nd − neff  . (6.7) ρ= nd + neff +  The maximum value of rd provided by (6.6) is rmax = 2ρ 1 + ρ 2 and the relative index and length of the defect layer are, respectively,  nd = neff

1 + rmax λ0 and ld = (2m + 1) 1 − rmax 4neff



1 − rmax , 1 + rmax

(6.8)

where m is an integer, preferably as small as possible to shorten the discontinuity. Substituting the appropriate embedding matrix SE into (6.4) and assuming a lossless coupler (for an analysis including a lossy coupler see [9]) the two distinct elements of the scattering matrix characterizing Fig. 6.2 are obtained as S11 =

rd K

√ (1 − K) exp ( − jkL) − exp (jkL) + j2td 1 − K

(6.9)

and √ [ exp ( − jkL) − exp (jkL)] 1 − K + jtd (2 − K) (6.10) S21 = √ (1 − K) exp ( − jkL) − exp (jkL) + j2td 1 − K √ √ Using the compact notation c = 1 − K and s = K, the reflected intensity is obtained as −1 



1 2 1 2 2 − 4rd + 4 sin (βL) − 4td cσ + , sin (βL) cσ + cσ cσ (6.11) which produces+ two maxima (splitting) in the undercoupled regime, i.e., when K < Kc = 2rd (1 + rd ) ∼ = 2rd . These maxima are centered at βL = (2n + 12 )π , upshifted by FSR/4 from the ring resonance due to the π /2 phase shift through the discontinuity represented by (6.5). For small values of rd the frequency separation between the maxima for a fixed K < Kc is proportional to rd , as confirmed by a recent study of the discontinuity-assisted ring resonator [10]. A plot of the transmitted intensity |S21 |2 , seen in Fig. 6.3, indicates two minima in the undercoupled regime. The frequency separation between these minima for a fixed rd is well r 2 s4 |S11 | = d 2 c 2

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0

I21 (dB)

-20 -40 -60 -80 0.03 0.025 0.02 0.015 K

0.02 0.01 0.01 0.005

0 -0.01 -0.02

-1 Δ (1/λ ) + 1.6129 (mm )

Fig. 6.3 Transmitted intensity through a lossless DARR coupled to a waveguide with coupling coefficient K. The ring mode number is N = 100 (L ≈ 34.45 μm), λ0 = 1.55 μm, neff = 4.5. Each division on the frequency scale represents 3 GHz, the maximum split at vanishingly small coupling is approximately 6 GHz. When K > Kc = 0.02 there is only one minimum whose depth reduces with increasing K. The reflection coefficient of the discontinuity is rd = 0.01018, and the center wavelength is shifted by FSR/4c = 1.6129 mm−1

$ + approximated by f ∼ = const rd2 − (K 2)2 . In the overcoupled K > Kc regime DARR resonance occurs at βL = (2n + 12 )π , where the magnitude of the reflected signal is at a maximum. The skirt selectivity, or shape factor of the mirror can be significantly improved by coupling several slightly undercoupled DARR’s to the same bus waveguide in parallel, as+demonstrated in [9]. The separation between them must be a multiple of π = λ0 2neff to achieve constructive interference. The group delay in reflection is an important attribute of the mirror. When the mirror is associated with a laser the group delay in reflection contributes to the total round trip time, which in turn determines the oscillator Q = 2π f0 τ p , where f0 is the resonance frequency and τ p is the photon lifetime, the time it takes for a round trip in the cavity, which includes the delay time within the laser plus the group delays in reflection of the mirrors. The characteristics of the latter are computed from (6.9) through [5]   ∂φij 1 ∂Sij = −Im , i,j = 1,2 , τij = − ∂ω Sij ∂ω

(6.12)

where φ ij is the phase of Sij . Due to reciprocity S21 = S12 , therefore τ21 = τ12 , moreover for a bilaterally symmetric network τ22 = τ11 . The transmission phase in a lossless reciprocal two-port is φ21 = 12 (φ11 + φ22 + π ± 2 Nπ ) [11], therefore

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τ21 = 12 (τ11 + τ22 ). This means that for a symmetrical network τ11 = τ21 as long as the network is lossless. We also conclude that while τ21 is independent of network asymmetry τ11 and τ22 are not. The group + delay in reflection, normalized to the delay in the ring at resonance τ0 = Lng c, as a function of detuning, is shown for a lossless DARR in Fig. 6.4, with rd as the parameter ranging from + 0.006 to 0.013 in steps of 1 × 10−3 . The solid line corresponds to rd = Kc 2 = 0.01. Notice that the group delay becomes a flat-top characteristic in the overcoupled regime, i.e., when K>Kc ∼ =2rd . As loss is introduced into the ring the double-hump τ 11 /τ 0 characteristic starts to flatten out until, at σ ≈ 0.993, it becomes practically + flat at τ11 τ0 = 88, subjecting the reflected signal to a 5.5 dB insertion loss (IL). This IL effectively reduces the reflection coefficient of the mirror, as we shall see later in this chapter where the efficiency of a number of reflector architectures is compared. Probably the first fiber-based DARR was introduced by Paschotta and coworkers [6] who used a circuit theory approach to calculate the device transmission and reflection coefficients in terms of the reflectivity of the discontinuity |rd |2 and the power coupling coefficient K. They also fabricated a fiber-based DARR by splicing the fiber loop and placing the facets into close proximity. By varying the distance between the facets Paschotta et al. could tune the reflection coefficient between near zero and ∼16%. The problems they encountered were mostly twofold. On the one hand the loop length could not be made sufficiently short (the FSR sufficiently large) to achieve single-mode operation, and on the other hand the splice gap length could not be stabilized to allow repeatable results. A study of a similar configuration, using σ = 1, K = 0.02, N r = 100

200

150

τ21 / τ0

0.006

100 0.013

50

0

–0.04

–0.02 0 0.02 –1 Δ (1/ λ )+1.6129 (mm )

0.04

Fig. 6.4 Normalized group delay in transmission as a function of detuning from resonance for a lossless DARR. Note, that for a lossless device the group delay in reflection τ 11 and in transmission τ 21 are identical. The power coupling coefficient is 0.02, the parameter is the reflection coefficient of the discontinuity rd , which varies from 0.006 to 0.013 in steps of 1 × 10−3 . The solid line corresponds to critical coupling at rd ∼ = 0.01. The vertical lines are computational artifacts

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flow-graph analysis and producing the same results as those in [6] although without experimental corroboration, was reported in [12]. A discontinuity-assisted Mach–Zehnder interferometer (DAMZI), where a discontinuity-assisted fiber ring was used as an external feedback loop to a Mach– Zehnder interferometer, as shown in Fig. 6.5, has been reported in [13]. The authors claim that this circuit has increased flexibility compared to the original DARR described above, in particular the extension of the FSR through the Vernier effect [14], obtained by judicious selection of the transmission line lengths L1 and L2 within the interferometer. Observe that there are two closed loops in the DAMZI, one through L2 and the direct connections of the couplers, the other through L1 and the coupled connections of the couplers. However, detailed calculations show that the suppression of the interstitial modes is only about 6 dB, which may turn out to be insufficient for the effective control of unwanted resonances. Another utility of this device, unexplored in [13], is the tunability of the resonance frequency through a fractional shift of the location of the discontinuity which can be effected by thermo-optic means. K1

K2

L1

1 o

2 o

L2

o 3

La

o 4

Lb

rd

Fig. 6.5 The discontinuity-assisted Mach–Zehnder interferometer (DAMZI) described in [13]. Near 100% reflection at port 1 can be obtained for rd  1 when either K1 + K2 ∼ = rd or K1 + K2 ∼ = 2 − rd . The two resonant loops, one through L1 the other through L2 allow Vernier operation of the device

The circuit illustrated in Fig. 6.5 can be analyzed by means similar to those described above using, instead of (6.3), the scattering matrix   0 SB (6.13) 23 , S = 23 T SB 0 where the superscript T represents transposition,  c1 c2 e−jθ1 − s1 s2 e−jθ2 js2 c1 e−jθ1 + js1 c2 e−jθ2 , SB = js1 c2 e−jθ1 + js2 c1 e−jθ2 c1 c2 e−jθ2 − s1 s2 e−jθ1 , , ci = 1 − Ki , si = Ki , θi = (β − jα)Li , i = 1,2, and 

⎡

23

1 ⎢0 =⎢ ⎣0 0

0 0 1 0

0 1 0 0

⎤ 0 0⎥ ⎥ 0⎦ 1

(6.14)

(6.15)

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is a permutation matrix needed because ports 2 and 3 have been interchanged. Application of this analysis produces results similar to those demonstrated above, namely close to 100% reflection in a narrow band of frequencies upshifted from the ring resonance by FSR/4 when K1 + K2 ∼ = rd , or in a narrow band of frequencies downshifted from the ring resonance by FSR/4 when K1 + K2 ∼ = 2 − rd , rd  1. The group delay characteristic in reflection of the DAMZI is much the same as that plotted in Fig. 6.4, including its dependence on the strength of coupling and on resonator loss. Recently, to be attached to fiber lasers, a new DARR mirror has been proposed by Sun and coworkers [15] that uses a high birefringence (HB) fiber in one section of the ring (from one end of the coupler to the discontinuity) and an ordinary singlemode (SM) fiber in the other section (from the other side of the discontinuity to the far end of the coupler). The waves polarized along the slow and fast axes of the HB fiber travel at slightly different velocity, imparting the ring two distinct resonance frequencies. In this way a Vernier effect is obtained in a single ring. An additional linear polarizer in the incident arm of the bus waveguide, aligned 45◦ with respect to the fast axis of the HB fiber, serves to further discriminate against those frequency components for which θfast − θslow = 2mπ . Using n s = 1.48 as the index seen by the slow wave and n f = n0 = 1.46 as the index seen by the fast wave and also by the SM fiber, a very large FSR enhancement of 292 was obtained, although again with only 6 dB side mode discrimination. A laser mirror fabricated on the same chip with the active medium, using a microring, a cleaved facet, and an absorbing pad, as shown in Fig. 6.6, has been described by Liu and coworkers [16]. Here the facet furnishes the reflective discontinuity, the ring the sharp wavelength selection necessary for single-mode operation, and the absorbing pad the removal of the unwanted off-resonant side modes. The gain and the absorption regions are materially identical with and without forward bias, respectively. To extract maximum transmission through the ring the coupling coefficients between the ring and the two passive waveguides must satisfy the

Fig. 6.6 Laser gain region joined to a mirror fabricated on the same chip using a microring resonator for wavelength selection, a cleaved facet reflector, and an absorber for spectral components lying outside the band of the ring resonance [16]. A similar mirror can be attached to the other end of the gain region

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relationship c1 = σ c2 , where σ is the per turn loss as defined above & [17]. The & √ large Q factor of the ring produces a large finesse F = FSR f = π G (1 − G) and a large+reflection + group delay enhancement at resonance, measuring approximately Leff L ∼ G (1−G), where G = σ c1 c2 and L is the physical ring perimeter = [17]. Considering that in practice G is slightly less than unity, both the finesse and the group delay enhancement can be very large. In fact the time delay in the reflector, or reflectors if there are two attached to both ends of the active region, can be significantly greater than that in the active region. The relatively large group delay increases the effective cavity length of the laser and thereby the photon lifetime. This in turn causes a reduction of the linewidth and the chirp in a semiconductor laser. When microring-based reflectors are attached to both ends of the gain region, as in [18], the opportunity arises +to fabricate rings with fractionally different mode numbers, defined by N = neff L λ0 , also known as azimuthal mode numbers, to exploit the Vernier effect [19], and extend the FSR of the reflectors. The principle of operation is illustrated in Fig. 6.7, showing the transmitted intensity through two series (direct) coupled rings as a function of detuning. The mode + numbers of the rings are N1 = 800 and N2 = 720, respectively, providing a N1 (N1 − N2 ) = 10 -fold extension of the FSR. Such a configuration allows the laser to be tuned over a much larger range of wavelengths than the one shown in Fig. 6.6, or one utilizing identical rings in the two reflectors. A grating-assisted single microring reflector has recently been described [20] that offers tunability coupled with an extremely sharp stopband characteristic and high sidelobe suppression (SS). The gratings provide the means to suppress repetition of the reflection spectrum at every FSR removed from the design wavelength. The architecture, shown+ in Fig. 6.8, features two identical quarter wavelength shallow gratings (|nb − na | na  1) each having NG periods and three alternative paths to

Fig. 6.7 A plot of the transmitted intensity through two coupled rings with fractionally different radii, illustrating the Vernier effect. The mode numbers of the rings are N1 = 800 and N2 = 720, respectively, resulting in a FSR enhancement of 10. Solid lines: resonant modes of ring #1, dotted lines: resonant modes of ring #2. Resonant enhancement requires an overlap of the resonances. The abscissa must be multiplied by 300 to obtain the detuning in GHz

6

Band-Limited Microresonator Reflectors and Mirror Structures Kc

1

(1- h ) L

149

2

hL

L

K NG

l3

l4

NG

o

o o

o o

o

o o

o o

type I

type II

o o

type III

Fig. 6.8 Three types of grating-assisted microring resonator reflectors. The direct and coupled connections of the coupler K, detailed at the bottom, are indicated by the solid and dotted lines, respectively. N G is the number of periods in each grating and h (0 < h < 1) locates the coupler K on the ring. The attenuation coefficient of the waveguides is α, the ring need not be circular

connect them to the microring. The arms l3 and l4 are characterized by their lengths in terms of mode numbers N3 and N4 , respectively, + while the gratings are characterized by the grating strength: GS = NG |nb − na | na . Balanced operation is ensured by selecting K ∼ = 0.5, while Kc must be small to prevent loading by the external circuitry and to preserve the resonance spectrum of the grating-assisted ring [21]. Here we remark that a type II mirror using Kc = 0.5 and 0.01 ≤ K ≤ 0.05 also produces a very high SS, relatively low return loss, or reflection loss, (RL) mirror. Thermo-optic tuning is available through length or refractive index adjustment affecting the mode numbers [22, 23]. Coupling scheme differences result in significant diversity in performance as well as demands on fabrication. Foremost among these is the extremely large SS and the heightened sensitivity to K in the vicinity of K = 0.5, observed in the type I mirror for a particular mode number combination, as described below. The device is analyzed by first establishing the 2 × 2 scattering matrix SK of the coupler K with arms l3 and l4 and the gratings attached. By building this result into the ring the relevant elements of the scattering matrix of the entire reflector can then be calculated, assuming α = 0 and h = 0.5, to yield

S11 =

  s2c SK,22 cc SK,21 ejβL/2 − cc SK,21 − ejβL 1 − 2cc SK,21 − c2c det (SK )

(6.16)

and

S21 = cc +

( ) 2 s2c cc SK,21 ejβL/2 − cc SK,11 SK,22 − SK,21 ejβL 1 − 2cc SK,21 − c2c det (SK )

(6.17)

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1 + 1 where cc = (1 − Kc ) 2 , sc = Kc2 , and det (SK ) = Γ3 Γ4 , (Γ3 Γ4 − c2 ) (1 − c2 Γ3 Γ4 ), + or (Γ3 Γ4 + s2 ) (1 + s2 Γ3 Γ4 ), for mirror types I, II, or III, respectively, with Γ 3,4 representing the complex reflection coefficients of arms 3 and 4, consisting of guide lengths l3 and l4 , respectively, terminated by their corresponding gratings of lengths N G , where  is the grating periodicity. Note that when dealing with lossy waveguides β must be replaced by β − jα Computed reflected intensity characteristics of the lossless type I mirror are shown for three azimuthal mode numbers, namely N = 60.25, 100.25, and 140.25 in Fig. 6.9, indicating a SS in excess of 60 dB. Also shown in the figure is the RL characteristic of the grating. The large SS is the result of the destructive interaction of the signals reflected from the gratings at the appropriate wavelengths, whose phase is determined by the mode numbers of the side arms, in this case N3 = N4 = 10.25. If, on the other hand, these mode numbers are selected as N3 = 10.25, N4 = 10, or vice versa, the SS drops to between –10 and –26 dB for the same size microrings due to the lack of destructive interaction at the sidelobe wavelengths. In both cases K = 0.4944 has been chosen, but while the high SS type I mirror is very sensitive to K as |K − 0.5| approaches zero, the low SS variety is practically insensitive to it. It is apparent that this grating-assisted reflector configuration, featuring three thermo-optically adjustable mode numbers, is very versatile with respect to tunability, parameter sensitivity, and group delay in reflection. To exploit the multifaceted characteristics of this device Vazquez and Schwelb offer a table of mode number selection rules to guide the designer. The BW of the resonant peak is affected by the combined value of the mode numbers (N + N3 + N4 ) and by the external coupling coefficient Kc . Waveguide attenuation, and/or grating loss through leakage, depresses the peak but does not directly affect the BW. Small changes in K have noticeable effect on the BW only in the high SS type I mirror, through overcoupling

Fig. 6.9 Reflected intensity of the type I mirror for three azimuthal mode numbers when N3 = N4 = 10.25. Also shown (dotted line) is the RL characteristics of the gratings. N G = 600, na = 4.5294, nb = neff = 4.5 (GS = 3.92), K = 0.4944, Kc = 0.02, λ0 = 1.5791 μm, and σ = exp(–αL). As |K – 0.5| approaches zero the SS increases

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151

or undercoupling. In general terms, increasing the combined mode number and/or decreasing K c reduces the BW, in accordance with Q=

2π N f0 2π N α=0 = −−−→ BW Kc + 2TG + 2αL Kc + 2TG

(6.18)

where TG denotes the power loss through a grating. A decrease in the value of K c also reduces the resonance peak, because with reduced coupling less external power is needed to maintain cavity oscillation. The RL depends on the leakage through the gratings determined by the GS, the resonator loss, 1 − σ and the power escaping through port 2, i.e., |S21 |2 . Resonator loss is determined by the per turn power loss 2αL, but reaches a peak at f0 due to resonant enhancement. If there is no leakage through the gratings due to a very large GS, the RL is a function of α, L, and K c . In contrast, when waveguide loss is negligible and the RL is driven by leakage through the gratings, only Kc and the GS affect the RL. Figs 6.10 and 6.11 display the dependence of RL on α and GS, respectively, at f0 for type I (solid lines) and type II (dash-dot lines) reflectors. The properties of the type III reflector are the same as those of the type II reflector. The diagrams show that type I mirrors have significantly smaller RL than the other types and that increasing Kc reduces the RL. (Note that positive numbers are used for loss, a negative level.) Coupling between counterpropagating waves on a microring resonator, obtained by a shallow grating fabricated onto the sidewall of the ring, has been reported recently [24]. Measured and calculated values of the splitting of the natural

Fig. 6.10 Computed RL at f0 in type I (solid lines) and type II (dash-dot lines) mirrors as a function of α. The nominal mode numbers are: N = 100, N3 = N4 = 0, GS = 13, K = 0.5. The parameter is K c . The characteristics of the type III mirror are the same as those of the type II mirror

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Fig. 6.11 Computed RL at f0 in type I (solid lines) and type II (dash-dot lines) mirrors as a function of the GS. The nominal mode numbers are: N = 100, N3 = N4 = 0, α = 0, K = 0.5, the parameter is Kc . The characteristics of the type III mirror are the same as those of the type II mirror

resonance of the resonator have shown good agreement. No results on the reflection coefficient of the device have been reported.

6.3 Multiring Reflectors With the trend toward miniaturization and advances in photonic waveguide fabrication the focus has increasingly shifted from fiber-based to integrated technology utilizing evanescent field coupled microrings and bus waveguides. As mentioned above Little et al. [7] suggested cutting a λ/4 wide shallow (h  λ) notch in a ring waveguide coupled to a bus waveguide to fabricate a second-order filter from a single microring. The reflection coefficient of this device was computed using coupling of modes in time formalism, as well as using a finite-difference time-domain (FDTD) simulator. Reasonable agreement was observed. It was pointed out in [7] that while the characteristics of the discontinuity-assisted single ring is identical to that of the double ring add/drop filter, it has the advantage of assured symmetry, as the same ring is used to furnish both resonators. The authors also suggested the use of the reflective properties of a resonator discontinuity, especially in an ultra-high Q microsphere configuration, to sense the presence of minute surface perturbations caused, e.g., by microparticles in the spirit of the biosensor suggested in [25]. One of the first multiring configurations proposed to be used as a mirror was the so-called figure 8 reflector, depicted in Fig. 6.12 [26]. This device can be viewed as a modified DARR where the discontinuity is replaced by the loop mirror of Fig. 6.1. When fabricated as an integrated device one must critically consider the small bending radii of the coupler region, and the associated potentially large radiation loss.

6

Band-Limited Microresonator Reflectors and Mirror Structures

Fig. 6.12 A figure 8 mirror described in [26, 27]. The partial lengths L1 and L2 are, in general, unequal. When fabricated using integrated technology large radiation loss can occur in the small bending radius coupler region unless high index contrast between core and cladding is maintained

1

Kc

o

153 2 o

L2

L

K

L

L1

Calculations that follow those outlined above to obtain (6.9) and (6.10), with the replacements rd → j2scσ e−jβL and jtd → (c2 − s2 )σ e−jβL , appropriate to Fig. 6.1, lead to the scattering matrix of the mirror [27]:  SM =

j2s2c cs exp [j(θ1 −θ2 )] D s2 [c2 −s2 −cc exp (−2jθ)] cc + c D

s2c [c2 −s2 −cc exp (−2jθ)] D j2s2c cs exp [j(θ2 −θ1 )] D

cc +

 ,

(6.19)

where θi = (β − jα)Li , i = 1,2 refer to the generally unequal resonator sections of the left ring, θ = θ1 + θ2 and D = 2(c2 − s2 )cc − σ −2 ej2βL − c2c σ 2 e−j2βL . In a lossless device, for any given Kc , there is an optimal value of K = Kopt that yields an optimal suppression level for the transmitted intensity I21 = 20 log10 |S21 |, or IL, and thus an optimal reflected intensity I11 = 20 log10 |S11 |, or RL, while assuring acceptable BW. These are related through   I11 = 10 log10 1 − 10I21 /10 .

(6.20)

Thus, to achieve a lossless passband reflection coefficient of, say, |S11 | = 0.9995, the couplers in the mirror must be adjusted to give IL = –30 dB, while a more modest reflection coefficient of |S11 | = 0.995 requires only IL = –20 dB. After inclusion of loss this optimized reflection will, of course, diminish because loss absorbs some of the intensity destined to be reflected. Typical reflected and transmitted intensity characteristics of the figure 8 reflector have been reported in [27] indicating significant improvement in selectivity, i.e., form factor, when two identical figure 8 mirrors are connected in parallel to the same bus waveguide. In the optimization process a K value that provided an IL of approximately –27 dB at f0 was selected. Computed values of the dependence of Kopt on K c , at IL = –27 dB, for a lossless mirror are plotted in Fig. 6.13. Optimization does not depend on L, but the BW is inversely proportional to it. Fig. 6.13 also plots the BW at IL = –15 dB, for L = 0.517 mm (N = 500), λ0 = 1.55 μm. To obtain the corresponding

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Fig. 6.13 The dependence of Kopt at IL = –27 dB, and bandwidth (BW), at IL = –20 dB, on Kc for the lossless figure 8 mirror shown in Fig. 6.12 (L = 517 μm, N = 500, neff = 1.5)

BW for, e.g., N = 200 multiply the BW values of Fig. 6.13 by the perimeter ratio: 500/200 = 2.5. An important figure of merit for microring-based reflectors is their sensitivity to resonator loss. This is measured by the reflection coefficient reduction ratio which compares the reflection coefficients with and without loss and can be evaluated directly from SM . For the figure 8 mirror, using the optimized value Kopt and assuming L1 = L2 the result is [27] 2(c2opt − s2opt )cc − 2 + s2c |SM11 |σ ,f =f0 = . |SM11 |σ =1,f =f0 2(c2opt − s2opt )cc − σ −2 − c2c σ 2

(6.21)

Computed values of this figure of merit are reported below in a universal plot as a function of Kc , using a K = Kopt appropriate for Kc , for four different multiring mirror configurations. A novel design of multiring wavelength selecting reflector was proposed by Poon and coworkers [28] employing a circular array of identical microrings coupled to a bus waveguide as shown in Fig. 6.14. It is immediately apparent that when an odd number ≥ 3 is assembled in such circular configuration there will be a reflected signal at port 1, on the other hand when the number of rings in the circular array is even and ≥ 4 there will be no reflected signal, provided there are no reflecting discontinuities within the device. The simplest reflector is then the one shown in Fig. 6.14. Its operation is analyzed in [28] by a modified transfer matrix method which is replaced here by a scattering matrix approach that (a) provides an explicit scattering matrix for Fig. 6.14 and (b) can be traced to the circuit schematic of the device, depicted in Fig. 6.15. The procedure is carried out in two steps. First the transmission between ports 1 and 2 of Fig. 6.15b is evaluated. We shall return to this later. Once the transmission

6

Band-Limited Microresonator Reflectors and Mirror Structures

Fig. 6.14 A multiring reflector consisting of three identical microrings [28]. The three sections between the couplers in the top ring are indicated, L1 and L2 are arbitrary. The coupling strengths between the rings are assumed to be identical

155

Kc

1

o L1

2

o L2

L K

L3

L

K

L K

Fig. 6.15 Schematic of the circular array of three microrings of Fig. 6.14, proposed in [28]

1 o

Kc

o

L1

1' o

L2

2' o

array 2 o

o

a) 1' o

K

3'

b)

K

K L

L

2' o 4'

L3

between 1 and 2 is known, the same steps as above examining Fig. 6.2 to calculate the transmission between ports 1 and 2 are followed. To find the transmission between ports 1 and 2 we recognize that Fig. 6.15b is a feedback-assisted fourport, consisting of five cascaded components, namely three couplers and two pairs of waveguides, where ports 3 and 4 are connected by a feedback waveguide of length L3 . Its scattering matrix, once again, can be written in the same form as (6.4). The details of the procedure are laid out in [5]. Finally, the scattering matrix of the mirror SM is obtained, observing that Fig 6.15a is similar to Fig. 6.2 except that here the discontinuity is replaced by the array. The result of this analysis is [27] ⎡ SM =

s2 U exp [−j2θ )] − c U 2 −V 2 2 ⎣ s2 V exp (−jθ ) cc + c U 2 −V 2 1+2

s2 V exp (−jθ ) cc + c U 2 −V 2 1+2 s2 U exp [−j2θ )] − c U2 −V 2 1

⎤ ⎦,

(6.22)

+ +



where U = −j2s3 A21 A231 t(A221 + s6 )2 , V = −cc e−j5θ/6 + A231 (A221 − s6 ) t(A221 + + s6 )2 , t = exp (−jθ 6), θ 1 and θ 2 are phase delays in sections L1 and L2 , respectively, θ1+2 = θ1 + θ2 with θ = (β − jα)L the phase delay in a closed ring, A21 = c[(1 + c2 ) − 2 cos (θ )], and A31 = 2c2 − (1 + c2 ) cos (θ ) − js2 sin (θ ) are elements of the 4 × 4 transfer matrix of the four-port network in Fig. 6.15b), and c, cc , s, and sc refer to couplers K and Kc , respectively.

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O. Schwelb and I. Chremmos

Assuming a lossless device and that the ring perimeter L has been selected to accommodate a required stopband width or FSR, one can find, as in the previous case, an optimal interring coupling K = Kopt for every value of Kc that assures an optimal suppression level for the transmitted intensity (IL), and thus an optimal reflected intensity (RL) associated with an acceptable BW. A typical passband reflection (solid line) and transmission (dashed line) characteristic for the mirror of Fig. 6.14 is plotted in Fig. 6.16 for λ0 = 1.55 μm, neff = 1.5, L = 0.517 mm, N = 500, Kc = 0.5, and Kopt = 0.0291, with (α = 2 dB/cm) and without loss. These intensity characteristics are independent of the location of K c in the top ring (L1 need not be equal to L2 ) and the BW scales with the inverse of N. Fig. 6.16 Reflection (I11 , solid lines) and transmission (I21 , dashed lines) characteristics for an optimized, lossless, and lossy (α = 2 dB/cm) three-ring mirror depicted in Fig. 6.14. Device parameters are given in the text. The total width of the diagram spans 60 GHz

A closed form expression of the reflection coefficient reduction ratio is rather cumbersome for the three-ring mirror, but results of a numerical computation, obtained directly from (6.22), are reported below, where it is compared to several other microring-based reflectors. Computed results for Kopt = Kopt (Kc ) and BW = BW(Kc ) of the three-ring mirror are plotted at IL=–20 dB in Fig. 6.17. The device parameters are the same as those listed above. Fig. 6.18 illustrates the dependence of the normalized group delay characteristics τ 11 /τ 0 on the coupling coefficient Kc using the optimized K. A very efficient and loss-tolerant feedback-assisted double microring reflector became the focus of interest by several independent research groups in the beginning of 2005 [29–32], although an early multiring version of it has been reported already in 2001 [33]. Adopting the name-coupled ring reflector (CRR) given to it in [29] the mirror is illustrated in Fig. 6.19. Notice that the circuit schematic of the CRR is the same as Fig. 6.15b, except that the coupling coefficients on the left and right are Kc , rather than K. Of the methods used to evaluate the characteristics of this device we follow the transfer matrix → scattering matrix procedure outlined above. Thus, after evaluating the transfer matrix of the five concatenated

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Band-Limited Microresonator Reflectors and Mirror Structures

157

Fig. 6.17 The dependence of Kopt , at IL = –27 dB, and bandwidth (BW), at IL= –20 dB, on K c for the lossless three-ring mirror of Fig. 6.14

Fig. 6.18 The effect of the coupling coefficient Kc on the τ 11 /τ 0 characteristics of the lossy three-ring mirror (α = 2 dB/cm and N = 500). For each Kc an optimized K value has been used

components in Fig. 6.15b using Kc as the external couplers, we proceed as in connection with (6.4) and described in detail in [5] to extract the scattering matrix of the CRR. The aforementioned transfer matrix is found to be ⎡

0 −A21 j ⎢ A 0 21 A=− 2 ⎢ sc s ⎣ A31 0 0 A42

⎤ A31 0 0 A42 ⎥ ⎥, 0 −A21 ⎦ A21 0

(6.23)

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O. Schwelb and I. Chremmos 1 o

L ext

Kc

L

K L

Kc

o2

Fig. 6.19 The CRR. Lext is arbitrary and there is no symmetry axis, indicating that the couplers Kc can be located anywhere on the rings. The couplers are assumed lossless

  where A21 = 1 + c2c c − 2cc cos (θ ), A31 = 2cc c − exp (jθ ) − c2c exp ( − jθ ), and A42 = −2cc c + exp ( − jθ ) + c2c exp (jθ ). The scattering matrix of the CRR mirror is  SM =

S11 S12 S21 S22

 =

tF A231



 j2s2c sA21 A221 − s4c s2 , A221 − s4c s2 j2s2c sA21

(6.24)

  where tF = exp −(α + jβ)Lext is the transmission coefficient of the external waveguide. Computed values of the reflectivity I11 = |S11 |2 as a function of K and detuning are plotted in Fig. 6.20, showing practically no reflectivity at negligibly small K, increasing reflectivity with rising K, until a centered maximum is attained at - + .2 √ Kcr,1 = 12 Kc 2 − Kc + 2(1 − Kc ) , followed by a split regime as K further  + 2 increases until at Kcr,2 = Kc (2 − Kc ) I11 sinks to a minimum and starts to rise .2 - + √ again to reach a centered maximum at Kcr,3 = 12 Kc 2 − Kc − 2(1 − Kc ) where a second splitting, giving rise to four maxima, begins. The above critical values of K exhibit an extremely weak dependence on waveguide and resonator loss [33]. Kopt = Kopt (Kc ) and BW = BW(Kc ) plots for the CRR, similar to those of Fig. 6.17, have been presented in [27], while further results on optimization and BW appear in the recent publication [34]. The distribution of field intensity in the CRR, both off resonance (left) and at resonance (right) is shown in Fig. 6.21, computed by an integral equation approach where the resonator field is expanded in terms of its natural cylindrical harmonics [35, 36]. In this example the radii and widths of the microrings were 2 and 0.2 μm, respectively, coupled to a 0.2 μm wide waveguide of neff = 2.25 and contrast ratio 3:1. The figure illustrates the nearly complete switching of the power flow between the ports of the mirror over a 1.4 nm wavelength span.

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159

Fig. 6.20 Reflected intensity of the CRR as a function of the ring-to-ring coupling coefficient K and detuning. Multiply the detuning axis by the velocity of light in mm/s to obtain the frequency in Hz. The ring-to-waveguide coupling is Kc = 0.2, the amplitude attenuation coefficient is α = 5 dB/cm, the center wavelength is λ0 = 1.55 μm, the effective guide index is neff = 4.5, and the mode number of the rings is N = 100

Fig. 6.21 Instantaneous field amplitude for TE polarization (electric field perpendicular to the image plane) at λ = 1.5745 μm (RL = –24 dB, IL = –0.026 dB, left) and at λ = 1.5759 μm (RL = –0.0087 dB, IL = –32.2 dB, right)

The reflection coefficient reduction ratio for the CRR, evaluated from 6.24, is |SM11 |σ ,f =f0 |SM11 |σ =1,f =f0

 2  copt (1 + c2c ) − cc (σ + σ −1 ) 2cc copt − 1 − c2c =  2 ,  copt (1 + c2c ) − 2cc 2cc copt − σ −1 − c2c σ 1

(6.25)

where copt = (1 − Kopt ) 2 . In evaluating (6.25) the loss in the feedback line Lext 2 , where can be immediately accounted for by simply multiplying the result with σext

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σext is the amplitude decrement along the feedback line [34]. However, this loss has been neglected in comparison to that in the rings due to their very different rate of influence. As a result of the large enhancement of signal amplitudes in the rings near resonance, losses in the rings are 20–200 times more harmful than those in the bus waveguides [37]. Off resonance, on the other hand, suppression levels are so high that waveguide attenuation has no perceptible effect. This justifies ignoring external waveguide losses altogether. Assuming a waveguide loss of 2 dB/cm and a mode number of N = 200 (L = 0.207 mm, αL = 0.041 dB/turn, σ = 0.995, neff = 1.5) Fig. 6.22 plots the reduced reflection coefficient of the four mirror configurations treated above as a function of Kc . As Kc decreases this reduction intensifies, because for smaller Kc the Q factor increases and the signal spends more time circulating in the rings. To obtain from Fig. 6.22 an updated value of the reflection coefficient for different values of either the attenuation coefficient or the ring perimeter, simply evaluate the new reflection coefficient from +   log10 (rnew ) = (αL)new (αL)old log10 (rold ),

(6.26)

where the subscript old refers to Fig. 6.22. According to (6.26) an increase in the αL product will reduce the reflection coefficient as expected.

Fig. 6.22 Mirror reflection coefficients as a function of Kc (N = 200, α = 2 dB/cm, L = 0.207 μm). Plot numbers 1, 2, 3, and 4 refer to the CRR, the DARR, the figure 8 shaped, and the triple-ring mirror configurations, respectively. Use (6.26) to obtain the reflection coefficient for a different αL product

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161

A microring-based inline reflector, a combination of a counterpropagating add/drop filter and a Mach–Zehnder interferometer, was proposed by Paloczi et al. also in 2005 [38]. A series (or direct) coupled add drop filter built with an odd number of rings gives rise to counterpropagating waves in the bus waveguides. If 3 dB couplers or Y junctions are attached to the input and the output of the add/drop filter, as shown in Fig. 6.23, a compact inline reflector is obtained which can be designed to exploit FSR extension by virtue of the Vernier effect if the ring radii are appropriately selected. As remarked in [38] these devices can be easily rendered tunable by the thermo-optic or electro-optic effect. An experimental reflector fabricated in optical polymer (SU-8, n = 1.565) over a silicon wafer with 5 μm thermal oxide (n = 1.445) substrate, using a single ring resonator (L = 678.6 μm, FSR = 292 GHz) with ring-to-waveguide power coupling coefficient K = 0.25, resulted in a reflector characteristic well approximating the theoretical [38].

Fig. 6.23 Inline reflectors built with three series coupled rings within the arms of a Mach–Zehnder interferometer: (a) identical rings, (b) rings with unequal radii to exploit the Vernier effect [38]

As an example we computed the performance characteristics of the reflector shown in Fig. 6.23a replacing the input and output Y junctions by 3 dB couplers. The device parameters were N = 100, neff = 4.5, L = 35.1 μm, α = 5 dB/cm, σ = 0.998, and FSR = 1.899 THz, or approximately 15 nm. The result is plotted in Fig. 6.24 showing flat-top reflection with RL = –0.25 dB and inband insertion loss ≤–28 dB. The external and internal coupling coefficients were chosen to be K1 = 0.33 and σ = 0.99798, K 1 = 0.33, K 2 = 0.025

σ = 0.99798, K = 0.33, K = 0.025 1 2

0

–5

–5

–10

–10 I11

I31, I41 (dB)

I11, I21 (dB)

0

–15 –20

–15 I41

–20

–25

–25

–30

–30

I31

–35

–35 I21

–40

-2

-1.5

-1

-0.5 0 0.5 Δ (1/λ ) + 0 (mm -1)

1

1.5

2

–40

–2

–1.5

–1

–0.5 0 0.5 Δ (1/λ ) + 0 (mm-1)

1

1.5

Fig. 6.24 Computed transmission and reflection intensities of the three-ring inline mirror of Fig. 6.23a. At the input and output the Y junctions were replaced by 3 dB couplers, N = 100, neff = 4.5, α = 5 dB/cm, FSR = 6.33 on the scale of the abscissa, 3 dB BW = 138 GHz, and RL = –0.25 dB

2

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K2 = 0.025, respectively, however, by varying K2 from 0.018 to 0.028 the 3 dB BW could be increased from 100 to 150 GHz maintaining the same RL.

6.4 Summary Recent advances in photonic reflectors have been described with emphasis on integrated microring-based devices. The review covers the range from the first fiber optic loop mirror through discontinuity- and grating-assisted configurations to compact multiring devices. After a brief analytic treatment the performance characteristics of each reflector is documented and an opportunity is provided to compare multiring reflectors on the basis of their effectiveness in a lossy environment. This is done through the introduction of a figure of merit, the reflection coefficient reduction ratio, which measures the effect of waveguide and radiation loss on reflectivity. Recognizing the impact of the group delay of the reflected signal on the laser Q factor and photon lifetime, the return group delay of the various configurations received special attention. Acknowledgments Ioannis Chremmos acknowledges the support of the Alexander Onassis Public Benefit Foundation. Otto Schwelb acknowledges the support of the Natural Sciences and Engineering Research Council of Canada.

References 1. Mortimore, D.B. Fiber loop reflectors. J. Lightw. Technol. 6, 1217–1224 (1988) 2. Urquhart, P. Fiber lasers with loop reflectors. Appl. Opt. 28, 3759–3767 (1989) 3. Cusmai, G., Morichetti, F., et al. Circuit-oriented modeling of ring-resonators. Optical Quant. Electron. 37, 343–358 (2005) 4. Brenner, E., Javid, M. Analysis of electric circuits (2nd ed.). McGraw-Hill, New York (1967), Section 15.11 5. Schwelb, O. Generalized analysis for a class of linear interferometric networks. Part I: Analysis. IEEE Trans. Microwave Theory Tech. 46, 1399–1408 (1998) 6. Paschotta, R., Brinck, D.J.B., et al. Resonant loop mirror with narrow-band reflections and its application in single-frequency fiber lasers. Appl. Opt. 36, 593–596 (1997) 7. Little, B.E., Chu, S.T, et al. Second-order filtering and sensing with partially coupled traveling waves in a single resonator. Opt. Lett. 23, 1570–1572 (1998) 8. Haus, H.A. Waves and fields in optoelectronics. Prentice-Hall, Inc., Englewood Cliffs (1984), Section 3.4 9. Schwelb, O., Frigyes, I. All-optical tunable filters built with discontinuity-assisted ring resonators. J. Lightw. Technol. 19, 380–386 (2001) ˇ 10. Ctyroký, J., Richter, I., et al. Dual resonance in a waveguide-coupled ring microresonator. Opt. Quant. Elec. 38, 781–797 (2006) 11. Collin, R.E. Foundations for microwave engineering (2nd ed.). McGraw-Hill, Inc., New York (1992), Section 4.8 12. Srivastava, S., Srinivasan, K. Coupled cavity analysis of the resonant loop mirror: closed form expressions and simulations for enhanced performance lasing. Appl. Opt. 44, 572–581 (2005) 13. Srivastava, S., Gopal, R., et al. Feedback Mach-Zehnder resonator with reflector: analysis and applications in single frequency fiber lasers. Appl. Phys. Lett. 89, 141118 (2006)

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14. Schwelb, O. The nature of spurious mode suppression in extended FSR microring multiplexers. Opt. Commun. 271, 424–429 (2007) 15. Sun, G., Moon, D.S., et al. High birefringence ring resonator with an inline reflector for singlefrequency fiber lasers. Opt. Commun. 280, 157–160 (2007) 16. Liu, B., Shakouri, A., et al. Passive microring-resonator-coupled lasers. Appl. Phys. Lett. 79, 3561–3563 (2001) 17. Schwelb, O. Transmission, group delay and dispersion in single-ring optical resonators and add/drop filters – A tutorial overview. J. Lightw. Technol. 22, 1380–1394 (2004) 18. Liu, B., Shakouri, A., et al. Wide tunable double ring resonator coupled lasers. IEEE Photon. Technol. Lett. 14, 600–602 (2002) 19. Rabus, D.G., Bian, Z., et al. A GaInAsP-InP double-ring resonator coupled laser. IEEE Photon. Technol. Lett. 17, 1770–1772 (2005) 20. Vázquez, C., Schwelb, O. Tunable, narrow-band, grating-assisted microring reflectors. Opt. Commun. 281, 4910–4916 (2008). Note two errors in Fig. 1: (1–hL) should be (1–h)L and in the Type III coupler the solid and dotted lines should be interchanged. 21. Schwelb, O. On the nature of resonance splitting in coupled multiring optical resonators. Opt. Commun. 281, 1065–1071 (2008) 22. Pruessner, M.W., et al. Thermo-optic tuning and switching in SOI waveguide Fabry-Perot microcavities. Opt. Exp. 15, 7557–7563 (2007) 23. Tsarev, A.V., De Leonardis, F., et al. Thin heterogeneous SOI waveguides for thermo-optical tuning and filtering. Opt. Exp. 16, 3101–3113 (2008) 24. Zhang, Z., et al. Resonance-splitting and enhanced notch depth in SOI ring resonators with mutual mode coupling. Opt. Exp. 16, 4621–4630 (2008) 25. Boyd, R.W., Heebner, J.E. Sensitive disk resonator photonic biosensor. Appl. Opt. 40, 5742–5747 (2001) 26. Zhang, J., Lit, J.W.Y. A symmetric figure-of-eight optical fiber resonator. Canadian J. Phys. 71, 20–24 (1993) 27. Schwelb, O. Band-limited optical mirrors based on ring resonators: Analysis and design. J. Lightw. Technol. 23, 3931–3946 (2005) 28. Poon, J.K.S., Scheuer, J., et al. Wavelength-selective reflector based on a circular array of coupled microring resonators. IEEE Photon. Technol. Lett. 16, 1331–1333 (2004) 29. Chung, Y., Kim, D.-G., Dagli, N. Widely tunable coupled-ring reflector laser diode. IEEE Photon. Technol. Lett. 17, 1773–1775 (2005) 30. Chremmos, I., Uzunoglu, N. Reflective properties of double-ring resonator system coupled to a waveguide. IEEE Photon. Technol. Lett. 17, 2110–2112 (2005) 31. Chung, Y., Kim, D.-G., Dagli, N. Reflection properties of coupled ring reflectors. J. Lightw. Technol. 24, 1865–1874 (2006) 32. Dagli, N., Chung, Y. Analytical analysis of coupled-ring reflectors based on symmetry arguments. Integrated Photonics Res. and Appl. April 24–28, 2006, Uncasville, CT, paper IWB7. Note that symmetry is not necessary for device operation 33. Schwelb, O. Some novel photonic bandpass and bandstop filters. 8th Int. Symp. Microwave Opt. Technol., Montréal, Canada, June 19–23 (2001), paper 105 34. Chremmos, I., Schwelb, O. Optimization, bandwidth and the effect of loss on the characteristics of the coupled ring reflector. Opt. Commun. vol. 282, 3712–3719 (2009) 35. Chremmos, I., Uzunoglu, N. Analysis of coupling between two slab waveguides in the presence of ring resonators. J. Opt. Soc. Am. A. 21, 267–279 (2004) 36. Chremmos, I., Uzunoglu, N. Transmission and radiation in a slab waveguide coupled to a whispering-gallery resonator: Volume integral equation analysis. J. Opt. Soc. Am. A. 21, 839–846 (2004) 37. Boscolo, S., Blow, K.J. Transfer characteristics and propagation effects of a multi-resonance ring resonator-based optical device. J. Modern Opt. 51, 559–573 (2004) 38. Paloczi, G.T., Scheuer, J., Yariv, A. Compact microring-based wavelength-selective inline optical reflector. IEEE Photon. Technol. Lett. 17, 390–392 (2005)

Chapter 7

Slow and Stopped Light in Coupled Resonator Systems Shanhui Fan, Sunil Sandhu, Clayton R. Otey, and Michelle L. Povinelli

Abstract In the first part of this chapter, a theoretical overview is presented on the different approaches to the use of dynamic tuning for coherent optical pulse stopping and storage in coupled resonator systems, which are amenable to fabrication in on-chip devices such as photonic crystals. The use of such dynamic tuning overcomes the delay-bandwidth constraint of slow-light structures. The second part of this chapter presents a discussion on recent experimental work that has demonstrated the possibility of such dynamic tuning in on-chip systems.

7.1 Introduction This chapter describes how coupled resonator systems can be used to stop light – that is, to controllably trap and release light pulses in localized, standing wave modes. The inspiration for this work lies in previous research on stopped light in atomic gasses using electromagnetically induced transparency (EIT) [1], in which light is captured in “dark states” of the atomic system via adiabatic tuning [2–4]. However, such atomic systems are severely constrained to operate only at particular wavelengths corresponding to available atomic resonances and have only very limited bandwidth. The coupled resonator systems described in this chapter are amenable to fabrication in on-chip devices such as photonic crystals (PCs) [5–9] or microring resonators [10]. As such, the operating wavelength and other operating parameters can be engineered to meet flexible specifications, such as for optical communications applications. The idea of using dynamic tuning in a coupled resonator system is to modulate the properties of the resonators (e.g., the resonator frequencies) while a light pulse is in the system. In so doing, the spectrum of the pulse can be molded almost arbitrarily, leading to highly non-trivial information processing capabilities. In past S. Fan (B) Ginzton Laboratory, Stanford University, Stanford, CA, 94305, USA e-mail: [email protected] I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_7,  C Springer Science+Business Media, LLC 2010

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work [11], it was shown that dynamic tuning can be used for time reversal of pulses. This chapter focuses on approaches to stopping light [12–17]. A wide variety of work has been done on slow-light structures employing coupled resonators [18–28]. However, in all such systems, the maximum achievable time delay scales inversely with the operating bandwidth [21]. As will be seen below, the use of dynamic tuning overcomes this constraint by manipulation of the photon spectrum in time. This chapter starts with an overview of theoretical work on light stopping in dynamically tuned coupled resonator systems. This is followed by a discussion on quite recent experimental results that have demonstrated the possibility of adiabatic tuning in on-chip systems and a review of the growing body of work inspired by dynamic optical modulation ideas.

7.2 Theory 7.2.1 Tuning the Spectrum of Light Here a simple example is provided to show how the spectrum of an electromagnetic wave can be modified by a dynamic photonic structure. Consider a linearly polarized electromagnetic wave in one dimension. The wave equation for the electric field is ∂ 2E ∂ 2E − (ε0 + ε (t)) μ0 2 = 0. 2 ∂x ∂t

(7.1)

Here ε (t) represents the dielectric modulation and ε0 is the background dielectric constant. Both ε0 and ε (t) are assumed to be independent of position. Hence different wave vector components do not mix in the modulation process. For a specific wave vector component at k0 , with electric field described by E (t) = √ f (t) ei(ω0 t−k0 x) , where ω0 = k0 / μ0 ε0 , we have  −k02 f − [ε0 + ε (t)] μ0

 ∂f ∂ 2f 2 − ω + 2iω f = 0. 0 0 ∂t ∂t2

(7.2)

By using a slowly varying envelope approximation, i.e., ignoring the ∂ 2 f /∂t2 term, and by further assuming that the index modulations are weak, i.e., ε(t) << εo , (7.2) simplifies to i

ε (t) ω0 ε (t) ω0 ∂f f ≈ f, = ∂t 2 [ε (t) + ε0 ] 2ε0

which has an exact analytic solution:

⎡

f (t) = f (t0 ) exp ⎣−iω0

t t0

  ⎤ ε t dt ⎦ , 2ε0

(7.3)

(7.4)

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where t0 is the starting time of the modulation. Thus the “instantaneous frequency” of the electric field for this wave vector component is

ε (t) . ω (t) = ω0 1 − 2ε0

(7.5)

Note that the frequency change is proportional to the magnitude of the refractive index shift alone. Thus, the process defined here differs in a fundamental way from traditional nonlinear optical processes. For example, in a conventional sum frequency conversion process, in order to convert the frequency of light from ω1 to ω2 , modulations at a frequency ω2 − ω1 need to be provided. In contrast, regardless of how slow the modulation is in the process described here, as long as light is in the system, the frequency shift can always be accomplished. Below, some spectacular consequences of such frequency shifts will be demonstrated, in particular when applied to stopping light pulses all optically in dynamic PC systems. The existence of the frequency shift in dynamic PC structures [29] and in laser resonators [30, 31] was also pointed out in a number of previous works. In practical optoelectronic or nonlinear optical devices, the achievable refractive index shift is generally quite small. Thus, in most practical situations the effect of dynamics is prominent only in structures in which the spectral feature is sensitive to small refractive index modulations. This motivates the design of Fano interference schemes described below, which are employed to enhance the sensitivity of photonic structures to small index modulations.

7.2.2 General Conditions for Stopping Light The aim of stopping light is to reduce the group velocity of a light pulse to zero, while completely preserving all the coherent information encoded in the pulse. Such ability holds the key to the ultimate control of light and has profound implications for optical communications and quantum information processing. There has been extensive work attempting to control the speed of light using optical resonances in static PC structures. Group velocities as low as 10−2 c have been experimentally observed at waveguide band edges [32, 33] or with coupled resonator optical waveguides (CROWs) [34–37]. Nevertheless, such structures are fundamentally limited by the delay-bandwidth product constraint – the group delay from an optical resonance is inversely proportional to the bandwidth within which the delay occurs. Therefore, for a given optical pulse with a certain temporal duration and corresponding frequency bandwidth, the minimum group velocity achievable is limited. In a CROW waveguide structure, for example, the minimum group velocity that can be accomplished for pulses at 10 Gbit/s rate at a wavelength of 1.55 μm is no smaller than 10−2 c. For this reason, static photonic structures can not be used to stop light. To stop light, it is necessary to use a dynamic system. The general condition for stopping light [12] is illustrated in Fig. 7.1. Imagine a dynamic PC system, with an

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Fig. 7.1 The general conditions for stopping a light pulse. (a) The large-bandwidth state that is used to accommodate an incident light pulse. (b) The narrow-bandwidth state that is used to hold the light pulse. An adiabatic transition between these two states stops a light pulse inside the system [63]. Reprinted with permission. Copyright 2006 IEEE

initial band structure possessing a sufficiently wide bandwidth. Such a state is used to accommodate an incident pulse, for which each frequency component occupies a unique wave vector component. After the pulse has entered the system, one can then stop the pulse by flattening the dispersion relation of the crystal adiabatically, while preserving the translational invariance. In doing so, the spectrum of the pulse is compressed, and its group velocity is reduced. In the meantime, since the translational symmetry is still preserved, the wave vector components of the pulse remain unchanged, and thus one actually preserves the dimensionality of the phase space. This is crucial in preserving all the coherent information encoded in the original pulse during the dynamic process.

7.2.3 Tunable Fano Resonance To create a dynamic PC, one needs to adjust its properties as a function of time. This can be accomplished by modulating the refractive index, either with electro-optic or with nonlinear optic means. However, the amount of refractive index tuning that can be achieved with standard optoelectronics technology is generally quite small, with a fractional change typically on the order of δn/n ≈ 10−4 . Therefore, Fano interference schemes are employed in which a small refractive index modulation leads to a very large change of the bandwidth of the system. The essence of a Fano interference scheme is the presence of multi-path interference, where at least one of the paths includes a resonant tunneling process [38]. Such interference can be used to greatly enhance the sensitivity of resonant devices to small refractive index modulation [14, 39, 40]. Here a waveguide side-coupled to two cavities is considered [41]. The cavities have resonant frequencies ωa,b ≡ ω0 ± δω/2. (This system represents an all-optical analogue of atomic systems exhibiting EIT [1]. Each optical resonance here is analogous to the polarization between the energy levels in the EIT system [26].) For simplicity, it is assumed that the cavities coupled to the waveguide with an equal coupling rate of γ , and the direct coupling between the side-cavities is ignored.

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Fig. 7.2 (a) Transmission spectrum through a waveguide side-coupled to a single-mode cavity. (b) and (c) Transmission spectra through a waveguide side-coupled to two cavities. The parameters for the cavities are ω0 = 2π c/L, γ = 0.05ω0 . The waveguide satisfies a dispersion relation β (ω) = ω/c, where c is the speed of light in the waveguide and L is the distance between the cavities. In (b), ωa,b = ω0 ± 1.5γ . In (c), ωa,b = ω0 ± 0.2γ [63]. Reprinted with permission. Copyright 2006 IEEE

Consider a mode in the waveguide passing through the cavities. The transmission and reflection coefficients for a single side cavity can be derived using the Green’s function method [42] and are used to calculate the two-cavity transmission spectrum via the transfer matrix method [41]. The transmission spectra of one- and two-cavity structures are plotted in Fig. 7.2. In the case of one-cavity structure, the transmission features a dip in the vicinity of the resonant frequency, with the width of the dip controlled by the strength of waveguide-cavity coupling (Fig. 7.2a). With two cavities, when the condition 2β (ω0 ) L = 2nπ

(7.6)

is satisfied, the transmission spectrum features a peak centered at ω0 . The width of the peak is highly sensitive to the frequency spacing between the resonances δω. When the cavities are lossless, the center peak can be tuned from a wide peak when δω is large (Fig. 7.2b), to a peak that is arbitrarily narrow with δω→0 (Fig. 7.2c). The two-cavity structure, appropriately designed, therefore behaves as a tunable bandwidth filter (as well as a tunable delay element with delay proportional to the inverse peak width [26]), in which the bandwidth can in principle be adjusted by any order of magnitude with very small refractive index modulation.

7.2.4 From Tunable Bandwidth Filter to Light-Stopping System By cascading the tunable bandwidth filter structure described in the previous section, one can configure a structure that is capable of stopping light (Fig. 7.3a). In

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Fig. 7.3 (a) Schematic of a coupled-cavity structure used to stop light. (b) and (c) Band structures for the system shown in (a), as the frequency separation between the cavities are varied, using the same waveguide and cavity parameters as in Fig. 7.2b and c, with the additional parameter L2 = 0.7L1 . The thicker lines highlight the middle band that will be used to stop a light pulse [63]. Reprinted with permission. Copyright 2006 IEEE

such a light-stopping device, the photonic band diagram becomes highly sensitive to small refractive index modulation. The photonic bands for the structure in Fig. 7.3a can be calculated using a transmission matrix method [13]. The band diagrams are shown in Fig. 7.3, in which the waveguide and cavity parameters are the same as those used to generate the transmission spectrum in Fig. 7.2. In the vicinity of the resonances, the system supports three photonic bands, with two gaps occurring around ωa and ωb . The width of the middle band depends strongly on the resonant frequencies ωa , ωb . By modulating the frequency spacing between the cavities, one goes from a system with a large bandwidth (Fig. 7.3b), to a system with a very narrow bandwidth (Fig. 7.3c). In fact, it can be analytically proven that the system supports a band that is completely flat in the entire first Brillouin zone [13], allowing a light pulse to be frozen inside the structure with the group velocity reduced to zero. Moreover, the gaps surrounding the middle band have sizes on the order of the cavity-waveguide coupling rate γ and are approximately independent of the slope of the middle band. Thus, by increasing the waveguide-cavity coupling rate, this gap can be made large, which is important for preserving the coherent information during the dynamic bandwidth compression process [12].

7.2.5 Numerical Demonstration in a Photonic Crystal The system presented above can be implemented in a PC of a square lattice of dielectric rods n = 3.5 with a radius of 0.2a (a is the lattice constant) embedded in air n = 1 [13] (Fig. 7.4). The photonic crystal possesses a band gap for TM modes with electric field parallel to the rod axis. Removing one row of rods along the pulse propagation direction generates a single-mode waveguide. Decreasing the radius of

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Fig. 7.4 Light-stopping process in a PC simulated using the FDTD method. The crystal consists of a waveguide side-coupled to 100 cavity pairs. Fragments of the PC are shown in part b. The three fragments correspond to unit cells 12–13, 55–56, 97–98. The dots indicate the positions of the dielectric rods. The black dots represent the cavities. (a) The middle dashed horizontal and vertical lines (with “pulsed released” and “pulse stopped” points indicated) represent the variation of ωa and ωb as a function of time, respectively. The far left pulse (solid line) is the incident pulse as recorded at the beginning of the waveguide. The pulse centered at ∼1.65. tpass (dotted line) and the far-right pulse (solid line) are both the output pulses at the end of the waveguide, in the absence and in the presence of modulation, respectively. tpass is the passage time of the pulse in the absence of modulation. (b) Snapshots of the electric field distributions in the PC at the indicated times [13]. Reprinted with permission. Copyright 2004 American Physical Society

a rod to 0.1a and the dielectric constant to n = 2.24 + provides a single-mode cavity with resonance frequency at ωc = 0.357 · (2π c a). The nearest neighbor cavities are separated by a distance of l1 = 2a along the propagation direction, and the unit cell periodicity is l = 8a. The waveguide-cavity coupling occurs through a barrier of one rod, with a coupling rate of γ = ωc /235.8. The resonant frequencies of the cavities are tuned by refractive index modulation of the cavity rods. The entire process of stopping light for N = 100 pairs of cavities is simulated with the FDTD method, which solves Maxwell’s equations without approximation [43]. The dynamic process for stopping light is shown in Fig. 7.4. A Gaussian pulse is generated in the waveguide (the process is independent of the pulse shape). The

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excitation reaches its peak at t = 0.8 tpass , where tpass is the traversal time of the pulse through the static structure. During pulse generation, the cavities have a large frequency separation. The field is concentrated in both the waveguide and the cavities (Fig. 7.4b, t = 1.0 tpass ), and the pulse propagates at a relatively high speed of vg = 0.082c. After the pulse is generated, the frequency separation is gradually reduced to zero. During this process, the speed of light is drastically reduced to zero. As the bandwidth of the pulse is reduced, the field concentrates in the cavities (Fig. 7.4b, t = 5.2 tpass ). When zero group velocity is reached, the photon pulse can be kept in the system as a stationary waveform for any time duration. In this simulation, the pulse is stored for a time delay of 5.0 tpass , and the pulse is then released by repeating the same index modulation in reverse (Fig. 7.4b, t = 6.3 tpass ). The pulse intensity as a function of time at the right end of the waveguide is plotted in Fig. 7.4a and shows the same temporal shape as both the pulse that propagates through the unmodulated system, and the initial pulse recorded at the left end of the waveguide.

7.2.6 Dispersion Suppression Through Dynamic Tuning The dynamic tuning scheme largely eliminates the dispersive effects associated with static delay lines. The time-varying dispersion relation ω(k,t) can be expanded around a central wave vector kc as (2)

(1)

ω(k,t) ≈ ω(kc ,t) + ωkc (t)(k − kc ) +

ωkc (t) 2

(k − kc )2 ,

(7.7)

(n)

where ωkc (t) ≡ dn ω(k,t)/dkn |k=kc . It can be shown [16] that the output width of the pulse in time ( tout ) after a total delay time τ is given by / τ 2

tout

=

2

tin

0

+

ωkc (t )dt (2)

2 ,

v2g (0) tin

(7.8)

where it is assumed that vg (τ ) = vg (0). For a static system, this reduces to the result  2

tout

=

2

tin

+

(2)

ωkc (0)τ v2g (0) tin

2 ,

(7.9)

and the pulse spreads with increasing delay. For the dynamic system, however, (2) ωkc (t) (and all higher order derivatives) are identically zero in the flat band state. If the bandwidth compression and decompression processes each occupy a duration T, ⎡

2

tout

⎤2 /T (2) 2 ωkc (t )dt ⎥ ⎢ ⎢ 0 ⎥ 2 = tin +⎢ 2 ⎥ . ⎣ vg (0) tin ⎦

(7.10)

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The pulse spreading is independent of the delay time τ , since it only occurs during spectrum compression and decompression. The delay can thus be increased arbitrarily without any additional increase in dispersion.

7.2.7 Capturing Light Pulses Using Few Dynamically Tuned Microresonators Instead of using the many resonators approach shown in Fig. 7.4, the capture and release of light pulses can also be performed using a dynamically tuned system with few resonators [44]. An example of such a system, shown in Fig. 7.5a, consists of two resonators coupled to a waveguide. The key feature to the pulse capturing/releasing process lies in the presence of a state that is decoupled from the waveguide, which we refer to as the dark state. When ω1 = ω2 = ω0 and γ1 = γ2 = γ0 , the system has an eigenstate, with eigenfrequency ω0 − β and resonator amplitudes α1 = −α2 , which does not leak into the waveguide. Starting from this dark state, if the resonators are tuned to ω1 = ω2 , the energy from the resonators leaks into the waveguide, generating a released pulse. Since the underlying physics of the system is time-reversal invariant, performing the time-reversed temporal detuning trajectory allows for the complete capture of the time-reversed pulse into the dark state. The entire pulse capture/release process is simulated using the system shown in Fig. 7.5b with the FDTD method [44]. The dynamic process for pulse capture/release is shown in Fig. 7.6. A Gaussian pulse is generated with carrier frequency ωc = ω0 − β = 200 THz and width T = 4 ps in the waveguide. These pulse parameters are for a system with lattice constant a = 370 nm. During the pulse generation, the resonators have zero detuning and the waveguide is decoupled from the resonators (dark state). As the pulse approaches the resonators are detuned by gradually tuning the dielectric constants within a region 1.25a around

Fig. 7.5 Schematic of double resonator system used for pulse capture/release process: (a) On the left is the waveguide with mode amplitude awg coupled to two resonators with modal amplitudes a1,2 ; γ 1,2 are the coupling constants between the waveguide and the resonators, while β is the coupling constant between the two resonators. (b) Actual structure used in FDTD simulations [44]. Reprinted with permission. Copyright 2009 American Institute of Physics

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Fig. 7.6 Dynamics of the pulse capture/release process simulated using the FDTD method. (a) The out-of-plane Hz field during the pulse capture process. (b) The Hz field in the dark state of the system. (c) The dielectric modulation curves for the two resonators, used in the simulation. ε0 is the dielectric constant in the absence of any dielectric modulation. (d) The pulse amplitude measured in the waveguide in a pulse release simulation from FDTD (solid line) and coupled mode theory (circles) [44]. Reprinted with permission. Copyright 2009 American Institute of Physics

the resonators in order to couple the incident pulse energy into the resonators. Figure 7.6c shows how proper tuning of the resonators results in the near complete capture (99.61%) of the incident pulse energy by the resonators. Details of the generation of a resonators tuning trajectory, which results in negligible reflection of the incident pulse, are discussed in [45]. The pulse field during and after this pulse capture process are shown in Fig. 7.6a, b. At the end of the pulse capture process, the pulse energy is in the resonators and the system can be kept in this dark state for any duration. In addition, there is high spatial compression of the pulse energy in the two resonators and consequently, very high nonlinearity enhancement may be achieved in this pulse capture regime [45]. In order to release the pulse energy trapped in the resonators, the time-reverse of the tuning trajectory shown in Fig. 7.6c is used. This results in the release of the Gaussian pulse that is the time-reverse of the captured pulse, and the near complete transfer of energy from the resonators to the waveguide. Figure 7.6d shows the pulse amplitude measured in the waveguide at the end of the release process. Here a brief comment is given on the differences between the use of a resonator array or only two resonators. In the case when an array is used, the dynamic modulation process can start after the entire pulse is contained in the array. As a result, the temporal profile of the modulation is independent of the pulse format, as long as the modulation remains adiabatic. Moreover, there is no spatial compression of electromagnetic energy during the light-stopping process. In contrast, with the use of two cavities, in order to completely capture a pulse, the temporal profile of the modulation is strongly dependent upon the format of the pulse.

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7.3 Experimental Progress 7.3.1 General Requirements for Microresonators The numerical examples above have demonstrated the use of PC microresonators for slowing and stopping light. However, the phenomena described are quite general and apply to arbitrary coupled resonator systems. To be useful for stopping light, the particular resonator implementation should satisfy several criteria. First, the resonator should be highly tunable on the time scale of operation of the device. The resonance frequency can be tuned by changing the refractive index of the material via electro-optic methods. For a small refractive index shift of δn/n = 10−4 , achievable in practical optoelectronic devices [46], and assuming a carrier frequency of approximately 200 THz, as used in optical communications, the achievable bandwidths are on the order of 20 GHz, which is comparable to the bandwidth of a single wavelength channel in high-speed optical systems. Second, the intrinsic quality factor of the resonator should be as high as possible, since it limits the delay time. Light stopped for longer than the cavity lifetime will substantially decay. However, the optical loss might be counteracted with the use of gain media within or external to the cavities. Third, small size of the resonator is generally desirable, since shorter length devices tend to consume less power. Moreover, for fixed device length, decreasing the size of the resonator increases the storage capacity [24].

7.3.2 Experiments with Microring Resonators Experiments with silicon microring resonators have demonstrated the use of a tunable Fano resonance in a double-resonator system [47] to controllably trap and release light pulses [48]. Initially, the frequencies of the two microring resonators are slightly detuned, as in Fig. 7.2b. In this state, input light couples into a “supermode” of the two resonators. The frequencies of the two resonators are then tuned into resonance with one another, as in Fig. 7.2c. In this state, the supermode is isolated from the input and output waveguides, and light is stored in and between the two resonators. After a given storage time, the resonator frequencies are again detuned to release the light. The resonators are tuned using the free-carrier dispersion effect in silicon [49] to blue-shift the resonant wavelength. In this experiment, an optical pump pulse at 415 nm was used to excite free carriers in the microrings. Electro-optic tuning of the ring resonances via built-in p-i-n junctions [50] should allow electrically controlled storage, with an expected bandwidth of over 10 GHz. In the experiment, the storage time was limited to < 100 ps by the intrinsic Q of the microresonators (Q = 143,000). However, the demonstration of Q ∼ 4.8 × 106 in a silicon ring resonator [51] suggests that storage times of several nanoseconds may be possible.

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The detailed theory for light pulse capturing in such double-resonator system has been discussed in [44, 45] and reproduced in a previous section in this chapter. One drawback of using this double-resonator system for pulse delay is that the pulse shape and spectrum are not preserved in the process. The information encoded in the shape of the original pulse can be retained using a cascaded multiresonator system [12, 13]. Nevertheless, this experiment represents a major first step toward the realization of the theoretical ideas for stopping light that was presented above.

7.3.3 Experiments with Photonic Crystals PC microcavities may represent the ultimate limit of miniaturization for resonator modes. Such microcavities have been demonstrated with Q up to 2 × 106 and modal volumes as small as a cubic wavelength [52]. A recent experiment has demonstrated the fundamental requirement for dynamic trapping and delay: the ability to tune between a supermode that is strongly coupled to an input waveguide, and one that is decoupled, or isolated [53]. The geometry used is shown schematically in Fig. 7.7. A single cavity is side-coupled to a waveguide that is terminated by a mirror. The coupling between the input waveguide and the supermode of the resonator–waveguide–mirror complex is determined by the reflection phase from the mirror. When the wave emitted from the cavity in the backward direction interferes constructively with the wave emitted from the cavity in the forward direction and reflected backwards by the mirror, light can easily couple from the supermode to the input waveguide. Conversely, when the waves interfere destructively, the coupling is reduced. We note that this structure is in fact conceptually very similar to the structure shown in Fig. 7.2b. The mirror, in essence, creates a mirror image of the first resonator. In the experiment, a pump pulse was used to dynamically tune the refractive index of the waveguide between the nanocavity and the mirror, adjusting the reflection phase. Pump-probe measurements of the power emitted from the cavity to free space show that the coupling properties of the supermode could be tuned on the picosecond timescale.

Fig. 7.7 Schematic of system used for experiments on dynamic light trapping in PCs

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7.3.4 Aligning Microresonator Resonances Using Differential Thermal Tuning An important pre-requisite for the experimental demonstration of multiple microresonators in PC structures that have been proposed for slowing and stopping light is the ability to tune the different microresonators to a desired resonance frequency. Due to geometrical errors during fabrication, it is generally not possible to fabricate two PC resonators with identical resonances. Hence, practical resonant frequency tuning methods are important for removing slight fabrication differences in nominally identical microresonators, relaxing fabrication tolerances required to realize multiple microresonators in PC structures. One example of such a post-fabrication tuning method is differential thermal tuning [54], which does not require any extra materials or structures and, consequently, avoids the potential quality degradation of PC microresonators and excess fabrication complexity. The differential thermal tuning technique generally involves focusing the output of a pump laser in the vicinity of a microresonator, which as a result experiences a shift in resonance frequency due to the induced thermal gradient. In the experiments demonstrated in [54], an initial difference of 3.15 nm in the resonant wavelength of two closely separated micro-resonators in a Si PC slab was decreased to zero using this thermal tuning technique.

7.4 Outlook and Concluding Remarks Beyond the work described above, the idea of using dynamic tuning of the refractive index for stopping, storing, and time-reversing pulses has sparked a wide range of research. For example, alternate dynamic tuning schemes that do not require translational invariance have recently been investigated [55]. Moreover, the generality of the physics governing coupled resonators has suggested the possibility of light stopping and time reversal in quite diverse physical systems. In semiconductor multiple quantum well structures, tuning of the excitonic resonance via the AC Stark effect can potentially flatten the photonic band structure to stop light pulses in a similar fashion as described here [56, 57]. In superconducting qubit systems, tuning of the qubit transition frequency can theoretically stop pulses on the single photon level [58]. Such an ability to manipulate single photons is of increasing interest for quantum information processing and quantum computing. The concept of using dynamic index tuning for frequency conversion is also being actively explored. Ideally, one could use a coupled resonator system to change the center frequency of a pulse while leaving its shape unchanged, a feat achieved via a uniform shift of the band structure [59]. While experiments are not yet feasible, a similar effect can be observed in single cavity systems. For a single cavity, changing the resonance frequency of the cavity mode on a timescale faster than the cavity decay time results in frequency conversion [60]. The frequency shift is linearly proportional to the index shift. The phenomenon has been demonstrated experimentally in both silicon microring resonators [61] and PC microcavities [62].

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In summary, dynamic tuning of coupled resonator systems opens the possibility for coherent optical pulse stopping and storage. More generally, dynamic processes in coupled resonator systems allow one to mold the spectrum of a photon pulse almost at will, while preserving coherent information in the optical domain. In the future, the use of dynamic photonic structures, as envisioned here, may provide a unifying platform for diverse optical information processing tasks. Acknowledgments The work is supported in part by NSF, DARPA, and the Lucile and Packard Foundation. The authors acknowledge the important contributions of Prof. Mehmet Fatih Yanik to the works presented here.

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22. Heebner, J.E., Boyd, R.W., et al. Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide. Phys. Rev. E 65, 036619 (2002) 23. Heebner, J.E., Boyd, R.W., et al. SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides. J. Opt. Soc. Am. B 19, 722–731 (2002) 24. Wang, Z., Fan, S. Compact all-pass filters in photonic crystals as the building block for high capacity optical delay lines. Phys. Rev. E 68, 066616 (2003) 25. Smith, D.D., Chang, H., et al. Coupled resonator-induced transparency. Phys. Rev. A 69, 063804 (2004) 26. Maleki, L., Matsko, A.B., et al. Tunable delay line with interacting whispering-gallery-mode resonators. Opt. Lett. 29, 626–628 (2004) 27. Poon, J.K.S., Scheuer, J., et al. Designing coupled-resonator optical waveguide delay lines. J. Opt. Soc. Am. B 21, 1665–1673 (2004) 28. Khurgin, J.B. Expanding the bandwidth of slow-light photonic devices based on coupled resonators. Opt. Lett. 30, 513–515 (2005) 29. Reed, E.J., Soljacic, M., et al. Color of shock waves in photonic crystals. Phys. Rev. Lett. 91, 133901 (2003) 30. Yariv, A. Internal modulation in multimode laser oscillators. J. Appl. Phys. 36, 388–391 (1965) 31. Siegmann, A.E. Lasers. University Science Books, Sausalito (1986) 32. Notomi, M., Yamada, K., et al. Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs. Phys. Rev. Lett. 87, 253902 (2001) 33. Vlasov, Y.A., O’Boyle, M., et al. Active control of slow light on a chip with photonic crystal waveguides. Nature 438, 65–69 (2005) 34. Altug, H., Vuckovic, J. Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays. Appl. Phys. Lett. 86, 111102 (2005) 35. Xia, F., Sekaric, L., et al. Coupled resonator optical waveguides based on silicon-on-insulator photonic wires. Appl. Phys. Lett. 89, 041122 (2006) 36. Huang, S.C., Kato, M., et al. Time-domain and spectral-domain investigation of inflectionpoint slow-light modes in photonic crystal coupled waveguides. Opt. Exp. 15, 3543–3549 (2007) 37. O’Brien, D., Settle, M.D., et al. Coupled photonic crystal heterostructure cavities. Opt. Exp. 15, 1228–1233 (2007) 38. Fano, U. Effects of configuration interaction on intensities and phase shifts. Phys. Rev. 124, 1866–1878 (1961) 39. Fan, S. Sharp asymmetric lineshapes in side-coupled waveguide-cavity systems. Appl. Phys. Lett. 80, 910–912 (2002) 40. Fan, S., Suh, W., et al. Temporal coupled mode theory for Fano resonances in optical resonators. J. Opt. Soc. Am. A 20, 569–573 (2003) 41. Suh, W., Wang, Z., et al. Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multi-mode cavities. IEEE J. of Quantum Electron. 40, 1511–1518 (2004) 42. Fan, S., Villeneuve, P.R., et al. Theoretical investigation of channel drop tunneling processes. Phys. Rev. B 59, 15882–15892 (1999) 43. Taflove, A. Hagness, S.C. Computational electrodynamics: The finite-difference time-domain method. Artech House, Norwood (2005) 44. Otey, C., Povinelli, M.L., et al. Capturing light pulses into a pair of coupled photonic crystal cavities. Appl. Phys. Lett. 94, 231109 (2009) 45. Otey, C., Povinelli, M.L., et al. Completely capturing light pulses in a few dynamically tuned microcavities. IEEE J. Lightwave Technol. 26, 3784–3793 (2008) 46. Lipson, M. Guiding, modulating and emitting light on silicon- challenges and opportunities. IEEE J. Lightwave Technol. 23, 4222–4238 (2005) 47. Xu, Q., Sandhu, S., et al. Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency. Phys. Rev. Lett. 96, 123901 (2006)

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48. Xu, Q., Dong, P., et al. Breaking the delay-bandwidth limit in a photonic structure. Nature Phys. 3, 406–410 (2007) 49. Soref, R.A., Bennett, B.R. Electrooptical effects in silicon. IEEE J. of Quantum Electron. 23, 123–129 (1987) 50. Xu, Q., Schmidt, B., Pradhan, S., Lipson, M. Micrometre-scale silicon electro-optic modulator. Nature 435, 325–237 (2005) 51. Borselli, M. High-Q microresonators as lasing elements for silicon photonics (Thesis). California Institute of Technology, Pasadena (2006) 52. Noda, S., Fujita, M., et al. Spontaneous-emission control by photonic crystals and nanocavities. Nat. Photonics 1, 449–458 (2007) 53. Tanaka, Y., Upham, J., et al. Dynamic control of the Q factor in a photonic crystal nanocavity. Nat. Mater. 6, 862–865 (2007) 54. Pan, J., Huo, Y., et al. Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning. Appl. Phys. Lett. 92, 103114 (2008) 55. Longhi, S. Stopping and time reversal of light in dynamic photonic structures via Bloch oscillations. Phys. Rev. B 75, 026606 (2007) 56. Yang, Z.S., Kwong, N.H., et al. Distortionless light pulse delay in quantum-well Bragg structures. Opt. Lett. 30, 2790–2792 (2005) 57. Yang, Z.S., Kwong, N.H., et al. Stopping, storing, and releasing light in quantum-well Bragg structures. J. Opt. Soc. Am. B 22, 2144–2156 (2005) 58. Shen, J.T., Povinelli, M.L., et al. Stopping single photons in one-dimensional quantum electrodynamics systems. Phys. Rev. B 75, 035320 (2007) 59. Gaburro, Z., Ghulinyan, M., et al. Photon energy lifter. Opt. Exp. 14, 7270–7278 (2006) 60. Notomi, M., Mitsugi, S. Wavelength conversion via dynamic refractive index tuning of a cavity. Phys. Rev. A 73, 051803(R) (2006) 61. Preble, S.F., Xu, Q.F., et al. Changing the colour of light in a silicon resonator. Nat. Photonics 1, 293–296 (2007) 62. McCutcheon, M.W., Pattantyus-Abraham, A.G., et al. Emission spectrum of electromagnetic energy stored in a dynamically perturbed microcavity. Opt. Exp. 15, 11472–11480 (2007) 63. Shin, J., Shen, J.T., Catrysse, P.B., Fan, S. Cut-through metal slit array as an anisotropic metamaterial film. IEEE J. Sel. Top. Quantum Electron. 12, 1116–1122 (2006)

Chapter 8

Processing Light in Reconfigurable Directly Coupled Ring Resonators Andrea Melloni

Abstract In this chapter, directly coupled ring resonator filters are considered. The state of the art achieved in glass and silicon technology is analyzed in detail, the discussion leading to the ring-size issue and to technological aspects. Advantages and disadvantages are pointed out. Key aspects related to structural disorder are then considered and developed. The impact of fabrication tolerances on the spectral characteristics and on the back-reflection is considered with great attention devoted to the effect of waveguide sidewall roughness enhanced by the ring resonance. The last section concentrates on the use of coupled ring resonator structures as delay lines and experimental results are discussed. In particular, a differential delay line for bit synchronization in DQPSK systems is presented. The chapter ends with an Appendix where the characteristic impedances of dielectric waveguides and coupled ring resonator waveguides are defined and calculated. It is shown that they are equal to the effective index and the group index, respectively.

8.1 Coupled Ring Resonator Structures Curiously, although ring resonators in integrated optics are known for a long time [1], their exploitation and application in more complex circuits has been considered only recently [2]. The ring is the best resonator realizable in passive integrated optics and can be combined with other rings or other structures to realize circuits with excellent flexibility and performance and is one of the most promising building blocks in integrated optics. Here, the spectral characteristics of directly coupled ring resonator filters are briefly summarized and the state of the art in glass and silicon technology reported and discussed. Design techniques, detailed theoretical aspects,

A. Melloni (B) Dipartimento di Elettronica e Informazione, Politecnico di Milano, Via Ponzio, 34/5, 20133 Milano, Italy e-mail: [email protected]

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and more complex structures are left to other chapters and publications (see, for example, [3–9]).

8.1.1 Structure and Spectral Characteristics This chapter focuses on directly coupled ring resonators filters where each ring is coupled only to the two adjacent rings or to a bus waveguide. In the literature this structure is named CROW (coupled resonator optical waveguide). CROWs have a periodic bandpass spectral response and are typically used as filters and delay lines. A schematic layout is shown in Fig. 8.1 where ports have been named according to the add-drop filter application. Each ring can be independently controlled by heaters allowing tuning and reconfiguration of the device. Typically, the spectral response shows a periodic passband transfer function with high extinction ratio and flat inband behavior, it does not possess transmission zeroes at the drop port. Such filters are very compact and have the advantage that the losses do not affect seriously the shape of the spectral characteristic. These, as well as other ring-based filters, can be exploited as demultiplexers, interleavers, channel extractors, or in turn they can be used as the building block of more complex devices such as add-drop filters and resonant routers [8, 10, 11]. Applications as sensors and switches in optical interconnects are also common.

Fig. 8.1 Coupled ring resonators structure. Heaters control independently the resonant frequencies

Filters realized as a cascade of directly coupled cavities are a classical solution in the microwave field and became popular in optics after the works reported in [3–9] which imported in the optics domain well-known synthesis techniques from microwaves and signal processing. Every cavity has the same circumference in order to resonate at the central frequency of the filter. The coupling coefficients determine the width and the shape of the bandpass. Their values decrease from the input/output sections toward the center in a suitable way so as to match the impedance of the external waveguides to the impedance of the ring chain. Simple closed form formulae are available in [3, 7, 9] to obtain Butterworth and Chebyshev transfer functions. Typical transmission, reflection, and group delay characteristics of filters designed according to [3] are shown in Fig. 8.2. The bandpass characteristic is very flat and the reflections, shown with dashed lines, can be quite weak but rarely below −25 dB, in spite of the apodization. A minimum level of reflection, or “return

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Fig. 8.2 Bandpass frequency responses for Chebyshev filters of order N = 3, 5, 7, bandwidth B = 20 GHz and FSR = 100 GHz. (a) Transmission and reflection; (b) group delay. The ring’s, optical radius is 477 μm and the coupling coefficients vary between 83%, in the first and last coupler, and 17% in the middle of the filter [3]. Reprinted with permission. Copyright 2002 Optical Society of America

loss”, toward the through port always exists, as stated by the Fano–Bode theorem [12], posing a universal constraint between the bandwidth (B) and the maximum achievable impedance matching (or minimum reflectivity) of a generic device. The shape of the group delay is flat in the center and has two peaks at the bandpass edges. The second-order dispersion vanishes at the central frequency, leaving a weak third–order dispersion that has a limited impact on the signal temporal shape if its spectrum is well confined inside the passband. By increasing the number of resonators the out-of-band rejection as well as the group delay increase. The values of the coupling coefficients are strictly related to the ratio between the free spectral range (FSR) and B, that is the finesse F [3, 13]. Narrowband filters require very weak coupling elements; hence, highly selective filters are more sensitive to fabrication tolerances and process variability. Coupling coefficients below 1% are difficult to control and a careful choice of the waveguide technology, materials, and fabrication processes is crucial. The spectral characteristic tolerates rather well the uncertainty on the coupling coefficients but is rather sensitive to the resonant frequency misalignments [14], especially for high finesses. Spectral distortions and increased back-reflections are the two principal detrimental effects and are thoroughly investigated in Section 8.2.

8.1.2 Technological Aspects Mature high index contrast waveguide technology is the necessary condition to realize a ring resonator. As a rule of thumb, the minimum bending radius that guarantees approximately 0.1dB/rad of radiation loss is Rmin  5 n−1.5 μm, n being the absolute index contrast. The minimum bending radius is 0.5 cm for n = 0.69%,

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0.4 mm for n = 4.5%, 20 μm for a silicon nitride (SiN) strip waveguide ( n = 35%), and a few microns for silicon-on-insulator (SOI) photonic wires. The ring radius R is related to the footprint of the circuits and the FSR is given by FSR = c/(2π Rng ), with ng being the waveguide group refractive index. Within a good approximation, the FSR can be expressed as FSR=29 n−1.5 nm for n > 0.1 [15].1 As an example, with a classical low index contrast technology ( n = 0.01), the achievable FSR is only 8 GHz while n = 0.28 (20%) is necessary to achieve a FSR = 6.4 nm (800 GHz), which is equal to eight channels in a 100 GHz-spaced WDM system. The entire C band (35 nm) can be covered using only silicon-wire waveguides. Few materials and technologies are available to fulfill such requirements. Siliconoxynitride (SiON) is an extremely flexible platform, having a refractive index in principle tuneable from 1.45 up to 2.0 [16–19] and propagation loss as low as 0.1dB/cm at 1550 nm. The main drawback of this material is the N–H absorption peak in the wavelength range between 1500 and 1510 nm that can be reduced by a proper annealing process or using deuterated precursors [20, 21]. Recently, it has been demonstrated that by eliminating the ammonia in the deposition process [22], SiON layers with index contrast of 2.5% can have very low losses down to α = 0.04 dB/cm at 1550 nm. Another well-established material is the stoichiometric Si3 N4 deposited by low-pressure chemical vapor deposition (LPCVD), with a refractive index n = 1.98 at wavelengths around 1.5 μm and losses around 0.1–0.2 dB/cm [23]. Unfortunately, the high tensile stress limits the maximum thickness to less than 0.4 μm [24], thus forcing the waveguide cross sections to strip shapes. The TE mode is well guided and bends down to few tens of microns but the TM mode is almost leaky. Finally, the SOI technology offers the highest index contrast enabling the most compact integrated optical circuits. Thanks to the low waveguide losses, 0.9 dB/cm in [25] and typically 2.5 dB/cm, unloaded quality factors above 105 have been reported in the literature and almost ideal transfer functions can be easily obtained. Silicon waveguides, however, are very small and defects, sidewall roughness and dimensions are extremely critical.

8.1.3 State of the Art As an example of CROW structures, two realizations in SiON and SOI technologies are discussed here, in order to point out the potential, the limits, and the most suitable applications of the two platforms. These two examples represent the state of the art of CROW up to the present time [26, 27]. Figure 8.3a shows a CROW made of 8 rings in 4.5% index contrast SiON technology [28]. The buried waveguide with cross Section 2.2 × 2.2 μm2 has 0.35 dB/cm propagation loss at a wavelength of 1550 nm, a bending radius of

1 This empirical expression comes from the fact that, as a rule of thumb, the minimum bending radius that guarantees roughly 0.1 dB/rad of radiation loss is given by R  5 n−1.5 μm.

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Fig. 8.3 (a) Photograph of reconfigurable 8-ring CROW in SiON. (b) Transmission of a 4-rings CROW

500 μm with negligible radiation losses and enables the realization of CROWs with FSR = 50 GHz. The device has a footprint of 1000 × 8000 μm2 . The CROW can be controlled and reconfigured by means of chromium heaters placed onto the waveguides of each ring whose resonant frequency can be controlled separately. The thermal reconfiguration requires 12.5 mW to shift the ring resonance by 1 GHz, achievable at a speed of about 150 μs; thermal crosstalk effects are acceptable if the distance between the heaters is greater than 200 μm. A Peltier cooler is used in order to keep the chip temperature fixed. Fig. 8.3b shows the transmission characteristic of a 4-rings CROW with FSR = 50 GHz and B = 7.5 GHz. The out-of-band rejection is higher than 40 dB and the in-band characteristic is almost ideal with an insertion loss of only 1.5 dB. As a comparison, Fig. 8.4a shows a top view photograph of an SOI CROW consisting of eight identical racetrack rings with a radius of 20 μm and a straight coupling section of 16 μm. The footprint (56 × 320 μm2 ) is about 500 times smaller than the glass CROW of Fig. 8.3. All the rings have the same shape because the coupling coefficients are optimized by changing the couplers gap distance. The silicon nano-waveguide of the bus lines and of the rings is 220 nm high, 480 nm wide, and is buried in silica. An optimized hydrogen silsesquioxane (HSQ) electronbeam-resist-based fabrication technology was employed to achieve low propagation losses (∼1 dB/cm) [25] and a high precision (<0.5 nm) on the waveguide width. Also in this case Ni–Cr microheaters are used to tune and reconfigure the device. The main advantages of SOI technology include wider passbands (>100 GHz), reduction of electric power dissipation (<60μW/GHz), and the possibility of fast activation by means of pn junctions embedded into the waveguides. The main drawback is the higher loss per unit delay (50 dB/ns [27]), suggesting that SOI CROWs could be competitive with glass ones at high bit rates where large B and lower absolute delays are of interest. The transfer function is reported in Fig. 8.4b obtained

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Fig. 8.4 (a) Photograph of reconfigurable 8-rings CROW in SOI technology and (b) its transmission spectral characteristic

after a careful individual tuning of the resonant frequency of the rings. B is as large as 112 GHz with an in-band insertion loss of only 4 dB. The extinction ratio is much higher than 45 dB, limited by the noise floor of the measurement setup. Figures 8.3 and 8.4 demonstrate the excellent performance and the potentialities of these structures, which are almost mature for filter applications in the telecommunications, sensors, and optical interconnects fields. Delay line applications are described in Section 8.3.

8.2 Structural Disorder In the real-world unavoidable fabrication tolerances, localized defects, layers thickness, and refractive index variations and imperfections affect the CROW spectral characteristics. All these randomly distributed imperfections go under the name “structural disorder.” This section is devoted to the three main imperfections: random deviations of the nominal values of the coupling coefficients between cavities, random deviations of the cavities resonant frequencies, and distributed back-scattering along the waveguide originated, for example, by sidewall roughness. The most evident effect is a distortion of the spectral in-band transfer function, as described in the next section. The second detrimental effect is the reflection of light back to the CROW input. Although it is generally a weak effect, with high index contrast technologies and under certain conditions it can become the main limiting factor for applications. A fairly convenient way to tackle these issues is based on the concept of characteristic impedance, which is well established in the transmission line theory. The characteristic impedance of both dielectric waveguides and CROW is derived in Appendix.

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8.2.1 Spectral Distortions In case of structural disorder, the transfer function of the CROW becomes distorted. Figure 8.5 shows 100 numerically simulated transmission and reflection transfer functions of a 4-ring filter with B = 25 GHz and FSR = 100 GHz with both random imperfections of coupling coefficients t (coupling disorder) and optical ring length disorder (phase disorder). Both disorders are assumed to obey a Gaussian statistical distribution with a standard deviation σt = 5% and σL = 2.5 nm, respectively. The coupling disorder is expressed as a percentage with respect to the nominal value while the perturbation of the ring length is absolute. The standard deviation σL can also be considered relative to the waveguide effective index. The transmission characteristic is slightly affected out of band and almost unaffected in the bandpass, demonstrating the robustness of these structures. When an imperfection in the CROW chain occurs, rings act as weak partial reflectors weakly perturbing the transmission behaviour. The reflectivity, instead, is highly sensitive and increases rapidly from the nominal value of −26 to −10 dB. The in-band back-reflected power adds to the nominal return loss at the through port, contributing to the crosstalk. Disorder-induced back-reflection is discussed in the next section.

Fig. 8.5 One hundred numerically calculated spectral transfer functions of a fourth order coupled ring resonators filter affected by structural disorder

The structural disorder affects also the group delay characteristic. Figure 8.6 shows the in-band group delay perturbation in case of (a) coupling disorder and (b) resonant frequency, i.e., ring length or phase disorder. The responses refer to a 60ring CROW suitable for delay line applications. The structural parameters have been randomly varied with a Gaussian distribution with standard deviations specified in

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Fig. 8.6 Group delay spectral characteristic of a coupled ring resonator filter affected by (a) coupling disorder and (b) resonant frequency disorder. The filter is composed by 60 rings and B = 20 GHz

the insets. The results refer to a filter with B = 20 GHz and are independent of the FSR. However, the higher the finesse, the smaller the coupling coefficients and hence their absolute accepted tolerance becomes very tight. Figure 8.6 evidences the robustness to the coupling disorder and the high sensitivity to misalignments in the resonant frequencies. In general, in order to obtain a nearly ideal filter response, each ring must be suitably tuned or trimmed [29, 30] to match the desired resonant frequency. The oscillations appearing in-band are due to multiple reflections of the propagating light and may seriously impact on the shape of the pulses propagating through the filter. Figure 8.7 shows the output pulses compared with the input pulse for two different phase disorders. The output pulse shape is not seriously affected but a long tail appears because of the group delay oscillations. These random “echoes” induce an interaction between adjacent bits called intersymbol interference, reducing the quality of the transmitted signals and the system bit error rate (BER). The intensity of the tail is related to the amplitude of the oscillations and the time duration is related to their spectral periodicity. Similar effects are observed in chirped fiber Bragg gratings used for chromatic dispersion compensation [31].

Fig. 8.7 Temporal response of the coupled ring resonator filter of Fig. 8.6b affected by structural phase disorder

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8.2.2 Back-Reflections As already observed in Fig. 8.5, the most evident effect of structural disorder is the increase of the back-reflected light. A defect in the CROW chain acts as shown in Fig. 8.8a. Both coupling and phase disorder produce a back-reflection that adds to the return loss and exits from the through port only because the disorder does not reverse the direction of light propagation. The back-reflection induced by both disorders can be analytically derived by considering the effects of a single defect and then statistically adding all the single contributions. Fig. 8.8 (a) Both coupling and phase disorder produce a back-reflection that adds to the return loss and exits from the through port. (b) Roughness induces a back-scattered wave component that exits from the input port and the isolated add port

Each discontinuity originates an impedance mismatch between the two sections of the chain with characteristic impedances Zcp and Zˆ cp , the caret in the notation referring to the impedance of the perturbed part. The back-reflected field can be determined by using the well-known relation ρ(λ) = (Zˆ cp − Zcp )/(Zˆ cp + Zcp ) where the perturbed characteristic impedance is readily calculated both for the coupling disorder and for the phase disorder (details are reported in [32]). The field backreflections induced by coupling disorder, ρt , and by phase disorder, ρϕ , result in 1 δtS(λ) 2 π δL ρϕ (λ) = −j S(λ), 2 λ ρt (λ) =

(8.1) (8.2)

where δt is the coupling coefficient perturbation, S(λ) = cos (kL/2)/[t2 - sin2 (kL/2)]1/2 [13, 32] is the CROW slowing ratio and δL is the absolute optical cavity length perturbation. These relations are valid over the whole passband, also very close to the band edges, provided that the perturbations Sδt and SδL/λ are small compared with unity. It is remarkable that the reflectivity ρt (λ) is proportional to the absolute value of the coupling coefficient, while ρϕ (λ) is proportional to the optical cavity length perturbation in wavelength units.

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The effects of random disorder distributed along the whole structure are calculated by assuming that the perturbations δt and δL are random variables. Considering normal distributions with standard deviations σt and σL around the nominal values, the mean power reflection coefficients become < ρt2 >= (σt S)2 /4 and < ρϕ2 >= (π σL S/λ)2 /4, respectively. Finally, the total mean back-reflected power < RD > of a CROW made by N cavities, where all the resonators are affected by statistically independent coupling and phase disorder, can be calculated by assuming the contributions given by each ring uncorrelated and added in intensity, while multiple reflections are neglected under the hypothesis of weak disorder. In the case of small random disorder and small loss the total back-reflection is < RD >= γ σt2 NS2 + γ (π σL /λ)2 NS2 ,

(8.3)

γ being the cavity round-trip power decrement. Both coupling and phase errors generate a back-reflected power that is proportional to the variances σt2 and σϕ2 of the disorder and to the square of the slowing ratio S2 , i.e., to the inverse of the group velocity squared 1/v2g , as predicted and observed by several authors in the case of back-scattering due to disorder and roughness in photonic crystal and dielectric waveguides [32–37]. Figure 8.9 shows the calculated average back-reflected power < RD > of a 50element CROW structure with FSR = 0.8 nm and B = 0.16 nm in case of pure coupling disorder, pure phase disorder, and with the two kinds of disorder acting simultaneously (total). A coupling disorder of σt /t = 3.5% and a phase disorder of σL /λ = 2.5 × 10−3 (corresponding to a resonance wavelength standard deviation of 2 pm) are considered. The disorder contribution to back-reflection is calculated as

Fig. 8.9 Average back-reflected power in a CROW affected by disorder. See the text for data. Back-reflections induced by pure coupling and pure phase disorder are reported together with the case of mixed disorder (total). Analytical results (dashed curves) agree with TMM 1D numerical simulations (solid curves) [32]. Reprinted with permission. Copyright 2009 Optical Society of America

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the difference between the perturbed and the ideal back-reflection. The field coupling coefficient in the central section of the chain is t = 0.3, and apodization according to [3] has been used. The comparison between the results obtained by applying the analytical model [dashed curves] and by averaging over 200 independent realizations of disorder – numerically calculated with the transmission matrix method (TMM) [38], solid curves – shows a very good agreement throughout the whole passband in all three cases. Experimental confirmations can be found in the literature [32].

8.2.3 Waveguide Roughness Effects Waveguide sidewall roughness is one of the main causes of attenuation in integrated optical circuits. Besides attenuation, light is also weakly scattered in the backward direction. In ring resonators this effect is enhanced at the resonance frequency and above a certain finesse back-scattering can be detrimental [39–41]. Figure 8.8b shows that roughness inverts the direction of light propagation and hence the back-scattered light exits from the input port and the isolated add port. To illustrate the importance of this phenomenon, let us consider the simplest example of a single phase-shifter ring resonator shown in Fig. 8.10a. The ring radius is 20 μm and is coupled to a bus waveguide by means of a directional coupler with power coupling ratio t2 . The bus and ring silicon nano-waveguides are 220 nm high, 480 nm wide, and are buried in silica. The measured effective group index of the

Fig. 8.10 Back-scattering in a ring resonator: (a) photograph of a SOI ring phase shifter; (b) measured group index of a straight SOI waveguide (dotted curve) and of a ring phase shifter with t2 = 0.75 (dashed curve) and t2 = 0.15 (solid curve); measured transmission (Tx ) and back−reflection (Rx ) of a 4-mm-long SOI waveguide (c1 , isolated waveguide) coupled with a ring phase shifter with t2 = 0.75 (c2 ) and t2 = 0.15 (c3 )

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device, defined as the group delay divided by the ring length times the speed of light, is given in Fig. 8.10b. The lower the coupling coefficient t2 , the higher the enhancement of the group index at the ring resonance with respect to the waveguide group index (ng = 4.2). This group index enhancement has a direct implication for the level of backscattering. The measured transmission (Tx ) and reflection (Rx ) of a 4 mm long waveguide is reported in Fig. 8.10c1 , showing that, within the considered spectral range, waveguide back-scattering appears as white noise (some degree of correlation is given by the finite length of the waveguide [33]) with a power level of about 32 dB lower than transmission (38 dB/mm). The transfer function of SOI phase shifters with t2 = 0.75 and t2 = 0.15 is shown in Fig. 8.10c2 and c3 , respectively. As it is well known, absorption inside the resonator can induce sharp notches at the resonant frequencies of the ring. In case of back-scattering, however, the light is mainly scattered backward. Back-scattering rapidly increases with the group index, so that, at smaller coupling coefficients, the back-scattered power can even exceed the transmitted power. For instance, in Fig. 8.10c3 at the wavelength 1550.5 nm, the transmission is Tx = −15 dB and the reflection is Rx = −5 dB. At different resonances, the back-scattered power (and hence the depth of the transmission notches) changes because the ring enhances the random back-scattering of the waveguide. Note also that the cavityenhanced back-scattered power does not appear as noise (i.e., with fast variations versus frequency) because the multiple round trips along the ring, or in other words its bandwidth, enhance also the degree of correlation. Off resonance, the effect of the resonator disappears, so that the phase shifter exhibits the same level and spectral features as the straight waveguide. Concerning the degree of correlation of back-scattering, one can ask if either a large or a small ring produces more back-scattering for a given group delay. Along the ring round trip the back-scattered contributions are not correlated (add as power) but every round trip is correlated with the previous one (add as field). Hence, for a given group delay, the smaller the number of round trips, the smaller the backscattering. This means that a high-finesse small cavity suffers from back-scattering more than a low-finesse large cavity. Figure 8.11 shows that the back-scattered power increases linearly with the length of the ring and quadratically with the group index of the ring. Dashed lines represent a linear and quadratic behavior and the marks are numerically obtained by simulating the ring using a transfer-matrix method.

8.3 Reconfigurable Delay Lines Bandpass filtering is the most straightforward application of CROW structures. In recent years, however, CROWs have been exploited also as tunable delay lines [28, 26, 42–44] and a dramatic progress at 10, 40, and even 100 Gbit/s has been reported with delays up to one entire byte [26, 27]. Here, tunable delay lines are

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Fig. 8.11 Back-scattered power at the input port of a single phase shifter versus the group delay. The linear behavior results when the ring length is varied and t is kept constant; the quadratic behavior is obtained by varying t for a constant ring length

described and discussed; their use for signal synchronization is described in the last section.

8.3.1 Tuning Techniques Synchronization, buffering, and storage of a wideband data stream in the optical domain are going to become mandatory tasks in optical networks, in order to remove the bottleneck of electronic processing speed and to avoid the cost of electro-optical format conversion. To this aim, optical devices capable of introducing arbitrarily large delays are required. However, in telecommunication applications, large delays are of interest only if associated with large bandwidth, low loss, and low signal distortion, and if the delay can be tuned finely, continuously, and easily with flexible devices. Furthermore, the miniaturization down to chip-scale and the integration with other functionalities undoubtedly represent valuable features. Nowadays, a chain of thermally activated tunable coupled resonators are able to manage, with a negligible impact on the signal quality, intensity, and phase of modulated bit streams up to 100 Gbit/s and beyond [27]. The CROW delay line can be reconfigured according to two basic principles. Owing to the CROW spectral characteristics, the group delay in transmission can be tuned by varying the ring coupling coefficients. The bandwidth changes too and the slowing factor, i.e., the group delay is modified as in Fig. 8.12a. There are two main disadvantages of such a tuning scheme. First of all, the coupling coefficients are difficult to be varied. Balanced Mach–Zehnder interferometers could substitute for the directional couplers, resulting, however, in a solution with an impractically large footprint. Second, by tuning the delay, the bandwidth changes, as well as the distortion of the group delay.

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Fig. 8.12 Spectral group delay characteristic versus (a) the coupling coefficients t, (b) the number of open rings N. Arrows indicate an increase of t and N

The second tuning scheme overcomes the previous drawbacks. It is a gating mechanism that introduces a continuously tunable time delay by controlling the individual resonances of the ring resonators, for example, through the thermo-optic effect. Consider the schematic shown in Fig. 8.1. An optical signal, incoming from the input port, can propagate along the chain of resonators only if its spectrum is entirely contained within the passband B = 2 FSR sin−1 (t)/π , t being the field coupling coefficient between adjacent resonators. On the contrary, if the signal spectrum lies outside B, i.e., if the rings resonate away from the signal carrier frequency, it is directly transferred to the through port without entering the structure and no significant delay is introduced. While propagating through the CROW, a pulse is slowed down by the factor S. With the aim to explain the working principle of the device, let us assume that the first M rings have been set to the same resonant frequency f0 . The optical signal can propagate along the CROW through the “open” rings only up to the Mth ring, because all the (M + n)-th (n ≥ 1) off-resonance rings (locked rings) act as mirrors. Once arrived at the Mth ring, the signal is reflected backward to the input/through bus waveguide. The delay experienced by the signal is roughly Td = 2M/π B, and Tr = 2/π B is the group delay given by the double transit through every elementary resonator of the folded structure. The number of rings resonating at f0 , and hence the overall delay, can be arbitrarily selected by controlling the round-trip phase shift of the rings. Depending on the waveguide technology, this can be conveniently obtained by a thermo-optic, electro-optic, or even all-optic control. By selecting M, a discrete, or digital, tunable delay between zero and Td,max = NTr = 2N/π B and with a minimum time-step equal to Tr can be achieved. Note that the folded layout enables an easy and complete reconfiguration of the structure as well as the reduction of the overall dimensions by a factor 2, improving the storage efficiency. A remarkable advantage of the proposed architecture is that, whatever the desired delay is, the CROW bandwidth is not affected by the tuning process. Moreover, the optical signal always propagates at the centre of

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the CROW bandwidth, where the effects of both dispersion [13] and disorder [45] have been demonstrated to be minimum. The discrete delay characteristic is shown in Fig. 8.12b. Actually, the delay introduced by the CROW can be tuned with resolution well below Tr . The (M + n)th ring resonator can be progressively opened by making its resonance frequency approach the signal carrier and an arbitrarily small delay, comprised between 0 and Tr can be added. The ultimate limit to delay resolution is imposed by the physical mechanism inducing the effective index change. In the case of thermal activation, 1 ps delay resolution can be achieved if the temperature of the heaters can be controlled within 0.1◦ C. This temperature accuracy is typically guaranteed by the state-of-the-art thermo-heaters, thus enabling both the stability and the fine adjustment of the delay. As disclosed in [28], continuous delay tunability in a CROW structure is allowed with no appreciable signal degradation.

8.3.2 Tunable Delay Lines Figure 8.13a shows the working principle of the tunable delay line and Fig. 8.13b its corresponding experimental time-domain behavior. The results refer to the SiON CROW shown in Fig. 8.3. When the first rings are locked, M = 0 and the output pulse experiences the minimum reference delay. By opening M = 4, 6, and 8 rings a

Fig. 8.13 (a) Tuning mechanism and (b) experimental time-domain behavior of the CROW of Fig. 8.3 with a 100 ps Gaussian input pulse. The time traces are reported in the same time (horizontal) and intensity (vertical) scale

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Fig. 8.14 BER measurements versus the OSNR at the receiver of an intensity-modulated 10 Gbit/s signal transmitted through the CROW: from left to right, back-to-back measurement, 0 ps (M = 0), 150 ps (M = 3), 200 ps (M = 4), and 300 ps (M = 6) delay. The eye diagrams acquired at error-free operation for 0, 2 and 3-bit delays are shown in the insets

delay of 400, 600, and 800 ps, respectively, is introduced, the latter corresponding to 8 bit lengths. Every ring provides up to Tr = 100 ps delay, that is a storage efficiency of 1 bit per ring. The measured fractional loss is 1dB/bit, very close to the theoretical limit αc/Bp ng = 0.7 dB/bit [26], α being the waveguide loss in dB per unit length and Bp the pulse bandwidth. The system performance of the CROW delay line was evaluated by means of BER measurements in a 10 Gbit/s system testbed. Figure 8.14 shows the BER curves of an intensity-modulated 10 Gbit/s non-return-to-zero (NRZ) optical signal (231 −1 pseudo-random binary sequence (PRBS)) versus the optical signal-to-noise ratio (OSNR) at the receiver. The leftmost line provides the back-to-back BER measurement of the transmission setup. The second line gives the reference BER, measured when all the ring resonators are locked and the signal propagates through the bus only. No OSNR penalties are observed due to fiber-to-waveguide coupling and propagation in the bus waveguide. As the delay increases, the BER performance slightly degrades, but error-free operation (BER < 10−9 ) is reached for any delay between 0 and 3 bits. The OSNR penalty, evaluated at the error-free point, is less than 1.5 dB for 150 ps delay, less than 2.5 dB for 200 ps delay, and about 4 dB for 300 ps delay, giving an average OSNR penalty of about 1.3 dB/bit. SOI CROW delay lines, instead, could be exploited at higher bit rates where large bandwidths and lower absolute delays are of interest. Figure 8.15 reports the experimental tunable delay of 1 byte at 100 Gbit/s. The results refer to the structure shown in Fig. 8.4. A fractional delay of 7.5 ps/ring with a fractional loss of 1.1dB/bit

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Fig. 8.15 Delay of 1 byte at 100 Gbit/s achieved in the SOI CROW shown in Fig. 8.4

has been achieved, a promising result for optical interconnects applications at very high bit rates. Also in this case the pulse shape distortion is acceptable and the main impairments are the back-scatter and the back-reflection contributions, clearly visible in the tail of the delayed pulses.

8.3.3 Pulse Synchronization An interesting and advanced exploitation of tunable delay lines is the capability of inducing and controlling a differential delay between two orthogonally polarized signals. In the polarization division multiplexing (PolDM) transmission systems, the management of the mutual delay between two orthogonally polarized data channels, transmitted on the same fiber, can be a crucial feature to optimize system performance [46]. If the reflective CROW architecture is realized with birefringent waveguides, the frequency bandwidths where delay occurs are different for TE and TM polarizations. A differential delay τd is introduced with no signal distortion provided that the TE and TM CROW passbands do not overlap. This implies that the relative spectral shift fTE,TM between the two must be in the range B < fTE,TM < FSR − B, a condition which is simple to obtain in both glass- and SOI-based technologies. Referring to the scheme of Fig. 8.16a, the TE mode resonates in the first M = 2 rings and is delayed by the double transit propagation inside the CROW. On the other hand, the TM mode does not match the CROW passband and propagates in the bus waveguide only with no significant delay. The induced signal distortion is generally negligible, for τd in the order of a time bit, and the polarization-dependent loss (PDL) is a fraction of a dB. The synchronization of two 50 Gbit/s data stream differential phase-shift keying (DQPSK)-modulated signals is one of the most advanced demonstrations of the potentiality of this device [47]. The structure in SOI technology shown in Fig. 8.4 is used because of its large bandwidth. The transmission performance was evaluated by means of BER measurements on two orthogonally polarized channels (27 − 1

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Fig. 8.16 (a) Differential delay between two orthogonally polarized signals induced in a birefringent CROW. (b) BER curves of TE-polarized signals versus OSNR for zero and one bit time delay. 100 Gbit/s DQPSK signals are used

PRBS) in a system testbed [27]. The two data streams were independently generated and combined in order to arrive time-overlapped at the CROW input with the same optical power. Back-to-back BER curves are reported in Fig. 8.16b for timeoverlapped and time-interleaved configurations, together with the eye diagrams for the TE polarization. The TM polarization is not affected by the induced delay and its BER curves overlap with the non-delayed TE one and are not shown for clarity. The power penalty between the non-delayed and delayed TE polarizations for τd = 50 ps, that is one time bit, is around 2 dB and reaches the error-free condition, demonstrating the low impact of the device on the signal integrity. Comparable results were obtained by launching single polarization signals separately, confirming negligible TE/TM mode coupling in the device.

8.4 Conclusion Directly coupled ring resonators structures own unique spectral properties and reconfiguration capabilities in a very compact footprint. The filter is designed with closed form formulae and the structure layout is straightforward. The structure is an infinite impulse response (IIR) filter and has a periodic spectral characteristic with flat passband, steep skirts, and high stopband rejection. Typical applications such as bandpass filters and delay lines can be tuned quite easily. However, this apparently uncomplicated structure is rather sensitive to fabrication tolerances, parameters nonidealities – generically named disorder – and must be carefully designed according to the well-known techniques. The effect of disorder is mainly a misalignment of the resonant frequencies of the coupled cavities and a non-perfect impedance matching along the chain of rings, inducing back-reflections and back-scattering that are

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responsible for the ultimate performance limits. The main advantage is the robustness of the intensity and group delay transfer functions to waveguide attenuation and coupling coefficient disorder. One of the most interesting exploitations of coupled ring resonators filters is the tunable delay line. Demonstrations of continuously tunable delay at 10 Gbit/s and even 100 Gbit/s up to 1 byte have been reported with an acceptable impact on the signal quality and integrity. The ability to operate at the same time on both TE and TM polarizations and with intensity- and phasemodulated signals suggests these structures as among the most promising devices for the management of high bit rate signals in actual future optical communication systems and optical interconnects.

Appendix Here, the characteristic impedances of dielectric waveguides and periodic waveguides are defined. The definition of the characteristic impedance of a guiding structure allows familiar concepts from classical transmission line theory [48] to be imported in a straightforward manner.

Characteristic Impedance of a Dielectric Waveguide The definition of the characteristic impedance of a dielectric waveguide is not unique and includes some arbitrariness [49, 50]. A possible definition of characteristic impedance is Zc = |V|2 /(2P), where V is the equivalent voltage across the equivalent transmission line that sustains the power flow P. In a dielectric waveguide the power flow is related to the phase velocity by the relation [51] P = (Wt − Wz )

c , neff

(8.4)

where Wt and Wz are the energies associated with the transverse and the longitudinal field components of the mode, respectively. The quantity (Wt −Wz ) can be identified as the electromagnetic momentum flow in the waveguide [51]. The characteristic impedance of a dielectric waveguide can then be written as

Zc =

|V|2 neff . 2c(Wt − Wz )

(8.5)

A convenient choice for the equivalent voltage is |V|2 = 2c(Wt − Wz ), so that the normalized characteristic impedance of the dielectric waveguide is equal to the effective mode index, that is, Zc = neff .

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Characteristic Impedance of a CROW In a periodic guiding structure the characteristic impedance can be related to the space-dependent reflection coefficient relating the forward and backward propagating waves. From the dispersion relation of the periodic structure and the transmission matrix of a single resonating cell, the reflection coefficient (z) within each resonator is [32, 48] (z) = exp ( − j2kz)

jt exp ( jβL/2) − exp ( jkL/2) , √ 1 − t2 exp ( − jkL/2)

(8.6)

t being the field coupling coefficient between resonators, L the geometrical roundtrip length, β the propagation constant of the Bloch modes [13], and k = 2π neff /λ the wave vector, with neff the waveguide effective index. The modulus of Γ is constant along z, while its phase is a periodic function. The higher || is, the more comparable are the amplitudes of the forward and backward waves and the slower is the mode propagation. The characteristic impedance is readily calculated as Zcp (z) = Zc (1 + )/(1 − ). The characteristic impedance Zcp of the periodic waveguide is therefore univocally defined with respect to Zc . Zcp is a wavelength-dependent complex quantity that varies with half-wavelength periodicity within each resonator and with L/2 spatial periodicity. Within the CROW passband, the characteristic impedance at the center of each cavity (z = L/4) is always real and simply equal to Zcp (L/4) = 2Zc S(λ)

(8.7)

where the slowing ratio is S(λ) = cos (kL/2)/[t2 − sin2 (kL/2)]1/2 [13]. The characteristic impedance of a periodic waveguide is thus inversely proportional to the group velocity vg of its modes. At the resonant wavelengths, Zcp reaches its minimum value equal to 2neff /t and tends to infinity on the edges of the passband.

Fig. 8.17 Modulus of the normalized impedance |Zcp /Zc | versus frequency at the reference planes located in the center of the cavities of a CROW. A structure with finesse F = 10 is considered. Zcp is real inside the passband B (solid curve) and purely imaginary in the stopband (dashed curves) [32]. Reprinted with permission. Copyright 2009 Optical Society of America

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In the stopband Zcp is purely imaginary, and propagation is not allowed. Figure 8.17 shows the modulus of the in-band (solid curve) and the off-band (dashed curve) normalized impedance Zcp /Zc .

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Chapter 9

Microresonators with Active Tuning Qianfan Xu

Abstract In this chapter, methods for actively controlling the resonant wavelength of photonic microresonators in order to manipulate the propagation of light are introduced. The presented tuning mechanisms are based on the free-carrier dispersion effect and the thermo-optic effect. The free carriers and the thermal power required for the tuning can be generated either electrically or optically, enabling different applications including electro-optic modulators, optical switches, optical routers, all-optical logic, and memories. These microresonator-based devices have small footprint, low power consumption, and the capability of large-scale integration. For a passive microresonator, all its parameters are fixed by design and fabrication, and so does the path of light at each wavelength. By adding active tuning capability, the path of light can be dynamically reconfigured, enabling many new functionalities and applications including modulators, switches, routers, tunable optical delay lines, dynamic spectral equalizers, optical signal processors. Tuning is mostly achieved by changing the refractive index of all or part of the material that forms the microresonator. This can be done thermally, as the refractive indices of most materials are temperature dependent. In a microresonator formed by semiconductor material, a much faster (nanosecond or shorter) tuning can be obtained through carrier injection. As discussed in other chapters in this book, the behavior of high-Q microresonators is very sensitive to the change in refractive index; a small change in the refractive index is sufficient to dramatically alter the path of light through the microresonator.

9.1 Electrically Tuned Microresonators Electrical tuning of semiconductor microresonators is based on the free-carrier dispersion effect in semiconductors, namely the change of the refractive index when Q. Xu (B) Department of Electrical and Computer Engineering, Rice University, Houston, TX, USA e-mail: [email protected] I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_9,  C Springer Science+Business Media, LLC 2010

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the electron or hole density varies. For silicon in particular, the change in refractive index at the telecommunication wavelength of 1.5 μm depends on the electron and hole density according to the following expression [1]:  

n = − 8.8 × 10−22 · N + 8.5 × 10−18 · P0.8 ,

(9.1)

where N and P are the densities of the electrons and holes (in cm−3 ), respectively. A change of refractive index in the microresonator results in a change in transmission (magnitude and phase) in the vicinity of the resonant wavelength of the microresonator and enables electrically controlled switching and modulation operations.

9.1.1 Electro-optic Modulators A silicon electro-optic modulator is a critical component for enabling optical interconnection systems on a microelectronic chip [2, 3]. Several silicon modulators based on the free-carrier dispersion effect in silicon have been demonstrated experimentally. The refractive index change in silicon is converted to amplitude modulation by either a Mach–Zehnder interferometer (MZI) [4, 5] or a microring resonator [6]. In comparison to the millimeter-long MZI-based modulators, the advantages of the microring-based modulator include its small size and low power consumption. In the microring resonator, in contrast to a single-pass device like the MZI, light at the resonant wavelength travels many round-trips and interacts repeatedly with the carriers. As a result, the total number of carriers needed to change the optical transmission of the resonator is much less than that needed in MZI-based modulators, and therefore much less electrical power is needed to drive these carriers in and out of the active region. The low power consumption and small size of the microring modulator is critical to the optical interconnection systems, since to handle the complex data flow of a computer system, thousands of modulators must be integrated on a chip. Another advantage of microring-based modulator is that, as will be shown below, it only modulates light near the resonant wavelength and allows light at other wavelengths to pass through without having been affected. This property allows multiple microring modulators with different resonant wavelengths to be cascaded along a single bus waveguide and modulates multiple data channels at different wavelengths, forming a simple wavelength–division-multiplexing (WDM) modulation system [7]. Figure 9.1 shows the schematic of a silicon microring modulator [6], composed of a silicon microring resonator with a built-in p–i–n junction side-coupled to a silicon waveguide. The inset of Fig. 9.1 shows the cross section of the ring. Both the bus waveguide and the ring consist of a silicon strip with a width of 450 nm and a height of 200 nm, sitting on top of a 50-nm-thick silicon slab. Since the thickness of the slab is much smaller than the wavelength of light propagating in the device

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Fig. 9.1 (a) Schematic layout of the microring resonator-based modulator. The inset shows the cross section of the ring. R: radius of ring; VF : voltage applied on the modulator. (b) Top-view microscopic image of the microring resonator [6]. Reprinted with permission. Copyright 2006 Nature Publishing Group

(∼ 1.5 μm), the mode profile of this rib waveguide structure is similar to the highly confined mode of a silicon strip waveguide [8]. The diameter of the ring is 12 μm, and the spacing between the ring and the bus waveguide is ∼ 200 nm. The device is fabricated on an SOI wafer using e-beam lithography and reactive ion etching. The highly p- and n-doped regions are defined around the ring with ion implantation. Figure 9.2a shows the TE-mode (major E-field parallel to the substrate [8]) transmission spectrum of a modulator with different bias voltages applied. The solid curve shows the spectrum when the forward bias on the p–i–n junction (0.59 V) is much lower than the build-in potential of the junction, and the current through the junction is below our detection limit (0.1 μA). The carrier density in the ring is very low at this state. The spectrum shows a 15-dB drop of transmission at the resonant wavelength. The carrier (electron-hole pair) density in the cavity increases as the forward bias on the p–i–n junction increases. The dashed and dotted curves in Fig. 9.2 show the spectra when the bias voltage is 0.87 and 0.94 V and the current is 11.1 and 19.9 μA, respectively. In both cases the resonance is blue-shifted due to the lowering of effective index caused by the increase of carrier density in the cavity according to (9.1). Figure 9.2b shows the transfer function of the device, i.e., the optical transmission of the modulator versus different bias voltages, at the initial resonant wavelength of λ0 . This transfer function shows a clear gate-like behavior. When the bias voltage is lower than the threshold of the diode, there is no carrier injection and the optical transmission is constantly low. When the bias voltage is high, the resonance is completely shifted away from the probe wavelength, and further shifting of resonance by applying higher bias voltage does not change the optical transmission significantly. Due to the high Q of this resonance, the transition from low transmission to

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Fig. 9.2 (a) The transmission spectra of the microring resonator at the bias voltages of 0.58, 0.87, and 0.94 V. Reproduced from [6]. (b) Transfer function of the modulator for light at the probe wavelength marked by the vertical dashed line in (a) [6]. Reprinted with permission. Copyright 2006 Nature Publishing Group

high transmission is very quick. One can see from Fig. 9.2b that the transmission changes from below 10% to over 90% with ∼ 0.15-V bias voltage change and with a current of below 15 μA. Besides the carrier-injection-based p–i–n modulator shown above, silicon modulators based on MOS capacitors [4] and depletion-based p–n diodes [9] have also been demonstrated. Among them, the carrier-injection-based p–i–n modulator can provide large changes of refraction index and high modulation depths because it does not suffer from either small optical confinement factor (as in the case of the MOS capacitor) or a low carrier density in the active region (as in the case of the depletion-based p–n diode). Conventionally, the operation speed of the carrierinjection-based p–i–n modulator is thought to be limited by the relatively slow diffusion of minority carriers, while the operation of the other two types of modulators is based on the drift of majority carriers, a much faster process. However, the gate-like transfer function of the microring modulator and the fast carrier extraction under reverse bias enables breaking the speed limit imposed by the slower carrier injection and experimentally achieving a modulation speed higher than 10 Gbit/s [10]. To show how this scheme works, let us first look at the nonlinear relationship between the optical transmission T at the wavelength λ0 and the total charge q in the junction when the microring is critically coupled to the waveguide. This is expressed by T =1−

1 , 1 + (q/q0 )2

(9.2)

e0 · ng VC , 2nf Q

(9.3)

where q0 =

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where e0 is the electron charge, ng is the group index of the ring, VC is the volume of the junction,  is the mode confinement factor, nf is the ratio between the change of refractive index and the carrier density, and Q is the quality factor of the microring resonator. Equation (9.3) also shows the importance of building a modulator with a small optical cavity; the required total charge injection and therefore the power consumption of the modulator are directly proportional to the volume of the cavity. When the p–i–n diode is forward biased, the electron and hole densities in the intrinsic region are equal, which means that the total charge carried by holes and electrons satisfies qh = –qe = q. Under dynamic driving voltage, the charge q in the junction obeys the following differential equation [11]: v(t) − vj (t) q(t) dq(t) q(t) = , = i(t) − − dt τC R τC

(9.4)

where v is the externally applied voltage, vj is the voltage drop on the junction, R is the serial resistance of the diode, determined primarily by the contact resistance at the metal–silicon interface, and τ C is the average carrier recombination lifetime in the device. The rise time of the optical transmission is much shorter than that of the charge injection, which is determined by the carrier dynamics with a positive bias voltage +V1 : dq(t) V1 − vth q(t) , = − dt R τC

(9.5)

where vth ≈ 0.7 V, close to the threshold condition. The steady-state value of the charge is qS = (V1 – vth)τ C /R. The rise time of the charge injection defined as the time needed for q to reach 0.9qS is 2.3τ C , which is ∼ 1 ns in the modulator used in the experiment [12]. On the other hand, from (9.2), one can see that the optical transmission quickly saturates to close to 100% as the charge q increases. Specifically, the transmission T > 90% as long as the charge q > 3q0 . Therefore, the optical rise time tr is only determined by the time needed to inject 3q0 charge into the junction. When 3q0  qS , the optical rise time is much shorter than the rise time of the charge injection. When the device is driven by an ideal NRZ signal shown in Fig. 9.3a, the carrier dynamics of the modulator and optical transmission are calculated based on (9.4) and (9.2) and plotted in Fig. 9.3b, c. By comparing the rising edge of the lines in Fig. 9.3b, c, one can clearly see that the optical rise time tr is much shorter than the electrical rise time determined by τ C . Since the optical rise time tr is determined by the rate of charge injection only, it can be further decreased by using a higher V1 . The fall time of the modulator is limited by the extraction time for the carriers. When the applied voltage on the device is switched to negative (v = −V2 ), the carriers are extracted out of the junction through drifting. This process can be very fast since the time for electrons to drift across the micrometer-wide junction can be as small as 10 ps. Given the finite serial resistance R, however, the total drift

Charge (10

-14

C)

Apllied Voltage (V)

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V 1'

2 0 –2

(a)

–4

–V 2

(d)

–V2

16 12 8 4

(b)

0 Optical Transmission

V1

V1

4

(e)

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tr

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t2

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tr

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(c)

(f)

0.0 0

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2 Time (ns)

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4

Fig. 9.3 Calculated dynamics of the microring modulator based on (9.2), (9.3), (9.4), and (9.5) (a) NRZ driving signal at 5 Gbit/s. (b) Total charge in the junction with driving voltage of (a). (c) Optical transmission of the microring resonator. (d) Pre-emphasized NRZ signal at 5 Gbit/s. (e) Total charge in the junction with driving voltage of (d). (f) Optical transmission of the microring resonator. The parameters used in the calculation are R = 7.7 k, VC = 4.5 × 10−12 cm3 ,  = 0.8, nf ≈ 4.3 × 10−21 cm3 , and Q ≈ 20,000 [10]. Reprinted with permission. Copyright 2007 Optical Society of America

current is limited by V2 /R, which limits the time to fully extract the excess carriers in the junction. The optical transmission of the modulator remains high until the charge q drops below 3q0 . The extra delay, which is called the charge storage time ts , is proportional to the excess charge injected into the cavity (qS – 3q0 ). This extra delay poses another limit to the modulation speed. When a higher driving voltage V1 is used to reduce the rise time tr , it leads to higher steady-state charge qS and longer charge storage time ts . In order to reduce ts without affecting tr , one can reduce the driving voltage immediately after q reaches the threshold 3q0 to avoid injecting excess carriers into the junction. As shown in Fig. 9.3e, f, by using a pre-emphasized NRZ driving signal with a high initial voltage immediately after a 0-to-1 transition, both the optical rise time tr and the charge storage time ts can be reduced, allowing for a higher operation speed. It is important to note that the pre-emphasis can also reduce the power consumption of the device by eliminating the injection and extraction of excess carriers.

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With the pre-emphasis, the power consumption is mostly determined by the power needed to swing the necessary amount of charge (∼ 3q0 ) in and out of the p–i–n junction during each bit-flip. The high-speed modulation has been demonstrated experimentally; the driving electrical signal (see Fig. 9.4a) is composed of a pseudo-random NRZ signal at 12.5 Gbit/s with Vpp ∼ 8 V and pre-emphasis pulses with a height of ∼ 4 V after each transition edges. The eye-diagram of the output optical signal is shown in Fig. 9.4b. One can see that the eye is completely open. The extinction ratio of the modulated signal is ∼ 9 dB. Note that the relatively high driving voltage used in the experiment can be greatly reduced if the contact resistance is reduced from that of the current device (∼ 7.7 k).

Fig. 9.4 (a) The waveform and eye-diagram of the electrical driving signal with pre-emphasis at 12.5 Gbit/s. (b) Eye-diagrams of the modulated optical output at 12.5 Gbit/s with pseudo-random binary sequence (PRBS) 210 –1 [10]. Reprinted with permission. Copyright 2007 Optical Society of America

With the pre-emphasized driving signal, the speed of the p–i–n microring modulator is fundamentally limited by the maximal drifting speed of electrons and holes in silicon, similar to the fundamental speed limit of the modulators based on MOS capacitor and depletion-type p–n junctions. When carriers are generated inside the microring, in addition to the resonance shift due to refractive index change, this also introduces an extra absorption loss α (in cm−1 ) which satisfies the expression [1]

α = 8.5 × 10−18 · N + 6.0 × 10−18 · P

(9.6)

This carrier absorption effect is responsible for the change in the depth of the resonance in Fig. 9.2a, because the microring resonator no longer satisfies the critical coupling condition with the additional absorption loss from the injected carriers. This extra absorption, however, does not affect the performance of the modulator significantly, because the signal is off-resonance when the carriers are generated and therefore does not interact strongly with the generated carriers.

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9.1.2 Electro-optic Switches A microring side-coupled to a single waveguide can provide the on–off modulation operation shown in the previous section. For more complicated operations, such as 1 × 2 or 2 × 2 switching, multiple waveguides and possibly multiple microresonators are required. The simplest form of a microresonator-based optical switch has two waveguides coupled to one microring, forming an add-drop configuration shown in Fig. 9.5. When it is on-resonance, light is directed from one waveguide to the other (to the drop port), while when it is off-resonance, light keeps propagating in the same waveguide (to the through port). By tuning the resonance of the ring, the path of light can be switched between these two output ports.

Fig. 9.5 SEM image of an SOI microring resonator coupled to two parallel waveguides

In the add-drop configuration shown above, light switched to the drop port propagates in the direction opposite to the direction in the input guide, which poses a complication for network design based on this device. To correct this problem, one can use two coupled microrings [13] instead of a single one. Coupled microrings also allow adjustment of the resonance shape, by varying the coupling strength and detuning between the microresonators. With multiple coupled microresonators, one can also obtain a resonance characteristic with a flatter top and steeper skirts than the Lorentzian line shape of a single ring. Microresonators can also be coupled to one arm of a MZI for either on–off modulation or 2 × 2 switching operation [14, 15]. The microresonator in this configuration is deeply over-coupled to the arm of the MZI; therefore its power transmission spectrum is essentially flat. However, the phase of the output light experiences an abrupt phase change across resonance. A microresonator in this state is usually referred to as an all-pass filter. The phase variation from index tuning is converted to an amplitude modulation or a switching operation by the MZI. With this structure, the linearity of a modulator can be improved [14, 15].

9.2 All-Optically Tuned Microresonators The small size and low power consumption of microresonator-based active optical devices make it possible to build large-scale optical systems on chip, for example, a multi-node communication network to connect different parts of a computer chip.

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One way of building this network is to use only point-to-point optical connections and put all the network functions in the electronic domain. This will, however, add a large latency and a high power consumption budget to the system. Another solution is to add basic network functions on the optical layer, so that the optics part can route the packages without the need to convert them to electronic signals and to convert back, thereby saving both time and energy consumption. This architecture requires the optical devices to make decisions, at least simple ones, based on the optical information received, and to control the flow of light based on these decisions. The optical devices also need to be able to handle situations like competition for an output port and congestions in the network. To fulfill these roles, all-optical devices like switches, logic gates, and buffers need to be developed on chip. Controlling the flow of light optically in a microresonator-based device involves changing the refractive index of a material optically. This can be achieved with the third-order nonlinear effect, or Kerr effect, in a dielectric material. Because the Kerr effect is almost instantaneous, the operation speed of such devices can be extremely fast. However, the Kerr effect is relatively weak in materials commonly used for integrated waveguides [16]; therefore, a CW all-optical device based on Kerr effect is hard to realize on chip. Another effect that can be used to control all-optical devices is again the freecarrier dispersion effect in semiconductors. Unlike the electro-optic modulators shown in the previous section, the carriers in the all-optical devices are generated optically through either linear absorption or two-photon absorption. In the first demonstration of such devices on silicon [17], the carriers were generated by an out-of-plane illumination. A short-pulse green laser was directed onto a silicon microring to create electron-hole pairs through linear absorption of the photons. The generated carriers shifted the resonance frequency of the ring and changed the transmission of an infrared probe light in the bus waveguide coupled to the ring. Because the control light and the probe light are vastly different in wavelength, this scheme does not allow cascading of devices and building of more complicated systems. However, this method allows for very fast generation of carriers, limited only by the duration of the control pulse, which can be well below a picosecond. This ultrashort transition time made the recent experimental demonstrations of wavelength conversion and stopping of light in microresonators possible [18, 19]. For most on-chip optical network applications, both the control and the signal (probe) light should be transmitting inside the integrated waveguide, which means the control light must be in the transparent spectral range of the waveguide medium and no linear absorption is tolerated. In this situation, one can rely on the twophoton absorption (TPA) effect [16] to generate the free carriers. When the energy of each photon is smaller than the bandgap of the semiconductor but larger than half of the bandgap, two photons can be absorbed simultaneously to create an electronhole pair in the material. The probability of this to happen, and thus the absorption rate, is proportional to the intensity of light (i.e., the photon density). The number of carriers generated by the TPA effect is proportional to the square of the optical power. The carriers generated by the TPA effect change the refractive index of the resonator and modify the transmission property of the device.

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While the TPA is normally quite weak in the waveguides and can only be excited efficiently with short high-power pulses, the optical intensity inside a highQ microresonator is significantly higher than that in the input waveguide when the resonator is on-resonance. This intensity enhancement effect in microresonators allows efficient carrier generation through the TPA effect even with a CW control light of moderate optical power. All-optical switches based on the TPA and carrier dispersion effects were first demonstrated in GaAs-InGaAs [20] and silicon microrings [12] using short pulses as the control light. Later, all-optical modulation was shown in a silicon microring with a higher quality factor using non-return-to-zero (NRZ)-coded CW control light [21].

9.2.1 All-Optical Modulation and Logic Figure 9.6 shows the input (control) and the output (probe) signal of an all-optical modulator based on a silicon microring resonator side-coupled to a bus waveguide. When the control light is on-resonance with the microring, its optical intensity inside the microring is (α2π R)−1 times higher than that in the input waveguide, where α is the attenuation coefficient of the waveguide forming the ring and R is the radius of the ring. This high optical intensity causes efficient carrier generation inside the microring and a strong modulation of the transmission of the probe light, whose

Fig. 9.6 (a) Input waveform of the control light, (b) output waveform of the probe light with positive modulation, and (c) output waveform of the probe light with negative modulation [21]. Reprinted with permission. Copyright 2007 Optical Society of America

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wavelength is in the vicinity of another resonance of the ring. Depending on the wavelength of the probe light, its transmission can either increase or decrease when the carrier is generated and the resonance is blue-shifted, resulting in either positive or negative modulations. The data initially carried on the control light can then be transferred onto the probe light.

Normalized Peak Transmission

1.0

0.8

0.6

0.4

0.2

0.0 0

2

4

6

8

10

Control Pulse Energy (pJ)

Fig. 9.7 Transfer function of the all-optical pulse modulation, showing the peak probe transmission immediately after the arrival of the control pulse, versus the control pulse energy, for the positive modulation (line with circles) and negative modulation (line with squares) [23]. Reprinted with permission. Copyright 2007 Optical Society of America

The required average input power for the control light to achieve efficient modulation of the probe light is found to be [21] 0 1 2 2 1 π ng neff ωC VC2 PC ≥ 2 , 2nf βλ22 Qτc

(9.7)

where nf is the ratio between the refractive index change and the carrier density N (nf ≈ 4.3×10−21 cm3 in silicon when N is of the order of 1017 cm−3 ),  is the mode confinement factor, β is the TPA coefficient (β = 0.8×10−9 cm/W in silicon [16]), τ c is the carrier lifetime, ng and neff are the group and effective indices of the waveguide forming the ring, respectively, ωC is the control photon energy, and Vc is the volume of the ring. In the experiment, the average power of the control light was only ∼ 4.5 mW. The all-optical modulation operation shown above can also be used to realize all-optical logic functions [22, 23]. This is based on the highly nonlinear (gate-like) transfer function of the modulation process. Using a short optical pulse as the control signal, the relationship between the transmission of the probe light and the control pulse energy is plotted in Fig. 9.7 for both positive and negative modulation in a silicon microring. Due to the nonlinear nature of this TPA-based process, there is a clear threshold energy that the control pulse has to pass to obtain large modulation.

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One can observe the gate-like behavior on the transfer functions with a sharp transition region sandwiched between two flat regions. Specifically, one can see a 10-dB increase of modulation depth on the positive modulation transfer function when the control pulse energy increases from 2.6 to 5.2 pJ. Similarly, one can see a 10-dB increase of modulation depth on the negative modulation transfer function when the control pulse energy increases from 4.1 to 8.2 pJ. Therefore, with properly chosen control pulse energy, one can obtain dramatically different modulations when two control pulses arrive simultaneously compared to when only a single control pulse arrives, enabling the logic operations of AND and NAND gates. Other logic functions can be realized by cascading the basic microresonator units [24]. To demonstrate the AND and NAND functions, a CW light from a tunable laser is modulated with PRBS 27 -1 return-to-zero (RZ) signal, which is split into two and sent into a silicon waveguide coupled to a silicon microring from two opposite directions. A CW probe light tuned at another resonance of the microring resonator is coupled with one of the control signals and sent into the device. The output of the probe is separated from the control using an optical circulator and an optical filter. The waveform of the probe is then observed on an oscilloscope. The two control signals have bit-rates of 310 Mbit/s and average optical power of ∼ 2 mW. The waveforms of these two control signals, synchronized at the device, are shown in Fig. 9.8a, b, respectively. When both control signals are in logic “1” state, which means that the two control pulses are coupled into the microring simultaneously, the total optical energy is higher than the threshold for obtaining large modulation, and a positive or negative modulation is imposed onto the probe light depending on the wavelength of the probe. When one or both of the control signals are in the “0” state, the total optical power is less than the threshold, and very little modulation is observed on the probe output. This results in the AND or NAND operations with extinction ratio ∼ 10 dB, as evident from Fig. 9.8c, d, respectively. The bit-rate is limited by the free-carrier lifetime in the resonator, which is ∼ 0.5 ns in silicon [12]. To increase the bit-rate, one can actively extract the carriers from the resonator, instead of waiting for the carriers in the ring to recombine at the Si/SiO2 interfaces. We have shown that carrier lifetime can be reduced to ∼ 30 ps by reversely biasing a p–i–n junction built across the microring [25], enabling logic operations at a rate of ∼ 5 Gbit/s.

9.2.2 Optical Bistability When the control light generates free-carriers inside a microresonator and tunes its resonance, it not only changes the transmission of the probe light but also the transmission of itself and its coupling strength to the ring. Depending on the wavelength of the control light, this self-acting effect can lead to a positive feedback and bistability [26, 27], which is the basis of sequential logic elements. In the combinational logic operation shown above, the output of the device is determined by a logic function of the two input signals within the same bit. Applications like the all-optical routing, where the route of the rest of the packet is determined by the header of

Microresonators with Active Tuning Control Power 2 (mW) Control Power 1 (mW)

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Fig. 9.8 Waveforms of the control light input and the probe light output. (a) Waveform of control input 1. (b) Waveform of the control input 2. (c) Waveform of the probe light with positive modulation, acting as the logic AND of the two control inputs. (d) Waveform of the probe light with negative modulation, acting as the logic NAND of the two control inputs [23]. Reprinted with permission. Copyright 2007 Optical Society of America

the packet, require the state of a switch to be controlled by previous information, and therefore the combinational logic is not sufficient. The bistability effect shown below adds memory to the system, and therefore allows sequential logic operations. The bistability effect is based on the TPA-induced carrier generation in the silicon microring resonator. The generated free carriers decrease the refractive index of silicon and cause a blue shift of the resonant spectrum of the microring. When the wavelength of the input light is shorter than the resonant wavelength with no free carriers, this blue shift enhances the optical coupling into the ring which, in turn, increases the carrier generation in the ring. The higher carrier generation then increases the blue shift of the resonance and causes a further increase in the coupling

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to the ring. This positive feedback process continues until the blue shift induced by carrier generation is so large that the wavelength of the input light becomes longer than the resonant wavelength of the ring. At this point, the previous positive feedback effect becomes a negative feedback effect, and the blue-shifting process reaches a steady state. To analyze bistability in a microring resonator analytically, we assume that the resonator is critically coupled to a single waveguide and its resonance follows an ideal Lorentzian line shape. We define the normalized initial detuning of the input optical wave as a=

λ − λ0 , 1

λ FWHM 2

(9.8)

where λ is the wavelength of the input light, λ0 is the initial resonant wavelength without carriers, and λFWHM is the bandwidth of the resonance. We define a normalized resonance shift as x=

δλ 1 2 λFWHM

< 0,

(9.9)

where δλ is the shift of the resonant wavelength caused by free-carrier generation. Under steady state, these quantities are related by the equation )2 ( P2in = K 1 + (a − x)2 x,

(9.10)

where Pin is the input optical power and K=

π 2 n3g λ20 nf

·

ω · VC2 1 · 3. βτc Q

(9.11)

For a given input wavelength detuning a and optical power Pin , the normalized steady-state resonance shift x can be obtained from (9.10). The transmission for the input optical wave can be calculated from the Lorentzian lineshape of the resonator as T=

1 Pout . =1−  Pin 1 + (a − x)2

(9.12)

Figure 9.9 shows the relationship between Pout and Pin for two different detunings. When the Pout versus Pin curve has a section with a negative slope as the solid line shows, the points on the negative-slope section are unstable. For each input power Pin within this range, there are two stable Pout solutions in addition to the unstable one. The transition between the two stable states follows the dotted arrows when the input power increases or decreases, respectively. The steady state of the device, therefore, follows different route when the input power increases or

Microresonators with Active Tuning

Fig. 9.9 The transfer function of a microring resonator with the normalized input wavelength detuning of a = −1 (dashed line) and a = −1.5 (solid line). The dotted arrows show the direction of transitions between the bistable states

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decreases, forming a hysteresis loop. When the input power lies within the region between the two dotted arrows, the state of the device depends on its previous state. The condition for the negative-slope section (and the bistability effect) to exist is √ a < − 5/2. When this condition is not satisfied, for example, when a = –1, the transfer function, shown as the dashed line in Fig. 9.9, shows no bistability. The transfer function (Pout versus Pin ) of the resonator measured experimentally shows a hysteresis loop as expected. Fig. 9.10 displays the TE-mode transmission spectrum of a 5-μm radius silicon microring around the resonant wavelength of λ0 = 1532.85 nm. This resonance, with ∼ 14 dB extinction ratio and a Q of ∼ 14,000, is used to demonstrate the bistability.

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1532.8 1532.9 Wavelength (nm)

1533.0

Fig. 9.10 Transmission spectrum of the microring resonator. The resonant wavelength λ0 and the wavelength used in the bistability measurements λ are marked with arrows on the spectrum [27]. Reprinted with permission. Copyright 2006 Optical Society of America

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Fig. 9.11 The transfer function of the microring resonator showing the bistability effect [27]. Reprinted with permission. Copyright 2006 Optical Society of America

In the experiment, a 4-ns-long optical pulse is sent into the waveguide coupled to the microring. Figure 9.11 shows the transfer function when the input wavelength is at λ marked on Fig. 9.10, corresponding to a normalized detuning a = −2. The two lines show the output power when the input power is increasing and decreasing. One can see a sharp transition of output power when the microring resonator changes from the off-resonance state (with high output power) to the on-resonance state (with low output power). This transition happens at two different input powers, namely at P1 when the input power is increasing and at P2 when the input power is decreasing, forming a hysteresis loop. This behavior agrees well with the calculated transfer function shown in Fig. 9.9. For the same input power of 6.2 mW, the output power of the resonator can be very different if the pulse follows a low input power (point A), or if it follows a high input power (point B). In order to reach point A, one needs to increase the input from 0 to 6.2 mW, which is done using a single square pulse with 6.2-mW peak power shown in Fig. 9.12a. Since the resonator is off-resonance before the pulse arrives, only a small percentage of the input power is coupled into the ring, which is not sufficient to trigger the positive feedback process of the resonance blue shift. Therefore, the resonator remains off-resonance as the pulse passes and the output power stays high, as shown in Fig. 9.12b. To reach point B on the curve, one must decrease the input power from a value greater than P1 . To achieve this, another high-power pulse (leading pulse), whose wavelength is at the short wavelength edge of another resonance of the ring, is sent into the waveguide right before the main pulse to be measured (see the gray line in Fig. 9.12c). This leading pulse blue shifts the resonances of the ring and causes the output power to reach point C on the curve in Fig. 9.11. Therefore, the resonator is on-resonance before the main pulse with 6.2 mW peak power arrives (black curve in Fig. 9.12c), and most of the main pulse is coupled into the ring.

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Input Power (mW)

(a)

(c) 12 8 4 0 0

2

4

6

8

10

Time (ns)

0.4

(b)

Output Power (mW)

Output Power (mW)

0.4 0.3

point A

0.2 0.1 0.0

(d)

0.3

point B

0.2 0.1 0.0

0

2

4

6

8

10

Time (ns)

0

2

4

6

8

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Time (ns)

Fig. 9.12 Waveforms of input and output pulses for obtaining points A and B on the transfer function curves in Fig. 9.10. (a) Input pulse for obtaining point A. (b) Output pulse for the input shown in (a). (c) Input pulses for obtaining point B. The gray line shows the leading pulse and the black line shows the main pulse, which is identical to the input pulse shown in (a). (d) The output pulse for the input shown in (c) [27]. Reprinted with permission. Copyright 2006 Optical Society of America

The optical power coupled into the ring generates enough carriers to keep the resonance shifted, so that the resonator is kept on-resonance and the output power stays low, as shown in Fig. 9.12d. The foregoing analysis shows that the state of the resonator depends on the previous state as well as the current input, forming the basis for building sequential logic on silicon for applications such as all-optical routing.

9.3 Thermally Tuned Microresonators While actively tuned microresonators based on the free-carrier dispersion effect can be operated at high speed, their tuning range is relatively small, usually below 1 nm in wavelength. In theory, it is possible to inject or generate enough carriers to shift the resonance for tens of nanometers. However, with a high carrier density, the optical loss caused by free-carrier absorption is significant, resulting in a significant drop of the resonator Q. In addition, the carriers are constantly recombining, generating heat in the material which shifts the resonance in the opposite direction due to the

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thermo-optical effect we shall discuss below. Thermo-optical tuning, on the other hand, has a larger tuning range, especially for materials with high thermo-optic coefficient like silicon (dn/dT = 1.86 × 10−4 K−1 [28]), is not encumbered with extra absorption loss, and can work with both dielectric and semiconductor materials. Therefore, for applications that only require moderate or low speed (MHz or below), thermo-optic tuning is the preferred choice. The resonant wavelengths of microresonators are very sensitive to dimensional change. Taking the silicon microring resonator as an example, if the average width of the ring changes by 1 nm, the resonant wavelength will shift ∼ 0.7 nm, much larger than the bandwidth of the microring we normally use. Therefore, with the current fabrication capability, where the critical dimension control is ∼ 5 nm (according to the International Technology Roadmap for Semiconductors), it is hard to fabricate a microresonator with a resonant wavelength right where it is designed to be. Under this situation, thermo-optic tuning can be used to compensate for fabrication variations and bring the resonant wavelength to where it is supposed to be. There are various ways to heat up the microresonators for thermo-optic tuning. It can be heated by an integrated electric heater [29], it can absorb energy from a laser source [30] or, in the case of semiconductor microresonators, it can be heated from inside with a current passing through [31]. There are numerous works on thermally tuned microresonators; here we show an example where integrated electric heaters are used to tune the bandwidth of the microring resonator [29].

9.3.1 Thermal–Optical Bandwidth Tuning The bandwidth of a microresonator, determined by its loaded Q, greatly affects its performance. While a narrow bandwidth is important for enhancing the optical intensity in the microresonator, the sensitivity to index tuning and the interaction between light and matter; a microresonator with narrow bandwidth blocks the transmission of ultrafast optical signals and is sensitive to small environmental variations or fabrication imperfections. Therefore, the optimal bandwidth of a microresonator can be vastly different depending on the applications, or on the operation modes of the device. With a tunable bandwidth, the microresonator can be adjusted to fit the requirements of different applications and can be fine-tuned to optimize its performance as required. Tuning the bandwidth requires varying the coupling between the resonator and the adjacent waveguide. This has been achieved with MEMS structures by physically moving the bus waveguide [32], or with coupled resonators to change the coupling between the coupled super-mode and the waveguides [18, 30]. Here, the directional couplers that couple the waveguides with the microring are replaced by MZIs [29], as shown in Fig. 9.13. Electrical heaters, in the form of wires patterned on a metal film composed of 100-nm-thick Ni and 150-nm-thick Au with a total resistance of ∼ 160 , are placed on the MZIs with a 1-μm-thick SiO2 layer between the metal and the waveguide. When electrical current is injected into the heaters, the effective index of the heated arms increases as temperature rises, which

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Fig. 9.13 (a) Optical and (b) scanning electron microscope (SEM) images of a microring resonator with thermal bandwidth tuning [29]. Reprinted with permission. Copyright 2007 Optical Society of America

changes the phase difference between the MZI arms and changes the coupling ratios between the waveguides and the ring. Figure 9.14 shows the measured transmission spectra of the resonator tuned to different bandwidths. The FWHM changes from 0.1 nm in curve (a) to 0.7 nm in curve (b), by adjusting the current on each heater.

Fig. 9.14 Tuning the bandwidth within “critical coupling” with two heaters. The dashed–dotted line marks the 3-dB level. The electrical powers absorbed by the two heaters are listed [29]. Reprinted with permission. Copyright 2007 Optical Society of America

9.3.2 Competition Between Carrier and Thermal Tuning Even though thermo-optic tuning can be used to actively tune them, it is adverse to many microresonator applications. The sensitivity to environmental temperature change is a severe problem which has not been satisfactorily solved to date. In the electro-optic devices based on semiconductor microresonators, whenever free carriers are generated, recombined, injected, or extracted, heat is also generated

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inside the resonator, causing further temperature shifts. For this reason the design of carrier-based optical devices must accommodate or avoid parasitic thermal effects. The competition between carrier-based and thermal-based effects can be seen, for example, in the following analysis of the carrier-induced and thermally induced bistability in silicon microring resonators [27]. In Section 9.2.2, carrier-induced bistability was demonstrated using a nanosecond pulse. The use of a short pulse instead of CW signal is necessary to avoid the thermal effect. In fact, if a CW wave were used, one would observe thermally induced bistability instead, which was demonstrated in an earlier paper [33]. For a quantitative analysis, let us assume that the optical power absorbed due to TPA is PA . Then the density N of the generated electron–hole pair is determined by the equation N dN PA + = . dt τc 2ω · VC

(9.13)

The refractive index of silicon changes with carrier density as n = nf N, where nf ∼ = 4.3×10−21 cm3 for N ∼ 1017 cm−3 . Since both the TPA and the carrier recombination are phonon-assisted processes, all the absorbed energy is eventually converted to thermal energy, and the temperature shift can be obtained from the equation

T d T PA + = , dt τθ ρCVC

(9.14)

where τ θ ∼ 1 μs [33] is the thermal dissipation time, ρ = 2.3 × 10−3 kg/cm3 is the density of silicon, and C = 705 J/kgK is the thermal capacity. The refractive index of silicon changes with temperature as nθ = nθ T, where nθ = 1.86×10−4 K−1 [28]. In steady state the ratio between the temperature-induced index change and the carrier-induced index change is

nθ 2ω · nθ τθ τθ = ≈ −0.022 .

nc ρCnf τc τc

(9.15)

With τ c = 0.5 ns and τ θ ∼ 1 μs, nθ / nc ∼ –44. Therefore, when the optical input is CW, the thermally induced effect dominates over the carrier-induced effect. Optical bistability based on the thermal effect has been observed in silicon microring resonators using CW optical input, with optical power of less than 1 mW [33], lower than that used here for carrier-induced bistability. When the optical input is a nanosecond pulse with pulse width T satisfying τ c  T  τ θ , the ratio between the temperature-induced index change and the carrier-induced index change becomes

nθ 2ω · nθ T T = ≈ −0.022 .

nc ρCnf τc τc

(9.16)

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Therefore, the carrier-induced effect is larger than the thermally induced effect when T is shorter than 46τ c = 23 ns. This time scale determines the lifetime of state B in Fig. 9.11. The carrier-based electro-optical and all-optical devices inevitably generate heat in the resonator and thereby excite parasitic thermal–optical effect. This has to be carefully considered when designing carrier-based devices. In some applications, an active feedback control mechanism is required to stabilize the temperature of the resonator.

9.4 Summary This chapter described several ways to tune the resonance of microresonators. The tuning can be either free-carrier based or thermal based, while the free carriers and the thermal power can be generated either electrically or optically. With the active tuning capability, microresonators can be used to manipulate the propagation of light at real time. Various optical devices, including electro-optic modulators, switches, all-optical logic, and memories, have been demonstrated experimentally. These microresonator-based devices have small footprint, low power consumption, and the capability of large-scale integration, that are unmatched by devices based on conventional low-index-contrast systems.

References 1. Soref, R.A., Bennett, B.R. Electrooptical effects in silicon. IEEE J. Quant. Electron. QE-23, 123–129 (1987) 2. Miller, D.A.B. Optical interconnects to silicon. IEEE J. Sel. Topics Quant. Electron. 6, 1312–1317 (2000) 3. Meindl, J.D., Davis, J.A., et al. Interconnect opportunities for gigascale integration. IBM J. Res. Dev. 46, 245–263 (2002) 4. Liu, A.S., Jones, R., et al. A high-speed silicon optical modulator based on a metal-oxidesemiconductor capacitor. Nature 427, 615–618 (2004) 5. Green, W.M.J., Rooks, M.J., et al. Ultra-compact, low RF power, 10 gb/s silicon MachZehnder modulator. Opt. Express 15, 17106–17113 (2007) 6. Xu, Q., Schmidt, B., et al. Micrometre-scale silicon electro-optic modulator. Nature 435, 325–327 (2005) 7. Xu, Q.F., Schmidt, B., et al. Cascaded silicon micro-ring modulators for WDM optical interconnection. Opt. Express 14, 9430–9435 (2006) 8. Vlasov, Y., McNab, S. Losses in single-mode silicon-on-insulator strip waveguides and bends. Opt. Express 12, 1622–1631 (2004) 9. Sadagopan, T., Choi, S.J., et al. Optical modulators based on depletion width translation in semiconductor microdisk resonators. IEEE Photon. Technol. Lett. 17, 567–569 (2005) 10. Xu, Q., Manipatruni, S., et al. 12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators. Opt. Express 15, 430–436 (2007) 11. Pierret, R.F. Semiconductor device fundamentals. Addison-Wesley, Reading, Mass (1996) 12. Almeida, V.R., Barrios, C.A., et al. All-optical control of light on a silicon chip. Nature 431, 1081–1084 (2004)

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13. Emelett, S.J., Soref, R. Design and simulation of silicon microring optical routing switches. J. Lightwave Technol. 23, 1800–1807 (2005) 14. Yang, J., Wang, F., et al. Influence of loss on linearity of microring-assisted Mach-Zehnder modulator. Opt. Express 12, 4178–4188 (2004) 15. Van, V., Herman, W.N., et al. Linearized microring-loaded Mach-Zehnder modulator with RF gain. J. Lightwave Technol. 24, 1850–1854 (2006) 16. Dinu, M., Quochi, F., et al. Third-order nonlinearities in silicon at telecom wavelengths. App. Phys. Lett. 82, 2954–2956 (2003) 17. Almeida, V.R., Barrios, C.A., et al. All-optical switching on a silicon chip. Opt. Lett. 29, 2867–2869 (2004) 18. Xu, Q., Dong, P., et al. Breaking the delay-bandwidth limit in a photonic structure. Nature Phys. 3, 406–410 (2007) 19. Preble, S.F., Xu, Q., et al. Changing the colour of light in a silicon resonator. Nature Photon. 1, 293–296 (2007) 20. Van, V., Ibrahim. T., et al. All-optical nonlinear switching in GaAs-AlGaAs microring resonators. IEEE Photon. Technol. Lett. 14, 74–76 (2002) 21. Xu, Q., Almeida, V.R., et al. Micrometer-scale all-optical wavelength converter on silicon. Opt. Lett. 30, 2733–2735 (2005) 22. Ibrahim, T.A., Grover, R., et al. All-optical AND/NAND logic gates using semiconductor microresonators. IEEE Photon. Technol. Lett. 15, 1422–1424 (2003) 23. Xu, Q., Lipson, M. All-optical logic based on silicon micro-ring resonators. Opt. Express 15, 924–929 (2007) 24. Ibrahim, T.A., Amarnath, K., et al. Photonic logic NOR gate based on two symmetric microring resonators. Opt. Lett. 29, 2779–2781 (2004) 25. Preble, S.R., Xu, Q., et al. Ultrafast all-optical modulation on a silicon chip. Opt. Lett. 30, 2891–2893 (2005) 26. Van, V., Ibrahim, T.A., et al. Optical signal processing using nonlinear semiconductor microring resonators. IEEE J. Sel. Top. Quant. Electron. 8, 705–713 (2002) 27. Xu, Q., Lipson, M. Carrier-induced optical bistability in silicon ring resonators. Opt. Lett. 31, 341–343 (2006) 28. Cocorullo, G., Rendina, I. Thermo-optical modulation at 1.5 mum in silicon etalon. Electron. Lett. 28, 83–85 (1992) 29. Chen, L., Sherwood-Droz, N., et al. Compact bandwidth-tunable microring resonators. Opt. Lett. 32, 3361–3363 (2007) 30. Xu, Q., Shakya, J., et al. Direct measurement of tunable optical delays on chip analogue to electromagnetically induced transparency. Opt. Express 14, 6463–6468 (2006) 31. Vlasov, Y.A., O’Boyle, M., et al. Active control of slow light on a chip with photonic crystal waveguides. Nature 438, 65–69 (2005) 32. Lee, M.C.M.,Wu, M.C. Tunable coupling regimes of silicon microdisk resonators using MEMS actuators. Opt. Express 14, 4703–4712 (2006) 33. Almeida, V.R., Lipson, M. Optical bistability on a silicon chip. Opt. Lett. 29, 2387–2389 (2004)

Chapter 10

Performance of Single and Coupled Microresonators in Photonic Switching Schemes Jacob B. Khurgin

Abstract Applications of single and coupled microresonators in various photonic switching schemes are considered. It is shown that both single and coupled resonators enhance the performance of the switches based on nonlinear optical or electro-optical effect, but the enhancement occurs at the expense of reduced bandwidth. The critical bit rates at which the enhancement is still possible are derived for single and coupled microresonators. It is shown that the choice of optimum microresonator scheme is determined most of all by the intrinsic switching speed of the nonlinear or electro-optic material.

10.1 Introduction The idea of photonic switching has been with us at least since the invention of laser and first tentative nonlinear optics experiments, but this idea became especially attractive as photonic technology turned into primary means of moving information at previously unimaginable speeds [1]. Over the course of the past two decades the world has become smaller as different parts of it have become interconnected by wide-bandwidth optical networks. Comparing the optical means of information transfer with electronic ones, in addition to the inherent bandwidth advantage, the optical means of communication are immune to electromagnetic interference, consume less power, and occupy less space; hence it is natural to think not only of transporting the information but also of processing and switching it in the optical rather than the electrical domain. Currently, no matter how fast information is transported optically, when it comes to switching, the information has to be transformed into the electrical domain and all the switching must be performed by electronic means, before the information is once again converted into an optical signal [2]. J.B. Khurgin (B) Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, USA e-mail: [email protected] I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_10,  C Springer Science+Business Media, LLC 2010

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These transformations, often referred to as OEO (optical-to-electronic-to-optical), are very inefficient as they limit the speed of the network and introduce additional costly, heavy, and energy-hungry components and sub-systems. Thus avoiding the OEO conversions altogether portends sophisticated reconfigurable networks unconstrained by the OEO bottlenecks and having higher capabilities and lower cost, dimensions, and energy consumption. Despite the enormous potential payoff expected from adopting photonic switching, it still remains largely within the domain of research rather than being widely used in industry. The reason behind this is well known: it is far more difficult to manipulate photons than electrons, because, unlike electrons, photons carry no charge and thus cannot be easily controlled by electric and magnetic fields.

10.2 Photonic Switching: Real and Virtual Processes In essence, photons interact with external fields and with each other only via the media in which they propagate, i.e., the photon interaction can be described as the next order of perturbation formalism when compared to interaction between the electrons [3]. In solid-state materials, dielectrics, and semiconductors (organic and inorganic), the interaction between two photons can be imagined as a twostep process. First the first photon excites an electron–hole pair. If the energy of the photon exceeds the bandgap of the material, as shown in Fig. 10.1a, or if there is an impurity level in resonance with that energy, then a real electron and a real hole are excited. The photoexcited carriers remain in the material until they either recombine or get swept away by the external fields, and for as long as the carriers remain the optical properties of the material (usually described by the complex refractive index n = n +jn

) change, thus influencing the propagation of the second photon. Therefore the speed of the effect is determined by the

ΔE τc

Eg hω Fig. 10.1 (a) Slow resonant nonlinear optical process with photoexcitation of real carriers. (b) Fast non-resonant nonlinear optical process with photoexcitation of virtual carriers

a

Eg

hω b

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carrier lifetime τ c , with the cut-off frequency fmax being of the order of τ c –1 . Now, if the power density of the light with frequency ω propagating in the material with frequency-dependent absorption coefficient α(ω) is Iω , the total density of photoexcited carriers is Nc = α(ω)Iω τc /(ω) and the change of the refractive index n(ω) = n2 (ω)Iω ∼ Nc ∼ α(ω)Iω τc /(ω) (n2 is the nonlinear refractive index) is proportional to the carrier lifetime. What is important is that the product of the nonlinear refractive index and the speed (or bandwidth) of the effect, n2 fmax ∼ α(ω)/ω, does not depend on the lifetime. Therefore, the nonlinearity can be either extremely strong and slow, as for instance the photorefractive effect in dielectric with traps such as LiNbO3 [4], where lifetime is measured in fractions of seconds, or much weaker and yet relatively fast as in direct bandgap semiconductors (e.g., GaAs) where the recombination times can be as fast as tens of ps [5]. In the indirect bandgap material Si the carrier lifetime is relatively long (microseconds) but the carrier sweep-out time can be as short as 100 ps [6]. Overall, though, one can loosely refer to the nonlinearities associated with excitation of real carriers as relatively strong, slow, and lossy since the real absorption is involved. When the photon energy is much less than the bandgap of the material that is free from impurities, as shown in Fig. 10.1b, the energy conservation does not allow excitation of the real carriers, but the electronic states in the valence band and the conduction band do get affected by the strong optical fields. This can be rather conveniently explained by the phenomenological picture of virtual carrier excitation. If the energy deficit between the energy required for the excitation of an electron–hole pair and the photon energy is E, according to Heisenberg’s uncertainty principle, one can say that the carriers excited by the off-resonant light have a virtual lifetime of τV ∼ ω−1 = / E, where the detuning ω = E/ can easily be of the order 10 THz or more [1]. This makes the virtual nonlinearities practically instant, meaning that they are much faster than most of the methods of detecting the optical radiation and that is why they are often called “ultrafast” in the literature. The high speed of virtual nonlinearity comes, of course, at the expense of its reduced magnitude, since the nonlinear refractive index is inversely proportional to the detuning

E. The product n2 fmax is once again independent of the detuning or the virtual carriers’ lifetime. The ultrafast nonlinear refractive index is quite small, ranging from less than 10−15 cm2 /W in glass [7] to less than 1013 cm2 /W in Si and other semiconductors [8, 9]. Thus, even with power densities approaching GW/cm2 , the index change will be less than 10−4 and, to achieve an optical switching condition

nL∼λ, the length L of the switch would have to be as large as a few centimeters. To summarize, when it comes to optical nonlinearities, nature gives us a choice of either weak ultrafast non-resonant nonlinearities or strong, slow, and lossy resonant nonlinearities based on real carrier excitations. The concept of photonic switching is not limited to nonlinear optical effects, or switching light by light. It also incorporates various electro-optical effects in which the complex refractive index experiences changes induced by the external low-frequency electric fields and currents [10]. Here the term “low frequency” is used in comparison to the optical frequencies; thus any frequency attainable by

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the state-of-the-art electronic devices up to hundreds of gigahertz can be considered “low frequency.” Just as in the case of nonlinear optical effects, electro-optical ones can also rely on real and virtual carriers. When real carriers are injected into the semiconductor [11–13] by current iinj , as shown in Fig. 10.2a, the dielectric constant at optical frequency ω decreases according to the expression [13] n2 (ω) = n20 (ω) −

e2 Nc , ε0 mc ω2

(10.1)

where mc is the carrier effective mass. In the most widely used silicon structures, the refractive index changes in the 1.3–1.5 μm range can be as high as 10−3 with carrier concentration of 1018 cm−3 . But the real carriers in Si have very long lifetime, on the microsecond range. Therefore one must employ a scheme in which the carriers are swept out from the device by applying reverse bias. In this way the carrier lifetime can be reduced to below 1 ns and modulators operating at up to 10 Gbps have been reported [14]. Fig. 10.2 (a) Slow electro-optical process with injection of real carriers. (b) Fast electro-optical process with field-induced modification of quantum states

eV

iinj

ΔE Eg

hω

τc

a

ΔE’ Eg

hω b

Another type of the electro-optic effect does not rely upon the injection of the carriers but rather on the modification of the electronic states in the material under the influence of external electric fields, as shown in Fig. 10.2b, where the externally applied voltage V causes a change of the energies of states. In all materials there exits a quadratic (or Kerr) electro-optic effect in which the change in the refractive index is proportional to the square of the external electric field:

n ∼

1 3 2 n sERF , 2

(10.2)

where s is a Kerr coefficient, which is, strictly speaking, a fourth-rank tensor. In the materials that do not posses inversion symmetry, such as zincblende (sphalerit) materials (GaAs, GaP, and other III–V semiconductors), perovskites (LiNbO3 ) [15],

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poled polymers [16], and many others, there also exists a much stronger linear or Pockels electro-optic effect:

n ∼

1 3 n rERF , 2

(10.3)

where r is a Pockels coefficient, which is in general a third-rank tensor. In a typical electro-optic material, LiNbO3 , the relevant element of the Pockels tensor r33 = 30 pm/V; hence with electric fields of 106 V/m one can obtain refractive index changes of the order of 10−4 – significantly less than what can be achieved with carrier injection. But since no real carriers are generated or injected, the Pockels effect is intrinsically instant, and the limitations usually come from the inability to apply sufficiently strong microwave signal at high frequencies due to high losses in both the electro-optic material and the metal electrodes. Electro-optic modulators operating at 40 Gbps [17] are available in production while modulation rates of higher than 100 Gbps have been reported in the labora tories [18]. To summarize, just as in the case of all-optical nonlinearities, the electro-optic effect can be either strong, relatively slow, and lossy, when associated with injection of real carriers, or it can be relatively fast, weak, and free of excessive loss.

10.3 Resonant Enhancement Given the limitations of available optical switching materials, it is no wonder that over the years numerous attempts have been made to synthesize new materials with better figures of merit. One of the most obvious ways this could be obtained is by using the resonant enhancement of nonlinear optical and electro-optical effect near the strong resonant optical transitions within the medium. This could be accomplished, for example, near the strong atomic resonances in metal vapors [19], or using exciton resonances in semiconductor quantum wells [20]. The results with metal vapors may be spectacular, but they do not lead to compact practical devices, and unfortunately all the resonances in solid-state materials are very broad; hence whatever resonant enhancement can be attained, its beneficial effect can be more than canceled by the detrimental effect of higher absorption loss. But the intrinsic material resonances, occurring when the frequency of the light is (nearly) resonant with the frequency of intrinsic transition, are not the only resonant optical phenomena available to the designer of optical switches. Another type of resonance occurs when the wavelength of the light matches some characteristic dimension of the structured medium leading to a constructive interference between the waves. Typical examples of these “photonic resonances” can be found in Bragg gratings [21] and Fabry–Pérot resonators [22]. The advantage of “photonic” over “intrinsic material” resonances is twofold. First of all, the photonic structure can be easily designed to be resonant at a given

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frequency, while in case of material resonances the choices are limited. The second advantage is the fact that the losses in photonic resonant structures are in general much smaller than in the intrinsically resonant materials. This remarkable feature of photonic resonant structures follows from the fact that the energy in them stays always stored in the form of electromagnetic radiation (photons), and, as we have already mentioned, the photons do not interact strongly with the environment. In case of the intrinsic material resonance the energy gets stored as material polarization – one often calls these material polarization excitation “polaritons” – a combination of electronic excitations and photons. The electronic excitations strongly interact with the environment and the energy is dissipated. In solid-state the electronic excitations strongly interact with lattice vibrations, namely phonons, leading to dephasing times, or material excitation lifetimes as short as hundreds of femtoseconds, and as a result the intrinsic Q of optical transitions there rarely exceeds 100. On the other hand – and there is ample evidence to it in the rest of this volume – Q factors of resonant photonic structures in excess of 106 have been demonstrated. This fact indicates that lifetimes of phonons inside these structures can exceed a nanosecond, leading to large enhancements of nonlinear optical phenomena. There exist a number of resonant photonic schemes, such as Bragg gratings, various photonic crystals of different dimensionalities, Fabry–Pérot interferometers, and microresonators (MRs), the subject of this volume. Recent years have seen spectacular progress in the development of both linear and nonlinear optical devices based on MRs. High-Q MRs made from various materials have been successfully fabricated and used for such diverse applications as delay lines [23], microwave photonics [24, 25], sensors [26, 27], and others. Especially impressive have been recent advances in active MRs made from electro-optic or nonlinear materials. Active MRs are expected to be employed as miniature optical modulators [28], switches [11], tunable filters [29], and delay lines [30] that are amenable to large-scale integration culminating 1 day in the development of photonic VLSIs. Yet, just like any other promising technology, the MR technology is not without limitations. As the term “microresonator” itself suggests, the ability of photon confinement rests upon existence of a strong resonance in its characteristics, with all the desirable properties confined to a relatively narrow band near this resonance. When it comes to photonic switching, the very fact that the photon lifetime τ p in them is long limits the speed with which the switching can be accomplished. The limitations of the switching speed in single and coupled MRs are the subject of this chapter. The goal is to examine the factors limiting the performance of photonic switching schemes relying on the specific switching material and to provide the guidelines for performance optimization. In order to achieve our goal we first review the basic properties of single and coupled MRs. Subsequently we introduce the figures of merit for various switching schemes and finally develop a set of simple principles to help the designer of photonic switches.

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10.4 Single and Coupled Microresonators 10.4.1 Single Microresonators We start by considering a simple MR introducing its most basic characteristics. We shall use the ring resonator as an example but one can consider many alternative schemes, such as Fabry–Pérot, photonic crystal, whispering gallery mode microdisk cavities. A ring resonator can be used in two configurations: as an all-pass filter [29] (APF) when coupled to a single optical input–output bus as in Fig. 10.3a, or with two coupled waveguides making it essentially an add-drop filter [31, 32] (ADF) as in Fig. 10.3b. The resonator is characterized by a circumference L, effective index neff , group index ng = neff + ωdneff /dω, and an amplitude coupling coefficient κ. Disregarding the loss, the transmission function of the APF is ρ − e−j(ω−ω0 )τ Eout (ω) = ≡ e−j(ω) , Ein (ω) 1 − ρe−j(ω−ω0 )τ

(10.4)

where the resonant angular frequency ω 0 is determined from the condition neff ω0 L/c=2mπ , ρ 2 =1–κ 2 and τ =Lng /c is the ring round-trip time. As the name “all pass” implies, the amplitude transmission of the loss-free MR is unity for all frequencies and only the phase is affected as tan (ω) =

κ 2 sin [ (ω − ω0 ) τ ] . (1 + ρ 2 ) cos [(ω − ω0 )τ ] − 2ρ

(10.5)

out

jκ

ρ a.

b.

jκ in

ρ

out

jκ in

ρ

through

Fig. 10.3 (a) An all-pass filter based on a single-ring microresonator. (b) An add-drop filter based on a single-ring microresonator

The group delay of the APF is Td (ω) = =

∂(ω) (1 − ρ 2 )τ = ∂ω 1 + ρ 2 − 2ρ cos (ω − ω0 )τ , (0) Td

(2) + Td (ω

− ω0 ) + · · · 2

(10.6)

234

J.B. Khurgin

where the group delay at the resonant frequency is (0)

Td = τ

1+ρ 1−ρ

(10.7)

and the group delay dispersion (GDD) is (2)

Td = −τ 3

) ( ρ(1 + ρ) ρ (0) 3 = − T . d 3 (1 − ρ) (1 + ρ)2

(10.8)

When a small loss per ring round-trip α R is included, which results from a combination of absorption, scattering, and bending losses in the ring, the expression for the transmission of the APF becomes √   2  Eout (ω) 2 1 − αR ρ cos (ω − ω0 ) τ  = 1 − αR + ρ − 2 √   E (ω)  2 1 + (1 − αR )ρ − 2 1 − αR ρ cos (ω − ω0 ) τ in . (10.9)   αR 1 − ρ 2 =1− √ 1 + (1 − αR )ρ 2 − 2 1 − αR ρ cos (ω − ω0 ) τ For an APF with small loss we can write from (10.9) and (10.6)    Eout (ω) 2 Td (ω)    E (ω)  = 1 − αR τ , in

(10.10)

which makes perfect sense indicating that the total loss is equal to the per ring round-trip loss multiplied by the average number of round trips the light makes inside the MR. In Fig. 10.4a,b the phase and group delay characteristics of an APF with circumference L=36 μm, effective index ng = 4.16 (τ = 500 fs), and κ = 0.3 [23] are shown, while Fig. 10.4c shows the transmission characteristic of the same APF with a 0.04 dB round-trip loss. As one can see, large group delays (exceeding τ by a factor of 40) can be achieved in a relatively narrow bandwidth near ω0 . In fact, one can estimate the full width at half maximum (FWHM) of the transmission and group delay characteristics as

ωAPF =

21−ρ 2 1+ρ = (0) 1/2 . 1/2 τ ρ Td ρ

(10.11)

This provides the simple relation between the group delay and the width of the resonance

ωAPF Td(0) = 2 (0)

1+ρ ρ 1/2

(10.12)

or simply fAPF Td ≈ 2/π ≈ 0.64 for resonators with small coupling coefficient κ. Note that using simple power series expansion in (10.6) one can obtain the expression for the FWHM of APF:

10

Performance of Single and Coupled Microresonators 3

235

18 16

2

0 –1 –2

12

Delay (ps)

Phase (rad)

14 1

10

6 4

a

–3 –200 –150 –100

ΔνAPF

8

b

2 –50

0

50

100

150

0 –200 –150 –100

200

Frequency (GHz)

–50

0

50

100

150

200

Frequency (GHz)

1

Transmission

0.95 0.9

ΔνAPF

0.85 0.8 0.75

c

0.7 –200 –150 –100

–50

0

50

100

150

200

Frequency (GHz)

Fig. 10.4 (a) Phase, (b) group delay, and (c) transmission characteristics of the APF of Fig. 10.3a



ωAPF ≈

2Td(0) (2) −Td

1/2 =

21/2 1 + ρ , (0) ρ 1/2 T

(10.13)

d

which differs from the exact equation only by a factor of 21/2 . Now, intuitively the strength of any optical effect, linear or nonlinear, increases with the interaction time, i.e., the time the light spends inside the resonator. Therefore the resonance enhancement in the MR can only come at the expense of the bandwidth, with the product of the two staying constant. Let us now turn our attention to the ADF of Fig. 10.3b whose transmission coefficient is −κ 2 e−j(ω−ω0 )τ/2 Eout (ω) = Ein (ω) 1 − ρ 2 e−j(ω−ω0 )τ

(10.14)

and has the Fabry–Pérot-like intensity spectrum shown in Fig. 10.5a. The corresponding FWHM is

ωADF = 2κ 2 τ −1 .

(10.15)

236

J.B. Khurgin 1

4.5

0.9

4

0.8

3.5 3

Delay (ps)

Transmission

0.7 0.6 0.5

Δν ADF

0.4 0.3 0.2 0.1 0

2.5

ΔνADF

2 1.5 1

a

0.5

–200 –150 –100 –50

0

50

100

150

0

200

b

–200 –150 –100

–50

0

50

100

150

200

Frequency (GHz)

Frequency (GHz)

Fig. 10.5 (a) Transmission characteristic and (b) group delay of the ADF of Fig. 10.3b

The delay time of the signal Td =

∂ (1 − ρ 2 )ρ 2 , = ∂ω 1 − 2ρ 2 cos (ω − ω0 )τ + ρ 4

(10.16)

i.e., the time spent inside the resonator is shown in Fig. 10.5b and reaches at resonance its maximum: (0)

Td = τ

ρ2 1 − κ2 =τ . 2 1−ρ κ2

(10.17)

From (10.15) and (10.17) we obtain the simple relationship between delay and passband width: Td(0) ωADF = 2(1 − κ 2 ).

(10.18)

(0)

For weakly coupled resonators fADF Td ≈ 1/π ≈ 0.32, i.e., one-half of the delay-bandwidth product of the APF. No matter what the application of MR is, the longer the time the light spends inside the resonator, the stronger is the effect one can impose on the light. Therefore (10.12) or (10.12) would lead to a gainbandwidth product for any application. In this chapter we shall limit ourselves to digital signals; therefore we shall consider bit rate limitations; the implications of bandwidth limitations on analog signals are well known. We consider an on–off-keyed signal with Gaussian profile whose FWHM is equal to one-half of the bit  interval 1/B. The temporal profile of such a signal is P(t) =  exp −16 ln (2)B2 t2 , while the power spectrum of the signal is also Gaussian with FWHM

ωsig = 8 ln (2)B.

(10.19)

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Performance of Single and Coupled Microresonators

237

Since one needs to maintain the transmission window wider than ωsig , ωsig ≤

ωADF . Then, according to the delay-bit rate product of (10.18), the number of stored bits in a single resonator is (0)

NADF = Td B ≤

1 − κ2 ≈ 0.36. 4 ln (2)

(10.20)

Although the APF is transparent at any frequency, one can still talk about useful bandwidth of the APF defining it as the bandwidth at which the group delay changes by a factor of two, i.e., ωAPF of equation (10.11). Then we obtain from (10.12) (0)

NAPF = Td B ≤

1+ρ ≈ 0.72 4 ln (2)ρ 1/2

(10.21)

indicating that one cannot possibly store even one bit of information in a single resonator without excessive distortion. What (10.20) and (10.21) – which are, by the way, independent on the circumference L of the resonator – indicate is that, as the bit rate of the signal increases, the interaction (delay) time decreases and the resonant enhancement also goes down accordingly. To break this relation one has to consider coupling the MRs. This arrangement would allow one to increase the delay (interaction) time without decreasing the FWHM of the structure.

10.4.2 Coupled Microresonators Just as with single resonators, two different coupled resonator arrangements can be made. The first arrangement is the composite all-pass filter, or SCISSOR (sidecoupled integrated spaced sequence of resonators) [33, 34] shown in Fig. 10.6a.

a in

out

out

in

b Fig. 10.6 Coupled resonator structures: (a) SCISSOR (b) CROW

238

J.B. Khurgin

The dispersion relation of a SCISSOR is the same as (10.5), except that the phase is multiplied by the number of resonators in the SCISSOR: (ω) = Nr tan−1

κ 2 sin (ω − ω0 ) τ . (1 + ρ 2 ) cos (ω − ω0 )τ − 2ρ

(10.22)

However, the relation between the group delay and GDD changes fundamentally from (10.7) and (10.8) to (0)

Td,S = Nr τ

4τ 1+ρ ≈ Nr 2 1−ρ κ

(10.23)

and

Td,S = −Nr τ 3 ρ(1+ρ)3 (1−ρ) ( )3 ( ) . (0) (0) 3 ρ 1 −2 −2 = −Nr Td,S Td,S 2 ≈ − 4 Nr (2)

(10.24)

(1+ρ)

The most important feature indicated by (10.24) is that as the number of resonators increases and the group delay stays constant, the GDD decreases as a square of the number of resonators. Therefore, using many small resonators in place of a single one will always yield less dispersion. Indeed, from the result (10.13) we obtain 

ωS ≈

(0)

2Td,S (2)

−Td,S

1/2 =

Nr (1 + ρ) (0)

Td,S

1/2 2 ρ

(10.25)

indicating that the delay-bandwidth product scales up with the number of resonators. As an alternative to the SCISSOR one can consider the different arrangement shown in Fig. 10.6b, namely the coupled resonator optical waveguide (CROW) [30, 35, 36] which, just like the single resonator of Fig. 10.3b, has a distinct transmission band. The dispersion relation of CROW can be written as sin (ωτ/2) = κ cos (/Nr ).

(10.26)

The dispersion curve is shown in Fig. 10.7a and consists of a series of passbands around resonant frequencies ωm = 2π m/τ separated by wide gaps. The width of the passband is

ωpass = 4τ −1 sin−1 (κ) ≈ 4τ −1 κ and the dispersion can be approximated as

(10.27)

10

Performance of Single and Coupled Microresonators

ω

239

5.5 5 4.5

Delay per ring (ps)

Δω pass ωm

π

4 3.5 3 2.5 2 1.5 1 0.5

a

b

0 –60

Φ

–40

–20 0 20 Frequency (GHz)

40

60

Fig. 10.7 (a) Dispersion and (b) group delay (per resonator) of a CROW

1∂  3  = 0 + ∂ ∂ω (ω − ω0 ) + 6 ∂ω3 (ω − ω0 ) + ... = , (0) (2) = 0 + Td,C (ω − ω0 ) + 13 Td,C (ω − ω0 )3 3

(10.28)

where the delay time at ω0 is (0)

Td,C =

∂ τ = Nr ∂ω 2κ

(10.29)

and the second-order dispersion of the delay time is (2)

Td,C =

 1 ( ) 1 ∂ 3 1  τ 3  (0) 3 2 −2 = N 1 − κ ≈ N T r d,C . 2 ∂ω3 2 2κ 2 r

(10.30)

Note that according to (10.29) the group delay per ring is increased by a factor of κ −1 relative to the simple waveguide with the same effective length – therefore the intensity of light inside the ring is increased by the same factor of κ −1 . The group delay per one ring in a CROW made of rings having the same τ = 500 fs and κ = 0.1 is plotted in Fig. 10.7b [23]. The relation (10.30) is almost identical in magnitude (except for the factor of 2) but opposite in sign to the SCISSOR relation (10.24). The difference in sign is actually important because it can be used to compensate GDD [37] but we shall here concentrate on the salient feature of the coupled resonators, i.e., the fact that the passband in either CROW or SCISSOR is independent of the (0) number of resonators. The delay time Td can be increased without affecting the width of the passband, and the delay-bit rate product and the delay-bandwidth product can be increased Nr -fold compared to a single resonator. However, this result assumes that as long as the bandwidth of the signal is less than the FWHM of the resonator, the signal remains undistorted. This assumption is obviously incorrect because even if the GDD is small relative to the delay, it does accumulate with the number of resonators and will eventually distort the signal below the recognition. This is what we shall now demonstrate.

240

J.B. Khurgin

There can be different ways to gauge the GDD limitation. The one we adopt here is rather simple. We want the differential group delay within the FWHM of the signal spectrum to be less than one-half of the bit interval B−1 , in order to avoid excessive inter-symbol interference. Therefore, the GDD limitation is (2)

Td,C(S)

ωsig 2

2 ≤

1 . 2B

(10.31)

For a CROW using (10.19) and (10.30) this relation immediately leads to ( ) (2) (0) 3 2 [4 ln (2)]2 Td,C B3 ≈ ρ 2 B3 [4 ln (2)]2 Nr−2 Td,C ≤ 1

(10.32)

while for a SCISSOR, using (10.24), we obtain a similar relationship, namely (2)

2 [4 ln (2)]2 Td,S B3 ≈

2ρ (1 + ρ)

2

( ) (0) 3 B3 [4 ln (2)]2 Nr−2 Td,C ≤ 1.

(10.33)

Assuming weakly coupled resonators and using the expression for the digital (0) delay-bandwidth product N = Td,C B, we obtain [4 ln (2)]2 Nr−2 N 3 ≤ M,

(10.34)

where M=2 for a SCISSOR and M=1 for a CROW. Note that (10.34) is independent of the bit rate and the per ring delay [38]. Thus, to store N bits of information in a CROW without incurring excessive distortion that can cause inter-symbol interference, one should use at least Nr,C ≈2.8N3/2 elements for a CROW and at least Nr,S ≈1.97N3/2 elements for a SCISSOR. The difference is insignificant so it makes sense to use a “rule of thumb” relation for the delay-bandwidth product attainable in coupled resonator structures: N = Td(0) B ≈ (Nr /3)2/3 .

(10.35)

In other words, the delay-bandwidth product does increase as more and more coupled resonators are used, but the increase is sub-linear, i.e., as the delay increases the bandwidth does goes down, but not as sharply as in the case of a single resonator.

10.5 Single Microresonators as Optical Switches In order to gauge the advantages offered by MRs in optical switching, one should define a point of reference with which the prospective MR-based switches and modulators can be compared. In this work we shall choose the most widely used switch based on a Mach–Zehnder interferometer (MZI) shown in Fig. 10.8. One or both arms of the MZI contain material whose refractive index can be varied by current

10

Performance of Single and Coupled Microresonators

241

Δϕ = π

L0

Δn

Fig. 10.8 Optical switch based on a Mach–Zehnder interferometer

injection, as in Si devices, by electric field (electro-optic effect), or by the optical field (nonlinear optical effect). With regard to the all-optical switches, we shall consider here only the case when the switching (pump) optical field is not resonant with the MR, while the signal is resonant. The double-resonant case, which is more difficult to implement in practice [39, 40], is treated in the next section. No matter what the mechanism causing the refractive index change n is, the effect can be characterized by the maximum attainable index change nmax and by the speed characterized by the cut-off frequency fmax . From the more fundamental point of view, the index change is limited by the material damage resulting from excessive electric or optical fields, or from extremely high-current injection level. In a more practical consideration the index change can be limited by the voltage, current, or optical power levels at the disposal of the designer. As we have already mentioned, the speed limitations can be intrinsic (recombination or sweep-out time in current injection schemes) or extrinsic (parasitic capacitances and losses at high frequencies in electro-optic devices). The cut-off frequencies extend from the gigahertz range in Si current injection devices [6, 13] to tens of terahertz in nonlinear optical devices based on the Kerr effect [3]. But as a rule, the faster is the speed of the effect the weaker it becomes, because the slow index change mechanisms usually rely upon accumulation of carriers (or temperature rise) while the faster mechanisms have to depend on the instantaneous values of the input. As a result, the product

nmax fmax which, in analogy to electronics one can call the “gain-bandwidth product,” does not vary much. We shall introduce the maximum bit rate that can be achieved from the speed limitation of the switching mechanism itself, which we shall call “intrinsic cut-off bit rate.” For a Gaussian signal (see 10.19) Bint = 2π fmax /8 ln (2) ≈ fmax .

(10.36)

To attain switching in the MZI of Fig. 10.8 an optical path length change equal to one-half wavelength must be induced in one of the arms and this gives us the switching condition for the required product of index change and length ( nL)MZI = λ0 /2.

(10.37)

242

J.B. Khurgin

If one considers other non-resonant switching schemes, for instance the ones relying upon directional couplers or Bragg gratings, they all have constraints on the length-index change product that are very similar to (10.37). Since the index change is limited, one can introduce the minimum switching distance in the MZI interferometer as L0 = λ0 /(2 nmax ).

(10.38)

The designer can be faced with different tasks when trying to improve the switch. Sometimes one wants to reduce the size of the switch, i.e., L, at other times the goal is to reduce the required index change n. Overall, however, the most meaningful optimization is aimed at the reduction of the entire nL product which can be loosely thought of as proportional to the switching power. Let us see how it can be done and what are the bit rate limitations. When the effective index of refraction is changed via either an electro-optic or a nonlinear optical process as n = n¯ + n

(10.39)

n n¯

(10.40)

the round-trip time changes as

τ = τ

and the resonance frequencies change according to

ω0 = −ω0

n n¯

(10.41)

as shown in Fig. 10.9. Consider now the single-resonator ADF of Fig. 10.3b (the results for the APF are similar). If the passband width ωADF , (10.15), is equal to the signal bandwidth ωsig , (10.19), which can be assured by selecting the coupling 1

Transmission

0.8

ΔωADF

0.6 0.4

Δωsig

0.2 Fig. 10.9 Principle of optical switching based on resonance shift in a single microresonator

0

ω 0 -Δω 0 ω 0

ω (arb. units)

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Performance of Single and Coupled Microresonators

243

coefficient to be , κ = 2 ln (2)Bτ

(10.42)

then shifting the resonance frequency by ω0 = ωADF = ωsig will cause switching in this resonance-shifting scheme and the required index change is

n = n¯

ωsig 8 ln (2)B = n¯ . ω0 ω0

(10.43)

Multiplying (10.43) by the circumference L = cτ/¯n one obtains the switching condition c λ0 8 ln (2)Bτ ≈ 1.7Bτ . ω0 2

( nL)r =

(10.44)

The gain in the resonance-shifting scheme of Fig. 10.9 can then be introduced as

Gr =

Bcut,r 0.6 ( nL)MZI = , ≈ Bτ B ( nL)r

(10.45)

where one can define the gain cut-off bit rate as Bcut,r ≈ 0.6τ −1 .

(10.46)

It would appear that one can increase the gain using smaller and smaller radii, but, since the maximum index change is limited, one can also define the maximum switching bit rate as Bmax ,r =

ω0 8 ln (2)



n n



= 1.1ν0

max

n n

.

(10.47)

max

The resonant gain at maximum switching bit rate in the resonance-shifting singleresonator scheme is then   Bcut,r

nmax −1 L0 ≈ 0.65 ν0 τ ≈ . Gr (Bmax ,r ) = Bmax ,r n¯ L

(10.48)

Since L0 , even for the largest value of nmax ∼10−3 , is in the millimeter range, for rings with circumference of less than a 100 μm, the maximum bit rate is always at least an order of magnitude smaller than the gain cut-off bit rate. In Fig. 10.10 we have plotted with dashed lines the resonant gain as a function of bandwidth for two different schemes. The first is a narrowband Si current injection scheme with

nmax ∼10−3 (L0 ∼0.7 mm at λ=1550 nm) [11] and Bint ∼10 Gbps. The second is a GaAs photonic switch based on the optical Kerr effect [9] with nmax ∼3×10−5 (L0 ∼2.5 cm at λ=1550 nm) and Bint >10 THz. We assumed that the ring resonators

244

J.B. Khurgin 4

4

10

10

Resonance shift scheme Nr =10

3

10

Bmax,r

Bint

Resonant Gain

Resonant Gain

Nr =10

3

10

2

10

Resonance shift scheme

Nr =100

(10) Bmax,r

Nr =1000

1

10

Bcut ,r

Bmax,r 10

Nr =100 Nr =1000

1

10

a

Bcut ,r b

0

10 8 10

(10) (100) Bmax,r Bmax,r

2

0

9

10

10

11

10

10

12

10

10 8 10

9

10

Bit Rate (Bps)

10

10

11

10

12

10

Bit Rate (Bps)

Fig. 10.10 Resonant gain (decrease in nL relative to MZI) for two different optical switches: (a) Si switch with current injection and (b) ultrafast GaAs all-optical switch

have circumference ∼36 μm or τ∼500 fs. In the Si scheme of Fig. 10.10a we obtain Bmax,r = 110 Gbps which is in fact much faster than Bint ; hence the speed of Si is intrinsically limited to about 10 Gbps (this is shown by the thick line). But even at this fairly low bit rate one can achieve the tremendous gain of about 100 compared to MZI. In the GaAs photonic switch (Fig. 10.10b), on the other hand, Bmax,r ∼3 GHz which is orders of magnitude less than its intrinsic potential of almost unlimited bandwidth. Therefore one should consider means to improve the speed of these devices using coupled MRs.

10.6 Photonic Switching with Coupled Microresonators It is obvious that combining MRs in a CROW or SCISSOR and still using the resonance-shifting scheme will simply use more rings to achieve the same shift and thus will only make things worse. For this reason one shall consider a different scheme that takes advantage of longer propagation times in the coupled MRs while still relying on the MZI geometry to achieve switching. We refer to this scheme, shown in Fig. 10.11a (SCISSOR case) and Fig. 10.11b (CROW case) as “slow-light scheme” [39, 40]. With the phase characteristic of CROW or SCISSOR described by 10.6 the phase change caused by index change is

 = −

n ∂ (0)

ω0 = Td,C ω0 =π ∂ω n¯

(10.49)

while the GDD limitation indicated in (10.34) imposes the condition ( ) (0) 3 B3 [4 ln (2)]2 Nr−2 Td,C ≤ M

(10.50)

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Performance of Single and Coupled Microresonators

245

L

Δϕ = π

Δn signal

a

Δϕ = π L

Δn signal

b Fig. 10.11 “Slow-light”optical switch based on multiple coupled microresonators inside an MZI. (a) SCISSOR and (b) CROW configuration

that must be enforced in order to avoid excessive inter-symbol interference. Then combining (10.49) with (10.50) we readily obtain the required index change B B

n [4 ln (2)]2/3 ≈ Nr−2/3 . = Nr−2/3 n¯ 2f0 M 1/3 f0

(10.51)

Thus the required index change is reduced compared to the case of a single res−2/3 , but the total length is increased by a factor of onator by a factor of roughly Nr 1/3 Nr . Therefore the length-index change product increases as roughly Nr c ( nL)Nr = Nr nτ n¯ λ0 cB 1/3 [4 ln (2)]2/3 τ ≈ 2Nr1/3 Bτ = Nr 1/3 2 2ν0 M 1/3

and the gain is actually reduced by a factor of Nr shifting scheme using a single resonator (10.45) r) = G(N r

(10.52)

compared to the resonance-

0.5 −1/3 ( nL)MZI ≈ ≈ Gr Nr−1/3 . N ( nL)Nr Bτ r

(10.53)

1/3

The gain cut-off bit rate is then also reduced by Nr : Bcutr ≈ 0.5τr−1 Nr−1/3 ≈ Bcut,r Nr−1/3 . (N )

(10.54)

246

J.B. Khurgin

But the main consequence is the fact that the maximum switching bit rate increases according to (10.51): (Nr ) Bmax

≈

Nr2/3 ν0

n n¯

≈ Nr2/3 Bmax ,r ,

(10.55)

max

which allows one to fully exploit the potential of fast-switching materials. In Fig. 10.10 we have plotted the gain versus bit rate in the “slow-light scheme” as solid lines for different numbers of resonators. As one can see from Fig. 10.10a for the Si scheme, where the index change is large and intrinsically slow, using coupled resonators offers no advantage relative to a simple resonance-shifting scheme with a single resonator. The increase in the number of resonators causes a drop in the resonant gain and thus raises the required switching power. This increased power demand comes with no switching speed benefit, because the switching speed is determined by the intrinsic speed of the device Bint = 10 Gbps. However, for the relatively weak and fast scheme, such as the GaAs nonlinear all-optical switch of Fig.10.10b, coupled MR schemes, in either SCISSOR or CROW configuration, greatly enhance the speed of the device, from Gbps for a single MR to hundreds of Gbps, and still reduce the power requirement compared to a simple MZI by a factor of up to 100 or more. Therefore, the decision of whether to use a single, or a few, MR scheme with resonance shift or a multiple coupled MR slow-light scheme should be based on the relation between the intrinsic limitations of the material and the required operational speed.

10.7 All-Optical Switching in Coupled MRs via Self-Phase Modulation We now turn our attention to the so-called double-resonant case where the switching is accomplished by the signal itself, as indicated by the term “self-phase modulation.” This type of nonlinearity is important for all-optical signal regeneration in communications [41], for limiting the signal to protect the detector [42], and also for nonlinear thresholding in various code-division-multiplexing techniques [43]. We call this case “double resonant” because not only the small changes of refractive index get translated into large changes in amplitude of the transmitted light, but, in addition, the intensity of light inside the resonator gets enhanced approximately by the same factor. Consider the ADF MR of Fig. 10.3b. According to (10.14) the intensity of the light inside the resonator is related to the intensity outside the resonator as Iins = I/κ 2 .

(10.56)

Iins = I/[4 ln (2)Bτ ].

(10.57)

Using (10.42) we readily obtain

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Performance of Single and Coupled Microresonators

247

The required index change from (10.43) is

n = n¯

n2 I 4 ln (2)B = n2 Iins = π f0 4 ln (2)Bτ

(10.58)

from which we obtain the switching intensity outside the MR I = n¯

(4 ln (2)B)2 τ . n2 π f0

(10.59)

The self-phase modulation in the unstructured MZI of Fig. 10.8 requires the intensity–length product to be (IL)MZI = λ0 /(2n2 ).

(10.60)

Then the gain for using a MR is

Gr,spm

π 0.2 (IL)MZI = = ≈ = 2 (IL)r 2 (4 ln (2)Bτ ) (Bτ )2



BSPM cut,r B

2 ,

(10.61)

where BSPM cut,r = 0.45/τ .

(10.62)

This value is not far from the cut-off bit rate of (10.46) but now the gain at low frequencies is much larger due to the double resonance. There is an additional limitation imposed by the fact that the intensity inside the resonator should not exceed the optical damage intensity Imax which, according to (10.58), leads to the maximum switching speed B(SPM) max ,r =

π n2 Imax π nmax f0 = f0 = Bmax ,r . 4 ln (2)¯n 4 ln (2)¯n

(10.63)

In Fig. 10.12 we have plotted with dashed lines the resonant gain as a function of bandwidth for the same two schemes as before. The first is a Si switch in which the real carriers are photoexcited with nmax ∼10−3 and Bint ∼10 Gbps while the second is a GaAs switch based on the optical Kerr effect [9] with nmax ∼3×10−5 and Bint >10 THz. For the Si scheme of Fig. 10.12a the speed of Si is intrinsically limited to about 10 Gbps (this is once again shown by the thick line), but at this bit rate the enormous gain of about 104 compared to MZI can be attained. In the GaAs photonic switch (Fig. 10.12b), on the other hand, Bmax,r ∼3 GHz and one should consider using coupled MRs to enhance it. In coupled resonators, in a CROW, for example, according to (10.29), the group velocity of the light is decreased by a factor of κ −1 , therefore intensity inside the structure is

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J.B. Khurgin 8

8

10

10

Resonance shift scheme

Resonance shift scheme

N r =10 6

Bmax,r

Resonant Gain

Resonant Gain

10

Bint 4

10

N r =10

6

10

Nr =100

( SPM ) Bmax,10

Nr =1000

2

10

Bcut ,r

( SPM ) Bmax, r

10

N r =100 Nr =1000

2

10

Bcut ,r

a

b

0

10 8 10

( SPM ) ( SPM ) Bmax,10 Bmax,100

4

0

9

10

10

11

10

12

10

10

10 8 10

9

10

10

10

Bit Rate (Bps)

11

10

12

10

Bit Rate (Bps)

Fig. 10.12 Resonant gain (decrease in IL relative to MZI) for two different optical switches based on self-phase modulation: (a) Si switch with photoexcitation of real carriers and (b) ultrafast GaAs all-optical switch (0)

Iins = I/κ =

2ITd,C Nr τ

.

(10.64)

Using (10.64), let us rewrite the switching condition (10.49) as (0)

2Td,C f0

) (

n n2 I (0) 2 = 4 Td,C f0 =1 n¯ n¯ Nr τ

(10.65)

and combine it with (10.51) and using ( ) (0) 3 B3 [4 ln (2)]2 Nr−2 Td,C ≤ 1

(10.66)

to exclude the delay time and obtain the expression for the minimum switching intensity I=

n¯ B2 [4 ln (2)]4/3 τ 1/3

4n2 f0 Nr

≈

n¯ B2 τ 1/3

.

(10.67)

n2 f0 Nr

This minimum switching intensity is obtained at the optimum coupling rate that 1/3 1/3 can be found as κopt = 0.5[4 ln (2)]2/3 Nr τ B ≈ Nr τ B. The gain of the resonant scheme can be found as (Nr ) GSPM

0.5 (IL)MZI = ≈ = 2/3 (IL)Nr (Bτ )2 Nr



BSPM cut,Nr B

2 ,

(10.68)

where BSPM cut,Nr =

0.6 −1/3 −1/3 N ≈ BSPM . cut,r Nr τ r

(10.69)

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Once again, the cut-off bit rate is reduced by a factor of Nr but it is the 2/3 maximum bit rate that actually experiences an increase by a factor of Nr . If we now plot the SPM gain for coupled resonators in Fig. 10.12a,b we see that in the end these figures represent scaled-up versions of Fig. 10.11a,b, respectively, with the same critical frequencies, but with the SPM gain being essentially a square of the single-resonant photonic switching gain. Once again we can conclude that, as long as a relatively slow nonlinear switching scheme based on real carriers is used, a single MR makes a good switch with a resonant enhancement of 104 at 10 Gbit/s. When faster rates are required one must use weak ultrafast nonlinearity and there it makes sense to use up to 100 coupled MRs and obtain modest improvements at rates up to 100 Gbps. It appears that with today’s technology the MRs cannot enhance the performance of any optical switch at bit rates above 100 Gbps.

10.8 Conclusion In this chapter we have studied the performance of MRs as optical switches operating at high bit rates. We have considered two alternative schemes. One is a single-resonator optical switch where the resonance frequency is shifted causing the change in transmission. The other is a “slow-light” scheme where the phase shift is accumulated as the light passes through a coupled resonator structure. We have demonstrated that for each scheme there exists a cut-off bit rate beyond which the MR offers no advantage over the standard MZI switch. We have shown that for relatively low-speed switches (up to 10 GBs) the single-resonator scheme works as well if not better than that with many coupled MRs, but for high bit rates the “slow-light” scheme has a distinct advantage. Our main conclusion is that for each material and each specific photonic switching application one can find the optimum MR enhancement scheme.

References 1. Mouftah, H.T., Elmirghani, J.M.H. (eds.) Photonic switching technology: Systems and Networks. Wiley, NY (1998). 2. Agrawal, G.P. Fiber optics communication systems. Wiley, NY (2002). 3. Boyd, R.W. Nonlinear optics. Academic Press, San Diego CA (2003). 4. Buse, K., Adibi, A., et al. Non-volatile holographic storage in doubly doped lithium niobate crystals. Nature 393, 665–668 (1998) 5. Mace, D.A.H., Adams, M.J. Carrier recombination time measurements of InAlAs/InGaAs multiple quantum wells using nonlinear frequency-dependent transmission. Semicond. Sci. Technol. 5, 105–107 (1990) 6. Jalali, B., Fathpour, S. Silicon photonics. J. Lightw. Technol. 24, 4600–4615 (2006) 7. Bach, H., Neuroth, N. (eds.) The Properties of optical glass. Springer, Berlin (1998) 8. Adair, R., Chase, L.L., et al. Nonlinear refractive index of optical crystals. Phys. Rev. B 39, 3337–3350 (1989) 9. Millar, P., Aitchison, J.S., et al. Nonlinear waveguide arrays in AlGaAs. J. Opt. Soc. Am. B 14, 3224–3231 (1997)

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10. Hunsperger, R.G. Integrated optics. Theory and applications. Springer, NY (2008) 11. Lipson, M. Compact electro-optic modulators on a silicon chip. IEEE J. Sel. Top. Quant. El. 12, 1520–1526 (2006) 12. Xu, Q., Manipatruni, S., et al. 12.5 Gbit/s carrier-injection-based silicon microring silicon modulators. Opt. Express 15, 430–436 (2007) 13. Soref, R.A., Bennett, B.R. Electro-optical effects in silicon. IEEE J. Quant. Electron. QE-23, 123–129 (1987) 14. Liao, L., Samara-Rubio, D., et al. High speed silicon Mach-Zehnder modulator. Opt. Express 13, 3129–3135 (2005) 15. Nikogosyan, D.A. Properties of optical and laser related materials. A handbook. Wiley, NY (1997) 16. Shi, Y., Zhang, C., et al. Low (sub-1-Volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape. Science 2888, 119–122 (2000) 17. Wooten, E.L., Kissa, K.M., et al. A review of lithium niobate modulators for fiber-optic communications systems, IEEE J. Sel. Top. Quant. Electron. 6, 69–82 (2000) 18. Chen, D., Fetterman, H., et al. Demonstration of 110 GHz electro-optic polymer modulators. Appl. Phys. Lett. 70, 3335–3337 (1997) 19. Kash, M.M., Sautenkov, V.A., et al. Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas. Phys. Rev. Lett. 82, 5229–5232 (1999) 20. Chemla, D.S., Miller, D.A.B. Room-temperature excitonic nonlinear-optical effects in semiconductor quantum-well structures. J. Opt. Soc. Am. B 2, 1155–1173 (1985) 21. Rosenthal, A., Horowitz, M. Analysis and design of nonlinear fiber Bragg gratings and their application for optical compression of reflected pulses. Opt. Lett. 31, 1334–1336 (2006) 22. Marburger, J.H., Felber, F.S. Theory of a lossless nonlinear Fabry-Perot interferometer. Phys. Rev. A 17, 335–342 (1978) 23. Xia, F., Sekaric, L., et al. Ultra-compact optical buffers on a silicon chip. Nat. Photon. 1, 65–71 (2006) 24. Tazawa, H., Kuo, Y.-H., et al. Ring resonator-based electro-optic polymer traveling-wave modulator. IEEE J. Lightw. Technol. 24, 3514–3519 (2006) 25. Cohen, D.A., Hossein-Zadeh, M., et al. High-Q microphotonic electro-optic modulator. SolidState Electron. 45, 1577–1589 (2001) 26. Bhola, B., Song, H.-C., et al. Polymer micro-resonator strain sensors. IEEE Photon. Technol. Lett. 17, 867–869 (2005) 27. Savchenkov, A.A., Matsko, A.B., et al. Ringdown spectroscopy of stimulated Raman scattering in a whispering gallery mode resonator. Opt. Lett. 32, 497–499 (2007) 28. Schmidt, B., Xu, Q., et al. Compact electro-optic modulator on silicon-on-insulator substrates using cavities with ultra-small modal volumes. Opt. Express, 15, 3140–3148 (2007) 29. Madsen, C.K., Lenz, G. Optical all-pass filters for phase response design with applications for dispersion compensation. IEEE Photon. Technol. Lett. 10, 994–996 (1998) 30. Poon, J. K., Zhu, L., et al. Transmission and group delay of microring coupled-resonator optical waveguides. Opt. Lett. 31, 456–458 (2006) 31. S. Xiao, M. H. Khan, H. Shen, and M. Qi, Multiple-channel silicon micro-resonator based filters for WDM applications, Opt. Express 15, 7489–7498 (2007) 32. Nawrocka, M.S., Liu, T., et al. Tunable silicon microring resonator with wide free spectral range. Appl. Phys. Lett. 89, 071110 (2006) 33. Heebner, J.E., Boyd, R.W. “Slow” and “fast” light in resonator-coupled waveguides. J. Mod. Opt. 49, 2629–2636 (2002). 34. Heebner, J.E., Boyd, R.W., et al SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides J. Opt. Soc. Am. B 19, 722–731 (2002) 35. Yariv, A., Xu, Y., et al. Coupled-resonator optical waveguide: a proposal and analysis. Opt. Lett. 24, 711–713 (1999) 36. Melloni, A., Morichetti, F., et al. Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures. Opt. Quant. Electron. 35, 365–378 (2003)

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37. Khurgin, J.B. Expanding the bandwidth of slow light photonic devices based on coupled resonators. Opt. Lett. 30, 513–515 (2005) 38. Khurgin, J.B. Optical buffers based on slow light in EIT media and coupled resonator structures-comparative analysis. J. Opt. Soc. Am. B 22, 1062–1074 (2005) 39. Soljaˇci´c, M., Johnson, S.G., et al. Photonic-crystal slow-light enhancement of nonlinear phase sensitivity. J. Opt. Soc. Am B 19, 2052–2059 (2002) 40. Khurgin, J.B. Performance of nonlinear photonic crystal devices at high bit rates. Opt. Lett. 30, 643–645 (2005) 41. Mamyshev, P.V. All-optical data regeneration based on self-phase modulation effect. ECOC, p. 475 (1998). 42. Goto, H., Konishi, T., et al. An all-optical limiter with high-accuracy thresholding based on self-phase modulation assisted by preparatory waveform conversion. J. Opt. A: Pure Appl. Opt. 10, 095306 (2008) 43. Sardesai, H.P., Weiner, A.M. Nonlinear fiber-optic receiver for ultra-short pulse code division multiple access communication. Electron. Lett. 33, 610–611 (1997)

Chapter 11

Single Molecule Detection Using Optical Microcavities Andrea M. Armani

Abstract Optical microcavities have successfully detected a wide range of analytes and analyte properties in a variety of conditions. There are several important parameters of an optical cavity which govern its operation and determine its applicability in a given sensing scenario. An overview of these conditions and a discussion of the primary microcavity geometries used in biological and chemical detection are given. Additionally, a brief explanation of the role of surface functionalization and several specific methods which have been demonstrated with optical resonant cavity sensors is discussed.

11.1 Introduction Currently, cell biologists use several different in vitro and in vivo techniques to extract information about a cell’s behavior: labeling several independent proteins and monitoring their behavior using microscopy [1–6], lysing cells and using complementary techniques like polymerase chain reaction (PCR) and enzyme-linked immunoassays (ELISA) to determine the cell’s components [7], and knocking out genes and watching for a loss of function or life (in some instances) [8, 9]. While all of these techniques have been very successful, they provide indirect evidence of the information that the biologists are looking for, in the previous examples, cell signaling pathways [10, 11]. To solve this problem, it is necessary to develop improved, multimodal label-free sensing technologies which can provide the direct evidence needed [12]. The new techniques need to be both sensitive and specific to the target molecule. Additionally, the detection method should be viable in multiple biological solutions.

A.M. Armani (B) Mork Family Department of Chemical Engineering and Materials Science and Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, CA, USA e-mail: [email protected] I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_11,  C Springer Science+Business Media, LLC 2010

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The majority of sensing experiments are performed in water or in buffer. However, a true test of a sensor’s viability is its threshold in a complex biological mixture. In healthcare applications, this would be whole blood or serum. Alternatively, in systems biology applications, the more applicable medium is cell culture media or cell lysate, the contents of a cell after it is lysed (cut) open. While the majority of biological research is still performed using microscopybased methods, the necessity for improved methods has spurred a surge in innovation in biosensors, pushing development in areas other than microscopy. Based on this recent growth, new techniques of detection are being developed based on optical, electrical, and mechanical principles [13]. A few examples of optical devices include optical waveguides [14, 15], surface plasmon waveguides and resonators [16], and split ring resonators. The majority of electrical sensors involve monitoring the current across a gap. Based on this simple principle, they are fabricated from a wide range of materials, including nanotubes and wires [17], nanocrystals, polymers, and conventional semiconductors [18]. Vibration-based sensors are classically formed from cantilevers, typically lithographically fabricated [19, 20].

11.1.1 Microcavities as Biosensors Optical microcavities are one of these new sensing modalities. While the fundamental sensing principle is not novel, the implementation and, more recently, the successful demonstration of single molecule and single virus detection are profound [21–31]. Sensitivity is inherent to microcavities because of the long photon lifetime (high Q factor) within the microcavity which results in an increase in sampling [32]. As the Q factor increases, the sensitivity increases. This ability allows microcavities to perform detection experiments without labeling the target molecules. All other sensors which are capable of performing single molecule detection experiments require that the target is labeled with a fluorescent probe [33]. Additionally, microcavity-based detection can be performed in real time, which allows for data to be taken continuously while other biologically relevant parameters (such as temperature, pH, salt concentration) are changed. Specificity is endowed to the microcavity through surface functionalization [34, 35]. Finally, several whispering gallery (WG) mode resonant cavities have been integrated with semi-automated or automated microfluidic delivery systems. Previous microcavity detection experiments have been performed using a wide range of geometries and materials [36–40]. An overview of the primary microcavity geometries used in biodetection along with the fundamental device performance characteristics is shown in Table 11.1. Additional information regarding other performance metrics which are critical to a microcavity’s suitability for performing biodetection experiments is also contained in this table.

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Table 11.1 Overview of Whispering Gallery Mode Cavity Geometries

Geometry

Microsphere [21, 28]

Microtoroid [25, 41]

Microring [31, 36, 41]

LCORR [12, 42]

Possible material

Silica

Silica

Silica

>108

Polymeric materials, silicon ∼103 –105

∼103 –105

>108

∼103 –105

∼103 –105

Quality factor (in >109 air) Quality factor (in >106 water) Detection Single virus, demonstrations cis/trans of protein, DNA Integrated with microfluidics Integrated with other optical components Feasibility of multiplexing

No, but possible

Single molecule Detection of in buffer and in bacteria, serum, glucose fluorophore No, but possible Yes

No, but possible

No, but possible

Difficult (sphere fabrication method is serial)

Straightforward, after waveguide is integrated

Detection of DNA, proteins

Inherent to device Yes (waveguides) No, but possible

Demonstrated

Difficult in compact footprint

In all of the devices shown in Table 11.1, detection occurs when a molecule interacts with the evanescent, WG optical field. However, the location and mode profile of the field is very different between the cavity geometries. Additionally, in the case of the microsphere, microtoroid, and microring, the biomolecules are flowed over the sensor surface and bind on it. In contrast, in the liquid core optical ring resonator (LCORR), the solution is flowed through the device. Microsphere resonant cavities were the first geometry used in biosensing and the majority of the initial theoretical analysis revolved around a spherical geometry. As seen in Table 11.1, silica resonant sensors fabricated from high-Q microspheres (Q∼2×106 ) have demonstrated the ability to distinguish between two strands of DNA, between cis/trans isomers, and even single virions based on a resonant wavelength shift in real time [21, 43, 44]. One of the inherent advantages of using a microsphere is also one of its fundamental limitations. Because a microsphere resonant cavity is fabricated by melting the tip of optical fiber, it is very straightforward to fabricate several microspheres and attach different molecules to each. However, the integration of the microsphere with microfluidics or multiplexing is difficult, because of this bulky platform.

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Microtoroid -based sensors have many of the advantages of microspheres, such as high Q, and enable the possibility of the development of a multiplexed device. The microtoroid sensor has demonstrated single molecule detection of signaling proteins along with chemical detection of heavy water. Finally, like the microsphere, it is fabricated from silica, making it compatible with a wide range of testing wavelengths. Polymer devices have also performed similar biological detection experiments. Polymer microring resonators have demonstrated detection of glucose [36, 45]. The techniques used to fabricate these devices enable integration and multiplexing [46]. Integrated polymer resonator sensors have also demonstrated detection of avidin [40]. The quality factors of the polymer devices were limited by the fabrication methods used [47]. While the Q of the polymer device does impact its sensitivity threshold, the ability to integrate this device with additional components such as microfluidics, lasers, and detectors, is a substantial benefit [48–50]. The planar platform of this device will enable multiplexing, which is difficult though not impossible with the microsphere. The final approach is based on the liquid core optical ring resonator [12, 51]. This distinctly different approach relies on creating a hollow cylinder from a microcapillary. The cross section of the capillary forms a ring resonator. Solutions are flowed through the capillary, directly interacting with the optical field. Using this truly integrated and encapsulated approach, detection of DNA and chemical vapors has been demonstrated [42]. Additionally, these devices have been integrated with waveguides and multiplexed to form more complex sensor arrays [52]. While the strength of the microcavity is its ability to perform label-free detection experiments, labeled experiments can also be performed [53, 54]. In these experiments the resonant wavelength of the cavity is used as the excitation source, and only molecules sufficiently close to the cavity are excited. These types of experiments are especially suited for Fabry–Perot resonators and for microdroplets resonators. Fabry–Perot resonators typically have lower quality factors and would otherwise have difficulty performing single molecule experiments without the availability of high-speed, single-photon cameras [54]. Similarly, microdroplets, which can be generated via electrospray or in microfluidic bubbler chips, must be protected from the environment to prevent evaporation in order to support a WG mode. However, unlike semiconductor or dielectric resonant cavities, the molecule of interest is injected into the cavity instead of attaching to the exterior. While this approach does impact the Q of the cavity, it also increases the interaction with the optical field, somewhat compensating [55].

11.2 Theoretical Analysis A resonant cavity can operate in either a linear (reactive) or a non-linear (thermooptic) regime, depending on the Q of the cavity and the input power [22]. The reactive effect is characterized by the optical field interacting with and polarizing

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each molecule. This process assumes that the refractive index of the resonant cavity is constant. In contrast, if the Q factor of the cavity is sufficiently high, then the binding of molecules to the surface of the cavity can heat the surface of the cavity, changing the refractive index. In either the reactive or the thermo-optic case, the detection is performed by monitoring the resonant frequency of the cavity. Typically, the position of the resonant frequency, which shifts upon the binding of biological molecules, is monitored. Alternatively, the shift of the resonant frequency [32] or the Q factor of the cavity can be monitored [37].

11.2.1 Reactive Regime In cavities with moderate Q, the interaction between the optical field and biological molecules can be described by applying Maxwell’s equations for optical microcavities. An optical cavity’s resonance condition is defined by the wavelength λ for which the cavity’s circumference is an integer multiple of the orbital wavelengths. Therefore, the resonance wavelength must change for any change in the cavity radius (R) or refractive index (n) according to the following:

R n

λ = + . λ R n

(11.1)

From this expression, it is straightforward to conclude that an increase in either the cavity radius (R) or the refractive index (n) will redshift the resonant wavelength. If you wish to extend this to the case in which the protein is non-uniformly distributed across the cavity surface, a more complex analysis is necessary. In this situation, each particle is polarized by the optical field, according to the excess polarizability of the particle (α ex ). Taking into account the global effect of all bound molecules by incorporating the surface density (σ ), the expression for resonant frequency shift in this domain, for the case of a sphere, is αex σ

λ , = 2 λ εo (ns − n2m )R

(11.2)

where ns and nm are the refractive indices of the cavity and medium, respectively, R is the cavity radius, σ is the surface density of bound molecules, and λ is the resonant wavelength. There are a couple of important points about this equation. First, although the magnitude of the shift increases as the radius of the cavity decreases, the radius of the cavity also determines the Q factor of the cavity. Therefore, a more useful expression is the threshold sensitivity or limit of detection. This metric is determined by setting the resonant frequency shift to the resonance linewidth (δλ = λ = λ/Q) [22]:

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σmin =

(n2s − n2m )RF , (αex /εo )Q

(11.3)

where F describes the ability to detect sub-linewidth shifts. This expression captures the lowest surface density which a given cavity can detect. It is dependent on the cavity Q factor, both for the recirculation of light and for the narrow resolution.

11.2.2 Thermo-optic Regime The advantage of using thermo-optic detection over reactive detection is the increase in the resulting signal. While both the silica microsphere [28] and microtoroid [25] resonators have achieved Q>108 in air, only the microtoroid has achieved these Q factors in aqueous environments [56]. From finite element modeling (FEM) of microtoroid resonators, it has been shown that most of the optical field intensity (over 90%) resides within the silica [23]. Additionally, the conductivity of water and silica is similar (0.6 and 1.38 Wm−1 K−1 ). Taking both of these into account and beginning with the wave equation and adding a perturbing thermal contribution to the susceptibility, the theoretical wavelength shift produced by a single bound molecule via the thermo-optic mechanism can be shown to be given by [22, 23] 

δλ λ

 SM

 + 

σ λ dn dT |u(r)|2 = dr, QP |r| + ε 8π 2 n2 κV

(11.4)

where λ is the wavelength, σ is the absorption cross section of a single molecule, dn/dT is the opto-thermal constant of silica (1.3×10−5 K−1 ), κ is thermal conductivity, n is the effective refractive index of the silica toroid, V is the optical mode volume, and P is the coupled optical power. The integral in this expression accounts for the spatial overlap of the WG mode field (u(r)) with the temperature profile created by the nearly point-like molecular heat source [41]. The parameter ε is on the order of the radius of the molecule and only becomes significant with larger structures. It is important to note that while all of the molecules studied to date with microtoroids are significantly smaller than the extent of the optical field, this term can play a significant role with larger biological materials. Using microsphere resonators, it has been shown that it is possible to probe inside a cell [23]. The magnitude of resonant frequency shift depends on where a molecule binds to the surface of the resonant cavity. If a molecule binds at a region of higher field intensity (at the equator), the shift will be larger than if it binds at a region of lower field intensity. Therefore, the single molecule resonant frequency shifts can always be plotted as a distribution of shifts, which directly relates to molecule binding throughout this field.

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If expression (11.4) is rearranged in terms of the threshold detection level, it becomes immediately apparent that the sensitivity benefits quadratically from the cavity Q factor. As an example, for a cavity Q factor of 250 million, a coupled power of 1 mW, a molecular radius in the range of 3–50 nm, a wavelength of 670 nm, a toroid of diameter of 80 μm, and using optical and thermal constants of silica, we would predict an absorption cross-section limit between 1.1×10−17 cm2 and 1.5×10−17 cm2 . Typical absorption cross-section values for antigens, which are some of the smallest molecules of interest to biologists, are ∼10−16 cm2 . Therefore, the absorption cross-section limit using the microtoroid resonator is well below these values and, furthermore, represents only the limit one would encounter using resonance-shift detection methods sensitive only to shifts of order of the resonator linewidth. If a cross section of 2×10−16 cm2 is assumed, then a single molecule wavelength shift of between 50 and 33 fm is predicted, which is easily detected using an ultra-high-Q toroidal microcavity [25, 56]. The small range of variation here (in predicted wavelength shift) illustrates the relative insensitivity of each of these quantities to the parameter ε. Often, however, reverse detection, or detection of the antibody, can be useful in revealing information about a biological system. These molecules are over an order of magnitude larger, typically ∼10−15 cm2 . Therefore, in generalizing this detection mechanism to other resonant cavities, it is important to determine the minimum requirements for detection of both antigens and antibodies. Additionally, the dependence on mode volume, thermal conductivity, refractive index, opto-thermal constant, and power in (11.4) can often compensate for a lower Q factor. In a sense, a resonator’s design can be optimized for detection , just as it would be for other applications. Two interesting examples are the ultra-high-Q microsphere and the high-Q polymer microring resonator. It might be assumed that the ultra-high-Q microsphere should also be able to achieve the same sensitivity levels as the microtoroid, because they have similar Q factors. Additionally, since they are both fabricated from silica, they have identical thermo-optic coefficients, refractive indices, and thermal conductivities. However, as alluded to previously, because of the increase in the mode volume (2000 μm3 ) due to the significant increase in diameter over the microtoroid, the sensitivity decreases by over an order of magnitude. Therefore, while the microsphere is still able to detect antibodies at the single molecule level using 1 mW input power, it requires 10 mW input power to detect antigens at the single molecule level, despite having Q>100 million. Conversely, because the cavity Q decreases to 105 when resonators are fabricated out of polymeric materials [36, 57] such as PMMA, single molecule sensitivity has never been proposed with these devices. However, it is possible to achieve single molecule antigen detection with 10 mW of input power, the same as the silica microspheres. This sensitivity is directly related to the low thermal conductivity (0.2 Wm−1 K−1 ), high refractive index (1.45), and high thermo-optic coefficient (1.0×10−4 K−1 ) of PMMA and the small mode volume (0.21 μm3 ) of the resonator.

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11.3 Experimental Considerations of Microcavity-Based Biosensors 11.3.1 Detection Technique There are several different methods of performing biological or chemical detection using optical resonant cavities. These are (1) resonant frequency shift, (2) change of peak transmission, and (3) Q factor change. To experimentally detect each of these events, it is necessary to very accurately identify the position of the resonant frequency and monitor it for changes (Fig. 11.1). This is accomplished by coupling the cavity to a single-mode, tunable external cavity laser using a tapered optical fiber waveguide. Tapered optical fibers are very low-loss/high-efficiency waveguides used for probing ultra-high-Q modes in microcavities in both air and water [56, 58].

Fig 11.1 Types of resonant cavity detection: (a) Resonant frequency shift, which occurs as a result of refractive index change or diameter change (less common), (b) quality factor change or bandwidth broadening which occurs as a result of material absorption change, and (c) peak transmission change which occurs as a result of optical coupling change

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By continuously scanning across the resonant frequency, the location of the resonance can be continuously determined and recorded, enabling the detection of the binding of biomolecules or changes in concentration of chemicals in real time (Fig. 11.1a). Similarly, transmissivity changes can be detected by monitoring the minimum level of the resonance (Fig. 11.1c). It is slightly more complex to perform measurements by monitoring the Q of the cavity. This requires calculating the intrinsic Q and recording the resonant wavelength while monitoring the power transmission spectra. The intrinsic Q factor is determined by scanning the wavelength of a single-mode laser and measuring both the resonant power transmission and the loaded linewidth (full-width halfmaximum) in the under-coupled regime. The intrinsic modal linewidth, and hence the intrinsic Q, can then be computed using a resonator-waveguide coupling model [58, 59] (Fig. 11.1b). Thus, limiting the sensor performance are the limits on the detector and the laser scan rate. The microcavity response rate is determined by the photon lifetime within the cavity (ns) which is significantly faster than the scan rate of the tunable laser (s).

11.3.2 Q Factor Optimization The quality factor is an expression for the losses, intrinsic and extrinsic, in the optical resonant cavity system. As this topic has been thoroughly discussed in previous chapters, it will be only briefly reviewed here for the specific case of silica WG mode optical cavities. The quality factor can be written as [28] −1 −1 −1 −1 −1 Q−1 tot = Qmat + Qss + Qrad + Qcoup + Qcont ,

(11.5)

−1 −1 where Q−1 mat is the material or absorption loss, Qss is the surface scattering loss, Qrad −1 is the WG or radiation loss, Q−1 coup is the coupling loss, and Qcont is the contamination loss [60]. To maximize the quality factor or the sensitivity of the cavity, all of these loss mechanisms must be minimized. Because the Q factor plays such a pivotal role in determining the sensitivity, it is necessary to maintain the Q throughout the experiment. This can be done with careful examination of the roles of all loss mechanisms, individually. All WG resonators experience a certain amount of radiation loss from the confined mode. This leakage or radiation loss increases as the diameter of the cavity is reduced (scaled by the wavelength). It is also dependant on the refractive index contrast between the resonator material and the surrounding medium. Q−1 rad is therefore easily minimized by optimizing the balance of the resonator diameter, the material refractive index, and the operational wavelength. Contamination loss can likewise be made small by fabrication and testing in a sufficiently clean environment.

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The expression for material or absorption loss is Qmat =

2π neff , αeff λ

(11.6)

where neff is the effective refractive index, α eff is the effective material or absorption loss, and λ is the resonant wavelength. The role of neff and α eff becomes significant when testing in water. While quality factors in excess of 108 are relatively easy to obtain with operation in air, in water it becomes more challenging due to the –OH overtones of the water molecule which increase the effective absorption, especially in the near IR. However, as will be discussed below, by appropriate wavelength selection it is possible to overcome this increase in absorption. Aqueous operation also has the side effect of increasing the radiation loss from the resonators at a given diameter. This occurs because the refractive index contrast between silica and water is lower than for silica-to-air operation. Nonetheless, operation in water is essential to keep biological species in their native state in order to maintain activity.

11.3.3 Experimental Verification of Ultra-High Q in Water As noted above, the radiation loss component [which scales as Q−1 ∝exp(-D/λ)] is dependent on diameter while the material absorption loss is strongly dependent on wavelength [28]. For a small enough microcavity diameter, the radiation loss is dominant, while at larger diameters, the material absorption loss (in the case of aqueous operation, this is typically the water loss) is dominant. This fact creates two clearly distinguishable operational regimes of loss controlled by the diameter, with the transition diameter between these regimes determined by the operational wavelength. To verify this theory, it was necessary to use two fluids which were able to accurately probe both regimes. As water and water-based solutions are the primary fluids used in biological detection experiments, it was desirable that water be one of the fluids, while D2 O ( heavy water ) was particularly useful for comparison. D2 O and H2 O have nearly identical refractive indices, but at select wavelength between 680 and 1550 nm, the absorption of D2 O is less than H2 O [61]. We can conclude that while the Qrad would be nearly identical (at smaller diameters) for operation in these fluids, the Qmat should diverge as diameter is increased. However, by proper optimization of both operational wavelength and microtoroid size, it is possible to recover the ultra-high-quality factors that are possible for air operation. To verify these two operational limits, ultra-high-Q silica microtoroid resonators were fabricated over a wide range (50–250 μm) of major diameters using the process described in Ref. [25] and outlined in Fig. 11.2. Experiments were performed in both water (H2 O) and heavy water (D2 O) to fully map out the inter-dependence of the Q factor on the refractive index and material absorption [56]. The measurements were performed from the visible through the near IR. The measurements were performed using a series of narrow linewidth, CW lasers. The resonant cavity was coupled to a tapered optical fiber waveguide which was contained inside a liquid-filled cell (Fig. 11.3a). Figure 11.3b and c presents

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Fig. 11.2 Overview of fabrication process for optical microtoroidal resonators. (a) Circular pads are lithographically defined in thermal oxide on a silicon wafer. (b) Using XeF2 , an isotropic etchant, to undercut the silica, high-Q silica microdisks are formed. (c) Using CO2 laser to reflow the microdisk, the microtoroid is formed. Scanning electron micrograph of an array of microtoroids

Fig. 11.3 Transmission spectra and quality factors of the resonator in an aqueous environment. (a) Fluid chamber for testing in fluid showing the tapered optical fiber waveguide coupled to the optical cavity, which was sandwiched between the silicon wafer and a glass cover slip. The intermediate region was filled with fluid, either H2 O or D2 O. (b) Transmission spectra of a microtoroid resonator in D2 O at 1300 nm. The resonator is highly under-coupled in the spectrum presented. (c) Quality factors measured and predicted in the 1300 nm band. In H2 O, the maximum quality factor measured is 8×105 . By changing to D2 O, the maximum quality factor increases to 2×107 . (d) Quality factors measured and predicted in the 680 nm band plotted versus toroid major diameter. The Q increases with major diameter over the range of diameters wherein radiation loss is the dominant loss mechanism. It then reaches a plateau at values set by absorption of the aqueous environment. Above 5×108 data taking is unreliable due to laser-linewidth stability limitations. The maximum quality factor measured in H2 O was 2.3×108 and in D2 O was 1.3×108 [56]. Reprinted with permission. Copyright 2004 American Institute of Physics

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broadband spectra and intrinsic Q factors at 1300 nm, for both water and heavy water, plotted against toroid major diameter (circles and triangles, respectively) [56]. The model-predicted values are also shown in the plot (dotted and dashed lines). Q factors tend to larger values with increasing toroid size and clearly exhibit separate material-loss- and radiation-loss-limited regimes. This behavior is in good agreement with the predictions of the model. At 1300 nm, D2 O has a lower optical absorption and hence exhibits an absorption-limited Q plateau that is significantly higher than for H2 O. The highest quality factors achieved in H2 O and D2 O at 1300 nm was 8×105 and 2×107 , respectively. Figure 11.3d shows the results when the same measurement is performed at 680 nm instead of 1300 nm. As a result of the lower optical absorption of water, the quality factor is significantly higher, achieving Q values above 108 . These values are notable as they represent the highest Q factors reported to date for operation in an aqueous environment. The highest aqueous Q factor reported previously was approximately 106 in a silica microsphere [43]. Measurements of Q factors beyond 5×108 were not possible in this experiment due to laser-linewidth stability. In principle, however, larger toroid diameters should exhibit quality factors as high as 1×109 in water and 1×1010 in D2 O.

11.4 Examples of Resonant Cavity-Based Detection 11.4.1 Chemical Detection by Q Change The change in Q factor that occurs when the microtoroid is immersed in either water or heavy water indicates that it may be possible to use this as a method to measure the concentration of heavy water in water [37]. To have the highest sensitivity, the difference between the Q in water and in heavy water must be maximized. Of the wavelengths studied, this difference was greatest at 1300 nm. To demonstrate this effect, a simple testing procedure was used: (1) immerse the microtoroid in 100% D2 O, (2) gradually increase the concentration of H2 O in D2 O until 100% H2 O is reached, and (3) return the concentration of D2 O to 100% [37]. The microtoroid diameter is chosen such that the quality factor (in H2 O and D2 O) was liquid or material limited [56]. The Q is determined, as before, by monitoring the linewidth and extinction of a particular optical mode. Further details on the measurement are contained in Ref. [37]. Two series of measurements were performed. In the first series of highconcentration measurements, the solutions were prepared in 10% increments (10% H2 O in D2 O, 20% H2 O in D2 O, etc.), starting with 100% D2 O. The quality factor was measured for a given concentration, and the chamber was then flushed five times with the next concentration solution before the new quality factor was determined. Figure 11.4a shows a series of Q factor measurements taken in this manner. As expected, when the concentration of D2 O was reduced, the quality factor decreased. The theoretical values for each concentration were calculated and are

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Fig. 11.4 Ultra-sensitive detection of heavy water. (a) The quality factor is decreased (circles) and recovered (triangles) as the D2 O and H2 O are exchanged repeatedly in 10% concentration increments. The measurement is cycled several times showing that the measurement is reversible. (b) Starting with 100% H2 O, the concentration of D2 O was gradually increased using lowconcentration solutions ranging from 1×10−9 % to 0.01%. The minimum detectable change in Q was at 0.0001% (1 ppmv), indicated by arrows [37]. Reprinted with permission. Copyright 2004 Optical Society of America

indicated by the dashed line. This Q decrease was reversible; by increasing the D2 O concentration, the quality factor is recovered as can be seen in the sequential runs. To determine the detection sensitivity or threshold, larger dilutions of D2 O in H2 O were prepared, ranging from 0.01% to 1×10−9 %. As can be seen in Fig. 11.4b, there is a strong signal at 0.001% D2 O in H2 O and a small, yet detectable, shift occurs with the 0.0001% D2 O solution [37]. These values are not believed to reflect the fundamental limit of the detection sensitivity of this device since no attempt was made to reduce operational sources of noise, such as waveguide jitter or the mechanical motion of the tapered optical fiber. Based on the difference in optical absorption between H2 O and D2 O, the ultrahigh-Q microcavity has demonstrated the ability to detect 0.0001% (1 ppmv) of D2 O in H2 O. This form of detection illustrates a mode of operation in which Q factor is directly varied by a substance. However, as discussed in previous sections, this is just one detection mode.

11.5 Biological Detection by Frequency Change 11.5.1 Surface Functionalization While sensitivity is a crucial component of any sensing platform, specificity plays an equally crucial role. Therefore, it is necessary to implement a surface functionalization protocol which endows the resonant cavity sensor with specificity. There are many different surface functionalizations which can be used with the silica microtoroid surface. Methods for attaching antibodies, DNA, antigens, and other

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proteins have all been previously developed for silica and can be applied directly [62, 63]. However, while the chemistry will work, the processing often negatively impacts the Q factor of the cavity through a variety of mechanisms (surface scattering, absorption, etc.). For this reason it is important to use a highly efficient surface functionalization method, which also minimally impacts the Q. Examples of successful surface functionalization methods include silane-based DNA techniques, as well as antibody–antigen-binding protocols. The biotin– streptavidin or biotin–avidin pair is also commonly used. In biological detection experiments, both specificity and sensitivity are important. While the ultra-high-Q toroidal optical resonator is inherently sensitive, as was demonstrated by the resonator’s Q values >108 , specificity is achieved by functionalizing the surface of the microtoroid. Because the microtoroid is fabricated from silica there are several different surface functionalizations which can be used. The most commonly used surface functionalization technique due to its robustness is biotin–streptavidin (Fig. 11.5a). This was the technique which was used in the initial demonstration in the first resonant cavity sensor based on a microsphere resonator [24]. As a result of the biotin -binding pocket on streptavidin which nearly completely intercalates the biotin protein, the interaction between biotin and streptavidin has a very high affinity under a wide range of environmental conditions. As a result, it is very commonly used by biologists and biochemists to functionalize sensor surfaces [64]. Additionally, because antibodies can be easily biotinylated, this technique creates a “self-passivating” surface or one where only the antibody with the biotin tag on it binds to the surface. Finally, studies have shown that the biotin–streptavidin pair correctly aligns and orients antibodies on a silica surface [34]. However, the immediate detection of biotin or streptavidin in “real-life” scenarios, such as lysates or cell monitoring, is not “scientifically interesting,” as biotin and streptavidin are not generated in cell proliferation nor act as intermediates in cell signaling. Therefore, this pair is more commonly used as a linker, where an antibody is biotinylated to attach it to a sensor surface [34] (Fig. 11.5b). Antibody surface functionalizations provide specificity for their associated antigen. Unlike the biotin–streptavidin, the antibody binds the antigen in a binding groove. Depending on the antibody, upon binding, the antibody may change shape or undergo a conformational change, potentially increasing the affinity of the antibody–antigen attachment. Additionally, the specific location where the antigen binds to the antibody or the binding site on the antibody may be dependent on the precise shape of the antigen, otherwise known as being conformationally dependent. In these cases, if the antigen is denatured or unfolded due to environmental changes, then the antigen will not bind to the antibody. These requirements also increases the specificity of the interaction. An alterative antibody attachment technique is to use a protein G surface functionalization (Fig. 11.5c). Protein G binds to the Fc region of the antibody or the “stalk” region [62]. Therefore, it orients the antibody toward the environment,

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Fig. 11.5 Surface functionalization techniques. (a) Biotin–streptavidin. The toroid surface is coated with biotin (V), then exposed to streptavidin (clover) which selectively binds to biotin. (b) Biotin–streptavidin-based antibody attachment. The toroid surface is coated with biotin (V), then exposed to streptavidin (clover) which selectively binds to biotin (V). Then the toroid is exposed to a biotinylated antibody (Y attached to V). Finally the toroid is exposed to the antigen of interest (circle). (c) Protein G-based antibody attachment. The toroid surface is coated with Protein G (bean) which selectively binds to the antibody (Y). Then the toroid is exposed to the antigen of interest (circle).

increasing the capture efficiency of the sensor. As seen in Fig. 11.5, this technique is much simpler that the biotin-linker technique, as it does not require biotinylating the antibody. To detect streptavidin, the surface of the toroid was functionalized with 0.1 μM of biotin. To implement this method, first the microtoroid was immersed in buffer, then the biotin (0.1 μM) was flowed across the surface. Immediately after, the detection experiments were performed. In the case of the experiments involving tryptophan (Trp), a blocking solution of pure Trp (1 μM) followed the initial biotin solution before the detection experiments. (Tryptophan is a commonly found amino acid in cell lysate.)

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11.5.2 Single Molecule Detection of Streptavidin in the Presence of Tryptophan To perform these single molecule measurements, a modification of the biotin–avidin method was employed (Fig. 11.6). Instead of a simple biotin surface functionalization, a secondary blocking step needed to be incorporated as well. First, the surface of the toroid was functionalized with 0.1 μM of biotin. Then, a blocking solution of pure Trp (1 μM) followed the initial biotin solution before the detection experiments.

Fig. 11.6 The surface functionalization used to attach streptavidin to the surface of the microtoroid. To attach streptavidin to the surface of the microtoroid (1) the microtoroid was immersed in buffer, (2) it was exposed to biotin (V), and (3) it was exposed to streptavidin (clover). Additional experiments involved an intermediate exposure to a pure solution of tryptophan (oval) to block any vacancies on the surface of the microtoroid.

To demonstrate that the microtoroid sensor’s single molecule detection capabilities are not negatively impacted by the presence of additional materials, a set of complementary single molecule detection experiments were performed using 300 aM streptavidin solution containing additional tryptophan (Sigma-Aldrich, 99.9% pure L-tryptophan) at either 1 nM or 1 μM. It is important to note that the testing was performed at 680 nm which is significantly displaced from the fluorescent maximum of Trp (fluorescence max=300 nm). As a result, the binding of the Trp to the surface of the toroid during the blocking rinse did not significantly impact the Q factor or change the sensitivity of the toroid. The first test was to flow a pure solution of Trp over the functionalized toroid to determine the noise floor of the device. The second injection of Trp induced a minimal shift, which is more accurately described as jitter, which is negligible in comparison with the streptavidin-induced shifts. It is of the same order of magnitude as noise-induced fluctuations generated by injecting pure buffer. The first 20 s of this data is shown in Fig. 11.7. It appears to be a flat line, but in Fig. 11.8a, it is apparent that the resonant frequency does change when the Trp solution is injected. In all subsequent data analyses, shifts of this level (shifts < 0.002 pm) are considered unreliable and are not included in the data set. Figure 11.7 also shows the resonance

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Fig. 11.7 The first 20 s of single molecule detection of pure streptavidin (squares), pure Trp (inverted triangles), and mixed solutions containing both streptavidin and Trp (circles, triangles) using the microtoroid sensor. The steps created by individual molecules binding throughout the WG mode are easily identifiable at this timescale

Fig. 11.8 A series of histograms created from the resonant wavelength shift data shown in Fig. 11.7. (a) In the 1 μM Trp data, all resonant wavelength shifts fell below what was typically classified as the noise threshold level. (b), (c), (d) The largest shift results from a molecule binding at the highest intensity region of the microtoroid. In the histograms containing streptavidin data, all resonant wavelength shifts below 0.002 pm were considered noise and not included. It is important to note that the largest resonant wavelength shift is the same in all of the streptavidin histograms. Note that approximately the same number of streptavidin molecules bind (note y-axis)

shifts which occurred as the microtoroid was exposed to the 300 aM streptavidin solutions with increasing amounts of Trp. It is important to compare the largest resonance shift induced for each of the different solutions. This occurs when the molecule binds at the equatorial region of

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the optical microcavity. As can be seen in Fig. 8b–d, the maximum shift is approximately the same, whether the toroid is exposed to pure streptavidin or a streptavidin solution containing additional Trp. It is also important to note that the number of molecules that bind is approximately constant, regardless of the amount of Trp in the solution. With larger molecules or cells in the solution, the capture efficiency most likely would decrease as a result of steric hindrance. However, Trp is a very small amino acid and therefore, it is not able to interfere as easily.

11.6 Summary Resonant cavity detection methods have demonstrated unprecedented sensitivity in many realms of detection, from chemical to biological. This chapter discussed a series of experiments which demonstrated various methods of performing chemical and biological detection as well as the fundamental hurdles to performing these experiments. Many of the inherent issues are currently being addressed to enable these unique devices to transition from research settings to real-world applications, such as systems biology and astrobiology settings. While optical resonators offer sensitivity enhancements over alternative techniques, they also offer numerous opportunities for integration, including fluidic, optical and electrical controls. For example, an area of future growth will be the incorporation of microfluidic controls which will allow for directed delivery of small volumes of reagents to the resonator’s surface. This technique will also significantly reduce the amount of liquid necessary for detection and the time required for individual experiments. In the present work, wavelength scanning of a high-Q resonance is accomplished by using a tunable laser. However, tunable high-Q resonators have been demonstrated [65–67] and, if used as biosensors, would enable a fixed wavelength optical source. This type of source could potentially be heterogeneously integrated onto the silicon chip, thereby further miniaturizing the chemical and biological detector. There is currently great interest in resonator-based detectors as they offer enhanced sensitivity in a microscale form factor. The work presented here has shown how silica toroidal devices can be used for both biological and chemical detection. Simultaneously, efforts are continuing in improving the performance of alternative cavities based on silicon, Six Nx , AlGaAs, and polymers [26, 27, 30, 47, 68]. We predict that over time, there will likely be many distinct device platforms on which ultra-high-Q biological and chemical sensors can be based. Acknowledgments A. M. Armani would like to thank Prof. Richard C. Flagan and Prof. Scott E. Fraser at the California Institute of Technology for numerous helpful discussions. Support for this work was provided by the University of Southern California Provost’s Initiative on Biomedical Nanoscience and the Women in Science and Engineering Program. Additional funding was provided by DARPA MTO, the Army Research Laboratory, and the Office of Naval Research.

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Chapter 12

Microfiber and Microcoil Resonators and Resonant Sensors Fei Xu and Gilberto Brambilla

Abstract The manufacture of tapers from optical fibers provides the possibility to get long, uniform, and robust micrometer- or nanometer-size wires. Optical microfibers are fabricated by adiabatically stretching conventional optical fibers and thus preserve the original optical fiber dimensions at their input/output pigtails, allowing ready splicing to standard fibers. Since microfibers have a size comparable to the wavelength of the light propagating in it, a considerable fraction of power can be located in the evanescent field, outside the microfiber physical boundary. When a microfiber is coiled, the mode propagating in it interferes with itself to give a resonator. In this chapter the latest results on the manufacture of optical microfiber resonators are presented. Optical microfibers can be used to fabricate single-loop and multiple-loop (coil) resonators with extremely high Q factors. High Q resonators can be used for refractometric biosensors and because of their design they provide an exceptionally high sensitivity.

12.1 Introduction Optical fibers have been the basis of the modern information technology since Kao and Hockham of Standard Telephones and Cables proposed glass waveguides as a practical medium for communication in 1965. For a number of sensing applications, it is important to reduce the fiber diameter as this increases the evanescent field allowing for strong interaction with the environment. A simple way to reduce the diameter of the fiber is to taper it. Optical fiber tapers are made by stretching a heated fiber and forming a structure comprised of a narrow stretched filament (the

F. Xu (B) Department of Materials Science and Engineering and National Laboratory of Solid State Microstructures, Nanjing University, Nanjing, Jiangsu 210093, People’s Republic of China e-mail: [email protected]

I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_12,  C Springer Science+Business Media, LLC 2010

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taper waist), each end of which is linked to an unstretched fiber by a conical section (the taper transition), as shown in Fig. 12.1.

Fig. 12.1 Illustration of an optical fiber taper

Generally, the uniform waist region with micrometer-/submicrometer-size diameters is called microfiber (MF). Applications of microfibers have previously been limited because of the difficulties in fabricating low-loss submicron structures. Prior to 2003, only two attempts to manufacture sub-wavelength wires using a top-down process have been reported in literature [1, 2]. In [1], Bures obtained a 400 nm diameter MF but the loss was very large and difficult to reproduce. In December 2003 a two-step process to fabricate low-loss sub-wavelength silica wires was reported [3]. This involved wrapping a pre-tapered section of standard optical fiber around a heated sapphire tip and drawing it into a nanowire. Although the measured loss was orders of magnitude higher than that achieved later with flame-brushing techniques [4], it was low enough to open the way to a host of new devices for optical communications, sensing, lasers, biology, and chemistry. Then single-step fabrication techniques were reported; researchers at the University of Southampton [4] pulled nanowires by using the standard manufacturing method of fiber tapers and couplers. This is based on a small flame moving under an optical fiber which is heated and stretched (flame-brushing technique). At the same time researchers at OFS Laboratories [5] replaced the small flame with a sapphire capillary tube heated by a CO2 laser beam. A modified version using a microheater was also presented to manufacture MF from compound glasses [6]. With the improvement of fabrication methods, MFs with length in excess of 100 mm, radii as small as 30 nm [7], and losses as small as ∼0.001 dB/mm [7, 8] have been produced. MFs offer a number of unique optical and mechanical properties, including large evanescent fields, high nonlinearity, extreme flexibility and configurability, and low-loss interconnection to other fibers and fiber-based components. MFs are fabricated by stretching optical fibers and thus preserving the original dimensions of the optical fiber at their input and output (Fig. 12.1) allowing ready splicing to standard fibers. This represents a significant advantage when compared to smallcore microstructured fibers (photonic crystal fibers) that always present significant insertion/extraction losses. Because of the large fraction of power traveling in the evanescent field, they can easily be used to manufacture resonators simply by coiling them. With the extreme flexibility and configurability, MFs can easily be bent

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and manipulated and yet remain relatively strong mechanically. Bend radii of the order of a few microns can be readily achieved with relatively low induced bend losses [3, 9] allowing for compact devices with complex geometry, e.g., 2D and 3D multiring resonators [10]. MF resonators can be classified according to their dimensionality: loop and knot resonators are example of 2D structures, while the microcoil resonator belongs to the group of a 3D assembly. In 2004, the self-touching optical microfiber loop resonator (OMLR) was first fabricated at OFS Laboratories [11] by bending a microfiber on itself and keeping two sections of a microfiber together by taking advantage of surface attraction forces (van der Waals and electrostatic). The diameter of the MF and the Q factor of the resonator were 900 nm and ∼120,000, respectively. A major drawback of the self-touching loop resonator in air is its geometrical stability; the coupling is strongly affected by the microcoil geometry and a small change in shape results in a large change in its transmission properties. A knot resonator has also been proposed and manufactured [3], but its fabrication requires the MF to be broken in the minimum waist region. Because of their fabrication methodology, knot resonators exhibit high input/output losses and relatively high difficulty to tune the resonance to a specified wavelength. In 2004 a 3D multiple-turn optical microfiber coil resonator (OMCR) was proposed [10]. OMCRs wrapped on a low index rod can overcome the problem of stability and have the potential of being used as a basic functional element for microfiber-based photonics. OMCR-based optical devices have two significant advantages over planar devices, namely smaller losses and greater compactness. The manufacture is still difficult because the MF is liable to be broken during wrapping. Prior to 2007, experimental demonstrations of OMCRs were reported in liquid [12] and in air [13]. Like all silica devices, bare MFs suffer from degradation [7] when uncoated; in order to obtain a practical device, the R [14]. Embedding made the OMCR was coated with low index polymer Teflon deployment of OMLR and OMCR as high-sensitivity microfluidic sensors possible [15, 16]. In 2008, the two sensors were experimentally demonstrated [17]. After a review of resonators based on microfiber loops and coils (Sections 12.2 and 12.3), a summary of recent results on the use of resonators for sensing will be presented in Section 12.4. Sensitivities as high as 700 nm/RIU were predicted [18] (RIU is the refractive index unit).

12.2 Optical Microfiber Loop Resonator (OMLR) The OMLR is a miniature version of a fiber loop resonator, which was first made in 1982 from a conventional single-mode fiber and a directional coupler [18]. Due to the bending losses of the weakly guiding single-mode optical fiber and to the dimensions of the fiber coupler, the maximum value of the free spectral range (FSR) of the fiber resonator was limited to the order of a gigahertz. Later, a 2 mm diameter self-coupling fiber loop resonator was fabricated from an 8.5 μm diameter optical fiber taper [19]. The fiber diameter was too large to ensure sufficient self-coupling,

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therefore the OMLR was embedded into silicone rubber having a refractive index close to the index of the fiber in order to enhance the coupling efficiency. Recently, OMLRs in air [5, 20, 21] were manufactured from an optical microfiber with a 650 nm diameter. Figure 12.2 shows the general shape of a microfiber loop with the input and output ends touching each other. Long-term contact was practically viable because of surface attraction forces (van der Waals and electrostatic), which kept the ends together and overcame the elastic forces that would otherwise straighten out the microfiber. The typical Q factor of the OMLR is in the range 1,500–120,000 [3, 5]. In many potential applications, operation often involves monitoring the resonance lines in the optical field. Since the quality factors are very high, even small scattering, coupling, or absorption out of the resonator mode set can be detected. Fig. 12.2 Illustration of the OMLR

The fabrication of an OMLR consists of drawing the optical microfiber and then bending it into a self-coupling loop. The characteristic radii of a MF used for the fabrication of an OMLR are in the range 300–500 nm, which is approximately uniform along several millimeters of its length. The setup for the OMLR fabrication is illustrated in Fig. 12.3. It consists of two XYZ translation stages, which can move the optical fiber ends in three directions. The MF loop is created by twisting the MF by hand and then manipulating the translation stages. With the aid of two stages and a stereo optical microscope, the MF can be coiled into a selftouching loop.

Optical fiber pigtails

Microfiber

Fig. 12.3 Illustration of the MF bending and looping in free space using XYZ translation stages

XYZ Stage XYZ Stage

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Figure 12.4 shows the transmission spectrum of an OMLR obtained at the University of Southampton by coiling a 750 nm MF. The resonator attenuation, FSR, full-width-at-half-maximum (FWHM), Q factor, and finesse (F) are ∼7 dB, ∼4 nm, ∼0.2 nm, 7,800 and 20, respectively. The analytical formulation of the resonator properties has been obtained by solving the coupled wave equations. The transmitted power T can be written as    e−j(β−jα)L−jξ − √K 2 (1 − σ 2 )(1 − K)   =1− , √ √ T=   1 − Ke−j(β−jα)L−jξ  1 + σ 2 K − 2σ K cos (θ + ξ )

(12.1)

where σ = exp ( − αL) is the fractional loss per turn, α is the amplitude loss coefficient, K is the power coupling coefficient, θ = βL is the round-trip phase, and ξ is the phase of the cross-coupling amplitude coefficient [21]. The MF is assumed uniform, the propagation constant β independent of the position, and the loop perimeter length is L = 2π R (where R is the loop radius). As for the resonators presented in other chapters, the resonance condition (minima of T) is θ +ξ = 2π or βR = neff

L ξ (N, the azimuthal mode number, is an integer), (12.2) =N− λ0 2π

an equation that governs the optimized design of the OMLR for several optical applications, including sensing, dispersion compensation, and time delay. For large mode numbers N the term ξ /(2π ) in (12.2) can be neglected. From (12.1) the minimum (resonant) transmission, namely T = 0, occurs when K = σ 2 . 0

Fig. 12.4 Transmission spectra of a 750 nm diameter OMLR Transmission (dBm)

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Note that another type of loop resonator exists, namely the knot resonator [22], fabricated by knotting a broken MF (Fig. 12.5). Knot resonators have the benefit of a long, stable coupling region but they present severe drawbacks including the complexity of fabrication, the need to break the MF to make a knot, the necessity for an additional coupler at the output or input of the resonator, high loss, and the presence of only one standard telecom fiber pigtail.

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Fig. 12.5 Illustration of a knot resonator

12.3 Optical Microfiber Coil Resonator (OMCR) Unlike resonators based on a simple loop (OMLR), the optical MF coil resonator (OMCR) has a 3D structure built up by self-coupling single-mode fiber helical microcoils. The 3D geometry can be created by wrapping an optical MF on a dielectric rod having a smaller refractive index, as illustrated in Fig. 12.6. Fig. 12.6 Schematic of an OMCR where an optical MF is coiled on a low refractive index support rod

The horizontal and vertical arrows show two possible directions of light propagation. Resonances are formed by the interference of light going from one turn to another along the MF (horizontal arrow) and returning back to the previous turn with the aid of weak coupling (vertical arrow). Compared to the OMLR, the OMCR has a more stable geometrical structure, making it suitable for applications requiring long-term stability. It is interesting to note that the OMLR can be described as the simplest OMCR with one single turn. In practical realizations, OMCRs with more turns require longer MFs and a high degree of accuracy in positioning the MF coils adjacent to each other. As a result they are much more difficult to fabricate. Because of the limitations imposed by transmission losses, in most cases only OMCRs with a relatively small number of turns are considered.

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The OMCR properties can be analyzed using coupled wave equations. If light propagates along an OMCR toward positive z (Fig. 12.7) and cylindrical coordinates are used, A(θ ) can be defined as the light amplitude at the position (θ , R(θ )), where R(θ ) is the distance from the z-axis and θ is the angular coordinate.

z

Fig. 12.7 Illustration of an OMCR in cylindrical coordinates

θ

R3(θ)

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2nd turn R2(θ)

θ 1st turn

R1(θ) θ

It is convenient to define the amplitude of the field at the mth turn as Am (θ ) and to consider θ as the common coordinate along turns, so that 0 < θ <2π . Ignoring coupling between turns that are not adjacent to each other, the propagation of light along the coil in a M-turn OMCR is described by the coupled wave equations for slowly varying R(θ ) [10]: ⎛ A1 ⎜ A2 ⎜ ⎜ . ⎜ . ⎜ . d ⎜ ⎜ ⎜ Am dθ ⎜ ⎜ . ⎜ ⎜ . ⎜ . ⎝ AM−1 AM

⎞

⎛ 0 R1 (θ ) χ12 (θ ) 0 ⎟ ⎟ 0 R2 (θ ) χ23 (θ ) ⎜ R2 (θ ) χ21 (θ ) ⎟ ⎜ ⎟ 0 R χ 0 (θ ) (θ ) ⎜ 3 32 ⎟ ⎜ ⎟ ⎜ . . . ⎟ ⎟ = −j ⎜ . . . ⎜ ⎟ ⎜ . . . ⎟ ⎜ ⎟ ⎜ 0 0 0 ⎟ ⎝ ⎟ 0 0 0 ⎠ 0 0 0

... ... ...

0 0 0

0 0 0

0 0 0

. . . . . . . . . ... ... 0 RM−2 (θ ) χM−1 M−2 (θ ) 0 ... RM−1 (θ ) χM−2 M−1 (θ ) 0 RM−1 (θ ) χM−1 M ... 0 RM (θ ) χMM−1 (θ ) 0

⎛ A1 ⎜ A2 ⎟⎜ ⎟⎜ . ⎟⎜ . ⎟⎜ . ⎟⎜ ⎟ ⎜ Am ⎟⎜ ⎟⎜ ⎟⎜ . ⎟⎜ ⎠ ⎜ .. (θ ) ⎜ ⎝ AM−1 AM ⎞

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(12.3) where

⎫ ⎧ θ

θ ⎨         ⎬ χpq (θ ) = κpq (θ ) exp j βp θ Rp θ dθ − j βq θ Rq θ dθ , (12.4) ⎭ ⎩ 0

0

β is the propagation constant, κpq is the coupling coefficient [12] between the pth and qth turns, and Rm+1 (0) = Rm (2π ). All βs are assumed independent of θ . If field continuity between the turns is to be satisfied, then

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⎧ 2π ⎫ ⎨ ⎬ Am+1 (0) = Am (2π ) exp −j βRm (θ ) ds , m = 1,2,...,M − 1. ⎩ ⎭

(12.5)

0

The average power coupling parameter is defined as K = (2π R0 κ)2 ,

(12.6)

where R0 is the average radius: M 1 R0 = Rm (θ ) dθ . 2π M 2π

(12.7)

m=1 0

The amplitude transmission coefficient can be written as ⎫ ⎧ 2π

⎬ ⎨ AM (2π ) exp −j βRM (θ ) dθ t= ⎭ ⎩ A1 (0)

(12.8)

0

The OMCR was fabricated using the setup shown in Fig. 12.8 [13]. The MF had its pigtails connected to an erbium-doped fiber amplifier (EDFA) and an optical spectrum analyzer (OSA) to check the resonator properties in real time during fabrication then, with the aid of a microscope, the MF was wrapped on a low refractive index rod while one of its ends was fixed onto a 3D stage. The close positioning of

Microscope to OSA

Microfiber from EDFA

Support rod

Fig. 12.8 Setup used to manufacture the OMCR [13]. A microfiber is wrapped on a support rod with the aid of three XYZ stages and a microscope. Light is launched into the OMCR from an EDFA and light is collected by an OSA to check in real time the OMCR spectral properties

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the MF coils resulted from a combination of manually applied longitudinal tension (which kept the relative position of the coils and avoided considerable overlapping) and gravity (which translated vertically the newly formed coil until it touched the coil beneath). Finally, the other MF end was fixed to another 3D stage and both the MF ends were gradually moved to find the optimum resonator spectrum. Because the coupling coefficient between two adjacent MFs is small, MFs need to be kept as close as possible. OMCRs were wrapped on a rod to maximize their temporal stability and robustness. For MFs with large evanescent fields or rods with high refractive indices, additional losses can arise because the fundamental mode becomes a leaky mode. When the loss becomes considerable, MF position and OMCR geometry are difficult to optimize. The overall propagation loss can be minimized by increasing the MF thickness and the rod diameter, by using a rod with low refractive index and a superior smooth surface. In these experiments a rod was R AF (DuPont, United States), to provide a low refractive index coated with Teflon (n∼1.3 at λ∼1.55 μm) at the interface with the MF. The coating thickness was a few tens of micrometers. Although OMCRs have also been demonstrated in a liquid [12], these present significant difficulties in packaging. Figure 12.9 shows the pictures of a two-turn, three-turn, and four-turn OMCR made from the same MF [13]. The MF radius and uniform waist region length were ∼1.5 μm and ∼4 mm, respectively. The rod diameter was ∼560 μm. The manual fabrication of the OMCR means that the pitches between adjacent turns are not uniform and the MF coils present some degree of twist.

Fig. 12.9 Pictures of OMCRs. The number of turns in the OMCR is two in (a), three in (b), and four in (c) [12]

Figure 12.10 shows the resonator spectra of a straight MF not in contact with the rod and of the three OMCRs shown in Fig. 12.9. The spectra in Fig. 12.10b–d show a complicated profile because coupling among the turns is irregular and non-uniform. The maximum extinction ratios for the two-, three-, and four-turn OMCRs are 3, 10, and 9.5 dB, respectively. While the spectrum (b) of the two-turn OMCR is simple, with a FSR∼0.86 nm, the spectra of three- and four-turn OMCRs show a complex profile. In particular, the spectrum of the three-turn OMCR (Fig. 12.10c) can be analyzed as a combination of two resonator modes (labeled A and B), with the same FSR (about 0.94 nm). There are two possible explanations for this double peak, which will be discussed later: (1) they are generated by X-polarized and Y-polarized modes [21] and (2) they arise from two different resonators. In fact, because of the irregular wrapping, microcoil turns can touch in several points and light traveling in the microcoil experiences two distinct resonators in two different parts of the microcoil. The four-turn OMCR (Fig. 12.10d) has an even more complicated

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Fig. 12.10 Spectra of OMCR with (b) two turns, (c) three turns, and (d) four turns. (a) represents the reference spectrum obtained for a free-standing MF [13]

spectrum characterized by three sets of peaks with different attenuation: (A) 3 dB, (B) ∼6 dB, and (C) 9.5 dB. As in the previous case, these groups of peaks can arise from different resonators or from a combination of two modes (X-polarization and Y-polarization modes) with the same FSR = 0.93 nm, namely A and B + C. All these modes are compatible with the complicated resonator structure. From [12], the difference between the resonator wavelengths of X- and Y-polarized modes can be estimated to be about 0.44 nm, which is similar to the 0.48 nm difference observed in Fig. 12.10b. It is therefore possible to ascribe the two types of peaks to X- and Y-polarized modes. The loss induced by wrapping the MF on the rod (estimated from the difference between spectra 12.10b–d and 12.10a for the two-, three-, and four-turn OMCR was ∼1, 2.5, and 5 dB, respectively. The loss is possibly induced by surface roughness, mode discontinuities at the input/output, and leaky modes associated with the rod. The largest Q factor obtained is ∼10,000, which is lower than that achieved with an OMCR immersed in a liquid [12]. This can be possibly explained by the small coupling strength associated with the short coupling distance (defined as the length along which two adjacent turns are touching) and thick MF or by the additional loss at the MF/rod interface. Although the structure of an OMCR exhibits a good temporal stability, submicrometer wires experience aging when left uncoated for some days [23].

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Moreover, uncoated OMCRs are not portable and can be easily damaged. The embedding of an OMCR in a low refractive index medium seems to be the best method to solve the reliability issues as it provides protection both from fast aging and geometrical and optical instability. Additionally, coated OMCRs wrapped on a rod make the fabrication of high-sensitivity biosensors possible. In 2007, a robust R [14]. The OMCR device was demonstrated by embedding an OMCR in Teflon was manufactured by wrapping an MF on a rod having a low refractive index with the same setup as in Fig. 12.8. The MF radius and uniform waist region length were ∼1.5 μm and 3.5 mm, respectively. The rod had three layers: the inner core (∼200 μm) was a multimode silica optical fiber which provided rigidity to the structure, the intermediate layer consisted of a coating polymer (Efiron UV373, manufactured by Luvantix, South Korea) with a low refractive index (n∼1.37 at R fluoropolymer resin coating, λ∼1.55 μm), and the outer layer, a thin Teflon with an extremely low refractive index (n∼1.3 at λ∼1.55 μm) providing a uniform refractive index surrounding to the MF. The total diameter of the rod was D∼700 μm. The embedded structure was manufactured as follows: at first, with the aid of a microscope, the MF was manually wrapped on the rod, then the R Teflon suspension was dropped onto the microfiber and the solvent was allowed to evaporate.

R Fig. 12.11 Microscope picture of an OMCR wrapped on a rod and embedded in Teflon . The MF and rod radii are 1.5 and 350 μm, respectively [14]

Figure 12.11a shows the three-turn OMCR prior to the embedding process; the main coupling region is between the lower two turns on the right, even though some small coupling regions may occur out of the field of vision. Because of R , the the large refractive index difference between the silica MF and Teflon loss observed during the OMCR fabrication was negligible. Indeed, this structure was also unaffected by air turbulences and preserved all the benefits of the self-touching loop resonator. The OMCR transmission spectrum in the wavelength interval 1525–1535 nm is shown in Fig. 12.12, the overall resonance depth is ∼7 dB. The resonator embedding was carried out with the use of the 601S1-100-6 soluR tion of Teflon AF amorphous fluoropolymer resin (DuPont, United States). The R R solution on a Teflon -coated substructure was repeatedly covered with Teflon strate, and the solvent was allowed to evaporate. The coupling process is extremely challenging because the solution reduces the effect of surface forces and if the MF is not tightly wrapped around the rod the resonator conditions are lost. Moreover,

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Fig. 12.12 Spectra of an OMCR before, during, and R after embedding in Teflon

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because of the rapid solvent evaporation, there is a constant solvent flow around the R MF, and Teflon particles and air bubbles move rapidly in the solution. In this situation a collision with the MF (especially with its pigtails) can displace the MF and change the OMCR transmission properties unexpectedly. Particular care has also to be taken to avoid any particle contamination into the solution because contaminants eventually come into contact with the MF and significantly alter its overall loss. R is shown in Fig. 12.11b. The picture of the OMCR after its embedding in Teflon R  Coating with Teflon provides a smooth surface and a homogeneous medium. Although some sporadic bubbles are observed in the polymer, their effect on the OMCR is negligible because none of them is in proximity of the MF. Figure 12 shows that the optical properties of the embedded resonator are similar to those of the OMCR in air. So far, the demonstrated OMCRs have had non-uniform pitch between turns and weak coupling because of the difficulties in handling and positioning small MFs. A considerable improvement should come from the use of precision XYZ stages.

12.4 Microfiber Resonant Sensors Because of their low cost, low loss, extended evanescent fields, and possibility to provide high Q microresonators, optical MFs are ideal sensor elements. In the

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past few years, evanescent-field-based optical resonators in the form of microspheres, photonic crystals, and microrings have been under intensive investigation for deployment as refractometric biological and/or chemical sensors [24–28]. For these applications, small size, high sensitivity, high selectivity, and low detection limits are the dominant requirements. Optical microresonators meet all these criteria; in particular, they can provide large evanescent fields for high sensitivity, high Q factors for low detection limits, and corresponding small resonant bandwidths for good wavelength selectivity. Moreover, because of their high Q, resonant sensors have an interaction length with the analyte which is considerably longer than their physical dimension, thus providing for high sensitivity and ultimate compactness. The drawback that all high Q resonators present relates to the difficulty of coupling light into and out of the resonator. The MF resonators presented in Sections 12.2 and 12.3 (OMLR and OMCR) do not have input/output issues because the fiber pigtails at the resonator extremities launch and collect the totality of the light. Since in free space the fabrication of these devices with high reliability is challenging due to problems of stability, degradation, and cleanliness, embedded devices have been considered; yet, the determination of the coating thickness is a challenge because a thick coating layer limits the sensitivity of the evanescent field, while a thin layer does not provide an appropriate protection to the device. In the next section the use of MF resonator for sensing is proposed with particular emphasis on the dependence of the device sensitivity on the coating thickness.

12.4.1 Sensors Based on a Microfiber Loop Resonator (SMLR) A sensor based on an embedded microfiber loop resonator (SMLR) can be fabricated as shown in Fig. 12.13[16]. Two substrates are fabricated from disposable materials (blue) such as polymethylmethacrylate (PMMA) and coated with a thin R layer of a low-loss low refractive index polymer such as Teflon (light gray). Next, the self-coupling MF loop resonator (gray) is fabricated on one of the substrates, then the other substrate is placed on top of the MF resonator. The use of a thick substrate allows for the easy handling of thin coating layers. The whole system is subsequently coated by a thick polymer layer and at last the expendable materials are removed, leaving a thin layer of low refractive index material on the MF. The final device is shown in Fig. 12.13b; a very thin polymer layer covers the loop of the SMLR while a thick coating deposit is used to fix the two fiber pigtails. The SMLR is a compact device with two sides exposed to the liquid to be sensed. The embedded MF has a considerable fraction of its mode propagating outside the embedding medium both in the upper and lower areas, thus any change in the analyte properties reflects in a change of the mode properties at the SMLR output. Assuming continuous wave (CW) input, a change of the analyte refractive index will lead to a change of the effective index neff of the propagating mode, thereby shifting the mode relative to the resonance and thus modifying the transmitted intensity T. Resonances (wavelengths where T is minimum) occur where β are near to any

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Fig. 12.13 (a) Schematic of the manufacturing process of an SMLR. (b) The final structure of the SMLR [16]. Reprinted with permission. Copyright 2008 Optical Society of America

of the values β n , solutions of (12.2), namely β n = 2π N/L. Because of the interface with the analyte, the mode propagating in the coated MF experiences a refractive index surrounding which is strongly affected by the analyte. Figure 12.14 shows examples of SMLR cross sections.

Fig. 12.14 Cross section of the SMLR: d1 and d2 represent the distance between the MF and the top and bottom surfaces, respectively. (I) d1 = d2 = d < 1 μm and (II) d1 →∞ and d2 = d < 1 μm [16]. Reprinted with permission. Copyright 2008 Optical Society of America

Three parameters have significant influence on the mode properties: the MF radius r, and the coating thicknesses d1 and d2 in the upper and lower layers. When one of the distances d1 or d2 is very large, the device resembles the case of a conventional D-shaped fiber [29, 30]. Two typical cases are investigated: (I) d1 = d2 = d < 1 μm and (II) d1 →∞ and d2 = d < 1 μm. The effective index has been evaluated by a finite element method using the commercial software COMSOL 3.3. Since higher order modes are significantly more confined into the MF, only the fundamental mode was considered. This fundamental mode has the largest propagation constant and is the only mode that is well bounded in the vicinity of the fiber core [29, 30].

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Figure 12.15 shows the dependence of neff on the analyte refractive index na assuming the refractive index of the MF and of the embedding low refractive index R polymer (Teflon ) to be nc = 1.451 and nt = 1.311, respectively. While r is assumed to be 500 nm, two values for d are used, namely 10 and 100 nm. Two different designs (I and II in Fig. 12.14) are considered. (a) 1.418

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1.3

na

Fig. 12.15 Dependence of the effective index of a coated MF neff on the index of the analyte na for nt = 1.311, nc = 1.451, r = 500 nm, d = 10 and 100 nm for small coating thicknesses (case I) and infinitely thick layer on one side (case II). The dotted line represents a bare MF. The wavelength of the propagating modes is (a) λ = 600 nm and (b) λ = 970 nm [16]

Generally, neff increases with na both in case I and II, and increases more quickly with smaller d because in this case a larger fraction of the mode is propagating in the analyte. Since the overlap between the propagating mode and the analyte in I is larger than in II, neff in I is larger than in II when na > nt , while it is smaller when na < nt . If na = nt , the propagating light does not see the boundary between R Teflon and the analyte, thus neff is independent of the coating thickness d. The dependence for the bare MF is also shown in Fig. 12.15 for comparison. Compared to the SMLR, the bare MF has the largest overlap between the propagating mode and the analyte, hence the strongest dependence on na . Therefore, the neff in bare MF is smaller than in SMLRs when na < nt and larger than in SMLRs when na > nt . Resonant sensors can be classified by their homogeneous sensitivity S, defined as the shift of the resonant wavelength λ0 corresponding to one of the propagation constants β n , in (12.2) with respect to a change in the analyte refractive index na [31]:

S=

∂λ0 ∂neff 2π ∂neff λ0 ∂neff ∂λ0 = = = . ∂na ∂neff ∂na βn ∂na neff ∂na

(12.9)

Since water is the solvent for most analytes and the device performance is affected by high losses, it is useful to work in a spectral region of low water absorption. S was calculated near na = 1.332 at 600 and 970 nm. Figure 12.16 shows the dependence of S on the MF radius r for different coating thickness d in both designs

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4

(a)

bare nanowire d=10nm at I d=100nm at I d=500nm at I d=10nm at II d=100nm at II d=500nm at II

Sensitivity (nm/RIU)

2

10

0

10

-2

10

10

Sensitivity (nm/RIU)

10

(b)

bare nanowire d=10nm at I d=100nm at I d=500nm at I d=10nm at II d=100nm at II d=500nm at II

2

10

0

10

-4

10

500

1000 1500 Radius (nm)

2000

500

1000 1500 Radius (nm)

2000

Fig. 12.16 Sensitivity of the SMLR versus MF radius for (a) λ = 600 nm and (b) λ = 970 nm and for different values of d. (I) and (II) represent the two designs reported in Fig. 12.13. Here, na = 1.332, nt = 1.311, and nc = 1.451

I and II of Fig. 12.13. S increases for increasing fractions of the mode field inside the fluidic channel, i.e., when d decreases λ increases or the MF radius r decreases. S reaches 500 nm/RIU at r∼200 nm for λ = 600 and 700 nm/RIU at r∼300 nm for λ = 970 nm. This is higher than what has been reported for most microresonator sensors. As expected, at the same wavelength and MF diameter, S in design I (two sensing surfaces) is larger than in design II (one sensing surface), because of the larger overlap between the evanescent field propagating in the MF and the analyte. For even smaller values of r, the S reaches a plateau because the fundamental mode is no longer confined and most of the evanescent field is in the analyte. In this case neff becomes linearly dependent on na and the derivative in the last term of (12.9) becomes a constant. In these simulations β, d, and the MF loss are assumed uniform, which in practice is difficult to realize. For example, where the MF loop closes, the SMLR height doubles, the overlay thickness can change, the two device/analyte interfaces are not parallel, and therefore d and β are not uniform. Additionally, when d is as small as 10 nm, the refractive index is close to the value of the surface refractive index rather than to the bulk refractive index. For theoretical estimation d, β, and n can be assumed to represent the averaged values over the MF loop and it is still possible to assume that they are uniform along the device length.

12.4.2 Sensors Based on Microfiber Coil Resonator (SMCR) Because of their intrinsically better stability, microfiber coil resonators are better suited than loop resonators for sensing applications. A coated MF coil resonator sensor (SMCR) can be fabricated as follows [15]: first, an expendable rod (made, for example, from PMMA, which is soluble in acetone) is initially coated with a layer of

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R thickness d of a low-loss polymer, such as Teflon , an optical MF is then wrapped on the rod as explained in Section 12.3, next the whole structure is coated with the same low-loss polymer, and finally the rod is removed. Figure 12.17 shows the final structure after the rod is removed: the MF and the analyte channel (in the space R in light previously belonging to the support rod) are shown in gray and Teflon gray. It is a compact and robust device with an intrinsic fluidic channel to deliver samples to the sensor. This represents a considerable improvement with respect to most ring or microsphere resonators which require an additional fluidic channel to deliver/collect the analyte. The embedded MF has a considerable fraction of its mode propagating in the fluidic channel, thus any change in the analyte properties is reflected in a change of the mode properties at the SMCR output. As for the SMLR, the SMCR is fabricated from a tapered optical fiber, thus light can be coupled into the sensor with essentially no insertion loss, a huge advantage over other resonating sensors.

Fig. 12.17 Schematic of the R Teflon -coated SMCR [15]. Reprinted with permission. Copyright 2008 Optical Society of America

As for the SMLR, a change of the analyte refractive index na leads to a change of the effective index neff of the propagating mode, thereby shifting the mode resonance and thus modifying the transmitted intensity T. Here only the simplest case of a two-turn resonator is investigated, although it can be easily extended to the case of multi-turn resonators. A two-turn SMCR can be easily evaluated using coupled mode equations as discussed in Section 12.3. Denoting β as the real propagation constant, α the loss coefficient, and K = (κL)2 the power coupling ratio for coupling coefficient κ and MF length L of a single turn, T can be obtained by (12.1). Resonances (minima of T) occur when K and β are near: K = σ 2 , βn =

2π neff 2π = N, λ0n L

(12.10)

where λ0n is the resonant wavelength, N is the azimuthal mode number, and σ = exp (− αL) is the fractional loss per turn. As shown for the previous sensor, the transmission properties are particularly affected by the MF radius r, the coating thickness between the MF and the fluidic channel d, and the wavelength. Since neff

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is a function of r and d, the resonance wavelength varies with r and d according to (12.10). Figure 12.18 shows the SMCR cross section. Its working principle is similar to that of SMLR, design II (Fig. 12.14 right): the field propagating in the MF is in contact with the analyte only on one side, toward the sensor central axis.

Fig. 12.18 The cross section R -coated SMCR of the Teflon [15]

Generally, neff increases with na and increases more quickly with smaller d since in this case a larger fraction of the mode is propagating in the analyte. Because modes propagating in a SMCR and in a SMLR (design II) have the same overlap with the analyte, they experience the same effective index dependence on r, d, and λ, thus the sensitivity S predicted for the SMCR is the same as that predicted for the SMLR (design II). Still, although the manufacture of an embedded SMCR is not easier than that of a SMLR or of a MF coil, the protected device with an intrinsic microfluidic channel is clearly advantageous. For a resonating sensor, the detection limit is another critical parameter to quantify the device sensing capability. The detection limit represents the smallest measurable analyte refractive index change and it can be defined as [31] DL =

δλ0 , S

(12.11)

where δλ0 is the smallest measurable wavelength shift. Ideally, δλ0 is determined merely by the instrument resolution and independent of the resonance shape or bandwidth. However, it is difficult for a broad resonator shape to achieve accurate detection because it is easy to be perturbed by noise and other external factors [31]. To enhance the accuracy in detecting the wavelength shift, a narrow resonance is required. Experimentally, δλ0 can be taken as a fraction of the BW of the monitored resonance [26]: δλ0 = 0.05BW.

(12.12)

The sharper is the resonance the smaller is the δλ0 which can be achieved. Generally, the sensor performance is mainly limited by its own properties, but the instrument resolution is also important, especially when the sensor Q factor is very large. Additionally, there are other limitations for the sensor arising from external factors, such as temperature, vibrations. It is possible to include the contribution of

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temperature, noise, and instrumental resolution in (12.11) by replacing δλ0 with the sensor resolution R defined as [32] $ 2 , R = 3 σN2 + σT2 + σSR

(12.13)

where σ N , σ T , and σ SR represent the standard deviations associated with amplitude noise, temperature, and detector spectral resolution, respectively (do not confuse R with the coil radius). For the resonating sensors presented in this chapter (SMCR and SMLR), the DL can be easily evaluated from (12.11). The BW depends on the resonator coupling and loss. In traditional microresonators, light is coupled into the resonator using prism coupling, anti-resonant reflecting waveguide coupling, or fiber taper coupling [24–28]. With the possibly only exception of fiber taper coupling, which has been proved to be reasonably efficient [33], coupling to microresonators has been found to be considerably more complicated in terms of device design and/or of increased overall loss. On the other hand, SMCR and SMLR have low insertion loss, because of their fiber-based pigtails with adiabatic profile. Resonant coupling and therefore high Q factors can also be achieved by optimizing the design [34, 35]. Loss in SMCR and SMLR arise from surface scattering, material (analyte, coating, and fiber) absorption, and bending losses. The smallest reported scattering loss of an optical MF is about α = 10−3 dB/mm with radii in the range of hundreds of nanometers [7, 8]. As far as material losses are concerned, water absorption can be reduced to levels well below 10−4 dB/mm by operating at short wavelengths. Cladding mateR or UV375) can be used for low coating loss; for water-core rials (such as Teflon Teflon waveguides losses of 10−3 dB/mm have been reported [36, 37]. Regarding bending losses, for a 200 nm radius MF and a 500 μm diameter coil at a wavelength of 600 nm, the estimated bending loss is 10−4 dB/mm, which quickly decreases further for increasing coil diameters. Thus, assuming the propagation loss to be α = 10−2 dB/mm, the other losses can be neglected. For a simple two-turn SMCR, the FWHM and the Q factor depend on the resonator coupling and loss. For resonant coupling, K = σ 2 in (12.10) and [15, 16] BW =

    2λ20 α λ20  π (σ 2 + σ −2 ) ∼  − arcsin .  2 −2 π neff L 2 2(σ + σ ) − 2  π neff

(12.14)

From (12.14), for a SMCR with r = 200 nm at λ = 600 nm, the bandwidth and detection limit are B∼4·10−4 nm and DL∼10−6 −10−7 RIU, respectively. Although the SMLR resonance bandwidth has a slightly different formula, the BW and the DL have virtually the same value as for SMCR. In the literature, microspheres and microcapillaries have been quoted having extremely high Q factors, and upon optimization these could provide sensitivities comparable to the one predicted here for SMCR. However, the ease of mode size control via the MF diameter and the lossless input and output coupling via the fiber pigtails are unique features of the SMCR. Additionally, although planar ring resonators are expected to have similar

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sensitivities when similar design parameters are used, from the experimental point of view it is challenging to achieve losses comparable to the loss achieved in MFs. A SMCR was first demonstrated in 2008 [17] fabricated from a MF with length and diameter of the uniform waist region of 50 mm and ∼2.5 μm, respectively. The MF was wrapped on a 1 mm diameter PMMA rod and then repeatedly coated by R solution 601S1-100-6, as shown in Section 12.3. The dried embedded the Teflon microcoil resonator was then submerged in acetone to dissolve the support rod. The whole PMMA rod was completely dissolved in 1–2 days at room temperature. At last the SMCR with a ∼1 mm diameter microchannel and two input/output pigtails was obtained. Figure 12.19 shows a microscope picture of the fabricated sensor.

Fig. 12.19 Picture of the SMCR [17]

The sensor consists of a MF resonator with five turns and a microfluidic channel inside. The adjacent MF coils are very close and the major coupling area is in the middle. Although some bubbles are left inside the SMCR during the drying process, these seem to be far from the MF and do not affect the overall sensor operation. The sensor was connected to an EDFA and an OSA and then immersed into the mixtures. Its sensitivity S was measured from the spectral shift (Fig. 12.20) observed inserting the sensor in a beaker containing different mixtures of isopropyl and methanol. In the seven mixtures the isopropyl fraction was gradually increased from 60 to 67.6% by adding small calibrated isopropyl quantities to the isopropyl/methanol solution at a location far from the sensor. The refractive indices of isopropyl and methanol at 1.5 μm were taken as 1.364 and 1.317, respectively [38]. Figure 12.20a shows that, as expected, the resonator peak shifts toward longer wavelengths for increasing isopropyl concentrations. The extinction ratio increases for increasing isopropyl concentrations until 64.4% and then decreases for higher concentrations. This can be explained by the change of the coupling coefficient with the refractive index. The dependence of the resonance peak shift on the mixture refractive index is summarized in Fig. 12.20b. Lines plotting the predicted sensor behavior for different coating thicknesses d are also reported. Simulations show that the slope increases when the thickness d decreases because a larger fraction of the evanescent field overlaps with the analyte [17]. The best fit occurs for d∼0 suggesting that

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0.50

a)

–4 –6

6 7

–8

0.2

Wavelength Shift (nm)

Output (dBm)

b) d=100 nm

–2

0

295

1 2 3 4 5

0.4 0.6 0.8 Wavelength Shift (nm)

0.45 d=500nm d=0 nm

0.40

0.35 Measurement Simulations

1

0.30 1.345

1.346

1.347

1.348

1.349

Analyte refractive index na

Fig. 12.20 (a) Output spectrum of the SMCR in mixtures of isopropyl and methanol; the isopropyl fraction is 60% (1), 61.5% (2), 63% (3), 64.3% (4), 65.5% (5), 66.7% (6), and 67.7% (7), respectively [17]. Reprinted with permission. Copyright 2007 American Institute of Physics. (b) Wavelength shift of the resonance peak versus the analyte refractive index R the Teflon layer is very thin. This is consistent with the experimental results since R can only the microfiber has been wrapped tightly onto the support rod. Teflon penetrate the small gaps provided by the support rod roughness, which is in average considerably smaller than 100 nm. Sensitivity S was evaluated from the slope of the line in Fig. 12.20b and found to be ∼40 nm/RIU. This value is comparable with those reported previously for other resonating sensors like microsphere and microring [24, 26, 27]. The relatively low value for the sensitivity can be attributed to the small overlap between the mode propagating in the MF resonator and the analyte. Another factor which has probably affected the sensor sensitivity is the lack of smoothness of the device surface in contact with the analyte; this is possibly caused by PMMA residues on the surface of the channel or it may originate from the original roughness of the PMMA support rod. This roughness produced the moderately low Q factor observed in the resonator (Q∼104 ), which limited the interaction length between the mode and the analyte. It is believed that the overall sensitivity can be considerably improved to ∼104 by using thinner MFs and by fabricating microcoil resonators with higher Q factors. The minimum detectable refractive index change for this sensor is limited by the accuracy in the measurement of the peak wavelength (15 pm). Assuming S = 40 nm/RIU, the minimum detectable refractive index change is obtained to be nmin ∼0.015/40∼4·10−4 . This value can be decreased by using better detection systems and more stable sources.

12.5 Conclusions In summary, optical microfibers (MF) are potential building blocks in future micro- and nanophotonic devices since they offer a number of unique enabling properties including large evanescent fields, extreme flexibility and configurability,

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and low-loss interconnection to other fibers and fiber-based components. MFs are manufactured by the so-called flame-brushing technique providing an exceptionally low loss and high strength. Resonators can be easily fabricated by knotting the MF or coiling it onto a support rod. Embedding in polymers provides a practical solution to long-term reliability of MF resonators and sensors. Embedded MF resonators are stable, portable, robust, and represent the key element for practical sensing applications. Refractometric resonant sensors can be easily manufactured from MF loop and coil resonators. The dependence of the MF effective index on the analyte and on the MF radius can be calculated from the coupled wave equations, thus the relation between the sensor sensitivity and the MF radius is easily obtained. A sensitivity close to 103 nm/RIU has been estimated for a 0.6 μm MF at a wavelength λ = 970 nm. The related estimated detection limit for the sensors was 10−6 –10−7 RIU. The sensor based on the MF coil resonator has been demonstrated experimentally and a sensitivity of ∼40 nm/RIU was measured for solutions with refractive index close to that of water. Improvements in the fabrication technique and additional surface functionalization of the sensor channel can increase the detection sensitivity for biological and chemical compounds to extraordinarily high levels. In fact functionalization on the channel surface allows to selectively immobilize chemical/biological compounds [39] and enhance the sensor sensitivity by orders of magnitude. Acknowledgments Gilberto Brambilla gratefully acknowledges the Royal Society (London, UK) for his research fellowship. The authors thank EPSRC (UK research council) for financial support.

References 1. Bures, J., Ghosh, R. Power density of the evanescent field in the vicinity of a tapered fiber. J. Opt. Soc. Am. A 16, 1992–1996 (1999) 2. Bilodeau, F., Hill, K. O., et al. Compact, low-loss, fused biconical taper couplers: Overcoupled operation and antisymmetric supermode cutoff. Opt. Letters 12, 634–636 (1987) 3. Tong, L.M., Gattass, R.R., et al. Subwavelength-diameter silica wires for low-loss optical wave guiding. Nature 426, 816–819 (2003) 4. Brambilla, G., Finazzi, V., et al. Ultra-low-loss optical fiber nanotapers. Opt. Express 12, 2258–2263 (2004) 5. Sumetsky, M., Dulashko, Y., et al. Fabrication and study of bent and coiled free silica nanowires: Self-coupling microloop optical interferometer. Opt. Express 12, 3521–3531 (2004) 6. Brambilla, G., Koizumi, F., et al. Compound-glass optical nanowires. Electron. Lett. 41, 400–402 (2005) 7. Brambilla, G., Xu, F., et al. Fabrication of optical fibre nanowires and their optical and mechanical characterisation. Electron. Lett. 42, 517–519 (2006) 8. Leon-Saval, S.G., Birks, T.A., et al. Supercontinuum generation in submicron fibre waveguides. Opt. Express 12, 2864–2869 (2004) 9. Tong, L.M., Lou, J.Y., et al. Assembly of silica nanowires on silica aerogels for microphotonic devices. Nano Lett. 5, 259–262 (2005) 10. Sumetsky, M. Optical fiber microcoil resonator. Opt. Express 12, 2303–2316 (2004)

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11. Sumetsky, M., Dulashko, et al. Demonstration of a microfiber loop optical resonator. OFC/NFOEC (2005) 12. Sumetsky, M., Dulashko, et al. Demonstration of a multi-turn microfiber coil resonator. OFC/NFOEC (2007) 13. Xu, F., Brambilla, G. Manufacture of 3-D microfiber coil resonators. IEEE Photon. Technol. Lett. 19, 1481–1483 (2007) 14. Xu, F., Brambilla, G. Embedding optical microfiber coil resonators in Teflon. Opt. Lett. 32, 2164–2166 (2007) 15. Xu, F., Horak, P., et al. Optical microfiber coil resonator refractometric sensor. Opt. Express 15, 7888–7893 (2007) 16. Xu, F., Pruneri, V., et al. An embedded optical nanowire loop resonator refractometric sensor. Opt. Express 16, 1062–1067 (2008) 17. Xu, F., Brambilla, G. Demonstration of a refractometric sensor based on optical microfiber coil resonator. Appl. Phys. Lett. 92, 101126 (2008) 18. Stokes, L.F., Chodorow, M., et al. All-single-mode fiber resonator. Opt. Lett. 7, 288–290 (1982) 19. Caspar, C., Bachus, E.J. Fibre-optic microring-resonator with 2 mm diameter. Electron. Lett. 25, 1506–1508 (1989) 20. Sumetsky, M., Dulashko, Y., et al. Optical microfiber loop resonator. Appl. Phys. Lett. 86, 161108 (2005) 21. Sumetsky, M., Dulashko, Y., et al. The microfiber loop resonator: Theory, experiment, and application. J. Lightwave Technol. 24, 242–250 (2006) 22. Jiang, X.S., Tong, L.M., et al. Demonstration of optical microfiber knot resonators. Appl. Phys. Lett. 88, 223501 (2006) 23. Xu, F., Brambilla, G. Preservation of micro-optical fibers by embedding. Jpn. J. Appl. Phys. 47, 6675–6677 (2008) 24. Adams, M., DeRose, G.A., et al. Lithographically fabricated optical cavities for refractive index sensing. J. Vac. Sci. Technol. B 23, 3168–3173 (2005) 25. Chao, C.Y., Fung, W., et al. Polymer microring resonators for biochemical sensing applications. IEEE J. Sel. Top. Quant. Electron. 12, 134–142 (2006) 26. Hanumegowda, N.M., Stica, et al. Refractometric sensors based on microsphere resonators. Appl. Phys. Lett. 87, 201107 (2005) 27. White, I.M., Oveys, H., et al. Integrated multiplexed biosensors based on liquid core optical ring resonators and antiresonant reflecting optical waveguides. Appl. Phys. Lett. 89, 191106 (2006) 28. White, I.M., Zhu, et al. Refractometric sensors for lab-on-a-chip based on optical ring resonators. IEEE Sens. J. 7, 28–35 (2007) 29. Dinleyici, M.S., Patterson, D.B. Vector modal solution of evanescent coupler. J. Lightwave Technol. 15, 2316–2324 (1997) 30. Marcuse, D., Ladouceur, F., et al. Vector modes of d-shaped fibers. sIEE Proc. Part J. Optoelectron. 139, 117–126 (1992) 31. Chao, C.Y., Guo, L.J. Design and optimization of microring resonators in biochemical sensing applications. J. Lightwave Technol. 24, 1395–1402 (2006) 32. White, I.M., Fan, X. On the performance quantification of resonant refractive index sensors. Optics Express 16, 1020–1028 (2008) 33. Knight, J.C., Cheung, G., et al. Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper. Opt. Lett. 22, 1129–1131 (1997) 34. Xu, F., Horak, P., et al. Optimized design of microcoil resonators. J. Lightwave Technol. 25, 1561–1567 (2007) 35. Xu, F., Horak, P., et al. Conical and biconical ultra-high-Q optical-fiber nanowire microcoil resonator. Appl. Opt. 46, 570–573 (2007)

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36. Altkorn, R., Koev, I., et al. Low-loss liquid-core optical fiber for low-refractive-index liquids: fabrication, characterization, and application in Raman spectroscopy. Appl. Opt. 36, 8992–8998 (1997) 37. Dress, P., Belz, M., et al. Physical analysis of teflon coated capillary waveguides. Sens. Actuators B 51, 278–284 (1998) 38. Kim, C.B., Su, C.B. Measurement of the refractive index of liquids at 1.3 and 1.5 micron using a fibre optic Fresnel ratio meter. Meas. Sci. Technol. 15, 1683–1686 (2004) 39. Balakirev, M.Y., Porte, S., et al. Photochemical patterning of biological molecules inside a glass capillary. Anal. Chem. 77, 5474–5479 (2005)

Chapter 13

Photonic Crystal Ring Resonators and Ring Resonator Circuits Weidong Zhou, Zexuan Qiang, and Richard A. Soref

Abstract We present the characteristics and applications of photonic crystal ring resonators (PCRR). Photonic crystal confinement can achieve very high cavity quality factor ring resonators. Diffraction-limited ultra-compact PCRRs are feasible due to the absence of size-dependent losses. The flexible modal properties can offer flexible design and integration schemes for either forward or backward propagating add-drop filters based on single- or dual-ring PCRRs. Hybrid-confined PCRRs also show great potential and useful characteristics. Furthermore, we report a few device and circuit configurations based on PCRRs for high-speed modulators, filters, delay lines, etc. Such PCRR structures can potentially provide a good alternative to the traditional microring resonators, as one of the key contributors to the emerging low-power nanophotonics technology.

13.1 Introduction Ring resonators, owing to their high spectral selectivity and large free spectral range (FSR), have been widely introduced in a wide range of device applications including linear, passive devices (such as filters [1], dispersion compensators [2], and sensors [3]) and active devices (such as modulators [4], switches [5], and lasers [6]), as well as non-linear devices for cavity quantum electrodynamics [7]. To date, ring resonators have been proposed and successfully realized in a variety of materials, as summarized in Table 13.1, including optical fibers [1, 8], low index contrast dielectric materials (with material index contrast n < 2, e.g., silica [9], polymer [10], LiNbO3 [11], Si3 N4 [12], SiON [13]), and high index contrast semiconductor materials ( n ≥ 2, e.g., SOI [14], InGaAsP [6], AlGaAs [15], GaAs [16]). Systems

W. Zhou (B) Department of Electrical Engineering, NanoFAB Center, University of Texas at Arlington, Arlington, TX 76019, USA e-mail: [email protected] I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_13,  C Springer Science+Business Media, LLC 2010

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Technology

Structure/material

Ring size

Loss

Integration

Optical fiber

Fiber [1, 8]

Large

Small

Poor

Low index contrast material ( n < 2), e.g., silica [9], polymer [10], LiNbO3 [11], Si3 N4 [12], SiON [13]

Medium

High index contrast material ( n = 2), e.g., semiconductor material SOI [14], InGaAsP [6], AlGaAs [15], GaAs [16]

Small

Hybrid, e.g., polymer/Si [19], Si/air slot [31]

Small

Hybrid PC waveguide [27]

Medium

Small

Poor

PC directional coupler loop [28, 29]

Medium

Small

Good

PC self-collimation loop [22]

Medium

Medium

Good

Ultra-compact PCRR [32–38]

Ultrasmall

Small

Good

Planar waveguide

Good

Loss increases with decrease of ring radius

Good

Good

Photonic crystal

based on traditional optical fiber ring technology are bulky and incompatible with photonic integration. Thanks to advanced microfabrication technology, planar waveguide-based integrated microring resonators and whispering-gallery-mode (WGM) microresonators have attracted great attention since they provide a most compact and efficient solution for the integration of these devices on the wafer scale. Among those materials mentioned above, silicon-based components are favored because they are capable of on-chip-network integration with CMOS electronics. For example, to achieve an FSR larger than the optical communication window (FSR > 30 nm for full C-band spectral coverage), a ring radius less than 5 μm in silicon is required [17]. Silicon-based resonators can be tuned effectively by free-carrier plasma effect-induced index changes [18]. However, Si does not have

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truly non-linear optical properties [11], which is highly desirable for optical modulations. To overcome this limitation, recently a hybrid silicon-polymer microring structure has also been proposed [19]. We note that for integrated waveguide ring resonators, based on the total internal reflection (TIR) confinement principle, propagation losses increase exponentially with reduction of the ring radii. This presents a problem or “roadblock” to scaling, which in practice sets a lower limit on the ring radius of a few micrometers (e.g., 3 μm). The performance of waveguide microring resonators is also highly sensitive to the surface roughness and the nanoscale gap between the ring resonator and the bus waveguide, which creates another challenge in manufacturing. On the other hand, photonic crystals (PC) offer great promise in ultra-compact photonic components due to the unique properties associated with the PC structure, such as photonic bandgap [20], guided mode resonance [21], self-collimation [22], and the superprism phenomena [23]. Within the realm of PC components, the smallest resonators reported in the literature are the point-defect resonator in a 2D crystal [24, 25] and the nanoscale slot within a 1D crystal [26]. While such devices are useful, they do not offer the versatility of a ring-shaped PC waveguide resonator that can be coupled to bus waveguides and other rings. Recently, several types of photonic crystal ring-shaped waveguide resonator structures have been demonstrated. In the hybrid PC waveguide structure [27], the 90◦ bend section of the ring resonator is replaced with PC reflectors to minimize the bending loss associated with the traditional waveguides. In the PC directional coupler loop structure [28, 29], the ring miniaturization is limited by the required long coupling length, e.g., 80a as reported in [30], where a is the lattice periodicity that constrains the ring size. In the PC self-collimated loop structure [22], self-collimation can only exist under special propagation directions and within a certain frequency range, in particular along the TM direction of the square lattice structure and within the frequency range 0.17a/λ and 0.19a/λ [22]. Otherwise, large scattering losses and reduced device efficiency will occur. PCRRs have attracted great attention for their potential in achieving ultracompact size and for flexible coupling and integration schemes [32–38]. Based on photonic bandgap (PBG) confinement, ultra-compact wavelength-scale ring resonators can be fabricated in cubic, hexagonal, and other complex photonic lattices [39]. The resonators – quasi-square, quasi-rectangular, hexagonal, or circular in shape – can have a diameter as small as 1 μm for the operation wavelength of 1550 nm, as reported in our previous publications [34, 40]. Ultimately, the smallest PCRR can be viewed as a single point-defect PC cavity, i.e., a 1a × 1a ring, which can exhibit very low loss with extremely high quality factor (Q), and with ultrasmall ring modal volume (V). Potentially, PCRRs present a solution to overcome the scaling obstacle of traditional WGM resonators. In this chapter, we first discuss the principles of the PCRR, its characteristics, and modal and spectral properties, followed by demonstrations of single- and dual-ring resonators. Size-independent loss analysis is given with a demonstration of highperformance ultra-compact add-drop filters based on single-ring PCRRs. A hybridconfined PC–TIR PCRR is subsequently introduced, along with a hybrid polymer/Si

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PCRR modulator structure. Finally, several applications based on PCRRs will be briefly described, such as N × N switching and cascaded PCRR circuits.

13.2 Basic Configurations of PCRRs Figure 13.1 shows some examples of PCRRs: a quasi-square ring in a square lattice, a hexagonal ring in a triangular lattice, and a circular ring in a quasi-PC lattice (shown is a quasi-PC with 12-fold octagonal symmetry). Compared to pointdefect or line-defect PC cavities, PCRRs offer scalability in size and flexibility in mode design due to their multimode nature. In waveguide microring resonators, the WGMs are supported by TIR on the outer boundary, which sets the ultimate limit for size reduction. By contrast, the resonant modes in PCRRs are supported by the PBG, which is more efficient in optical confinement, especially for the wavelengthscale ultra-compact cavities. The smallest PCRR can be a single point-defect cavity, which can be very low loss with extremely high Q and ultra-small volume. For devices, the choice of the ring size is determined by the desired resonant wavelength and the trade-off between the cavity Q and the modal volume V. For PCRR cavities with two different sizes shown in Figs. 13.2 and 13.3, simulated cavity Q factors from 4 × 103 to 3 × 105 were obtained. This large Q feature is highly desirable for spectral selectivity and wavelength tunability in filter and switch design.

Fig. 13.1 Photonic crystal ring resonators (PCRR): (a) quasi-square ring PCRR in square lattice, (b) hexagonal ring PCRR in triangular lattice, (c) circular PCRR in quasi-photonic crystal structure (12-fold symmetry is shown) [34]. Reprinted permission. Copyright 2007 Optical Society of America

13.3 Single-Ring PCRRs A traditional add-drop filter (ADF) device is typically in the form of a microring closely coupled to a bus waveguide operating in single mode, and its behavior is usually based on phase-matched coupling between orbiting waves in the ring and propagating modes in the waveguide. Similarly, an ultra-compact PCRR ADF can be formed by removing one row of PC posts (W1 line-defect waveguide) to perform as a bus waveguide and a ring (or racetrack) shape of posts to perform as a compact

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Fig. 13.2 Simulated field patterns for the cavity modes in square lattice dielectric rod-based quasisquare PCRRs. (Modified from [34], with permission from the Optical Society of America)

Fig. 13.3 Resonant mode wavelengths and Q factors corresponding to the modes of both the 3 × 3 and 2 × 2 PCRRs shown in Fig. 13.2. The different modes are monopole (MO), dipole (DI), quadrupole (QUAD), hexapole (HEX), octupole (OCT), and decapole (DEC). The modal degeneracy is denoted with the subscript ‘0’ and ‘90.’ (Modified from [34], with permission from the Optical Society of America)

ring resonator as shown in Fig. 13.4. In general, dielectric-rod-type PC waveguides can be easily designed to operate in single mode while air-hole-type PC waveguides tend to function in multimode regime without any other structural modifications or optimizations. Thus, for simplicity, we first consider the 2D array of posts within a low-index slab, i.e., dielectric-rod-type PC waveguides. Their design principles can be extended to air-hole-type waveguides. However, the determining principle for wavelength-size photonic crystal cavities, in general, can no longer be characterized

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Fig. 13.4 Basic configurations of ring resonator-based add-drop filters: (a) rib waveguide-based ring resonator and (b) photonic crystal ring resonator

as the propagating states, and the idea of phase-matched coupling may not be applicable here [24]. Instead, photonic crystal resonant cavity coupling can be analyzed based on the resonant state symmetry and degeneracy in both the real and imaginary parts of the frequency plane, as proposed by Fan et al. [24]. The response of this new PCRR is numerically analyzed here using a finite-difference time-domain (FDTD) approach. Unless otherwise noted, all the discussions here are based on the 2D square lattice photonic crystal structure as shown in Fig. 13.1a, where the refractive index of the dielectric rods, nh , is 3.48 (silicon), surrounded by air background (nl = 1.00); the ratio of the rod radius to the lattice period is r/a = 0.185. This choice is meant to simulate the behavior of a silicon-on-insulator (SOI) PC structure operating at telecommunication wavelengths. Although the real SOI structure would, in practice, require 3D numerical analysis (3D FDTD), typically highly time and memory consuming, our 2D approach gives a general indication of the expected 3D behavior. Two-dimensional analysis carried out here allows us to identify qualitatively many of the issues in ring cavity design, such as mode control and cavity Q, as well as the coupling scheme design, such as the placement of the ring and the W1 defect waveguide, the relative coupling, and the impact of symmetry. This can offer us the design trade-offs and guidelines for the real structure design based on a complete 3D FDTD technique. The photonic bandgap and the dispersion curve for the guided defect mode in the single line-defect waveguide W1 (missing one row of rods along the X direction) are shown in Fig. 13.5. There exists a (normalized) frequency range from 0.303a/λ to 0.425 a/λ that supports a single mode. To be used in the 1550 nm communication window, the lattice constant, a, is set to 540 nm. Thus the W1 PC waveguide is broadband, with a guided single-mode span from 1270 to 1740 nm indicated by the vertical dashed lines of Fig. 13.3. The PCRR cavity, on the other hand, supports one or at most a few cavity, mostly whispering-gallery-like, multipolar modes with their specific spectral locations controlled by the structural parameters of the cavity

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Fig. 13.5 (a) Single line-defect (W1) photonic crystal waveguide. (b) Dispersion plot and the corresponding defect mode shown in the photonic bandgap region [34]. Reprinted with permission. Copyright 2007 Optical Society of America

design. The resonant wavelengths are tunable by means of a localized change in the dielectric lattice’s refractive index. A typical single-ring PCRR ADF is schematically shown in Fig. 13.6. The rod radius to lattice constant ratio is 0.185, and the incident and exit ports are labeled A, B, C, and D, respectively. The number of periods surrounding the W1 bus waveguide and the ring resonator are, respectively, 5 and 10, i.e., d = 4a and L = 9a. The coupling strength between the bus waveguide and the PCRR is determined by the number of periods between them, i.e., by Lc , where Lc = 0a for the PCRR ADF shown in Fig. 13.6; the distance between the two bus waveguides is 6a. For the PCRR, four extra scattering rods, labeled ‘S’, are introduced to improve the spectral selectivity and give a near-ideal drop efficiency. Each of them is located in the center of its four nearest-neighbor rods with exactly the same diameters and refractive indices as all other dielectric rods in the PC structure.

Fig. 13.6 Schematic of single-ring PCRR-based ADFs with coupling section Lc = 0a (one period of rods)

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The transmission characteristics were simulated with the 2D FDTD technique using perfectly matched layers (PML) as the absorbing boundary condition. A Gaussian optical pulse, covering the whole frequency range of interest, is launched at the input port A. Power monitors were placed at each of the other three ports (B, C, D) to collect the transmitted spectral power density after Fourier transformation. All the transmitted spectral power densities were normalized to the incident spectral power density. Shown in Fig. 13.7 are the normalized transmission spectra for the three output ports (B, C, D) in the single-ring PCRR shown in Fig. 13.6. Note that for a ring cavity without scatterers, a low drop efficiency (ratio of the dropped power to the input power) with poor spectral selectivity was obtained. By simply introducing four scatterers, significantly improved spectral selectivity with close to 100% (>99%) drop efficiency can be achieved at the resonant frequency, in this case at 1555 nm, of the ADF. Snapshots of the electric field distribution in the ADF for the through (off-resonance: λ0 =1524 nm) and drop (on-resonance: λ1 =1555 nm) channels are shown in Fig. 13.8.

Fig. 13.7 Normalized transmission spectra at the three output ports B, C, D for PCRR ADFs with and without scatterers (‘S’)

Fig. 13.8 The electric field patterns for (a) the through (off-resonance: λ0 = 1524 nm) and (b) drop (on-resonance: λ1 = 1555 nm) channels.

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It is worth mentioning that the Q of this single-ring ADF shown in Fig. 13.6 is only ∼130 (Q = λ0 /δλ), where λ0 and δλ are the center wavelength and the linewidth (FWHM) of the drop channel, respectively. For optical communication applications, it is desirable to have a much higher Q, on the order of 1500 or higher, to achieve a linewidth of 1 nm or less. Theoretically, the Q value for the ideal single-ring (3a × 3a) PCRR can be greater than 105 [34], as shown in Fig. 13.3. Q values greater than 2,000 were obtained experimentally in an air-hole-based hexagonal ring cavity laser [32]. The discrepancy between the ideal PCRR Q and the ADF Q is mainly caused by the coupling strength determined by Lc in the coupling sections between the W1 waveguide and the PCRR ring. As shown in Fig. 13.9, higher spectral selectivity can be achieved with Q greater than 1,000 in a single-ring PCRR-based ADF by simply increasing the Lc from Lc = 0a in Fig. 13.6 to Lc = 1a in Fig. 13.9a. However, the drop efficiency decreased at the same time from 100 to 80%.

Fig. 13.9 (a) Schematic of a single-ring PCRR-based ADF with increased coupling section width Lc = 1a (two rows of rods) and (b) normalized transmission spectra with Lc = 1a

To achieve both high drop efficiency (>98%) and high spectral selectivity, a further optimization process can be considered, such as coupling section engineering, adjustment of the size and location of the coupling rods and scatterers, and adjustment of the surrounding periods d and L, as defined in Fig. 13.6. The impact of some of these adjustments is shown in Fig. 13.10. Plots of the normalized intensity transmission in channel D (drop channel) are shown in Fig. 13.10a and c, and the drop efficiency and quality factor plots are shown in Fig. 13.10b and d. While the impact of the width of the surrounding region d on the drop efficiency is negligible for d > 4a (Fig. 13.10b), spectral selectivity does increase with the increase of d, due to the improved confinement. On the other hand, both the drop efficiency and the quality factor Q increase significantly with the increase of the length of the surrounding period L (Fig. 13.10d). The principle of the add/drop properties in PCRRs can be explained with the mode superposition principles [24]. As we know, when a cavity supports degenerate modes and possesses mirror symmetry with respect to the plane perpendicular or

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Fig. 13.10 The effect of surrounding region widths d and L on the drop efficiency and spectral selectivity (Q). (a) Normalized transmission intensity in the drop channel D for L = 9a and d = 14a, (b) drop efficiency and Q for drop channel spectra when d changes with L = 9a fixed, (c) normalized transmission in the drop channel for L = 28a and d = 4a, and (d) drop efficiency and Q factor in the drop channel when L changes with d = 4a fixed

parallel to the waveguides, it is feasible to obtain an add-drop filter [24, 39]. This 3a × 3a PCRR supports two degenerate hexapole modes, namely HEX0 and HEX90 , with 90◦ phase difference as illustrated in Fig. 13.11, where the HEX0 mode is even with respect to the perpendicular plane and odd with respect to the parallel plane of waveguide, while the HEX90 mode is odd with respect to the perpendicular plane and even with respect to the parallel plane. At resonance, the resonant field couples to the drop and the input channels. The ring cavity power that is coupled back into the input waveguide does so in antiphase with respect to the input signal, resulting in cancellation at the throughput port thus making a complete power transfer from the input to the drop port possible.

13.4 Dual-Ring PCRRs By cascading and coupling multiple resonators, the ADF transfer characteristics can be flexibly engineered. Forward dropping can be achieved in dual-ring configurations, similar to the demonstrated schemes in the microring-based optical

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Fig. 13.11 The electric field patterns of a PCRR cavity supporting degenerate hexapole modes at λ = 1557.6 nm: (a) HEX0 and (b) HEX90

add-drop multiplexers (OADM). Here, based on the basic building blocks of single-ring PCRRs, dual-ring PCRR-based OADMs were analyzed for both forward and backward dropping by controlling mode symmetry and ring coupling. For the weakly coupled dual-ring PCRRs, shown in Fig. 13.12, where the total length of bus waveguide (LT ), width of region external to the waveguide (d), coupling ‘strength’ (Lc ), and number of coupling periods (defined as the number of the shared dielectric rods) were 28a, 4a, 0a, and 2a, respectively, backward dropping was achieved. Comparing the backward drop characteristics, shown in Fig. 13.12b, with the characteristics obtained from a single-ring PCRR shown in Fig. 13.7, we found higher spectral selectivity obtained through higher Q, approaching 260.

Fig. 13.12 Backward-dropping dual-ring PCRR ADF: (a) Schematic showing the weakly coupled dual PCRR rings with coupling period of 2a, (b) normalized transmission spectra, (c) field patterns of electric field distribution for through (off-resonance at λ0 = 1500 nm) and backward drop (onresonance at λ1 = 1555 nm) channels. (Modified from [34], with permission from the Optical Society of America)

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Fig. 13.13 Dual-ring PCRR ADF for forward dropping: (a) Schematic showing the strongly coupled dual PCRR with coupling length of 4a, (b) normalized transmission spectra, (c) field patterns of electric field distribution for two forward drop channels (on-resonance at λ2 = 1545 nm and at λ3 = 1561 nm). (Modified from [34], with permission from the Optical Society of America)

On the other hand, for the strong coupling case, as shown in Fig. 13.13, where the coupling length is 4a, forward drop was obtained. Here the drop channels switched to the other two resonant cavity modes with different symmetry properties at 1545 and 1561 nm. Note that this flexibility in backward and forward dropping is one of the advantages of the PCRR. Based on the aforementioned mode superposition, resonant coupling occurs due to frequency and phase matching between the propagating waveguide mode and the resonant cavity mode. The coupling direction is mainly determined by the modal symmetry and the relative modal coupling between the rings. The direction of propagation is the same for the wave in the waveguide and the coupled wave inside the ring. However, the direction may be the same or reverse for the coupling between the rings, depending on the coupling length and the modal symmetry [41, 42]. Both forward and backward drop can be obtained depending on the properties of mode symmetry with respect to the coupling configuration. We attempted to understand this with the mode propagation animation shown in Fig. 13.14, which shows the propagation directions for three different modes in a dual-ring PCRR ADF. In Fig. 13.14a, the maximum coupling efficiency occurs at λ1 = 1555 nm, along the GM direction, with the field minimum ‘–’ and field maximum ‘+’ alternatively coupled. Modes propagate clockwise in both rings, which results in backward dropping. On the other hand, a pair of doubly degenerate modes at λ2 = 1545 nm and at λ3 = 1561 nm appeared to have maximum coupling efficiency in the strongly coupled dual-ring cavities along the GX direction, with both ‘even-mode-like’ (‘+’ to ‘+’ in Fig. 13.14b) and ‘odd-mode-like’ (‘+’ to ‘–’ in Fig. 13.14c) coupling. The coupled mode in the second PCRR ring propagates counter-clockwise, which leads to a forward drop. This opens up additional design flexibility with the desired or optimal coupling attained simply by adjusting the coupling strength and modal engineering the multi-PCRR-based ADF.

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Fig. 13.14 Comparison of backward and forward dropping. Shown are the different modal propagation directions due to differences in coupling strength and modal symmetry. The coupling field relations are labeled ‘+’ and ‘–’ for even and odd modal coupling, respectively. (Modified from [34], with permission from the Optical Society of America)

13.5 Estimates of Size-Dependent Insertion Loss In this section, the PCRR size-dependent loss characteristics will be discussed. For traditional microring resonators the relationship between FSR and R is [43]: λ2 λ2  FSR( λ) =  , = ∂n ng · 2π R neff − λ ∂λeff · 2π R

(13.1)

where λ is the resonant wavelength, while neff and ng are the effective index and the group index of the ring, respectively. The group index can be further derived from ng =

c c , = vg dω/dk

(13.2)

where c is the speed of light in vacuum and vg = dω/dk is the group velocity. We here attempted to use the same equation (13.1) to analyze the spectral characteristics of PCRR structures. The group velocity vg can be obtained from the dispersion curve for the confined defect mode in the single line-defect bus waveguide [30] from which the group index ng is found. We first introduce the effective radius concept for the PCRR structure based on the equivalent area concept, as shown in Fig. 13.15a at right, with √ Reff = (m + 1)a/ π ,

(13.3)

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Fig. 13.15 (a) Schematic of photonic crystal ring resonators with effective radius Reff defined as shown on the right and (b) microring resonators and the associated radius R. (Reprinted from [40], with permission from the International Society for Optical Engineering)

where m is the number of dielectric rods or air holes enclosed in the PCRR resonator (m = 3 for the structure shown). By changing m the effective radius changes accordingly. Finally, we can get the FSR of the PCRR by substituting the defined effective radius and the derived group index into (13.1). The PCRR structure parameters of Fig. 13.15a are the same as previously discussed, that is, square lattice silicon rods surrounded by air background with r/a = 0.185. For comparison, a 2D single-ring silicon waveguide microring having similar configuration parameters are used to simulate the transmission and loss properties, as shown in Fig. 13.15b, where the effective index and group index are both equal to 2.44, with the guide width and gap size of W = 500 nm and g = 100 nm, respectively. The lengths of the waveguide buses are kept the same for all cases in these two structures, namely LT = 33.5 μm (or 62a for the PCRR). In the size-dependent loss analysis, PCRR ADFs with different ring radii R were investigated. Based on the discussions in Section 13.3, to maintain high drop efficiency and spectral selectivity, the coupling ‘strength,’ Lc , was adjusted accordingly from 1a for small ring sizes (e.g., the one shown in Fig. 13.15a) to 2a when the Reff of the PCRR increased to 2.44 μm and greater (m > 7). Numerical simulations were carried out for PCRR and microring resonator. Shown in Fig. 13.16a are the normalized transmission spectra for three output ports (B, C, D) in the single-ring PCRR. Close to 100% drop efficiency at the drop channel at 1557.5 nm was obtained with a high spectral selectivity: Q > 1319, in the single-ring PCRR-based ADFs with 1.2 μm effective radius. We determined that the spectral performance reported here is comparable to, or better than, that

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Fig. 13.16 Spectral response of single-ring add-drop filters based on (a) PCRRs, (b) microring resonators, (c) the field patterns for the corresponding drop channels with different ring sizes for PCRRs and microring resonators. (Reprinted from [40], with permission from the International Society for Optical Engineering)

of the microring resonator with even larger ring radius (R = 3 μm), as shown in Fig. 13.16b. The field patterns for the drop channels with different ring sizes for PCRRs and for the microring resonator are shown in Fig. 13.16c. From Fig. 13.16c it is obvious that some energy radiates out of the 3 μm microring resonator. Therefore, we used the total normalized power of the three output ports (B +C + D)/A to derive the total bending loss in dB for the ADF devices with different ring radii. We note that the total loss in the ADF devices derived here includes both the bending loss associated with small ring radius PCRRs and the coupling loss between the waveguide buses and the rings. The size-dependent loss of the single-ring ADF and the PCRR is shown in Fig. 13.17. The inset magnifies the lower portion of the graph. As expected, the bending loss increases drastically in microring resonators for ring radii less than

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Fig. 13.17 Size-dependent loss in PCRR and microring resonator-based ADFs. The low bending loss in PCRRs is shown in the inset. (Reprinted from [40], with permission from the International Society for Optical Engineering)

5 μm. On the other hand, we did not see any systematic size-dependent bending loss in PCRRs, which is a great advantage of the proposed device. We believe that the different loss behavior is due mainly to the differences in the mode confinement mechanism. The size-dependent FSR was investigated and compared for both PCRRs and microring resonators as shown in Fig. 13.18. One set of the data plotted in Fig. 13.18a is derived from the FDTD simulated spectral response curves, as shown in Fig. 13.16, for different radii. It is interesting to note that the change in FSR

Fig. 13.18 (a) Size-dependent FSR in microring resonators and PCRRs based on FDTD simulation and theory, (b) group index in PCRR structures based on the defect mode dispersion curve shown in Fig. 13.5b. (Reprinted from [40], with permission from the International Society for Optical Engineering)

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associated with the effective radius in PCRRs follows closely the FSR change for microring-based resonators. By using (13.1) and (13.2), theoretical values were obtained for each radius with different resonator wavelength and the corresponding group index. As shown in Fig. 13.18a, very good agreement was obtained between theory and FDTD simulation in each case.

13.6 Hybrid-Confined PC–TIR PCRRs Another type of PCRR is shown schematically in Fig. 13.19, where the outer ring boundary is defined by the air hole PC and the inner boundary by TIR. As discussed earlier, ring resonators with photonic crystal confinement can approach diffraction-limited wavelength-sized rings. However, there is a 1/D size-dependent loss in waveguide ring resonators due to the TIR waveguide confinement mechanism. Further investigation based on ray optics and TIR principles in waveguide ring resonators can lead to the conclusion that it is feasible to have ultra-compact PCRRs with only PC confinement at the outer boundary. As shown in Fig. 13.19, either an air disk or air gap with a central high index island can provide sufficient inner boundary confinement. Such configurations can have an advantage for integration, where the inner circles can be used for the placement of metal contacts and other structures for the control of ring resonators.

Fig. 13.19 Hybrid-confined PCRRs based on PC and TIR confinements (a) without and (b) with center high index island

Based on this type of hybrid confinement a 2D ADF is constructed with a single PCRR ring, as shown in Fig. 13.20. Note that the forward-dropping scheme is feasible even with a single ring, which is impossible for conventional waveguide-based ring resonators. This is due to the modal confinement and coupling mechanism in the PCRR and it further extends the design freedom associated with the PCRR, namely to have either forward or backward dropping with both single- and dual-ring configurations.

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Fig. 13.20 Simulated forward-dropping properties based on single PC–TIR hybrid PCRRs

13.7 Hybrid Polymer/Silicon PCRRs Thin-film electro-optic materials (mostly polymers) are fast in response, with over 100 GHz modulation speed and sub-1-volt halfwave voltage reported in polymerbased modulators [44–47]. Additionally, E-O coefficients [48], as high as 353 pm/V, can be achieved in ceramic thin films, e.g., La-modified Pb(Mg1/3 Nb2/3 )O3 -PbTiO3 or PLMNT developed by Boston Applied Technologies, Inc. (BATI). This value, almost 20 times that of LiNbO3 crystals, enables modulators with very low bias voltage and high speed. However, the size of polymer-based photonic devices is mostly limited by the relatively low refractive index. With the incorporation of a high index Si dielectric rod-based PCRR as the core of the modulator configuration, diffraction-limited modulators can be realized based on standard CMOS-compatible processing. These PCRRs can be extremely small. A functional polymer modulator configuration with embedded Si rod PCRR add-drop filter [40] is shown schematically in Fig. 13.21, where a disruptive silicon nanomembrane (SiNM) transfer process [49] is used to transfer the modulator structure onto the low index indium tin oxide (ITO)-coated glass substrate. The vertical waveguiding structure consists of the polymer-filled Si rod PCRR core region, sandwiched between the low index polymer cladding layers. The device design parameters used in our design is also shown in Fig. 13.21. Electrical control is through the bottom transparent ITO electrode and the top high-speed electrode (RF frequency can be greater than 100 GHz). The ON/OFF modulation is due to electric field-induced index change, causing the resonance shift. We assume the polymer index at zero bias voltage as 1.785, based on the data for high index polymer materials from BATI. For an optical modulator operating at the 1.55 μm telecommunication window, the Si rod radius r and the lattice constant a of the PC are set to 100 and 379 nm, respectively. Based on (13.3), the effective radius of this PCRR is only 2.687 μm, which reduces the complexity of the RF/microwave electrode design because the optical–microwave interaction region here is much less

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Fig. 13.21 Hybrid integration of functional polymers with Si PCRRs for a RF photonic modulator with ultra-compact size; 3D cross-sectional and top views of the device structure. (Reprinted from [50], with permission from the International Society for Optical Engineering)

than the RF wavelength – thus permitting high modulation speed. The optical signal propagating through the W1 PC waveguide can be modulated by controlling the index of the coupled PCRR cavity region through electrical contact on top. At the ON state of the modulator, light confined by the PC structure travels along the W1 waveguide, while at the OFF state, the PCRR is tuned to resonance at the working frequency and no light passes through the W1 waveguide. By using 2D FDTD, we qualitatively confirm that very high spectral selectivity can be achieved for different indices of polymers, as shown in Fig. 13.22. For an input signal at 1565.2 nm, the modulation depth is about 25 dB with virtually no insertion loss. The corresponding propagation field patterns are shown in the inset, for ON (n = 1.792) and OFF states. The drop channel wavelength changes from 1565.2 nm for a polymer index of 1.785 (with zero bias voltage) to 1571.8 nm for a polymer index of 1.805 (with bias voltage of a few volts). The Q factor of the filter, as shown in Fig. 13.23 remains high for these different indices, with values greater than 105 . Such high Q factors are easily achievable for different modes of resonators, as shown in Fig. 13.24, where the Q factors are greater than 105 for all three types of resonant modes. These results illustrate the significant potential associated with PCRRs for ultra-compact ring resonators and for diffraction-limited photonic devices with high spectral selectivity, low switching/modulation power, and high modulation speed.

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Fig. 13.22 Simulated spectral tuning characteristics for a single-channel PC waveguide coupled to a single-ring PCRR embedded in an index-tunable polymer material. The propagating field profiles for ON and OFF states are shown in the insets for the channel with center wavelength of 1565.2 nm at polymer indices of 1.805 and 1.785, respectively. (Reprinted from [40], with permission from the International Society for Optical Engineering)

Fig. 13.23 Simulated quality factor and resonant wavelengths for different polymer indices. (Reprinted from [40], with permission from the International Society for Optical Engineering)

13.8 PCRR Circuits and Applications As discussed above, PCRRs offer a potential substitute for traditional microring resonators due to their flexible dropping schemes and size-independent loss for ultra-compact ring dimensions. In this section, we will provide conceptual nanophotonic applications such as N × N switching and cascaded PCRR ring circuits.

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Fig. 13.24 Simulated quality factors for three different resonant modes in a poly-Si PCRR. (Reprinted from [50], with permission from the International Society for Optical Engineering)

13.8.1 N × N Switching The small ‘footprint’ of the PCRR allows dense integration, i.e., closely packed arrays of ring devices. A prime example of such integration is the N × N switch, known as a cross-connect, or matrix switch (see Figs. 6 and 7 of [51]). In order to illustrate the potential of PCRRs for dense integration, we present here the example of a 4 × 4 permutation-matrix switch in SOI, a matrix discussed for microstrips in [51]. The integration approach can be monolithic or hybrid; we shall focus here on the hybrid. The effective dual-ring approach [52, 53] is assumed here. Starting with a SOI wafer, the first construction step is to define lithographically a set of linedefect and ring-defect waveguides within a cubic PC lattice as shown in Fig. 13.25.

Fig. 13.25 A 4 × 4 permutation-matrix switch in silicon-on-insulator based on dual-ring PCRRs and line-defect waveguides in a square lattice PC structure

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The next step is to bond, at low temperature, an array of active III–V semiconductor dual rings, as shown in Fig. 13.26, with these III–V PIN-diode gain media exactly aligned to the Si guiding regions. The cross section of one pair of rings is shown in Fig. 13.27, where the III–V ring guide is evanescently coupled to the Si PC guide, so that 80% of the fundamental mode is in Si and 20% in the III–V active cladding. The Si/III–V hybrid in Fig. 13.27 is unusual because it differs from the prior art in which the hybrid served as a forward-biased gain medium for lasing [54, 55] or as a reverse-biased absorber for photodetection [56]. In the present switching case, the optical property of a ring pair is altered by bias switching from a high absorption

Fig. 13.26 An array of active III–V semiconductor dual rings bonded to a 4 × 4 permutationmatrix switch in silicon-on-insulator. The electrodes for the switch control are shown

Fig. 13.27 The cross section of one ring of a pair is shown, where the III–V ring guide is evanescent wave coupled to the Si PC guide

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condition to a condition of optical gain. That is how the two switching states of each 2 × 2 switch within the 4 × 4 matrix are attained. The III–V medium presents strong absorption of light at zero PIN bias. Then this cladding swings into a gain state at forward bias. In the resonant switching art, it is known that highly absorbing rings will produce the low-loss ‘bar state’ of the 2 × 2 switch, while gain within the rings provides the ‘cross state’ also having low insertion loss [57]. The example of Fig. 13.26 shows three of the six 2 × 2 switches forward biased, with the other three unbiased, leading to the four input-to-output pairings shown in Fig. 13.27. Waveguides in PCs suffer from optical scattering loss associated with wall roughness that is created during fabrication of the PC structure. This loss is ‘technological’ rather than fundamental [58, 59]. If present-day PC processing and smoothing techniques can be improved to the point where the waveguide losses go below 1 dB/cm, then the PCRRs described here will have performance superior to that of microring resonators in the small diameter range of 4–10 λ/neff .

13.8.2 Cascaded Ring Devices Having just discussed hybrid integration, we shall now explore what can be done with monolithic integration in which electrodes contact the monolithic P- and N-doped regions of a 2D photonic crystal slab structure in order to make an active device. In particular, we are going to look at the higher order geometric arrangements of PCRRs that are feasible and useful. These cascade devices use multiple coupled rings and usually include multiple resonant stages. A variety of passive and active waveguided devices become enabled when several PCRRs are clustered or are arrayed along one or two bus waveguides as illustrated in this section. The passive devices include optical filters, delay lines, buffers, and wavelength division multiplexers/demultiplexers. These cascaded ring devices have optical passbands that can be shaped in sophisticated ways. The application-specific passbands have several degrees of freedom in their design. The actuation and control of the active devices is accomplished by electro-optic, thermo-optic, opto-optic, or electro-mechanical means. The active structures are tunable filters, variable delay lines, switchable optical buffers, reconfigurable add-drop multiplexers, modulators, switches, comb switches, logic elements, photodetectors, amplifiers, and light emitters. Active devices often have a geometric layout similar to that of the passive devices. The lateral PIN diode is one example of an electro-optic device. Here P and N regions bracket appropriate waveguide regions in the cascade for the desired carrier injection or depletion. Thermo-optic control is offered by miniature heater segments deposited upon appropriate cascade areas. Opto-optic or all-optical control results from intense pump light impinging from free space upon the cascade or from light that is end-fire launched into an appropriate PC waveguide. To effect electro-mechanical control of certain waveguides, it is possible to bring ultra-small metal arms or disks in and out of close proximity to the waveguide ‘top’ to provide variable cladding or ‘loading.’ The drawings below illustrate passive devices. To

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each passive device shown, there corresponds an active one in which the effective index of the rings is perturbed. In the next three figures, we propose higher order filters that are the ‘PC lattice analogs’ of dielectric-strip structures that have already been discussed in the optics literature. If we compare the PC technology with the non-lattice strip technology, we can say that the optical performance physics differs in the two cases. However, the geometric layout of the cascades is roughly the same in the two technologies. Let us first consider the coupled-resonator optical waveguide (CROW). This is illustrated in Fig. 13.28a. The CROW is the series version of a related parallel device, the side-coupled integrated spaced sequence of resonators (SCISSOR) [60]. In both cases, these cascades offer tailoring of the passbands to attain a flat top and steep skirt selectivity for the filter characteristic [60]. Figure 13.28b presents a slowwave filter that is engineered to give a low group velocity for light emerging from the bus waveguide outputs. The clustered arrangement of PCRRs is analogous to the ring grouping shown by Van [61].

Fig. 13.28 (a) Four-ring photonic crystal CROW, (b) slow-wave filter based on a circular array of coupled PCRRs

A representative SCISSOR is illustrated in Fig. 13.29a. Light incident in the first bus waveguide is usually dropped in the backward direction in the second bus. Figure 13.29b presents a novel concentric ring filter that allows optical engineering of the filter shape to give enhanced notch depth in the transmission spectrum of this band reject filter. The layout of the quasi-rings is analogous to that in Fig. 7 of Zhang [62]. In Fig. 13.30 we offer a compact configuration for segregating the ‘colors’ of a multi-wavelength input beam. Figure 13.31 shows an advanced SCISSOR that gives more degrees of freedom in designing its filter shape than does the device shown in Fig. 13.29a.

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Fig. 13.29 (a) Four-ring photonic crystal SCISSOR, (b) four-port PCRR filter utilizing nested concentric ring resonators Fig. 13.30 Three-color PCRR wavelength division demultiplexer with multi-spectral input and monochrome output

Fig. 13.31 Three-stage photonic crystal SCISSOR employing two coupled PCRRs per stage

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13.9 Conclusions In conclusion, we have reviewed the progress of photonic crystal ring resonators and ring resonator devices. Emphasis has been on the principles and applications of ultra-compact photonic crystal ring resonators. We proved that there are no sizedependent bending losses in PCRR-based structures. We also demonstrated that the FSR in a PCRR follows relationships similar to those applicable to the microring in relation to the effective ring radius and the dispersion-related group velocity. These findings make the PCRRs an alternative to current microring resonators for ultra-compact WDM components and applications in high-density photonic integration. Acknowledgments The authors appreciate helpful discussions with Drs. Z. Ma, Z. Sheng, and K. Zou. The authors also acknowledge the help and support of Dr. Gernot Pomrenke. This work was supported in part by the US Air Force Office of Scientific Research under Grant 07-SC-AFOSR1004 and in part by the National Science Foundation under Grant DMI-0625728.

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Chapter 14

High-Q Photonic Crystal Microcavities Qiang Li and Min Qiu

Abstract Over the past few years, much work has been devoted to the study of microcavities in photonic crystals (PCs). High-quality factor (Q) PC microcavities exhibit attractive properties since they can confine light in wavelength-scale dimensions, making them potentially useful not only for photonic integration but also in quantum optics. Most of the high-Q PC microcavities are realized in slabs (PCSs), which are two-dimensional (2D) PCs with a high refractive index core layer providing light confinement in the third dimension. The key to design high-Q cavities is to reduce the radiation losses, i.e., to minimize the spatial Fourier components above the light line of the PCS. This chapter will focus on high-Q PC microcavity, including design rules, characterization methods, device demonstrations, and applications.

14.1 Design Rules of High-Q Microcavities in PC Slabs 14.1.1 2D PCSs and PCS Microcavities A typical dielectric slab consists of a layer with a high refractive index surrounded by low-index materials. The light is confined in the high-index slab by total internal reflection (TIR). A 2D PC structure is then patterned on such a dielectric slab to form a PCS, as shown in Fig. 14.1. PCSs with air holes (or holes with a low-index material) are the most common structures studied so far. Such a PCS usually has a photonic band gap (PBG) for the TE-like modes (electric field vector in the slab plane). Usually, the size of the PBG increases with increasing index contrast and air-hole radius. Without loss of generality, in this chapter we restrict ourselves to PCSs with air-hole-based PCs utilizing TE-like modes. M. Qiu (B) Photonics and Microwave Engineering, Royal Institute of Technology (KTH), 164 40 Kista, Sweden e-mail: [email protected] I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_14,  C Springer Science+Business Media, LLC 2010

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Fig. 14.1 A schematic picture of a PCS microcavity. The PC is a triangular lattice of air holes in a dielectric slab and the cavity is formed by removing one air hole from the center. The positions and the radii of the neighboring air holes are also usually modified

In a PCS microcavity the in-plane light confinement is provided by the PBG effect. For resonant modes whose frequency is within the band gap, the in-plane leakage (loss) can be neglected in most cases if the surrounding PC has a sufficiently large extent. Thus the out-of-plane radiation loss plays a key role for the performance of the microcavity.

14.1.2 Q Factor and Loss Mechanisms in PCS Microcavities In PCS microcavities, light is confined within the defect region by two combined mechanisms: the distributed Bragg reflection due to the PBG in the in-plane direction and the TIR in the out-of-plane (vertical) direction. The in-plane confinement is determined by the number of periods of the host lattice surrounding the cavity, with the assumption that the resonant frequency of the defect mode lies within the inplane PBG. Thus, the more periods surrounding the defect, the stronger the in-plane light confinement and the larger the in-plane Q (Q// ). To analyze the vertical loss mechanism, we use spatial wave vector analysis. The energy–wave vector dispersion relationship for a homogenous dielectric cladding (refractive index n) of the 2D PCS waveguide is (nω/c)2 = (k// )2 + (kz )2 , where ω is the angular frequency, k// is the wave number parallel to the slab, kz is the wave number normal to the slab, and c is the speed of light in vacuum. For a PCS with an air cladding, the expression (k// )2 = (ω/c)2 defines a cone in the (kx , ky , ω) space, commonly referred to as the light cone (see Fig. 14.2). Modes that radiate vertically will have small in-plane wave vector components that lie within the light cone of the cladding. The larger the fractions of in-plane wave vector components that lie within this light cone, the larger the vertical radiation loss and consequently the smaller the vertical Q value (Q⊥ ). In many cases, Q⊥ becomes the limiting factor of the total quality factor Qt defined as 1/Qt =1/Q// +1/Q⊥ .

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Fig. 14.2 Two-dimensional hexagonal PCS waveguide structure and cladding light cone [1]. Reprinted with permission. Copyright 2002 Optical Society of America

In order to achieve high-Q values we have to reduce the k// components of the cavity mode that lie inside the leaky region (above the lightcone). The first step is to apply symmetries to enforce special boundary conditions on the wave vector space representation of the mode, so that the in-plane electric field polarizations at k// = 0 are eliminated. To further reduce the in-plane wave vector components inside the leaky region, the geometry of the lattice structure surrounding the defect can be modified. In essence, this changes the Bragg reflection condition at the cavity edges. By doing so, light is allowed to penetrate deeper inside the surrounding lattice to be reflected more gradually. The spatial variation of the envelope function is tailored to obtain a Gaussian profile, which significantly reduces the wave vector components inside the leaky region while keeping the modal volume small [1, 2]. For the k-space analysis, we take the 2D (kx , ky ) Fourier transform (FT) of Ex , Ey and Hx , Hy field distributions in the air layer above the slab at z = h (h > t/2, t being the slab thickness), denoted by F(Ex ), F(Ey ), F(Hx ), and F(Hy ), respectively. We then follow the deduction process in Ref. [3] to determine the Q factor. In cartesian coordinates (kx , ky ), the Fourier transform of component Ex is written as

F(Ex ) =

Ex (x,y)ei(kx x+ky y) dx dy

(14.1)

Here, expressing the wave vector in polar coordinates (k, α) allows a convenient integration of the leaky components inside the light cone. Thus we have

F(Ex ) = Subsequently we define

Ex (x,y) · ei·(k cos α·x+k sin α·y) dx dy

(14.2)

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Nx (k|| ) = −F(Hy ), Ny (k|| ) = F(Hx ) Lx (k|| ) = F(Ey ), Ly (k|| ) = −F(Ex ) k|| = k cos α · xˆ + k sin α · yˆ = k0 sin θ · ( cos ϕ · xˆ + sin ϕ · yˆ )

(14.3) (14.4)

where k0 = 2π /λ0 = ω0 /c and ω0 is the resonant angular frequency of the mode. Wave number k0 is essentially the radius of the light cone at frequency ω0 . Note that (14.4) expresses the in-plane wave vector k// in spherical coordinates. After these definitions, the total averaged power radiated into the half-space z > 0 is given by

π/2

P1 =

2π

dθ sin θ

0

dϕ · K(θ ,ϕ)

(14.5)

0

where K(θ , ϕ) is the radiation power per unit solid angle and can be denoted by K(θ ,ϕ) =

η 8λ20

  Nθ +

 Lϕ 2 η 

  + Nϕ −

2

Lθ  η 

Nθ = (Nx cos ϕ + Ny sin ϕ) cos θ , Nϕ = −Nx sin ϕ + Ny cos ϕ Lθ = (Lx cos ϕ + Ly sin ϕ) cos θ , Lϕ = −Lx sin ϕ + Ly cos ϕ

(14.6)

and η = (μ0 /ε0 )1/2 is the wave impedance of free space. In polar coordinates (k, α) we obviously have k = k0 sin θ

α = ϕ,

(14.7)

Thus (14.5) evolves into η P1 = 8π 2

k0

,

0

k 1 − (k/k0 )2

dk

π/2

K(k,α)dα

(14.8)

0

where    2

   L 2 K(k,α) = Nθ + ηϕ  + Nϕ − Lηθ  , (14.9) Nθ = (Nx cos α + Ny sin α) · ,1 − (k/k0 )2 , Nϕ = −Nx sin α + Ny cos α Lθ = (Lx cos α + Ly sin α) · 1 − (k/k0 )2 , Lϕ = −Lx sin α + Ly cos α With the assumption that most of the radiated power is collected at vertical incidence (along the z-direction and in the vicinity of very small angles θ ), P1 can be simplified into P2 = where

η 8π 2

k0 0

π/2 0

I(k,α) · k · dkdα

(14.10)

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 2  2     1 1 I(k,α) = F(Hy ) + F(Ex ) + F(Hx ) − F(Ey ) η η

(14.11)

The k-space intensity I is much easier to calculate than K(k,α) of (14.9) and thus (14.11) is frequently used to investigate the leaky properties of the cavity modes. The radiation factor (RF) of the cavity mode is defined as RFi =

Pi , i = 1, 2 W

(14.12)

where W is the total energy stored between the two boundary layers (xy planes) z = 0 and z = h: 1 W= 4



h

dz 0

μ0 ε(x,y,z) · |E(x,y,z)| dxdy + 4

2



h

dz

|H(x,y,z)|2 dxdy

0

(14.13)

where ε(x,y,z) is the spatial distribution of the dielectric constant and  2 |E(x,y,z)|2 = |Ex (x,y,z)|2 + Ey (x,y,z) + |Ez (x,y,z)|2  2 |H(x,y,z)|2 = |Hx (x,y,z)|2 + Hy (x,y,z) + |Hz (x,y,z)|2

(14.14)

It makes more sense to compare the radiation factor of different modes than the absolute radiated powers (P1 or P2 ) due to the fact that different cavity modes store different energies. RF is inversely proportional to the vertical Q factor of the same mode. In fact Q⊥ = ω0 ·

W ω0 = , i = 1, 2 Pi RFi

(14.15)

From (14.8) and (14.10) it is clear that only the wave vector components lying inside the light cone contribute to the radiation losses, in agreement with the previous discussion. The Q factor can also be calculated directly using other methods, e.g., finite-difference time-domain (FDTD) simulations. The values should be similar to those obtained with (14.15).

14.2 Measurement of High-Q Factors 14.2.1 Ultrahigh-Q Measurements in the Spectral Domain Conventionally, light spectra are measured by sweeping the laser frequency to determine the Q factor of a cavity. For a high-Q PC microcavity with a Q factor of 106 and resonance wavelength around 1550 nm, the spectral width is about 200 MHz. However, for the existing commercially available swept tunable laser diodes which can support a wide tuning range, the frequency resolution is usually limited. To achieve simultaneously a wide tuning range and a high-frequency resolution, a

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single-sideband (SSB) electro-optic modulator has been proposed to be exploited as an optical frequency shifter to measure the spectra [4]. The radio-frequency generator can support a frequency resolution of several hertz. Therefore, the resolution of the system is only limited by the linewidth of the laser, which can be below 100 kHz for the fiber laser. Figure 14.3a gives the output spectrum of the SSB modulator at a radio frequency of 10 GHz. The laser diode is set to operate at 1564.85 nm. A first-order SSB is used for transmission measurement with an extinction ratio over 20 dB. The radio-frequency output is scanned from 2 to 12 GHz in steps of 1 MHz. The measured spectrum of a high-Q PC cavity is shown in Fig. 14.3b. By fitting the measured data to a Lorentzian function, a resonance at λ = 1564.77 nm and a Q = 1.2×106 are obtained. It can be seen that the SSB measurement system is a powerful tool for measuring the spectra of a PC with a Q factor above 105 .

Fig. 14.3 (a) Output spectrum from a SSB modulator at a radio frequency of 10 GHz. (b) Measured spectrum obtained using the first-order SSB light scan method [4]. Reprinted with permission. Copyright IEEE 2007

14.2.2 Ultrahigh-Q Measurements in the Time Domain Time-domain measurements can directly view the dynamic behavior of photons in a cavity. For high-Q cavities, time-domain measurement are potentially more accurate than spectral-domain measurements since the photon lifetime (τ ph ) is proportional to the Q factor (Q = ω0 τ ph ). Ring-down measurement is a direct and intuitive way of measuring the photon lifetime and has been widely used for the characterization of ultrahigh-Q cavities [5, 6]. Figure 14.4a illustrates the principle of ring-down measurement of the Q factor of a cavity. The cavity is charged by a square-shaped input pulse and is discharged after the pulse ends. When the laser wavelength is set to the cavity resonance, the power transfer continues until the resonant mode is fully charged. Then the input pulse is abruptly terminated. By monitoring the cavity discharge waveform (ring-down process), the cavity lifetime can be computed. Figure 14.4b shows the measured waveform, where a smooth exponential decay is observed. The photon lifetime obtained by fitting the data is 1.12 ns, corresponding to a Q factor of about 1.34 × 106 .

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Fig. 14.4 Schematic diagram of the ring-down measurement of the Q factor [5]. Reprinted with permission. Copyright 2007 Optical Society of America

14.3 High-Q 1D PC Cavity In 1997, a 1D PC resonator was demonstrated with a Q factor of only 265 [7]. Great progress has been made since the introduction of new design rules for high-Q PC cavities. These apparently simple structures actually have high Q/V factors (V is the modal volume) rivaling even the best 2D PC cavities.

Fig. 14.5 SEM photo of 1D PC resonator. N is the number of holes forming the periodic mirrors, NTI and NTO are the number of tapered holes within and outside the cavity, respectively [8]. Reprinted with permission. Copyright 2008 Optical Society of America

Figure 14.5 gives an example of a 1D PC cavity in silicon-on-insulator (SOI) technology [8]. To enhance the Q factor, tapering within the cavity is adopted. The cavity is formed by two mirrors, each including five periodically spaced holes with diameters of 182 nm and a periodic center-to-center spacing of 350 nm. The width of the waveguide is 500 nm. The transition section within the cavity is composed of four holes with diameters of 170, 180, 166, and 131 nm and center-to-center distances of 342, 304, 310, and 290 nm, respectively. This tapering inside the cavity can reduce losses induced by abrupt changes in the electrical mode distribution at the interfaces between the periodic mirror sections and the cavity section. The transition section outside the cavity comprises three holes with diameters of 131, 166, and 185 nm and center-to-center distances of 275, 305, and 314 nm, respectively. The tapering outside the cavity can greatly reduce the coupling loss and thus enhance the peak transmission at resonance. For this structure, the theoretical Q factor is 1.77 × 105 and the transmission is 0.48. In the experiment, the Q factor approach 1.47 × 105 at λ = 1479.705 nm and the transmission is around 0.34.

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Fig. 14.6 (a, b) Ey profiles of the cavity fundamental and second-order modes, respectively, s denotes the minimum edge-to-edge distance between the inner two air holes. (c) SEM photo of a fabricated 1D PC cavity [9]. Reprinted with permission. Copyright 2009 American Institute of Physics

By adopting the membrane structure, the Q factor can be further enhanced. Figure 14.6 shows the structure of the 1D PCS cavity [9]. The silicon slab thickness is t = 220 nm. The radius for the air holes in the periodic mirror sections is r = 120 nm. Inside the cavity, linear tapering is applied over five air holes from a = 430 nm to a = 330 nm at the cavity center (a is the pitch). The corresponding hole radius is r = 0.28a. Figure 14.6a, b shows the calculated Ey distributions of the fundamental and second-order modes, respectively. The light confinement is achieved by index guiding in the y- and z-directions and by Bragg scattering from the 1D PC mirror in the x-direction. The mechanism of light confinement can be interpreted in terms of the mode-gap effect. The calculated Q factor for the fundamental mode is 1.4 × 107 and the mode volume is V = 0.39(λ0 /n)3 , where λ0 is the resonance wavelength in vacuum and n the refractive index of silicon. Figure 14.7 shows the

Fig. 14.7 (a) Experimental spectra and (b) Q factor for a range of cavities with different s, For (a), the s from left to right are 116, 126, 136, 146, 156, 176 nm, repectively. For (b), the crossing and dot denote the experiment and simulation values, respectively [9]. Reprinted with permission. Copyright 2009 American Institute of Physics

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experimental results for cavities with different s. The Q factor reaches a record high value of 7.5 × 105 when s = 146 nm.

14.4 High-Q Point-Defect Cavities in a 2D PCS 14.4.1 High-Q L3 Point-Defect PC Cavity Figure 14.8a depicts a cavity with three missing holes in a 2D silicon PCS with a triangular lattice of air rods [10]. This kind of cavity is termed as L3 cavity. The slab thickness and the air-hole radius are t = 0.6a and r = 0.29a, respectively, a being the lattice constant. The two air holes at the cavity edges are shifted by 0.2a out of the cavity to avoid an abrupt change in the electric field envelope. Figure 14.9a, e provides the Ey profile of the cavity fundamental resonant mode without and with air-hole shift, respectively. The mode field profile is fitted with the product of a sinusoidal wave and a Gaussian envelope, as shown in Fig. 14.9b and f. Without the shift of the air holes, the envelope of Ey decreases more rapidly than the Gaussian function due to the strong reflection at the two edges of the cavity. The FTs of the electric field profiles are shown by the solid line in Fig. 14.9c, g. Figure 14.9d, h provides the 2D FT spectra. The gray area around |k// | = 0 denotes the leaky region, which leads to out-of-slab leakage of light. Fewer components are confined in the leaky region for the shifted air-holes case, thus increasing the Q factor of the cavity. The calculated Q of the cavity shown in Fig. 14.8c is 105 , which is 20 times larger

Fig. 14.8 (a) Perspective view of an L3 cavity in a 2D silicon PCS. (b) and (c) are the zoomin views of the cavity without and with a 0.2a air-hole shift, respectively [10]. Reprinted with permission. Copyright 2003 Macmillan Publishers Ltd: Nature

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Fig. 14.9 (a) The calculated Ey field distribution of the cavity fundamental resonant mode without air-hole shift at both edges. (b) The profile of (a) along the center line of the cavity and the fitted curve corresponding to the product of a fundamental sinusoidal wave and a Gaussian envelope. (c) The 1D FT spectra of (b). (d) The 2D FT spectra of (a). (e)–(h) are corresponding counterparts for a cavity with 0.20a air-hole shift at both edges [11]. Reprinted with permission. Copyright 2005 Optical Society of America

than that without air-hole shifts. When the air-hole shift is 0.15a, the experimental Q and the modal volume V are 4.5 × 104 and 0.69(λ0 /n)3 , respectively [10]. Figure 14.9f shows that there is still some mismatch between the mode profile and the fitted curve with a 0.20a air-hole shift. The Q factor of the L3 cavity in the PCS can be further improved by fine-tuning not only the nearest-neighbor air

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Fig. 14.10 Schematic of L3 cavity in a 2D silicon PCS. An even higher Q factor can be obtained by fine-tuning the positions of six air holes near both cavity edges compared with the two air-holes shift case [11]. Reprinted with permission. Copyright 2005 Optical Society of America

holes at the cavity edges, but also the second and third nearest-neighbor air holes [11]. Figure 14.10 provides the schematic of this structure. In simulation, a high-Q factor of 2.6 × 105 is obtained when the shifts of holes A, B, and C are dA = 0.2a, dB = 0.025a, and dC = 0.2a, respectively. The corresponding V is 0.69(λ0 /n)3 . In experiment, a PC microcavity with a high-Q factor of 105 was fabricated with dA = 0.176a, dB = 0.024a, and dC = 0.176a. The mode volume calculated for the cavity structure is as small as 0.71(λ0 /n)3 . Thus, Q/V = 1.4 × 105 (λ0 /n)−3 , i.e., twice as large as that for the cavity with two air-holes shift [10]. This result clearly demonstrates that the Q of the PC cavity can be enhanced significantly by tailoring air holes A in combination with air holes B and C, while still keeping a small cavity volume.

14.4.2 High-Q L1 Point-Defect PC Cavity The PCS cavity with only one missing hole, termed as L1 cavity, can also be used to realize high-Q factor [12–14]. Figure 14.11a shows the corresponding schematic. The six innermost holes in the cavity are slightly shifted (by ck ) outward in order to achieve highQ for the hexapole mode, which exhibits perfect hexagonally symmetric field distribution. This symmetry can lead to a significant reduction in the vertical radiation loss due to destructive interference and far-field cancellation effects [15]. Figure 14.11b shows the Q versus the shift ck . The highest theoretical Q is 1.4 × 106 is associated with the small mode volume of 1.18(λ0 /n)3 for ck = 1.23. The measured transmission spectrum is shown in Fig. 14.11c. The spectral width is 4.8 pm, corresponding to a total Q of 3.2 × 105 , and a photon lifetime of 260 ps. The derived intrinsic quality factor Qintr is ∼4 × 105 . Figure 14.11d shows the result of the time-domain ring-down measurement. The photon lifetime obtained from the fitted exponential curve is 300 ps.

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Fig. 14.11 (a) Schematic diagram of a hexapole mode L1 PC cavity; ck is the normalized center-to-center distance between defect hole and shifted holes. (b) Theoretical Q versus ck for the hexapole mode. (c) and (d) are frequency-domain and time-domain measurement results, respectively [14]. Reprinted with permission. Copyright 2009 American Institute of Physics

14.4.3 High-Q L0 Point-Defect PC Cavity A PCS cavity involving no missing holes, termed as L0 cavity, has been proposed in [16]. The corresponding geometric configuration is shown in Fig. 14.12a. The structure is very simple, with only two adjacent holes (A) shifted outward along the x-direction by a distance d, and the radii of the two air holes nearest to the cavity center in the y-direction (B) are modified to R1 . The theoretical Q factor is 1.35 × 105 with resonant wavelength at 1564 nm for a = 450 nm, d = 0.14a, and R1 = 0.27a. The Hz field distribution is shown in Fig. 14.12b. The Hz field profile is now very simple and the number of the nodal points is much reduced. Figure 14.12c shows the k-space electric intensity profile. There are very few k// components inside

Fig. 14.12 (a) L0 cavity with d = 0.14a and R1 = 0.27a. (b) Hz field distribution of the L0 mode. (c) k-space intensity profile [16]. Reprinted with permission. Copyright 2004 Optical Society of America

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the leaky region, which accounts for the large Q⊥ (1.9 × 105 ) of this mode. The effective modal volume for the L0 mode is extremely small, only 0.29(λ0 /n)3 .

14.5 High-Q Line-Defect PC Cavities in 2D PCS 14.5.1 High-Q Heterostructure Line-Defect PC Cavity In 2005, Song B.S., et al. proposed the concept of photonic multi-heterostructures (HS) as a more comprehensive method to tune the field envelope function of a cavity [17]. The basic PC environment exploited to construct the photonic double-HS is a 2D silicon PCS with a triangular lattice and a line-defect waveguide formed by a missing row of air holes in the Γ –J direction, as shown in Fig. 14.13a. The corresponding band diagram is shown in Fig. 14.13b. It can be seen that a waveguide mode is formed within the bandgap region. The upward arrow indicates the transmission region, where propagation is allowed through the waveguide, and the downward arrow indicates the mode-gap region, where propagation is prohibited. Figure 14.13c shows the double HS formed by two triangular-lattice PCs (I and II) with different lattice constants. The lattice constant of PC I is a1 . For PC II, the lattice constant is a2 (a2 > a1 ) in the Γ –J direction but still a1 in the Γ –X direction. This lattice constant difference between PCs I and II leads to different transmission and mode-gap regions, thus forming a region where photons are confined in the waveguide of the PC II, as is shown in Fig. 14.13d.

Fig. 14.13 (a) A 2D PCS with a triangular lattice and a line-defect waveguide formed by a missing row of air holes in the Γ –J direction. (b) The calculated band structure of (a). (c) Photonic double HSs formed by assembling the basic PC structures I and II with different lattice constants. (d) Schematic of the band diagram along the waveguide direction [17]. Reprinted with permission. Copyright 2005 Macmillan Publishers Ltd: Nature

Figure 14.14a, c shows the calculated electric field distribution of the photonic double-HS cavity and its profile along the waveguide direction, respectively. The lattice constants for PCS I and II are, respectively, a1 = 410 nm and

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Fig. 14.14 (a), (c) Calculated electric field distributions for the photonic double-HS cavity and the L3 cavity with a shift of 0.2a for two neighboring air holes, respectively. (b), (d) Corresponding electrical field profiles (solid line) and the Gaussian profile (dashed line) along the waveguide direction [17]. Reprinted with permission. Copyright 2005 Macmillan Publishers Ltd: Nature

a2 = 420 nm, and the slab thickness is t = 0.6a2 . The results for the point-defect cavity with air-hole tuning are also provided for comparison. Figure 14.14b, d shows the corresponding electric field profiles, which clearly demonstrate that the electric field profile of the photonic double-HS cavity is much closer to the ideal Gaussian curve than that of the point-defect cavity. The theoretically calculated Q factor of the double-HS cavity is ∼106 , one order of magnitude higher than that of the shifted point-defect cavity. The corresponding modal volume is as small as 1.2(λ0 /n)3 . The experimentally demonstrated Q factor of this photonic double-HS cavity was 6 × 105 [18]. A multi-step HS can be utilized to further increase the Q factor, where the outer part of the envelope function is divided into multiple parts and each part can be optimized to best approximate the Gaussian function. For the triple-step HS shown in Fig. 14.15a, the lattice constants of PCS0, PCS1, and PCS2 are 420, 415, and 410 nm, respectively [19]. The theoretical Q can be as large as 1.6 × 107 . Figure 14.15b provides the measured emission spectrum of the fabricated cavity. The fitted λ is in the range of 0.4–0.7 pm; therefore, the corresponding Q is between 2.3 × 106 and 4 × 106 . In the time-domain measurement, for a rectangular input pulse with a 4 ns width and a 140 ps decay time, the corresponding output pulse is shown in Fig. 14.15c. The photon lifetime is as long as 2.13 ns, indicating a Q factor of 2.52 × 106 , which is the largest recorded Q-factor in PC cavities so far. The modal volume is 1.4(λ0 /n)3 .

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Fig. 14.15 (a) SEM photo of a triple-step HS PCS cavity. (b) Frequency-domain and (c) timedomain measurement results, respectively [19]. Reprinted with permission. Copyright 2005 Optical Society of America

A six-step HS cavity with a Q factor of 109 has been designed analytically [20]. Figure 14.16a shows the schematic of this structure, in which six PCSs with different lattice constants are concatenated. The band structure of this defect is given in Fig. 14.16b. To achieve a Gaussian mode fielddistribution, the lattice constants   2 should satisfy the law an = a0 1 − C n + 14 , where C is a coefficient related to the cut-off frequency and a0 is the lattice constant of PC0. For a cavity with the following characteristic parameters: C=1.45×10−3 , a1 = 0.998a0 , a2 = 0.993a0 , a3 = 0.985a0 , a4 = 0.974a0 , a5 = 0.96a0 , r = 0.29a0 , and slab thickness t = 0.6a0 , the calculated Ey field distribution and the profile along the centerline are shown in Fig. 14.17a, b, respectively. It can be seen that the envelope of the mode field

Fig. 14.16 (a) Schematic of multi-HS microcavity and (b) its band diagram along the waveguide direction [20]. Reprinted with permission. Copyright 2008 IEEE

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Fig. 14.17 (a) Calculated Ey distribution of the fundamental resonant mode of the multi-HS microcavity. (b) Ey profile along the centerline of the cavity in the x-direction (solid line) and fitted Gaussian envelope function (dashed line). (c) Two-dimensional Fourier transform spectra of the Ey electric field profile for this cavity mode [20]. Reprinted with permission. Copyright 2008 IEEE

closely matches the Gaussian function. The calculated Q factor and the modal volume are 5 × 108 and 1.3(λ0 /n)3 , respectively. Figure 14.17c shows the calculated 2D FT of the electric field of this cavity, demonstrating that almost no wave vector components exist inside the leaky region.

14.5.2 High-Q Width-Modulated Line-Defect PC Cavity In 2006, Kuramochi E., et al. proposed a high-Q factor PCS cavity created by the local width modulation of a line defect [21]. This kind of ultrahigh-Q PC cavity is based on a line defect in a PCS with a triangular lattice of air holes and air cladding. The design concept is to increase the waveguide width at the center of the cavity by shifting the surrounding holes away from the waveguide. The corresponding design is shown in Fig. 14.18a [22]. Here inset A1 shows the normal PC line-defect waveguide and the insets A2–A4 show three different PC cavity designs. Simulations have shown that the A4 cavity has the highest theoretical Q factor of 7 × 107 and a mode volume of 2.5(λ0 /n)3 . Given the following parameters for a designed A3 cavity: lattice constant a = 420 nm, √ hole radius r = 108 nm, slab thickness t = 204 nm, waveguide width w = 0.9 3a, hole shift s = 8 nm for the nearest-neighbor holes and s = 4 nm for the second nearest-neighbor holes, the experimental transmission spectrum is presented in Fig. 14.18b. The spectral width is 1.22 pm, which corresponds to a Q factor of 1.28 × 106 . Figure 14.18c shows the ring-down measurement results from which a photon lifetime of 1.12 ns is deduced.

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Fig. 14.18 Width-modulated line-defect PC cavities. (a) Cavity design. (b) Frequency-domain and (c) time-domain measurement results for a A3 cavity, respectively [22]. Reprinted with permission. Copyright 2007 Optical Society of America

Based on the proposed structure, width-modulated line-defect microcavities in Si PCS with very thin barriers have been demonstrated despite the common belief that high-Q PC microcavities require relatively thick in-plane barriers [23]. Figure 14.19 provides the schematic of such a structure. The parameters are the same √ as those of A3 in Fig. 14.18 except that the line-defect width is now w = 0.98 3a. All the air holes outside the air slots are removed. In this design, q is the lateral thickness of the PC barrier and m is the length of the air slots. The holes labeled A, B, and C are shifted away from the line defect by 9, 6, and 3 nm, respectively, in order to form the cavity. The air holes in the outmost barrier line are cut in half at the center.

Fig. 14.19 (a) PC microcavity with a pair of air slots. The dashed rectangle in the center denotes the location of the microcavity. (b) Zoom-in of the cavity [23]. Reprinted with permission. Copyright 2008 American Institute of Physics

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Fig. 14.20 SEM photos and measured cavity resonances for a PCS cavity with thin barriers [23]. Reprinted with permission. Copyright 2008 American Institute of Physics

Figure 14.20b, c is SEM photos of fabricated devices showing also the corresponding transmission spectra. For q = 4, the Q factor is as high as 1.3 × 106 . For q = 3, the Q is still as high as 5.2 × 105 . These results prove that a thick PCS barrier or a nearly complete 2D PBG are not necessary in order to fabricate high-Q PCS cavities, with respect to the current design.

14.6 High-Q Surface-Mode Cavities Surface-mode cavities are a special kind of PC cavities, where light is confined at the interface between the PC and the dielectric. These open cavities are formed by terminating the two ends of the surface edge with reflecting mirrors. Surface-mode cavities can be considered as conventional Fabry–Pérot resonators whose resonant wavelengths are given by λn = 2π /kn , with the wave vector kn = (Nπ – ϕ)/L, where N is the mode number, L is cavity length, and ϕ is the phase shift associated with reflection from the boundary [24]. Consider a PC formed by a 2D triangular lattice of air holes patterned on a SOI structure. The height of the Si layer is 0.6a and the silica substrate is assumed to be of infinite thickness. Two reflecting mirrors are introduced by enlarging the radii of two semi-infinite lines of surface air holes from 0.3a to 0.31a, thus forming a cavity between them. Figure 14.21a, b shows the structure and the surface band structure of this surface-mode cavity. The shadowed regions are the projected band structure of the TE-like slab modes. It can be seen that such a surface structure supports one surface mode, the group velocity of which is nearly zero for the wave vector k0 = π /a. Figure 14.22 shows the resonance frequency and the Q factor of this resonant mode as a function of the cavity length. It is inferred that there is only one high-Q resonant mode as the cavity length varies and the resonant mode is related

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Fig. 14.21 (a) Schematic of a surface-mode cavity in a 2D SOI PC. The reflecting mirrors in the y-direction are introduced by enlarging the radius of the surface air holes outside the cavity. (b) Corresponding K-direction band structure [24]. Reprinted with permission. Copyright 2007 Optical Society of America

Fig. 14.22 Frequencies and Q factors of the resonant mode as a function of the cavity length. The microcavity is composed of a 2D SOI PC with air hole height of 0.6a [24]. Reprinted with permission. Copyright 2007 Optical Society of America

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to the same zero-group velocity surface mode. As the cavity length increases, the Q increases and the frequency of the resonant surface mode converges to ω0 = 0.2518(a/λ), corresponding to that of the nearly zero-group velocity surface mode of the PCS. For L = 12a, the Q factor is approximately 4.2 × 104 . A surface-mode cavity has been fabricated in an amorphous SOI structure [25]. Figure 14.23 shows the schematic of the surface-mode cavity, side coupled to a silicon wire waveguide. The height and the width of the Si waveguide are 280 and 425 nm, respectively. The lattice constant is a = 380 nm and the regular airhole diameter is r = 228 nm. The scattering loss of the reflecting mirrors in the x-direction is reduced by slightly enlarging the radii of six surface holes close to the boundary. The resonance wavelength is around 1580 nm. The extinction ratio of the resonance is larger than 10 dB and the intrinsic Qintr of the surface-mode cavity is approximately 2 × 103 .

Fig. 14.23 (a) SEM photo and (b) normalized transmission spectrum of the fabricated device described in the text [25]. Reprinted with permission. Copyright 2008 American Institute of Physics

Fig. 14.24 Normalized transmission spectra. (a) Broad transmission spectrum corresponding to a cavity length of 28a. (b) Transmission spectrum and fitting curve around one of the peaks of (a) [26]. Reprinted with permission. Copyright 2008 IEEE

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A surface-mode cavity has also been fabricated on a crystalline SOI structure [26]. Figure 14.24 shows the experimental results for the surface-mode resonance with a = 390 nm, r = 122 nm and a mirror with length L = 28a, and mirror hole radius r = 122 nm. After curve fitting, the Q is found to be about 6.9 × 103 and the intrinsic Qintr is 1.37 × 104 at λ = 1573.8 nm.

14.7 Dynamic Control of High-Q PC Cavities 14.7.1 Dynamic Control of the Resonance Frequency Tanabe T., et al. have predicted that the frequency of trapped photons can be tuned by changing the cavity resonance in a time scale, which is shorter than the photon lifetime of a small high Q cavity [27, 28, 29]. The experimental setup used for this adiabatic frequency shifting is shown in Fig. 14.25a. Figure 14.25b shows the SME photo of the corresponding PC cavity. The lattice constant is a = 420 nm, the slab thickness t = 204 nm, and the neighboring air holes have been shifted by 3, 6, and 9 nm, respectively. The input and output ports are line-defect waveguides with a width of w = 1.05a. The input waveform is an on-to-off step input signal with a central wavelength of 1608.34 nm. A 775 nm pump pulse with a width of 14 ps is injected onto the PCS immediately after the signal light is turned off. The carriers generated through the single-photon absorption effect reduce the refractive index of silicon by virtue of the free-carrier dispersion effect. When no pump pulse, the emission spectrum exhibits a single peak at the input wavelength as seen in Fig. 14.26a. However, when a pump pulse is injected, a new peak appears at a shorter wavelength (1608.19 nm). Without the pump pulse, the decaying waveform of the output signal yields a photon lifetime of 0.32 ns, which corresponds to a Q factor of 3.7 × 105 . With the pump pulse, the output power increases immediately about two times and releases a short pulse as shown in Fig. 14.26c. This is because the coupling quality factor Qc is dynamically tuned and reduced by a factor of 2.

Fig. 14.25 (a) Experimental setup. (b) SEM photo of a width-modulated line-defect silicon PC cavity [28]. Reprinted with permission. Copyright 2009 American Physical Society

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Fig. 14.26 (a) Time-resolved transmission spectra. The carrier lifetime obtained from the slow red-shift after application of the pump is 2 ns. (b) Spectrum obtained by taking a cross-sectional image along the wavelength axis of the 2D map at t = 0 and t = 200 ps as shown in (1) and (2). (c) On-to-off step input, output from the PC waveguide with and without a pump pulse [28]. Reprinted with permission. Copyright 2009 American Physical Society

14.7.2 Dynamic Control of Q Factor The dynamic control of Q factor within the photon lifetime can be of great significance. Figure 14.27 shows the system that was used by Tanaka et al. to demonstrate the dynamic control of the Q factor [30]. This system is composed of a high-Q microcavity, a waveguide, and a hetero-interface mirror. After the probe pulse is injected into the cavity, half of the light leaving the cavity travels in the left direction of the waveguide, while the other half travels in the right direction, and is then reflected backward by the hetero-interface mirror. The interference between these two paths determines the in-plane quality factor of the system Q// . The vertical quality factor Q⊥ is fixed and determined by the TIR mechanism. Fig. 14.27 Schematic of system used for the dynamic Q factor control of a microcavity [30]. Reprinted with permission. Copyright 2007 Macmillan Publishers Ltd: Nature

The in-plane quality factor is Q// in the absence of a mirror and is modified to Q// /(1 + cosθ ) in the presence of the mirror. The phase term is θ = 2βd + , where β is the propagation constant of the waveguide, d is the distance between the right edge of the waveguide and the point defect, and is the phase-shift generated by the reflection at the mirror [31]. Thus, the total quality factor (Qt ) of the system can

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be expressed by [31]: 1 + cos θ 1 1 = + Qt Q Q⊥

(14.16)

If Q⊥ >> Q// is satisfied, the total quality factor can be changed from Q// /2 to Q⊥ by varying θ from 0 to π . Since θ is related to β, it can be varied by changing the refractive index of the waveguide between the microcavity and the mirror. Injecting an optical pulse into the waveguide can lead to a change in the refractive index through the free-carrier dispersion effect. In the experiment, the pump and probe method was used to demonstrate dynamic control of Qt [30]. The microcavity comprises three missing air holes with two air holes at the cavity edges shifted by 0.15a. The calculated Q⊥ and Q// of this microcavity are 5.5 × 104 and 3 × 103 , respectively. A 4 ps pump pulse with a central wavelength 775 nm is injected onto the waveguide; at the same time, a 4 ps probe pulse with a central wavelength of 1550 nm is sent through the waveguide to the microcavity. The initial θ is set at zero giving the lowest Qt , (∼Q// /2) and the pump pulse energy is adjusted to set the final θ to be π , which gives the highest Qt (∼Q⊥ ). Figure 14.28a provides the output power from the microcavity to free space versus the time difference t between the arrival of the pump and probe pulses. Fig. 14.28b provides the spectra of the emitted probe pulse for different t. When the probe pulse arrives before the pump pulse (A), the corresponding spectrum has a line width of ∼0.50 nm, which corresponds to a Qt of 3 × 103 . In this case, Qt is fixed at the lowest possible value and the probe pulse interacts easily with the microcavity, as θ is set to be 0. When the probe and pump pulses arrive

Fig. 14.28 Experimental results of pump and probe measurements of dynamic Q factor control. (a) Output power of probe pulse emitted from the cavity to free space as a function of t. (b) Spectra of the emitted light for (A) t = −20 ps, (B) t = 20 ps, and (C) t = 0 ps [30]. Reprinted with permission. Copyright 2007 Macmillan Publishers Ltd: Nature

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simultaneously (C), the output power from the cavity into free space increases significantly and the corresponding line width reduces to 0.12 nm, corresponding to Qt ∼1.2 × 104 . When the probe pulse reaches the microcavity, it couples efficiently into the cavity since Q// is low. The pump pulse increases the Q// and thus prevents the leakage of the trapped probe pulse to the in-plane direction, leading to an increase of output power from the cavity into free space and spectral narrowing. When the probe pulse arrives after the pump pulse (B), the output power from the cavity into free space is quite small. This is because the pump pulse leads to a high Q// , making it difficult to couple the probe light into the cavity and thus resulting in the reduction of power transfer into free space.

14.8 Applications of High-Q PC Cavities 14.8.1 Optical Delay Slow light based on a single-photonic cavity offers a group delay as large as the photon lifetime while, at the same time, it allows to achieve zero-group velocity dispersion at the resonance frequency. Coupled-resonator optical waveguides (CROWs) provide the same feature with the only exception that the delay is multiplied by the number of the cavities [32]. Figure 14.29a shows a schematic of a PC

Fig. 14.29 (a) Design of a PC CROW based on width-modulated line-defects. (b), (c) Transmission spectra for N = 150 and 200, respectively [33]. Reprinted with permission. Copyright 2008 Macmillan Publishers Ltd: Nature

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CROW, consisting of width-modulated line-defect cavities [33]. The lattice constant is a = 420 nm and the slab thickness is t = 0.5a. The nearest-neighbor and second nearest-neighbor holes √ defect √ are shifted by 8 and 4 nm. The base width of a line for a cavity is 0.98 3a. The width of the input/output waveguides is 1.05 3a and the lengths involved are Lcc = 7a and Lcw = (3.5a, 5a, 6a). The cavity-waveguide coupling strength is varied by the cavity-waveguide distance Lcw . Figure 14.29b, c shows the transmission spectra for N = 150 and 200 cavities, respectively. Note that the transmittance is extremely high. Figure 14.30a shows the slowest pulse propagation through a sample with Lcc = 10a and N = 30, where the delay is 72 ps and the estimated vg is c/170. Figure 14.30b illustrates the result of slow-light propagation through a sample with Lcc = 10a and N = 150 with the largest fractional delay of 5, which is defined to be the ratio of delay with respect to the pulse width (D/W). The slow light based on PC cavities has a great advantage compared to that based on microring resonators in terms of size and footprint. IBM has demonstrated slow light based on ring-resonator CROWs, with N = 100, total length 1.9 mm, and a footprint of 5 × 10−2 mm2 [34]. Slow light based on PC cavity also has great advantages over that based on PC waveguide. The slowest vg obtained in experiments is c/25 for PC waveguides and c/42 for dispersion-compensated chirped PC waveguides [35, 36].

Fig. 14.30 (a) Pulse propagation at the lowest group velocity (N = 30). (b) Pulse propagation experiment with the longest delay (N = 150) [33]. Reprinted with permission. Copyright 2008 Macmillan Publishers Ltd: Nature

14.8.2 All-Optical Switching All-optical switching has been demonstrated by utilizing microring resonators [37, 38] and PC microcavities [39, 40]. The power required for switching is inversely proportional to the Q/V ratio. However, the switching speed is limited by the effective carrier lifetime. Figure 14.31 shows the switching behavior of a PC resonator upon the incidence of a 6 ps control pulse [39]. Clear all-optical switching from state OFF to ON (ON to OFF) for the detuning of 0.45 nm (0.01 nm) can be observed. The required switching energy is only a few hundred femtojoules and is much smaller than that of silicon all-optical switches based on microring cavities (0.9 pJ) [38]. The

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Fig. 14.31 All-optical switching in a silicon PC microcavity realized by carrier-plasma nonlinearity induced by two-photon absorption in silicon. (a) ON-to-OFF switching (b) OFF-to-ON switching [39]. Reprinted with permission. Copyright 2008 American Institute of Physics

switching speed can be improved by ion implantation. Indeed, when a high dose of Ar+ (2.0 × 1014 cm−2 ) is introduced, the switching time (∼70 ps) is three times smaller than that without ion implantation (∼220 ps), while the required operating energy remains almost the same (<100 fJ).

14.8.3 Channel-Drop Filters Over the past few years, there have been different designs of channel-drop filters in 2D PCS. One design is the surface-emitting type, which consists of a waveguide coupled to a cavity [41, 42]. Input signal launched in the waveguide tunnels into the cavity and is emitted in the vertical direction. Though in principle all light can be dropped in the vertical direction by using a cavity system with two degenerate modes, this design poses a real challenge while collecting light vertically. Another design involves two waveguides (bus and drop) and a cavity system in a plane [43, 44]. The channel to be selected comes from the bus waveguide, tunnels into the cavity and is eventually transferred to the drop waveguide [45]. Though there have been designs of cavities in 2D PCS with Q⊥ larger than 106 , there is a tremendous deterioration when they are used in an in-plane channel-drop filter system. The reason

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lies in the presence of the waveguides. When the waveguides are placed close to the cavity, the boundaries of the cavity are changed and so is the k-space distribution of the cavity mode. As a result, more k components are lying inside the light cone and thus Q⊥ is severely reduced [46]. An in-plane channel-drop filter in a 2D silicon PCS with a single cavity that supports two high-Q modes has been proposed in [46]. The two cavity modes (even and odd) have to be tuned into exact frequency degeneracy with similar Q⊥ and Q// . The configuration of the filter is shown in Fig. 14.32a. The cavity is formed by a central large air hole with radius R0 , surrounded by four periods of air holes with radii decreasing along the outward direction, denoted by R1 , R2 , R3 , and R4 , respectively. The parameters are as follows: a = 420 nm, air-hole radius R = 0.30a, slab thickness t = 0.6a, R0 = 0.38a, R1 = 0.372a, R2 = 0.364a, R3 = 0.34a, R4 = R. The two modes have very close resonance wavelengths λ1 =1554.51 nm, λ2 = 1554.52 nm as well as Q factors, Q1 = 3.06 × 103 and Q2 = 3.02 × 103 . Note that the separate Q factors due to vertical and in-plane radiation are Q⊥1 = 4.05 × 104 , Q//1 = 3.1 × 103 for the even mode and Q⊥2 = 3.5 × 104 , Q//2 = 3.05 × 103 for the odd mode. The transmission spectrum is shown in Fig. 14.32b. At resonance, 78% of light power is transferred along the forward direction of the drop waveguide, 1.75% is transferred along the backward direction of the drop waveguide, and 1.6% remains in the bus waveguide. The Q factor measured from the full width at half maximum (FWHM) of the transmission spectrum is around 3 × 103 . Figure 14.32c, d shows the Hz field distributions at the central slab plane for the even and odd modes, respectively. By carefully tuning the boundary of the drop waveguide, the light remaining in the bus can be further reduced to below 0.4% and thus raise the channel isolation over 22 dB. Figure 14.33 shows the steady-state field distribution (Hz ) at the resonant frequency after fine-tuning the drop waveguide boundary. In-plane multi-channel-drop filters based on HS PCS have also been proposed and demonstrated experimentally [47–49]. Figure 14.34a shows the schematic of such a proposed filter. This device consists of multiple PCSs, each of them has a different lattice constant and resonance wavelength. The HS interfaces between two adjacent PCSs act as wavelength-selective mirrors and reflect back the light that is at resonance with the corresponding cavity. When multi-wavelength light is incident on this device through the input waveguide, the light that is resonant with the pointdefect cavity of each PCS is trapped by that cavity and then dropped to a separate output waveguide. The non-resonant light is directed to the next PCS. In experiment, an in-plane four-channel-drop filter has been demonstrated in a 2D silicon PCS. The device consists of a total of five PCSs, named PC1–PC5, each with a different lattice constant: a1 = 420 nm, a2 = 415 nm, a3 = 410 nm, a4 = 405 nm, and a5 = 400 nm. Figure 14.34b shows the drop spectra for fourchannel-drop operation with the drop wavelengths located at 1516, 1536, 1559, and 1583 nm, respectively. The corresponding quality factors are almost the same (Qt = 900−1000) [49].

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Fig. 14.32 (a) Top view of the channel-drop system. The selected channel is transferred along the forward direction of the drop waveguide. (b) Intensity spectra. (c), (d) Hz field distributions at the central slab plane for the even and odd mode, respectively [46]. Reprinted with permission. Copyright 2005 Optical Society of America

Fig. 14.33 Steady-state Hz field oscillation at resonant frequency for the structure shown in Fig. 14.32a after carefully tuning the boundary of the drop waveguide [46]. Reprinted with permission. Copyright 2005 Optical Society of America

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14.8.4 PC Lasers There is a widespread consensus that silicon processing technology is superior to that of III–V semiconductors and therefore there is little chance that PCs based on III–V semiconductors will approach the state-of-the-art. On the other hand, III–V compounds (GaAs or InP) are the materials of choice for active devices such as lasers or coupled-cavity emitter systems. An experimental Q factor of 2.5 × 105 has been reported in a 2D GaAs PCS by using a HS line-defect cavity [50]. Several PC lasers have been demonstrated based on the InGaAs/InP material system, where the PC resonator can provide both a high-Q factor and a small modal volume, which can significantly decrease the lasing threshold [51]. We will here briefly present several kinds of PC resonators exploited for realizing PC lasers. Figure 14.35 shows the experimental setup of an electrically driven, single-mode, low-threshold current (∼260 μA) PBG laser operating at room temperature [51]. As shown in Fig. 14.35a, a sub-micrometer-sized semiconductor post is placed at the center of the single-cell PC resonator. Electrons are supplied laterally from the top electrode, whereas holes are injected directly through the bottom post. Figure 14.35b shows the cavity configuration. The modified single-cell PC cavity is surrounded by five heterogeneous PC lattices with the same lattice constant but different air-hole sizes in order to improve the position and size of the central InP post. The lattice constant is a = 510 nm and the radii of the air holes in regions I, II, III, IV, and V are

Fig. 14.34 (a) Schematic of a multi-channel-drop filter based on HS PCSs. (b) Drop spectra of the fabricated multi-wavelength drop filter [49]. Reprinted with permission. Copyright 2006 Optical Society of America

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Fig. 14.35 (a) Schematic diagram of current injection. (b) Top view of fabricated sample. (c) Typical P–I curve of the monopole-mode laser. The threshold current is ∼260 μA and the output power indicates the peak value measured at the spectrometer. The inset is the spectrum at 700 μA [51]. Reprinted with permission. Copyright 2004 AAAS

0.28a, 0.35a, 0.385a, 0.4a, and 0.41a, respectively. The measured Q factor for the monopole mode of a cold cavity as estimated from the spectral line width associated with a transparent current of ∼225 μA, is ∼2500. Room temperature, optically driven continuous-wave operation and controlled spontaneous emission has also been demonstrated in ultra-small PC nanolasers [52].

Fig. 14.36 CW lasing characteristic of L0 nanolaser. (a) SEM photo of fabricated device. (b) Calculated Hz field distribution. (c) Intensity characteristic and lasing spectrum above the lasing threshold. (d)–(f) Corresponding information for the L1 nanolaser [36]. Reprinted with permission. Copyright 2007 Optical Society of America

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Figure 14.36a, b shows the top view of the L0 nanolaser and its modal distribution (the L0 cavity has been detailed in Section 14.4.3). The L0 nanolaser exhibits a clear threshold of Peff = 1.2 μW, above which the single-mode spectrum of the monopole mode has an extinction of 40 dB and spectral width ( λ) of 0.06 nm. At 0.8 times the threshold, λ = 0.08 nm and the corresponding Q is 2 × 104 . Figure 14.36c, d shows the corresponding results for the L1 nanolaser. In this case, the diameter of the innermost holes is 85% of that of the outer holes. The L1 nanolaser exhibits a threshold of Peff = 2.4 μW and a dipole-mode operation with an extinction of 25 dB. In the transparent state, λ is estimated to be 1.0 nm and the corresponding Q is 1.5 × 103 .

14.9 Conclusion This chapter has summarized the state-of-the-art technology of high-Q PC microcavities. The key to the design is the reduction of radiation loss through the reduction of the spatial Fourier components lying above the light line of the PCS. For a practical design, the envelope of the localized field distribution should be as close as possible to the Gaussian function. The Q factor can be measured both in the spectral as well as in the time domain. To date, the maximum experimental Q factor is above 2 × 106 and has been realized in HS line-defect PC cavities. The minimum mode volume is 0.29(λ0 /n)3 and has been realized by the L0 cavity. The cavity characteristics including the photon frequency inside the cavity and the Q factor can be dynamically tuned in a time scale shorter than the photon lifetime. High-Q PC resonators find useful applications as multi-functional devices, ranging from passive devices such as optical buffers, channel-drop filters, optical switches, sensors, to active devices such as low-threshold lasers, single-photon emitters, and many others. Applications in other fields of science also benefit from high-Q PC cavities.

References 1. Srinivasan, K., Painter, O. Momentum space design of high-Q photonic crystal optical cavities. Opt. Express 10, 670–684 (2002) 2. Srinivasan, K., Painter, O. Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals. Opt. Express 11, 579–593 (2003) 3. Vuˇckovi´c, J., Lonˇcar, M. et al. Optimization of the Q factor in photonic crystal microcavities. IEEE J. Quant. Electron. 38, 850–856 (2002) 4. Tanabe, T., Notomi, M., et al. Measurement of an ultra-high-Q photonic crystal microcavity using a single-side-band frequency modulator. Electron. Lett. 43, 187–188 (2007) 5. Tanabe, T., Notomi, M., et al. Large pulse delay and small group velocity achieved using ultrahigh-Q photonic crystal nanocavities. Opt. Express 15, 7826–7839 (2007) 6. Armani, D., Kippenberg, T., et al. Ultra-high-Q toroid microcavity on a chip. Nature 421, 925–928 (2003) 7. Foresi, J.S., Villeneuve, P.R., et al. Photonic-bandgap microcavities in optical waveguides. Nature 390, 143–145 (1997)

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8. Zain, A.R.M., Johnson, N.P., et al. Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in SOI. Opt. Express 16, 12084–12089 (2008) 9. Deotare, P.B., McCutcheon, M.W., et al. High quality factor photonic crystal nanobeam cavities. Appl. Phys. Lett. 94, 121106 (2009) 10. Akahane, Y., Asano, T., et al. High-Q photonic microcavity in a two-dimensional photonic crystal. Nature 425, 944–947 (2003) 11. Akahane, Y., Asano, T., et al. Fine-tuned high-Q photonic-crystal micro-cavity. Opt. Express 13, 1202–1214 (2005) 12. Ryu, H.Y., Notomi, M., et al. High quality-factor and small mode-volume hexapole modes in photonic crystal slab nano-cavities. Appl. Phys. Lett. 83, 4294–4296 (2003) 13. Kim, G.H., Lee, Y.-H., et al. Coupling of small, low-loss hexapole mode with photonic crystal slab waveguide mode. Opt. Express 12, 6624–6631 (2004) 14. Tanabe, T., Shinya, A., et al. Single point defect photonic crystal microcavity with ultrahigh quality factor achieved by using hexapole mode. Appl. Phys. Lett. 91, 021110 (2007) 15. Ryu, H.-Y., Notomi, M., et al. High quality-factor whispering-gallery mode in the photonic crystal hexagonal disk cavity. Opt. Express 12, 1708–1719 (2004) 16. Zhang, Z., Qiu, M. Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs. Opt. Express 12, 3988–3995 (2004) 17. Song, B.-S., Noda, S., et al. Ultra-high-Q photonic double-heterostructure microcavity. Nature Mater. 4, 207–210 (2005) 18. Asano, T., Song, B.-S., et al. Analysis of the experimental Q factors (∼1 million) of photonic crystal nanocavities. Opt. Express 14, 1996–2002 (2006) 19. Takahashi, Y., Hagino, H., et al. High-Q nanocavity with a 2-ns photon lifetime. Opt. Express 15, 17206–17213 (2007) 20. Tanaka, Y., Asano, T., et al. Design of photonic crystal nanocavity with Q-factor of ∼109 . J. Lightwave Technol. 26, 1532–1539 (2008) 21. Kuramochi, E., Notomi, M., et al. Ultrahigh-Q photonic crystal nano-cavities realized by the local width modulation of a line defect. Appl. Phys. Lett. 88, 041112 (2006) 22. Notomi, M., Tanabe, T., et al. Nonlinear and adiabatic control of high-Q photonic crystal nanocavities. Opt. Express 15, 17458–17481 (2007) 23. Kuramochi, E., Taniyama, H., et al. Ultrahigh-Q two-dimensional photonic crystal slab nanocavities in very thin barriers. Appl. Phys. Lett. 93, 111112 (2008) 24. Xiao, S., Qiu, M. Optical microcavities based on surface modes in two-dimensional photonic crystals and silicon-on-insulator photonic crystals. J. Opt. Soc. Am. B, 24, 1225–1229 (2007) 25. Zhang, Z., Dainese, M., et al. Optical filter based on two-dimensional photonic crystal surfacemode cavity in amorphous silicon-on-silica structure. Appl. Phys. Lett. 90, 041108 (2007) 26. Jing, W., Min, Q. High-Q optical filter based on photonic crystal surface-mode microcavity. 2nd IEEE LEOS Winter Topicals, WTM 2009, art. no. 4771635, pp. 22–23 27. Tanabe, T., Notomi, M., et al. Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal microcavity. Nat. Photonics 1, 49–52 (2007) 28. Tanabe, T., Notomi, M., et al. Dynamic release of trapped light from an ultrahigh-Q microcavity via adiabatic frequency tuning. Phys. Rev. Lett. 102, 043907 (2009) 29. Notomi, M., Mitsugi, S. Wavelength conversion via dynamic refractive index tuning of a cavity. Phys. Rev. A 73, 051803 (2006) 30. Tanaka, Y., Upham, J., et al. Dynamic control of the Q factors in a photonic crystal microcavity. Nature Mater. 6, 862–865 (2007) 31. Song, B.-S., Asano, T., et al. Role of interfaces in heterophotonic crystals for manipulation of photons. Phys. Rev. B 71, 195101 (2005) 32. O’Brien D., Settle M. D., et al. Coupled photonic crystal heterostructure nanocavities. Opt. Express 15, 1228–1233 (2007) 33. Notomi, M., Kuramochi, E., et al. Large-scale arrays of ultrahigh-Q coupled nanocavities. Nature Photon. 2, 741–747 (2008)

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34. Xia, F.N., Sekaric, L., et al. Ultracompact optical buffers on a silicon chip. Nature Photon. 1, 65–71 (2007) 35. Settle, M.D., Engelen, R.J.P., et al. Flatband slow light in photonic crystals featuring spatial pulse compression and terahertz bandwidth. Opt. Express 15, 219–226 (2007) 36. Baba, T., Kawasaki, T., et al. Large delay-bandwidth product and tuning of slow light pulse in photonic crystal coupled waveguide. Opt. Express 16, 9245–9253 (2008) 37. Almeida, V.R., Barrios, C.A., et al. All-optical control of light in a silicon chip. Nature 431, 1081–1084 (2004) 38. Almeida, V.R., Barrios, C.A., et al. All-optical switching on a silicon chip. Opt. Lett. 29, 2867–2869 (2004) 39. Tanabe, T., Nishiguchi, K., et al. Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities. Appl. Phys. Lett. 90, 031115 (2007) 40. Tanabe, T., Notomi, M., et al. All-optical switches on a silicon chip realized using photonic crystal nanocavities. Appl. Phys. Lett. 87, 151112 (2005) 41. Chutinan, A., Mochizuki, M., et al. Surface-emitting channel drop filters using single defects in two-dimensional photonic crystal slabs. Appl. Phys. Lett. 79, 2690–2692 (2001) 42. Min, B.-K., Kim, J.-E., et al. High-efficiency surface-emitting channel drop filters in twodimensional photonic crystal slabs. Appl. Phys. Lett. 86, 11106/1–3 (2005) 43. Qiu, M., Jaskorzynska, B. A design of a channel drop filter in a two-dimensional triangular photonic crystal. Appl. Phys. Lett. 83, 1074–1076 (2003) 44. Qiu, M. Ultra-compact optical filter in two-dimensional photonic crystal. Electron. Lett. 40, 539–540 (2004) 45. Manolatou, C., Khan, M.J., et al. Coupling of modes analysis of resonant channel add-drop filters. IEEE J. Quant. Electron. 35, 1322–1331 (1999) 46. Zhang, Z., Qiu, M. Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs. Opt. Express 13, 2596–2604 (2005) 47. Song, B.-S., Asano, T., et al. Transmission and reflection characteristics of in-plane heterophotonic crystals. Appl. Phys. Lett. 85, 4591–4593 (2004) 48. Takano, H., Song, B.-S., et al. Highly efficient in-plane channel drop filter in a twodimensional heterophotonic crystal. Appl. Phys. Lett. 86, 241101 (2005) 49. Takano, H., Song, B.-S., et al. Highly efficient multi-channel drop filter in a two-dimensional hetero photonic crystal. Opt. Express 14, 3491–3496 (2006) 50. Weidner, E., Combrí, S. Achievement of ultrahigh quality factors in GaAs photonic crystal membrane micro-cavity. Appl. Phys. Lett. 89, 221104 (2006) 51. Park, H.-G., Kim, S.-H., et al. Electrically driven single-cell photonic crystal laser. Science 305, 1444–1447 (2004) 52. Nozaki, K., Kita, S., et al. Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser. Opt. Express 15, 7506–7514 (2007)

Chapter 15

Radial Bragg Resonators Jacob Scheuer and Xiankai Sun

Abstract Circular resonators are promising candidates for a wide range of applications, ranging from optical communication systems through basic research involving highly confined fields and strong photon–atom interactions to biochemical and rotation sensing. The main characteristics of circular resonators are the Q factor, the free spectral range (FSR), and the modal volume, where the last two are primarily determined by the resonator radius. The total internal reflection (TIR) mechanism used for guidance in “conventional” resonators couples these attributes and limits the ability to realize compact devices exhibiting large FSR, small modal volume, and high Q. Recently, a new class of annular resonator, based on a single defect surrounded by radial Bragg reflectors, has been proposed and analyzed. The radial Bragg confinement decouples the modal volume from the Q and paves the way for the realization of compact, low-loss resonators. These properties as well as the unique mode profile of these circular Bragg nanoresonators (CBNRs) and nanolasers (CBNLs) make the devices within this class an excellent tool to realize nanometer scale semiconductor lasers and ultrasensitive detectors, as well as to study nonlinear optics.

15.1 Introduction Circular resonators, ring and disk shaped, are key components in the realization of numerous basic devices used in advanced optical communication systems. During the last decade, numerous circular resonator-based applications such as lasers, filters [1], add/drop multiplexers [2], modulators [3], and delay lines [4] have been suggested and demonstrated. In addition, the applicability of

X. Sun (B) Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA e-mail: [email protected]

I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_15,  C Springer Science+Business Media, LLC 2010

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circular resonators was shown to extend beyond telecommunication, in particular to the fields of sensing [5], spectroscopy, and biophotonics [6], as well as to basic research in quantum electrodynamics, nonlinear optics, and other related fields [7]. For many of these applications, the circular resonator is required to provide low losses or, equivalently, a high quality factor (Q). In addition to high Q, small resonator dimensions are also desirable or, equivalently, that the resonator exhibits a large free spectral range (FSR). Unfortunately, for conventional resonators, which utilize total internal reflection (TIR) as the radial confinement mechanism, these requirements are contradictory. To exhibit large FSR, a circular resonator is required to have a short circumference, i.e., a small bending radius. As a result the efficiency of the TIR confinement mechanism is significantly impaired, leading to larger power dissipation and lower Q factor [8]. Recently, we have proposed utilizing Bragg reflection instead of TIR as the radial confinement mechanism in order to break the linkage between the FSR and loss and thus facilitate low-loss, large-FSR circular resonators [9–11]. This concept is illustrated in Fig. 15.1. A pillar (Fig. 15.1a) or an annular defect (Fig. 15.1b) is located within a medium which consists of annular Bragg layers. The confinement of the modal field in the disk (Fig. 15.1a) or within the defect (Fig. 15.1b) is accomplished by Bragg reflection. Unlike conventional resonators, the reflectivity of the Bragg mirror can be increased simply by adding more layers. As a result, the radius of the disk (or the annular defect) can be reduced almost arbitrarily without inducing higher bending losses. Such microcavities employing distributed Bragg reflection as the transverse confinement mechanism, e.g., photonic crystal (PC) defect cavities [12, 13] and annular Bragg resonators [9–11], exhibit unique characteristics (e.g., mode profile, dispersion relation) which are advantageous for numerous applications as discussed in the following sections.

Fig. 15.1 SEM images of (a) a disk CBNR and (b) a ring CBNR [50]. Reprinted with permission. Copyright 2009 Springer Science and Business Media

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15.2 Theoretical Framework We consider a radially symmetric structure as illustrated in Fig. 15.2. The guiding defect, consisting of a material of refractive index ndefect , is surrounded by distributed Bragg reflectors on both sides, where the reflectors’ layers are of refractive indices n1 and n2 . All electromagnetic field components can be expressed in terms of the z component of the electric and magnetic fields [14]. These components satisfy the scalar Helmholtz equation which in cylindrical coordinates is given by  

  2 ∂ ∂2 Ez 2 ∂ 2 2 2 + k (ρ) − β ρ + ρ = 0, +ρ 2 2 H ∂ρ ∂ρ ∂φ z

(15.1)

where ρ and φ are the radial and azimuthal coordinates, respectively, β is the propagation coefficient in the z direction, k(ρ) = k0 ·n(ρ), k0 is the wavenumber in vacuum, and the refractive index n(ρ) equals ndefect , n1 , or n2 according to the radius ρ. The general solution of (15.1) can be expressed by a superposition of the Bessel functions of the first and second kind: $ $ (  ) Ez = AJm kj2 − β 2 ρ + BYm kj2 − β 2 ρ cos (βz) exp (imφ), $ $ (  ) Hz = CJm kj2 − β 2 ρ + DYm kj2 − β 2 ρ sin (βz) exp (imφ),

(15.2)

where kj is the material wavenumber in the jth layer and m is the azimuthal wavenumber. The other field components can easily be derived from Ez and Hz [14]. Introducing (15.2) into (4–7) of [14] yields all the field components in the jth layer.

Fig. 15.2 Illustration of a ring-shaped CBNR structure. Reprinted from [9]

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The field components Ez , Hz , Eφ , and Hφ must be continuous at the interfaces separating successive layers. This requirement can be written in the form of a transfer matrix, connecting the amplitude vector [A B C D]T in the jth and (j + 1)th layers: ⎡ ⎤ ⎤ A A ⎢B⎥ ⎢B⎥ −1 ⎢ ⎥ ˜ (ρj+1 ) · M ˜ j (ρj ) · ⎢ ⎥ , =M j+1 ⎣C⎦ ⎣C⎦ D j D j+1 ⎡

(15.3)

˜ j is given by where M ⎛ ⎜ ⎜ ˜ Mj = ⎜ ⎜ ⎝

J(γj ρ)

n2j γj

Y(γj ρ)

n2j

J (γj ρ)

γj

0 2 J(γj ρ)

mβ ρωμγj

0

Y (γj ρ) 0 2 Y(γj ρ)

mβ ρωμγj

0

⎞

⎟ mβ J(γj ρ) mβ 2 Y(γj ρ) ⎟ ρωε0 γj2 ρωε0 γj ⎟ J(γj ρ) 1 γj J (γj ρ)

⎟, Y(γj ρ) ⎠ 1 Y (γ ρ) j γj

(15.4)

ε and μ are the dielectric and magnetic susceptibilities, ω is the optical angular frequency, and the primes indicate derivatives with respect to the function argument. Assuming strong vertical confinement (i.e., β  kj ) the modal field solutions can be separated into two distinct polarizations: TE consisting of Hz , Eρ , and Eφ and TM consisting of Ez , Hρ , and Hφ . The z component of the relevant field for each polarization can be described by two coefficients in each layer, namely Aj and Bj for TM and Cj and Dj for TE. For each polarization, the boundary conditions at the interfaces between successive layers can be represented by a simpler 2 × 2 transfer matrix  ˜ jTM = M

J(γj ρ)

Y(γj ρ)

n2j n2j γj J (γj ρ) γj Y (γj ρ)



 ,

˜ jTE = M

J(γj ρ) Y(γj ρ) 1 1 γj J (γj ρ) γj Y (γj ρ)

 .

(15.5)

Relation (15.3) and the transfer matrices (15.5) can be used to “propagate” the field components from the inner layers to the outer layers. The boundary condition at ρ = 0 is the finiteness of the field so that B1 = D1 = 0. The second boundary condition is that there is no inward-propagating field beyond the last layer, so that BN+1 = iAN+1 for TM and DN+1 = iCN+1 for TE, where N is the number of layers.

15.3 Design Rules The transfer matrix formalism enables us to find the modal field profile in the case of an arbitrary arrangement of annular concentric dielectric rings. However, we are especially interested in structures that can confine the modal energy near a predetermined radial distance, i.e., within the defect.

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It has been shown that the best strategy to attain an exponential decrease (or increase) in the field intensity in the grating region is to position the interfaces of the layers at the zeros and extrema of the z component of the field [14]. The index profile and the field are calculated simultaneously, using the index to find the field and the field to determine the position of the interfaces. The resulting optimal index profile exhibits an inherent resemblance to the conventional (Cartesian) Bragg reflector. The optimal Cartesian Bragg reflector can be designed in a similar way, leading to layers that are quarter-wavelength thick [14]. Here the resulting layers are also “quarter-wavelength” thick but in the sense of the quasi-periodicity of the mth-order Bessel function. The defect (again, as in the Cartesian case) should be “half-wavelength” wide, i.e., its interfaces should be located at successive zeros of the field. In order to attain a transverse field profile which is confined within the defect, the profile must exponentially increase for ρ < ρ def and exponentially decrease for ρ > ρ def . This requirement determines which index interfaces (low → high or high → low) should be positioned at zeros of the field and which at the extrema of the field. The constraints on the index profile are similar to the Cartesian case and differ for the TE and TM polarizations. For the TE polarization, the interfaces for decreasing (increasing) field should be at the zeros (extrema) of Hz if n(ρ − ) > n(ρ + ) at the interface and at the extrema (zeros) of Hz if n(ρ − ) < n(ρ + ) at the interface. For the TM polarization, the interfaces for decreasing (increasing) field should be at the extrema (zeros) of Ez if n(ρ − ) > n(ρ + ) at the interface and at the zeros (extrema) of Ez if n(ρ − ) < n(ρ + ) at the interface. The interfaces of the defect must be located at zeros of Hz for TE and of Ez for TM polarization. Figure 15.3a depicts the refractive index and the TE modal field profile of a CBNR designed for a 0.55-μm-thick InGaAsP layer suspended in air. The device is designed to have a mode with an angular propagation coefficient of m = 7 at λres = 0.852 μm. The effective index approximation in the vertical dimension is used to reduce the 3D problem to an equivalent 2D one. As can be seen in the figure, the field is primarily confined in the defect and it decays while oscillating in the Bragg reflectors. While the overall envelope of the radial modal profile (averaging over the oscillations) is quite similar to that of a conventional ring resonator, the oscillatory behavior of the field in the reflector region is the fingerprint of the distributed feedback mechanism. Figure 15.3b shows the widths of the Bragg layers as a function of the layer number of the CBNR depicted in Fig. 15.3a. There are two notable properties of the Bragg layers: (1) the width of the high-index layers is smaller than the width of the low-index layers and (2) the width of the layers decreases exponentially as a function of the radius, converging asymptotically to a constant value. The first property exists in conventional Bragg reflectors as well and stems from the dependence of the spatial oscillation period, or the “wavelength,” on the index of refraction. The second property is unique to the cylindrical geometry and arises from the unperiodic nature of the solutions of the wave equation (Bessel or Hankel functions) in this geometry.

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Fig. 15.3 (a) Refractive index and TE-polarized modal field profile of a CBNR designed for m = 7, λres = 0.852 μm, with 5 internal and 10 external Bragg layers. (b) A non-periodic distribution of high- and low-index material is required in the CBNR [50]. Reprinted with permission. Copyright 2009 Springer Science and Business Media

At large radii the widths of the Bragg layers converge asymptotically to the conventional (Cartesian) quarter-wavelength condition. Mathematically, this can be explained by noting that for large radii the Bessel function can be approximated by a sinusoidal function divided by square root of the radius. From the physical point of view, at large radii the curvature of the Bragg layers is small and, therefore, the structure of a radial reflector at large radii is similar to that of a Cartesian reflector.

15.4 Advantages and Applications The Bragg confinement mechanism allows great flexibility in engineering the radial mode profile. For example, it is possible to design a resonator in which the light is confined within an annular defect composed of low refractive index material or even air. By contrast, this would be impossible in the case of conventional TIRbased resonators. Such configuration is useful for sensing applications as discussed below. The Bragg reflection concept allows one to tailor the reflector structure to obtain a desired radial field profile and decouple between the modal volume (or cavity dimensions) and the bending losses. In fact, the bending losses can be reduced

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almost arbitrarily, without significantly changing the modal volume, simply by adding more layers to the (external) Bragg reflector. Therefore, resonators exhibiting both high Q and small modal volume can be engineered and their modal field profile can be tailored to specific applications. Another interesting and important property of the CBNR structure is that it supports slow-light propagation along the circumference of the defect. Intuitively, this property can be understood as follows. Because the Bragg reflector does not require the incidence angle of the optical wave to be larger than some “critical” angle (as in TIR-based devices), it is possible to design a CBNR in which the incidence angle of the wave is almost perpendicular to the Bragg layers. The angular component of the wavevector in this case is very small and thus the angular propagation velocity is very low. The slower group velocity is formed by the fact that the dispersion curve becomes flatter for lower angular numbers – a property that has been shown and explored in [9]. Slow whispering gallery modes have several advantages, especially for applications requiring strong atom–field interactions. For a given power level in the guiding channel, the corresponding intensity of the electric field is inversely proportional to the group velocity |E|2 ∼ P/vg . Therefore, such modes can be exploited to attain large gain (if the device is a laser), large and controllable dispersion, and enhanced nonlinear response. More specifically, the unique modal field profile and characteristics of the CBNR structure are advantageous not only for surface emitting lasers but also for biochemical and rotation sensing applications and for studies involving strong atom–field interactions such as nonlinear optics and cavity quantum electrodynamics.

15.4.1 Surface Emitting Lasers Surface emitting circular grating lasers with large emission apertures have attracted considerable attention because of their capabilities in producing circularly symmetric, narrow-divergence laser beams [15–19]. With optical gain, radial Bragg resonators (RBRs) can easily be designed for surface emitting laser applications to efficiently emit light perpendicular to the surface, similar to vertical cavity surface emitting lasers (VCSELs), by employing a second-order Bragg reflection scheme. In the Bragg reflectors, the first Fourier harmonic of the structure reflects the incident wave in the vertical (out-of-plane) direction while the second Fourier harmonic is responsible for the in-plane reflection. Based on the various occasions and requirements for different applications, these lasers can be designed to have a compact or a broad emission aperture. Therefore, CBML (circular Bragg microlaser), which will be used mostly in this section, refers to the same structure as CBNR or CBNL and these terms are interchangeable. We primarily consider two CBML configurations: disk- and ring CBMLs. In order to demonstrate their advantages as surface emitting lasers, the conventional configuration of circular DFB laser is also studied as a comparison. Figure 15.4 is an illustration of these structures. In these lasers, the gratings serve two purposes:

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Fig. 15.4 Surface emitting circular grating lasers: (a) disk CBML; (b) ring CBML; and (c) circular DFB laser. Laser radiation is coupled out of the plane in vertical direction via the gratings [23]. Reprinted and modified with permission. Copyright 2008 Optical Society of America

first providing feedback for the in-plane fields to form a radial resonator and second acting as a broadside output coupler. In this section, we consider the case of shallow circular gratings and thus can introduce the coupled mode theory for the modal analysis. The coupled-wave equations are solved under two typical circular boundary conditions obtaining two types of reflectivities for the circular Bragg gratings, which are employed to derive the laser oscillation conditions. Then the modal properties of disk- and ring CBMLs and circular DFB lasers are obtained, compared, and discussed.

15.4.1.1 Resonance (Laser Oscillation) Conditions With inclusion of the effect of resonant vertical radiation, a comprehensive coupled mode theory for the circular grating structures in active media was developed in [20]. The vertical radiation is incorporated into the coupled in-plane wave equations by a numerical Green’s function method. The in-plane (vertically confined) electric (1) (2) (1) (2) field is expressed as E(x) = A(x)Hm (x) + B(x)Hm (x) where Hm (x) and Hm (x) are the mth-order Hankel functions representing the in-plane outward- and inwardpropagating cylindrical waves. A set of evolution equations for the amplitudes A(x) and B(x) is obtained [20]: 9 dA(x) dx dB(x) dx

= uA(x) − vB(x)e2iδx = −uB(x) + vA(x)e−2iδx .

(15.6)

In the above equations, x is the normalized radius x = βρ where β here denotes the in-plane propagation constant and ρ is the radial coordinate. δ = (β design –β)/β denotes the normalized frequency detuning factor, which is the relative frequency shift of a resonant mode from the designed value. The coefficients u and v are defined as u = gA − h1 , v = h1 + ih2 , where gA is the normalized gain coefficient. The minimum value of gA required to achieve laser action will be determined by the resonance conditions; h1 and h2 are, respectively, the radiation- and feedbackcoupling coefficients which can be precalculated based on the wafer layer structure and the circular grating profile [20].

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The radiation field is associated with the in-plane fields by the relation   (1)  

E(x,z) = s1 (z)A(x,z)e−iδx + s−1 (z)B(x,z)eiδx Hm (x) ,

(15.7)

where s±1 can be calculated by an integral involving the numerical Green’s function and the Fourier component of the grating profile. A and B in (15.7) are z dependent since they also contain the vertical modal profile determined by the wafer layer structure. The radiation pattern at the emission surface (known as “near-field”) is

E(x, z = 0) if the surface plane of the laser resonator is set to be z = 0. It should be noted that (15.6) implicitly includes the effect of vertical radiation due to h1 . When obtaining the thresholds for the laser modes, vertical radiation can be treated as a loss term [20]. Solved from (15.6), A(x), B(x), and E(x) have the following analytic expressions: sinh [S(x − L)] + C cosh [S(x − L)] , − sinh [SL] + C cosh [SL] A(0)e−iδx [(u − iδ) − CS] sinh [S(x − L)] + [C(u − iδ) − S] cosh [S(x − L)] B(x) = , v − sinh [SL] + C cosh [SL]

A(x) = A(0)eiδx

(1)

(2)

E(x) = A(x)Hm (x) + B(x)Hm (x), (15.8) , where S ≡ (u − iδ)2 − v2 . C is a constant to be determined by specific boundary conditions, and L is a normalized length parameter. The determination of the constant C in (15.8) requires the specific boundary conditions to be applied. We investigate two typical boundary conditions to obtain C and the corresponding field reflectivity in each case. 1. As shown in Fig. 15.5a, the grating extends from the center x = 0 to x = L. An inward propagating wave with amplitude B(L) impinges from outside on the grating. The reflectivity is defined as r1 (L) = A(L)/B(L). In this case the boundary sinh [SL]+S cosh [SL] condition A(0) = B(0) leads to C = (u−v−iδ) S sinh [SL]+(u−v−iδ) cosh [SL] and to r1 (L) =

(u − v − iδ) sinh [SL] + S cosh [SL] A(L) = e2iδL . B(L) −(u − v − iδ) sinh [SL] + S cosh [SL]

(15.9)

2. As shown in Fig. 15.5b, the grating extends from x = x0 to x = L. An outward propagating wave with amplitude A(x0 ) impinges from inside on the grating. The reflectivity is defined as r2 (x0 , L) = B(x0 )/A(x0 ). In this case the boundary condition B(L) = 0 leads to C = S/(u − iδ) and to r2 (x0 ,L) =

v sinh [S(L − x0 )] B(x0 ) = e−2iδx0 . A(x0 ) (u − iδ) sinh [S(L − x0 )] − S cosh [S(L − x0 )] (15.10)

It should be noted that, as seen from their definitions, the above reflectivities include the propagation phases. With the reflectivities for different boundary

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A(L) A(0) B(0) B(L)

L

r1

x0

A(L) A(x0)

r2

B(x0) L

(a)

B(L) = 0

(b)

Fig. 15.5 Two types of boundary conditions for calculating reflectivities. (a) A(0) = B(0), r1 (L) = A(L)/B(L) and (b) B(L) = 0, r2 (x0 , L) = B(x0 )/A(x0 ) [23]. Reprinted and modified with permission. Copyright 2008 Optical Society of America

conditions, it is easy to derive the resonance conditions for the following laser configurations. 1. Disk CBML: Considering the radially propagating waves in the disk and taking the unity reflectivity at the center, the resonance condition is 1·e2gA x0 ·r2 (x0 ,xb ) = 1, which reads e2(gA −iδ)x0

v sinh [S(xb − x0 )] = 1. (u − iδ) sinh [S(xb − x0 )] − S cosh [S(xb − x0 )]

(15.11)

2. Ring CBML: Considering the radially propagating waves in the annular defect, the resonance condition is r1 (xL ) · e2gA (xR −xL ) · r2 (xR ,xb ) = 1, which reads (u − v − iδ) sinh [SxL ] + S cosh [SxL ] −(u − v − iδ) sinh [SxL ] + S cosh [SxL ] v sinh [S(xb − xR )] × = 1. (u − iδ) sinh [S(xb − xR )] − S cosh [S(xb − xR )]

e2(gA −iδ)(xR −xL )

(15.12)

3. Circular DFB laser: The limiting cases r1 (xb ) → ∞ or r2 (0, xb ) = 1 lead to the same result

tanh [Sxb ] =

S . u − v − iδ

(15.13)

The above resonance conditions govern the modes of the lasers of each type and will be used to obtain their threshold gains (gA ) and detuning factors (δ). Substituting the obtained gA and δ into (15.8) yields the corresponding in-plane modal field patterns. These resonance conditions were shown to be equivalent to the characteristic equations in [20, 21], which were derived by matching the tangential component of electric fields and their first derivatives at the interfaces of grating and no-grating regions.

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15.4.1.2 Modal Properties Modal Field Patterns, Threshold Gains, and Detuning Factors For numerical demonstration, we assume all the lasers possess a vertical layer structure as described in [22] which was designed for 1.55-μm laser emission. We focus our analysis on a case of shallow grating etch so that the coupled mode theory applies. The effective index neff is calculated to be 2.83 and the in-plane propagation constant β = k0 neff = 11.47 μm−1 . The grating is designed such that m = 0 in order to favor a circularly symmetric field distribution. The coupling coefficients are chosen to be h1 = 0.0072 + 0.0108i and h2 = 0.0601 for a quarter-duty-cycle Hankel-phased circular grating which was shown to be able to provide both large feedback for in-plane waves and a considerable amount of vertical radiation [20]. Since we would like to compare the modal properties of lasers with the same footprint, a typical device size of xb = 200 (corresponding to ρ b ≈ 17.4 μm) is assumed for all types of lasers. For the disk CBML, the inner disk radius x0 is assumed to be xb /2 = 100. For the ring CBML, the annular defect is assumed to be located at the middle xb /2 = 100 and the defect width is set to be a wavelength of the cylindrical waves therein, yielding xL + xR = xb = 200 and xR − xL = 2π . The calculated modal field patterns, along with the corresponding threshold gain values (gA ) and frequency detuning factors (δ), of the disk- and ring CBMLs and circular DFB laser are listed in Table 15.1. Table 15.1 Modal field patterns, threshold gains (gA , in unit of 10−3 ), and frequency detuning factors (δ, in unit of 10−3 ) of the disk CBML, ring CBML, and circular DFB laser with exterior boundary radius xb = 200 [23]. Reprinted and modified with permission. Copyright 2008 Optical Society of America Laser type Disk CBML

Ring CBML

Modal field gA

δ

Circular DFB laser

Modal field gA

δ

Modal field gA

δ

Mode 1

0.127 49.8

0.457 55.9

0.283 61.8

Mode 2

0.288 21.2

1.06

66.9

1.03

66.6

Mode 3

0.454 −8.09

1.92

71.0

2.04

74.1

Mode 4

0.690 −37.4

3.14

84.4

3.11

83.6

Mode 5

1.21

−66.5

4.09

91.6

4.12

94.6

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The following is a summary of the modal properties of these laser structures: 1. Disk CBML: All the displayed modes are confined to the center disk with negligible peripheral power leakage and thus possess very low thresholds and very small modal volumes. 2. Ring CBML: With the exception of mode 1 (the defect mode), all the modes resemble their counterparts of the circular DFB laser. The defect mode has a larger threshold gain than mode 1 of the circular DFB laser, but the former possesses a much higher emission efficiency as will be shown subsequently. 3. Circular DFB laser: All the displayed modes are in-band modes on one side of the bandgap, which is due to the radiation coupling-induced mode selection mechanism [21]. Higher gain results in the excitation of higher order modes. Far-Field Patterns Due to the radiation coupling by the gratings, the near-field pattern can be calculated using (15.7) for m = 0. The coefficients s1 and s−1 are calculated to be 0.1725 – 0.0969i at the surface [20]. Using the Huygens–Fresnel principle, the diffracted farfield pattern can be obtained under the parallel ray approximation (|r|  |r |) [24]: eikr U(r) ∝ 2r

ρb

E(ρ,z = 0)J0 (kρ sin θ )ρdρ,

(15.14)

0

where r = r sin θ cos ϕ xˆ + r sin θ sin ϕ yˆ + r cos θ zˆ is the field point and r = ρ cos φ xˆ + ρ sin φ yˆ is the source point. The far-field intensity pattern is then given by I(r) = U ∗ (r)U(r) = |U(r)|2 and plotted in Fig. 15.6 for the fundamental modes of disk- and ring CBMLs and circular DFB laser.

Far-field intensity (a.u.)

1

1

(a)

0.9 0.8

1

(b)

0.9

0.8

0.8

0.7

0.7

0.7

Disk CBML

0.6

Ring CBML

0.6 0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1 5

10

15

20

θ (degree)

25

30

Circular DFB laser

0.6

0.5

0

(c)

0.9

0.1 0

5

10

15

20

θ (degree)

25

30

0

5

10

15

20

25

30

θ (degree)

Fig. 15.6 Far-field intensity patterns of the fundamental modes of (a) disk CBML, (b) ring CBML, and (c) circular DFB laser (xb = 200) [24]. Reprinted and modified with permission. Copyright 2009 Optical Society of America

As expected, the different lobes correspond to different diffraction orders of the light from the circular emission aperture. The disk CBML has most of the energy concentrated in the zeroth-order Fourier component, thus its peak is located at the center. The ring CBML and the circular DFB laser have their first-order lobe

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dominating, thus exhibiting donut shapes. These calculation results are similar to previously published experimental data for circular DFB and DBR lasers [16, 25]. Quality Factors, Emission Efficiencies, and Modal Areas In what follows we vary the device size (xb ) to investigate the size dependence of the quality factors, emission efficiencies, and modal areas. We still keep x0 = xb /2 for the disk CBML and xL + xR = xb , xR − xL = 2π for the ring CBML. As is well known, larger devices are able to lase with lower threshold levels but they suffer from smaller modal discrimination, which is unfavorable for single-mode operation. As a result, there exists a range of xb values for which single-mode operation in each laser configuration can be achieved. This single-mode range is calculated to be 60–140 for disk CBML and 50–250 for both ring CBML and circular DFB laser. Since users are mostly interested in single-mode laser designs, we will limit xb to remain within each single-mode range and focus on the fundamental laser mode only. The quality factor Q for optical resonators is usually defined as ω0 E/P where ω0 denotes the radian resonance frequency, E the total energy stored in the resonator, and P the power loss. In the surface emitting lasers investigated here, the power loss P has two contributions: vertical radiation coupled out of the resonator due to the first-order Bragg diffraction, and peripheral leakage due to the finite radial length of the reflector. Considering that the energy stored in a volume is proportional to / Bragg / |E|2 dV and that the outflow power through a surface is proportional to |E|2 dS, we define an unnormalized quality factor /D

Q

= // grating

/ 2π

/ ρb

|E(ρ,z)|2 ρ dρ , / / 2π D | E(ρ,z = 0)|2 ρ dρdφ + 0 dz 0 |E(ρ = ρb ,z)|2 ρb dφ 0

dz

0

dφ

0

(15.15)

where D denotes the thickness of the resonator. The unnormalized quality factor Q in (15.15) is proportional to the usually defined Q, but the former is easier to understand and more convenient for purposes of calculation. Figure 15.7a displays Q of the fundamental modes for the three laser configurations. As expected, increases in the device size (xb ) result in enhanced Q values. Additionally, the disk CBML exhibits a much higher Q than the other two laser structures of identical dimensions. As an example, for xb = 100, the Q value of the disk CBML is approximately three times greater than those of the ring CBML and the circular DFB laser. This is consistent with their threshold behaviors shown in Table 15.1. The emission efficiency η is naturally defined as the fraction of the total power loss which is represented by the useful vertical radiation. Figure 15.7b depicts η of the fundamental modes for the three laser configurations. As can be seen, all the lasers possess a larger η with a larger device size. Comparing devices of identical dimensions, only the disk- and ring CBMLs achieve high emission efficiencies, which is because of their fundamental modes being bandgap modes. The fundamental mode of the circular DFB laser is a band-edge (in-band) mode. Bandgap

J. Scheuer and X. Sun 250

1

Disk CBML

0.9 200

Circular DFB laser

150 100

Disk CBML

Ring CBML

50

Emission efficiency η

Unnormalized quality factor Q'

374

100

150

200

0.7 0.6

Circular DFB laser

0.5 0.4 0.3

(a) 0 50

Ring CBML

0.8

(b)

0.2 50

250

Exterior boundary radius xb

100

150

200

250

Exterior boundary radius xb

106

Modal area Aeff mode

Top surface area πx 2b 105 104 Ring CBML 103

Circular DFB laser

102 Disk CBML 10

(c)

1

50

100

150

200

250

Exterior boundary radius xb

Fig. 15.7 (a) Unnormalized quality factors, (b) emission efficiencies, and (c) modal areas of disk CBML (dashed), ring CBML (dotted), and circular DFB laser (dash dot) [23]. Reprinted and modified with permission. Copyright 2008 Optical Society of America

modes experience much stronger reflection from the Bragg gratings, yielding less peripheral power leakage than in-band modes. Based on the definition of modal volume [26], // the2 effective  modal2 area in a 2D case is similarly defined as Aeff = |E| xdxdφ /max{|E| }. The mode modal area is a measure of how the modal field is distributed within the resonator. A highly localized field can have a small modal area and therefore strong interaction with the emitter. Figure 15.7c plots Aeff mode of the fundamental modes for the three laser configurations. As a reference, the top surface area of the devices (π xb2 ) is also plotted. The modal area of the disk CBML is found to be at least one order of magnitude lower than those of the ring CBML and the circular DFB laser. This is not surprising and can be inferred from their unique modal profiles listed in Table 15.1. Furthermore, our experimental work has demonstrated single-mode lasing with ultrasmall modal volume in a nano-sized disk Bragg resonator in InGaAsP quantum well active membrane [11]. Clearly, a disk configuration is preferable in ultracompact laser design. Alternatively, ring CBMLs have a large modal area with

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high emission efficiency, rendering them suitable for high-efficiency, high-power, large-area laser applications.

15.4.2 Rotation Sensors Rotation sensors and gyroscopes have numerous applications ranging from inertial navigation systems in aircrafts and vessels, stabilization and positioning systems for robotics, virtual reality applications, and more. While rotation sensing can be based on both mechanical and optical effects, it is the optical gyros that provide both higher sensitivity and lower drift rates. Although the details may vary according to the specific implementation, all optical rotation sensors exploit the Sagnac effect [27] – the different phase shifts accumulated by waves propagating in a circular path along or against the rotation direction. The difference between these phase shifts is then converted to measurable optical power using interferometry. The Sagnac phase shift exhibits some interesting characteristics; in particular, it is determined solely by the angular velocity, the optical frequency, and the area circumscribed by the closed-loop optical path, while completely independent of the medium index of refraction and dispersion properties [28, 29]. This surprising property of the Sagnac effect has been the subject of ongoing debates for several decades [28] not only due to its fundamentality but also because it inherently limits the possibilities to design highly sensitive miniaturized rotation sensors. Recent studies pointed out the advantages of using the Sagnac effect in slow-light medium generated by electronic (such as electromagnetically induced transparency) [30] or photonic (e.g., coupled resonator waveguides) [31–33] resonances for the realization of an ultrasensitive optical gyroscope. Particularly, the studies involving coupled resonator slow-light structures revealed new manifestation of the Sagnac effect enabling the enhancement of the phase shift (and hence, the sensitivity) without enlarging the dimensions of the device. A different approach, based on rotation-induced bandgap in the passband of a coupled resonator optical waveguide, was also suggested and studied [34]. More recently, it has been shown that rotation can substantially modify the lasing properties of CBNL, offering a completely new approach for rotation sensing [35]. When a dielectric medium is rotating at a slow angular velocity  = zˆ , satisfying ||  c/L where c is the speed of light and L is the maximal dimension of the CBNL, mechanical relativistic effects are negligible and the wave equation describing the TM-polarized electric field in the structure is given by [36–38]: ∇ 2 Ez = −ε(r)

ω2 ω2  ∂ E + 2i Ez , z c2 c2 ω ∂φ

(15.16)

where Ez is the z component of the electric field, ε is the (space-dependent) dielectric coefficient, ω is the optical frequency,  is the angular rotation velocity, and φ is the angular coordinate. Assuming a solution in the form Ez = ψ(ρ) exp(imφ), where m is an integer and ρ is the radial coordinate, (15.16) can be rewritten as

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1 ∂ ρ ∂ρ



   ∂ 1 2 2 2 ρ ψ(ρ) − 2 m ψ(ρ) + k0 n (ρ) + 2m ψ(ρ) = 0. ∂ρ ω ρ

(15.17)

This equation yields the interesting result that the existence of rotation induces an effective change of the dielectric coefficient of the medium. This change depends linearly on the rotation velocity, where the dielectric coefficient effectively increases if the medium and the wave are co-propagating and decreases if they are counterpropagating. The effective index change corresponds exactly to the difference between the phase shifts accumulated by waves propagating in a closed-loop path (i.e., the Sagnac effect). In a CBNR, changes in the refractive index affect the reflectivity of the radial Bragg mirrors and thus directly affect the Q factor of the cavity. It should be mentioned that a similar effect exists also in conventional TIR-based circular cavities. Nevertheless, in such cavities the impact of the index change on the Q factor is completely negligible, especially when compared to the phase shift. In CBNRs, on the other hand, the rotation-induced changes in the Q factor could be significant. The solutions of (15.17) are a superposition of the mth-order Hankel functions of the first and second kind: (1) (2) (x) + B(x)Hm (x), ψ(x) = A(x)Hm

(15.18)

where x = k0 neff ρ, k0 = ω/c, and neff is the rotation-dependent effective refractive index given by n2eff = n2 + 2m

 . ω

(15.19)

A and B are (space-dependent) coefficients describing the amplitudes of the Hankel functions and are assumed to be piecewise constant. The change in the effective index, caused by the rotation, shifts the resonance frequency of the resonator from its stationary resonance frequency, according to | ω/ω| = | n/n| ≈ m/n2 ω. To illustrate the new effect, we consider the case of shallow circular gratings. This structure has been studied in depth [22, 39, 40], showing that for enhanced confinement in radial geometry a non-periodic grating profile is desired. In particular, grating profiles whose oscillations are determined by the phase of the mth-order Hankel function were found to provide better reflectivity than periodic gratings, i.e.,

(2)

ε(x) = α

Hm (x) (1)

Hm (x)

+ α∗

(1)

Hm (x) (2)

,

(15.20)

Hm (x)

where ε is the perturbation profile of the dielectric coefficient and α is the complex amplitude of the perturbation. Consider a disk resonator with radius ρ 0 surrounded by a Bragg reflector with an external radius ρ R (see Fig. 15.4a). The unperturbed

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index of refraction is n0 and the perturbation profile is assumed to consist of alternating high-index and low-index layers with dielectric coefficients εH,L = n20 ± ε0 . The interfaces between the layers are determined by (15.20) where the amplitude is given by α = 2 ε 0 /π [39]. The (stationary) cavity is designed to resonate at the mth-order mode with optical frequency ω0 . The cavity Q factor or, equivalently, the photon lifetime in the cavity τ exhibits an exponential dependence on the length of the Bragg reflector and the perturbation strength [40]:  

ε0 ω0 τ ∝ exp 2 (ρR − ρ0 ) . π n0 c

(15.21)

Under rotation, i.e.,  = 0, (15.19) modifies ε H,L . However, because the rotation effectively increases (or decreases) the dielectric coefficient in the structure in the same magnitude, ε 0 remains unchanged and (15.21) is modified only through the average (unperturbed) refractive index n0 : ⎡ τ () ∝ exp ⎣2

⎤

ε0 ω0

$ (ρR − ρ0 )⎦ . 2 π c n0 + 2m/ω0

(15.22)

For small rotation rates, i.e., ||/ω0  1, the photon lifetime in the cavity is given by  τ () ≈ τ ( = 0) exp −

2m ε0 (ρR − ρ0 ) π n30 c

  .

(15.23)

Equation (15.23) indicates the profound difference between the classical Sagnac effect and the novel effect discussed here: the photon lifetime in the cavity (and the Q factor) exhibits exponential dependence on the rotation velocity. Therefore, small changes in the rotation velocity will be translated to relatively large changes in the cavity Q factor and, as a result, in the emitted intensity of CBNLs. This is in contrast to the linear dependence of the Sagnac phase shift on . Therefore, rotation sensors based on Q factor modification in circular Bragg cavities may exhibit significantly enhanced sensitivity compared to conventional gyros employing the Sagnac effect. Figure 15.8 depicts a comparison between the photon lifetime calculated according to (15.23) and that calculated by FDTD simulations. The simulated structure is based on a disk-like CBNR designed to resonate at λ = 1.55 μm with m = 8. The central disk consists of material with refractive index n = 2.0 and the index modulation in the grating region is n = 0.15. The relatively strong perturbation was chosen in order to allow for a small device and to reduce the computation load. The index profile is determined according to the design rules outlined in [22]. The radius of the central disk is determined by the first zero of the Bessel function of order 8 (ρ 0 = 1.5 μm), the external radius of the grating is ρ R = 10 μm, and the index

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Fig. 15.8 Dependence of the cavity photon lifetime (τ ) on the normalized rotation rate ( ≡ /ω0 ) [35]. Reprinted and modified with permission. Copyright 2007 Optical Society of America

perturbation is calculated according to (15.20). As expected, the lifetime τ decreases exponentially when  is increased. The exponential coefficient extracted from the ¯ = −22.1 which is quite close to the predicted value slope of the curve is d ln (τ )/d of −25.5 given by (15.23). We attribute the difference between the analytical and the numerical results to the fact that the perturbation is relatively strong ( n/n ∼ 0.08). Equation (15.23) provides the impact of the various parameters of the CBNR on the sensitivity for rotation sensing. There are several important properties and trends that must be emphasized: (1) the impact of the rotation on the Q factor increases for higher reflection from the external Bragg mirror, i.e., stronger index contrast ε0 and larger external radius ρ R , and (2) the sensitivity increases for higher angular modal number m. This is similar to the increased Sagnac phase shift in loops with larger radii. However, larger m implies a larger central disk (larger ρ 0 ), which effectively reduces the length of the Bragg reflector, thus reducing the reflectivity. If the external dimensions of the CBNR are fixed (i.e., a given footprint), the sensitivity of the device is determined by the expression m(xR – x0 ) where x is the normalized radial coordinate. The central disk radius x0 , which is determined by the position of the first zero of the mth-order Bessel function, can be approximated by x0 (m) = a0 + a1 m where a0 ∼ 4.25 and a1 ∼ 1.05 [41]. Therefore, for a given footprint there is an optimal m for which the sensitivity is maximum: mopt =

xR − a0 . 2a1

(15.24)

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One of the important outcomes of the small perturbation analysis is the enhanced responsivity of the device for stronger perturbation. However, once the perturbation cannot be considered “small,” the approximate derivation is inappropriate and such devices must be analyzed using a different approach. The transfer matrix method (TMM) can be used to accurately analyze passive as well as active (i.e., possessing optical gain) CBNRs. A detailed TMM and coupled mode theory analysis of such passive and active CBNR structures are presented in [9, 22, 40]. The underlying concept of the TMM is to represent the relation between the tangential components of the electromagnetic field (Ez , Hz ) in adjacent Bragg layers in a form of a transfer matrix and then to multiply the matrices to obtain a relation between the field components of the innermost and outermost layers. Applying the appropriate boundary conditions (finite field at ρ = 0 and no incoming wave at ρ > ρ R ) yields the characteristic equation whose solution provides the resonance frequency and Q factor of the cavity. In the case of an active device, the solution of the characteristic equation provides the lasing threshold of the CBNL. Similar to the resonance wavelength, the threshold gain level and the lasing intensity of the clockwise (CW) and counterclockwise (CCW) waves are identical for a stationary CBNL. However, when the device is rotating the degeneracy of the Q factor of these modes is lifted and each possesses not only a different resonance frequency but also a different lasing threshold (or equivalently different lasing intensity at the same pumping level). Generally speaking, there are two approaches to exploit the rotation-dependent Q factor for rotation sensing: (1) identifying the threshold levels of the CW and CCW rotating modes by increasing the pumping level from zero while monitoring the emitted pattern and (2) measuring the different power levels in each mode when both of them lase (i.e., well above threshold). Essentially, both methods extract the photon lifetime (or the Q factor) of each mode and thus they are equivalent. Figure 15.9 illustrates a highly important phenomenon: the relative rotationinduced change in CBNL threshold gain level (i.e., the imaginary part of the refractive index) is larger by more than an order of magnitude than the change in the lasing wavelength (i.e., the real part of the refractive index) [35]. This behavior is

¯ Fig. 15.9 (a) Lasing wavelength and threshold level dependence on the normalized rotation rate  and (b) field profile of stationary CBNL [35]. Reprinted and modified with permission. Copyright 2007 Optical Society of America

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completely opposite to what is observed in conventional circular resonators where the rotation-induced change in the cavity Q factor is completely negligible compared to Sagnac effect. Therefore, the new effect offers an alternative and potentially more sensitive method to detect rotation.

15.4.3 Biochemical Sensors Among the most straightforward approaches for optoelectronic (bio)chemical sensing is to detect the change in the refractive index or the absorption caused by the presence of a (bio)chemical agent. Several schemes have been suggested to detect these types of changes, employing directional couplers [42], Mach–Zehnder interferometers (MZIs) [43], or high-Q optical resonators [44]. The detection mechanism underlying these sensors is the modification of the phase accumulation rate (i.e., the propagation coefficient) of the field due to the interaction of the evanescent tail of the field with the analyte. The primary disadvantage of these detection methods is that the interaction of the field with the surrounding environment is weak, and therefore, the influence of the analyte on the propagation coefficient is small. As a result, achieving high sensitivity requires large interaction length leading to the requirement of long interferometers or very high-Q resonators. In addition, the MZI-type sensors might have difficulties detecting small numbers of (or single) molecules, regardless of their length. On the other hand, the RBRs and their mode profile (see Fig. 15.3) allow for the interaction with the non-evanescent part of the field, especially when the device is designed to include an air defect. As a result, RBR-based sensors are expected to offer significantly enhanced sensitivity compared to those based on conventional resonators of similar dimensions and materials. Figure 15.10 shows a comparison between the shifts of the resonance frequency of a CBNL and a conventional ring laser due to changes in the refractive index of the surroundings. The CBNL consists of alternating layers with refractive indices of 1.545 and 1.0 and an air defect. The conventional ring laser consists of a core with n = 1.545 surrounded by air cladding. Both resonators are approximately 16 μm in diameter and designed to resonate in the visible wavelength regime. The sensitivity of each device is indicated by the slope of the curves shown in Fig. 15.10. The resonance wavelength of the conventional ring laser shifts by approximately 0.007 nm for an increase of 10−3 in the ambient refractive index. For the same index change, the CBNL’s resonance wavelength shifts by 0.4 nm, i.e., a factor of 60 higher than that of the conventional ring laser.

15.4.4 Other Telecommunication Applications For telecommunications, low loss and small dimensions are desired for any resonator-based application. For example, the strong dispersion characteristics of

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Fig. 15.10 Comparison of the sensitivity of a CBNL and a conventional ring laser to changes in the refractive index of the surroundings [22]. Reprinted and modified with permission. Copyright 2005 IEEE

the RBR can be exploited to realize a compact dispersion compensator. Because of the low group velocity of the field in an RBR, the achievable intensity inside the cavity is much higher compared to conventional resonators with the same quality factor [10, 11]. This is because for a given power level in the guiding channel, the corresponding field intensity is inversely proportional to the group velocity. Therefore, an RBR can exhibit a strong nonlinear response even for relatively modest input power levels and can be used to realize integrated all-optical processing elements.

15.5 Fabrication CBNRs of several geometries were fabricated within a thin membrane of InGaAsP semiconductor material [22]. As is illustrated in Fig. 15.11, several epilayers were grown by metal-organic chemical vapor deposition (MOCVD) on a (100) oriented InP substrate. This structure consists of a 500-nm-thick InP buffer layer, followed by a 50-nm InGaAsP (λg = 1.3 μm) etch-stop layer, and a 250-nm InP sacrificial layer. A 250-nm membrane composed of 60.5-nm InGaAsP (λg = 1.1 μm) layers sandwiching six 7.5-nm quantum wells (1% compressive strain) separated by 12-nm InGaAsP barriers (λg = 1.2 μm, 0.5% tensile strain) completes the structure. Photoluminescence from the quantum wells peaks at λ = 1559 nm. The CBNRs were designed to have resonant wavelengths between 1.5 and 1.6 μm, for large overlap with the gain spectrum of the multi-quantum-well (MQW) material. The optical gain of the semiconductor epitaxial structure favors the TE polarization (Hz ) because of the optical properties of compressively strained quantum wells [45]. Therefore, the design of the fabricated devices was optimized for this

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Fig. 15.11 InGaAsP/InP multi-quantum-well semiconductor structure fabrication process. (a) SiO2 etch mask deposition; (b) PMMA spin coating; (c) Electron-beam lithography and development; (d) SiO2 etch; (e) PMMA stripping; (f) InGaAsP membrane etch; (g) SiO2 stripping; (h) chip flipping and adhesive-bonding to sapphire; (i) InP substrate etch; and (j) adhesive etch [50]. Reprinted with permission. Copyright 2009 Springer Science and Business Media

polarization. In order to simplify the design calculations, we employed the effective index approximation in the vertical dimension. An effective index neff = 2.8 was found by solving for the TE-polarized mode of the transferred InGaAsP membrane. To facilitate the device fabrication, a mixed Bragg order scheme was used, with second-order (3λ/4 ∼ 430 nm) high-index layers and first-order (λ/4 ∼ 400 nm) low-index layers. This mixed Bragg order implementation also induces a coherent diffraction component in the vertical direction [46]. Although this mechanism reduces the overall Q of the cavity, it facilitates the observation and measurement of the resonator emission. The procedure followed to fabricate the CBNRs (or CBNLs) is documented in Fig. 15.11. First, an etch mask consisting of 120 nm SiO2 was deposited by plasma-enhanced chemical vapor deposition (PECVD). A 550-nm-thick layer of polymethylmethacrylate (PMMA) electron-beam resist was spin-coated onto the chip, and electron-beam lithography was used to define the resonator patterns in the PMMA. The patterns were then transferred to the SiO2 layer with a reactive ion etching (RIE) step after which the remaining PMMA was stripped using O2 plasma. Then, a second RIE step was used to transfer the SiO2 pattern through the 250-nmthick InGaAsP membrane. Finally, the remaining SiO2 etch mask was removed with wet chemical etching. The high refractive index of the InP substrate (n = 3.17 at λ = 1.55 μm) reduces the vertical confinement of the guided optical mode within the membrane. Therefore, to achieve strong vertical optical confinement and improve the quality factor Q of the resonators, the patterned InGaAsP membrane must be surrounded by low-index material. One method for achieving this is to generate an air-suspended membrane by selective removal of the InP underneath, as is commonly done in photonic crystal devices [46–48]. However, this technique is not applicable to the resonators studied here, because the concentric ring structure would collapse if

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the solid substrate were removed. Therefore, the resonators were flip-bonded to a double-side polished sapphire substrate with a thin layer of UV curable optical adhesive (Norland NOA 73, n ≈ 1.54 at λ = 1.55 μm). After fully curing the adhesive under a UV lamp, the InP substrate was removed using a combination of mechanical lapping and selective chemical etching. Note that the excellent thermal properties of sapphire can improve the conduction of heat generated during optical pumping away from the membrane. In addition, the large bandgap energy of sapphire and the associated transparency at visible and infrared wavelengths permit optical pumping and imaging of the resonator luminescence from either the top or the bottom of the device. Transfer of the semiconductor membrane to a transparent substrate also facilitates use of the CBNRs for sensing applications. For instance, a solution containing an analyte whose concentration is to be monitored by the resonator could be introduced via the exposed topside of the device, while optical pumping of the resonator and imaging of the sensor signal could be accomplished through the transparent substrate.

15.6 Experimental Results We measured and analyzed the vertical emission from the resonators under pulsed optical pumping. The experimental setup is illustrated in Fig. 15.12a. A Ti:sapphire mode-locked laser was used to optically pump the devices at a center wavelength of 980 nm, repetition rate of 76.6 MHz, and pulse duration of approximately 150 fs. A variable attenuator was used to control the pump power. The average pump power and center wavelength were monitored by a wavemeter, through a 50/50 beamsplitter. The pump beam was focused on the back side of the sample with a ×50 objective lens. A ×20 objective lens was used to collect the vertical emission from the sample and focus it on an IR camera to obtain the near-field intensity pattern or couple the light into a multi-mode fiber to obtain the emission spectrum. A CCD camera was used to image the resonators and the pump spot, for alignment purposes, using a white light source. When the unpatterned MQW layer structure is pumped, the emission spectrum consists of a wide peak centered at 1559 nm. As the pump power is increased from 1 to 20 mW, the FWHM of the photoluminescence broadens from approximately 70 to 110 nm and the peak shifts toward longer wavelength due to heating. No significant shift is observed when the pump power is below 5 mW, indicating that heating is of less significance at these pump levels. When a CBNR is pumped, the emission characteristics change significantly. While the specific details (threshold levels, emission wavelengths, etc.) vary from device to device, the overall behavior is similar. Once a certain pump intensity threshold is exceeded, clear and narrow (∼0.5-nm FWHM) emission lines appear in the spectrum (see Fig. 15.12b). As the pump intensity is increased, these emission lines broaden toward shorter wavelengths with their intensity increasing as well. Further increase in the pump power results in the appearance of additional emission lines.

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Fig. 15.12 (a) Experimental setup. (b) Optical spectra of a CBNL under different pump levels. Inset 1: Integrated emission power vs. pump power, showing laser threshold at ∼800 μW. Inset 2: Lasing pattern [50]. Reprinted with permission. Springer Science and Business Media Copyright 2009

Depending on the defect width and design, the CBNLs (or the pumped CBNRs) can have emision patterns with very small angular modal number (see Fig. 15.13). These modes correspond to an effective index which is lower than 1 and thus cannot be sustained in conventional TIR-based microring lasers. The ability to support such lasing modes provides substantial flexibility and control over the properties of the modal field, allowing for engineering field profiles that are highly susceptible to changes in the ambient refractive index. To demonstrate the unique advantages of the CBNR structure we focus on two applications, namely, surface emitting lasers and biochemical sensors.

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Fig. 15.13 IR images of emission patterns with small angular modal number (a) m = 7 and (b) m = 6 [50]. Reprinted with permission. Copyright 2009 Springer Science and Business Media

15.6.1 Surface Emitting Nanolasers Figure 15.14b depicts the emission spectra from the m = 0 laser for various pumping levels above the lasing threshold. An SEM image of the resonator is shown in Fig. 15.14a. The emitted light consists primarily of a single wavelength at λ = 1.56 μm, very close to the target design wavelength of 1.55 μm. The inset of Fig. 15.14b shows a L–L curve of the same device, indicating a threshold at Pth = 900 μW. It should be noted that the pump powers quoted indicate the overall power carried by the pump beam while the actual power absorbed by the quantum wells is significantly lower. At high pump levels (∼1.5Pth ), two additional low-intensity emission lines appear at longer wavelengths (∼1.59 μm) although the main emission line remains the dominant one. We attribute these modes to emission from the external Bragg grating region. Figure 15.14c shows a contour plot of the nanolaser index profile superimposed on a cross section of the modal field intensity profile in the center of the active medium. As shown in the figure, the mode of the nanocavity is confined almost completely to the 300-nm-wide central pillar with a modal volume of 0.213(λ/n)3 (0.024 μm3 ) – only 1.75 times the theoretically possible limit of a cubic halfwavelength. This modal volume is 30% lower than those demonstrated in photonic crystal defect cavities [47, 48]. This non-negligible improvement stems from the optimized match between the cavity dimensions and the quasi-periodicity of the grating, made possible by the cylindrically symmetric geometry. The employment of second-order radial Bragg gratings in the nanolaser provides, simultaneously, the radial feedback necessary for lasing and the vertical output coupling. Compared to conventional VCSELs, these circular Bragg grating-based surface emitting lasers are simpler to fabricate and are expected to provide higher output powers because of the larger field–gain overlap in the active material.

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Fig. 15.14 (a) SEM image of a circular Bragg nanocavity designed to support the m = 0 mode in the 300-nm-wide central pillar. (b) Evolution of the emission spectrum from the device shown in (a) as a function of pump intensity. Inset: L–L plot, indicating a lasing threshold of Pth = 900 μW. (c) Calculated modal intensity profile of the cavity. (d) IR image of the emission beam profile [50]. Reprinted with permission. Copyright 2009 Springer Science and Business Media

15.6.2 Biochemical Sensors Figures 15.15a and b show, respectively, SEM micrographs of a CBNR- and a conventional ring-based sensor, realized within a thin membrane of InGaAsP active material. Figures 15.15c and d depict, respectively, the cross-sectional mode profiles of the CBNR and the conventional ring. The index structures are superimposed on the calculated mode profiles. The field profile of the CBNR peaks in the defect and decays with oscillation in the reflector regions. Figures 15.15c and d also illustrate the advantages of the CBNR structure for sensing applications. Even though the main lobe of the mode profile does not interact with the device surroundings, a significant part of the field within the trenches between the semiconductor rings is exposed to the surroundings, thus yielding large interaction volume (compared

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Fig. 15.15 SEM images of (a) a CBNR sensor and (b) a conventional ring sensor and their corresponding modal field profiles (c, d) [50]. Reprinted with permission. Copyright 2009 Springer Science and Business Media

to a conventional resonator). Thus, a CBNR-based sensor is expected to exhibit enhanced sensitivity compared to a conventional resonator of similar dimensions. To test the influence of changes in the ambient refractive index on the resonance properties, the optically pumped CBNR (or, named more appropriately, the CBNL) was immersed in a specially designed beaker containing index-matching fluids with different indices of refraction. For each fluid, the emission spectrum was measured (Ppump ∼ 2.5 mW). To demonstrate the advantages of CBNL sensors in terms of sensitivity and resolution, we compared the performance of our device with that of a conventional ring resonator with 5-μm radius consisting of the same material [49]. The conventional ring was tested under the same conditions (pump power, spot size, etc.) that were used to characterize the CBNL sensor. Figure 15.16 depicts the relative shifts in the resonance wavelength of the CBNL and the conventional microring as a function of the ambient refractive index. The sensitivity of the conventional resonator is approximately dλ/dn ∼ 33 nm/RIU (star) and the resonance FWHM is ∼1.4 nm wide, while the sensitivity of the CBNL is dλ/dn ∼ 130 nm/RIU (circle) and the resonance FWHM is ∼1 nm wide. As a result, the advantage of the CBNL as a sensor is twofold: (1) the sensitivity of the CBNL is four times larger than that of the conventional device because of the larger interaction volume and (2) the conventional resonator has wider linewidth because of its lower Q, which reduces the ability to resolve the resonance wavelength. For the 1-nm linewidth of the CBNL, shifts in the resonance wavelength of 0.1 nm can easily be resolved, thus allowing the detection of ambient index changes of n ∼ 5×10−4 , which to our knowledge is one of the best resolutions ever demonstrated using integrated optical nanosensors [13]. As a comparison, for the 1.4-nm linewidth of the conventional ring, the ability to resolve the resonance

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Fig. 15.16 Comparison between the shifts in the emission wavelength of a CBNL (circle) and a conventional ring laser (star) in response to the change in the ambient refractive index [50]. Reprinted with permission. Copyright 2009 Springer Science and Business Media

wavelength is limited to ∼0.14 nm, which when combined with the lower sensitivity of the conventional device yields a minimum detectable index change of n ∼ 4 × 10−3 , approximately an order of magnitude larger than that of the CBNL sensor.

15.7 Conclusions The employment of circular Bragg gratings for lateral light confinement offers novel physical mechanisms and features which can be used for enhancing performances in various applications as well as for basic research. For applications in surface emitting lasers, the CBNL structures offer tight confinement, low lasing thresholds, and controllable out-of-plane coupling mechanism. Compared to conventional VCSELs, the CBNLs exhibit simpler fabrication flow, more wavelength flexibility, improved beam profile, and, in principle, higher output power due to the larger gain per (lateral) round trip. With a comparative study, we numerically demonstrated that, under similar conditions, disk CBNLs have the highest quality factor, the highest emission efficiency, and the smallest modal area, indicating their suitability in low-threshold, high-efficiency, ultracompact laser design, while ring CBNLs have a large single-mode range, high emission efficiency,

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and large modal area, indicating their suitability for high-efficiency, large-area, highpower applications. Experimentally, we have demonstrated single-mode lasing at telecom wavelengths from a CBNL with an ultrasmall modal volume. For rotation sensing applications, the CBNLs exhibit a novel sensing mechanism based on direct conversion of rotation rate to lasing power. Unlike the conventional Sagnac effect, the sensitivity of this structure is exponential and can exceed that of conventional optical gyros. For biochemical sensing applications, the CBNRs offer significant enhancement of the ability to detect small changes in the ambient refractive index, compared to conventional microrings, because of their larger interaction volume and higher Q factor. We have also demonstrated, experimentally, the sensitivity and resolution enhancements in CBNRs compared to conventional ring resonators. Further enhancement of the CBNR sensor sensitivity can be achieved by employing an airdefect design. The resonators’ highly compact dimensions and enhanced resolution make them excellent candidates as key components for the realization of compact, integrated biochemical sensor arrays.

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Chapter 16

Photonic Molecules and Spectral Engineering Svetlana V. Boriskina

Abstract This chapter reviews the fundamental optical properties and applications of photonic molecules (PMs) – photonic structures formed by electromagnetic coupling of two or more optical microcavities (photonic atoms). Controllable interaction between light and matter in photonic atoms can be further modified and enhanced by the manipulation of their mutual coupling. Mechanical and optical tunability of PMs not only adds new functionalities to microcavity-based optical components but also paves the way for their use as testbeds for the exploration of novel physical regimes in atomic physics and quantum optics. Theoretical studies carried on for over a decade yielded novel PM designs that make possible lowering thresholds of semiconductor microlasers, producing directional light emission, achieving optically induced transparency, and enhancing sensitivity of microcavitybased bio-, stress-, and rotation sensors. Recent advances in material science and nano-fabrication techniques make possible the realization of optimally tuned PMs for cavity quantum electrodynamic experiments, classical and quantum information processing, and sensing.

16.1 Introduction As other chapters in this volume clearly demonstrate, optical micro- and nanocavities offer tremendous opportunities in creating, studying, and harnessing confined photon states [1–6]. The properties of these states are very similar to those of confined electron states in atoms. Owing to this similarity, optical microcavities can be termed ‘photonic atoms.’ Taking this analogy even further, a cluster of several mutually coupled photonic atoms forms a photonic molecule [7]. As shown

S.V. Boriskina (B) Department of Electrical and Computer Engineering, Boston University, Boston MA, USA e-mail: [email protected] I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_16,  C Springer Science+Business Media, LLC 2010

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in Fig. 16.1, typical PM structures consist of two or more light-confining resonant cavities such as Fabry–Pérot resonators, microspheres, microrings, point-defect cavities in photonic crystal (PC) [8]. The first demonstration of a lithographically fabricated photonic molecule (Fig. 16.1a) was inspired by an analogy with a simple diatomic molecule [7]. However, other nature-inspired PM structures (such as ‘photonic benzene’) have been proposed and shown to support confined optical modes closely analogous to the ground-state molecular orbitals of their chemical counterparts [9]. The most celebrated example of a coupled-cavity structure is a coupled-resonator optical waveguide (CROW), which is formed by placing photonic atoms in a linear chain [10]. The energy transfer along the chain can be achieved through nearest neighbor interactions between adjacent cavities (photon hopping), and the unique dispersion characteristics of CROWs make possible the realization of ultra-compact on-chip optical delay lines [11–13]. Optical properties of more complex PM structures considered in this chapter depend on mutual coupling between all the cavities forming the PM and can be optimally tuned by adjusting the sizes and shapes of individual cavities as well as their positions. In this chapter, we first

Fig. 16.1 Typical configurations of PMs: (a) square-shaped photonic dots coupled via a semiconductor bridge [7], (b) planar Fabry–Pérot cavities coupled through a partially transparent Bragg mirror [14]. Reprinted with permission. Copyright 1998 The American Physical Society, (c) whispering-gallery mode microdisks side-coupled via an airgap [15]. Reprinted with permission. Copyright 1998 The American Physical Society, (d) closely located defects in a photonic crystal membrane [16]. Reprinted with permission. Copyright 2008 Optical Society of America, (e) triangular photonic molecule composed of touching microspheres [17]. Reprinted with permission. Copyright 2004 The American Physical Society, (f) vertically coupled microrings [18]. Reprinted with permission. Copyright 2006 Nature Publishing Group

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review the mechanism of mode coupling and splitting in microsphere, microdisk, and point-defect diatomic PMs and introduce classification of the PM super-modes. We then demonstrate various ways of engineering the PM super-modes spectra and explore the opportunities of post-fabrication tuning and dynamical modulation of PM structures. Unique optical properties of photonic atoms, including light confinement in compact structures that enable modification of the optical density of states and enhancement of nonlinear material properties, ultra-high-quality factors, and sensitivity to environmental changes have made them attractive building blocks for a variety of applications in basic science, information processing, and biochemical sensing. Mechanical tunability and increased design flexibility offered by PM structures make possible not only enhancing but also adding new functionalities to microcavity-based optical devices. For example, a proposal to use the effect of position-dependent splitting of whispering gallery (WG) modes in evanescently coupled microsphere resonators as a basis of a high-sensitivity coordinate meter dates back to 1994 [19], even before the term ‘photonic molecule’ was introduced. In recent years, PM structures have also entered the field of sensing as optical transducers for high-sensitivity stress [20], rotation [21–23], and refractive index [24, 25] measurements. Lineshape- and bandwidth-tuning capabilities of PM structures drive their applications as optical filters and switches [26–30] and also help to increase sensitivity of PM-based sensors [31, 32]. Furthermore, the optical interactions between photonic atoms may be tuned to enhance select modes in PM structures and to shape their angular emission profiles [8, 33–36], paving the way to realizing low-threshold single-mode microlasers with high collection efficiency. It should be noted that many of the above-mentioned applications of PMs to be discussed in this chapter have well-known analogs in the field of microwave and millimeter-wave engineering [37–41]. However, it was recently discovered that PM structures may also serve as simulators of quantum many-body physics, yielding unique insights into new physical regimes in quantum optics and promising applications in quantum information processing [42]. A few examples of these exciting new opportunities will also be discussed [14, 43–47].

16.2 Optical Properties If individual photonic atoms are brought into close proximity, their optical modes interact and give rise to a spectrum of PM super-modes. Adopting the terminology used in the studies of localized plasmonic states coupling, this mode transition and splitting can be called mode hybridization [48]. The electric field patterns of the six lowest energy super-modes in the doublecavity PM shown in Fig. 16.1a are plotted in Fig. 16.2a. The super-modes can be characterized by their parities, either even (++) or odd (+–) with respect to the PM axes of symmetry. The observed splitting of localized modes of individual photonic atoms into PM super-modes is analogous to that of the electron states of a diatomic molecule. For example, the even-(odd-)parity super-modes shown in the top row

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Fig. 16.2 (a) Calculated electric field distributions of the six lowest energy super-modes in the two-atom PM shown in Fig. 1a [7]. Reprinted with permission. Copyright 1998 The American Physical Society; (b) diagram of bonding and anti-bonding molecular orbitals in a diatomic chemical molecule

in Fig. 16.2a, which arise from the constructive (destructive) interference of lowest energy non-degenerate modes in individual photonic atoms, correspond to bonding and anti-bonding σ-like molecular orbitals (MO) formed by superposition of s atomic orbitals (see Fig. 16.2b). The other four electric field patterns in Fig. 16.2a, which correspond to σ-like and π-like MOs formed by superposition of p atomic orbitals, are the result of the interference of two double-degenerate higher energy optical modes of individual photonic atoms. Electromagnetic coupling between point-defect PC cavities [16, 20] and WG mode microdisk, microring, and microsphere resonators [19, 49–57] also results in the splitting of individual cavity modes into blue-shifted anti-bonding super-modes and red-shifted bonding super-modes (see the super-mode field distributions on the right of Fig. 16.3). The shift of the resonant frequency depends on the strength of the coupling between individual atoms. Furthermore, the values of the super-mode Qfactors also change (which may be referred to as ‘loss splitting’ [16]) and vary with the change of the inter-cavity coupling efficiency. These two effects are illustrated in Fig. 16.3a, b for the case of two microdisk resonators laterally coupled via an airgap. WG modes in individual microdisks (WGmn ) are double degenerate and are classified by two indices that correspond to the number of azimuthal (m) and radial (n) field variations, respectively. Breaking the symmetry of the structure results in lifting of the WG mode degeneracy; however, bonding and anti-bonding super-modes form closely located doublets in the PM optical spectrum and may be indistinguishable experimentally [53]. It should be noted that PMs of symmetrical configurations can support both non-degenerate and degenerate super-modes [8, 18, 25, 33, 35]. Bonding modes, which have enhanced field intensity in the inter-cavity gap, are more sensitive to the gap width than anti-bonding ones, which have a nodal plane (a plane of zero intensity) between the microcavities (see Fig. 16.3a). Figure 16.3a also demonstrates that, differently from splitting of electronic energy levels in chemical molecules, the parity of the blue- and red-shifted PM super-modes is not conserved with the change of the coupling distance. Oscillations of the super-mode resonant

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Fig. 16.3 Splitting of WG6,1 modes in a double-microdisk PM: wavelengths (a) and quality factors (b) as a function of the airgap width between microdisks (TE polarization, disks radii 0.9 μm, effective refractive index n= 2.63). The inset shows the magnetic field distributions of bonded and anti-bonded WG super-modes. (c) Bisphere PM internal energy spectrum normalized to the vacuum electromagnetic energy within the sphere (U0 double =(2/3)r3 ) vs spheres size parameter (S= kr). The PM is composed of two identical touching spheres with n= 1.59 excited by the plane wave incident at 10◦ to the PM major axis [49]. Reprinted with permission. Copyright 2000 The American Physical Society

frequencies around that of the isolated cavity reflect the oscillatory behavior of the modes evanescent field. In microsphere resonators, the WG modes are classified by angular (l), azimuthal (m, −l ≤ m ≤ l), and radial (n) mode numbers and are (2l+ 1)-fold degenerate. The most interesting for practical applications are the fundamental modes (with n = 1 and m = l), which feature the highest Q-factors and the smallest mode volumes. Optical coupling between microspheres lifts the WG mode degeneracy and leads to a multi-peak spectral response of the PM structures [49, 52, 54, 55, 58, 59]. The spectrum of the internal energy in a bisphere PM, calculated by using a tight-binding approximation [49], is presented in Fig. 16.3c and shows a fine structure of the WG super-modes. The fine structure of the bisphere super-modes has also been observed experimentally by studying the photoluminescence from melamine–formaldehyde spherical microcavities with a thin shell of CdTe nanocrystals [52, 58]. In some situations, only two broad peaks can be observed in a bisphere-measured spectrum (as in the case of coupled microdisk resonators), which represents broadened envelopes of unresolved multiple resonances [60].

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Intermixing between WG super-modes in microdisk and microsphere PM structures that are characterized by different values of radial and azimuthal quantum numbers can also occur [61, 62]. For example, if the wavelengths of the first- and second radial order WG super-modes have a crossing point when plotted as a function of a tuning parameter (e.g., airgap width), the induced anti-crossing coupling between these modes produces PM super-modes with greatly reduced Q-factors [61]. Furthermore, even in the case of ‘weak’ mode coupling (when the mode dispersion curves do not cross each other) the interband coupling can significantly modify the super-mode dispersion curves [62]. If N photonic atoms are side-coupled to form a linear chain, the optical spectrum of the resulting PM features multiple super-mode peaks, with the number of peaks proportional to the number of photonic atoms and to the degeneracy number of the modes in the stand-along atom. This situation is illustrated by Fig. 16.4a, which shows the splitting of WG modes in a finite-size linear chain composed of

Fig. 16.4 Splitting of WG12,1 modes in a linear chain of seven 2.8-μm diameter microdisks with the decrease of the inter-cavity airgap width w: (a) shift of resonant wavelengths and (b) degradation of the mode Q-factors [72]. Reprinted with permission. Copyright 2000 Optical Society of America; (c) experimentally observed optical band formation in linear chains of coupled squareshaped photonic atoms: energies of optical modes versus the angle along the chains for varying numbers of coupled cavities [63]. Reprinted with permission. Copyright 1998 The American Physical Society

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seven identical microdisks with effective refractive indices neff = 2.9 and diameters d = 2.8 μm with the change of the width of the airgaps between the disks. In the structures composed of a few strongly coupled resonators, such as that shown in Fig. 16.4a, the modes splitting can be resolvable experimentally [53]; however, excessive mode splitting in long microcavity chains results in the appearance of discrete photonic bands [10, 63–67]. This effect (which is demonstrated in Fig. 16.4b [63]) is analogous to the appearance of electronic bands in semiconductors as a result of the overlap of atomic wave functions and multiple atomic-level splitting. In the limit of an infinite number of coupled photonic atoms (infinite-length CROWs), the discrete band becomes a continuum, making possible an analytical description of the CROW dispersion properties [10]. It has also been shown, both theoretically and experimentally, that coupling of many photonic atoms in chains [10, 68, 69] or two-dimensional (2D) arrays [70, 71] can yield flat optical bands, which enable slow light engineering and enhancement of light–matter interactions. If an isolated photonic atom is driven by external excitation field (plane wave, dipole, or optical waveguide evanescent field), its spectral response usually features a series of Lorentzian peaks corresponding to the excitation of its morphologydependent resonances. Coherent interference of optical modes in PMs makes possible the manipulation of the bandwidth and/or shape of the resonant peaks. One interesting phenomenon that can be observed in PM structures composed of two or any even number of photonic atoms is coupled-resonator-induced transparency (CRIT) or resonant cancellation of absorption caused by mode splitting and destructive interference [60, 73–75]. Once again, an analogy can be drawn between PMs and atomic systems, where a phenomenon of electromagnetically induced transparency (EIT) can be observed under coherent excitation by an external laser. EIT in atomic systems is a result of destructive quantum interference of the spontaneous emission from two close energy states that reduces light absorption (see Fig. 16.5a). CRIT is manifested as a sharp dip in the PM absorption spectrum as shown in

Fig. 16.5 (a) EIT in a three-level atomic system: destructive interference of the transition probability amplitudes between two nearly degenerate energy states (split by a strong resonant pump) and a common ground state (see [73] for details); (b) CRIT in a diatomic PM structure: absorptance in the PM structure excited by a bus waveguide exhibits a sharp dip when plotted as a function of the single-pass phase shift (an analog of the frequency detuning) [73]. Reprinted with permission. Copyright 2004 The American Physical Society

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Fig. 16.5b (or, alternatively, a sharp peak in the PM transmission spectrum) caused by the destructive interference between PM optical modes. Unlike EIT effect, which can only be achieved by making use of limited number of accessible transition frequencies and requires a high-intensity drive laser, CRIT is a result of classical optical interference. Thus, a need for a powerful drive laser is eliminated and the PM spectral lineshape can be flexibly manipulated by tuning the PM geometry (e.g., the inter-resonator coupling efficiency can play the role of the strength of the pump field in EIT). It should also be noted that CRIT can be realized not only in PM structures composed of identical photonic atoms as shown in Fig. 16.5b, but also in asymmetrical PMs composed of size-mismatched optical cavities [73, 76]. In this case, the cavity sizes and the inter-cavity gap width should be adjusted to ensure efficient coupling between their optical modes. The strength of the optical coupling between modes of individual microcavities depends on the difference between their resonant wavelengths. For example, the coupling strength between two side-coupled microdisk resonators can be tuned by changing the diameter of one disk while keeping the diameter of the other fixed, as shown in Fig. 16.6 [77]. It can be seen that when the wavelengths are tuned closer to each other, the PM super-modes strongly couple with the avoided crossing scenario (Fig. 16.6a). Frequency anti-crossing is accompanied by crossing of the corresponding widths of the resonance states (see Fig. 16.6b), and at the points of avoided frequency crossing the modes interchange their identities, i.e., Q-factors and field patterns. If however the wavelengths differ significantly, the coupling between the modes is weak, and the intensity of the PM super-modes is mostly localized in one of the cavities. Similar anti-crossing coupling of super-modes can be observed if the sizes of two photonic atoms are only slightly detuned from each other [15, 77, 78]. At the point of avoided frequency crossing, the super-mode eigenvectors represent a symmetric and an anti-symmetric

Fig. 16.6 Wavelengths migration (a) and Q-factors change (b) of diatomic PM super-modes as a function of the radius of one of the disks (r2 = 1.1 μm, gap = 400 nm, n= 2.63). The insets demonstrate mode switching at the anti-crossing point (the modal near-field distributions shown in the upper (lower) insets correspond to the points labeled as A and D (B and C), respectively) [77]. Reprinted with permission. Copyright 2007 Optical Society of America

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superposition of the eigenvectors of the uncoupled system [41]. In the simplest case of an asymmetric double-cavity PM this effect causes formation of bonding (the symmetric superposition) and anti-bonding (the anti-symmetric one) PM supermodes at the point where both cavities are identical. If the microcavities are severely size-mismatched, their WG modes couple with the frequency crossing scenario [77], which results in significant spoiling of their Q-factors. However, weak coupling between severely detuned modes of photonic atoms can produce sharp asymmetric features in the PM transmission spectrum, similar to the Fano effect in atomic systems [79], which occurs as a result of the interference between a discrete autoionized state with a continuum (Fig. 16.7a). Tunable Fano interference effect in an asymmetrical bisphere PM is illustrated in Fig. 16.7c. Reshaping of the resonant peak of a single sphere by tuning it in and out of resonance with the second one can be observed. If the resonant frequencies of two spheres are tuned to the same value, CRIT phenomenon is observed resulting in appearance of a narrow transparent window (Fig. 16.7c2) In the case of larger frequency detuning, this symmetrical transmission peak reshapes into a sharp asymmetric feature (Fig. 16.7c3). This feature is a manifestation of a Fano-type resonance caused by coupling between a discrete energy state (a high-Q WG mode in one of the spheres) and a continuum of ‘quasi’ WG modes with non-circular shapes and severely reduced Q-factors [80, 81]. The field intensity distribution

Fig. 16.7 (a) Schematic illustration of the Fano effect in atomic transition: coupling between a discrete energy state and a continuum ionization level produces a sharply varying asymmetric structure [79]. Reprinted with permission. Copyright 2009 Optical Society of America. (b) Intensity distribution in a bisphere PM in a weak coupling regime: coupling of a high-Q WG mode to a continuum of quasi-WG mode states [81]. Reprinted with permission. Copyright 2006 American Institute of Physics. (c) Experimental (solid) and theoretical (dashed) transmission spectra of a single fiber-coupled microsphere (c1), and of a PM composed of two size-mismatched microspheres with the WG mode frequencies detuned from each other (c2–c4). The spectrum is centered at the WG mode frequency of the first sphere, and the arrows indicate the resonance frequency of the second one [79]. Reprinted with permission. Copyright 2009 Optical Society of America

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of the modes excited in bisphere PMs under such non-resonance condition is shown in Fig. 16.7b. Excitation of bonding and anti-bonding super-modes in PM structures also creates attractive and repulsive forces between photonic atoms [18, 82, 83]. The magnitude of this force scales linearly with the mode Q-factor, and also is a function of the coupling-induced frequency shift. The optical forces generated by different super-modes in a bisphere PM are plotted in Fig. 16.8a [82]. It can be seen that a bonding super-mode generates an attractive force, while an anti-bonding one generates a repulsive one. Figure 16.8a also demonstrates that weak coupling between tightly confined WG modes of higher radial number result in smaller forces. The forces are large enough to cause displacements of photonic atoms [82, 84] and may be harnessed for all-optical reconfiguration of photonic devices. Coupling-induced optical forces also give rise to strong and localized optomechanical potential wells, which enable all-optical self-adapting behavior of PMs. This effect has been demonstrated in a diatomic PM structure composed of two vertically coupled microring resonators, one static and one mobile (shown in Fig. 16.1f). The inter-resonator coupling can be modified by tuning the airgap q between the microrings and causes WG mode splitting into bonding and antibonding PM super-modes (Fig. 16.8b). If the PM structure is illuminated by the external laser light at a fixed frequency ωL , either a bonding or an anti-bonding mode is excited depending on the inter-cavity coupling strength. As a result, either attractive or repulsive forces are generated (Fig. 16.8c) leading to the creation of an effective potential between points (2) and (3) [18]. Resonant synthesis of potential wells can be used to obtain dynamical self-alignment between a PM super-mode and external laser light over a wide frequency range [18].

Fig. 16.8 (a) Optical force generated by bonding and anti-bonding WGlm1 super-modes in a bisphere PM as a function of inter-cavity separation [82]. Reprinted with permission. Copyright 2005 Optical Society of America. (b) Bonding and anti-bonding mode splitting in a vertically coupled double-ring PM as a function of the inter-disk coupling, and (c) forces versus ring-to-ring separation for a fixed wavelength excitation indicated by a dotted line in (b) [18]. Reprinted with permission. Copyright 2007 Nature Publishing Group

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16.3 Spectral Engineering of Photonic Molecules We have already observed that even in the simplest case of diatomic PM structures interference between PM super-modes can result in linewidth narrowing (Q-factor boost) and amplitude enhancement of certain super-modes (see Fig. 16.3, [50, 60, 77, 85]). More complex PM structures can be engineered with the aim to improve spectral characteristics of isolated optical microcavities and add new functionalities to microcavity-based optoelectronic devices. Many approximate techniques have been developed and applied to design and investigate PM structures, including perturbation methods [19, 82] and tight-binding approximation [10, 13]. These approaches, however, are only applicable to the analysis of non-degenerate modes and should be used with caution in the simulations of coupled microresonators that support double- or multiple-degenerate WG modes, even in the case of weak coupling [54]. Accurate analytical methods such as multi-particle Mie theory in 2D and 3D [50, 62, 86–88] and integral equation techniques [25, 89, 90] need to be used to properly reproduce the fine structure of the PM spectrum [54], account for intermixing between WG modes characterized by different values of the azimuthal quantum number [62, 77], and accurately consider coupling that occurs between non-neighboring cavities in the photonic molecule [8, 33–35, 72, 89]. It should be noted that extended multi-cavity coupling has been demonstrated both experimentally and theoretically not only in complex-shaped PMs such as triangles, squares [8, 25, 33, 66, 89, 91], but also in slightly bent linear arrays of photonic atoms [66, 89]. The use of rigorous techniques becomes essential when the sizes of interacting optical atoms are on the scale of the wavelength. Mutual coupling between such atoms can significantly modify field distributions of WG modes and causes significant detuning of super-mode frequencies from their single-cavity counterparts [25, 89]. Among the advantages offered by PM structures over isolated microcavities is improved robustness against disorder. Rotational symmetry of microdisk or microsphere resonators can be broken by fabrication imperfections, resulting in splitting the double or multiple WG modes degeneracy and spoiling their high-quality factors [92, 93]. Degenerate mode splitting in microcavities also occurs due to coupling to fibers and bus waveguides [94]. Optical response of PM structures can be more robust against disorder as the mode degeneracy is already lifted due to inter-cavity coupling [8, 77]. Furthermore, a quality factor of an optical mode is a very important parameter, which has to be maximized in order to enhance light–matter interactions in microcavities, to provide higher sensitivity of resonator-based optical sensors, to increase optical forces, etc. However, while the Q-factors of WG modes in microdisks and microspheres grow exponentially with the cavity size, the cavityfree spectral range (FSR, spectral separation between neighboring high-Q modes) shrinks. Higher azimuthal-order WG modes in large microcavities are also more sensitive to Q-factor degradation caused by the cavity surface roughness [93]. For many applications, such as low-threshold microlasers, single-photon sources, and optical biosensors, however, single-mode (or rather quasi-single mode) cavities with a narrow mode linewidth are required. Coupling cavities into PM structures offers

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a design strategy for the controlled manipulation of their optical spectra through destructive and constructive optical interference. Thus, only one mode (e.g., the one that has a resonant frequency within the material gain bandwidth) may be enhanced, and all the other modes (considered parasitic) are suppressed. An example of a wideFSR quasi-single mode square-shaped PM structure is shown in Fig. 16.9 [8, 25, 33]. Optical coupling between microdisks in such a configuration splits their degenerate WG modes into four non-degenerate super-modes with even/odd symmetry along the square diagonals and the x- and y-axes (termed as EE, EO, OE, and OO super-modes) as well as two double-degenerate modes. The modes with the highest Q-factors are the OE and OO super-modes (their near-field patterns are shown in the inset to Fig. 16.9). A dramatic 23-fold enhancement of the non-degenerate OE mode (Fig. 16.9b) in the optimal square-molecule configuration can clearly be observed. Note that all the other modes have significantly lower Q-factors at this point. Other symmetrical PM geometries such as equilateral triangles [33], circles, and hexagons [8, 35, 95] can also be tuned to provide selective mode enhancements. Furthermore, the FSR of a PM structure can be increased by using the Vernier effect [36, 96, 97].

Fig. 16.9 Shift and splitting of wavelengths (a) and change of Q-factors (b) of the TE-polarized WGE6,1 modes in a square 4-disk PM (disk radii 0.9 μm, n= 2.63) as a function of the inter-cavity gap width. The inset shows the magnetic field distribution of two PM super-modes whose Q-factors are boosted at certain values of the gap width (symmetry-enhanced super-modes) [33]. Reprinted with permission. Copyright 2006 Optical Society of America

Many practical applications require directional emission from microcavity structures. One of the disadvantages of conventional WG mode circular or spherical microcavities is their multi-beam emission patterns. However, proper engineering of inter-cavity optical coupling enables shaping far-field emission characteristics of PMs. One possible PM configuration featuring narrow-beam emission pattern is shown in Fig. 16.10a [97]. Here, emission directionality was achieved by coupling a microring resonator to a spiral-shaped microcavity, which generates directional emission pattern owing to its highly asymmetrical shape [98]. Simultaneously, the Vernier effect was used to obtain a single-mode emission spectrum [97]. However, highly directional emission can also be achieved in PMs composed of circularly

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Fig. 16.10 (a) Far-field emission pattern of the quasi-single-mode ring-spiral coupled PM structure [97]. Reprinted with permission. Copyright 2008 American Institute of Physics; far-field patterns and near-field distributions in coupled-microdisk PM structures shaped as (b) equilateral triangle [8], (c) asymmetrical cross [34]. Reprinted with permission. Copyright 2006 IEEE, and (d) asymmetrical dumbbell [8, 77]. Reprinted with permission. Copyright 2007 Optical Society of America

symmetrical microdisk resonators (Fig. 16.10b–d). Emission into four well-defined beams has been demonstrated in the simplest case of a double-disk PM [8, 36, 50, 57, 77]. The number of beams can be reduced (and directivity of the device improved) by using more complex PM configurations. For example, the EE supermode supported by the triangular PM shown in Fig. 16.10b yields directional emission pattern with three narrow beams [8]. Emission directionality can also be improved by introducing position and/or size disorder in the PM structure. Emission patterns featuring just two narrow beams have been engineered in asymmetrical cross-shaped photonic molecules (Fig. 16.10c [34]) and asymmetrical diatomic molecules with cavity size mismatch (Fig. 16.10d) [8, 77]. Furthermore, emission into a pre-defined number of narrow beams can be achieved through the mechanism of multiple-nanojet formation in PM structures [99]. Optical microcavities arranged in linear chains can be used for light guiding by virtue of light propagating via evanescent coupling of microcavity modes. Such structures, termed coupled resonator optical waveguides or CROWs [10], provide new ways of controlling the group velocity of light on the optical chip and have been

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extensively studied in the literature (see e.g., [100] and references therein). Here, we will focus on a specific application of CROWs, namely, a possibility of achieving very efficient or even complete transmission through sharp bends. In the pioneering paper introducing the CROW concept [10] it was suggested that if a CROW is composed of resonators that operate in an optical mode with an n-fold rotational symmetry, perfect transmission can be achieved through 2π /n bends. Efficient transmission through a microsphere CROW section with a bend angle of 90◦ has later been observed both theoretically [101] and experimentally [102]. Accurate integralequation full-wave analysis of mode coupling and light transfer through CROW bends has revealed that this prediction was indeed valid for weakly coupled microcavities supporting high azimuthal- order WG modes [89]. This effect is illustrated in Fig. 16.11a, which shows evolution of Q-factors of two nearly degenerate highQ WG20,1 super-modes in the bent finite-size section of microdisk CROW with the change of the bend angle [89]. It can be seen that the Q-factors oscillate with the period 2π /n= 18◦ . Furthermore, because the Q-values of two nearly degenerate super-modes oscillate in anti-phase, this CROW can efficiently transfer light through π /n bends. However, bending of CROW sections composed of smaller size strongly coupled microcavities can result in strong extended multi-cavity coupling [66] that may significantly disturb WG mode field patterns and render the above estimation of the optimal angles inapplicable [89]. In this case, efficient transmission through any pre-defined angle can still be achieved by tuning the CROW structural parameters, e.g., the radius of the microdisk positioned at the bend (Fig. 16.11b) [89]. Furthermore, tuning sizes of individual resonators in linear CROWs may provide an additional degree of freedom for designing CROW band structures [103]. With the above examples we have demonstrated a variety of novel optical functionalities offered by PMs, both composed of identical microcavities and of microcavities with either slight or significant size mismatch. These functionalities

Fig. 16.11 Quality factors change of two finite CROW section super-modes forming a close doublet in the CROW spectra as a function of (a) the bend angle (disks radii 3.65 μm, ε = 2.5) and (b) the central resonator radius for the bend angle of 90◦ (disks radii 0.9 μm, ε = 7.0). The insets show the near-field portraits of high-Q super-modes in the optimally configured CROWs [89]. Reprinted with permission. Copyright 2007 IEEE

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together with higher design flexibility and tunability of PM structures over isolated photonic atoms pave the way for their use in various technological areas ranging from biotechnology to optoelectronics to quantum computing. In the next section, we will review some of the devices and applications based on spectrally designed photonic molecules.

16.4 Devices and Applications Among the most promising potential applications of PM structures in integrated optics, signal processing and quantum cryptography are engineering of single-mode high-power microlasers and single photon sources. A single-mode microlaser can be engineered by optimally coupling two size-mismatched photonic atoms to yield selective enhancement of a single optical mode [36, 77, 97]. An example of such a device composed of two side-coupled microring resonators fabricated by coating glass fibers with thin layers of hybrid organic–inorganic materials is shown in Fig. 16.12a. Proper adjustment of the microring radii selectively suppresses all the non-spectrally overlapping modes of individual cavities and effectively increases the FSR of the resulting PM structure by means of the Vernier effect [36]. Furthermore, the structure asymmetry helps to improve the emission directivity, and thus boosts the microlaser collection efficiency (a portion of the emitted light that is redirected to the desired point or direction). Other examples of PM microlasers include chain-like coupled-cavity structures (Fig. 16.12b) [67] and 2D arrays of coupled PC defect nano-cavities (Fig. 16.12c), [70]. In the structure shown in Fig. 16.12b, coupling of the individual defect modes creates mini-bands within the bandgap of the InGaAsP/InP PC membrane, and stable single-mode lasing occurs on the mini-band mode with the lowest group velocity [67]. The 2D array of PC nano-cavities showed in Fig. 16.12c operates on a single high-Q super-mode and enables two orders of magnitude increase in the output power over a single-defect device, with a comparable low-threshold power [70]. On the other hand, tunable dual- (or multiple-) wavelength laser sources are desirable in several applications such as two-wavelength interferometry for distance measurements [104], terahertz signal generation [105], and non-linear optical frequency mixing [106]. Two-wavelength laser emission has been successfully demonstrated in various types of vertical cavity surface emitting lasers (VCSELs) composed of two coupled microcavities containing multiple quantum wells (Fig. 16.13a), [107]. It has also been shown that to achieve stable dual-frequency lasing in such doublecavity sandwiches it is enough to pump only one of the cavities, whose emission then acts as an optical pump for the quantum wells in the other cavity [108, 109]. Coupling-induced splitting of the cavity optical modes in multi-atom photonic molecules leads to the appearance of multiple peaks in their lasing spectra, as demonstrated in Fig. 16.13b, for the case of a microdisk PM structure. A possibility of manipulating the spectral response of PM structures by tuning the inter-cavity coupling strength also facilitated their use as multi-functional

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Fig. 16.12 (a) Single-mode lasing from a diatomic PM structure composed of size-mismatched laterally coupled microring resonators (n= 1.52, ring thickness 2 μm, radii 115 and 125 μm) [36]. Reprinted with permission. Copyright 2008 Optical Society of America. (b) Laser spectrum of the PC coupled-cavity chain featuring quasi-single-mode lasing with side mode suppression greater than 42 dB [67]. Reprinted with permission. Copyright 2003 American Institute of Physics. (c) Single-mode emission from a 2D array of coupled identical defect nanocavities in an InGaAsP PC membrane [70]. Reprinted with permission. Copyright 2005 Optical Society of America

components for all-optical on-chip interconnection networks. By adapting the microwave circuit design principles, higher order bandpass and add/drop filters can be engineered with cascaded microdisk resonators (Fig. 16.14a). As shown in Fig. 16.14b, higher order optical filters offer flatter-top passbands with sharper rolloff than their single-disk counterparts, together with greater out-of-band rejection [26] (see also [5, 28, 30], Chapter 1 of [2], and references therein). On the other hand, filters with ultra-narrow passband (much narrower than the linewidth of each or the microcavities) can be engineered by making use of the EIT effect [27, 75] (see Fig. 16.5b). Furthermore, the shapes of the transmission characteristics of the multi-cavity structures are very sensitive to the detuning of any of the cavities, making them attractive candidates for designing optical switches, routers, and tunable delay lines. For example, high-bandwidth optical data streams can be dynamically routed on the optical chip by tuning one or more microcavities in the cascaded high-order filter configuration out of resonance [110]. Detuning of the resonator frequency can

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Fig. 16.13 (a) Pump-energy dependence of time-averaged spectra of a VCSEL composed of two Fabry–Pérot-type Al0.2 Ga0.8 As/Al0.5 Ga0.5 As microcavities coupled via a common semitransparent Bragg mirror shown in the inset [107]. (b) Top view and multimode-lasing spectra of PM structures composed of several side-coupled 3-μm diameter GaInAsP microdisk resonators [53]. Reprinted with permission. Copyright 2005 American Institute of Physics

Fig. 16.14 (a) Scanned electron microscopy (SEM) micrographs of first- and second-order AlGaAs–GaAs microring racetrack resonator filters; (b) measured transmission spectra at drop port of single- and double-resonator filters, with frequency scale normalized to 3 dB widths [26]. Reprinted with permission. Copyright 2000 IEEE

be achieved all-optically by a pump laser [110] or electrically using a PIN diode. Alternatively, microcavities made of electro-optic crystals may be used [111]. The large thermo-optic effect of silicon and polymers makes possible tuning of resonant wavelengths of the microdisks by changing their temperature [112]. Selective opto-fluidic tuning of an individual cavity in a PM structure has also been demonstrated [113]. PM-based switches can be designed to feature a tailorable asymmetric Fano resonance response (see Fig. 16.7c) which yields high extinction ratio, large modulation depth, and low switching threshold [114]. The linewidth of the narrow

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passband filter based on the CRIT effect can also be manipulated by detuning the frequencies of the coupled photonic atoms away from each other [75]. It should finally be noted that all the above-discussed functionalities enabled by PMs can be realized not only with identical or size-mismatched cavities of the same type (e.g., coupled microrings) but also with completely different cavities (such as a microdisk coupled to a Fabry–Pérot-type resonator formed by two partial reflectors in a bus waveguide) [75, 115]. Sharp and asymmetric Fano resonance features in the spectral characteristics of PMs can be useful for refractometric sensing applications, because the steeper slope between zero and unity transmission translates into a higher sensitivity to environment-induced resonant frequency shifts [31, 113, 115–117]. PMs can also exhibit large sensitivity to the change in the intrinsic loss in one of the optical cavities [60]. Another advantage offered by the PM structures is that different PM configurations exhibit distinct optical mode spectra, which may be considered as PM optical fingerprints. These fingerprints can be used for tracking the PM response to the change of the ambient refractive index or to the adsorption of the molecules on the PM surface, even without the knowledge of the exact PM positions [24]. Such sensing scheme is highly promising for the in situ multiplexed detection in the array format. Finally, using PMs supporting high-Q symmetry-enhanced super-modes (such as those shown in Fig. 16.9) can further improve sensitivity of microcavity-based optical biosensors [25]. Being collective multi-cavity resonances, super-modes of optimally tuned PMs provide better overlap of the modal field with the analyte (see insets to Fig. 16.9) accompanied by the dramatic increase of their quality factors. Figure 16.15 demonstrates the performance improvement offered by the PM-based sensor for the detection of the changes in the ambient

Fig. 16.15 Comparison of the wavelength shifts (a) and Q-factors change (b) as a function of the ambient refractive index for a single 2-μm diameter microdisk (n = 2.63) operating on a WG7,1 mode, a square 4-disk PM, and a 16-disk super-PM structure operating on the symmetry-enhanced WG4,1 super-mode (n = 2.63, radius = 0.67 μm). Magnetic field portraits of the operating modes are shown in the inset [25]. Reprinted with permission. Copyright 2005 Optical Society of America

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refractive index over sensors based on individual microcavities with comparable Q-factors. PM structures with 4 and 16 disks can be tuned to resonate on symmetryenhanced WG4,1 super-modes and feature larger frequency shifts caused by the change in the environment than the higher radial order mode of comparable Q-factor in an isolated larger size microdisk. The capability of coupled-microcavity structures to slow light also facilitates their use as ultra-compact optical gyroscopes for numerous practical applications that require detection and/or high-precision measurement of rotation. The principle of operation of optical gyroscopes is based on the Sagnac effect, i.e., the accumulation of the additional phase by an electromagnetic wave propagating in a moving medium [21, 22]. One possible configuration of a PM-based rotation sensor is shown in Fig. 16.16a and consists of several laterally coupled microdisk resonators. The relative phase difference between clockwise and counter-clockwise propagating signals in such a structure is a function of its angular velocity and can be measured to detect the device rotation [22]. It can be seen in Fig. 16.16a that the sensitivity of the device can be enhanced by increasing the number of coupled microcavities. It has already been mentioned that the sensitivity of the PM super-mode frequencies to the width of the inter-cavity coupling gap enables the use of PMs as optical parametric transducers for measuring small displacements [19]. Ultra-small sizes of PM structures offer unique opportunities for such ultra-sensitive applications as ponderomotive quantum nondemolition photon number measurements. If a position displacement in a PM structure is caused by the applied mechanical stress, after a proper calibration, it can be used as a stress sensor. A highly sensitive stress sensor design based on a PM structure composed of two photonic-crystal cavities vertically

Fig. 16.16 (a) Coupled-microring slow-light rotation sensor: the device schematic and the output signal intensity as a function of the PM angular velocity for structures composed of various numbers of ring resonators [22]. Reprinted with permission. Copyright 2006 The American Physical Society. (b) Optical stress sensor: wavelength shift of the bonding super-mode in a double-layered PC PM caused by the change of the width d of the vertical airgap between coupled cavities [20]. Reprinted with permission. Copyright 2009 Optical Society of America. The inset shows the schematic of the double-layer PM structure

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coupled via an airgap is shown in Fig. 16.16b [20]. The resonant frequencies of the bonding and anti-bonding super-modes supported by such a double-layer structure shift with the change in the airgap width (Fig. 16.16b). Measurements of this wavelength shift can be translated into measurements of the applied stress. Furthermore, by geometrical tuning of the double-layer PM structure, selective enhancement of one of the super-modes may be achieved, leading to the improved sensor spectral resolution and sensitivity [20]. Spatial re-distribution of the optical modes inside PM structures occurring in the vicinity of the points of strong mode coupling with the avoided crossing scenario (see Fig. 16.6) offers exciting new prospects for switching optical intensity between coupled photonic atoms and realization of optical flip-flops. If the frequencies of two modes are detuned from each other (e.g., by cavity size mismatch), the optical fields of the resulting super-modes are mostly localized in different parts of the PM structure [15, 77, 78, 109]. Tuning the frequencies of individual cavities (e.g., by optical pumping) can result in the lasing bistability and mode switching in PMs [15, 78]. Bistable lasing achieved in a diatomic PM structure composed of two side-coupled GaInAsP microdisks (Fig. 16.1c) is demonstrated in Fig. 16.17. The PM-lasing spectrum features two narrow peaks corresponding to the unresolved bonding (S1 ) and anti-bonding (S2 ) super-mode doublets (Fig. 16.17a). The lasing characteristics of the highest Q anti-bonding super-mode exhibit bi-stable behavior when the structure is pumped non-uniformly by an external laser (Fig. 16.17b) [15, 78]. Pumping changes the degree of microdisk frequency detuning and can be used to switch the optical intensity between microdisks. Optical intensity switching can also be realized in more complex PM configurations consisting of several microcavities by selectively tuning the refractive index of

Fig. 16.17 (a) Lasing spectrum of a size-mismatched twin microdisk PM shown in Fig. 16.1c [15]. Reprinted with permission. Copyright 2005 American Institute of Physics [78]. Reprinted with permission. Copyright 2006 IEEE (b) Dynamics of the lasing characteristics of the anti-bonding super-mode (S2 ) under pumping of the entire area of the smaller disk and only a small part of the larger one. Open and closed circles correspond to the measurements with increasing and decreasing pump intensities, respectively. The inset shows the super-mode field profile and the pumping area

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a single cavity. For example, transmission through a CROW bend can be enhanced (see Fig. 16.11b) by tuning the resonator material parameters rather than the geometrical parameters. Another useful example of a switchable coupled-cavity structure is illustrated in Fig. 16.18 showing a branched CROW section that enables splitting of waves or pulses into two output waveguide ports or routing all the energy into a single port. As demonstrated in Fig. 16.18, switching between two high-Q supermodes of a branched CROW structure can be achieved by tuning the refractive index of the central disk [72].

Fig. 16.18 Switching in the branched CROW section. Change in the high-Q CROW super-modes resonant wavelengths (a) and Q-factors (b) with the change in the effective refractive index of the microdisk positioned at the CROW branching point. The inset shows a sketch of the branched CROW section (neff = 2.9, diameters 2.8 μm, airgaps 100 nm). The magnetic field distributions in the CROW correspond to the three different values of the central disk refractive index marked with circles in (a) and (b) [72]. Reprinted with permission. Copyright 2007 Optical Society of America

It should finally be noted that dynamical intensity switching between different parts of PMs can be used to coherently transfer excitation between quantum dots (QD) [118, 119] or quantum wells (QW) [14, 120] embedded in different cavities. Controllable interaction between bonding and anti-bonding PM super-modes and degenerate QW exciton states confined in separate cavities makes possible coupling between excitons over very large macroscopic distances [14]. Overall, the possibility to selectively address individual cavities in the PM structures doped with atoms or containing quantum wells/dots (e.g., by an external laser light) makes them very attractive platforms for simulating complex behavior of strongly correlated solidstate systems [42, 43, 46, 121]. Controllable interaction of PM modes with atoms or QDs also paves the way to engineering devices for distributed quantum optical information processing applications [42, 122]. For example, PMs can form a useful platform for efficient optical probes that reveal an interplay between macroscopic coherence and interactions in strongly correlated photonic systems [44]. A possible design of an optical analog of the superconducting Josephson interferometer (quantum-optical Josephson interferometer) is shown in Fig. 16.19a [44]. It consists of a central cavity with nonlinearity

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Fig. 16.19 (a) Schematic of a quantum-optical Josephson interferometer with three coupled cavities and a possible device design with PC nanocavities, where the central cavity has an embedded QD or a QW strongly coupled with a high-Q optical mode [44]. Reprinted with permission. Copyright 2009 Nature Publishing Group. (b) A schematic of a quantum logic gate composed of four CROWs representing two qubits, and a possible practical implementation of the device using coupled defects in a PC doped with atoms or QDs [45]. Reprinted with permission. Copyright 2007 Elsevier

(e.g., caused by the light–matter interaction with an embedded QD), which is coupled to two other cavities driven by external lasers. Interactions between QDs and optical modes can be controlled by manipulating coupling between different cavity modes [122]. The interplay between interactions and inter-cavity photon tunneling can be explored by observing the light emitted from the central cavity under the change of the inter-cavity coupling strengths. A cross-over from the coherent regime (with optical intensity delocalized over all three cavities) to the strongly correlated regime can be detected. Furthermore, many other basic elements needed for quantum information processing, including state transfer [123, 124], entanglement generation [47, 125], and quantum gate operations [45, 126], can be realized in spectrally engineered PM structures. In particular, logic operations can be performed by exploiting strong coupling between atoms and optical super-modes in coupled-cavity PM structures. One possible design of a quantum gate is shown in Fig. 16.19b. Two waveguides composed of coupled PC defect cavities doped with atoms or QDs are brought close to each other to enable optical coupling between them. These two waveguides represent two different qubits (units of quantum information). The dopant atoms or QDs can be tuned on- or off-resonance with propagating light by applying an external electric field. Interplay between (resonant) two-photon and (dispersive) one-photon transitions leads to phase shifts required both for single-qubit phase gates and two-qubit controlled-phase gates [45]. Mature modern nano-fabrication technologies make possible practical realization of the above PM-based device designs and open new exciting opportunities in the fields of quantum cryptography and quantum information processing.

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16.5 Conclusions and Outlook Optical and light–matter interaction in PM structures composed of coupled microcavity resonators give rise to amazingly rich physics and can be harnessed for a variety of practical applications. PM structures already find their way into commercial applications for high bit-rate linear signal processing devices [2, 5, 12]. They also hold high promise for chip-scale integrated sensing systems [116] and tunable optical buffers and delay lines [100, 110]. One of the emerging highly promising areas of photonic molecules research and application is cavity optomechanics [127, 128]. Coupling of optical and mechanical degrees of freedom in microcavities and photonic molecules provides unique mechanism of control and measurement of mechanical motion with micro- and nano-scale structures. Optical forces of switchable polarity that can be reversibly generated in coupled-cavity structures [18, 82–84, 129] offer exciting opportunities for the design of novel tunable lasers and filters, optomechanical signal processing, high-speed all-optical switching, wavelength conversion, and dynamic reconfiguration of optical circuits [130–132]. Furthermore, tunable interaction of light forces generated by photonic molecules with small biological objects paves the way to the development of ultra-sensitive devices for mass, size, and orientation spectroscopy. Recent experiments have demonstrated low-power optical trapping of nanoparticles in the orbit around a whispering-gallery-mode microsphere (Carousel trapping) [133]. Fluctuations in the microsphere resonance frequency were measured and used to estimate the size and mass of the trapped nanoparticle in the solution without binding it to the resonator surface. Coupled-cavity configurations are expected to provide additional degrees of freedom for detecting, capturing, and manipulating biological nano-objects. For example, a PM sensor capable of measuring the dipole moment orientation of nanocrystals or molecules by tracking the dipole-induced spatial field distribution in the photonic molecule has already been theoretically demonstrated [134]. Finally, controllable manipulation of interactions between quantized PM optical modes and trapped atoms or embedded quantum dots opens up the possibility of using photonic molecules as quantum simulators of many-body physics and as building blocks of future optical quantum information processing networks [42]. Acknowledgments The author would like to thank Dr. Vladimir Ilchenko, Dr. Frank Vollmer, Dr. Sunil Sainis, Prof. Vasily Astratov, and Prof. Kerry Vahala for useful discussions. Support from the EU COST Action MP0702 ‘Towards functional sub-wavelength photonic structures’ and from the NATO Collaborative Linkage Grant CBP.NUKR.CLG 982430 ‘Micro- and nano-cavity structures for imaging, biosensing and novel materials’ is gratefully acknowledged.

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Chapter 17

Fundamentals and Applications of Microsphere Resonator Circuits Vasily N. Astratov

Abstract Dielectric microspheres, with sizes on the order of several wavelengths, support high-quality whispering gallery (WG) modes and provide nonresonant focusing of light into tiny spots termed nanoscale photonic jets. In this chapter, we review properties of more complicated multiple-cavity systems that are formed by microspheres assembled in chip-scale structures. The resonant optical properties of such systems can be engineered on the basis of tight-binding WG modes in photonic atoms. In practical systems of coupled cavities, the optical transport properties are strongly influenced by disorder effects, leading to scattering, localization, and percolation of light. The desirable tight-binding properties require selecting more uniform spheres, which can be achieved by novel methods based on using sizeselective radiative pressure. Due to controllable dispersions for photons, collective emission and absorption, and enhanced light–matter coupling, such structures can be used for developing coupled arrays of microlasers, ultracompact high-resolution spectrometers, and sensors. The nonresonant properties of such systems are connected through subwavelength focusing of light in chains and arrays of microspheres that can be used in a variety of biomedical applications including ultraprecise laser tissue surgery.

17.1 Introduction The area of coupled microresonators experienced a significant boost in 1990s when the tight-binding approximation well known in solid-state physics [1, 2] was applied to optical cavities. Analogous to electrons, the tight-binding approximation for photons [3–7] is based on resonant coupling of electromagnetic fields confined to identical “atoms” formed by optical cavities. Several proposals of coupled cavity V.N. Astratov (B) Department of Physics and Optical Science, University of North Carolina at Charlotte, Charlotte, NC 28223-0001, USA e-mail: [email protected] web:http://maxwell.uncc.edu/astratov/astratov.htm

I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_17,  C Springer Science+Business Media, LLC 2010

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devices appeared at that time, including high-order optical filters [8], coupled resonator optical waveguides (CROW) [6], side-coupled integrated spaced sequences of resonators (SCISSOR) [9], and more sophisticated structures [10–16]. Another important factor was connected with the possibility of fabricating coupled cavities with sufficiently high-quality factors (Q>>104 ) of individual resonances integrated on the same chip. This included patterned semiconductor microcavities [17], photonic crystal cavities [18–20], and microrings [21–23]. Due to controllable dispersion relations for photons, these structures can be used for developing chip-scale delay lines, spectral filters, and sensor devices. Although, theoretically, coupled cavity structures can provide almost lossless optical transport [6] in the passbands under the condition κ>>1/Q, where κ is a coupling constant, this is only true with the assumption that the cavities and their coupling conditions are identical. In real physical structures, the disorder [24–26] becomes to be a dominant factor of optical losses. This remains true even for structures obtained by the best semiconductor technologies [23], such as CMOS, where the standard deviation δ of the individual cavities’ frequencies is typically limited to ∼ 0.1% over the millimeter scale distances. An ultracompact optical buffer on a silicon chip [23] is illustrated in Fig. 17.1a. For such structures, losses around 0.3 dB per microring, have been reported [23] for a chain of 100 coupled microrings. There are several techniques that can in principle be used for fine tuning of the individual resonances, such as free carrier injection in p-i-n junctions in semiconductor structures [27] or the use of electro-optical [28] or thermo-optical [21, 29] effects. Application of these techniques to a large number of resonators integrated on a single chip, however, still remains a challenge. From the time CROW devices were initially proposed [6], it has been well recognized that there is an alternative way of making such structures based on microspheres. Conventional technologies for the fabrication of microdisks, rings, and toroids are essentially planar (2D) technologies, whereas the microspheres can be assembled in arbitrary 3D structures. In addition, the microspheres are characterized with extremely high whispering gallery (WG) mode Q-factors [30–35]

Fig. 17.1 (a) Ultracompact optical buffer on a silicon chip [23] obtained by CMOS technology [23]. Reprinted with permission. Copyright 2009 Nature Publishing Group and (b) chain of microspheres with coupled WG modes [53] assembled in a groove [53]. Reprinted with permission. Copyright American Physical Society 2009

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(>104 for 5 μm spheres and up to 1010 for submillimeter spheres). The inexpensive microspheres can be manufactured using a variety of materials with different indices of refraction, such as silica [36–38] (n = 1.45), borosilicate glass (n = 1.47), PMMS [39] (n = 1.48), soda lime glass (n = 1.50), polystyrene [36–38] (n = 1.59), melamine resin (n = 1.68), barium titanate glass (n = 1.9), titania [39] (n ∼ 2.0), and even silicon [40] (n = 3.5). These properties of individual microspheres have attracted significant interest in studies of the optical effects in more complicated structures formed by spherical cavities, such as bispheres [41–49], linear chains [50–60], 2D clusters [61, 62], and 3D crystal structures [63, 64]. The strong coupling between WG modes in microspheres has been observed [42–47] along with a number of interesting optical effects. At the same time, it was demonstrated [47, 48, 52] that the standard deviation (δ ∼ 1–3%) of the sphere’s diameters provided by various manufacturers is too high to achieve efficient WG mode-based coupling. One way of overcoming this problem is to use the spectroscopic characterization of the WGM peak positions to select size matched microspheres individually [42, 44–46, 53, 54]. This approach has achieved an unprecedented uniformity (δ ∼ 0.05%) of WG mode resonances. Most recently, even higher uniformity (δ ∼ 0.03%) was achieved [65, 66] in bispheres. Future development of this technology is likely to be connected with massively parallel sorting of spheres [67, 68] based on using size-selective optical forces. It has been theoretically demonstrated [68] that when a microsphere is illuminated by an evanescent wave, the optical forces on- and off-WG mode resonance can differ by several orders of magnitude. The force can be extremely size selective (∼1/Q), and as such, it allows for parallel particle-sorting according to their resonant frequency. In recent years, the assembly techniques for microspheres have also been dramatically improved. Generally, these techniques belong in three areas: self-assembly, holographic tweezers, and micromanipulation. These include use of self-assembly in pipe-like flows [57–59, 69, 70] for producing long chains of touching cavities, template-directed self-assembly [52, 60, 71, 72] for making arrays and clusters of spheres on grooved substrates, and hydrodynamic flow-assisted self-assembly [64, 73] for synthesizing 3D lattices of spheres. Holographic tweezers [74–79] use a computer-controlled hologram to produce multiple optical traps for microspheres, which can be manipulated to create arbitrary 3D structures. The capability of massively parallel manipulation with hundreds of cavities has already been demonstrated using holographic tweezers, as well as using a method [80] based on optical image-driven dielectrophoresis forces. The last technique requires much smaller optical intensity than the standard optical tweezers. Direct assembly by using a manipulator represents one more possibility for constructing 3D arrays of spheres [81]. It was demonstrated that the gaps between the spheres can be precisely controlled by using stretchable substrates [48]. Particularly interesting results have been obtained in the area of optical transport properties of microsphere resonator circuits (MRCs), where two new concepts were proposed. The first concept concerns the mechanism of transport of WG modes

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in 2D and 3D coupled cavity systems with disorder. It was argued [64] that such optical transport can be considered based on an analogy with percolation theory [82, 83] where the sites of the lattice (spheres) are connected with optical “bonds,” which are present with probability depending on the spheres’ size dispersion. It was predicted that the percolation threshold [64] for WG mode-based transport should be achievable in slightly disordered MRCs. Similar optical percolation effects have been recently observed [84, 85] in porous ceramics. The second concept is connected with the light-focusing properties of microspheres. It has been shown [86–91] that, under plane wave illumination, a single microsphere produces a focused spot, termed “nanoscale photonic jet,” with subwavelength dimensions. Small sizes of such photonic nanojets can be used for detecting nanoparticles [87] and for developing high-resolution optical data storage [92]. It has also been suggested [55] that the focusing effects can be periodically reproduced in chains of spheres. Recently, periodical focusing modes, termed “nanojet-induced modes” (NIMs), were directly observed [57–60] in long chains of polystyrene microspheres (>100 cavities). Thus, MRCs emerged as a novel way of integrating cavities on the same chip, which can provide advantages over conventional structures (microrings, disks, etc.) in specific applications. As an example, size-matched MRCs allow the control of tight-binding photonic dispersions in slow-light devices [53], filters, and arrayresonator LEDs. On the other hand, MRCs with size variations can be used for developing novel spectrometers [93] and sensors [94, 95]. The high sensitivity of the WG mode-based sensor devices [96–102] is well known. Multiple-cavity systems can extend the functionality of such sensors. The focusing properties of microsphere chains and arrays can be used for developing biomedical microprobes and laser scalpels with subwavelength spatial resolution. In this review, we present state-of-the-art technology, theory, and applications in the area of MRCs. In Section 17.2, we consider the tight-binding approximation when applied to microspheres. Section 17.3 is devoted to WG modes in microspheres. Section 17.4 describes the fabrication of MRCs. Section 17.5 explores the optical properties of MRCs. In Section 17.6, we consider applications of MRCs. Finally, in Section 17.7, we draw some conclusions.

17.2 Tight-Binding Approximation for Microspheres The text book description of tight-binding approximation [1, 2] is based on degeneracy of energy levels in identical uncoupled atoms. For photonic cavities, this translates into the requirement that the dimensionless constant κ, describing the coupling between the identical cavities, should be larger than 1/Q, where Q is the quality factor of the cavities. As illustrated in Fig. 17.2a, the dispersion relation for photons [6, 7] in infinitely long chains of cavities is given by ωK =  [1 + κ cos (K)], where K is the wavevector along the chain,  is the uncoupled cavity resonance frequency, and  is the distance between neighboring cavities. The bandwidth of

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the CROW structure can be calculated as ω = 2κ. The group velocity can be obtained from this dispersion relation as vg = ∇K ωK = −κ sin (K). As shown in Fig. 17.2b, the group velocity is reduced to κc at the center of the passband, and it approaches zero at the edges of the dispersion diagram. This dispersion is calculated for κ = 0.0034, which corresponds to the case of touching polystyrene spheres [42, 65] with 4–5 μm diameters. In the case of chains formed by a finite number (M) of cavities, the tight-binding approximation can be reformulated [24], leading to a discretization of the allowed values of the vector: Ki = iπ/(M + 1), where i = 1,. . ., M. Such discrete Ki values have been directly observed in experiments [53] with chains of six size-matched (δ = 0.05%) polystyrene microspheres. The photonic figure-of-merit [103] describing a possibility of achieving a coherent tight-binding transport of WG modes in chains of spheres is presented in Fig. 17.2c. In this figure, our experimentally measured [42, 65] values of parameter κ for the polystyrene spheres in air are shown. Coherent coupling phenomena in such systems are achievable under two conditions: (i) κ>>1/Q and (ii) κ>>δ. It is seen that the first condition is well satisfied for 2–10 μm spheres. This remains true despite the fact that, in real physical structures, the Q-factors of WG modes in microspheres are significantly reduced in comparison with the estimations (Qrad ) based on pure curvature leakage [32, 104, 105], which is shown in Fig. 17.2c. The curvature leakage is a factor limiting Q values only for very small spheres with diameters around 2–3 μm. For larger spheres, the Q values are determined by other factors, such as surface imperfections and material absorption. It should also be noted that if spheres are assembled on the substrate, there are additional factors [104] that affect the broadening of the WG mode resonances, which are discussed in Section 17.3. However, even in this case, as illustrated in Fig. 17.2c, κ exceeds 1/Q by approximately an order of magnitude.

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Unfortunately, the second condition of coherent transport of WG modes, κ>>δ, cannot be satisfied with the commercially available microspheres where δ ∼ 1%, as also illustrated in Fig. 17.2c. In order to achieve tight-binding properties, the spheres should have size variations at least at the level comparable to the dimensionless coupling constant κ. This problem can be solved by using special techniques of selecting size-matched spheres, described in Section 17.5.5.

17.3 Whispering Gallery Modes in Microspheres Light waves can be trapped in cavities with circular symmetry, such as microrings [21–23], microcylinders [14], and microspheres [41–56], due to total internal reflection of light. Below, WG modes are considered in spherical resonators. Due to spherical symmetry, WG modes in microspheres are characterized [34, 35] by radial n, angular l, and azimuthal m mode numbers. The radial number, n, indicates the number of WG mode intensity maxima along the radial direction, whereas the angular number, l, represents the number of modal wavelengths that fit into the circumference of the equatorial plane of the sphere. Such waves propagate inside the sphere close to its surface so that they traverse a distance of about 2π a in a round trip, where a is the sphere radius. The condition of constructive interference of such waves in the cavity with circular symmetry can be approximated as 2π a ≈ l(λ/N), where λ/N is the wavelength in the medium with refractive index N. Under the constructive interference condition, standing WG modes are formed [106] in the cavity. This condition can be expressed in terms of the size parameter, X = 2π a/λ ≈ l/N. In an ideal free-standing sphere with a perfect shape, the azimuthal modes represented by m numbers are degenerate. It should be noted, however, that in practice, this degeneracy is lifted by small deformations of the microspheres from the spherical shape. As illustrated in Fig. 17.3, a single sphere on the substrate is rotationally invariant around the z axis playing the part of the polar axis. In this geometry, the fundamental WG modes with m = l are defined [104] in the equatorial plane of spheres parallel to the substrate. Such fundamental modes have the highest Qfactors, due to the fact they are separated from the surface of the substrate by the radius of the sphere. In contrast, the modes with m< 105 ) for perfect 5 μm spheres.

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540

Fig. 17.3 (a–c) Schematics of a single sphere on a substrate illustrating intensity maxima distributions for azimuthal modes with different m numbers. (d) Fluorescence spectral image and emission spectrum of a single green fluorescent 5.0 μm sphere with TEn l and TMn l WGM peaks measured through the transparent substrate [65, 66]. Reprinted with permission. Copyright 2009 Optical Society of America

The positions of fundamental WG mode peaks can be derived from Maxwell’s equations solved in spherical coordinates by using the Mie scattering formalism [107, 108]: P

J l+1/2 (NX) N l+1/2 (X) = , Jl+1/2 (NX) Nl+1/2 (X)

(17.1)

where P = N for TE polarization (P = N−1 for TM), and the (l + 1/2) term appears, due to translating the spherical Bessel and Neumann functions to their cylindrical counterparts. By expanding the quantities in (17.1) as an asymptotic series in powers of (l + 1/2)−1/3 , it is possible to express first terms of the WG mode resonances [108] in terms of the size parameter: NXn,l

1 =l+ − 2



l + 1/2 2

1/3 αn − √

P N2 − 1

+ ...,

(17.2)

where n is the radial number, and α n are the roots of the Airy function Ai(–z).

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This leads to a quasi-periodic spectrum of WG mode resonances versus l with the frequency separation between the peaks with the same polarization represented by a free spectral range (FSR): √  −1 N2 − 1 c tan .

ω = √ 2π a N2 − 1

(17.3)

The frequency spacing between WG modes having the same n and l but different polarizations can be approximated [109] according to the following formula: TE−TM

ωn,l =

c 2π aN

$

N 2 − 1/N.

(17.4)

For applications in cavity quantum electrodynamics studies, the figure-of-merit +√ for photonic cavities is usually estimated as ∼Q V, where V is the modal volume. The basic property of WG modes is connected with the relatively small volume occupied by the electromagnetic field of the mode in comparison to the total volume of the cavity [34, 110]:

Fig. 17.4 (a) Modeling of the TE1 30 WG mode for a polystyrene sphere with a = 2.25 μm. Linewidth variation versus the azimuthal mode number m for different index of refraction ns of the substrate. (b) Spectrum resulting from a shape deformation of |ε| ∼ 0.01 in absence of the substrate. (c) Normalized spectrum of even m modes and ε = 0 in presence of the substrate [104]. Reprinted with permission. Copyright 2009 Optical Society of America

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Fundamentals and Applications of Microsphere Resonator Circuits

V = 3.4π 3/2

λ 2π N

3

√ l11/6 l − m + 1,

431

(17.5)

The degeneracy of azimuthal (m) modes in perfect spheres can be removed by deformations from the spherical shape [111, 112], leading to the appearance of quasi-equidistant closely spaced peaks. An exact solution [104] for deformed (|ε| ∼ 0.01) polystyrene microspheres with a = 2.25 μm is presented in Fig. 17.4b. The WG mode broadening effect caused by the interaction with the substrate [104] is illustrated in Fig. 17.4a, c. It should be noted that the shape deformation of the sphere breaking the symmetry around the z-axis can mix the modes of different m, thus giving rise to a broadening of all modes, due to tunneling to the substrate. These effects are likely to be responsible for the WG linewidth broadening observed in the experimental spectra of microspheres.

17.4 Fabrication of Microsphere Resonator Circuits In contrast to “hard” semiconductor in-plane fabrication techniques, the synthesis of MRCs relies on a variety of soft condensed matter methods. Soft matter structures and devices are often inexpensive which allows one to quickly iterate and modify designs [113, 114]. These methods can be divided into three major categories considered above: self-assembly, optical tweezers, and micromanipulation.

17.4.1 Self-Assembly During the last two decades, the techniques of self-assembly of spheres with submicron dimensions ∼0.2–0.5 μm have been developed for obtaining 3D photonic crystal structures. These techniques include slow sedimentation in gravity for growing synthetic opals [115–118], vertical deposition techniques [119, 120] for obtaining well-ordered layers of spheres, template-directed colloidal deposition techniques [71, 72, 121–123] for synthesizing well-ordered arrays of spheres on patterned substrates, techniques of 3D self-assembly in hydrodynamic flow [73], and the techniques of self-assembly in pipe-like flows [69, 70] for producing long chains of touching cavities. The size of the spheres used in MRCs (∼2–10 μm) is larger than in opal-like structures by an order of magnitude. For such enlarged spheres, the sedimentation times are orders of magnitude shorter than those for conventional opals. This does not allow obtaining ordered structures by conventional sedimentation techniques on flat substrates. The quality of ordering can be improved by using dents, grooves, or other patterns [52, 71, 72] fabricated in the substrate. The spheres tend to occupy lowest energy positions in the patterned templates which results in better ordered structures. The ordering of spheres can be further improved [60] by using effects of

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(a) Pinning point

(b)

(c)

30 μ m

(d)

(e)

Fig. 17.5 (a, b) Schematic of the self-assembly by slowly moving front of liquid and SEM image of the microspheres on a patterned substrate [60]. Reprinted with permission. Copyright 2009 Optical Society of America. (c–e) Self-assembly in pipe-like flows on a flat substrate [57–58]. Reprinted with permission. Copyright 2009 American Institute of Physics

packing of microspheres by a slowly moving front of evaporating liquid illustrated in Fig. 17.5a, b. Another technique of self-assembly [57–59] suitable for fabrication of straight chains of spheres in a touching position on a flat substrate is illustrated in Fig. 17.5c–e. In this method, the liquid film with spheres is sandwiched between two hydrophilic glasses. The evaporating liquid film forms pipelike flows, stretching from massive deposits of microspheres to “pinning” points nucleated by bigger spheres or by the defects on the substrate. Evaporation of microflows in a lateral direction leads to the formation of straight and long chains of touching spheres. As shown in Fig. 17.5d, e, this technique allows obtaining perfectly aligned chains [57, 58] with less than 1 defect per 100 spheres. Similar results can be obtained by a modified technique [124] where the self-assembly of microspheres occurs in the proximity of a glass rod placed on a flat substrate. In this technique, during the evaporation of the solvent, the spheres are dragged into the meniscus of the suspension between the glass rod and substrate. Ordered chains of microspheres have been obtained after complete solvent evaporation, [124]. Fabrication of 3D lattices of spheres can be achieved by using hydrodynamic flows of microspheres in confined geometries [64, 73]. In this technique, the suspension of microspheres is injected into a cell with the porous walls allowing the liquid portion of the suspension to drain through, while retaining the spheres inside the cell. The realization of this method is straightforward for enlarged spheres [64] used in MRCs, since the size of the pores in the walls should be smaller than the size of the spheres. The thickness of the 3D lattice of spheres is determined by the thickness of the cell. An ultrasonic bath is used to speed up the synthesis of 3D structures. This process usually yields polycrystalline samples containing sizable (∼100 μm) domains with face-centered-cubic (FCC) packing of the microspheres.

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17.4.2 Optical Tweezers An optical trap can be created [125] by focusing a laser beam to a diffraction-limited spot using a high numerical aperture objective lens. As schematically illustrated in Fig. 17.6a, the strong light gradient near the focus creates a potential well, in which a particle with a refractive index higher than that of the surrounding aqueous medium is trapped [126]. By using multiple optical traps, up to 400 particles can be simultaneously manipulated [126, 127]. An example of manipulation with multiple polystyrene microspheres [127] is shown in Fig. 17.6b. The arrays of microspheres can be held in position using holographic tweezers [74, 75]. After that, the surrounding medium can be gelled to fix the spheres in place [76]. Alternatively, the spheres can be forced together using holographic tweezers and caused to adhere by controlling the colloidal interaction in the system [79]. An advantage of using multiple traps is connected with a possibility of direct assembly of 3D structures with arbitrary configurations of microresonators [78, 79]. An example of a FCC crystal structure [79] formed by 3 μm polystyrene microspheres is illustrated in Fig. 17.6c. An interesting method [80] has been developed recently in which particle manipulation is achieved, due to the combined effect of an electric field and illumination. This optical image driven dielectrophoresis technique permits (a)

(b)

(c)

Fig. 17.6 (a) Schematic of the focused beam with a colloidal particle drawn toward the focus [126]. Reprinted with permission. Copyright 2009 Nature Publishing Group. (b) Water-borne 0.8 μm polystyrene spheres trapped in a plane and reconfigured with dynamic trapping patterns [127]. Reprinted with permission. Copyright 2009 Elsevier. (c) Three layers of FCC structure made from 3 μm silica spheres [79]. Reprinted with permission. Copyright 2009 Optical Society of America

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high-resolution patterning of electric fields on a photoconductive surface for manipulating single particles. It requires 105 times less optical intensity than optical tweezers.

17.4.3 Micromanipulation Small-scale structures containing a limited number of microresonators can be fabricated by micromanipulation. The mechanical microassembly requires the position of the spheres on the substrate to be stable. This can be achieved by using patterned substrates, as shown in Fig. 17.7a. The nanorobotic manipulation of 0.9 μm spheres using SEM can obtain perfectly ordered structures as seen in Fig. 17.7a. The disadvantage of nanorobotic manipulation is connected with the fact that it is a time-consuming process.

(a)

(b)

Fig. 17.7 (a) Nanorobotic manipulation in SEM showing six layers of body-centered-cubic structure formed by 0.9 μm spheres assembled on a patterned substrate [81]. Reprinted with permission. Copyright 2009 Wiley-VCH. (b) First Korean letter formed by 5 μm polystyrene microspheres manipulated by a tapered fiber [65]. Reprinted with permission. Copyright 2009 Optical Society of America

It should be noted that micromanipulation is greatly simplified when using large spheres as in MRCs. The micromanipulation can be provided by tapered fibers or metallic micro-needles connected with hydraulic micromanipulators. An example of a structure formed by 5 μm polystyrene microspheres on a flat glass substrate [65] is represented in Fig. 17.7b. The spheres are attached to the substrate, due to a surfactant with gluing properties used by the manufacturer of microspheres. These techniques are convenient for building bispheres or small clusters of spheres. They can also be used for sorting size-matched spheres as described in Section 17.5. The variation of the inter-sphere gaps can be achieved by using the substrates with elastomeric properties [48]. This technique was developed for PDMS substrates obtained with a reduced concentration of a cross-linker to provide increased elasticity. Using this technique the inter-resonator gaps were controlled [48] in the 0–2.5 μm range with 20 nm accuracy.

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17.5 Optical Properties of Microsphere Resonator Circuits In recent years, the focus of theoretical studies of coupled cavities has been on their spectral properties. Coupling between identical cavities is well studied at present time, especially for 2D systems, such as coupled microrings and microdisks [8–16]. The degeneracy of WG modes can be lifted in such systems that result in the spectral properties depending on the configuration of cavities and on the strength of their coupling. It is shown that various types of coupled microresonators demonstrate common spectral properties. Examples of such coupled systems include biatomic photonic molecules formed by planar semiconductor microcavities [128], patterned microcavities [17], photonic crystal cavities [129], microrings [11], microdisks [130], and microspheres [41–49]. Integration of several identical cavities in symmetric photonic molecules leads to the formation of high-Q supermodes [131]. It has been shown that the photonic dispersions can be modified in infinite 2D arrays of coupled microrings [132]. In structures with positional disorder of cavities, the collective resonances of multiple cavities can be calculated using multipole formulation [133]. Some of these results are reviewed in more detail in Chapter 16 of this book. In this section, we focus on the optical properties of coupled microspheres, including both spectral and transport effects. We consider effects which are essential for developing MRCs, such as the role of the substrate, optical losses in size-disordered structures, efficiency of coupling between size-mismatched cavities, percolation of WG modes in 2D and 3D structures, effects of periodical focusing of light in chains of spheres, and methods of selecting spheres with particularly high size uniformity.

17.5.1 Role of Substrate: Fundamental WG Modes Similar to individual spheres considered in Section 17.3, in coupled cavity structures, the interaction with the substrate plays an important role in the optical transport properties. Contact with a high-index substrate leads to a damping of many WG modes in individual spheres. This factor complicates the theoretical understanding of the WG tight-binding effects in such structures. Indeed, in the absence of the substrate, a bispherical system (as the simplest example) is characterized [43, 49] with a polar axis connecting the centers of two cavities, which means that the single cavity fundamental WG modes are defined in the planes perpendicular to such polar axis, as shown in Fig. 17.8a. If a bisphere is assembled on a substrate, however, such fundamental modes should be damped, due to their leakage in the substrate. It is well known [65, 66, 104] that only WG modes with the intensity maxima separated from the substrate have sufficiently high Q-factors. In the case of flat substrates, this favors WG modes located in the equatorial plane of spheres parallel to the substrate, shown in Fig. 17.8b. In the case of spheres assembled in V-grooves, this factor favors WG modes in a plane perpendicular to the substrate [53], as illustrated in the inset of Fig. 17.1b.

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Fig. 17.8 (a) Fundamental modes for free-standing bispheres and (b) bisphere on a substrate [134]

(b)

y

In this review, the uncoupled WG eigenstates of the coupled cavity system are introduced in a similar manner as in the case of a single sphere on the substrate, as illustrated in Fig. 17.8b. This approach describes a weak coupling between the cavities when the perturbation of WG modes caused by their interaction is small. It should be noted that, in a strong coupling regime, the hybridization effects for WG modes in the presence of the substrate are not well studied at present.

17.5.2 Optical Transport in Disordered Structures As shown in Section 17.2 in overcoupled structures (κ>>1/Q) formed by relatively uniform (κ>>δ) resonators, the optical transport should be very efficient. The last condition, however, is not satisfied in real physical assemblies of microspheres. This problem is generic for all types of chip-scale coupled cavity structures including microrings obtained by the precise CMOS technology [23]. The cavities’ size variations reduce the efficiency of evanescent coupling between their WG modes. As a result, WG modes are partially reflected at the contact points between the spheres, leading to the localization of light [24–26] and scattering losses. The average losses (∼0.3 dB per ring) have been reported [23] for silicon microrings with ∼0.1% standard size variations. In MRCs, the transport of WG modes is expected to be less efficient due to larger standard deviation of the spheres’ diameters ∼1%. The measurements of losses were performed [52, 58] in long chains of 5 μm polystyrene microspheres, as illustrated in Fig. 17.9. In these experiments, a dye-doped sphere was used as a source (S) of evanescently coupled WG modes, as shown in Fig. 17.9a. The transport of light at different wavelengths was studied using spectral scattering images, shown in Fig. 17.9b. As illustrated in Fig. 17.9c, in the first few spheres adjacent to the source, the average propagation loss at the WG mode wavelengths is found to be ∼3 dB per sphere. The optical transport was found to be more efficient at the wavelengths away from the resonance with WG modes. This nonresonant mechanism of optical transport is considered in Section 17.5.6.

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< 1 dB/sphere

~ 3 dB/sphere

S (a)

(b) 520

530

540

Wavelength (nm)

(c)(c) 1

10

x5 2

10

3

10

4

10

10

5

Intensity (a.u.)

Fig. 17.9 Scattering spectroscopy [58]: (a) image of the chain in contact with the dye-doped S sphere with the central section indicated by the dashed line, (b) scattering spectral image at the central section in the spheres equatorial plane, and (c) intensity distribution measured at 527.3 nm TE1 41 WG mode peak (solid line) and away from the WG peaks at 525.0 nm (distribution with the filled area under the curve). The dashed lines are to guide the eye [58]. Reprinted with permission. Copyright 2009 American Institute of Physics

17.5.3 Efficiency of Coupling Between the Size-Mismatched Cavities The simplest system for studying the efficiency of evanescent coupling between microcavities is represented by biatomic photonic molecules. It is well known that the classical system of strongly coupled cavities can be considered by analogy to a quantum mechanical strongly driven two-level system [135]. One of the examples of such systems is represented by microcavity polaritons [136], where excitons confined in quantum wells are strongly mixed with the optical cavity modes. Each polariton mode, |p, is a linear combination of an exciton, |e, and a cavity photon, |l: |p = ce |e + cl |l. Exactly at resonance, both polaritons consist of equal mixtures of the exciton and photon states, so ce = ±cl = 2−1/2 . The symmetric and antisymmetric modes are then split by a finite energy, known as the vacuum Rabi splitting [137]. When the system is detuned from resonance, the mixture is not equal, with one mode predominantly excitonic in character (ce → 1, cl → 0) and the other predominantly photonic (ce → 0, cl → 1) [138, 139]. A similar approach can be used for photonic molecules, where the fractions of individual cavity modes in the strongly mixed molecular states define the coupling efficiency between photonic atoms. It should be noted that the behavior of coupled cavities in the vicinity of energy level anticrossing exhibits several interesting effects, including the formation of so-called exceptional points [140], where the complex eigenvalues of the

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Fig. 17.10 (a) Antibonding and (b) bonding states. (c–h) Spectral energy density in receiving (R) sphere for different gaps (d). (i, j) spectra of WG eigenstates [47]. Reprinted with permission. Copyright 2009 American Institute of Physics

(b)

(a)

R

z

RR

S

SS

y x

Δ~0 Δ>0

Δ<0

NMS

(c)

d = 0 μm

Spectral energy density IR (a.u.)

(d) 0.1 (e)

0.2

(f) 0.3 (g) 0.5 (h) Δ

TE 1 22 TE 1 21

TE 1 23

TE 1

TE 117

18

1.0

(i)

TE 1 16

2.0 (j)

2.0

530 540 550 560 570 580 Wavelength (nm)

NMS (nm)

Weak Coupling

Strong

6

(a)

4 1

2

Linewidth

Coupling Efficiency η (a.u.)

2 0 1

1

0.1

(b)

(c)

x 30

2

R

3 0.01

0.0

0.2

0.4

0.6

0.8

S

1.0

Inter-Resonator Gap d (μm)

Fig. 17.11 (a) Normal mode splitting (NMR) in resonant case as a function of d (line 1). (b) The coupling efficiency (η) as a function of d for different detuning ( ) [47]: 1– = 0.5 nm, 2– = 5 nm, 3– = –7.5 nm. (c) Distorted mode in the R sphere in detuned cases [47]. Reprinted with permission. Copyright 2009 American Institute of Physics

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439

corresponding levels coalesce. It also includes the formation of long-lived, scarlike modes [141]. For coupled microdisks, the behavior in the vicinity of resonance crossing has been studied using numerical modeling [142, 143] and also experimentally [144, 145]. For size-mismatched (3.0–2.4 μm) spheres with index 1.59, the coupling effects were studied [47] using 3D FDTD modeling, as illustrated in Fig. 17.10. One of the spheres was used as a source (S) of WG modes. The spectra of electromagnetic energy (IR ) were calculated in a second receiving (R) sphere. The effect of the substrate was mimicked by launching the WG modes in the plane containing the axis of the bisphere. Due to the fact that each resonator produces its own comb of uncoupled WG mode peaks in Fig. 17.10i, j, it was possible to study coupling phenomena for various detuning ( ) between closest resonances. The regime of strong coupling is evident in all cases under the spheres’ touching condition (d = 0), as illustrated by classical bonding and antibonding molecular states in Fig. 17.10a, b. Increasing the separation between the cavities leads to a gradual transition to weak coupling shown in Fig. 17.11a. As a measure of the total energy (ER ) deposited in the R cavity, the area under the spectral peaks was used. A rough estimate of the coupling efficiency (η) was obtained by normalizing ER by similarly estimated energy (ES ) in the S sphere: η = ER /ES. The results of calculating η for different detuning, , as a function of d are summarized in Fig. 17.11b. It is seen that, in a contact position, the coupling efficiencies are typically ∼0.1–0.2, regardless of detuning. In strongly detuned cases, the modes induced in the R sphere exhibit an irregular noncircular shape, as illustrated in Fig. 17.11c. Such weakly coupled modes have a shape that can be continuously adjusted to variations in frequency of the WG mode in the S sphere [47]. Such coupling between a discrete energy state (true WG mode in the S cavity) and a continuum of irregular modes in the R cavity resembles Fano resonance [146] phenomena observed in photonic crystal waveguides [147] and side-coupled waveguide-cavity [148] systems.

17.5.4 Percolation of WG Modes in 2D and 3D Structures The optical transport in 2D and 3D structures [61–64] is a much more complicated phenomenon compared to that in single chains of cavities [50–60]. In close-packed 2D arrays of spheres, each cavity has six neighbors. In close-packed 3D lattices, each cavity has 12 neighbors. This means that the probability of finding a neighbor with highly efficient WG mode coupling is higher in the 2D case, and much higher in the 3D case, compared to bispheres (one neighbor) or chains (two neighbors). The optical transport in such 2D and 3D systems can be considered as analogous to the bond percolation theory [82, 83]. This is illustrated in Fig. 17.12a for a square lattice of sites connected with bonds with probability p [83]. If p is smaller than a critical value, pc , termed the bond percolation threshold (pc = 0.5 for the lattice of square sites), only finite clusters of connected sited can be formed. However, if p exceeds this critical value, an infinite or giant cluster is formed, which means that it

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(a)

(c) d

OPO I scatt

(b) (d) 10000 Pump

Iscatt (arb. units)

1000 100 10 1 Background level 0.1 510 520 530 540 550 560 570 Wavelength (nm)

Fig. 17.12 (a) Illustration of bond percolation in 2D [83]. Reprinted with permission. Copyright 2009 American Physical Society. For p = 0.315 (left), which is below the percolation threshold, pc = 0.5, the connected nodes form isolated clusters. For p = 0.525 (right), the largest cluster percolates. (b) Optical passband and two peaks of the normalized group delay in a chain of uniform microresonators [10]. Reprinted with permission. Copyright 2009 Optical Society of America. (c) Optical set up for FL transmission measurements in 3D lattices of dye-doped microspheres. (d) FL transmission spectra demonstrating double peak structures at high pumping conditions [64]. Reprinted with permission. Copyright 2009 Optical Society of America

should be possible to propagate the entire thickness of the lattice through the system of connected bonds. In the case of optical cavities, the probabilities of the bonds are represented by the WG mode coupling efficiencies. As it was already stated, the transport between identical cavities can be very efficient, leading to the formation of a passband in transmission [6, 7, 10] shown in Fig. 17.12b. It also leads to the formation of two peaks of normalized group delay [10, 12] at the edges of the passband. Such double peak structures appear due to increased dwelling time of light in the structure, which is caused by the partial reflections of light between the cavities. These phenomena were studied [64] in 3D lattices formed by closely packed dye-doped polystyrene microspheres. By using optical pumping, the WG modes

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were locally excited in multiple spheres located in the near-surface region of the sample, as illustrated in Fig. 17.12c. At small pumping levels these WG modes can be detected from the opposite face of the sample as inhomogeneously broadened peaks superimposed on a broad background, as shown in Fig. 17.12d. At high pumping levels, each peak was found to split into a double peak structure, very similar to that is shown in Fig. 17.12b. The magnitude of splitting was found to be similar to that in the calculated [43, 47] and experimental [42, 44–46, 65, 66] spectra of resonant bispheres with comparable size. These results can be explained [64] by the presence of finite clusters of size-matched spheres inside otherwise disordered samples, which are well connected at WG mode wavelengths. These results also suggest the possibility of achieving an “optical percolation threshold” for WG mode-based transport in such systems by selecting spheres with smaller standard size variation δ. It is interesting that a similar approaches to understanding optical transport properties have been applied to pores ceramics [84, 85], where the optical percolation threshold has been experimentally observed.

17.5.5 Selection of Size-Matched Spheres The straightforward way of selecting size-matched spheres is based on their micromanipulation, controlled by spectroscopy [42, 44–46, 53, 54]. Sorting of microspheres with δ ∼ 1/Q can be obtained by this technique. For diatomic molecules formed by almost indistinguishable classical “atoms,” a series of optical coupling effects has been observed. First, this includes observation of normal mode splitting for fundamental WG modes [42, 44–46] with the orientation determined by the interaction with the substrate, as it is discussed in Section 17.5.1. Second, the lifting of the degeneracy of azimuthal WG modes can be directly observed in such structures. For photonic molecules (PMs) with δ ∼ 0.07% assembled in microwells, it was observed in experiments with off-axis excitation and detection [46], as illustrated in Fig. 17.13a, b. For PMs with δ ∼ 0.03% assembled on the substrate [65, 66], the lifting of degeneracy of azimuthal modes results in peculiar shapes resembling kites in the spectral images, as illustrated in Fig. 17.13c. Such kites give direct spectral evidence for maximal coupling of WG modes in the equatorial plane of spheres determined by the substrate. The coupling constant κ was quantified [65, 66] for such modes with respect to sphere sizes and modal numbers. Future development of MRC technology is likely to be connected with massively parallel sorting of spheres [67, 68] based on using size-selective radiative pressure. It has been theoretically predicted [67] that when a microsphere is illuminated by an evanescent wave, the optical forces on- and off-WG mode resonance can differ by several orders of magnitude, as illustrated in Fig. 17.14. This technique allows for potentially accurate size-selective manipulation, as well as parallel particle-sorting according to their size or resonant frequency with δ ∼ 1/Q size

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(a) Substrate Lens Slit (b) TM236

520 Wavelength (nm)

2

TM2 35 TM 35

(c) 2

TM2 34 TM 34

540 530 Wavelength (nm)

Fig. 17.13 (a) The photoluminescence (PL) spectra of the diatomic PM (δ ∼ 0.07%) in a microwell measured with off-axis excitation and detection [46]. Reprinted with permission. Copyright 2009 American Physical Society. Inset: microscope image of the PM. The dark cross indicates the excitation position. (b) The spacing between neighboring peaks. (c) Diatomic PM (δ ∼ 0.03%) assembled near the edge of the substrate, experimental configuration and spectral image illustrating shapes resembling kites near each WG mode resonance. For clarity this is illustrated by dashed boxes for the second-order TM peaks [65, 66]. Reprinted with permission. Copyright 2009 American Physical Society

Fig. 17.14 The optical force acting on a microsphere with a ∼ 2.317–2.375 μm at λ = 520 nm. (a) A schematic illustration of how the evanescent wave excites a WG mode. (b) For the sphere’s dielectric permittivity ε sphere = 2.5281 + 10−5 i. The incident wave is an s-polarized evanescent wave with k|| /k = 1.25 and I0 = 104 W/cm2 , where k|| is the component of the wavevector parallel to the wave’s propagating direction [67]. Reprinted with permission. Copyright 2009 American Institute of Physics

uniformity. One of the challenges in developing this technique is due to the reduction of Q-factors for WG modes in spheres immersed in a liquid. Developing these techniques will have a profound impact on modern photonic integration technologies, due to controllable tight-binding photonic dispersions in such structures and a wide variety of structures and devices which can be assembled using size-matched microspheres.

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17.5.6 Nonresonant Light-Focusing Properties of Chains of Spheres While previous sections considered spherical cavity resonant properties, these cavities can also be used to focus light having a broad range of wavelengths. For sufficiently large (a>>λ) microspheres, the focusing effects follow the laws of geometrical optics [149]. For spheres with radii comparable to the incident radiation wavelength, near-field optical effects become an important which result in extremely small sizes of the focused spots [150–154]. It has been demonstrated that when the focus point is close to the surface of the sphere, the beam width in the focus area becomes smaller than the wavelength over a distance of a few wavelengths [86–91]. This unique distribution of electromagnetic intensity has been termed a photonic nanojet [86] based on its appearance. It has been shown that the nature of electromagnetic field outside the sphere is strongly dependent on its index of refraction [55]. For index N = 1.59, the focused beam possesses not only evanescent fields, but also radiative-mode components, as illustrated in Fig. 17.15a for a plane wave-illuminated (λ = 400 nm) sphere with a = 1.5 μm. In contrast, for N = 1.8, the optical power within such a microsphere is totally internally reflected, and it can tunnel out of the sphere through evanescent coupling, as illustrated in Fig. 17.15b. It has been suggested that, in chains of coupled cavities, this should lead to two different mechanisms [55] of optical transport: nonresonant micro-lensing, Radiative photonic nanojet

Evanescent field (b)

(a)

N = 1.59 –3 –2 –1 0 1 Z (µ m)

N=1.8 2

Nonresonant micro-lensing

(c)

3

–3 –2 –1 0

1

2

3

Z (µ m) Resonant WG mode coupling

(d)

Fig. 17.15 Visualization of a photonic nanojet [55] emerging at the shadow-side surface (circle) of a dielectric microsphere (a = 1.5 μm) with the refractive index N of (a) 1.59 and (b) 1.8. The sphere is illuminated by a plane wave with λ = 400 nm. Electric field intensity distributions of a chain of five touching microspheres that have a refractive index of (c) 1.59 at λ = 429.069 nm and (d) 1.8 at λ = 430.889 nm [55]. Reprinted with permission. Copyright 2009 Optical Society of America

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illustrated in Fig. 17.15c, and resonant WG mode-based coupling, shown in Fig. 17.15d. Such periodical focusing effects have been directly observed [57–60] in the long chains of touching polystyrene microspheres with 2–10 μm sizes. As illustrated in Fig. 17.16a, b, the quasi-periodical series of nanojets was visualized [57] by using imaging spectroscopy and microscopy. One of the interesting results, which can lead to applications of coupled microspheres, is the progressively smaller focused spots’ sizes observed along such chains, as illustrated in Fig. 17.16b,c. It was shown that the diffractionlimited spot sizes can be obtained in such structures for chains containing less than ten spheres [57]. The period of these quasi-periodic nanojet-induced modes (NIMs) propagating in such structures was found to be equal to the size of two spheres with index 1.59 [58], as illustrated in Fig. 17.16d. These modes are most efficiently coupled to the collimated incident beams. For noncollimated incident beams, the maximum losses ∼2–3 dB per sphere occur in the first few spheres, as shown in Fig. 17.16e. Further along the chain, the losses become progressively smaller, reaching a very small level ∼0.08 dB per sphere [58]. For this reason, NIMs are the most favored modes surviving in the long straight chains of coupled microresonators with disorder. Such focusing effects take place in a broad range of wavelengths, which can be used in a variety of biomedical applications requiring a combination of tight focusing and high optical output.

(d)

(e)

2 μm

3 μm

10 μm

Fig. 17.16 (a) Coupling of light in the chain using FL spheres as a local source [57]. Reprinted with permission. Copyright 2009 Optical Society of America. (b) Scattering image illustrating periodical focusing by the spheres with a = 1.45 μm with progressively smaller spot sizes. (c) Cross-sectional width of the spots illustrating their diffraction-limited dimensions. (d) Periodical focusing in 5 μm spheres with N = 1.59 obtained by FDTD numerical modeling. (e) Attenuation of the nanojets intensities along the chains formed by 2, 3, and 10 μm spheres [58]. Reprinted with permission. Copyright 2009 American Institute of Physics

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17.6 Applications of Microsphere Resonator Circuits The existing methods of fabrication of coupled cavities have a number of drawbacks, which limit the functionality of the resulting structures. For example, while lithography is a flexible technique for generating different optical structures, the technique is essentially two-dimensional, meaning that many steps must be repeated to create 3D structure. Microspheres, on the other hand, can be assembled in arbitrary 3D structures using a variety of inexpensive methods such as self-assembly, optical tweezers, or micromanipulation. Most importantly, the microspheres can be pre-sorted with the size uniformity [42, 44–46, 65–68], δ ∼ 1/Q, which is not available for conventional techniques of fabrication of microrings, disks, toroids, or photonic crystal defects [21–23, 27–29, 155]. This extraordinarily high level of size uniformity allows developing novel structures and devices with unique functionality. A particularly interesting area is represented by the resonant optical forces in systems of size-matched spheres. The use of optical forces in optomechanical cavities [156] and integrated photonic circuits [157, 158] has attracted significant attention recently due to their novel physical properties and applications in nanomechanical resonators. It is known that even size-disordered microspheres can be assembled in relatively regular arrays [159] using evanescent optical fields. In size-matched microspheres, the resonant optical forces between coupled WG resonators are expected to be much stronger [160], which should result in the development of new methods of assembling optically bounded structures and devices. Resonant positions of WG modes in size-sorted cavities open new possibilities for developing laser applications with advanced functionality. Examples include bistable [161] and single-mode [162] lasing, observed in different coupled cavity structures. Another example is represented by coherently driven high-power laser arrays, such as VCSELs [163–165]. A monolayer of close-packed microspheres behaves as an array of laterally coupled microlasers [64]. Such arrays of active (dyeor Er-doped) size-matched microspheres can be integrated with the semiconductor lasers, planar waveguides, or fibers, in order to develop coherently driven sources of light with applications in communications, imaging, and other fields of optical engineering. Tight-binding photonic dispersions can be used for developing high-order spectral filters [8] and tunable delay lines [6]. It should be noted that, in recent years, there has been significant progress in developing designs [166, 167] and applications [21–23, 168, 155] of coupled microrings. As an example, the thermal tuning of silicon oxinitride (SiON) microrings permitted the realization of an error-free, continuously tunable delay at 10 Gbit/s in a reconfigurable on-chip delay-line [168]. The practical realization of such structures is, however, complicated by the necessity to provide a final tuning of the individual cavities. Size-sorted microspheres can provide advantages in such applications. The control of the tight-binding photonic dispersions has been demonstrated [53] in linear chains of size-matched (δ ∼ 0.05%) spheres.

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Another interesting area of applications of MRCs is connected with the control of light–matter interaction in coupled cavities [169] through constructive interference of the resonator decays. It has been shown [169] that the temporal decay rate of a parallel resonator chain is proportional to the number of resonators, while the intensity of the decay is proportional to the square of the number of resonators, when all the resonators are prepared in the same state. These superradiance properties can be used in life science research and clinical diagnostics dealing with small concentrations of analyte molecules, due to enhanced fluorescent properties of surface-bound molecules. The physical mechanism of such collective emission and absorption effects is based on the interference of resonator decays, which plays an important part in other related effects, such as coupled resonator-induced transparency (CRIT) [170–173] and Bragg reflection of light from multiple quantum-well structures [174, 175]. The superradiance occurs when the individual cavity resonances are weakly coupled through waveguides or fibers under the condition, L = 2π a, where L is the path length between the resonators. In order to observe these effects in practical structures, the size uniformity of the cavities should be at the level of δ ∼ 1/Q, which in many cases exceeds the requirements for observation of tight-binding WG mode transport in overcoupled structures (δ ∼ κ). MRC technology can provide the required level of uniformity of cavities, thus opening the prospect for achieving superradiance properties in massive numbers of spheres with potential applications in ultrasensitive fluorescent sensors. Due to high-Q factors of WG modes in spherical, cylindrical, or toroidal cavities, they are widely used in lab-on-chip and sensor applications [96–102]. In such microresonators, WG modes circulate thousands of times inside the cavity, which increases their interaction length with the nanoparticles, such as DNA, molecules, colloidal particles, or quantum dots, located in the vicinity of the resonators’ surface. The conventional spectroscopic sensors [96–98, 100, 101] operate based on measurements of a spectral shift or broadening of corresponding WGM resonances. These studies have made it possible to develop sensors with extraordinarily high sensitivity, enabling single molecule detection [98]. Most recently, a new type of sensor device, based on the measurement of a fine splitting of WG mode peaks by the nanoparticles embedded in the cavity, has been proposed [99, 102]. In comparison with conventional spectroscopic sensors, such devices are less sensitive to the local heating effects and other environmental factors that give them a significant performance advantage in practical applications. Clusters of microspheres have several important advantages over single spheres in sensor applications. Usually, a regime of weak coupling between WG modes in individual spheres is provided in such sensing clusters. It can be achieved either by separating cavities with resonant WG modes by a submicron gap [47, 95] or by assembling size-mismatched spheres [47, 48, 62, 64, 94] in a contact position. In weakly coupled clusters of identical spheres, the sensitivity of the molecule detection can be considerably enhanced, because a background-free detection can be implemented by either a spectral or spatial shift between the object and sensor

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[95]. This allows achieving novel sensor functionalities, such as detection of the orientation of the dipole moment of the molecules attached to the spheres’ surface [95], as illustrated in Fig. 17.17. On the other hand, the clusters formed by size-mismatched spheres usually display spectra containing very dense groups of WG mode peaks, due to contributions of all constituting cavities [62, 64, 94], as illustrated in Fig. 17.18b. Such spectra, compared to the single sphere emission spectra [94] in Fig. 17.18a, are more suitable for developing multi-wavelength sensors. In addition, clusters of microresonators offer the advantage of exhibiting specific WGM spectra, which can be considered as their fingerprint [94]. Therefore, individual clusters can be traced throughout an experiment even without the knowledge of their precise positions. Besides sensor applications, arrays of size-mismatched microspheres can be used for developing compact high-resolution spectrometers [93]. The principle of operation of these devices has some similarity with multimodal-multiplex spectroscopy [176] in disordered photonic crystals, where wavelength-dependent patterns of scattered light have been used for identifying different wavelengths. However, the devices based on mesoscale (a ≥ λ) spheres have better spectral resolution due to their high-Q WG mode resonances. The input light from a waveguide is resonantly coupled to such arrays of size-mismatched microspheres creating

Fig. 17.17 Relative intensity distribution for the three-sphere (a = R = 1.4 μm, N = 1.59) cluster calculated for λ = 572 nm. Orientation of the dipole moment is (a) along x-axis, (b) along y-axis, and (c) along z-axis. Inset: Layout of the sensing cluster of three identical weakly coupled spheres. The dipole source (asterisk) is located on a line connecting the centers of spheres 1 and 2. Sphere 3 is the sensing sphere and probes the dipole moment orientation [95]. Reprinted with permission. Copyright 2009 American Institute of Physics

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Fig. 17.18 WGM spectra of (a) a single microsphere (a = 5 μm, N = 1.59) in an aqueous environment and (b) a cluster of three microspheres, respectively, upon sequential deposition of polymer layers [94]. Reprinted with permission. Copyright 2009 American Institute of Physics. Spectra 1–4 correspond to increasing thickness of the polymer layers

Fig. 17.19 Intensity distribution on the resonator array for three different wavelengths. From left to right: λ = 685.3980, 685.3569, 685.3189 nm [93]. Reprinted with permission. Copyright 2009 Optical Society of America

wavelength-dependent intensity distributions, as shown in Fig. 17.19. A wavelength resolution of λ/ λ = 7 × 107 has been demonstrated [93] for such devices. Finally, the microspheres can be used for developing applications based on their nonresonant properties of focusing light. This area was stimulated by recent observations of photonic nanojets [86–92] and near-field effects in the focusing properties [150–154] of mesoscale spheres. In such photonic nanojects, the intensity of light in the middle of the focused spot exceeds the intensity of the illuminating wave by a factor of ∼ 103 . The focused spot has the full-width-half-maximum (FWHM) beamwidth between one-third and one-half of the wavelength in the background

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medium and is only weakly dependent on the size of the dielectric sphere. The light waves propagate in the focal spot area with little divergence for several wavelengths. These properties have been used in several application areas, such as nanoparticle detection [87, 90], detection of subwavelength pits in the optical data-storage disks [92], locally enhanced Raman scattering [177], high-resolution Raman imaging [178], and subwavelength direct-write nanopatterning [179]. Large-area arrays of closely packed microspheres can be used for laser-induced periodical surface patterning [151]. In this case, the illumination is provided perpendicular to the plane of array, and the spheres behave as individual microlenses. A particularly interesting case is represented by illumination along the long chains of microspheres, leading to the periodical focusing of light effects [57–60] described in Section 17.5.6. Propagation of light in such chains leads to the formation of photonic nanojets with progressively smaller sizes, which can be used for developing a variety of biomedical applications, including ultra-precise laser tissue surgery. The optical transport losses of such systems can, in principle, be reduced by assembling chains of spheres inside hollow waveguides. For a certain range of parameters of spheres and waveguide the resulting transmitted and reflected power has been analyzed [180] using a rigorous integral equation formulation.

17.7 Conclusions The studies of different types of coupled cavities, microrings, disks, toroids, photonic crystal defects, and microspheres are based on common ideas of tight-binding of photonic atoms [3–7], critical coupling [181], impedance matching [182], Fano resonances [47, 146–148], optical supermodes [131], disorder-induced scattering [24], localization [25, 26], and percolation [64] of light. From an engineering perspective, some of these structures show great potential for use in device applications. Examples include ultracompact buffers on a silicon chip [23] and error-free continuously tunable delay lines at 10 Gbit/s based on silicon oxynitride microrings [168]. Along with already established devices, this area shows significant potential for discovering new physical effects and designing structures with unique functionality. Examples of this include novel structures harnessing optical forces in integrated photonic circuits [156–160], effects of collective emission and absorption [171], and effects of dynamical tuning of cavities [183]. The resonant optical properties of coupled cavities can be engineered on the basis of tight-binding WG modes in photonic atoms. In practice, however, this requires an independent tuning of the individual cavity resonances, which is a rather difficult problem in chip-scale structures. Microspheres, in this regard, are unique cavities since they can be pre-sorted with the size uniformity, δ ∼ 1/Q, which is not available for other types of cavities fabricated by the well-established lithographic and etching techniques. Small quantities of size-matched spheres can be sorted by spectroscopically controlled micromanipulation. Future development of MRC technology is likely to be connected with massively parallel sorting of spheres based on using size-selective radiative pressure [67, 68].

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This extraordinarily high level of size uniformity allows the development of novel structures and devices with unique functionality. Due to controllable dispersions for photons, collective emission and absorption, and enhanced light–matter coupling, such structures can be used for developing coupled arrays of microlasers, ultracompact high-resolution spectrometers, and sensors. Cavities with high-Q WG modes are widely used in lab-on-chip and sensor applications [96–102] for detecting nanoparticles, such as DNA, molecules, colloidal particles, or quantum dots, located in the vicinity of the resonators’ surface. Sensors based on clusters of microspheres with weakly coupled WG modes can significantly extend the functionality of singlecavity devices, adding such features as multi-wavelength [94] and background-free [95] detection. Another unique property of mesoscale (a ≥ λ) microspheres is connected with their ability to focus collimated beams into subwavelength spots termed nanoscale photonic nanojets [86–92]. This property has been used in a variety of applications [150–154, 177–179]. In linear chains of microspheres, such focusing effects can be periodically reproduced giving rise to novel NIMs [57–60] with small propagation losses and progressively smaller sizes of the focused spots along the chains. These focusing effects occur in a broad range of wavelengths in comparison with the resonant WG coupling effects. Integrated chains of microspheres can be used for developing novel optical scalpels and optical microprobes with subwavelength spatial resolution. Acknowledgments The author would like to thank S.V. Boriskina, A.B. Matsko, M. Sumetsky, U. Woggon, A. Taflove, S. Arnold, A. Melloni, Y.A. Vlasov, M.S. Skolnick, J.J. Baumberg, and M.A. Fiddy for useful discussions. The author thanks M.D. Kerr and R. Hudgins for a critical reading of the manuscript and useful comments. This work was supported by the US Army Research Office (ARO) under grant No. W911NF-09-1-0450 (J.T. Prater), by the National Science Foundation (NSF) under grant ECCS-0824067, and, in part, by funds provided by The University of North Carolina at Charlotte.

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Chapter 18

MEMS-Tuned Microresonators Ming-Chang M. Lee, Ming C. Wu, and David Leuenberger

Abstract Micro-electro-mechanical systems (MEMS) is a key enabling technology to realize scalable and reconfigurable optical components. Since Peterson (1982) demonstrated the first MEMS scanning mirror, many free-space optical MEMS including digital micromirror devices (DMD), micromirror switches, and optical scanners have been accomplished. In guided-wave optics, MEMS have demonstrated the ability for controlling evanescent and butt coupling between waveguides. This chapter investigates the integration of MEMS actuators with optical microresonators and introduces a new family of tunable photonic devices. Most tunable microresonators rely on electro-optics, thermo-optics, free-carrier dispersion or gain/loss control to vary the index of refraction, which can be classified as “material modulation.” For MEMS-tuned microresonators, however, tunability comes from the structure or the configuration of microresonators modified by mechanical actuation, which is termed as “configuration modulation.” Using MEMS to alter the configuration of microresonators has various advantages. First, the functionality of photonic devices based on the resonators is adjustable or tunable. Second, unintentional variation of spacing between components after fabrication may be ameliorated by fine-tuning the configuration. The content of this chapter is organized as follows. Section 1 introduces control factors to implement tunable microresonators and the model of corresponding optical transfer functions. A summary of the state-of-the-art tuning mechanisms is also given. Section 2 demonstrates MEMS variable couplers for tunable microdisk resonators. Several functions, including tunable coupling regimes, tunable slow light, dynamic add-drop filters, and dynamic bandwidth (BW) allocation, are presented. Section 3 shows that the quality factor of microresonators can be reduced by attaching an absorber through MEMS actuators. A wavelength switch is demonstrated.

M.-C.M. Lee (B) Department of Electrical Engineering and Institute of Photonics Technologies, National Tsing Hua University, HsinChu, Taiwan e-mail: [email protected]

I. Chremmos et al.(eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_18,  C Springer Science+Business Media, LLC 2010

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Section 4 gives an example of that the resonant wavelength tuned through varying the cavity length of microresonators. Based on this idea, tunable filters and lasers are accomplished by incorporating MEMS actuators.

18.1 Control Factors for Tunable Microresonators In modern integrated photonic circuits, microresonators are important building blocks for optical signal processing because of compact size, scalability, and integration compatibility with other integrated devices. Many optical functions such as filtering [1, 2], multiplexing [3], demultiplexing, phase engineering [4] are successfully demonstrated by incorporating a single microresonator or cascaded multiple microresonators. Active devices such as integrated lasers are intrinsic microresonators in combination with gain materials. Tunable microresonators, on the other hand, are attractive because they enable dynamic functions such as signal modulation [5], switching, reconfigurable add-drop multiplexing, wavelength tuning, and variable dispersion compensation. To realize these functions thermo-optics and electro-optics are often used methods to vary the refractive index of the microresonator. Alternatively, for semiconductor microresonators, plasma dispersion [6], gain trimming [7], and electroabsorption [8] are feasible approaches, by applying voltage or injecting current into the resonators through microelectronic devices such as p−n junctions [5] or metal-oxide-semiconductors (MOS) [9]. In this chapter, we focus on the topic using MEMS for tuning the microresonators and discuss the related functions. In general, tunable microresonators can be obtained by adjusting any of their three fundamental characteristics, namely, the resonant frequency (or wavelength), the round-trip coupling ratio, and the round-trip resonator loss. The round-trip coupling ratio is defined as the ratio of power coupled between the cavity and the external medium, such as the bus waveguide, in a complete cavity round trip. Similarly, the round-trip resonator loss is the optical power dissipated inside the cavity during a round trip, mainly resulting from scattering or absorption, but excluding the contribution from coupling. Varying any of the above three characteristics can alter the optical functions. To underline this idea, two simple microresonators are considered as an example. One is a Gires–Tournois interferometer (GTI) and the other is an analogous microring coupled to a waveguide, as shown in Fig. 18.1. The optical transfer function of each device can be characterized by three variables: the resonant frequency ω0 , the round-trip resonator loss , and either the roundtrip coupling ratio K or the reflectivity of the interferometer R. These variables are subject to the following properties: ne Lω0 /c = 2mπ, m = 1,2,3,...  = 1 − exp(−2αL) = 1 − σ 2 , 0 <  < 1,0 < K < 1, R < 1,

(18.1)

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(a)

ω0, Γ

R

461

(b)

Waveguide

E1

E1 E2

K

d

100% reflection

r

E2 ω 0, Γ

Fig. 18.1 Two types of microresonators: (a) Gires–Tournois interferometer and (b) microring

where ne is the effective refractive index inside the cavity, c is the speed of light in free space, α is the amplitude attenuation coefficient in the resonant medium, m is the mode number, and L corresponds to a round-trip propagation length, which can be further expressed by  L=

2d (Gires − Tournois interferometers) 2π r (microrings)

(18.2)

In (18.1), the resonance occurs when the wave propagating inside the cavity undergoes a 2mπ angular phase shift. The round-trip resonator loss is defined by the product of the length of the resonator and the attenuation coefficient. According to the lumped element model of a single resonator, coupled to an external source [1], and under the assumptions that , K  1 and (ω − ω0 ) T  1, the optical power transfer |E2 /E1 |2 can be expressed as  2  2  E2      =  j(ω − ω0 )T + ( − K)/2  E   j(ω − ω )T + ( + K)/2  1 0

(18.3)

 2  2  E2      =  j(ω − ω0 )T + ( + R − 1)/2  E   j(ω − ω )T + ( − R + 1)/2 

(18.4)

for microrings or

1

0

+ for the (GTI), where T = L vg is the round-trip propagation time depending on the group velocity of the optical wave and the path length of the cavity. The power coupling ratio for microrings and the reflectivity at the interface(s) of the (GTI) is related to the power exchange between the cavity and the external medium. Expressions (18.3) and (18.4) are equivalent if R is replaced by 1−K. In addition to the transfer function, the other important characteristic, the quality factor (Q), can be modeled through these variables and is given by Q = ω0

T +K

(18.5)

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In (18.3) and (18.4), the optical transfer function is primarily governed by K, , and T at resonance (ω = ω0 ). The resonator loss  is mainly caused by interband absorption, free carrier absorption, bending (radiation) loss, scattering loss, and nonlinear loss such as two-photon absorption. The power coupling ratio K is subject to device geometry and is usually fixed once the device has been fabricated. The roundtrip propagation time T is determined by the cavity size and the refractive index of the material. Analogous to the microwave resonator, the optical resonator can operate in either one of the three coupling regimes: undercoupled (K < ), critically coupled (K = ), and overcoupled (K > ). By examining (18.3), the corresponding transmission spectra in these three coupling regimes are schematically plotted in Fig. 18.2. At resonance the optical transmittance is zero if the microresonator operates in the critical coupling condition. In this case, all the optical power is transferred to the cavity and dissipates entirely. On the other hand, the transmittance in the undercoupled and overcoupled regimes ranges between 0 and 1, depending on the K to  ratio. Although the transmittance spectra are similar for the undercoupled and overcoupled regimes, the phase responses are actually different by a π -shift, as indicated by (18.3).

Fig. 18.2 Transmission spectra of microresonators operating in (a) undercoupled, (b) critically coupled, and (c) overcoupled regime. The insets are the corresponding phase spectra

Table 18.1 summarizes the mechanisms for tunable microresonators. Most of the tuning mechanisms reported in the literature focus on varying the resonant frequency and the resonator loss. The resonant frequency can be tuned by changing the refractive index through local heating, electro-optics, or free-carrier injection. Rabiei et al. [10] reported electro-optically tuned polymer microrings for signal modulation, reaching a speed of 1 Gb/s. Djordjev et al. [11] demonstrated tunable

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Table 18.1 Comparison of tuning mechanisms for optical microresonators

Free-carrier injection Electro-absorption Gain trimming Thermo-optics Electro-optics MEMS

Resonant frequency (ω0 )

Resonator loss ()

Coupling ratio or reflectivity (K or R)

++ + + ++ ++ ++

+ ++ ++ – – +

– – – – – ++

++, very effective; +, effective; –, moderately effective

InP microdisk resonant filters which employ a p−i−n active region and utilize freecarrier injection to alter the index of refraction. Xu et al. [5] presented a 12.5 Gb/s carrier injection-based silicon microring modulator. In III–V compound microresonator systems, resonator loss can be tuned through electro-absorption or trimmed by optical gain [7, 8]. In general, the tuning mechanisms introduced above are all based on the variation of material properties and have minimal effect on the power coupling ratio. MEMS technology, on the other hand, takes a different approach by directly modifying the microresonator structure or configuration. MEMS have been successfully utilized for tuning the resonant frequency, the resonator loss, and the power coupling ratio of microresonators.

18.2 MEMS-Tuned Resonator Coupling One of the distinguished features of MEMS-tuned microresonators is the capability of adjusting the coupling ratio between the cavity and the external guided-wave medium, which is strongly configuration dependent. Generally speaking, as little as only one wavelength of spatial variation can cause dramatic change in the coupling ratio. Two examples of tunable coupling by MEMS actuation are shown in Fig. 18.3. The first is a mechanically tunable reflector for a GTI and the other is a gap-controlled variable coupler for a microdisk resonator. Madsen et al. [12] demonstrated a tunable dispersion compensating filter through a GTI integrated with MEMS-tuned reflectors. The GTI is based on a Fabry–Pérot cavity consisting of a silicon substrate, an air gap, and a quarterwave thick silicon-nitride membrane actuated by applying voltage on the electrodes. The voltage creates an electrostatic force that pulls the membrane closer to the substrate surface, varying the reflectivity of light entering the cavity. Later, Lee et al. [13] proposed the first gapcontrolled variable coupler for microdisk resonators. A microdisk is positioned near one or between two deformable silicon waveguides, which can be pulled toward the microdisk by electrostatic gap-closing actuators. Since the coupling coefficient between the waveguide and the microdisk is an exponential function of the gap spacing, the coupling ratio can be dramatically tuned over many orders of magnitude.

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Suspended Waveguide (Straight)

Electrode

Disk

Electrode

Suspended Waveguide (Deformed)

Electrode

Vs

(a)

Disk

Electrode

Vs

(b)

Fig. 18.3 Examples of MEMS tunable coupling: (a) mechanically tunable reflector for Gires– Tournois interferometer [12] and (b) gap-controlled variable coupler for microdisk resonator

The device was fabricated on a silicon-on-insulator (SOI) wafer. The radius of the microdisk was 10 μm and the initial gap spacing between the waveguide and the disk was 1.4 μm. Using this design, the coupling gap can be continuously reduced until physical contact without experiencing instability.

18.2.1 Tunable Coupling Conditions By varying the power coupling ratio between the cavity and the external guidedwave medium the resonator can be operated either in the undercoupled, critically coupled, or overcoupled regime. A high-Q, vertically coupled MEMS-tuned microdisk resonator [14] was demonstrated with the capability of operating the device in any of the three coupling regimes. Figure 18.4 shows the scanning electron micrograph of the device and lists the associated dimensions of the microdisk and waveguides. The coupling gap spacing is tunable by applying a DC voltage between the waveguide and the four electrodes underneath. The electrostatic actuator functions as a vertical comb-drive actuator with one movable finger and two fixed comb fingers. This design avoids the pull-in instability and permits the waveguide to be pulled down continuously from an initial gap spacing of 1 μm to almost touching. The fabrication process is simply described as follows. The microdisk (R = 20 μm) and the MEMS electrodes are patterned on a 0.25 μm thick single crystalline silicon layer, while the waveguides are fabricated on another SOI and stacked on the top of the microdisk. The waveguides and the microdisk are separated by a 1 μm-thick SiO2 spacer, leading to complete decoupling without actuation on the waveguides. A novel hydrogen-assisted annealing process [15] was employed to reduce the rootmean-square (rms) sidewall roughness to be less than 0.26 nm. The intrinsic quality

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• Microdisk

Left Electrode Waveguide

– Radius: 20 μm

Right Electrode

– Thickness : 0.25 μm

• Waveguides – Width: 0.8 μm Microdisk Resonator

– Thickness: 0.25 μm – Length: 100 μm

• Initial gap spacing Waveguide

– WG/Disk: 1 μm

Electrode

Waveguide

Fig. 18.4 Scanning electron micrograph of a MEMS-tuned microdisk resonator [14]. Reprinted with permission. Copyright 2006 Optical Society of America

factor (Qint ) of the microdisk resonator was measured to be in excess of 105 . A more detailed description is given in [14]. We present below tunable coupling regimes for the microdisk resonator when only one waveguide is actuated. The transmission spectra of the integrated device were characterized by scanning wavelengths within the optical telecommunication band (1520–1565 nm). The measured spectral response is plotted in Fig. 18.5. One of the resonant wavelengths occurs at 1549.37 nm. Without bias, the straight waveguide keeps the initial gap spacing of 1 μm from the microdisk and no optical power is transferred to the resonator because of negligible coupling (Fig. 18.5a). As a result, all the optical wavelengths pass through the bus waveguide to the output end. As the bias increases, the waveguide is bent toward the microdisk, leading to increased coupling. When the coupling ratio is equal to the round-trip resonator loss, the transmission is shown to drop swiftly at 1549.37 nm (Fig. 18.5b). In this case, the microdisk operates in the critical coupling condition. If the gap spacing further decreases by continuously increasing the voltage, the coupling ratio exceeds the resonator loss, the microdisk is in the overcoupled regime, and the transmission returns to unity (Fig. 18.5c). Figure 18.6 plots the detailed transmission curve versus the applied voltage for the MEMS microdisk resonator at the resonant wavelength of 1549.37 nm. At the bias of 17 V, the measured optical transmission is –30 dB, corresponding to the critical coupling condition. The gap spacing is estimated to be about 0.37 μm, according to the simulation result of waveguide deformation calculated using a commercial software (ANSYS). The voltage required to have the gap spacing for critical

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(a)

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Fig. 18.5 Transmission spectra measured on the microdisk operating in (a) decoupling, (b) critically coupled, and (c) overcoupled regimes. The resonant wavelength shown in these spectra is around 1549 nm [14]. Reprinted with permission. Copyright 2006 Optical Society of America Gap Spacing (µm) 1

0.8

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Input

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Fig. 18.6 Transmission curve of a microdisk-coupled waveguide at the resonant wavelength [14]. Reprinted with permission. Copyright 2006 Optical Society of America

coupling can be engineered by designing the length and thickness of deformable waveguides, and the distance between the waveguide and the bottom electrodes. In addition to microdisks, other MEMS-tuned microresonators such as microtoroids [16] and microrings have also been reported.

18.2.2 Tunable Slow Light All-optical packet switches are foreseen in future optical networks. To synchronize the different data channels with each other, data packets are temporarily stored in buffers. As opposed to readily available electrical buffers (e.g., random

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access memory), optical buffers still present a bottleneck [17]. Recent progress in compact solid-state systems can be categorized into (i) approaches based on material dispersion engineering such as electromagnetically induced transparency (EIT) [18], coherent population oscillation (CPO) [19], and spectral hole burning [20]; (ii) waveguide dispersion and cavity resonances [21]; and (iii) combinations thereof such as injection-locked VCSELs [22]. A method based on wavelength conversion by means of four-wave mixing has achieved the longest delay (around 80 pulsewidths [23]), but it is also one of the most complex and power-hungry solutions. With respect to optical resonators, several authors have pointed out that it would be advantageous to combine a number of resonators to form delay lines such as the side-coupled integrated spaced sequence of resonators (SCISSOR), two-channel SCISSORs [24], or coupled-resonator optical waveguides (CROW) [25]. The idea behind SCISSOR is that each resonator generates a certain phase response φ(ω) and the net group delay TD basically increases linearly with the number of resonators in the chain, i.e., TD = −N

dφ dω

(18.6)

The fractional group delay, defined as the group delay normalized to one pulse width is a convenient figure of merit characterizing a resonator-based delay line. A fractional group delay near unity corresponds to a delay by one pulse width. This is the greatest delay that can be achieved with a single resonator without introduction of higher-order dispersive effects. Fractional group delays greater than unity can be obtained by using many resonators. Dispersive effects, however, accumulate and eventually severely distort the pulses. An open question is whether it is more advantageous for slow-light applications to use a single low-loss (high-Q) resonator or a larger number of moderately high-Q resonators. An example is a large-scale (N > 100) array of coupled resonators based on ultrahigh-Q (Q > 106 ) photonic crystal nanocavities with experimentally observed pulse propagation at group velocities well below c/100 [26]. Despite recent advances in slowing light down, many existing schemes either suffer from small time delay, small BW, high power consumption, or lack of tunability. MEMS-tuned microresonators seem to be ideal candidates for providing controllable slow light on the silicon platform with a small form factor. The fabrication is CMOS compatible, potentially allowing for large scale integration. Figure 18.7 schematically illustrates the launch pulse being delayed through a single waveguidecoupled microdisk resonator. By adjusting the gap spacing between the waveguide and the microdisk, the delay time can be conveniently varied. The pulse delay can be thought of as a consequence of optical phase dispersion in the vicinity of the resonant frequency of the microresonator. Consider, e.g., a Gaussian pulse 

t2 si (t) = exp − 2 2T0

 · exp (jωc t)

(18.7)

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Fig. 18.7 Schematic of a MEMS-tuned slow-light device. For the slow-light application only one of the waveguides is actuated while the other remains uncoupled

with a full width at half maximum (FWHM) √ FWHM = 2 ln 2 · T0  1.665T0

(18.8)

launched into this microresonator slow-light device at resonance (ωc = ω0 ). The Gaussian pulse at the output port can be analyzed by taking the inverse Fourier transform of the input pulse multiplied by the transfer function (18.3) in the frequency domain. We assume that the pulse width T0 is much longer than the round-trip propagation time (T) of the resonator. The pulse delay TD is approximately given by TD ∼ =

4K T − 2

K2

(18.9)

It can be easily shown that the delay is theoretically maximized for critical coupling K = , i.e., when the round-trip loss equals the power coupling ratio. However, the transmittance is also reduced to zero under this condition, as it has already been discussed in the previous section. On the other hand, the delay time gradually decreases with the coupling ratio in the overcoupled regime (K > ). To achieve a widely tunable range of delay, without losing transmittance low resonator loss  is necessary. The experimental setup for slow-light measurement is shown in Fig. 18.8. The continuous wave of the tunable laser is modulated with a Mach–Zehnder type electro-optic (EO) modulator. The electrical modulation signal is provided by a 10 Gb/s bit-error rate tester (BERT). A custom bit sequence allows launching single pulses with a half-maximum pulse width of 100ps. The modulated signal goes through an erbium-doped fiber amplifier (EDFA) to compensate for fiber-coupling loss. A polarizer and half-wave plate allow launching linearly TE polarized light pulses into the device under test. The light is coupled into the submicron waveguide by means of a polarization-maintaining AR-coated lensed fiber with 2.5 μm spot size and 14 μm working length. The output of the slow-light device is boosted by another EDFA and the amplified spontaneous emission (ASE) noise is filtered out

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Fig. 18.8 Setup for time-domain pulse delay measurement. A tunable laser (TL) and an electrooptic 10 GHz modulator (MOD) generate single pulses. The pulses are amplified by an EDFA. A polarizer (POL) and a half-wave plate ensure efficient coupling of TE polarized light into the slow-light device. The output signal is amplified (EDFA), filtered (Etalon), and detected with a fast sampling oscilloscope

by a 1 nm-BW etalon filter. The optical signal is detected by a 20 GHz HP lightwave detector and a 40 GHz BW sampling oscilloscope. The MEMS-tuned microdisk resonator is capable of operation in the undercoupled, critically coupled, and overcoupled regimes. For this test device, 48 V corresponds to the required bias to achieve the critically coupled condition. The measured transmitted pulses in the time domain for variable actuation voltages are shown in Fig. 18.9. A benefit of this tunable microresonator is that by choosing a voltage with no coupling, the pulse propagates through the waveguide undisturbed, therefore providing a timing reference for calibrating the relative delay time. In this experiment, a bias of 40 V is utilized for a reference pulse. Around the critical voltage (48 V), the pulse experiences maximum positive delay, and the time delay gradually reduces with increasing voltage. In the opposite case, in the undercoupled

Fig. 18.9 Transmitted pulses in the time domain for variable coupling/actuation voltages. The case of 40 V applied voltage (highest peak) has negligible waveguide coupling and therefore constitutes a time reference for the pulse delay measurement

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regime (bias <48 V), the pulse is actually slightly advanced with respect to the reference pulse, thus providing “fast light.” One thing worthy to note is that there is a tradeoff between delay and amplitude for a single microresonator. As the microresonator operates close to the critical coupling condition, the delay increases while the transmission decreases. In addition to tunable time delay the transmission BW of the resonator is also affected by the waveguide-disk coupling. As the signal band is broader than the resonator bandwidth, the pulse shape is distorted due to the phase dispersion. In this experiment, under certain coupling conditions, the incoming pulse splits into two sub-peaks. This is mainly due to the fact that in the undercoupled and critically coupled regimes, the spectrum of the incoming 100 ps pulse is broader than the resonance width of the microdisk. In order to remove any ambiguities a Gaussian curve-fitting is applied in order to determine the exact center position of the pulses in the time domain. Experimentally observed pulse delays versus actuation voltages are shown in Fig. 18.10. Based on the measured pulse BW of 5.4 GHz, the delay– BW product, a key parameter in comparing different slow-light devices, yields 0.5. In addition, the measured delays exhibit a sign change during the transition from undercoupling to overcoupling. This discontinuity is caused by the fact that as the coupling regime varies across the critical coupling condition, the phase response introduces a π-phase shift. A maximum delay of 94 ps is estimated.

Fig. 18.10 Experimental pulse delays versus actuation voltages. The data points enclosed by a circle in this figure are estimated delays for the double-peak distorted pulses via Gaussian curve-fitting. The inset shows the transformed pulse at the actuation voltage of 47 V

In summary, the main advantages of the described MEMS-tuned microdisk resonator is the continuously variable coupling ratio in combination with high intrinsic quality factor, which allows to alleviate the insertion loss for a given BW in the overcoupling all-pass regime. Additionally, it seems quite straight forward to extend the concept of MEMS tunability to SCISSOR structures.

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18.2.3 Reconfigurable Add-Drop Filters Wavelength-selective devices are essential building blocks for many types of wavelength-division-multiplexed (WDM) optical systems. Since wavelength determines the routing in a WDM network, the ability to manipulate the spectral characteristics of in-line devices turns out to be very important. These devices commonly include optical filters, wavelength multiplexers, and add-drop multiplexers. Reconfigurable add/drop multiplexers are especially vital for dynamically directing wavelength channels in and out of the data buses. This function can be realized by an array of switchable micromirrors connected with a grating spectrometer. An integrated solution was also proposed by cascading multiple tunable microresonators on a single chip [27]. MEMS-tuned microresonators are ideal for implementing reconfigurable adddrop multiplexing. A simple idea of device operation is illustrated in Fig. 18.11. Provided that the two waveguides are actuated simultaneously, signals bandcentered at the resonant wavelengths can be transferred from one waveguide to the other through proper control of the coupling ratio, while off-resonant signals are not affected. In this way a dynamic add-drop filter can be accomplished. Four transmission ports, namely, input port, through port, add port, and drop port, according to the destination of the signal, are labeled here for better illustrating the function. When the two waveguides are far from the microdisk, in the so-called off-state, no optical signal is transferred from the input port to the drop port or from the add port to the through port. On the other hand, the device can be switched to an on-state by moving the two waveguides close to the microdisk. Such an integrated switchable add-drop filter is convenient to implement a large-scale reconfigurable add-drop multiplexer. Through

Resonator

Resonator

Input

(a)

Add

Through

Add

Input

Drop

Drop

(b)

Fig. 18.11 Schematic of a reconfigurable add-drop filter: (a) off-state (decoupling) and (b) onstate (coupling)

To better understand the add-drop function, the lumped element resonator model is again used for examining the characteristics of optical transmission among these ports. The launch and probe waveguides are coupled to the microdisk via two couplers with power coupling ratios K1 and K2 , respectively. The microdisk is

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modeled to have a round-trip resonator loss . Similarly to the analysis of single waveguide-coupled microdisks and under the assumptions that Γ , K1 , K2  1 and (ω − ω0 ) T  1, the power transfer functions, from the input port to the through and drop ports are, respectively, given by      Ethrough (ω) 2  j(ω − ω0 )T + ( + K2 − K1 )/2 2   =   j(ω − ω )T + ( + K + K )/2   E  input (ω) 0 2 1

(18.10)

√    2  Edrop (ω) 2   K1 K2   =  . E   j(ω − ω0 )T + ( + K2 + K1 )/2  input (ω)

(18.11)

and

In (18.10), the optical transmission from the input port to the through port is very similar to (18.3) if the round-trip resonator loss Γ in (18.3) is replaced by the roundtrip resonator loss plus the power coupling ratio of the probe waveguide Γ + K2 . This can be explained from the lumped element model; since the power flow from the microdisk to the probe waveguide is unidirectional without reverse coupling, the coupled power can be considered as an additional loss for the microdisk cavity. From (18.11), the maximum transferred power from the input port to the drop port occurs when the power coupling coefficients K1 , and K2 , are commensurate and are much larger than the resonator loss. On the other hand, to get the maximum power at the through port, the power coupling ratio K1 should be equal to zero, corresponding to a decoupled state. A dynamic add-drop filter was demonstrated by actuating both suspended waveguides of the MEMS-tuned microdisk resonator with equal bias. A broad-band light source was launched at the input port and the transmitted powers at the through port and the drop port were measured via an optical spectrum analyzer. Figure 18.12a presents the spectral responses at the through port and the drop port without applying voltage. Since the coupling ratio is minute for both waveguides due to a large gap (1 μm), all the input power remains in the launch waveguide and exits at the through port. As the applied voltage increases, inducing sufficient coupling at 30 V, the input power at the resonant wavelength is diverted to the drop port through the microdisk. The results are shown in Fig. 18.12b. Figure 18.13 presents the optical transmission versus the applied voltage for both the through port and the drop port at the resonant wavelength. A 20 dB extinction ratio is achieved by varying the bias from 0 to 35 V. We note that at the threshold voltage of 19 V, the sum of the powers at the through port and the drop port is not equal to the input power, accounting for about 28% insertion loss. This inequality is due to the significant power dissipation inside the resonator, engendered by the high optical intensity in the microdisk operating near the critical coupling condition. This loss decreases dramatically as the coupling is changed to be very strong or to be very weak. Provided that the bias voltage is sequentially switched between 0 and 30 V, this device can be used as a building block for a large-scale reconfigurable add-drop multiplexer if multiple devices with respect to different wavelength channels are

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Fig. 18.12 Spectral response of transmission at the through port and the drop port: (a) 0 and (b) 30 V. The free spectral range (FSR) is about 3.7 nm [28]. Reprinted with permission. Copyright 2006 Optical Society of America 1

0.6 0.4 0.2 0

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cascaded. A salient feature of MEMS-tuned microresonators is the almost zero loss from the input port to the drop port if the device is not enabled. This is especially important for a cascaded configuration in practical applications.

18.2.4 Bandwidth-Tunable Filters To build a tunable spectral width integrated band-pass filter is an extremely challenging task. However, if it can be achieved, many laudable applications of dynamic BW allocation in WDM systems could be realized. One example is matched optical filtering for a system that has different channel data rates, where there would be the ability to optimally filter the single channel to minimize the power penalty without wasting excess BW. Another example is reconfigurable “channel bending” with the

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ability to dynamically reroute either a single data channel or a contiguous set of data channels to specific destinations. Directly tuning the power coupling ratios of a two waveguide-coupled microresonator is an effective way for implementing an integrated tunable filter not only in switching wavelength channels but also in engineering the filter BW. However, for most microresonators, the coupling ratio is very difficult to be tuned once the device is fabricated. Using a MEMS-tuned microdisk, the gap spacing between the waveguide and the disk can be easily adjusted, resulting in a variable coupling ratio. Figure 18.14 displays the operating schemes of microdisk resonators with different gap spacing and the corresponding signal transmission. When the waveguides are strongly coupled with the microdisk the transmission BW broadens. Through this mechanism a tunable BW is accomplished. Through

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Fig. 18.14 Operation schemes of MEMS-tuned microdisk resonators and the corresponding signal transmission spectra [28]. Reprinted with permission. Copyright 2007 Optical Society of America

The lumped element resonator model is used for analyzing the BW of MEMStuned microresonator by assuming variable power coupling ratios and a fixed resonator loss. As discussed in the previous section, the maximum transferred power from the launch waveguide to the probe waveguide occurs when the coupling ratios are much larger than the resonator loss. Under this condition, if the coupling is further enhanced, the intensity of power transmission only slightly increases while the BW can significantly expand. From (18.5) and (18.10), the Q factor of the two waveguides-coupled microresonator can be derived and is given by Q

−1

=

K2 K1 + + 1 Q−1 int  

(18.12)

where Qint = ω0 T/  represents the intrinsic quality factor. Since the transmission spectrum actually follows the Lorentzian-shaped function, the FWHM is expressed by

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ωFWHM = ω0

K2 K1 + + 1 Q−1 int  

(18.13)

according to the relation Q = ω0 / ωFWHM . Equation (18.13) indicates that the BW increases with the power coupling ratio. The first application of tunable BW is presented for matched optical filtering [29]. This function is especially important if multiple signal formats co-propagate on a WDM system. To examine this function, a 5 Gb/s NRZ data train was tested on the MEMS-tuned microdisk resonator which is able to alter the BW from 5 to 12 GHz via actuating the coupled waveguides. The experimental result on the power penalty versus the filter BW is plotted in Fig. 18.15. The optimum filter BW was found to be 10 GHz. Selection of a smaller BW leads to intersymbol interference due to pulse broadening and therefore results in a higher bit-error rate (BER). For a larger BW, the BER also increases because of the non-ideal filter shape and containing more amplified spontaneous emission (ASE) noise.

1.2 Power Penalty (dB)

Fig. 18.15 Optimal filter BW for a 5 Gb/s NRZ data channel [29]. Reprinted with permission. Copyright 2007 IEEE

BW = 10 GHz 0.8

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12

The second experiment demonstrates the ability of reconfigurable “channel bending” on a WDM system. Three 2.5 Gb/s NRZ data channels separated by a fixed channel spacing of 7.5 GHz were combined and launched into the MEMS-tuned microdisk filter. The setup is shown in Fig. 18.16. The microdisk filter can either demultiplex a single channel (“Band 1”) or a group of channels (“Band 2”), depending on the adjustable BW which is set to be 6.5 and 9.5 GHz, respectively, in this experiment. The measured signal spectra are shown in Fig. 18.17. By adjusting the filter BW to 6.5 GHz and tuning the center wavelength of the filter through the thermo-optics effect, it is possible to drop the second channel with an extinction ratio of 14.5 dB. As the filter BW is opened up to 9.5 GHz and the center wavelength is tuned to the middle of channels 1 and 2, both channels are selected with an extinction ratio of 13.9 dB.

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Input

λ/4 Plate

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Optical Preamp

Mod TL 3 BERT

Limiting Amplifier

att

Tunable BW Filter

Rx 1 nm

Fig. 18.16 Experimental setup of a three channel WDM system using a MEMS tunable-BW microdisk resonator [29]. Reprinted with permission. Copyright 2007 IEEE

Fig. 18.17 Experimental results on the reconfigurable “channel bending.” The microdisk filter is tuned to drop either a single data channel, or a group of two channels by varying the BW from 6.5 to 9.5 GHz [29]. Reprinted with permission. Copyright 2007 IEEE

18.3 MEMS-Tuned Resonator Loss Varying the resonator loss is another way to alter the functionality of microresonators. From (18.3), (18.10), and (18.11), if the resonator loss is tunable, the optical transmission is also affected. Through this mechanism, for example, dynamic adddrop filtering and tunable coupling conditions can be realized. Electro-absorption and gain trimming were already reported for III–V semiconductor microresonators, however, it is difficult to apply these mechanisms in other dielectric materials. A cheap and widely applicable approach to vary the resonator loss is attaching an optical absorber close to a microresonator, which is called “Q factor spoiler” since the quality factor is decreased by the external absorber.

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Researchers at the MIT/Pirelli Labs presented a MEMS wavelength-selective switch via actuating a parallel-plate absorber, made by aluminum, on the top of a microring resonator [30]. The device structure is shown in Fig. 18.18. The metal plate is suspended and can be pulled down through an electrostatic force if voltage is applied to the metal plate. When the lossy plate is far from the evanescent field of the microring, the resonant wavelength is switched from the through port to the drop port. On the other hand, when the metal plate bends down and couples to the evanescent field, the free-carrier absorption induces resonator loss, spoiling the resonance condition and all the wavelengths pass by the ring into the through port.

Fig. 18.18 Schematic illustration of a MEMS-actuated absorber on a microring switch: (a) on state and (b) off state [30]. Reprinted with permission. Copyright 2005 IEEE

The microring and waveguides are made by silicon nitride and are fabricated on silicon dioxide as the bottom cladding. The detailed process flow and device dimensions are described in [30]. The measured transmission spectra at the drop port and the through port are shown in Fig. 18.19. A 15 dB extinction ratio is observed at the drop port with/without MEMS actuation. Although the insertion loss is relatively high due to the unintentional deformation of the metal plate, using MEMS to control the resonator loss shows the potential of a low-cost solution in resonator-based integrated optics. Fig. 18.19 Measurement on the spectra of the MEMS microring switch with and without actuation [30]

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18.4 MEMS-Tuned Resonator Cavity Length Perhaps the most widely applied mechanism for MEMS microresonators is tuning the cavity length or, correspondingly, the resonant frequency. Tuning of resonant frequency is essential in many practical applications, for example, compensation for wavelength shift of optical filters due to thermal instability or in wavelengthselective switches. Although electro-optics is mainly used for tuning the resonator wavelength via altering the refractive index, the variation is usually small so the device must be relatively large. Directly adjusting the cavity length is very efficient since a couple of microns change is sufficient for hopping over many resonant modes (resonant frequencies). The use of MEMS actuators is advantageous for precisely controlling the cavity length on a nanometer scale. Here the cavity, or partial cavity, usually consists of one or several air gaps that can be decreased or expanded through MEMS actuation. In this section, two applications of MEMS-tuned microresonators with a varying cavity length are addressed; one is a tunable optical filter, the other is a tunable laser.

18.4.1 Tunable Optical Filter Irmer [31] presented a tunable micro-machined Fabry–Pérot filter consisting of two InP-air-gap distributed Bragg reflectors (DBR) separating with an air cavity. The

Fig. 18.20 Device schematic and SEM picture of MEMS tunable Fabry–Perot filters [31]. Reprinted with permission. Copyright 2003 IEEE

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Fig. 18.21 Experimental setup and measured tunable reflectance spectra [31]. Reprinted with permission. Copyright 2003 IEEE

device structure is shown in Fig. 18.20. The top DBR mirror is p-doped and the bottom one is n-doped. An electrostatic force is induced via reverse bias between the top and bottom mirrors, which causes a decrease of the air cavity length. The experimental setup and the measured reflectance spectra are shown in Fig. 18.21. A continuous tuning range of 142 nm was demonstrated through a bias voltage variation of 3.2 V. Similar ideas using MEMS actuators were also presented by several other research groups [32–35].

18.4.2 Tunable Laser If the microresonator is applied for an integrated laser, tuning the cavity length can be used for adjusting the emitted laser wavelength. Researchers at the University of California, Berkeley, demonstrated a micromachined tunable VCSEL which has a wide and continuous wavelength tuning range [36]. The laser cavity comprises an air gap sandwiched between a top and bottom Al0.6 Ga0.4 As-GaAs DBR mirror, embedding three In0.15 Ga0.85 As quantum wells serving as the gain medium. Through electrostatic actuation of the top mirror, the central air gap decreases, resulting in a wavelength blue shift up to 19.1 nm. With a similar structure but replacing the top DBR mirror by a sub-wavelength high-index-contrast grating (HCG), the same research group also demonstrated a nano-electro-mechanical tunable VCSEL [37, 38]. The device structure is illustrated in Fig. 18.22. Compared with DBR, this HCG top mirror has the advantages of better control over laser polarization and has an intrinsically small footprint, significantly increasing the tuning speed and lowering the actuation voltage. The experimental results are shown in Fig. 18.23. A continuous wavelength tuning range of 2.5 nm as well as a low actuation power (<50 nW) has been achieved.

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Sacrificial Layer

Air gap

Top mirror

Active Cavity Bottom DBRs

(a)

(b)

Fig. 18.22 Nano-electro-mechanically tunable VCSEL. (a) Schematic cross-section, and (b) SEM of the device (top view). The inset shows the grating structure [37]. Reprinted with permission. Copyright 2007 Optical Society of America

Fig. 18.23 (a) Measured emission spectra of tunable MEMS VCSEL under various external applied voltages across the HCG cantilever. (b) Observed laser wavelengths and corresponding spectral intensity as a function of the applied voltage [37]

18.5 Conclusions Integrated optical microresonators are essential for the implementation of a variety of WDM photonic integrated circuits (PICs). To make the optical function reconfigurable, tunable microresonators are desired. Conventional tuning mechanisms including free-carrier injection and the use of electro- and thermo-optical effect are employed to modulate the resonant frequencies through altering the refractive index of the material. Gain trimming and electro-absorption are used to modify the resonator loss. By employing MEMS actuators, however, not only the wavelength and loss but also the coupling coefficients are adjustable. The latter is especially difficult to be accomplished by other tuning mechanisms since the resonator coupling is mainly determined by the device structure or configuration which is normally fixed once the device is fabricated.

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The first example of MEMS-tuned microresonators was the MEMS-actuated microdisk with variable coupling ratio. Through varying the gap spacing between the microdisk and the waveguides, the microdisk can operate in the undercoupled, critically coupled, or overcoupled regime, depending on the bias voltage applied to the actuators. Due to the variable coupling ratio, a dynamic add-drop filter with an extinction ratio of more than 20 dB was demonstrated. Moreover, once the filter function is turned on, a slight increase of the coupling ratio can expand the BW without decreasing the extinction ratio. Two applications for matched optical filtering and reconfigurable “channel bending” were presented via this BW-tunable microdisk filter. Additionally, since the optical transfer function of a high-Q resonator coupled to a single waveguide is similar to an all-pass filter, it can be used as an optical delay line or dispersion compensator. In this chapter, a tunable optical delay for a Gaussian pulse was realized through tuning the coupling ratio of the MEMS-actuated microdisk. The tunable optical delay is about 80 ps. More sophisticated optical functions can be realized by cascading multiple MEMS-actuated microdisks in a serial or parallel configuration. Tuning the resonator loss through MEMS actuators is attainable by attaching an absorber close to a resonator cavity. Unlike III–V semiconductor microresonators using electro-absorption or gain trimming for directly tuning material absorption, passive microresonators such as silicon or silicon nitride microrings can utilize this mechanism to indirectly vary resonator loss. However, the loss is usually increased rather than decreased. Based on this idea, a wavelength-selective switch was demonstrated. If parts of a resonator cavity comprise free space, MEMS actuators can be incorporated for changing the size of the cavity. In such a case, the resonant wavelength can be shifted accordingly. Generally speaking, the tuning range of resonant wavelength by MEMS is broader than by other tuning mechanisms, such as electro-optic or thermo-optic, since the variation of cavity length via MEMS can be over several optical wavelengths. MEMS-tuned microresonators with variable resonant wavelength have been demonstrated on various passive and active photonic devices. One prominent example is the integrated tunable laser. The tuning range of laser wavelength in previous experimental reports was up to 19 nm. The dimensions of MEMS actuators and microresonators are of the same scale and are possibly fully integrated on a single chip. The variable structure or configuration of the microresonator results in significant modification of its optical characteristics. Additionally, other researchs show that optical waves resonating inside a resonator can also induce mechanical force, for example, optical pressure. Such a mutual interaction stimulates a new study on optomechanics. Therefore MEMS-tuned microresonators are key elements both in optical signal processing as well as in ongoing fundamental research. Acknowledgments The authors would like to acknowledge Prof. A. Uskov from the P.N. Lebedev Physical Institute, Moscow, for helpful discussions and the support of DARPA for the work on tunable slow light. Furthermore they would like to acknowledge the contribution by Dr. B. Zhang and Prof. A.E. Willner from USC in the experiment of bandwidth-tunable filters.

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References 1. Little, B.E., Chu, S.T., et al. Microring resonator channel dropping filters. J. Lightw. Technol. 15, 998–1005 (1997) 2. Chu, S.T., Little, B.E., et al. Second-order filter response from parallel coupled glass microring resonators. IEEE Photon. Technol. Lett. 11, 1426–1428 (1999) 3. Little, B.E., Chu, S.T., et al. Microring resonator arrays for VLSI photonics. IEEE Photon. Technol. Lett. 12, 323–325 (2000) 4. Lenz, G., Eggleton, B.J., et al. Dispersive properties of optical filters for WDM systems. IEEE J. Quant. Electron. 34, 1390–1402 (1998) 5. Xu, Q., Manipatruni, S., et al. 12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators. Opt. Exp. 15, 430–436 (2007) 6. Sadagopan, T., Choi, S.J., et al. Carrier-induced refractive index changes in InP-based circular microresonators for low-voltage high-speed modulation. IEEE Photon. Technol. Lett. 17, 414–416 (2005) 7. Djordjev, K., Choi, S.J., et al. Gain trimming of the resonant characteristics in vertically coupled InP microdisk switches. Appl. Phys. Lett. 80, 3467–3469 (2002) 8. Djordjev, K., Choi, S.J., et al. Vertically coupled InP microdisk switching devices with electroabsorptive active regions. IEEE Photon. Technol. Lett. 14, 1115–1117 (2002) 9. Liu, A., Jones, R., et al. A high-speed silicon optical modulator based on a metal-oxidesemiconductor capacitor. Nature 427, 615–618 (2004) 10. Rabiei, P., Steier, W.H., et al. Polymer micro-ring filters and modulators. IEEE J. Lightw. Technol. 20, 1968–1975 (2002) 11. Djordjev, K., Choi, S.J., et al. Microdisk tunable resonant filters and switches. IEEE Photon. Technol. Lett. 14, 828–830 (2002) 12. Madsen, C.K., Walker, J.A., et al. Tunable dispersion compensating MEMS all-pass filter. IEEE Photon. Technol. Lett. 12, 651–653 (2000) 13. Lee, M.C.M., Wu, M.C. MEMS-actuated microdisk resonators with variable power coupling ratios. IEEE Photon. Technol. Lett. 17, 1034–1036 (2005) 14. Lee, M.C.M., Wu, M.C. Tunable coupling regimes of silicon microdisk resonators using MEMS actuators. Opt. Exp. 14, 4703–4712 (2006) 15. Lee, M.C.M., Wu, M.C. Thermal annealing in hydrogen for 3D profile transformation on silicon-on-insulator and sidewall roughness reduction. J. Microelectromech. Syst. 15, 338–343 (2006) 16. Yao, J., Leuenberger, D., et al. Silicon microtoroidal resonators with integrated MEMS tunable coupler. IEEE J. Select. Topics Quant. Electron. 13, 202–208 (2007) 17. Tucker, R.S., Ku, P.C., et al. Slow-light optical buffers: Capabilities and fundamental limitations. J. Lightw. Technol. 23, 4046–4066 (2005) 18. Hau, L.V., et al. Light speed reduction to 17 metres per second in an ultracold atomic gas. Nature 397, 594 (1999) 19. Bigelow, M.S., Lepeshkin, N.N., et al. Superluminal and slow light propagation in a roomtemperature solid. Science 301, 200–202 (2003) 20. J. HegartyEffects of hole burning on pulse-propagation in GaAs quantum wells. Phys. Rev. B 25, 4324–4326 (1982) 21. Heebner, J.E., Chak, P., et al. Distributed and localized feedback in microresonator sequences for linear and nonlinear optics. J. Opt. Soc. Am. B; Opt. Phys. 21, 1818–1832 (2004) 22. Zhao, X., Palinginis, P., et al. Tunable ultraslow light in vertical-cavity surface-emitting laser amplifier. Opt. Exp. 13, 7899–7904 (2005) 23. Sharping, J.E., Okawachi, Y., et al. All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion. Opt. Express 13, 7872–7877 (2005) 24. Pereira, S., Sipe, J.E., et al. Gap solitons in a two-channel microresonator structure. Opt. Lett. 27, 536–538 (2002)

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25. Poon, J.K.S., Zhu, L., et al. Transmission and group delay of microring coupled-resonator optical waveguides. Opt. Lett. 31, 456–458 (2006) 26. Notomi, M., Kuramochi, E., et al. Large-scale arrays of ultrahigh-Q coupled nanocavities. Nature Photon. 2, 741–747 (2008) 27. Chu, S.T., Little, B.E., et al. Eight-channel add-drop filter using vertically coupled microring resonators over a cross grid. IEEE Photon. Technol. Lett. 11, 691–693 (1999) 28. Lee, M.C.M., Wu, M.C. Variable bandwidth of dynamic add-drop filters based on couplingcontrolled microdisk resonators. Opt. Lett. 31, 2444–2446 (2006) 29. Zhang, B., Leuenberger, D., et al. Experimental demonstration of dynamic bandwidth allocation using a MEMS-actuated bandwidth-tunable microdisk resonator filter. IEEE Photon. Technol. Lett. 19, 1508–1510 (2007) 30. Nielson, G.N., Seneviratne, D., et al. Integrated wavelength-selective optical MEMS switching using ring resonator filters. IEEE Photon. Technol. Lett. 17, 1190–1192 (2005) 31. Irmer, S., Daleiden, J., et al. Ultralow biased widely continuously tunable Fabry-Perot filter. IEEE Photon. Technol. Lett. 15, 434–436 (2003) 32. Aziz, M., Meissner, P. et al. WDM system integration of micromachined tunable two-chip Fabry-Perot filters. Opt. Comm. 208, 61–68 (2002) 33. Correia, J.H., Bartek, M., et al. Bulk-micromachined tunable Fabry-Perot microinterferometer for the visible spectral range. Sensor Actuators A 76, 191–196 (1999) 34. Peerlings, J., Dehé, A., et al. Long resonator micromachined tunable GaAs-AlAs Fabry-Perot filter. IEEE Photon. Technol. Lett. 9, 1235–1237 (1997) 35. Spisser, A., Ledantec, R., et al Highly selective and widely tunable 1.55-μm InP/air-gap micromachined Fabry-Perot filter for optical communications. IEEE Photon. Technol. Lett. 10, 1259–1261 (1998) 36. Vail, E.C., Li, G.S., et al. High performance and novel effects of micromechanical tunable vertical-cavity lasers. IEEE J. Select. Topics Quant. Electron. 3, 691–697 (1997) 37. Huang, M.C.Y., Zhou, Y., et al. Nano electro-mechanical optoelectronic tunable VCSEL. Opt. Exp. 15, 1222–1227 (2007) 38. Huang, M.C.Y., Zhou, Y., et al. A surface-emitting laser incorporating a high-index-contrast subwavelength grating. Nature Photon. 1, 119–122 (2007)

Chapter 19

Microresonators for Communication and Signal Processing Applications Lin Zhang and Alan E. Willner

Abstract Photonic microresonators exhibit a great potential for various advanced functions for communication and signal processing applications. In this chapter, we briefly review recent advances in achieving space-, power-, and spectrally efficient chip-scale optical devices and subsystems using microresonators. With an emphasis on signal integrity and system performance, we describe microresonator-based channel adding/dropping in time-division-multiplexing systems, signal generation and demodulation for advanced optical data formats, and frequency comb generation for arbitrary waveform generation.

19.1 Introduction Optical communication has fueled the global telecom and information technology revolution in the past two decades [1]. Nowadays, worldwide transport of huge volume of data relies on optical communication systems. Furthermore, optical signal processing technology has been aggressively pursued [2], aiming at higher operation speed and enriched functionality. This holds a great potential to diminish the need for O-E-O conversion in the data transmission links in order to upgrade system capacity, reduce latency and cost, and enable a wavelength- and dataformat-transparent optical network. Integrated photonics has attracted a great deal of attention in recent years not only because it allows for more cost-effective production and easier packaging but also because smaller chip size assists in realizing faster and less power-consuming photonic devices to facilitate “green” information technology. Integrated photonics also plays an increasingly important role in datacom systems. Optical communication has been investigated in almost every layer of L. Zhang (B) Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089, USA e-mail: [email protected] I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7_19,  C Springer Science+Business Media, LLC 2010

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information technology infrastructure: continent-to-continent, city-to-city, serverto-server, computer-to-computer, board-to-board, chip-to-chip, and finally intrachip. For example, in contemporary high-performance computing systems, global interconnection between electronic computation units may cause long latency and high power consumption as clock rates increase. This has been identified to be one of bottlenecks in the future development of the integrated circuit industry, according to the International Technology Roadmap for Semiconductors (ITRS). To alleviate these challenges, many efforts from both academia and industry have been pursued to explore the possibility of employing optical interconnections. Advanced integrated photonics is an enabling technology platform to build space-, power-, and spectrally efficient on-chip photonic interconnection networks that could be seamlessly integrated with CMOS electronics [3–7]. In general, optical communication and signal processing represent different categories of devices and subsystems, although it is hard to clearly separate them from each other. In communication, optical subsystems include, but are not limited to, signal generation, detection, amplification, filtering, and signal degrading effects’ monitoring and compensation. In signal processing, functional units would range even more widely from pulse manipulating, data routing to optical logic, as listed in Fig. 19.1. Most of these functions have been demonstrated using integrated devices.

Fig. 19.1 Some functions needed in optical communication and signal processing

As a key element in integrated photonics, microresonators exhibit small chip size, resonance-enhanced operation efficiency, low power consumption, and high wavelength selectivity. Various types of the microresonator-based devices have been proposed and demonstrated for communication and signal processing applications. Those include lasers [8–11], modulators [12–16], filters [17–20], amplifiers [21–24], switches [25–27], logic gates [28–30], wavelength converters [31–33], delay elements [34–35], dispersion compensators [36–38], demodulators [39–40], format converters [41], and sensors [42–46].

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In practice, an advanced communication or signal processing subsystem may require a cascade of many optical and electrical devices, which raises at least three system design concerns: compactness, power consumption, and signal quality. As an example, an agile and ultra-fast arbitrary waveform generator [47] may consist of multi-wavelength optical source, amplitude and phase modulators, couplers and electrical and optical amplifiers, and filters. Even if each component has very little performance imperfection, it is still possible that the whole system creates considerable data degradation. Although microresonators are potentially able to provide more compact and power-efficient solutions than waveguide- and Mach– Zehnder-based devices, they do not necessarily produce a satisfactory signal quality. Therefore, from a system point of view, an emphasis is placed on the signal quality in microresonator research. In one chapter, it is virtually impossible to present a complete discussion on every kind of microresonator-based devices in terms of their system applications. Selected topics include channel adding/dropping in time-division-multiplexed (TDM) systems, modulation and demodulation for advanced data formats, and optical frequency comb generation for high-speed optical arbitrary waveform generation. We tend to provide a detailed discussion on how the microresonators’ temporal and spectral responses affect the devices’ system performance and signal quality, with an emphasis on impact of carrier and cavity dynamics in semiconductor microring resonators. State-of-the-art results are outlined. It is believed that building a generic system-level model of microresonator-based devices helps deepen our understanding on the resonators’ system performance. We start with a description of such a model.

19.2 Modeling of System Performance A comprehensive system model of the microresonator devices includes signal generation in pseudo-random bit sequence (PRBS), simulation of device characteristics, and signal detection and evaluation. For a passive device, typically only a static model is needed, which is detailed in other chapters of this book. For an active device, there could be different driving and controlling techniques, using all-optical, electro-optic, or thermo-optic effects. Here, we describe a model for an electro-optic semiconductor microring resonator. Generally speaking, the resonance is tuned when a drive voltage or current is applied onto an electrode integrated with the resonator, which changes carrier density inside the cavity and thus the effective index of refraction of the guided mode. The index change is translated into a resonance shift in frequency domain. Note that the carrier density variation may also cause a change in propagation loss of the guided mode, which results in the reshaping of the resonance profile. At the beginning of the modelization, logic data in PRBS at a bit-rate of interest is encoded according to the specific modulation format [this is needed for a format other than binary on–off-keying (OOK) signals]. A long PRBS word is desirable for

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system-level evaluation but also requires strong computation capability. The PRBS length is 213 –1=8191, which is usually long enough to examine data pattern dependencies induced by the resonator devices. Then, the data stream is sent to a driver, and a square-wave voltage signal in a non-return-to-zero (NRZ) format is generated, which may need to be biased according to the type of electrode used. The signal then passes through an electrical filter with a relatively large bandwidth. The filter bandwidth is chosen depending on whether or not the driving circuit’s speed limitation needs to be considered. It is normally set to be 1.5∼3 times the signal bit-rate so that the circuit limitation becomes negligible. In this way, we obtain a driving signal. There have been many ways to convert a drive voltage signal to a carrier density variation in the cavity, based on different materials (e.g., III–V compound semiconductor or silicon) and electrode designs. For silicon-based devices, typical electrode designs include metal–oxide–semiconductor (MOS) capacitors [12, 48–51], p–n junctions [52–54], and p−i−n diodes [14, 16, 55–59]. These have different carrier transit times and RC time constants. In particular, parasitic capacitances may change with drive voltage. Therefore it is difficult to build a uniform electrical model to describe the behavior of these devices. Here, we focus on the MOS capacitors operated at hole accumulation regime, since there is low dc power consumption and no carrier confinement required. A typical MOS capacitor electrode structure can be found in [51]. In the time domain, time-varying capacitances of the accumulation layer in the MOS capacitor are negligible [51], with only the fixed oxide gate capacitance left. The carrier density in the electrode thus follows an exponential transition over time. The carrier transit time, from 10 to 90% of the maximum carrier density, is ranged from 12 to 100 ps [12, 48–51]. The dependence of the transit time on drive voltage is ignored. We also assume that the transit time at the rising and falling edges is the same [12, 50]. Silicon-based microresonators could have a radius below 10 μm, and the RC time constant is estimated to be <3 ps for the proposed structure in [51] as an example, which is negligible as compared to the carrier transit time. Carrier-induced index change in silicon is caused mainly by the plasma dispersion effect [60], because unstrained pure crystalline silicon has no linear electro-optical (Pockels) effect due to crystal symmetry and has a weak Franz– Keldysh effect [61]. Spatially, the carrier density is a 2D distribution over the cross section of the electrode. It can be simulated based on specific MOS structures to obtain the effective index change of an optical guided mode using commercially available mode solvers [12, 15, 49–51]. In a simplified model, a spatially averaged carrier density with a steady-state value linearly proportional to the peak-to-peak value Vpp of the drive voltage is used, so that the effective index change is calibrated to be 2.4×10−4 at Vpp = 5 V, which can be obtained using several proposed electrode designs [48–51]. The imaginary part of the index change causes dynamic loss, which is also calculated from [60] at the wavelength of interest. The loss variation results in a dynamically changed cavity Q-factor and thus a reshaped resonance profile when it is shifted [14, 15, 54]. Note that, for other material systems such as polymer, there may be no dynamic process of carriers. In that case, the index change can be obtained directly from the electro-optic effect [62, 63].

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The effective index change over time causes a shift of the resonance (angular) frequency, which is reflected in the coefficients in a set of dynamic coupled mode equations describing the microresonator system in the optical domain. The dynamic model can accurately predict a transient evolution of the optical field in the cavities to account for the effect of the cavity photon lifetime on output waveform and frequency chirp. The model given in [17, 64] abstractly treats an optical wave traveling around the cavity as a whole, and it is applicable to both traveling-wave [17] and standing-wave cavities [65]. One may solve the dynamic equations for the energy amplitude that represents the total energy stored in the ring [17, 64], with a time step not limited by cavity round-trip time. The model works only when one has no need to examine the walk-off of optical waves traveling along different paths in ring resonators [14, 66]. Another model adopted in [67], in a more intuitive way, tracks the optical wave along its path in a cavity and is inherently suitable for cases, where the optical fields at different parts of the rings have to be resolved to examine the interference and evolution of the optical waves [16, 68, 69]. However, this means that a ring resonator is divided into different parts separated by ring-to-ring and ringto-waveguide couplers, requiring a time step less than the cavity round-trip time in simulations, which implies the need for more memory and computation time. For the devices discussed in this chapter, the two models would generate simulation results in good agreement with each other [70]; we use the first model here. For a specific resonator system, one can write corresponding dynamic equations based on the first modeling technique [17, 64]. Figure 19.2 shows an example, which is a triple-ring structure with only the middle ring being active. The dynamic equations are

1 1 ∂a1 (t) a1 (t) + jμ1 Ein (t) + jμ2 a2 (t) − = −jωr1 − ∂t τl1 τc1

(19.1a)



∂a2 (t) 1 = −jωr2 (t) − a2 (t) + jμ2 a1 (t) + jμ3 a3 (t) ∂t τl2 (t)

(19.1b)

∂a3 (t) 1 1 a3 (t) + jμ3 a2 (t) − = −jωr3 − ∂t τl3 τc4

(19.1c)

Eout1 (t) = Ein (t) + jμ1 a1 (t)

(19.1d)

Eout2 (t) = jμ4 a3 (t)

(19.1e).

The energy amplitude is defined as ai (t) = Ai (t)e−jωt , and ω and ωri are the carrier and the ith ring’s angular resonance frequencies, respectively. Ein , Eout1 , and Eout2 represent the input and output signals, τ li is the ith ring’s amplitude decay

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Fig. 19.2 A triple-ring structure with the middle ring only being active, as an example of building a dynamic model of a microresonator-based device for system-level characterization

rate due to cavity loss, and τ c1 and τ c4 are the amplitude decay rates due to ringto-waveguide coupling that represents equivalently a loss to the ring [17]. In the jth coupling area, coupling factor μj satisfies μj 2 = κ j 2 vg /2π R = 2/τ cj , where κ j 2 is the power coupling coefficient and vg is the group velocity in a ring with a radius R [17]. Since only the middle ring is driven, only its resonance angular frequency and loss are temporal functions. The differential equations can be solved using the implicit Euler method with a time step of ∼1 ps, which is much longer than a round-trip time in a ring resonator with a radius of a few micrometers. Note that, for a resonator, there are three different operating regimes, overcoupled, critically coupled, and under-coupled, depending on whether the coupling to the resonator is stronger, equal, or weaker as compared to its round-trip loss [71, 72]. Each regime features a unique phase transition in the vicinity of the resonance frequency and may have dramatically different dynamic properties [73]. Although one can use a uniform dynamic model as described above to simulate all these types of devices, a static model is still valuable for an active device [71], since it indicates the operating regime of a resonator by showing its phase profile. Moreover, a static model provides information of steady-state operations of the device (e.g., extinction ratio) and allows one to estimate, say, pump power, drive voltage, or heater requirement when the device employs all-optical, electro-optic, or thermo-optic effects. In the described model, nonlinear Kerr and Raman effects and two-photon absorption (TPA) of the traveling waves are ignored. Therefore, the model works for an all-optical operation only when a pump is launched in the direction normal to the resonator (or wafer) plane [27, 69, 74], because in this case, although pump-induced carrier density variation by TPA is considered, the pump wave is not traveling around the cavity, and there is no need to consider its waveform change caused by Kerr and TPA effects. At the receiver end, we typically set the receiver sensitivity to be –25 dBm for a BER at 10−9 in the back-to-back case. There is no optical amplifier (and thus no optical filter) used. The received power at photodiodes is not held fixed, which is typical for on-chip communications. For performance evaluation in fiber transmission, we set the receiver sensitivity to be –20 dBm. Pre-amplification is used to compensate for the fiber loss and to keep the same received power. An optical filter with four times bit-rate bandwidth is built into the receivers to remove amplifier

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noise. In all cases, an electrical filter with a bandwidth of 0.7 times the bit-rate is used for NRZ signals in the receivers. System performance of the microresonator devices can be evaluated using a signal Q-factor, eye-opening penalty (EOP), and power penalty [75]. We usually use signal Q-factor and eye-opening penalty, defined in Fig. 19.3, in the back-to-back case and power penalty in fiber transmission. Fig. 19.3 Signal quality is evaluated from the eye diagram in terms of signal Q-factor and eye-opening penalty. Dashed line shows a sampling instant in signal detection

19.3 Channel Adding/Dropping in TDM Systems In a TDM system, independent data channels at the same optical carrier frequency are multiplexed into a TDM signal and their pulses are interleaved in the time domain. This is a way to combine several low-rate signals into a high-speed channel [76]. Multiplexing these low-rate signals is quite straightforward, which can be done, for example, using an N×1 coupler. However, de-multiplexing of a TDM signal requires certain signal processing functions. At ultra-high signal rates (>160 Gb/s), all-optical approaches based on four-wave mixing (FWM) [77] and cross-phase modulation (XPM) [78] have been demonstrated to achieve demultiplexing. However, efficient nonlinear operation requires relatively high power. In on-chip scenario, the signal bit-rate may be just 5∼10 Gb/s [6], and an electrooptic switch operated in a bit-by-bit manner would be capable of dropping a low-rate signal from a TDM channel. After the drop, there are some time slots available to accommodate a new signal. A microresonator-based E/O switch can be used to do channel adding/dropping simultaneously. Figure 19.4a shows a dual-waveguide single-ring electro-optic switch, integrated with an MOS capacitor electrode. When the ring is off-resonance (with no drive voltage applied), the original TDM signal coming from the input port goes directly through the waveguide on the left side. As the ring resonator is turned on, pulses in the TDM channel are dropped down to the waveguide on the right side. At the same time, the signal coming from the add port can be uploaded to the TDM channel. Here, the carrier transit time is 16 ps. The amplitude coupling coefficient between the ring and the waveguide is 0.0678, and the ring round-trip amplitude attenuation coefficient is 0.01. Cavity Q-factor is 19,000, with 10-GHz resonance linewidth and

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Fig. 19.4 (a) A dual-waveguide single-ring E/O switch for channel adding/dropping in TDM systems. (b) A dual-waveguide triple-ring structure to improve signal quality, in which only the middle ring is driven

Fig. 19.5 Generated signal waveforms and eye diagrams at both through and drop ports for the single-ring channel add/drop switch shown in Fig. 19.4a. Horizontal axis shows time and vertical axis shows instantaneous power

5-μm radius. A 5-Gb/s RZ signal with 50% duty cycle is launched as the input. A half-rate electrical clock with 37.5% duty cycle and a voltage of 5 V is employed to drive the resonator. Figure 19.5 shows the generated signal waveforms and eye diagrams, which are significantly degraded, with jitter up to 63 ps and 11 dB extinction ratio (ER). The signal degradation can be explained as follows. With the electrode RC time constant negligible, there are three factors determining the temporal response of the device. The carrier transit time and cavity photon lifetime govern dynamic properties of the resonator being switched on and off, while the cavity-induced group delay adds time misalignment to the signals at steady state. It is noted that the added pulses experience the group delay before combined with those coming from the input port. Thus, two data streams have a time offset relative to each other due to the unbalanced group delay, which causes an overlap between the data bits in the TDM signal at the through port, as shown in Fig. 19.5. In addition, the transfer function of a single ring (first-order Lorentzian profile) has long roll-off edges and introduces large cross talk and interference into the combined signals. Both the walk-off and interference contribute to an increased jitter, while the cross talk degrades the signal ER. Channel add/drop performance can be improved using a higher order ring resonator switch shown in Fig. 19.4b. It is a symmetric structure with three equally sized rings and an electrode integrated onto the middle ring. When the middle ring is not driven (i.e., all three rings are on-resonance), the signal from the add port goes across the three rings to the through port, experiencing a group delay of τ . When the middle ring is tuned off-resonance, the left ring is still on-resonance and

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becomes over-coupled [71, 72], since its equivalent power loss due to its coupling to the middle ring is reduced. Therefore, the input signal also experiences a delay comparable to τ when arriving at the through port. In other words, the input signal is a slowed light because of the over-coupling between the waveguide and the left ring. The two signal delays are designed to be almost identical by changing the ring structures. On the other hand, the transfer function of the three-ring switch becomes more box-like, which reduces the aforementioned cross talk as well. We set the waveguide-to-ring coupling coefficient to be 0.1265 and the interring coupling coefficient to be 0.00762, with the round-trip amplitude attenuation coefficient for each ring the same as the single-ring switch. As a result, the 3dB bandwidth of the on-resonance transfer function increases to ∼25 GHz, which enables the three-ring switch to be operated for 10-Gb/s RZ signal with 50% duty cycle. Using a static model based on coupled mode theory [71, 79], we calculate the amplitude response and group delay of the three-ring switch, as shown in Fig. 19.6. The “add” signal experiences a bandpass filtering effect at the on-resonance state, while the input signal experiences a shallow-notch profile at the off-resonance state with nearly the same average delay of ∼30 ps. This way, balanced delays are obtained for the two signals in the TDM system.

Fig. 19.6 The amplitude and group delay responses of a dual-waveguide triple-ring switch, when it is operated (a) on-resonance and (b) off-resonance, at both the through (notch) and drop (bandpass) ports

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The triple-ring switch exhibits much better channel add/drop performance as shown in Fig. 19.7. All bits in the output signals are timed well, and the jitter is reduced to 6.4 ps, as seen from the eye diagrams at the through and drop ports, due to the balanced group delay. Drive voltage is 8 V. The residual cross talk exists in the dropped signal with an ER of 8 dB, because of unsuppressed coupling between the ring resonators. Fig. 19.7 Generated signal waveforms and eye diagrams at both through and drop ports for the triple-ring channel add/drop switch shown in Fig. 19.4b. Horizontal axis shows time and vertical axis shows instantaneous power

19.4 Generation and Demodulation for Advanced Data Formats In telecom systems, advanced modulation formats such as differential phase shift keying (DPSK) and differential quadrature phase shift keying (DQPSK) have exhibited great advantages [80, 81]. Those include better receiver sensitivity, better tolerance to chromatic dispersion and nonlinearity in optical fibers, and higher spectral efficiency (for DQPSK), as compared to OOK signals [80]. Conventionally, DPSK modulation is based on a Mach–Zehnder modulator (MZM), and the laser carrier experiences a phase jump of π across the minimum point in the MZM’s transfer function, in which a drive voltage of 2 Vπ is needed [81]. For demodulation, the phase of a bit is compared to its adjacent bit in a delay-line interferometer (DLI), and the differential phase-encoded signal is converted to intensity-modulated signals called duobinary (DB) and alternate-mark inversion (AMI) for balanced detection [81]. A DQPSK signal contains two independent DPSK signals that are orthogonal in the phase domain and is generated using a pair of MZMs and a 90◦ phase shifter, all nested in an interferometric structure [80, 81]. The DQPSK demodulator has two DLIs that are phase shifted relative to each other. Therefore, transmitters and receivers for DQPSK are more than twice as complex [80, 81]. In a short-haul communication environment, the common method for carrying information has been intensity modulation; data is carried by OOK signals. There has been little attention paid to whether or not other modulation formats can improve the system performance of the optical interconnection. In general, the electrical signal can be modulated onto amplitude, phase, frequency, or polarization of an optical wave [81]. Potentially, utilizing phase-modulated data formats may help achieve

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lower power consumption and higher spectral efficiency for data center-wide and chip-scale optical interconnects. When the phase-modulated signals are introduced into short-haul optical interconnects, the standard signal generation and demodulation scheme (MZM + DLI) in telecom systems would require too much “real estate” and unaffordable power dissipation. These devices are fairly large, on the order of a square centimeter. A phase modulator and some other demodulators [82–84] tend to be large as well. More importantly, MZMs may not be power efficient, with a drive voltage of 2 Vπ , normally ranging from 5 to 15 V. We believe that this is an important issue since E/O conversion power consumption is dominant for most optical interconnection applications [6]. In addition, the MZMs and DLIs are built using interferometric structures and would not enable highly wavelength-selective operation in dense wavelength-division-multiplexing (WDM) systems. It would be desirable to design compact microring-based DPSK modulators and demodulators compatible with SOI platforms, with two potential advantages: (i) cost-effective fabrication into arrays and (ii) reduced electrical power consumption. For resonator-based modulators, a sharp phase transition occurs across resonance and a π phase shift can be obtained by shifting the spectral response of a resonator using a low voltage. A single-ring modulator shown in Fig. 19.8 is considered for phase modulation. To obtain a phase shift of π , an over-coupled resonator [71, 72] is better than an under-coupled one, because the phase shift in the under-coupled resonator is not monotonic and has a theoretical maximum value of π that is sensitive to small variations of coupling and loss coefficients. In contrast, the over-coupled ring has a 2π phase shift across resonance, as shown in Fig. 19.8, which is monotonic and much more tolerant to a structural perturbation. The laser carrier is biased to be off-resonance originally. Applied voltage shifts the resonance profile toward higher frequency so that the optical carrier experiences the phase shift across resonance. Note that Fig. 19.8 only shows a conceptual principle of NRZ-DPSK signal generation based on a microring. In reality, the resonance profile is dynamically reshaped due to the carrier-induced loss, and the optical bias needs to be modified to obtain a constant pulse peak power in bit durations.

Fig. 19.8 Microring-based NRZ-DPSK modulation and demodulation. In modulation, the continuous wave (CW) laser carrier experiences a phase shift of π across resonance, using a over-coupled ring resonator. A double-waveguide microring filter enables demodulation of both DB and AMI signals for balanced detection

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Microring-based DPSK demodulation is achieved in the frequency domain [83, 84] instead of operation in the time domain as in a DLI. A double-waveguide singlering filter shown in Fig. 19.8 has two output ports delivering bandpass and notch filtering functions, demodulating DPSK into DB and AMI signals for balanced detection. The frequency domain scheme removes the need for a one-bit delay, which ensures a greatly reduced chip size. Otherwise, a multiple-ring delay element [34, 85, 86] may have to be used to build a microring-based version of a DLI. The symbol constellation diagram in Fig. 19.9a shows a comparison of different DPSK generation schemes, in which the electric field of the modulated signals is mapped onto the complex plane. The microring DPSK modulator (dash) is compared to a phase modulator (dash dot) and an MZM (dot). The modulated optical field transits between two symbols (black dots), following different tracks. In theory, an MZM has a track exactly along the real axis, so it adds no frequency chirp to the signal. On the other hand, a phase modulator has constant output power, and the symbol goes back and forth along the arc in Fig. 19.9a, producing stronger chirp [80]. The microring-based DPSK modulator induces both frequency chirp and intensity dips (Lorentzian-shaped notch profile). However, the frequency chirp and intensity dips always occur together, which means that the chirped frequency components have low power. Therefore, the microring-based DPSK signal has a transition track close to that given by the MZM (on the real axis).

(a)

(b)

Fig. 19.9 (a) Symbol constellation diagrams for a microring-based modulator (dash), a phase modulator (dash dot), and an MZM (dot). (b) EOP of the modulated DPSK signal vs. carrier transit time. Eye diagrams are plotted in the same scale

An over-coupled ring modulator with a radius of 5 μm has a cavity Q-factor of 10000, which is used to generate DPSK signals at 10 Gb/s. The drive voltage is 1.88 V. The dynamic equations described above are solved to allow evaluating the signal quality. With a varied carrier transit time from 12 to 50 ps, we calculate the EOP as shown in Fig. 19.9b. A DLI is used as a demodulator to isolate the microring-based modulation effect. The EOP increases with transit time. The microring modulator produces good DPSK signals with no significant imperfections, as long as the transit time is shorter than 30 ps, and accordingly the EOP

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is less than 0.5 dB. Inserted eye diagrams are plotted in the same scale and corresponding to transit times of 16 and 50 ps, respectively. Timing jitter in the modulated signal also becomes worse for a long transit time, as seen in the eye diagrams. Different than an MZM, in which the phase shift can only be π , the microring DPSK generator can provide a continuously tunable phase shift by changing drive voltage. We examine the microring-based demodulator’s tolerance to a non-π phase shift. As an example, DPSK signals at 10 Gb/s are produced using a microring modulator with a cavity Q-factor of 22000, and drive voltage increases from 0.77 to 2.2 V to introduce a phase shift from exactly π to 1.41π . Accordingly, the output signal power increases by almost 5 dB. In this case, the carrier transit time is set to be 23 ps, and the microring demodulator has a cavity Q-factor of 22000 (i.e., 3-dB bandwidth of 8.7 GHz). Fig. 19.10a shows a comparison of the microring-based demodulator and a DLI in terms of demodulation performance. Over-driving of the microring modulator improves eye-opening dramatically by 7 dB for microringbased demodulation and by 5 dB for a DLI. It is important to mention that the DLI demodulator is based on interference of light waves and usually requires an accurate π phase shift. When the phase shift is changed, the DLI demodulator would have a relatively low extinction ratio that discounts the benefit from the signal power increment. In contrast, a varied phase shift does not change the spectral distribution of DB and AMI signals in DPSK signal spectra. Thus, the microring demodulator operating in the frequency domain releases the requirement for an exact π phase shift. It should be noted that, for telecom applications, the phase shift in DPSK signals is desired to be exactly (or close to) π , which makes the signal more tolerant to phase noise induced by in-line amplifiers [80]. In this case, one may not want to over-drive the modulator too much. For short-reach or on-chip interconnects, system performance would benefit from over-driving the microring modulator.

Fig. 19.10 (a) Eye-opening improvement by driving the ring modulator harder, as phase shift is more than π. (b) Eye-opening penalty is examined as a function of demodulator frequency offset. The microring-based demodulator is more tolerant to the offset than a DLI. Eye diagrams are in the same scale for each figure

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The quality of the demodulated DPSK signal at 10 Gb/s is examined when a frequency offset between the demodulator resonance frequency and the signal carrier is increased from 0 to 3 GHz. Here, an MZM is used as a DPSK modulator, and the demodulator bandwidth is 8.7 GHz, corresponding to a cavity Q-factor of 22000. As shown in Fig. 19.10b, the microring-based DPSK demodulator is much more tolerant to demodulator frequency offset than a DLI, because DLI demodulation scheme is based on interference of optical waves and relies on accurate phase control that is reflected as a frequency shift of the DLI transfer function. Since a one-bit-time DLI has a periodic transfer function with a free spectral range (FSR) of 10 GHz, the demodulated DPSK signal will be the same if the DLI has a frequency shift of one FSR. Similarly, the demodulated signal will be inverted for a frequency offset of FSR/2, but demodulation is still successful. However, the demodulation will be completely degraded with FSR/4 offset (i.e., 2.5 GHz), and this is why we see a dramatic EOP in Fig. 19.10b for the DLI-based demodulation. In contrast, the microring demodulator works in the frequency domain and has a huge FSR (e.g., tens of nanometers for a ring radius of 1.5 μm [87]) and a relatively stable performance with much smaller EOP, compared to a DLI. In addition to the DPSK format, other phase-modulated data formats include DQPSK and DB. As a multi-level phase shift keying signal, DQPSK allows to double the volume of information that is being transmitted without extending signal spectral width. Microring-based DQPSK generators and demodulators have been proposed [88], but high drive voltage is needed to achieve low bit-error-rate detection. Another way to increase the interconnection capacity is to employ DB modulation format, which occupies only half-rate bandwidth and can be detected with no need for a demodulator [89].

19.5 Frequency Comb Generation for Arbitrary Waveform Generation Based on the discussion above, we understand that a resonator-based device has an operation speed that is limited by cavity photon lifetime t, because it determines how fast photons can be coupled in and out of the resonator. In the frequency domain, this limitation is translated to the resonance linewidth f by the relationship f = 1/2π t. When a high-speed signal with a signal bandwidth much larger than f is launched into the resonator, it sees a filtering effect given by the resonator, and its high-frequency components are removed. For an active resonator device, the operation speed may be more limited, for example, due to carrier dynamics in semiconductors. A natural impression is that the resonator-based devices may not be suitable to handle high-speed signals. Such a concern would be true when a single resonance peak of a resonator is manipulated for communication and signal processing applications. However, a resonator has many resonance peaks that are almost equally spaced in the frequency domain. A joint control over all these tones can be utilized to achieve signal processing functions at an ultra-high speed. In this section, we discuss the nonlinear

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frequency comb generation using microresonators, which holds a promising future for high-speed arbitrary waveform generation and pulse shaping. Recent years have witnessed significant scientific and technological advances in optical frequency combs [90, 91]. Several applications of the optical frequency combs have been explored, including optical frequency metrology [92], gas sensing [93], astrophysical spectrometer calibration [94], and femto-second optical pulse generation and reshaping [95, 96]. Employing the nonlinear Kerr effect such as FWM in a high-Q microresonator, one has a chance to achieve wideband optical comb generation with ultra-low power consumption [97, 98]. Figure 19.11 shows broadband parametric frequency conversion using a microtoroid resonator on a silicon chip. The toroidal resonator with a diameter of 75 μm has a cavity Q-factor of >108 , which allows for a strong nonlinearity enhancement. An optical pump from a CW laser with 60 mW output power is launched into the cavity, producing an optical intensity exceeding 1 GW/cm2 inside the cavity. As shown in Fig. 19.11, over a 300 nm wavelength range, the optical frequency comb is generated with an FSR of 7 nm. The uniformity of the comb spacing has been experimentally verified to be 7.3×10−18 relative to the optical carrier frequency.

Fig. 19.11 Optical comb generation using parametric four-wave-mixing in a microtoroid resonator. Inset is a scanning electron microscope image of the toroid resonator on a silicon chip

There are two advantages to using the microtoroid resonator. First, the material is silica, having no TPA and thus being able to handle high optical power inside the cavity, where the Q-factor is very high. Second, the silica resonator has relatively low material dispersion and waveguide dispersion, which is responsible for extending the parametric interaction bandwidth and improving nonlinear efficiency. However, even for high-speed signal processing application, one may not need such a wide bandwidth comb with nearly 1 THz FSR. For ultra-fast pulse shaping and arbitrary waveform generation [47, 95, 96], the optical frequency comb is first de-multiplexed into different waveguides. Then, each of the frequency tones is individually modulated in both amplitude and phase, as

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Fig. 19.12 Optical arbitrary waveform generation scheme. Microresonators play a critical role in achieving an on-chip version of this signal processing unit

shown in Fig. 19.12. Note that amplitude modulation (AM) and phase modulation (PM) can also be realized using microresonators, as discussed above. After multiplexed, a new pulse waveform is generated based on how its frequency components have been modulated in amplitude and phase. More specifically, we note that the FSR in the generated optical comb is inversely proportional to the cavity length of the resonator. A larger device will produce a smaller FSR for the signal processing applications. Moreover, a resonator in silicon or III–V compound semiconductor might be desired for the optical comb generation due to the following reasons: (i) obtaining a better material compatibility to on-chip phase and amplitude modulators; (ii) benefiting from strong nonlinear Kerr effects; and (iii) taking advantage of the higher effective index of the guided mode compared to silica, allowing for a resonator size reduction to produce the same FSR. However, there are also two other issues to consider when using semiconductor resonators. First, the TPA and carrier-induced absorption may limit optical pump power and thus the frequency comb bandwidth. This would not be a severe problem as long as the resultant bandwidth is more than a THz. Second, a strong chromatic dispersion [99, 100] in a semiconductor waveguide makes a nonlinear parametric process inefficient. Recently, silicon waveguides with a flat and low dispersion over a wide band have been demonstrated [101, 102], potentially mitigating the dispersion challenge. It is expected that, in the near future, an on-chip high-speed optical arbitrary waveform generator based on microresonators will be experimentally demonstrated.

19.6 Summary In this chapter, we have briefly outlined the potential communication and signal processing applications of photonic microresonators. A system-level numerical model has been described for evaluating system performance of the resonatorbased functional devices, with an emphasis on signal quality. We have addressed several specific applications of electro-optic and all-optical resonators, including channel add/drop filtering in TDM systems, modulation and demodulation for advanced data formats, and optical frequency comb generation for high-speed optical arbitrary waveform generation. Our goal is to link the device physics to their

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system performance and provide potential on-chip solutions to some challenging problems in optical communication and signal processing. Acknowledgment: The authors would like to thank Dr. Raymond G Beausoleil, Dr. Muping Song, Jeng-Yuan Yang, and Yunchu Li for helpful discussions. This work was sponsored by HP Laboratories and the DARPA SPAWAR program (San Diego) with contract number N66001-08-1-2058.

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Index

A Absorption cross section, 258–259 Active tuning, 205–225 Add-drop filter, 6, 182, 233, 301–302, 304, 308, 313, 316, 459, 471–473, 481 Adhesive-bonding, 382 Adiabatic tuning, 165–166 Admittance inverter, 125–126 matrix, 127–128 Airy function, 429 All-optical devices, 213, 225 All-optical logic, 215 All-optically tuned microresonators, 212–221 Allpass filter, 119 All-pass microring filters, 116, 118–122 Alternate-mark inversion (AMI), 494–497 Amplifier noise, 497 AND gate, 216 Anisotropic etching, 18 Annular Bragg resonators, 362 Anti-bonding super-modes, 396, 402, 412 Anti-crossing, 398, 400 Antigens, 259, 265–266 Apodization, 109–110, 182, 191 Arbitrary waveform generation, 487, 499–500 Arrays of microlasers, 450 Asynchronously tuned microrings, 121 Azimuthal mode numbers, 148, 150, 279, 291, 430 B Backreflection, 33, 189 Backscattering, 192 correlation of, 192 Backward dropping, 309, 310, 315 Bandgap engineering, 14 Bandwidth decompression, 172 Beat length, 66, 70

Bending loss, 67, 77, 99, 234, 277, 293, 301, 313–314, 362, 366–367 radius, 9, 22, 62, 65, 67–68, 76, 82–83, 153, 183–185, 362 Bend modes, 30, 32, 34, 37–41, 43, 46–48, 55–56, 62–63, 66, 71, 79–80 Bent waveguide attenuation constant, 32, 37–39 2D analytical model, 31–35 3D film mode matching (FMM), 54 phase constant, 32, 39, 66, 71 propagation constant, 30–31, 38, 41, 45, 64 transverse electric (TE) mode, 38 transverse magnetic (TM) mode, 38 Bessel equation, 38 functions, 363 Bessel–Thompson filter, 107 Bilaterally symmetric, 142, 144 Bilinear transformation, 118, 121 Biodetection, 254 Biological detection, 256, 262, 265–270 Biosensors, 254–256, 260, 285, 403, 410 Biotin, 266–268 Biotin–streptavidin, 266–267 Birefringence, 62, 64–65, 67, 69–70, 72, 77–78, 80, 82, 84, 140–141, 147 Birefringent fiber, 141 Bispheres, 425, 434, 436, 439–441 Bistability, 216–221, 224, 412 thermally induced, 224 Bit error rate (BER), 188, 196–198, 468, 475, 490, 498 Bit rate limitations, 236, 242 Bonding super-modes, 396, 402, 411–412 Bond percolation, 439–440 Boundary conditions, 38, 49, 54, 329, 364, 368–370, 379

I. Chremmos et al. (eds.), Photonic Microresonator Research and Applications, Springer Series in Optical Sciences 156, DOI 10.1007/978-1-4419-1744-7,  C Springer Science+Business Media, LLC 2010

507

508 Box-like filter response, 95–97, 109 Bragg layers, 362, 365–367, 379 Bragg reflection, 2, 328–329, 362, 366–367, 446 Bragg reflector, 363, 365, 367, 376–378, 478 Brillouin zone, 170 Broadband microring filters, 118 Buffer, 1, 8, 193, 213, 254–255, 267–268, 321, 381, 415, 424, 449, 466–467 Build-up factor, 6–7 Buried waveguide, 84, 184–185 Bus waveguide, 3, 5, 10, 29–32, 34, 50, 52, 54, 70–71, 80, 89, 97, 103, 109–110, 123, 132–133, 141, 152–154, 160–161, 182, 191, 194, 196–197, 222, 301–302, 311, 352, 460 Butterworth filters, 96–97 C Canonical coupling topology, 129 Capturing light pulses, 173–174 with few resonators, 173 Carousel trapping, 415 Carrier dynamics, 209, 498 injection, 205, 207–208, 231, 321, 424, 462–463, 480 lifetime, 215–216, 229–230, 348, 351 transit time, 488, 491–492, 496–497 Cascade, 29, 77, 79, 87–89, 111, 118, 120–121, 132–135, 155, 176, 182, 206, 302, 318, 321–323, 408, 460, 487 Cavity finesse, 4 lifetime, 5–6, 123, 175, 332 photon lifetime, 378, 489, 492, 498 -waveguide coupling, 170, 351 C band, 184, 300 Circular Bragg microlaser (CBML) advantages and applications, 366–381 biochemical sensors, 380 coupled-wave equations, 368 design rules, 364–366, 377 detuning factor, 368, 370–371 disk, 368, 370–374 emission efficiency, 372–375 experimental results, 383–385 fabrication, 382, 388 far-field patterns, 372–373 lasing pattern, 384 modal areas, 373–374 modal field patterns, 370–371 modal field profile, 365–367, 387

Index quality factor, 373–374 resolution, 387, 389 ring, 368, 370–375 sensitivity, 375, 377, 380–381, 387–389 single-mode range, 373, 388–389 surface emitting lasers, 367–375 telecommunication applications, 380–381 theoretical framework, 363–364 threshold gain, 371–372, 379 Chains of resonators, 398, 426–427, 431, 435–436, 449 Channel adding/dropping, 487, 491–494 bending, 473–476, 481 Characteristic impedance of CROW, 200–201 of waveguide, 186, 199 Charge storage time, 210 Chebyshev filters, 183 polynomial, 95 Chemical etching, 382–383 Chirp, 148, 489, 496 Circuit schematic, 154, 156 Circular array of identical microrings, 154 Circular Bragg nanolasers (CBNL), 375, 377, 379–381, 384–389 Circular Bragg nano-resonators (CBNR), 362–363, 365–367, 376–379, 386–387 Circular DFB laser, 367–368, 370–374 detuning factor, 368, 370–371 emission efficiency, 372–375 far-field patterns, 372–373 modal areas, 373–374 modal field patterns, 370–371 quality factor, 373–374 single-mode range, 373 threshold gain, 370–372 Circular microresonators 2D, 29 3D, 30, 56 Cladding index, 9, 69 CMOS, 11, 13, 19, 300, 316, 424, 436, 486 Coherent coupling, 427 Coherent population oscillation (CPO), 467 Collective absorption, 449 Collective emission, 446, 449 Colloidal deposition, 431 Compensate fabrication variations, 23 Complex bend mode propagation constants, 38 III–V compound semiconductor, 488, 500 Configuration modulation, 459 Conformal mapping, 9

Index Continued fraction, 94 Coprimes, 99 Core index, 9, 69 Counterpropagating waves, 140–141, 151, 161 Coupled cavities, 30, 182, 198, 398, 411, 424, 435, 437–438, 443, 445–446, 449 Coupled mode equations, 30, 42–44, 55, 123, 291, 489 Coupled mode theory, 30–31, 40, 56, 62, 64–66, 118, 174, 368, 371, 379, 493 Coupled resonator induced transparency (CRIT), 399–401, 410, 446 Coupled resonator optical waveguide (CROW) group velocity, 167, 190, 247, 350 slowing ratio, 189 tuning, 406 Coupled ring reflector (CRR), 156–160, Coupled ring resonators, 77–78, 88–99, 106–111, 181–201, 495 Couplers 2D bent–straight waveguide, 123, 301, 304, 328, 339, 344, 352, 399 3D bent–straight waveguide, 97, 304, 439 scattering matrix, 33, 43–44, 55 transfer matrix, 43, 45, 90, 155–157 Coupling angle, 129, 142 Coupling coefficients, 46–48, 55, 65, 69, 82–83, 95, 122–123, 128–132, 134–136, 161, 182–183, 185–188, 192–194, 368, 371, 472, 480 Coupling matrix diagonalization, 79 optimization, 125, 128 synthesis, 127 Coupling of modes in time, 152 Coupling rate, 168, 170–171, 248 Critical coupling, 4–5, 72–74, 91–92, 145, 211, 223, 449, 462, 465–466, 468–470, 472 Critically coupled regime, 470 Cross-phase modulation (XPM), 491 Crosstalk, 185, 187 D Dark states, 165 Datacom system, 485 Data pattern dependence, 488 Deep etching, 11 Deep ultra-violet (DUV) lithography, 19–20 Degenerate resonances, 141 Degenerate super-modes, 396, 404, 406 Delay-bandwidth product constraint, 167 Delay-line interferometer (DLI), 494–498

509 Delay lines, 89, 116, 121, 172, 182, 186–187, 192–198, 232, 321, 361, 408, 415, 445, 449, 481, 494 Dense wavelength division multiplexing (DWDM), 87, 495 Detection limit, 207, 292–293, 296 Detection sensitivity, 265, 296 Detuning, 123, 145, 148, 158–159, 173, 212, 218–220, 229, 370–371, 399, 401, 408, 410, 412, 438–439 Diagonal matrix, 79, 124, 127 Dielectric modulation, 166, 174 Dielectrophoresis, 425, 433–434 Differential group delay, 240 Differential phase shift, 133–134, 140, 375 Differential phase shift keying (DPSK), 197, 494–498 Differential quadrature phase shift keying (DQPSK), 197–198, 494, 498 Differential thermal tuning, 177 Digital micromirror devices (DMD), 459 Direct-coupled microring filters, 116, 123–132, 135–136 equivalent electrical network model, 125–126 Directional coupler, 33, 70–71, 75, 77, 80, 82, 139, 191, 193, 222, 242, 277, 300–301, 380 Directly coupled ring resonators, 77–78, 181–201 Discontinuity-assisted ring resonator, 141, 143 Discontinuous Galerkin FEM, 31, 258 Disorder effects, 423 Dispersion compensation, 188, 279, 460 relation flattening, 168 suppression, 172–173 Distributed Bragg reflectors (DBR), 362–363, 373, 478–479 Double-resonant, 241, 246 Double-resonator system, 175–176 Duobinary (DB), 494–498 Dynamic coupled mode equations, 489 Dynamic photonic structure, 166 frequency shift in, 167 E Effective refractive index, 4, 258, 262, 376, 397, 413, 461 Electrical tuning of microresonators, 205 Electro-absorption (EA), 14, 90, 93, 108, 460, 463, 476, 480–481 Electromagnetically-induced transparency (EIT), 165, 168, 375, 399–400, 408, 467

510

Index

Electron beam lithography (EBL), 20–21, 382 Electro-optical effects, 229, 231 Electro-optic modulators, 206–211, 213, 231, 332 Electro-optic switches, add-drop configuration, 212 Elliptic filter, 135 Embedding matrix, 142–143 Emission directionality, 404–405 Energy coupling analysis, 118, 123 coefficient, 123, 125, 128, 132 Excess polarizability, 257 Exciton, 231, 437 Excitonic resonance, 177 Exciton resonances, 231 Extension of the FSR, 99, 146, 148 Extinction ratio (ER), 76, 82, 182, 186, 211, 216, 219, 283, 294, 332, 346, 409, 472, 475, 477, 481, 490, 492, 497 Extrinsic losses, 12 Eye-diagram, 211 Eye-opening penalty (EOP), 491, 496–498

Flame-brushing technique, 276, 296 Fluorescence, 15, 268, 428–429 Fluorescent probe, 254 Focusing of light, 435, 449 Forward dropping, 308–311, 315–316 Four wave mixing (FWM), 467, 491, 499 Franz–Keldysh effect, 488 Free-carrier dispersion effect, 175, 205–206, 221, 347–349 injection, 462–463, 480 lifetime, 216 Free-carrier absorption (FCA), 11, 14, 221, 477 Free spectral range (FSR) extension, 2–6, 23, 36, 72–78, 81–84, 87–111, 118, 128, 131, 143, 146–148, 161, 183–188, 190, 197, 277, 283–284, 299, 311–312, 362, 403, 430, 473, 498–500 Frequency comb generation, 487, 498–500 Full width at half maximum (FWHM), 5–6, 36–37, 104, 106–107, 218, 223, 234–237, 239–240, 279, 293, 353, 383, 387, 448, 468, 474–475 Fused silica, 12–15

F Fabrication tolerances, 7, 57, 88, 177, 183, 186, 198 Fabry–P´erot (FP), 2–4, 23, 142, 231–233, 235, 256, 344, 394, 409–410, 463, 478 cavity, 142, 463 interferometer, 3–4, 232 resonator, 23, 231, 256, 343, 394 Face-centered-cubic (FCC), 432 Fano–Bode theorem, 183 Fano interference, 167–168, 401 Fano resonance, 168–169, 175, 409–410, 439, 449 Fast light, 470 Feature size, 18, 20 Feedback-assisted four-port, 142 Feldtkeller relation, 117 Figures of merit, 231–232 Figure 8 reflector, 152–153 Filter poles and zeros, 116–117 transfer functions, 118, 130, 134, 136 Finesse, 2–4, 6–7, 20, 23, 36–37, 104, 148, 183, 188, 191–192, 279 Finite difference time domain (FDTD) method, 31, 49–50, 52–53, 57, 152, 171, 173–174, 304, 306, 314–315, 317, 331, 377, 444 Finite element method, 288

G Gain cut-off bit rate, 243, 245 Gain trimming, 460, 463, 476, 480–481 Gaussian pulse, 171, 173–174, 467–468, 481 Gires–Tournois interferometer, 3–4, 460–461, 463–464 quality factor (Q), 4, 461 transfer function, 461 Glass waveguide, 83, 275 Goos–Hanchen shift, 7 Gram–Schmidt orthogonalization, 128, 130 Grating -assisted microring reflector, 148 strength, 149 Green’s function method, 169, 368 Group delay, 6–7, 105–107, 116–117, 121–122, 145, 147–148, 156, 182–183, 187–188, 192–194, 233–240, 440, 467, 492–494 Group delay dispersion, 105–107, 234 delay enhancement, 148 delay in reflection, 144–145, 150 index, 5, 36–37, 191–192, 209, 233, 311–312 refractive index, 184 velocity, 5, 123, 167–168, 170, 172, 190, 247, 311, 322, 344, 350, 367, 381, 405, 427, 490

Index Guided mode resonance, 301 Guided modes, 32, 34, 43, 61, 65 Gyro, 375, 377, 389 H Heater, 15, 222–223, 321, 490 nickel–chromium (Ni–Cr), 185 Heavy water, 256, 262, 264–265 Helmholtz equation, 363 Higher order effects, 8 High index contrast semiconductor materials, 299 waveguide, 63, 183 High-order filters, 115 High-speed modulation, 211 Hot-embossing technique, 21–22 Huygens–Fresnel principle, 372 Hybrid modes, 53, 84 Hybridness, 62–63, 79 Hydrogen-assisted annealing process, 464 Hydrogen silsesquioxane (HSQ) electron-beam-resist, 185 Hysteresis loop, 219–220 I Impedance matrix, 127 mismatch, 189 Imperfections, 4, 186–187, 222, 227, 403, 496–497 Index contrast, 7–8, 10–14, 20, 22, 30, 39, 63–64, 68–69, 74, 153, 183–184, 186, 261–262, 299–300, 479 profile, 54–55, 62, 64–66, 69, 81, 365, 377, 385 Indirect bandgap, 13, 229 Indirect-bandgap material, 13 Inductively coupled plasma, 118 Infinite impulse response (IIR) filter, 198 Inline reflector, 161 Integral equation approach, 158 methods, 31 Integrated microring-based devices, 162 Integrated photonics, 115, 485–486 Interconnection networks, 408, 486 Interleaver, 75, 84, 107–109, 111, 182 International technology roadmap for semiconductors (ITRS), 222, 486 Interstitial modes, 146 Inter-symbol interference, 245, 240 Intrinsic cut-off bit rate, 241

511 Intrinsic quality factor (Q int ), 175, 337, 470, 474 Isotropic etching, 13, 16, 18, 22 ITU Grid, 110 J Jacobi rotations, 129 Jitter, 265, 268, 492, 494, 497 Jones calculus, 141 Josephson interferometer, 413–414 K Kerr coefficient, 230 effect, 14, 213, 241, 243, 247, 499–500 Knot resonator, 277, 279–280 L Lab-on-chip, 446, 450 Label-free, 253, 256 Laser linewidth stability, 263–264 tissue surgery, 449 Lateral and the vertical coupling, 7 Leakage through the gratings, 151 Light cone, 328–331, 353 Light–matter coupling, 450 Linearly polarized electromagnetic wave, 166 Linear-phase flat-top filter, 122 Linewidth, 4–5, 16, 248, 257–259, 261–264, 307, 387–388, 403, 408–409, 430–431, 438, 491–492, 498 Liquid core optical ring resonator, 255–256 Localization of light, 436 Loop mirror, 139–141, 152 Lorentzian function, 87–88, 97 profile, 492 -shape, 3–4, 474, 496 Loss, 4–6, 8–15, 22–23, 71–73, 77–78, 92–94, 122, 131–132, 135, 140, 145, 148–154, 183–187, 196, 231, 233–234, 260–263, 283–287, 301–302, 311–315, 337, 436, 460–463, 470, 472, 490, 493, 495 Lossless ring, 72–73 Low -dispersion filters, 121 index contrast dielectric materials, 299 -pressure chemical vapor deposition (LPCVD), 184 M Mach–Zehnder modulator (MZM), 494–498 Matched optical filtering, 473, 475, 481

512 Material absorption, 4, 8, 260, 262, 427 modulation, 459 Matrix eigenvalue decomposition, 124 vector, 79, 400 Maximally flat response, 73, 82, 121 Maximum switching bit rate, 243, 246 Maximum switching speed, 247 Maxwell equations, 42 Micro-electro-mechanical systems (MEMS), 222, 459–481 bandwith-tunable filters, 460, 473–474 reconfigurable add-drop filters, 471–473 tunable laser, 464, 476, 478 tunable VCSEL, 467, 479–480 Micro-electro-mechanical systems (MEMS) tuned microdisk resonator, 464–465, 469–470, 472, 474–475 microresonators, 459–481 resonator cavity length, 478–480 resonator coupling, 463–476 resonator loss, 476–477 Microspectrometers, 426, 447, 450, 471 Metal-oxide-semiconductor (MOS), 460, 488 Method of variation, 110 Microcavities, 1, 3, 29, 32, 176–177, 253–270, 327–357, 362, 396–397, 400–401, 403–409, 411–412, 435 Microdisks, 263, 394–399, 403–404, 409, 412, 435, 439, 466, 472, 481 Microdroplets, 256 Microfiber, 275–296 Microfiber resonant sensors, 286–295 Microfluidic, 255–256 channel, 292, 294 controls, 270 sensors, 277 Microlasers, 2, 395, 403, 407, 445, 450 Micromanipulation, 425, 431, 434, 441, 445, 449 Microresonators, 1–23, 29–57, 173–174, 177, 205–225, 227–249, 293, 433–435, 446, 447, 459–481, 485–501 analysis, 15, 31–141, 155 arrays, 435, 447 resonances alignment, 177 Microring ladder filters, 116, 132–135 transfer function, 116, 132–133, 135–136 Microring modulator, 206, 208, 210–211, 463, 496–497 fabrication, 206, 208, 210–211, 495–497

Index transfer function, 115, 136, 187 Microring quadruplets, 128 Microrings, 99–100, 105, 109–111, 118, 120, 122–123, 128–132, 135–136, 154–155, 212, 287, 394, 402, 410, 424, 426, 428, 435–436, 445, 461–462 Microsphere, 3, 11–15, 17, 23, 255–256, 293–295, 394, 397, 401, 403, 424–436, 441–449 resonators, 15–16, 258, 291, 395–397, 403 linear chains, 425, 445, 450 Microtoroid, 3, 7, 11, 13, 16–18, 23, 255–256, 258–259, 262–269, 466 Microtoroid sensor, 256, 268–269 Mie scattering, 429 Mie theory, 403 Minimum phase reflection response, 117 Microsphere resonator circuits (MRC) applications, 425–426, 431, 435, 441, 446, 449 fabrication, 426, 431–432 optical properties, 426, 435, 449 role of substrate, 435–436 Modal volume, 2, 8, 14, 17, 176, 301–302, 329, 333, 336, 339–340, 342, 355, 366–367, 372, 374, 385, 389, 430 Mode -conversion loss, 10 coupling, 63, 198, 394–395, 398, 406, 412, 439, 443 hybridization, 395 orthogonality, 33, 63 splitting, 399 superposition principle, 307–308 volume, 258–259, 334, 337, 338, 342, 357, 397 Molecule detection, 253–270, 446 Monomodal resonators, 35–37 Morphology-dependent resonances, 399 Multimodal resonators, 31, 48, 447 Multiple patterning techniques, 20 Multi-step lithography, 18 N NAND gate, 216 Nanoimprinting lithography (NIL), 21–23 Nanojet-induced modes (NIM), 426, 444 Nanoparticle detection, 449 Nanopatterning, 449 Nanoscale photonics jets, 426 Negative coupling coefficients, 129, 136 N–H absorption peak, 184 Noise floor, 186, 268

Index Nonlinear refractive index, 229 Non-realizable coupling topology, 128–129 Nonresonant light focusing, 443–444 Nonresonant micro-lensing, 443 Non-return-to-zero signal (NRZ), 196, 209–211, 214, 475, 488, 495 Normal mode splitting, 438, 441 O On-chip devices, 165 On–Off-Keying (OOK), 487–488, 494 Optical buffers, 1, 321, 415, 467 Optical cavities, 261, 400, 410, 423–424, 440 Optical communication, 165, 167, 175, 199, 276, 300, 307, 361, 485–486 Optical cross-connects, 116 Optical fiber taper, 275–277 Optical flip-flops, 412 Optical forces, 402–403, 415, 425, 441, 445, 449 Optical gyroscopes, 411 Optical interconnection, 206, 486, 494–495 Optical microfiber coil resonator (OMCR) bandwidth, 287 coupled wave equations, 281 detection limit, 287 effective index, 287 fabrication, 277, 282, 285, 287 loss, 277, 280, 283–287 Q factor, 277, 284, 287 sensitivity, 277, 285 sensors, 277 transmission, 277, 280, 282, 285 Optical microfiber loop resonator (OMLR) fabrication, 277–278 finesse, 279 full-width-at-half-maximum (FWHM), 279 loss, 277, 279–280, 287 Q factor, 278 resonance condition, 279 sensitivity, 277, 287 sensors, 277 transmission, 277, 279, 280 Optical microfiber loop resonator (OMLR), 277–280, 287 Optical microprobes, 450 Optical mode of a toroid, 17 Optical polymer, 161 Optical rise time, 209–210 Optical scalpel, 450 Optical sensors, 403 Optical signal-to-noise ratio (OSNR), 196, 198 Optical supermodes, 449

513 Optical transport, 424–426, 435–436, 439, 441, 443, 449 Optical tweezers, 425, 433–434, 445 Optimal interring coupling, 156 Optimum reflection coefficient, 141 Optomechanical cavities, 445 Order reduction technique, 134 Orthogonal matrix, 79 Overcoupled, 144–145, 436, 446, 462, 464–466, 468–470, 481, 490 Over-coupled regime, 212, 493, 495–496 Overcoupling, 91, 150, 466, 470 P Para-Hermitian conjugates, 117 Parallel-coupled ring resonators, 88, 107–111 interleaver, 107–109 transfer function, 88 Parallel coupling, 88, 108–111 Parallel ray approximation, 372 Parasitic thermal effect, 224 Peltier cooler, 185 Percolation threshold, 426, 439–441 Perfectly matched layers (PML), 49, 306 Perturbation methods, 403 Phase matching, 7, 41, 62, 78–79, 84, 109, 310 Phase-mismatch coupling, 8 Photoluminescence, 381, 383, 397, 442 Photon hopping, 394 Photonic atoms, 393–396, 398–403, 407, 410, 437, 449 Photonic band, 170, 177, 327 uniform shift of, 177 Photonic band gap (PBG), 2–3, 301–302, 327–328, 344, 355 Photonic benzene, 394 Photonic crystal (PC), 2–3, 6, 167–168, 170–172, 175–177, 190, 215, 233, 299–323, 327–357, 362, 394, 396, 407–408, 411, 414, 435, 439, 445, 447, 467 cavities, 303–304, 411, 424, 435 dynamic, 167 Photonic crystal (PC), microcavities all-optical switching, 351–352 applications, 350–357 channel-drop filters, 352–354 coupled resonator optical waveguide (CROW), 350–351 1D, 333–335 2D, 327–328, 335–344 delay line, 350–351 design rules, 327–331

514 Photonic crystal (PC), microcavities (cont.) dynamic control, 347–350 dynamic control of Q factor, 348–350 dynamic control of the resonance frequency, 347–348 heterostructure line-defect, 339–342 L0 point-defect, 338–339 L1 point-defect, 337–338 lasers, 355–357 membrane structure, 334 out-of-plane radiation loss, 328 Q factor, 327 surface-mode, 344–347 width-modulated line-defect, 342–344 Photonic crystal (PC), nano-cavities, 407 Photonic crystal ring resonators (PCRR) add-drop filter (ADF), 302, 304, 308, 313, 316 basic configurations, 302 cascaded-ring devices, 321–323 circuits and applications, 318–323 coupled-resonator optical waveguide (CROW), 322 dual-ring, 308–311 effective radius, 311–312, 315 free spectral range (FSR), 299–300, 311–312, 314–315 hybrid confined PC–TIR, 315–316 hybrid polymer/silicon, 316–318 NxN switching, 319–321 permutation-matrix switch, 319 polymer modulator, 316 side-coupled integrated spaced sequence of resonators (SCISSOR), 322 single-ring, 302–308 size-dependent insertion loss, 311–315 wavelength division demultiplexer, 323 Photonic dispersions, 426, 435, 442, 445 Photonic filter design, 115–135 Photonic figure-of-merit, 427 Photonic molecules, 393–415, 435–437 Photonic nanojets, 426, 448–450 Photonic resonances, 231 Photonic switching, 227–249 Photon lifetime, 4, 144, 148, 232, 254, 261, 332, 337, 340, 347–348, 350, 357, 377–379, 489, 498 Photoresist, 18 p–i–n junction, 175, 206–207, 211, 216, 424 Plasma dispersion effect, 488 Plasma enhanced chemical vapour deposition (PECVD), 18, 382 pn junction, 185–186

Index Pockels effect, 14, 231, 488 electro-optic effect, 231 Polariton, 232, 437 Polarization dependence, 110, 140 polarization division multiplexing (PolDM), 197 rotation efficiency, 61–84 rotators, 61–62, 78, 82–84 Poles-zeros method, 62 Pole-zero diagram, 130 Polymer material, 11, 14, 316, 318 microring resonators, 256 mold, 21 Polymethylmethacrylate (PMMA), 21, 259, 287, 290, 294–295, 382 Potential wells, 14, 402 Power coupling analysis, 118 coupling coefficient, 5, 80, 83, 128, 131, 140–141, 145, 161, 279, 472, 490 penalty, 198, 473, 475, 491 Pre-emphasis, 210–211 Prism-coupling, 7 Probe pulse, 348–350 Projections, 31, 44 Propagation constant, 4, 9, 15, 30–32, 34, 38, 41, 45, 48–52, 63–64, 89, 141–142, 200, 279, 281, 288–289, 291, 371 Protein G, 266–267 Prototype filter, 128 Pseudo-elliptic filter, 135 Pseudo-random binary sequence (PRBS), 196, 211 Pseudo-random bit sequence (PRBS), 487 Pulse shaping, 499 spreading, 173 synchronization, 197–198 width, 224, 350, 467–468 Q Q factor, 4–6, 8, 11–13, 15–17, 19–20, 22–23, 341, 396–404, 406, 410–411, 413, 424, 427–428, 435, 442, 491 measurements, 264 Quantum computing, 177, 407 dots, 14, 413, 415, 446, 450 information processing, 167, 177, 395, 414–415

Index wavelength shallow gratings, 148 wells, 14, 231, 381, 385, 407, 413, 437, 479 Quasi WG modes, 401 R Racetrack cavity, 10 geometry, 8 resonators, 30 ring resonators, 185 Radiation factor (RF), 18, 230, 258, 316–317, 331, 476 Radiation loss, 8–10, 20, 152–153, 183–185, 261–264, 328, 337, 357, 462 modes, 65 Radiative pressure, 441, 449 Raman effect, 490 imaging, 449 scattering, 15, 449 Rayleigh scattering, 12 RC time constant, 488, 492 Reactive ion etching, 18, 207, 382 Receiver sensitivity, 490, 494 Reciprocity, 30, 33, 42, 47–48, 55 identity, 42 of scattering matrix, 30, 33, 47–48 Recirculating cavity modes, 3 Reflected intensity, 143, 150, 153, 156 Refractive index, 2, 4, 9, 12–15, 32, 37–39, 54–55, 64, 67–69, 81, 84, 141, 143, 167–171, 175–177, 184, 206, 209, 211, 224, 228–231, 240, 257–262, 280, 282–283, 287–292, 294–296, 304–305, 327–328, 347–349, 365, 376, 410–413, 462, 480 change, 206, 211, 215, 230–231, 241, 260, 292, 295 tuning, 168 Remez exchange algorithm, 118 Resonance condition of cavity, 3, 16, 35–36 enhancement, 231–232 -shifting, 243–244, 246 Resonant cavity detection, 260, 270 Resonant enhancement, 148, 151, 231, 237, 249 Resonant tunneling, 168 Resonator–waveguide–mirror system, 176 Return loss, 77, 149, 187, 189 Reverse detection, 259

515 Ring resonators equivalent circuit, 70–71, 80–82 phase shifters, 70–74 resonance, 3, 110, 175, 191, 301 Ring resonator types, 300 Rotation sensing, 367, 375, 378–379 sensors, 375–380 Roundtrip amplitude attenuation, 5, 120, 461, 291, 493 coupling ratio, 5, 120, 123, 140, 159, 461, 491, 493 loss, 462, 472 resonator loss, 460–461, 465, 472 time, 4–5, 120, 144, 233, 242, 489–490 Runge–Kutta scheme, 44 S Sagnac effect, 375–377, 380, 411 photon lifetime, 377 Q factor, 376–377, 380 rotation dependent effective refractive index, 376 Scattering of light, 192 Scattering matrix, 30, 33–34, 43–52, 56, 140–143, 146, 149, 153–158 s-domain transfer functions, 117–118 Second order filter, 152 Sedimentation, 431 Self-assembly, 425, 431–432, 445 Self-collimation, 300–301 Self-passivating, 266 Self-phase modulation, 246–249 III–V semiconductors, 14, 230–231, 355 compounds, 14 Sensors, 182, 186, 232, 254–256, 260, 270, 275–295, 375–380, 386, 403, 410–411, 426, 446–447, 450, 486 Sequential logic, 216–217, 221 Serially-coupled microring filters, 98 Series-coupled ring resonators, 88–99, 101, 106–107 adjacent channel crosstalk, 99, 101 anti-symmetric arrangement, 400–401 asymmetric arrangement, 102, 104 crosstalk reduction, 99–103 group delay dispersion, 105–107, 234 high-order, 93–103, 106–107 optimum condition, 91–93 physical insight, 99 symmetric arrangement, 102–103 transfer function, 88–91

516 Shape factor, 96–97 Short-circuit admittance matrix, 127–128 Side-coupled integrated spaced sequences of resonators (SCISSOR), 237–240, 244–246, 322–323, 424, 467, 470 Sidelobes, 110 Sidewall roughness, 8, 11, 20, 184, 186, 191, 464 Signal processing, 1, 182, 407, 415, 460, 485–501 Silicon, 9, 11–13, 16–19, 21–22, 175, 177, 181–182, 184–185, 206–209, 214–217, 224, 263, 270, 300, 312, 320, 334–335, 347, 351–353, 424–425, 449, 463–464, 481, 488, 499–500 Silicon on insulator (SOI), 11–14, 17–21, 115, 184–186, 191–192, 196–197, 207, 212, 299–300, 304, 319–320, 333, 344–346 Silicon nitride (SiN, Si3 N4 ), 12–13, 18, 21, 183–184, 300, 463, 477, 481 Siliconoxynitride (SiON), 12–13, 184–185, 195, 299–300, 445 Similarity transformations, 128–129 Single mode waveguide, 65, 69, 170 Single molecule measurements, 268 Single-sideband (SSB) electro-optic modulator, 33–36, 332 Single virus detection, 254 Size dispersion, 426 Size-mismatched optical cavities, 400 Slow light, 1, 166, 244–246, 350–351, 375, 411, 426, 466–470 with cascade tunable bandwidth filter, 169–170 with coupled resonators, 166 scheme, 244, 246, 249 Slowly varying envelope approximation, 166 Small bending radii, 61, 81, 83, 152 Spectral characteristic, 77, 82, 110, 117, 182–183, 186, 188, 198, 311, 403, 410, 471 Spectral distortions, 183, 187–188 Spectral engineering, 393–415 Spectral resolution, 393, 412, 447 Spectroscopy, 15, 362, 415, 437, 441, 444, 447 Spectrum evaluation, 48–50, 57 constant scattering matrix method, 50 direct method, 48 reduced scattering matrices method, 49 Spin-coating, 18 Spiral-shape microcavity, 404 Standard resonator model, 31 Standing-wave resonators, 489

Index Stark effect, 14, 177 Static model, 487, 490, 493 Stopped light, 165–178 in atomic gasses, 165 conditions for, 167–168 Storage time, 4, 175, 210 Straight waveguide, 9–10, 16, 32–34, 41, 43–48, 51–52, 55, 62–67, 78–79, 81, 83–84, 109, 192, 465 modes, 44, 62–64, 79 Streptavidin solution, 268, 270 Stress sensor, 411 Stripline pedestal anti-resonant reflecting optical waveguide, 15–16 Strong coupling, 310, 414, 425, 436, 439 Structural disorder, 186–192 standard deviation of, 190 Sub-wavelength silica wires, 276 Sum-difference all-pass microring filters, 116, 118–122 Superconducting qubit systems, 177 Super-modes, 222, 394–398, 400, 402–404, 406–407, 410–414 Q factors, 396–398, 400, 402–404, 406, 410–411, 413 Superprism, 301 Superradiance, 446 Surface emitting nano-lasers, 367–375, 385–386, 388, 407 functionalization, 254, 265–268 Switching, 1, 12, 14, 206, 212, 227–249, 317–321, 351–352, 409–410, 412–413, 460 Symbol constellation diagram, 496 System eigenstates, 436, 438 T Tapered optical fibers, 260, 262–263, 291 Tapered silica fiber, 15 Teflon, 277, 283, 285–287, 289, 291–295 Thermal bandwidth tuning, 223 Thermal oxidation, 11, 19 Thermal reflow, 11 Thermally tuned microresonators, 221–225 Thermo-optic detection, 258 effect, 13, 194, 409, 487, 490 tuning, 149, 222–223 Thermo-optical effect, 222, 480 Threshold sensitivity, 257 Tight-binding approximation, 397, 403, 423, 426–428

Index Time-division-multiplexing (TDM), 487, 491–494, 500 Time domain cavity modes, 331, 343 coupled-mode equations, 123 coupled mode theory, 118 Time reversal invariance, 173 of pulses, 166 Time-varying dispersion relation, 172 TM mode, 63, 65–66, 71, 73, 82, 170, 184, 197–198 TM polarization, 49, 74, 197–199, 365 Total internal reflection (TIR), 2, 7, 9, 301–302, 315–316, 327–328, 348, 362, 367, 384, 428 Transfer matrix commutative property, 135 method, 89, 154, 169, 192, 379 Translational invariance, 168, 177 Transmission matrix method (TMM), 66, 70–71, 78, 170, 191, 200 Transmission spectra, 169, 208, 223, 261, 263, 306–307, 309–310, 343, 346–347, 350, 462, 465–466, 474, 477 Transverse electric (TE) mode, 11, 53, 63, 82, 184, 197, 207, 219, 307, 310, 344, 396, 400, 403–404 polarization, 50, 66, 159, 198, 365, 382, 429 Tryptophan, 267–270 Tunable fano resonance, 168–169 Tunable microresonators, control factors, 460–463 Tunable refractive indices, 13 Tunable slow light, 459, 466–470 Tuning optical resonances, 4, 167–168 Tuning the spectrum of light, 166–167 Two-photon absorption (TPA), 11–12, 14, 213–215, 217, 224, 352, 462, 490, 499–500 Two-wavelength interferometry, 407 U Ultrahigh-Q measurements ring-down measurement, 332–333, 337, 342 spectral domain, 331–332 time domain, 332–333 Ultra-high-Q microsphere, 259

517 Undercoupled, 143, 462, 464, 469–470, 481 regime, 261, 270 Undercoupling, 91, 151, 466, 470 Uniform asymptotic expansions, 37–38 V Van der Waals forces, 277–278 Variational formulation, 30 Vernier effect, 98–105, 146–148, 161, 404, 407 Vertical-cavity surface-emitting laser (VCSEL), 367, 385, 388, 407, 409, 445, 467, 479–480 Vertical deposition, 431 Vertical light confinement, 17–18 Virtual lifetime, 229 Virtual nonlinearity, 229 Voltage node equations, 142 W Wafer-based fabrication, 17 Wave equation, 166, 258, 365, 375 Waveguide cross section, 11, 54, 77, 184–185 refractive index contrast, 84 refractive index profile, 81 rib waveguide, 69, 84, 207, 304 shape, 62 side-coupled to two cavities, 168 single-mode waveguide, 65, 69, 170 with slanted side-walls, 67 Wavelength-division multiplexing (WDM), 1, 75, 84, 121, 184, 206, 471, 473, 475–476, 480, 495 Wavevector component, 367, 442 Weak coupling, 118, 183, 280, 286, 401–403, 436, 438, 446 Wet etching, 18 Whispering gallery modes (WGM), 2, 39–40, 51–52, 233, 254, 300–301, 304, 367, 394–395, 415, 424, 428–431, 446–447 microdisks, 394, 415 microspheres, 428–431 Writing time, 20 X XYZ translation stages, 278 Z Z-domain transfer functions, 117–118, 121 Zero-birefringence waveguide, 65, 67, 77, 84