EDITORIAL ADVISORY BOARD G. S. AGARWAL,
Ahmedabad, India
T. ASAKURA,
Sapporo, Japan
A. ASPECT,
Orsay, France
M.V. BERRY,
Bristol England
V. L. GINZBURG,
Moscow, Russia
E GORI,
Rome, Italy
A. KUJAWSKI,
Warsaw, Poland
J. PE(~NA,
Olomouc, Czech Republic
R. M. SILLITTO,
Edinburgh, Scotland
H. WALTHER,
Garching, Germany
PREFACE It is a pleasure to record that Progress in Optics is commencing the fifth decade of its existence. The first volume was published in 1961, only a few months after the invention of the laser. This event triggered a wealth of new and exciting developments, many of which were reported in the 240 review articles which were published in this series since its inception. The present volume contains seven articles covering a wide range of subjects. The first article, by M.H. Fields, J. Popp and R.K. Chang, presents a review of various optical effects in spherical and circular micro-cavities capable of supporting high-Q resonant modes (commonly referred to as morphology-dependent resonances (MDRs) or whispering gallery modes (WGMs)). The article treats the theory of symmetrical and deformed micro-cavities and describes recent research and development in the areas of quantum electrodynamics, lasers, optical spectroscopy, and filters for telecommunications. The second article, by J. Carriere, R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi and M. Mansuripur, presents a comprehensive review of the theory and practice of optical disk data storage. It covers the major characteristics of the storage media and the recording and signal detection schemes used in current optical disk systems. In the third article I. Ohlidal and D. Franta present an account of ellipsometry of thin films including a review of the theory and of main ellipsometric methods used to characterize such systems in practice. The Jones formalism is employed to describe the principles of ellipsometry and the basic techniques of ellipsometfic measurements. The uniform metric approach is utilized to express ellipsometric parameters of various thin-film systems containing isotropic or anisotropic materials. Considerable attention is devoted to thin-film systems that exhibit effects such as boundary roughness and optical inhomogenities. The review also outlines a classification system of ellipsometric methods. The fourth article, by R.T. Chen and Z. Fu, describes optical true-time delay control systems for wideband phased array antennas. A brief review is given of the basic principles and the optical technology relating to phased array antennas, including Fourier optics beam forming, optical RF phase shifters, and optical true-time delay. Different options and the status of the photonic true-time delay devices reported to date are described in detail.
vi
PREFACE
The following article, by J. Pe~ina Jr. and J. Pefina, deals with the quantum statistical properties of optical beams interacting in nonlinear couplers. Second-harmonic and subharrnonic generation, nondegenerate optical parametric processes, the Kerr effect and Raman (Brillouin) scattering are discussed. Mode coupling in waveguides, phase mismatching effects and losses are taken into account. Particular attention is paid to generation, amplification, and transmission of nonclassical light. The sixth article, by A. Luis and L.L. Sfinchez-Soto presents a review of recent advances made in the description and measurement of the quantum optical relative phase difference. Recent theoretical and experimental work on the subject demonstrates that relative phase circumvents some of the difficulties that quantum phase has encountered from the beginning of quantum theory. It is shown that the Stokes operators make it possible to draw parallels with results from precision spectroscopy. The problem of the ultimate limit in the detection of small phase shifts is also addressed. The concluding article, by C. Etrich, E Lederer, B.A. Malomed, T. Peschel and U. Peschel, presents a review of the theory of solitons in quadratically nonlinear media. Much of this research was stimulated by recent advances in fabricating periodically poled materials, that allow for quasi-phase matching. The formation, stability and interaction of various types of localized solutions, i.e. spatial, temporal, spatio-temporal, discrete and Bragg solitons, are treated and basic methods for dealing with the underlying non-integrable system are discussed. Readers may note a change in the style of numbering of these volumes. The first forty used Roman numerals. For the sake of simplicity Arabic numerals will be used from now on. Emil Wolf
Department of Physics and Astronomy University of Rochester Rochester, New York 14627, USA August 2000
E. WOLF, PROGRESS IN OPTICS 41 9 2000 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED
NONLINEAR OPTICS IN MICROSPHERES
BY
MITCHELL H. FIELDS, JORGEN POPP AND RICHARD K. CHANG
Department of Applied Physics and Center for Laser Diagnostics, Yale University, PO. Box 208284, New Haven, CT 06520-8284, USA
CONTENTS
PAGE w 1.
INTRODUCTION
. . . . . . . . . . . . . . . . . . .
w 2.
CAVITY MODES OF MICROSPHERES
w 3.
PERTURBATION EFFECTS ON MICROSPHERE RESONANCES
w 4.
CAVITY-MODIFIED OPTICAL PROCESSES IN
. . . . . . . . . .
MICROSPHERES
. . . . . . . . . . . . . . . . . . .
w 5.
FLUORESCENCE
A N D L A S I N G IN M I C R O S P H E R E S
w 6.
NONLINEAR
w 7.
CONCLUSION
REFERENCES
4 20
40 . .
OPTICAL PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ACKNOWLEDGEMENTS
3
.
53 69 89
. . . . . . . . . . . . . . . . . .
89
. . . . . . . . . . . . . . . . . . . . . . .
89
w 1. Introduction
Dielectric microparticles, particularly in the form of spheres and cylinders with the radius larger than the wavelength, have attracted a diverse group of scientists and engineers. They are theorists and computationalists and also experimentalists in fields that include quantum optics, nonlinear optics, linear optics, electromagnetics, combustion diagnostics, fuel dynamics, colloid chemistry, atmospheric science, telecommunications, industrial hygiene, and pulmonary medicine. Most of the earlier research on dielectric microparticles concentrated on elastic scattering and internal absorption by spheres and infinite cylinders by using the well-developed Lorenz-Mie formalism. With the advent of modern computers and development of computationally intensive approaches (such as the T-matrix and the generalized Lorenz-Mie techniques), the determination of the scattered and internal field distributions was extended to spheroids, spheres with inclusions and finite length cylinders when these microcavities with high symmetry are illuminated by plane waves or tightly focused beams at or outside the rim. In the late 1970s, Ashkin and Dziedzic [ 1977] reported on the observation of optical resonances in the radiation levitation forces exerted on evaporating liquid droplets. Soon afterwards, resonance peaks were observed in the fluorescence spectra of fluorescent polystyrene latex spheres (pls) and fluorescence and Raman scattering from silica fibers. These optical resonances are referred to now as morphology-dependent resonances (MDRs) and whispering-gallery modes (WGMs). After realization that the dielectric sphere (cylinder) was acting as a 3-d (2-d) microcavity with Q values around 108-109, lasing and a series of nonlinear optical experiments rapidly ensued on pls as well as liquid droplets, columns, and within capillary tubes. Some precaution is required in the adaptation of standard laser and nonlinear optics formalisms to such interactions in microcavities. For example, in these microcavities the concept of the phasematching condition for plane waves needed to be recast into spatial overlap of various MDRs or WGMs, which consist of countercirculating waves within the spherical (circular) dielectric surface. Nevertheless, the standard Lorenz-Mie theory can readily calculate the wavelengths and Q of the resonances, even for
4
NONLINEAR OPTICS IN MICROSPHERES
[1, w 2
the large size parameters (ratios of the circumference to the wavelength) of the microparticles used in the experiments. Some current research is directed toward nonspherical (noncircular) dielectric microparticles. When the shape distortion amplitude is small, the powerful 1st and 2nd order time-independent perturbation theory can be used to predict the frequency splitting of the degenerate azimuthal modes and their precession frequency. When the shape distortion amplitude is large, however, the perturbation theory fails and the T-matrix method is too computer intensive even for the modern supercomputers. The recently introduced ray-dynamics approach to these resonances, where the rays become chaotic in a manner determined by the Kolmogorov-Arnold-Moser (KAM) theory of classical Hamiltonian dynamics, provides clear physical insights and can predict the Q of the deformed cavity, the directionality of the refractively leaked radiation, and the location on the deformed interface where the leakage predominantly occurs. Recent experiments with deformed quantum-cascade and liquid-droplet lasers have motivated and benefited from this ray-dynamics approach. This chapter will briefly review the essential characteristics of all the aforementioned topics. Our work on lasing and nonlinear optical effects in microdroplets is emphasized. Because of space limitations, we have not emphasized the elegant developments of a new type of optical resonator (with Q > 101~ for cavity quantum electrodynamics (CQED) experiments and thresholdless lasers. We also have not emphasized the many current applications of microcavities in combustion diagnostics of burning fuel droplets, in telecommunication of add/drop filters for WDM systems, in chemistry of reactions without containers, and in biological airborne particle detection. We apologize to many authors and groups for leaving out some of articles because of page restrictions.
w 2. Cavity Modes of Microspheres In this section we review several treatments that explore the physical and mathematical properties of the resonance modes and interaction of light with dielectric microspheres. These modes of microspheres are commonly referred to as morphology-dependent resonances (MDRs) (Hill and Benner [1988]), whispering-gallery modes (WGMs) (Garret, Kaiser and Long [ 1961 ]) and quasinormal modes (QNMs) (Ching, Leung and Young [1996]). Many of the novel optical properties of microspheres are associated with the electromagnetic modes of the cavity. Significant confinement of the electric field occurs at specific
1, w 2]
CAVITY MODES OF MICROSPHERES
5
resonance frequencies that satisfy the appropriate boundary conditions. The modes of microspheres are confined in three dimensions, whereas the modes of Fabry-Perot cavities are confined in one dimension. 2.1. RAY AND WAVE OPTICS
The most intuitive picture describing the optical resonances of microspheres is based upon ray and wave optics. A ray of light propagating within a sphere of radius a and index of refraction re(to) will undergo total internal reflection if the angle of incidence with the dielectric interface, Oinc, is 0inc ) 0 c -- arcsin(1/m(to)). The rays of a mode have the property that all subsequent bounces have the same angle of incidence. Hence, the light is confined to a band within the great circle of the sphere. A 'caustic region' can be defined as an inner-sphere region within the dielectric sphere to which the propagating bouncing rays are tangent. The radius of the caustic sphere is approximately the radial distance to a cord defined by a ray with 0inc ' ~ 0c. A small fraction of the light on an MDR is contained in the caustic sphere. (Note that for the case of a perfect sphere, geometric optics does not provide a method for the light to escape as long as 0inc ~> 0c. This problem is resolved by wave theory; diffraction due to the curvature of the sphere surface causes light to leak tangentially from the sphere rim.) For the case of a sphere with circumference 2:ra >> ~, and light propagating with 0inc ~ 90 ~ the resonance condition is that the optical path length is approximately equal to the circumference of the sphere. The permitted limits of n wavelengths in the dielectric is the path length for one roundtrip with wavelengths for waves that are either confined mostly within the dielectric (~,/m(to)) or extended mostly into the surrounding air (Z): 2:ra
2:va <~ n <~ ~,/m(to)"
(2.1)
Using the dimensionless size parameter x = 2r
(2.2)
the resonance condition occurs for integer n in the range x <~ n <~ m ( t o ) x .
(2.3)
The phase-matching condition for the n = re(to)x mode and the external wave corresponds to the case where an internal ray completes one roundtrip as the n~, of the plane wave reaches the sphere surface.
6
NONLINEAROPTICSIN MICROSPHERES
[1, w2
The integer n can be identified as the angular momentum of the mode by equating hn ,~ ap ,-~ a h k
(2.4)
for the case of near-glancing incidence of the ray (0inc ~ 0c). In many papers the symbol g is used instead of n. The great-circle orbit of the rays need not be confined to the x - y plane (e.g., the equatorial plane). If the normal to the orbit is inclined at an angle 0 with respect to the z-axis, the z-component of the angular momentum of the mode is m = n cos 0.
(2.5)
For a perfect sphere, all of the m modes are degenerate (with 2n + 1 degeneracy). The degeneracy is partially lifted when the cavity is axisymmetrically (along the z-axis) deformed from sphericity. For such distortions the integer values for m are +n, + ( n - 1),... 0, where the • degeneracy remains because the resonance modes are independent of the circulation direction, i.e., clockwise or counterclockwise. Highly accurate measurements of the clockwise and counterclockwise circulating m-mode frequencies reveal a splitting due to internal backscattering that couples the two counterpropagating modes (Lef~vreSeguin, Knight, Sandoghdar, Weiss, Hare, Raimond and Haroche [ 1996]). The general features of plane-wave illumination of a microsphere can be described by ray optics. The curvature of the illuminated face focuses the incident light to a cigar-shaped 'hot spot', 'hot region', or 'hot line' (with length ~ 0. la and width ~ 2/2) within the shadow face of the microsphere. Subsequent reflections lead to a weaker hot line near the illuminated face. The intensity of light in the main hot line can be greater than 100 times the intensity of the incident field, for lossless spheres with x >> 10. Thus, the hot lines can serve as pumping regions for nonlinear optical processes and lasing. Rays of the plane wave that tangentially strike the sphere surface can excite resonance modes only if 2 satisfies the resonance condition of eq. (2.3). The ray and wave optics description of light interaction with a microsphere has several limitations. As previously mentioned, no means of escape for light on an MDR exists (for perfect spheres), and hence the characteristic leakage rates cannot be calculated. By reciprocity, ray and wave optics provides no possibility for incident light to couple into an MDR. Furthermore, the polarization of light is not taken into account. Finally, the radial character of the optical modes cannot be determined by ray and wave optics. All of these issues are resolved by properly
1, w 2]
CAVITY MODES OF MICROSPHERES
7
treating the modes with the Maxwell wave equations with appropriate boundary conditions. 2.2. ELECTROMAGNETIC THEORY
A complete description of the interaction of light with a dielectric is given by electromagnetic theory (Stratton [1941], Haus [1984], Jackson [1975], Hill and Benner [1988]). The spherical geometry suggests expanding the fields in terms of vector spherical harmonics. Characteristic equations for the resonance modes are derived by requiring continuity of the tangential components of both the electric and magnetic fields at the boundary of the dielectric sphere and the surrounding medium. Internal-intensity distributions are determined by expanding the incident wave (plane-wave or focused beam), internal field, and external field, all in terms of vector spherical harmonics and again imposing appropriate boundary conditions.
2.2.1. Properties of MDRs In the case of spherical symmetry (i.e., only radial variations are in the dielectric constant), the modal expansion of the electric fields can be expressed in terms of TE modes (no radial component of the electric field) described by ETE = z~(m(w)kr)Xnm(O, r
(2.6)
and TM modes (no radial component of the magnetic field) described by ETM = 27 • {zn(m(to)kr)Xnm(O,
q~)}.
(2.7)
The wave vector in free space is k = m/c. The zn(m(w)kr) characterizes the radial dependence of the fields, zn(m(w)kr) is the spherical Bessel functionjn(m(co)kr) for incident fields (here re(w) is the refractive index of the surrounding medium) and internal fields (here m(o9) is the refractive index of the sphere), z,(m(m)kr) is the spherical Hankel function h(~2)(m(w)kr) for external fields (here re(w) is the refractive index of the surrounding medium). The angular distribution of the electric field is characterized by the vector spherical harmonics X,m(O, r defined by Xnm(O, r = where
-i
v/n(n + 1)
r X VYnm(O, r
Ynm(O,r are spherical harmonics.
(2.8)
8
NONLINEAR OPTICS IN MICROSPHERES
[1, w 2
The vector spherical harmonics often used in scattering theory Mlmn, Mlemn,
Nlomn, and Nlm,, are related to the Xnm(O, O) by ._ /
jn(kr)Xnm(O, ~) + Xn*m(O, r . / ( 2 n + 1)(rt - m)I 1 -Momn, 2 g4Jrn(n + 1)(n + m)!
(2.9)
jn(kr)Xnm(O'O)-Xn*m(O'O) 2 = i ~ 4Jrn(n (2n+ +l)(n ! l ) -( n+m)m)! Melmn
(2.10)
'
and the Nm,, = V x Mm,,/k. The index n is referred to as the mode angular momentum, or mode number. The index m is the z-component of the mode angular momentum or the azimuthal mode number (distinguished from the refractive index m(~o) by bold). The time variation of the fields is assumed to be e -k~ throughout the chapter. To obtain the characteristic equations, the tangential components of E and H are matched at the boundary. For a sphere of radius a and index of refraction m(og) = mr(og)+ imi(~o) in air (m(~o) can also represent the ratio of the refractive index of the sphere to that of the surrounding medium), the characteristic equation for TM resonances is
~p"(m(~o) x) _ m(~o) r (x)
~p,,(m(og)x)
g,,(x) - 0,
(2.11)
where x - ka = 2~a/A is the size parameter. For TE resonances the characteristic equation is
m(oo) ~p~(m(oa)x) ~p,(m(~o)x)
g~(x) _ O. ~.(x)
(2.12)
The functions ~p,(z) = zj,,(z) and gn(Z) = zh(n2)(z) are the Ricatti-Bessel and Ricatti-Hankel functions, respectively. Even when the dielectric is lossless (mi(~o) - 0), the solutions to eqs. (2.11) and (2.12) are the complex resonance size parameters xa. The real part of the size parameters designate the resonance frequencies for a fixed radius sphere. The imaginary part of the size parameter is related to the leakage loss (previously referred to as diffractive-related escape) of the mode out of the absorptionless dielectric cavity. Both eqs. (2.11) and (2.12) represent several independent equations, one for each mode number n. The resonances, or MDRs, are found by solving these transcendental equations numerically. In addition, for each n, there is a set of solutions labeled by the radial mode order g. For a fixed n and m(~o) the first
1, w 2]
CAVITY MODES OF MICROSPHERES
9
Table 1 Common notations Quantity
Our notation (EE)
Alternate notation (QM)
Refractive index
re(to)
n(to)
Angular momentum or mode number
n
g
z-component of angular momentum or azimuthal mode number
m
m v
Radial mode order Vector spherical harmonic
Xnm
X~m
solution is labeled g = 1 and the second is labeled g = 2. The physical picture is that the mode has g intensity maxima along r = a/m(o9) to r = a. For the case of a perfect sphere, the resonance size parameters are independent of m and are labeled xn,e. The resonance modes are also commonly denoted by their polarization as TMn,e and TE~,~. The notation for mode numbers in this chapter is the notation that is most common in the electromagnetic-wave community. However, another notation, which is similar to the notation used in quantum mechanics, is also used. Table 1 shows the correspondence between the two notations. The spectral and spatial characteristics of MDRs are now presented in the context of wave theory. The real part of xa is related to the resonance frequency by toa = cRe(xa)/a. The FWHM of the resonance, denoted Axa or Aoga, is twice the imaginary part of xa. The quality factor, Q, of a resonance is defined as Re(xa) _ toa Q = 2I--m-~-a) Atoa - ~oar,
(2.13)
where r is the lifetime of a wave on an MDR. In a perfectly smooth homogeneous lossless sphere the Q values are limited by diffractive leakage losses (also referred to as tunneling leakage losses from the centrifugal barrier, w and can be as high as 10 ~~176 In reality, volume inhomogeneities, surface roughness, and absorption restrict the maximum Q values to be less than 1010. Local or global shape deformations and nonlinear effects can further reduce the maximum Q value. For frequencies near an MDR, the electric field inside the cavity varies as
(
E(t) = Eo exp - i t o 0 t - ~-0t
.
(2.14)
10
NONLINEAR OPTICS IN MICROSPHERES
[1, w 2
The decay term leads to a broadening of the resonance linewidth, giving a Lorentzian lineshape for the energy distribution Ig(~o)l 2 ~
1
(2.15)
( t o - 0)0)2 + (~Oo/2Q)2"
A cavity mode is uniquely defined by its polarization (TE or TM), angular momentum or mode number n, azimuthal mode number m, and radial mode order g. A physical interpretation of n, m, and g was given in the sections on wave and geometric optics and on electromagnetic theory. Computer codes for calculating the resonance frequencies and linewidths of MDRs are well documented (Barber and Hill [1990]). For perfect spheres the m-modes are frequency degenerate (i.e., the characteristic equations do not depend on m). For a given mode number n the g = 1 modes have the highest Q (smallest Ax), with a peak intensity located closest to the surface and the evanescent wave penetrating shortest into the surrounding medium. As increases, the Q value decreases, the peak intensity moves away from the surface, and the evanescent wave penetration into the surrounding medium increases. For a fixed radial mode order g, modes with higher angular momentum or higher mode number n have higher Q values. For spheres with large x, several authors derived expressions to determine the spectral location, separation, and width of MDRs. Using asymptotic analysis (Schiller and Byer [1991], Lam, Leung and Young [1992]), the positions of MDRs are approximated by P
m(tO)Xng, = V + 2-1/30tgV 1/3 - - - + ( - - 3 0 2 - 2 / 3 )
P _ 2p2/3)
_ 2-1/3p(m2(to)
0~V- 1/3
(2.16) orgy-2~3 +
O(v -1)
p3 where P =
re(og) for TE modes, 1/m(o9) for TM modes,
(2.17)
v = n + 1/2, p2 = m2(to)_ 1 and at are the roots of the Airy function. The resonance positions approximated by eq. (2.16) have excellent agreement with exact calculations (Lam, Leung and Young [ 1992]) and experiment (Schiller and Byer [ 1991 ], Eversole, Lin, Huston, Campillo, Leung, Liu and Young [ 1993]). Asymptotic expressions for the linewidth have also been derived (Lam, Leung and Young [1992], Johnson [1993]).
1, w 2]
CAVITY MODES OF MICROSPHERES
11
Another approximation in the limit x ~ n for the resonance positions, which is slightly more accurate for g ~> 5 but less accurate for g < 5, is given by the solution to (Probert-Jones [1984])
tanr
PP
E(2u) 11 -~
,218,
- 4u
'
U2
a'g
2vp
4
where r = v ( p - tan -1 p ) - p u -~
(2.19)
and u = v - x. The advantage of eq. (2.16) is that it is an explicit formula and is accurate for the high-Q resonances (i.e., low-order modes that are relevant for most nonlinear optics experiments). However, eq. (2.18) is more accurate for the broader resonances with high-order g (Lam, Leung and Young [ 1992]). For many purposes (e.g., to determine the approximate sphere size and approximating mode numbers), the separation between resonances Ax~,e = X~+l,e- xn,e is more useful than the absolute mode positions. Asymptotic analysis gives (Lam, Leung and Young [ 1992]) m(o)Ax~,e
2-1/3 2-2/3 = 1 + - - - ~ o t e v -2/3 - ~ O l ~
10
+
V-4/3
[233p(m2(oo)-2p2/3)2-1/3 194/3 9
otgv -5/3
+ O(v-2).
(2.20) Although eq. (2.20) is more accurate, a simple approximation to &~c is given by (Ch~,lek [1990])
Ax =
x tan -1 [ ( m ( m ) x / n ) 2 - 1 ]1/2 n [ ( m ( m ) x / n ) 2 - 111/2 for I x - nl >> 1/2, tan -~ p p for x / n ~ 1.
(2.21)
The density of MDRs (defined as the number of resonance modes per frequency or size-parameter interval) was calculated (Hill and Benner [1986]). The mode density is an important parameter in the description of many nonlinearoptical processes and lasing within microcavities. For example, multiple-order, stimulated Raman scattering (SRS) is observed in microspheres, even for species with narrow Raman bands (Qian and Chang [ 1986]). For this nonlinear optical cascade process to occur, high-Q MDRs must exist within the Raman-gain profile
12
NONLINEAR OPTICS IN MICROSPHERES
[1, w 2
Fig. 1. Intemal-intensity distribution in the equatorial plane for a TE M D R with n = 9 and g = 2.
at the various Raman shifted frequencies to provide the feedback for SRS. Mode-density calculations indicate that the mode-density of high-Q modes is sufficiently dense so that there always exist some high-Q MDRs to support multiorder SRS, as high as twentieth order (Leach, Chang, Acker and Hill [1993]) in microspheres. More than 15 high-Q MDRs per 10cm -1 frequency interval are present in a 35 ~tm radius sphere at 2, = 633 nm. An approximation to the mode density of high-Q MDRs is (Hill and Benner [1986]) x p 2( p -
MDR density =
tan- 1p) yr
.
(2.22)
Equation (2.22) implies that the mode density increases rapidly as the refractive index increases. The spatial distribution of MDR internal intensities can be readily calculated, and give further physical insight into the properties of MDRs. Figure 1 shows the internal intensity distribution in the equatorial plane for a TE9,2 mode of a sphere with index of refraction ratio re(co) = 2.5. Clearly, n - 9 • 2 intensity peaks are around the equator (corresponding to n = 9 wavelengths within the circumference and to an effective angular momentum of 9). There are g = 2 intensity peaks along the sphere radius. Figure 2 shows the angleaveraged radial intensity distribution for TE30,1, TE30,2, and TE30,3 modes, further demonstrating that the mode order g corresponds to the number of intensity peaks along the sphere radius. The dependence of the internal intensity distribution on the azimuthal mode number m is shown in fig. 3, in which the angular internal intensity distribution
1, w 2]
CAVITYMODESOF MICROSPHERES
13
g= 3
g= 2
•
g= 1 .
.
.
.
!
0.5
9
9
9
I
1.5
1.0
r/a Fig. 2. Angle-averaged intensity as a function of r/a for TE MDRs with n = 30 and g = 1, 2, and 3. n=50
.~ "~
1.0 = 1
m = 25
m = 50
m =1
o.8
/
~ - 2 s
/
0.6
.~ 11.4 0.0
i "~ 0 10
i 20
i 30
40
50
60
70
80
90
0
Fig. 3. Internal-intensity distribution as a function of 0 for TE MDRs with n = 50 and m = 1, 25, and 50. The m = 1 mode is mostly confined to the polar region, and the m - 50 mode is confined near the equatorial region. is a function o f 0. T h r e e M D R s for m = 0, 25 and 50 w i t h n = 50 are illustrated as the a n g l e 0 varies f r o m 0 ~ to 90 ~ T h e s e m o d e s have the s a m e r e s o n a n c e frequency, but the m a x i m u m
intensity for e a c h m is inclined at an angle
0 = sin -1 (m/n). T h e m a x i m u m intensity p e a k a g r e e s w i t h the ray optics picture o f an m - m o d e circulating in a c o n f i n e d orbit inclined at 0 = sin -1(m/n) and w i t h its n o r m a l inclined at an angle 0 = cos -1 (m/n). T h e m - m o d e c o r r e s p o n d s to the z - c o m p o n e n t o f the m o d e a n g u l a r m o m e n t u m n.
2.2.2. Excitation of MDRs M a n y applications o f m i c r o s p h e r e s require c o u p l i n g o f light into M D R s f r o m external light sources. Several m e t h o d s can be u s e d for exciting M D R s . T h e m o s t
14
NONLINEAROPTICSIN MICROSPHERES
[1, w 2
Fig. 4. Surface plots of the internal intensity for (a) resonant and (b) nonresonant incident planewave illumination of a sphere with index of refraction m(~o) = 2.5. The resonant size parameter in (a) is x0 = 7.53207, which corresponds to a TE MDR with n = 11 and g -- 2. In (b) the size parameter is x -- 7.5. Note that the figures do not have the same scale.
simple case is plane-wave illumination. The internal-intensity distributions in the equatorial plane o f a sphere illuminated by resonant and nonresonant incident wavelengths with TE polarization are shown in figs. 4a and 4b, respectively. In both cases two internal hot lines (predicted by ray optics) can serve as pumping regions for nonlinear optical processes and lasing. W h e n the input wavelength corresponds to an M D R (referred to as an input resonance), fig. 4a shows the intensity distribution in the equatorial plane. The intensity of the M D R is mainly confined to around the rim of the sphere. The spatial overlap between the two hot lines and the M D R s is small. Even if the intensity of the excited M D R is less
1, w 2]
CAVITY MODES OF MICROSPHERES
rE~8
TE276
redo TE~26
r~2
lO
15
TE94
lO
TEl4 i
QleakXlO"3 n~ 7 8 9
i
~
22 24
0.2 1.1 0.5
26 8.4 2.3 28 27 5.6 30 96
~
\ 9
i 9
. I
22.0
22.5
23.0
23.5
I
24.0
-
24.5
Mie size parameter Fig. 5. Computed elastic scattering spectrum for a sphere with index of refraction re(o)) = 2.5 in the size-parameter range 22.0 < x < 24.5 and at a -~ ~ / 2 scattering angle. The incident plane wave is perpendicularly polarized to the scattering plane. Odd mode numbers n are not observable at ~/2. The resonances are labeled as TE~n, and the corresponding Q values are shown in the table.
than that of the two localized hot lines, the MDR at the pump wavelength can have a large spatial overlap with the MDR at the shifted wavelength and thereby at input resonance the incident beam more efficiently pumps nonlinear optical processes and lasing. At an input resonance the effective pumping path length approaches 2Jva, and is much greater than that for nonresonant pumping. The generated fields (e.g., those that are of the SRS or lasing) depend on feedback provided by another MDR (referred to as an output resonance), which can have almost perfect spatial overlap with the input resonance. In w an effective-average gain model is described that quantifies how spatial overlap between the pumping field (on or off input resonance) and the generated field (on resonance at a different wavelength) relates to the effective gain for the optical process. Figure 5 displays an elastic scattering spectrum for plane-wave illumination of a sphere with index of refraction re(co)- 2.5 in the size range 22.0 < x < 24.5. The incident light is perpendicularly polarized to the scattering plane, and the scattering angle is 0 ~ zc/2. MDRs with Q values greater than 105 are generally not observable in elastic scattering spectra because, when convolved with the spectrometer spectral function, the integrated intensity of these high-Q peaks is too low (because the peak linewidth decreases as Aw whereas the peak scattering
16
NONLINEAROPTICSINMICROSPHERES
[1, w2
amplitude cannot exceed unity). Similarly, MDRs with Q values less than 102 a r e difficult to observe because they are too broad and blend into the background. To improve the input coupling of the incident pump and the MDR, a focused Gaussian beam is aimed at the equatorial edge of a sphere and cylinder. The interaction of focused Gaussian beams with dielectric spheres was studied both theoretically (Barton, Alexander and Schaub [ 1988, 1989], Gouesbet, Maheu and Grrhan [1988], Khaled, Hill, Barber and Chowdhury [1992], Khaled, Hill and Barber [1993], Gouesbet and Lock [1994], Lock and Gouesbet [1994], Lock [1995]) and experimentally (Lin, Eversole, Campillo and Barton [1998]). By focusing inside the sphere and away from the edge, even if the wavelength of the incident light corresponds to an MDR, the internal intensity distribution approximately preserves the Gaussian profile except for the refraction and reflection at the sphere surface. By focusing outside the sphere, an optimum distance exists between the beam center and the sphere edge for maximum coupling of the input-focused Gaussian beam with a particular MDR with lowradial mode order g or with high-Q value. For the modes with highest Q, however, the coupling is most efficient for a beam located within the sphere radius (Lin, Eversole, Campillo and Barton [1998]). A clever way to excite an MDR is to use evanescent coupling with a prism or partially unclad (stripped) optical fiber (Serpengiizel, Arnold and Griffel [1995], Dubreuil, Knight, Leventhal, Sandoghdar, Hare and Lef~vre-Seguin [1995], Knight, Cheung, Jacques and Birks [ 1997]). It is easier to match the wavevector of the evanescent field of the MDR with that of the prism or stripped fiber. Near equatorial modes (m ~ n) are excited by placing the microsphere equator in the evanescent field of the fiber. The coupling efficiency is typically less than 20%. The fiber coupling method has many potential applications in electro-optics and optical communications. The time dependence of internal fields has been calculated for both pulsed plane-wave and pulsed Gaussian-beam illumination for pulses with a Gaussian time dependence (Chowdhury, Hill and Barber [ 1992], Khaled, Chowdhury, Hill and Barber [1994]). The time dependence of the internal field is a function of both, the frequency of the incident field and the spatial location within the sphere. If the incident pulse has significant spectral overlap with an MDR, near the sphere edge the time dependence of the internal intensity shows a transient buildup and exponential decay with time constant 1/A~oa, where Acoa is the width of the nearby MDR with frequency ~oa. The time dependence of the internal intensity at spatial locations away from the sphere edge in the resonant case and for all spatial locations in the nonresonant case essentially follows the time profile of the incident pulse. To couple into an MDR efficiently, the linewidth of
1, w2]
CAVITYMODESOF MICROSPHERES
17
the laser should be equal to or narrower than the MDR width. In other words, the input-laser pulse duration must be longer than the cavity lifetime r = Q/oo. 2.3. QUANTUMMECHANICAL ANALOG Substituting E(r) = ~)n(r)Xnm(O, ~) into the Helmholtz-wave equation, a scalar equation describing TE modes is derived to be (Johnson [ 1993]) d2dr------5-q~(r) + Ik 2 m 2 (o), r) -
n(n+ r2 1)1 q~(r) = 0,
(2.23)
where
9 (r) = rG(r ).
(2.24)
For the case of a homogeneous re(co, r) = m(oJ) (constant refractive index from r = 0 to r = a), q~(r) = ~p~(m(co)kr).
(2.25)
Equation (2.23) is analogous to the radial Schr6dinger equation (in the units h2/212 = 1) d21p(r) ~- V(r) + n(nr2+ dr 2 where V(r) substitutions re(to, r) = 1 Schr6dinger
1)1~p(r) = E~p(r),
(2.26)
is the potential energy and E is the total energy. By using the V(r) = k2[1 - m2(~o, r)] (where m(~o, r) = m(~o) for r ~< a and for r > a) and E = k 2, eq. (2.23) can be recast in the form of the equation. The total potential energy is
Vn(r) = { n(n k2(l+-lm2(~)))/Y2
-}- n(n
+ 1)/r 2 rr >~
(2.27)
The part n(n + 1)/r 2 of the potential is often referred to as the centrifugal or angular-momentum barrier. The main difference between the Schr6dinger and Helmholtz equations is that the potential for the electromagnetic Helmholtz equation depends on k 2. The potential Vn(r) in the quantum-mechanical analog is plotted for the case n = 40, m - 1.47, and k = 33 in fig. 6.
18
NONLINEAR OPTICS IN MICROSPHERES
(a)
Quantum Mechanical Analog
\-
!i'-.. ....... ": i ""-. i "-Oo.~
=
(b)
[1, w 2
Ray Picture
3 ...""'" "'"...
=1
""..... ................................
!
~...
i
..........................
...:-
i
. ~ . . \
u
o
""......
...............
aim(co) "~
o.s
i
f-1
r/a
i.5
Fig. 6. (a) Effective potential (dotted curve) associated with a dielectric sphere of constant refractive index m(o9) = 1.47 and the radial wavefunctions (solid curve) for TE M D R s with n = 40 and g = 1, g = 2 and g = 3. (b) Ray picture of the angle of incidence 0inc > 0c for the g = 1 and g = 3 radial modes. For MDRs, all reflections have the same 0inc and form a caustic sphere within.
Classically, the quantity pZ(r)
= E - V~(r) =
k2m2(o))
- n(n + 1)/r 2
(2.28)
must be larger than zero, i.e., p~(r) >~ O. The classical turning points rl and r2 are determined by solving p](r) = 0 (escape through the potential barrier is forbidden classically). This condition, along with the substitution x = ka, imposes a restriction on the allowed values of the size parameter to n/m(~o) < x < n.
(2.29)
Physically, if x > n, the energy of the wave is above the top of the well and no confinement occurs. If x < n/m(~o), the energy of the wave is below the bottom of the potential well. Equation (2.29) is equivalent to eq. (2.3) and to
x < n < m(w)x.
(2.30)
The inequality (2.30) can be derived from electromagnetic theory by considering the behavior of the fields in the two cases n < x and m(oo)x < n. For n < x
1, w 2]
CAVITY MODES OF MICROSPHERES
19
the external field is sinusoidal and not exponentially decreasing. This behavior is consistent with a traveling wave rather than an evanescent field. Physically, the light rays are no longer totally internally reflected. The case for which n > m ( t o ) x is nonphysical because p2 has negative values. It was shown that the discrete levels in the well Vn(r) correspond to MDRs with a fixed n (the angular momentum n determines the centrifugal barrier), but with different radial mode order g (Johnson [ 1993]). Quantum tunneling out of the well through the centrifugal barrier is related to the leakage rate or lifetime of the MDR. Therefore, resonances near the bottom of a well have longer lifetimes because they see a larger and wider centrifugal barrier. Calculations indicate that the low radial mode order (low-g) modes lie near the bottom of the well, and thus are predicted to have longer lifetimes than higher-g modes (agreeing with the electromagnetic wave calculations). The tunneling leakage is equivalent to the diffraction leakage of a total internal reflected wave with the angle of incidence 0inc larger than the critical angle 0c incident on a curved surface. At the well bottom the rays have the largest 0inc, whereas at the top of the well the rays have 0inc ~'~ 0c. 2.4. QUASI-NORMAL M O D E S - LEAKY CAVITIES
Dielectric microspheres with no absorption loss are nonconservative open systems. Each resonance has a finite lifetime associated with tunneling or diffractive leakage of energy from the cavity to the external. Other open systems include linear laser cavities and black holes. The modes of these leaky systems are referred to as quasi-normal modes (QNMs), which have complex eigenfrequencies and are outgoing waves far from the cavity. The mathematics of QNMs can be complicated, however, for microspheres; the spherical symmetry permits calculation of optical properties of the leaky cavity (Lai, Leung, Young, Barber and Hill [ 1990], Lai, Lam, Leung and Young [ 1991], Leung and Young [ 199 lb], Leung, Liu and Young [ 1994a,b], Leung, Liu, Tong and Young [ 1994], Ching, Leung and Young [1996]). If a leaky microsphere and its external bath are considered to be an entire system (the universe), the whole system is conservative. The modes of such a conservative system (called normal modes of a Hermitian system) are eigenfunctions of a Hermitian operator and form a complete orthonormal basis. Completeness and orthogonality of the normal modes imply that any function of the same variables can be expanded and the dynamics of the system can be expressed in terms of the normal modes. The Hermitian feature of the sphere-bath universe has been used to calculate transition rates of
20
NONLINEAROPTICSIN MICROSPHERES
[1, w3
molecules in microspheres (Ching, Lai and Young [1987a,b]) by performing the necessary expansions in terms of the normal modes of the universe. Among the disadvantages of using the Hermitian universe approach is that the modes of the universe are continuous and the optical properties of the cavity are not obviously independent of assumptions about the universe. The QNMs of leaky cavities (a non-Hermitian system) do not form a complete orthonormal basis. However, the QNMs provide an intuitively appealing set of discrete functions with which to describe the optical properties of the cavity. Fortunately, under fairly general conditions and with a suitable redefinition of the inner product, the QNMs do form a complete and orthonormal basis (Leung, Liu and Young [ 1994a]). Then, as long as the general conditions are satisfied, the usual formalisms of Hermitian systems can be applied to leaky optical cavities, provided that the new norm and inner product are used. The redefinition of a norm and inner product is necessary because the wavefunctions extend outside the cavity (to infinity) and are growing exponentials at infinity (the outgoing wavefunctions are proportional to exp (io)r)/r with Im(o)) < 0). The QNM analysis has tremendous utility in determining the effects of perturbations on both, the resonance frequencies and linewidths, as will be discussed in w The analysis can become complicated because of the redefined norm and inner product. For further details on QNM analysis as it applies to microspheres, as well as other systems, the reader is referred to the papers from the Chinese University of Hong Kong group. In w we present an alternative approach to calculating the effects of perturbations on microsphere resonances (called the effective-average model) that simplifies calculations by cutting off all integrals beyond the first zero of the Hankel function (Chowdhury, Hill and Mazumder [ 1993]), thus defining an effective volume of the sphere Vm. w 3. Perturbation Effects on Microsphere Resonances
The microspheres encountered in most experimental situations are neither perfectly spherical nor homogeneous. Perturbations due to shape deformations from sphericity, index of refraction gradients, and small inclusions all affect the resonance frequencies and linewidths of the cavity modes. For example, liquid droplets falling through air generally have quadrupolar shape deformations resulting from the drag force exerted by air currents (Taylor and Acrivos [ 1964]). Index of refraction perturbations in microspheres are caused by temperature gradients, composition gradients, and small inclusions (such as polystyrene latex spheres or dust particles), either deliberately or unavoidably placed in the microsphere.
1, w 3]
PERTURBATION EFFECTS ON MICROSPHERE RESONANCES
21
Several methods are used to calculate the effects of shape and index of refraction perturbations on MDRs. Separation of variables techniques can be used in certain circumstances, which preserve spherical symmetry (Hightower and Richardson [1988], Chowdhury, Hill and Barber [1991]). For more general perturbations, three methods are noteworthy: (1) T-matrix calculations (Barber and Yeh [1975]); (2) the time-independent perturbation theory (TIPM) using QNMs for slightly deformed spheres (Lai, Leung, Young, Barber and Hill [1990], Lai, Lam, Leung and Young [1991]); and (3) the effective-average model (Chowdhury, Hill and Mazumder [1993]). Several experiments have confirmed the predictions of these models and illustrated their usefulness for characterization of shape deformation. Recently, a chaos theory approach to describe light leakage from highly distorted cavities (asymmetrical resonant cavities, or ARCs) appeared in print (N6ckel and Stone [1996, 1997]). 3.1. RADIALLY I N H O M O G E N E O U S SPHERES - SEPARATION OF VARIABLES
Two general classes of radial variation have been studied. One case is that of a layered sphere with an abrupt index of refraction boundaries at the interface between each layer. The other case has a smoothly varying radial index of refraction. For radially inhomogeneous spheres the separation-of-variables method of solving the wave equation in spherical coordinates results in solutions with the same angular part as for homogeneous spheres (w but with different radial functions (Kerker [1969]). For a layered sphere the radial solutions in each layer and the far scattered field are linear combinations of the Ricatti-Bessel functions ~Pn,Zn and ~ (note that ~n = ~P, + iz~). In each region the field must be finite. Therefore, only /Pn may be used to describe the incident field and the field in the core. The scattered field is described by ~, because it drops off properly at r = oo. In the layers both ~pn and Z, are well behaved and the fields are described by linear combinations of them. Equating the fields at each boundary leads to a set of linear equations that are solved for the scattering coefficients (Kerker [1969]). Hightower and Richardson [1988] performed a systematic numerical study of the resonant response of two-layered spheres to incident light. The effects on resonance location, energy density, and scattered light intensity of the TE39 mode were determined as a function of core and layer radius and index of refraction. For example, the index of refraction of the core has little influence on the resonance size parameter of the TE39 mode when the core radius is sufficiently smaller than the particle radius (rcore/rparticle < 0 . 8 5 ) because the energy distribution of the mode resides primarily in the layer. When
22
NONLINEAR OPTICS IN MICROSPHERES
[1, w 3
rcore/rparticle > 0.85, the resonance size parameter decreases as mcore(tO) increases relative to inlayer(tO) because the wavelength of the field is effectively longer. The numerical results were tested in an experiment measuring the elastic scattering intensity of an HeNe laser from glass spheres coated with glycerol as the glycerol layer evaporated (Hightower, Richardson, Lin, Eversole and Campillo [ 1988]). Excellent agreement between theory and experiment was observed. The applicability of the layered sphere with an abrupt index of refraction boundary problem is practically limited to cases of spheres with solid cores and either a solid or liquid shell. A more realistic situation considers a smooth radial variation in the refractive index, for example, liquid droplets with a radial temperature gradient, which induces a radial re(to, r) gradient because of the temperature dependence of the refractive index, or multicomponent microspheres with radial composition gradients. Assuming that a function describing the radial dependence of the index of refraction m(w,r) can be formulated, the separation of variables technique may be used to derive differential equations for the radial functions (Kerker [1969]). Chowdhury, Hill and Barber [ 1991 ] investigated the change in MDR location and Q value for two types of functions m(w,r): with a smooth roll-off at the surface (simulating droplets under high pressure and temperature that approach critical conditions), and with a smooth increase near the surface (simulating droplets with a positive intensity-dependent index of refraction and high-internal fields due to incident light being on an input resonance). For the case of a smooth roll-off of the index of refraction, the MDRs are blue shifted in wavelength and the Q values decrease. For a spherically symmetrical increase in re(w, r) near the sphere surface, the MDRs are red shifted and the Q values increase. When the maximum increase in m(w,r) is 0.3%, the size parameters decrease by 0.13% and the Q values increase by 11.0%.
3.2. T-MATRIX METHOD
For an arbitrarily shaped object the T-matrix method can be used to calculate the internal and scattered fields, as well as the resonance wavelengths and Q values of the resonances (Barber and Yeh [1975], Barber and Hill [ 1990]). However, the T-matrix method is most useful in treating axisymmetrical or layered particles (Wang and Barber [ 1979]). The method requires that the dielectric be piecewise homogeneous, and therefore cannot be used to determine the resonances of a particle with a continuously varying index of refraction.
1, w 3]
PERTURBATION EFFECTS ON MICROSPHERE RESONANCES
23
The T-matrix method relates the scattered field to the incident field by a transformation matrix, which is dependent on the morphology of the scattering object. After expressing the incident field, internal field, scattered field and surface currents in terms of vector spherical harmonics, the equivalence theorem is applied to express the scattered field in terms of a set of surface currents on the object that exactly cancel the internal field. An A-matrix is computed that relates the coefficients of the internal field to those of the incident field via the surface currents. A B-matrix is computed that relates the coefficients of the scattered field to those of the internal field. The T-matrix [r] = [B] [A]-1
(3.1)
then relates the scattering coefficients to the incident-field coefficients. The matrix elements of the A- and B-matrices are integrals of products of vector spherical harmonics over the surface of the dielectric. For arbitrarily shaped objects the T-matrix is a (2M x 2N) x (2M x 2N) matrix, where N and M are the maximum values of the mode number n and azimuthal mode number m required for convergence. The T-matrix for axisymmetrical objects can be separated into M separate (2M x 2N) T-matrices. The M T-matrices are all equal and diagonal for the case of a homogeneous sphere. The resonance size parameters and Q values are determined from the diagonal elements. For an object with a large size parameter, the T-matrix approach is still too computer intensive even for present-day microcomputers. For a spherical object with small perturbations, the T-matrices remain predominantly diagonal and the resonance characteristics of a particular (n, m) mode can still be determined from the diagonal elements. This assumption is valid as long as the maximum contribution of an off-diagonal element to the scattering coefficient is less than 5% of the contribution of the diagonal element. The off-diagonal elements couple the different n-modes of the dielectric cavity. The T-matrix method has calculated the scattering from layered spheres and the resonances of nonspherical particles. T-matrix calculations on spheroidal particles showed that the frequency degeneracy among the m-modes of a particular n-mode of a sphere split into n + 1 multiple resonances (Barber and Hill [ 1987]). This prediction was confirmed by experiment and used to measure the surface tension of oscillating spheroids (Tzeng, Long, Chang and Barber [1985]). 3.3. TIME-INDEPENDENT PERTURBATION METHOD (TIPM)
A time-independent perturbation theory (up to second order) based on the quasi-
24
NONLINEAROPTICSINMICROSPHERES
[1, w3
normal mode (QNM) analysis of open systems (Ching, Leung and Young [ 1996]) was developed to calculate the effects of perturbations of dielectric microspheres on the resonant frequencies (Lai, Leung, Young, Barber and Hill [1990]) and resonant widths (Lai, Lam, Leung and Young [ 1991 ]). For small distortions the results from the time-independent perturbation method (TIPM) and T-matrix approach agree remarkably well (see w The application of perturbation theory to open systems requires careful consideration of the imaginary part of the eigenfrequencies and the completeness of the QNMs (Leung, Liu and Young [1994a,b]). Fortunately, the usual perturbation methods of Hermitian systems apply to microspheres, but a new norm must be defined. The unperturbed system is a homogeneous sphere with permittivity er = m 2 and radius a. The perturbed refractive index is
re(r) =mo + Am(r),
(3.2)
where Am(r) is the perturbation to the real or imaginary part of the refractive index. Ignoring terms of order Am2(r), the perturbation in the relative permittivity is el (r) = m 2(r) - m 2 = 2moAm(r).
(3.3)
For the case of axisymmetrical perturbations, the perturbed complex resonance size parameter x no can be expressed in terms of the unperturbed resonance size parameter X n00, using first-order perturbation theory as
o 00(l
Xn -- Xn
~
34,
'
where, as before, n is the mode number of the MDR. V is the overlap of the permittivity perturbation with the energy density in the mode,
v =/i
dye,(,)
[En+mEn,m]
,11i
(3.5)
= ~ hn(l)(.~n~.00~ , /
(j.(,,,oxOO)
2 fVs el(r) Ijn(moknr)X,,ml 2 d g
where G is the normalization integral
G = R~lim
d Veo(r)
[e.,me.,m] +
-
dS o(R)
[en,me ,m] (3.6)
a3
=
2 000
,
k, = 2:r/~,n, &n is the resonance wavelength, and Vs is the volume of the sphere. The second line in eqs. (3.5) and (3.6) is specifically for TE modes.
1, w 3]
PERTURBATION EFFECTS ON MICROSPHERE RESONANCES
25
In calculating the conjugate v e c t o r E+m, the complex conjugate of the field En, m is taken without complex conjugating the radial function. From second-order perturbation theory the expression for the linewidth Ax, is derived to be (Lai, Lam, Leung and Young [1991 ]) Ax,
~_00,
1
x, ~Q0 + C1 + C2),
(3.7)
where Q0 is the Q value of the unperturbed sphere and C1 and C2 are the firstand second-order corrections. C1 is related to the spatial overlap of the n-mode intensity distribution with the perturbation el (r). C2 describes how strongly the perturbation couples two different n-modes (Lai, Lam, Leung and Young [ 1991 ], Mazumder, Hill and Barber [1992]). In all cases G / > 0,
(3.8)
independent of the type of perturbation. As long as terms of O(1/Qo) can be neglected (true for perturbations to high-Q modes), C1 << C2. Hence, the width of a narrow resonance is increased (Q is spoiled) by perturbations. Interpreted another way, the leakage rate from one MDR into other QNMs is enhanced by the perturbation. The approximation C1 << C2 is valid for small shape deformations, but is generally not valid for perturbations to the real (mr) or imaginary (mi) part of the refractive index. In particular, for perturbations to mi the second-order correction to the width is of order m 4 and is usually negligible compared with the first-order correction calculated from eq. (3.4). The applicability of TIPM for several classes of perturbations is explored further in {}3.5. When a sphere is shape distorted into a spheroid (a quadrupolar deformation that is axisymmetrical about the z-axis), the 2n + 1 degeneracy of each n-mode is lifted to n + 1 distinct modes. For a QNM with mode numbers (n, m), the fractional size-parameter shift due to the distortion calculated by first-order perturbation theory is (Lai, Leung, Young, Barber and Hill [ 1990]) x.00 - x . 0 x"o
e -
g
[
3m 2
1-n(~T1)
1
(3.9)
'
where the distortion amplitude e = (rp- r~)/a, rp is the polar radius, re is the equatorial radius, and a is the equivalent radius of a sphere that is equivolume with the spheroid. Equation (3.9) yields an m-dependence of the frequency splitting of
{ e[ 1 - n(n3m2 1} + l)
~o(m) = ~Oo 1--6
'
(3 0,
where 6o0 is the unperturbed (degenerate) resonance frequency and m = n cos 0 is the azimuthal mode number (0 = cos-l(m/n) is the azimuthal angle from
26
NONLINEAR OPTICS IN MICROSPHERES
[1, w 3
Fig. 7. CCD image of five lasing microdroplets as they appear at the entrance slit of a spectrograph (left panel) and the spatially preserved wavelength-dispersed data recorded by a CCD camera (right
panel) (Chen, Mazumder, Chemla, Serpengiizel, Chang and Hill [1993]). the z-axis). The functional form of eq. (3.10) was confirmed experimentally by measuring along the rim of distorted droplets the frequency shift of both SRS (Chen, Chang, Hill and Barber [ 1991 ]) and lasing (Chen, Mazumder, Chemla, Serpengiizel, Chang and Hill [1993]). The shape distortion amplitude of the droplet can be determined to one part in 103 from eq. (3.10). The sign of e was measured by noting the wavelength shift of the light emitted from the equator and the poles of the spheroid (Chen, Mazumder, Chemla, Serpengiizel, Chang and Hill [ 1993]). This point is illustrated in fig. 7, which shows spatially preserved lasing spectra from lasing microdroplets undergoing quadrupolar shape deformations. A D-shaped wavelength dispersion curve is expected from each n-mode of oblate droplets, whereas a c-shaped wavelength dispersion curve is expected from each n-mode of prolate droplets. Thus, it is possible to deduce from eq. (3.10) whether the deformed object is a prolate (e > 0) or an oblate (e < 0) spheroid. The m-dependence of the perturbed linewidth is (Lai, Lam, Leung and Young [ 1991 ]) Ax,, oc ( 1 -
m2/n2) 2.
(3.11)
Another result from perturbation theory using QNMs is the prediction of precession of tilted MDR orbits in spheroids (Swindal, Leach, Chang and Young [ 1993]). Figure 8 illustrates an MDR in a spheroid with a nonequatorial orbit (m < n). For a spheroid the z-component of the angular momentum Lz is still conserved, but Lx and Ly are not and the L+ and L_ cause mixing among the m and m + 1 modes. Hence, the total angular momentum vector L must
1, w 3]
PERTURBATION EFFECTS ON MICROSPHERE RESONANCES
27
Fig. 8. Precession of an MDR orbit. The shaded plane represents the plane to which the internal intensity of an m-mode is mostly confined. precess about the z-axis. The precession frequency of an MDR with effective quantum numbers (n, m) is calculated to be (Swindal, Leach, Chang and Young [1993]) s
m(m + 1 ) - oo(m) ,.~ doo/dm
= 090 le[
m o)0 lel ~ cos 0. n(n + 1) m(oo) x
(3.12)
The assumptions n(n + 1) ~ n 2 (true for microdroplets larger than 20 ~tm) and n ~ m ( m ) x (true for high-Q modes) are made to derive the final relation. The precession of MDR orbits was observed experimentally by Swindal, Leach, Chang and Young [ 1993]. Dispersive optical bistability in microspheres was modeled by using the firstorder perturbative relation depicted by eq. (3.4) with m(r) = m0 + m2/(r) (Mazumder, Hill, Chowdhury and Chang [1995]). The effect of the mzI(r) perturbation on the Q values of the MDRs was shown to be negligible. The scattering intensity from a microsphere (e.g., liquid droplets of CS2, which is known to have a large m2 value) is predicted to exhibit a bistable behavior versus incident intensity for near-resonant illumination, as shown in fig. 9. Recent experiments of dispersive bistability have been reported (Treussart, Ilchenko, Roch, Hare, Lef+vre-Seguin, Raimond and Haroche [1998]).
28
NONLINEAR OPTICS IN MICROSPHERES
~
200
[1, w 3
~_-------'-~-~-'~gh-intensity ( Istate
~100
~ = -5
~
nt 0
100
200
I inc (KW/cm 2) Fig. 9. Scattering intensity/scat as a function of incident intensity iinc for the case of a droplet with refractive index m = 1.62 and nonlinear refractive index coefficient m2 = 10-20 (m2/volt2). The diameter of the sphere is ~ 7 wavelengths. The input laser is detuned from an MDR with Q = 5.5 • 106 by t5 = - 5 linewidths (Mazumder, Hill, Chowdhury and Chang [1995]).
3.4. EFFECTIVE-AVERAGE METHOD: MODAL GAIN OR LOSS
When the dielectric sphere is treated as an open cavity, the usual perturbation theory does not apply (Lai, Leung, Young, Barber and Hill [1990], Lai, Lam, Leung and Young [ 1991 ]). It is necessary to define a new norm to avoid integrals from diverging (the essence of TIPM using QNMs). If the sphere can be treated as a closed cavity with mode volume Vm (to be specified later), however, the usual first-order perturbation method for Hermitian systems may be applied to calculate the changes in MDR frequency and linewidth resulting from a perturbation, such as in the refractive index. The accuracy of the perturbation calculation depends on the validity of the closed-cavity assumption (Chowdhury, Hill and Mazumder [1993]). The effective-average method is a perturbation procedure that is useful in the closed-cavity approximation (Chowdhury, Hill and Mazumder [ 1993]). It is an extension to spherical cavities of the modal-gain theory for semiconductor waveguide lasers (Yariv [ 1989]). In semiconductor waveguide lasers the electric field is confined in one dimension because of the thinness of the lasing layer. The confinement factor F,, for mode/t is defined to be
F F, =
d/2
~
J~
,
(3.13)
levi 2 dx (x)
where d is the thickness of the lasing layer and Ev is the transverse electric field. F~ is the ratio of the energy flux of the radiation confined within the central layer to the total energy flux of the radiation generated in the laser. The modal gain g~
1, w 3]
PERTURBATION EFFECTS ON MICROSPHERE RESONANCES
29
is dependent on the spatial overlap of the local gain (g(x)) and the mode intensity, and is defined as
fd/2g(x) fE f 2 dx d/2
9
(3.14)
le dx O(3
The imaginary part of the index of refraction mi,# is related to g~, by a factor of 4:r/Z (if mi,# < 0 the modal gain is referred to as modal loss). The calculation of g~, and F~ for a semiconductor waveguide laser is not difficult because the transverse electric field decays rapidly in the direction perpendicular to the propagation direction. The denominators in eqs. (3.13) and (3.14) are therefore easily evaluated. In addition, there is no evanescent wave in the beam leaving laser. The trapped fields and the propagating waves are easily distinguished. The propagating laser beam is not trapped and has no effect on the gain and should not be included in the normalization integral. In the case of a spherical cavity, however, the evanescent fields and propagating waves both extend outside the dielectric object, overlap spatially, and are not easily distinguished. Therefore, care must be used in evaluating the normalization integrals. The evanescent fields of a dielectric sphere are described by Neumann functions Yn. The boundary of the integration can be approximated by the radius rm that yields the first zero of the Neumann function (i.e., with yn(2~rm/Z) = 0). The radius rm defines the mode volume Vm. For spherical dielectric cavities, analogously to linear cavities, the effective average imaginary part of m(r) (for loss and gain within the cavity) and filling factor F~ of a/t-mode are defined as (Chowdhury, Hill and Mazumder [1993]):
mi,# =
~s
mi(r)6_or(r) IE~(r)l 2 dv /. s IE"(r)l 2 dr)
(3.15)
L and
/Vs
C.r(r) E ' ( r ) l 2 do
Y~ = c
L
'
(3.16)
e~(r) iE,(r )[2 do
where Vm is the bounded sphere and Vs is the actual cavity volume. E~(r) is the unperturbed electric field of mode/~ (/~=TE or TM,n,g,m), e0r(r)= Re{e(r)} = Re{m2(r)} and too(r) is the unperturbed refractive index.
30
NONLINEAR OPTICS IN MICROSPHERES
[1, w 3
By considering the spherical cavity as a closed cavity with mode volume Vm, the usual first-order perturbation scheme for a Hermitian system may be applied. The sphere is initially considered to be lossless (mi = 0) and homogeneous (e0(r) = m2(r) = m2). The unperturbed complex frequency and Q of the MDR are too = (-/)Or- ito0i and Q0 = tO0r/2t~i. The perturbed refractive index is
re(r) = mr + Amr(r)+ imi(r),
(3.17)
where Amr(r), mi(r) << mr and the perturbation in the dielectric constant is = m2(r) - m~(r) ~ 2mrAmr(r) + i2mrmi(r).
s
(3.18)
Using first-order perturbation theory (in the closed-cavity approximation), the perturbation to the complex MDR frequency is (Harrington [ 1961 ]) P
0)1
O ) l r - iO)li -
I -
(DOr
O)Or
JVs
s
IE"(r) le do .
( 3 . 1 9 )
2 ~Vm Co(r)IE"(r)l e dv
: the changes to the Substituting eq. (3.18) into eq. (3.19) and using e0(r)= mr, real and imaginary part of the complex MDR frequency due to the perturbation are derived to be A/~r, tt
tOlr = - ~
ntr
tO0r
(3.20)
and (-Oli =
~i,ll
(3.21)
(/)Or-
mr
Amr,F~is analogous to mi4~; it is the effective change in the real part of the index of refraction for mode It and is defined as
fzs
Arnr(r)eo~(r) IEl'(r)l 2 dv
Alttr,lt =
.
vms
(3.22)
IE~'(r)l 2 do
The Q value of the MDR after the perturbation (assuming tOlr << tO0r) is calculated from ,~ 2tO0i ~ 2tOli -- 1 ~ 1 1 _ 2(to0i + O)li) __ Q to0r + tOlr (/)Or (/)Or Q0 Q a '
(3.23)
where Qa = mr/2mi,~ is the Q value associated with the change in width of the MDR due to absorption or gain.
1, w 3]
PERTURBATION EFFECTS ON MICROSPHERE RESONANCES
31
If the perturbation is homogeneous (Amr(r) = Amr and mi(r) = mi), the changes in the real and imaginary part of the MDR frequency are F~Amr O)lr --
--600r /'?/r
(3.24)
and F~mi O)li -- ~ O ) 0 r , mr
(3.25)
and Qa = mr/2F~mi. In the approximation Vm = Vs, F~ = 1 and Qa = mr/2mi. The effective-average model mi,~ has been used to calculate the effect of twophoton absorption on an MDR with/t designated mode numbers. The ratio of the back-scattered elastic scattering intensity to the incident intensity from a sphere containing a two-photon absorber was also calculated (Chowdhury, Hill and Mazumder [ 1993]). A model of absorptive optical bistability in a microdroplet, doped with a two-level saturable absorber, was also developed (Mazumder, Chowdhury, Hill and Chang [1996]). The saturable absorber results in an intensity-dependent effective-average mi,~t. The energy stored in a mode and the Q value of the mode show bistable behavior as a function of input intensity. The effective-average model is also useful for computing the nonlinear gain for SRS in microdroplets (Mazumder, Schaschek, Chang and Gillespie [ 1995]). Calculations indicate and experiments verify that the effective-nonlinear gain for SRS, assumed to be on an output MDR, is larger with pumping by SRS on another MDR (the resonant case) than with pumping by the nonresonant input beam (the nonresonant case). The larger-effective gain for the resonant case is related to the large spatial overlap between the pumping MDR and the SRS-supporting MDR. A nonresonant input pump has most of its intensity confined to the two hot lines, and consequently has a much smaller spatial overlap with the SRS supporting MDR. Figure 10 displays the spatial overlap factor f pertinent to SRS pumped by radiation in both the resonant and nonresonant cases. For the resonant pumping the input MDR is a TE36,5 mode. f , and hence the nonlinear gain, is largest for SRS supported by ~ = 5 MDRs. Experimentally, it was found that SRS from the minority species of a multicomponent microdroplet was more efficiently pumped by the majority species SRS, which is on an output resonance and therefore provides resonant pumping, than by the incident pump radiation, which may not be on an input resonance.
32
NONLINEAR OPTICS IN MICROSPHERES
0.07
9
non-resonant
9
resonant
[1, w 3
pump
pump
0.06 0.05 0.04 o 0.03 0.02
e•sonant
0.01
9
. non-resonTa~.
0.00 0
1
2
3
4 5 6 7 mode order (1)
A .... 8
9
10
Fig. 10. Spatial overlap factorf for a plane wave (on or off an input resonance of n = x x and g = yy) as a function of g for TE MDRs with (n,g) as follows: (44,1), (39,2), (35,3), (31,4), (28,5), (25,6), (22,7), (20,8) and (17,9). The nonlinear gain is proportional to f (Mazumder, Schaschek, Chang and Gillespie [1995]). For the resonant case the input MDR is a TE36,5 mode (Mazumder, Schaschek, Chang and Gillespie [1995]).
3.5. COMPARISON OF THE PERTURBATION METHODS
Frequency shifts and linewidth changes of MDRs predicted by the TIPM and effective-average method have been compared to the results of T-matrix calculations and other analytical methods, such as separation of variables. The comparisons indicate that the perturbation methods are accurate as long as the perturbations are small. The fractional frequency shift Aw/w for quadrupolar shape deformations with e = - 0 . 0 3 calculated by eq. (3.9) of the TIPM and by T-matrix calculations are in excellent agreement (Lai, Leung, Young, Barber and Hill [1990]). The small differences between the results are due to truncation of the second-order terms in the TIPM calculations of Ato. The change in Q value for several values of m for the n = 10 mode versus deformation calculated by the second-order TIPM also agrees well with T-matrix calculations (Lai, Leung, Young, Barber and Hill [ 1990]). The accuracy of the TIPM for calculating frequency shifts and Q value changes due to one spherical inclusion (with different mr(O)) but mi(o)) = O) in a microsphere was also checked (Mazumder, Hill and Barber [1992]). The frequency shifts are well approximated by the TIPM. However, when low Q modes were considered, the change in Q value was poorly approximated by
1, w 3]
PERTURBATION EFFECTS ON MICROSPHERERESONANCES
33
eq. (3.7) if the C1 terms were neglected. Even when the C1 terms are considered, the accuracy of the TIPM in calculating changes in Q became poor as the magnitude of the perturbation increased. As was previously discussed, the TIPM compares favorably to analytical methods for calculating the effects of perturbations to the real part of the refractive index on both the MDR frequencies and Q values. By using eqs. (3.20) and (3.21), the accuracy of the effective-average model can also be checked. For perturbations to only m r ( O ) , eq. (3.15) implies that mi,~ = 0, and the perturbation has no effect on the Q value of MDRs (calculated by eq. 3.23). This inaccuracy is a shortcoming of the effective-average method. However, the accuracy of this method in predicting the frequency shift of an MDR due to a smooth roll-off in mr(co, r) is excellent (Mazumder, Chowdhury, Hill and Chang [1996]). For perturbations to mi(co, r), the fractional change in Q value of an MDR as determined by the TIPM, effective-average method, and T-matrix calculations have been compared (the three methods do not predict any change in MDR frequencies for such a perturbation) (Chowdhury, Hill and Mazumder [1993]). The second-order term C2 in the expansion of 1/Q in eq. (3.7) is small compared with the first-order correction C~ for small perturbations to mi(co, r) because C2 oc m 4(co, r), whereas the first-order term C1 oc m 2(co, r). Therefore, when considering mi(co, r) perturbations, first-order perturbation theory is sufficient for the TIPM (usually, second-order perturbation theory is necessary for MDR widths) (Mazumder, Hill and Barber [1992], Chowdhury, Hill and Mazumder [1993]). The percent error in Q values calculated by the TIPM and effective-average method (compared with T-matrix calculations) were computed for homogeneous and inhomogeneous perturbations to the imaginary part of the index of refraction of a microsphere (Chowdhury, Hill and Mazumder [1993]). Agreement between the two perturbation models and exact results is excellent. The utility of the TIPM and effective-average method depends on the type of perturbation. For perturbations to the imaginary part of the index of refraction, both methods calculate equally accurate results. This similarity is not surprising because both methods weight the effect of the perturbation by the spatial overlap of the perturbation and the energy density of the mode. For situations involving gain (e.g., SRS gain or laser gain), the effective-average method gives physical insight into the effect of the mi perturbation on the resonances. However, this method is not useful for calculating the effect of mr-type perturbations (e.g., shape perturbations and layered spheres) on the Q value of the resonances.
34
NONLINEAROPTICSIN MICROSPHERES
[1, w3
Fig. 11. (a) Shadow graphs and (b) simultaneously recorded lasing images of three lasing droplets falling in air taken at different phases of oscillation: prolate (top), spherical (middle) and oblate (bottom). The light regions in (b) indicate lasing (Mekis, Nrckel, Chen, Stone and Chang [1995]). 3.6. CHAOTICLIGHT- ASYMMETRICALRESONANTCAVITIES A fundamental difference between the Fabry-Perot type of microcavities and microspherical cavities is the directional output of the former. The lack of directionality of the output from microspherical cavities is a distinct disadvantage to incorporating microspheres (or microcylinders, microdisks and microrings) in many optical systems. Observation of the laser emission patterns from deformed microcylinders (Nrckel, Stone, Chen, Grossman and Chang [1996]) and microspheres (Mekis, Nrckel, Chen, Stone and Chang [1995]) shows evidence indicating that large deformations of the cavity can induce directionality on the laser output. Recent theoretical and experimental developments are beginning to provide quantitative predictions and measurements of Q spoiling and directionality of laser output from deformed dielectric cavities (Nrckel and Stone [1996, 1997], Gmachl, Capasso, Narimanov, Nrckel, Stone, Faist, Sivco and Cho [1998]). Good agreement between the directionality of emission predicted by the ray-chaos model and the measured far-field pattern of laser output from a quantum cascade laser shaped with a quadrupolar deformation and having a stable orbit consisting of a four-bounce 'bow tie' mode has been reported (Gmachl, Capasso, Narimanov, Nrckel, Stone, Faist, Sivco and Cho [ 1998]). Further theoretical and experimental development is necessary because quantitative measurements of Q spoiling and directionality of laser output from a noncircularly deformed liquid column indicate that the threshold deformation for directionality may be larger than first predicted (Moon, Ko, Noh, Kim, Lee and Chang [ 1997]). Figure 11 shows the shadowgraph image and the laser emission from prolate, spherical, and oblate microdroplet lasers of ~70 ~tm diameter (Mekis, Nrckel,
1, w3]
PERTURBATIONEFFECTSONMICROSPHERERESONANCES
35
Fig. 12. Polar coordinates systemused in the 2-d asymmetricalresonant cavities (ARCs). The angle of incidence Z is measured from the normal of the cavity boundary at each reflection. Each point in an SOS correspondsto a reflection at angle q~of the 2-d cavity with sinZ. Chen, Stone and Chang [1995]). The droplet size and deformations from sphericity are too large for the T-matrix method or any of the perturbation methods to be useful. A chaos-dynamic method (for both ray and wave optics) was developed to explain the emission pattern from these largely deformed cavities (henceforth termed asymmetrical resonant cavities, or ARCs). The properties of ARCs cannot be derived from the properties of spherical cavities in a simple manner (Mekis, N6ckel, Chen, Stone and Chang [ 1995], N6ckel and Stone [ 1996, 1997]). As the cavity is smoothly deformed, the ray trajectory of a mode with a stable orbit that has 0inc > Oc at each internal reflection (assumed to be 100% in the billiard-ball model) becomes chaotic in a manner determined by the Kolmogorov-Arnold-Moser (KAM) theory of classical Hamiltonian nonlinear dynamics. For ARCs, eventually somewhere along the trajectory, the incident angle satisfies 0inc < 0c, and the ray leaves the cavity with 100% transmission (in the billiard-ball model). Combining the general understanding of the chaotic nonlinear dynamics given by the KAM theory and simulations of the trajectory of the rays within the deformed cavity, this approach predicts a sharp onset of Q spoiling as a function of deformation. The Q spoiling is accompanied by highly directional output from specific locations of the cavity. If ARCs can be designed that combine the advantages (particularly the low lasing threshold) of microsphere and microdisk lasers with directional output of Fabry-Perot cavities, the applicability of these microlasers may become more widespread. The situation is most easily described in the ray optics model for microcylinders and deformed microcylinders, which are 2-d cavities (N6ckel, Stone and Chang [ 1994], N6ckel and Stone [ 1996, 1997]). A natural coordinate system in which to study these deformed 2-d structures is shown in fig. 12. If the cylinder
36
NONLINEAR OPTICS IN MICROSPHERES
[1, w 3
has a circular cross section, a ray propagating within the cavity with initial angle of incidence measured from the surface normal 2'0, larger than the critical angle for total internal reflection 2"c, is trapped forever in the microcylinder with a cavity lifetime r = ~ . Stable orbits are formed when the angle of incidence 2" at each total-internal reflection is identical. When the circular cross section has a quadrupolar deformation, 2" is no longer a constant but oscillates about 2"0 for these quasi-stable orbits. For large deformations the angle of incidence after several bounces can become less than 2"c (particularly near regions of high curvature), and the ray can refractively or classically escape from the cavity. The word 'classically' is used to assure the reader that no tunneling (or diffractive) escape is included in this model. The direction of refractive escape is usually tangential to the surface, but could be at an angle consistent with the Snell law of refraction. The Q of this mode is proportional to the number of roundtrips around the ARC (hence the length of time) before the ray escapes. The procedure of the ray optics model (analogous to a billiard ball) is most informative and can be described as follows (Nrckel and Stone [1996, 1997]). For a given shape (deformation) begin with an ensemble of rays (or balls) that is uniformly distributed in the starting position r and in the starting angle 2"0 > 2"c, which is determined from the eikonal rule sin 2"0 = n/m(to)ka. The ARC is assigned a mode number n, which corresponds to the angular momentum number of the mode in the undeformed cylinder. The ray ensemble is then propagated in time with a velocity c/m(to). From the total trajectory length before refractive escape, the mean refractive escape rate r -1 is calculated. The classical escape probability is zero for sin 2" > 1/m and one for sin 2' < 1/m. The quality factor of the mode is Q = ckr. A Poincar6 surface of section (SOS) is computed by plotting in the phasespace coordinates (sin X, r the angular position r around the boundary and the sine of the angle of incidence sin 2" at each bounce at the surface. For the billiardball model, the reflected angle is equal to the incident angle. The trajectory to the next bounce is computed, and another point added to the SOS. For various degrees of deformation, their SOS are shown in fig. 13 (Nrckel and Stone [1996]). The most important features in the SOS are the horizontal lines, the grainy points, and the closed curves forming islands. The horizontal lines are KAM or MDR curves. It is forbidden for a trajectory from high sin 2" to cross these MDR curves to get to lower sin 2". Therefore, escape is impossible until the deformation is large enough so that sin 2"0 is less than the smallest sin 2" of an MDR curve, which can be an unbroken oscillatory curve in the SOS. The grainy points indicate regions of chaos. A ray with phase-space coordinates in the grainy region can diffuse to smaller values of sin 2", i.e., below sin 2"c, and hence
1, w 3]
PERTURBATION EFFECTS ON MICROSPHERE RESONANCES
37
Fig. 13. Poincar6 surfaces of section for (a) the circle as well as quadrupolar deformations with eccentricities (b) e = 0.51 and (c) e = 0.63. Vertical and horizontal two-bounce orbits in (a), shown as crosses and stars, are depicted in the schematic below the surface of section (SOS). Shown to the right of each SOS are trajectories starting at sin 2'0 = 0.7 in all cases. In the schematics below (b), trajectories close to the horizontal and vertical diametric orbits are plotted (each below its bounce position in the SOS) (N6ckel and Stone [ 1996]).
refractively escape. As long as no K A M curve is p r e s e n t to stop the diffusion, the value o f sin Z can reach sin Zc and the ray can refractively escape. O n c e the ray has escaped, the trajectory calculation o f this ray stops. T h e directionality o f the e s c a p e d ray outside the cavity obeys Snell's law o f refraction. The star and plus in fig. 13a indicate 2 - b o u n c e orbits, the triangle a 3 - b o u n c e orbit, and the square a 4 - b o u n c e orbit. The closed curves in figs. 13b and 13c are islands, w h i c h also result f r o m quasi-periodic orbits. I f an island intersects
38
NONLINEAR OPTICS IN MICROSPHERES
[Near the Pole l
IEquator --> Pole I [curvature'[']
lntemal angle :X < ~c
Internal angle $ :~ --->2c
Refractive escape
Diffraclion or tunneling escape1" (Good output-coupling) (Tangent emission)
(Good output-coupling) (Non-tangent emission)
[1, w 3
[ Near the Equator[ [nearly circle] Internal angle : ~ > ~c Low diffraction or tunneling escape (Poor output-coupling) (Tmlgent emission) Fig. 14. Images of lasing prolate droplets (aspect ratio ~ 1.3) taken with a CCD camera at two different inclination angles 0D = 90 ~ and 0D - 122 ~ with f/# = 16. Bright light patches in the images correspond to the laser light emission. The two light-leakage mechanisms are responsible for laser emission at different locations of the droplet.
sin Xc, escape from the cavity is forbidden at the values of r for which the island spans. These quasi-periodic orbits, corresponding to islands in the SOS, give rise to 'dynamical eclipsing' (N6ckel and Stone [1997]), which (Chang [1998] and Chang, N6ckel, Stone and Chang [1999]) experimentally observed as dark zones. This ray optics model in 2-d and 3-d has been used to calculate the far-field emission pattern, location of emission, and dark zones (associated with the existence of stable-orbit islands in SOS that cause what is referred to as dynamical eclipsing) from deformed microcylinder-cavity lasers. The results from this ray optics model compare well with those of a wave optics chaos model. The situation in deformed spheres (3-d) is slightly more complicated than the case of cylinders (2-d), because of the extra degree of freedom. For each axisymmetrical (z-axis) deformation amplitude, SOS need to be computed for different values of the conserved quantity Lz, which is the z-component of angular momentum n of the ARC orbit. The 3-d ARC with Lz - 0 is the polar orbit. The images in fig. 11 are explained by computing SOS for oblate and prolate spheres with varying degrees of a quadrupole deformation. Lasing is assumed to occur on resonant modes with sufficient feedback. Both tunneling loss (for sinx > 1/m(og)) and refraction loss (for sin z ~< l/re(w)) and dynamical eclipsing are needed to explain the observed lasing emission images and angular profile (Mekis, N6ckel, Chen, Stone and Chang [ 1995], N6ckel and Stone [1996], Chang [1998]). In fig. 14 lasing prolate droplets imaged at different inclination angles 0D = 90 ~
1, w 3]
PERTURBATION EFFECTS ON MICROSPHERE RESONANCES
39
Fig. 15. Schematics for dynamical eclipsing of lasing rays that are from near-periodic and near-polar orbits in prolate (aspect ratio ~ 1.3) droplets. Quasi-4-periodic orbits (thin solid or dashed lines) have angle of incidence 2" < 2"c at the north and south poles and, therefore, are too leaky to provide sufficient feedback for lasing. Chaotic orbits cannot have the same 2" as quasi-4-periodic orbits at the two poles. Therefore, chaotic orbits that can be trapped inside the cavity long enough to lase have to escape refractively away from the two poles and, hence, dynamical eclipsing chaotic rays from entering this region of space. (along the equatorial plane) and 0o = 122 ~ (32 ~ below the equatorial plane) are shown with the explanation of the leakage mechanisms. The bright light patch in the 0o = 90 ~ image is tangential emission, and is caused by diffractive leakage (sin Z > 1/m(~o)) of lasing light from the precessing orbits. The observed bright emission at the top and bottom of the image at 0o - 122 ~ is nontangential emission, and is caused by refractive leakage (sin z < l/re(co)) from chaotic near-polar orbits. The fact that no emission occurs near the north and south poles (dark zones at the north and south poles of the image at 0D = 90 ~ but does occur away from the two poles (the bright regions at the top and bottom of the image at 0D = 122 ~ can be explained by dynamical eclipsing. For the prolate deformation, with aspect ratio ~1.3, o f the lasing droplet, near-polar orbits are chaotic except for quasi-n-periodic orbits (or islands of period n), which occur around stable orbits with period n. The quasi-4-periodic orbits shown in fig. 15 have 2' ~< Zc at the two poles (thus too leaky to lase) and form islands near sin 2, ~ sin 2,c in SOS. Because chaotic orbits and regular orbits are disjoint in SOS, the nonlasing islands of period 4 effectively block the emission from lasing chaotic orbits at the north and south poles.
40
NONLINEAR OPTICS IN MICROSPHERES
[1, w 4
w 4. Cavity-modified Optical Processes in Microspheres Fluorescence spectra (Benner, Barber, Owen and Chang [ 1980]) and spontaneous Raman spectra (Owen, Chang and Barber [1982]) from small microspheres display sharp peaks at wavelengths corresponding to MDRs that are not observed in large-sample spectra. The enhanced emission at MDRs (and partial suppression of radiation at frequencies not corresponding to MDRs) is attributed to cavity-induced modifications to the transition rates of molecules within and near spherical dielectric cavities. These transition-rate modifications are predicted by both classical electromagnetic theory (Chew [ 1987]) and quantum mechanics (Ching, Lai and Young [1987b]). The field of cavity-QED, perhaps spawned by the seminal paper by (Purcell [ 1946]) considers cavity effects on the radiative properties of atoms and molecules. This section reviews several methods that have been used to calculate the cavity-QED modifications. Several experiments that have directly measured the modification of molecular transition rates in small microspheres are also described. Novel optical effects, such as thresholdless lasing and cw-stimulated Raman scattering, can be explained in terms of cavity-QED. Several conclusions are common to all methods used to calculate cavitymodified emission rates. A molecule must have spectral and spatial overlap with an MDR to ensure significant enhancement to the transition rate. Enhancements as large as 1000x the free space transition rate are predicted (Chew [ 1987]). If the emission frequency of the molecule coincides with an MDR but the molecule is located at an intensity minima of the mode, the transition rate can be reduced to be a fraction of the free space transition rate (inhibited emission). Most theoretical treatments apply only in the weak-coupling regime (Chew [1987], Ching, Lai and Young [1987b]), for which r < r0,
(4.1)
where r is the cavity lifetime and r0 is the excited state lifetime of the molecule. In addition, the number density of the molecules must be small enough to ensure that spontaneously emitted photons are not reabsorbed by the molecules. In the Fourier-transform limit the weak-coupling condition is stated Ato0 < Aog, where Ato0 is the homogeneously broadened spectral width of the emission and Ato is the spectral width of the cavity mode. Calculations of transition rate modifications that assume the weak-coupling limit also can include cases where the emission is not Fourier transform limited (Yokoyama and Brorson [1989]). A theoretical description of electromagnetic decay in the
1, w 4]
CAVITY-MODIFIED OPTICALPROCESSESIN MICROSPHERES
41
strong-coupling limit has been formulated (Lai, Leung and Young [1988]), and experiments demonstrating modified transition rates in the strong-coupling limit (r > r0) are in progress (Lef~vre-Seguin, Knight, Sandoghdar, Weiss, Hare, Raimond and Haroche [1996], Lef~vre-Seguin and Haroche [1997], Vernooy, Ilchenko, Mabuchi, Streed and Kimble [1998], Vernooy and Kimble [1998]). 4.1. CLASSICAL ELECTROMAGNETIC THEORY
Raman and fluorescence scattering from molecules within dielectric spheres has been modeled using classical electromagnetic theory (Chew, McNulty and Kerker [1976], Kerker, McNulty, Sculley, Chew and Cooke [1978], Chew, Sculley, Kerker, McNulty and Cooke [1978], Chew [1988a]). The transition rates of molecules near the surface of spherical particles (both inside and outside) were also calculated, using the same model (Chew [1987, 1988b]). According to the model, a molecule at position r' radiates a dipole field Edip(r, fOt) at frequency oo' due to an incident field Einc(O)). The induced dipole p(r') at frequency of is proportional to the molecular polarizability and the internal field ir
A
Eint(r', 0 ) ) = Z-~-T-jS~,~bE(n'm)v' • [jn(klrt)Ynnm(rt)]
n,m "'~1 w
(4.2)
+ bM(n, m) jn(klr')Ynnm(]'t), where
bE(n, m) = imZaE(n, m)/[lt2x2Dn((O)],
(4.3)
bM(n, m) = iltlaM(n, m)/[x2D1n(OO)].
(4.4)
Medium 1 is the spherical dielectric and medium 2 is the surrounding infinite region. The index of refraction of each medium is ma = (ItaCa) 1/2, a = 1,2. The size parameter Xa = kaa, where/ca is the wavevector and a is the sphere radius. The expansion coefficients of the incident field are aE(n, m) and aM(n, m). The resonance factors
Dn(OO) = 6_ljn(Xl )[X2h(1)(X2)] t - 6~2h(1)(x2)[Xljn(xl )] ,,
(4.5)
D',,(w) = Dn(w) with (ea ---+/ta)
(4.6)
are small when the frequency w is resonant with an MDR. The internal field at frequency w' is the sum of Edip(r, 03t) and Edielectric(r, C0'), where the latter field is due to the dielectric (to account for the effect of the
42
NONLINEAROPTICSIN MICROSPHERES
[1, w4
boundary on the dipolar field). Each of the internal fields and the scattered field external to the dielectric Escat(r, tot) are expanded in terms of vector spherical harmonics. The expansion coefficients are determined by applying the appropriate boundary conditions. The result is (Chew, McNulty and Kerker [1976])
m11i22,CE(n'm)~7 • [h(nl)(k~r)Ynnm(r)]
Escat(r' to) = Z
(4.7)
II,m
+ CM(n,m)k(~l)(k~r) Y..m(r), where the scattering coefficients are - - - - . ,2,_,2
r E ( n , m) =
14"Tglrn2r~l
' 'D.(oo')
/tlxl
/
,\1/2
~ -5gl ) el
• p(r'). {
V'
r . "k'r'" * ^'
x tJ, t 1 ) Ynnm( r )]}
(4.8)
and
CM(n, m) -
4~/~k[3 jn(k~r')p(r'). Yn*m(k'). I I I
I
(4.9)
6-1XlDn(to )
The primed quantities are evaluated at the frequency to'. Equations (4.8) and (4.9) indicate that the inelastically (to ~ to') scattered field (e.g., fluorescence or spontaneous Raman scattering) is enhanced when the frequency to' is resonant with an MDR (on a output resonance where the Dn(to') and D',(to') in the denominators of CE(n,m) and CM(n,m) are small). The emission is enhanced most if the molecule is at a position r', where the MDR at to' has a high internal intensity, as well as ifp(r') is large (because the molecule is located in a hot line or a maxima of a resonant incident field, which is on an input MDR) (Pendleton and Hill [1997]). The time-averaged power radiated per solid angle (dP/df2 e( IEscat(r)l2) and the total power (P o( f df2[Escat(r)l 2) radiated by an individual dipole have been calculated (Chew, McNulty and Kerker [1976]). Incoherent (Kerker, McNulty, Sculley, Chew and Cooke [ 1978]) and coherent (Chew, Sculley, Kerker, McNulty and Cooke [ 1978]) summing of the distributed dipoles contributing to the inelastic scattering power have also been modeled. For the case of incoherent scattering (such as fluorescence and spontaneous Raman scattering), the time average power radiated per solid angle for each molecule is multiplied by the volume distribution function of the molecules and integrated over the sphere volume. For coherent scattering (such as lasing, stimulated Raman scattering, and coherent anti-Stokes Raman scattering), the electric field for each molecule
1, w 4]
CAVITY-MODIFIED OPTICAL PROCESSESIN MICROSPHERES
43
is multiplied by the molecular distribution function and integrated before the radiated power is calculated. The normalized transition rate of a dipole inside or near the surface of a dielectric sphere is calculated from the ratio of the total radiated power of the dipole to the total radiated power of the same dipole in the absence of the dielectric (Chew [1987, 1988b]). The transition rate ratio is divided into the component due to radial (_1_) and tangential (ll) dipole oscillations: /-'_k
Fox
"2 t 1) - 3~Xtl~l' ~C ~ ) 1/2 cx~ J"(klr 2 \ t~2 Z n(n + 1)(2n + 1) k[2r,2 ]Dn(to,)12
(4.10)
n=l
and FII _ 3 e l m 1 F~ [
4 Xtl2
s ~-~
~(2n+ n= 1
1)
kfrtDn( tot )
+ ~ 1 ~ Jn(kl .2 t r t ) ~ ~ [Otn(tot)[ 2
"
(4.11) The normalized rate is dependent on both the frequency of the dipole oscillation (to') and the spatial location of the dipole within the sphere (rl). When to / does not coincide with an MDR, the normalized transition rate is mostly unaffected, but does display some variation about unity with radial position. The normalized transition rate can be enhanced by several hundred times when to' coincides with an MDR and the dipole overlaps the spatial distribution of the MDR. The normalized transition rate may be partially suppressed in the resonant case when the dipole has poor spatial overlap with the MDR. These classical calculations of transition rates are consistent with quantum mechanical calculations (Chew [ 1987]). The model of a dipole radiating within a spherical dielectric is analogous to the classical example of an antenna in a metal cavity. If the wave is resonant with the cavity and the antenna is at an antinode, the antenna broadcasts more strongly, whereas if an antenna is at a node of the cavity, broadcast is less efficient. In the case of a sphere, if the field emitted from the dipole can reflect from the surface and return to the dipole to drive it in phase with the original field, the transition rate can be enhanced. If the dipole is at null locations of the internal field (even if its frequency is on resonance), the transition rate can be partially suppressed. This classical approach is valid only in the weak-coupling limit. Another restriction is that the density of atoms within the sphere must not be too large because effects such as Rabi oscillations and superradiance are not accounted for in the classical model.
44
NONLINEAR OPTICS IN MICROSPHERES
[1, w 4
4.2. DENSITY OF STATES FOR L E A K Y CAVITIES
The transition rates of molecules within microspherical cavities can be expressed (in the weak-coupling limit) in terms of the density of states of the cavity pc(tO). For a closed system Pc(tO) is known to be a series of delta functions, the spacing of which decreases as the cavity size increases. Extending the idea of a density of states for a leaky cavity is not straightforward because the eigenvalues for the cavity modes are complex numbers. It has been shown for open systems with small leakage, however, that Pc(tO) consists of narrow peaks, which are well approximated by narrow Lorentzians (Ching, Lai and Young [ 1987b]). Two important results describing properties of the density of states for an open system have been formulated in terms of two sum rules (Ching, Lai and Young [1987a], Ching, Leung and Young [1996]). (1) Strength o f a resonance: the frequency integral of the density of states of one n mode (the so-called weight of the resonance) is approximately equal to the degeneracy D of the mode, which is 2n + 1 for a sphere. For low-Q modes, the weight of the resonance is slightly less than 2n + 1. For more highly confined modes (i.e., high-Q modes), the weight of the resonance approaches 2n + 1. (2) Asymptotic sum rule: the total number of modes of a cavity is the same as that in an extended medium. In a cavity these modes are, however, spectrally redistributed. The utility of the sum rules is that they allow us to predict general features about the spontaneous emission spectrum and lifetimes of excited systems within microspheres. For example, when the emission spectrum covers many MDRs (e.g. a fluorescent dye), the second sum rule predicts that the emission spectrum would be modified such that emission occurs predominantly at MDRs. However, assuming that the total emission rate is proportional to the frequency integral of Pc(tO), the second sum rule also predicts that the overall spontaneous emission rate is unchanged by the cavity (Lai, Leung, Liu and Young [1992]).
4.3. GENERALIZED F E R M I ' S GOLDEN RULE
A generalization of Fermi's golden rule to microsphere cavities was formulated by (Ching, Lai and Young [1987a,b]). The theory provides insight into the physical parameters responsible for emission rate modification of dipoles within microspheres. However, the theory applies only in the weak-coupling regime, for which the cavity lifetime (e.g., r can be longer than 10ns) is shorter than the excited state lifetime of the molecule (e.g., "Co is typically 1 ns).
1, w 4]
CAVITY-MODIFIED OPTICAL PROCESSES IN MICROSPHERES
45
Fermi's golden rule states that the transition rate F for a two-level atomic system in free space to emit a photon of frequency to is F0 = _~_2r IMI2 Po(co),
(4.12)
where IMI 2 is the matrix element for the transition and Po(co) is the photon density of states of an extended vacuum. Purcell [1946] formulated a heuristic argument to generalize Fermi's golden rule for use with atoms within cavities, where the photon density of states is position dependent. Then, 2r
/'c-- -~-IM{ 2 pc(r, co),
(4.13)
where pc(r, co) is the cavity-modified, spatially dependent density of states. Experimentally, the measured rate is from a collection of atoms in the cavity, in which case pc(r, co) is replaced by its volume average
'/ drpc(r, co),
Pc(CO) = ~c
(4.14)
where Vc is the cavity volume. The enhancement of the transition rate due to the cavity is written as K -
rc pc(~O) /5 p0(~o)'
(4.15)
The enhancement ratio K depends only on the properties of the cavity (the modified density of states Pc(co)) and not on any molecular properties (IM[2). The interesting physics lies in the modified density of states Pc(co). The density of photon states in an extended medium (vacuum) is the smooth function 09 2
P0(CO) - :r 2c3.
(4.16)
The density of states in a closed cavity consists of delta functions. However, microspheres are open cavities because some energy leakage always occurs from the cavity modes. The density of states therefore consists of peaks with halfwidth Aco ~ l / r , where r is the decay time of the cavity mode. For states of degeneracy D, the density of states is assumed to be (Purcell [1946]) density of states =
VcPc(co) ~ D/2Aco.
(4.17)
Using the previous three equations, the enhancement factor is D
602
3
~3=
K ~ 2Aco(4Jra3/3)/~2c3 ~ 3-~DQ(a)
3
~3
- ~ D Q Vc'
(4.18)
where Q = co/2Aco is the quality factor of the resonance. The enhancement is large for small cavities and high-Q modes.
46
NONLINEAR OPTICS IN MICROSPHERES
[1, w 4
A quantum mechanical calculation of the density of photon states in spherical microcavities was made by Ching, Lai and Young [1987a,b] and Ching, Leung and Young [ 1996]. The ratio K shows sharp peaks at specific Xa that correspond to MDRs. The spontaneous and stimulated emission rates of molecules within dielectric microspheres are expected to be modified because of the significant difference between the density of states of the leaky cavity and of vacuum. Calculation of the Einstein A and B coefficients of atoms within microspheres shows enhancements when the emission frequency coincides with an MDR (Ching, Lai and Young [1987b]). The ratio K for spontaneous emission is
K(r, tot) = ( f
,,,(~o'))2 hE( r-'a~t~
'
(4.19)
where to' is the transition frequency of the atom a n d f - 3m2(to')/(2m2(to ') + 1) relates the field seen by the atom to the macroscopic field. The function hE ..~ pc(r, to')/Po is equivalent to the enhancement factor K in eq. (4.15). Plots of hE versus size parameter (or equivalently, to') at a fixed position show narrow peaks at MDRs (Ching, Lai and Young [ 1987a], Ching, Leung and Young [ 1996]). hE ranges from values less than one (when co' is far from a resonance) to greater than several hundred (when to' coincides with a resonance). The dependence of hE on the position of the atom within the sphere (with a fixedsize parameter) is similar to that of the internal-intensity distribution calculated from electromagnetic models (Hill, Barnes, Whitten and Ramsey [1997], Hill, Saleheen, Barnes, Whitten and Ramsey [1998]). When the atomic resonance coincides with an MDR (to = 09'), the ratio K is well approximated by (Ching, Lai and Young [ 1987a], Ching, Leung and Young [1996])
)3
K(r, oJ) ,~
32~2
re(to,)
The effective sphere radius
aefr(r) is
Vc
pc(r, co) =_ Pc(to) (4Jr/3)ae3fr(r).
aeff(r) implies that the transition rate
aeff(r)
(4.20)
defined by (4.21)
of the atom depends on its spatial location within the sphere. The result in eq. (4.20) is similar to the Purcell result in eq. (4.18), with a replaced by aeff(r) and the additional factor (f/re(to)) 2
1, w 4]
CAVITY-MODIFIED OPTICAL PROCESSES IN MICROSPHERES
47
associated with the local field correction, aeff(r) differs from a of the Purcell formalism because the MDR has an evanescent field, which extends outside the microsphere cavity. The enhancement ratio in eq. (4.19) is valid only under the assumption that the cavity resonances are broader than the atomic or molecular emission linewidth (i.e., in the weak-coupling regime). In vacuum the condition in eq. (4.1) is always valid, since P0 is completely smooth. The application of eq. (4.19) to calculate the enhancement to an optical transition with a 1 ns lifetime is limited to MDRs with Q values less than 104. The resonances observed in microspheres have Q values as high as 108 or 109. This strong-coupling regime is summarized in w A general expression for the enhancement factor K, which is applicable in the weak-coupling limit (r < r0) but when the emission bandwidth is not Fourier transform limited (e.g. broadened by many different final vibrational states allowing r < r0 but Am < Am0), is (Yokoyama and Brorson [1989])
j(o ~ p c ( (.o)R ( o) ) d co K --
~
,
(4.22)
~ po( m)R( o))d (.o
where R(co) is the spontaneous transition rate per mode. Although exact calculations using eq. (4.22) are difficult, a useful approximation in the limit of zero absorption and Amc > Amo > Ao) is K ~
Ao)c A~o0'
(4.23)
where Amc is the cavity mode separation and A~o0 is the spectral width of the emission. Equation (4.23) can also be derived from the sum rules given in w Because A coc becomes further separated as the microsphere radius decreases, the enhancement effects are expected to be largest in smaller microspheres. For broadband fluorophores, such as organic dyes, Ao30 > Acoc, and no enhancement to the overall emission rate is expected, although spectral modifications are possible. Using a mode-density argument, the ratio of the cavity-stimulated emission rate to the free space-stimulated emission rate is shown to be (Campillo, Eversole and Lin [ 1992a]) Ao9c K'stim --
A~o0
.
(4.24)
The enhancement ratio K'stim is surprisingly identical to K. Again, eq. (4.24) implies that cavity-QED effects are most prominent in smaller microspheres.
48
NONLINEAR OPTICS IN MICROSPHERES
[1, w 4
However, in very small microspheres with a ~ 4~tm and re(to) ~ 1.5, the maximum Q of a mode is less than 103, and hence little modification is expected based on eq. (4.18). K'stim can be -10 for lasing in liquid droplets (e.g., with europium Aco0 - 50 cm-1), -200 for SRS (Aco0 ~ 1 cm-1), and as large as 105 in low-temperature crystalline solids (Ato0 - 10-3 cm-1). 4.4. S T R O N G - C O U P L I N G R E G I M E
A general theory of electromagnetic decay into a narrow resonance of spherical dielectrics that uses the Hermitian modes of the sphere-bath universe rather than the QNMs of the leaky cavity was presented by Lai, Leung and Young [ 1988]. The theory is equally applicable in the weak-, strong-, and intermediatecoupling regimes. In the weak-coupling limit the general theory reproduces the generalized Fermi golden rule, for which the spontaneous decay rate is proportional to the cavity Q. In the strong-coupling limit, which is applicable to extremely high-Q modes, the long lifetime of the cavity mode allows spontaneously emitted photons to be reabsorbed by the atoms. Hence, the observed decay rate (external to the cavity) is the mode leakage rate, which is proportional to Q-l. Therefore, the most significant modifications to transition rates are expected to occur with modes having an intermediate value of Q ~ 104. This result appears to be reasonable because the decay from the cavity can be viewed in two steps: emission of the photon (K cx Q) and leakage from the cavity (at rate cx Q-l). For small Q the emission process is the rate-determining step, and for large Q the leakage process is the rate-determining step. For Q ~ 104 the observed decay is independent of Q. 4.5. EXPERIMENTALOBSERVATIONSOF CAVITYQED EFFECTS IN MICROSPHERES Most observable cavity-QED effects in microspheres are seen as spectral, temporal, or gain modifications (Campillo, Eversole and Lin [1996]). Most experiments performed to date have been in the weak-coupling regime (experiments with ionic species within microspheres and some experiments with fluorescent dyes) or the intermediate-coupling regime (experiments with fluorescent dyes). Although it is not strictly correct to calculate enhancements to emission rates by K from eq. (4.23) in the intermediate regime, authors have used this expression for K as an approximation. Spectral effects of cavity QED are commonly observed in fluorescence spectra (Benner, Barber, Owen and Chang [1980]) and spontaneous Raman spectra
1, w 4]
CAVITY-MODIFIED OPTICAL PROCESSES IN MICROSPHERES
49
(Owen, Chang and Barber [1982]) from microspheres illuminated with lowpower lasers. The spectra consist of sharp peaks superimposed on top of a broad background. The peaks in the spectra can be associated with specific MDRs, which allows accurate identification of the size, composition, and index of refraction of the particle (Eversole, Lin, Huston, Campillo, Leung, Liu and Young [1993]). The height of a peak is independent of the Q value of the MDR when the width is spectrally unresolved and correlates with the spatial overlap between the pump-laser internal-intensity distribution (possibly an input resonance) and the output resonances. The intensity of the broad-band emission is independent of the input laser frequency because this emission is from the region of the microsphere in which the MDR fields are not dominant, e.g., within the caustic sphere region (w2.2). In Nd:glass microspheres, fluorescence from the 740 and 810nm bands of Nd can be much larger than for a large rectangular sample when the pump laser beam is focused near the edge of the sphere because of cavity-QED enhancements to the transition rates relative to the cavity-independent nonradiative rates for these upper-state transitions (Wang, Lu, Li and Liu [ 1995]). When the pump laser beam illuminates the center of the sphere, however, fluorescence spectra from Nd:glass microspheres are similar to the rectangular sample spectra. The emission spectra from microspheres become similar to rectangular sample emission spectra when the radius of the sphere is larger than 250 ~xm because the MDRs are closely spaced and the emission enhancements are small for such large Vc. A surprising observation was a 2 x enhancement of the average fluorescence yield from 4 to 5 ~tm radius rhodamine 6G-doped microdroplets when compared with the yield from droplets with radii greater than 6 ~tm (Barnes, Whitten and Ramsey [1994]). The discrepancy is not entirely surprising because the experiment is not strictly in the weak-coupling regime. The sum rules do predict that fluorescence is spectrally redistributed by cavity-QED effects, but that the total fluorescence yield is not modified. The 2x enhancement in the total fluorescence yield implies that rate enhancement at MDRs and rate suppression away from cavity resonances do not exactly cancel in dye-doped droplets with less than 4 to 5 ~m radius. The observations are consistent with a model of spectral diffusion in which emission frequencies are not fixed but are perturbed by solvent-fluorophore interactions on very fast time scales (Barnes, Whitten and Ramsey [ 1994]). In this way the spectral width of the fluorescence becomes inhomogeneously broadened and can overlap with an MDR. Enhancement by this spectral diffusion process may be useful in detecting single molecules in small microspheres. Enhanced energy transfer between donor and acceptor molecules was also
50
NONLINEAR OPTICS IN MICROSPHERES
[1, w 4
observed in microdroplets (Arnold, Holler and Druger [1996]). The average transfer efficiency between donor and acceptor molecules was found to be as high as 50%, which is ~1000x greater than expected for F6rster-type transfer. This MDR-enhanced energy transfer is explained by a quantum mechanical model based on weak coupling and irreversibility (Leung and Young [1988]). Physically, the long-range energy transfer process can be viewed as a twostep process: (1) cavity-QED enhanced emission of photons by excited donor molecules and (2) cavity-QED enhanced absorption of these photons by the acceptor molecules. Cavity-QED effects in small microdroplets enhance the probability that excited donor molecules will emit photons into MDRs, as well as enhance the probability that acceptor molecule will absorb the emitted photons at MDRs. These two cavity-modified emission and absorption processes result in the observed enhanced donor-acceptor energy transfer. Emission rate modification due to cavity QED in microspheres has also been observed. Lifetime measurements of europium ion emission from doped liquid microdroplets in the weak-coupling regime (r0 "~ 0.5 ms, mo ,~ 50cm -1) (Lin, Eversole, Merritt and Campillo [1992]) showed little change from bulk values when the droplet radius was 24~tm. In 10~tm radius droplets, however, the MDR features showed a 2.5x enhanced emission rate and the off-resonant emission rate was inhibited by a factor of 1.5. The dependence of transitionrate modification on molecular orientation has been measured in both the weak- and intermediate-coupling regimes (Barnes, Kung, Whitten, Ramsey, Arnold and Holler [ 1996]). In the weak-coupling limit the decay rates scaled as expected (~ I/a), but in larger spheres (a > 15 ~tm) an anomalous decrease in the decay rate was observed and attributed to intermediate coupling and molecular oriemation effects (Arnold, Holler and Goddard [ 1997]). Cavity QED also enhances stimulated emission rates for processes such as lasing and stimulated Raman scattering in microspheres. The enhanced gain is inferred from the much lower pumping threshold required for the onset of the stimulated process when compared with the threshold required in large rectangular samples. The cavity-QED enhancement of lasing gain has been measured in small liquid droplets. Initial experiments using cw excitation to pump lasing in rhodamine 6G-doped droplets indicated a 100x cavity enhancement to the lasing gain (Campillo, Eversole and Lin [1991]), which exceeds the prediction of unity from eq. (4.24). A second investigation under identical conditions but using pulsed laser excitation showed no enhancement to the lasing gain (Campillo, Eversole and Lin [1992b]). Gain enhancements of 6x the rectangular-cell gain were observed from microdroplets doped with europium, a narrow-band emitter that is consistent with the weak-coupling
1, {} 4]
CAVITY-MODIFIED OPTICAL PROCESSES IN MICROSPHERES
51
regime (A~o0 ~ 50 cm -1 and A~oc ~ 130 cm -1). The enhancement is twice that predicted by eq. (4.24) (Campillo, Eversole and Lin [1992b]). Significant SRS gain enhancement was also measured by Lin, Eversole and Campillo [1992b]. The most dramatic demonstration is the observation of cwSRS, which can occur when the pump laser is resonant with an input MDR (Lin, Eversole and Campillo [1992b]). The pump power required for cw-SRS in microdroplets can be as low as 5 mW. When the peak of the Raman gain is coincident with an MDR, the threshold can be reduced even further. Under these conditions, thresholdless cw-SRS (three photons of pump laser on an input MDR) was observed in 4 ~tm radius CS2 microdroplets (Lin and Campillo [1997]). The cavity enhancement to the SRS gain in this case was estimated to be greater than 100. Enhancements to the gain for other nonlinear optical processes (such as stimulated Rayleigh wing scattering and four-wave parametric oscillation) were also reported by Lin and Campillo [1994]. The reduced lasing and SRS thresholds of microdroplets can also be interpreted as enhanced coupling of spontaneous emission into the cavity modes. The parameter /3 is the spontaneous emission (fluorescence or spontaneous Raman) coupling efficiency, and is defined as the ratio of the rate of spontaneous emission into an MDR to the total rate of spontaneous emission. Equivalently,/3 is the fraction of energy that is spontaneously emitted into an MDR divided by emission into all other modes. The spontaneous-photon coupling efficiency/3 can be approximated by (Yablinovich [ 1994]) 1
/3 = K _ I + 1'
(4.25)
where K is from eq. (4.20) or (4.22). Values of fi near unity are also responsible for thresholdless lasing (Lin and Hsieh [1991]) and thresholdless cw SRS (Lin and Campillo [1997]) in microspheres. The physics of the threshold reduction originates from the relation between spontaneous and stimulated emission rates. The spontaneous emission rate can be considered to be the stimulated emission rate with one photon in the cavity mode. High spontaneous emission coupling efficiency (i.e., fi ~ 1) therefore results in a high stimulated emission rate, which is proportional to the gain. Thus, a value of fi near unity enhances gain and reduces the threshold for stimulated processes. Provided that no significant absorption (or other photon-loss mechanisms) occurs at the lasing wavelength, as is the case for organic-dye lasers, the enhanced gain can result in thresholdless lasing as/3 --+ 1. However, in solid-state laser systems there can be significant absorption at the lasing wavelength and, hence, the necessity of a threshold for the laser system. The latter is identified by fluctuations in the output,
52
NONLINEAROPTICSINMICROSPHERES
[1, w4
even for fl ~ 1. Numerical simulations of the effect of [3 on laser threshold and operation in linear microcavity lasers (Yokoyama and Brorson [1989]), microsphere lasers (Lin and Hsieh [1991]), and microdisk lasers (Slusher and Mohideen [ 1996], Ho, Chu, Zhang, Wu and Chin [ 1996]) have been presented. Microdisk (Chu, Chin, Bi, Hou, Tu and Ho [1994]) and photonic-wire (Zhang, Chu, Wu, Ho, Bi, Tu and Tiberio [1995]) semiconductor lasers are being designed with near unity values of fl for use in optical communications (Ho, Chu, Zhang, Wu and Chin [ 1996]). Some of these structures have realized values of 13 as high as 0.85. Cavity-QED experiments in the strong-coupling limit by using fused-silica microspheres attached to the end of an optical fiber have been proposed (Haroche [1992]), and several laboratories are actively pursuing this line of research (Lefbvre-Seguin, Knight, Sandoghdar, Weiss, Hare, Raimond and Haroche [1996], Vemooy, Ilchenko, Mabuchi, Streed and Kimble [1998], Vernooy and Kimble [1998]). To achieve strong-coupling conditions, very high Q values (109-1010) and homogeneously broadened resonances are necessary. In addition, the thermal bistability of microspheres at room temperature due to heating of the microsphere by a probe laser (Braginsky, Gorodetsky and Ilchenko [1989], Collot, Lef~vre-Seguin, Brune, Raimond and Haroche [1993]) needs to be mitigated before observing intensity squeezing in a Kerr bistable device and subshot-noise effects such as quantum nondemolition (i.e., using the nonlinear coupling of photons between MDRs to determine the number of photons in one MDR by the frequency shift of another mode). Recently, bistable behavior due to the intrinsic Kerr nonlinearity of the fused silica microsphere (instead of thermal bistability) at liquid helium temperatures was observed by Treussart, Ilchenko, Roch, Hare, Lefbvre-Seguin, Raimond and Haroche [ 1998]. If the high O of the modes of the liquid-helium temperature larger microsphere can be maintained with the presence of single molecules within the sphere, the energy exchange rate between the molecule and an empty MDR will be comparable with the loss rate of the molecule-cavity system and the strong-coupling limit would be realized. In addition, at a liquid helium temperature the homogeneous linewidth of the Nd 3+ transition is 2 Mhz, and the strong-coupling between several Nd 3+ ions and photons in an MDR can be achieved (Lefbvre-Seguin and Haroche [ 1997]). Thresholdless microsphere lasers may also be possible in these low-temperature microspheres (Lefbvre-Seguin and Haroche [1997]). Other proposed experiments of cavity-QED in the strong-coupling limit include measuring the quantized atom-field force experienced by an atom in the evanescent field of an MDR (Treussart, Hare, Collot, Lefbvre-Seguin, Weiss, Sandoghdar, Raimond and Haroche [1994]) and preparation of atom galleries
1, w 5]
FLUORESCENCE AND LASING IN MICROSPHERES
53
around microspheres (the atoms are trapped by the WGM of the sphere and the radiative properties of the cold atom are modified) (Vernooy and Kimble [1997]). In preparation for the latter experiment, the effects of coupling single atoms (from a dilute atomic vapor) and the external evanescent field of a high-Q MDR of a fused silica microsphere (populated by 'several' photons) were measured by Vernooy, Ilchenko, Mabuchi, Streed and Kimble [1998]. Although the experiment satisfies the conditions for strong coupling, any observables (such as vacuum-Rabi splitting) were masked by the thermal distribution of atoms in the atomic vapor. Future experiments using cold atoms in a magnetooptical trap, instead of an atomic vapor, will enable realtime observation of the interaction of single atoms with the evanescent field of an MDR.
w 5. Fluorescence and Lasing in Microspheres This section reviews some properties and applications of fluorescence and lasing in microspheres and the important role that MDRs play in altering these effects. We do not intend to give a detailed review of all important contributions that various research teams have made on these subjects. However, references are cited with the intent of pointing the reader to some representative literature. Other reviews of microsphere lasers (Lefbvre-Seguin, Knight, Sandoghdar, Weiss, Hare, Raimond and Haroche [ 1996], Campillo, Eversole and Lin [1996], Armstrong [1996]) and other shaped microcavity lasers (Slusher and Mohideen [1996], Ho, Chu, Zhang, Wu and Chin [1996]) are noteworthy.
5.1. F L U O R E S C E N C E IN M I C R O S P H E R E S
Fluorescence spectra from microspheres consist of sharp MDR peaks superimposed on the broad fluorescence spectra (Benner, Barber, Owen and Chang [1980]). As described in w molecules located where the energy density of an MDR is high can exhibit an enhanced Einstein A coefficient and, hence, increased spontaneous emission rate (Chew [1988b], Ching, Lai and Young [1987a,b]). Which MDRs are observed in fluorescence spectra depend on several factors, such as the leakage rate of photons, absorption (Ch~lek, Lin, Eversole and Campillo [1991]) and scattering from small inclusions (Lin, Huston, Eversole, Campillo and Ch~lek [1992]). For small absorption and scattering losses a
54
NONLINEAR OPTICS IN MICROSPHERES
[1, w 5
scattering efficiency r can be defined for each MDR of a sphere as (Lin, Huston, Eversole, Campillo and Ch~,lek [1992]) =
1/Qo + 1/Q/~
(5.1)
1/Qo + 1/Q. + 1/Q~' where Qo 1 is the rate at which light leaks out tangentially from the cavity in the absence of other loss mechanisms, Q~I = [2;rmr(,~)/Aa(A)]-i = [2mr(,~,)/mi(,~,)]-I is the rate that light is absorbed within the cavity (a(A)= wavelength-dependent absorption coefficient, mr(,~,) = wavelength-dependent real part of the index of refraction and mi(,~,) = wavelength-dependent imaginary part of the index of refraction), and Q~l = [2~mr/Afl]-I is the rate at which light is internally scattered (,6 = scattering coefficient). MDRs with ~ on the order of unity are readily observed in fluorescence spectra as long as their Q is not so small that the peaks are too broad to be distinguished from the background. Internal scattering causes light leakage in all directions, not just tangentially. The overall Q of an MDR is well approximated by 1
Q
1
Q0
+ ~
1
~a
+ ~
1
+
1
Qy
(5.2)
where the term 1/Qy is due to other possible perturbations to the MDR quality factor (such as surface roughness). The spectral-integrated intensities of all MDRs in fluorescence spectra from homogeneous lossless spheres are predicted to be equal (Ching, Lai and Young [ 1987a,b]) because the QED enhancement is proportional to Q0 and the spectral width is proportional to 1/Qo. Experiments have confirmed this prediction in moderate-sized microspheres (Ch~,lek, Lin, Eversole and Campillo [1991 ]). Fluorescence-emission spectra from 15 ~tm diameter, rhodamine 6G/ethanol microdroplets for four different concentrations of the absorber nigrosin in the solution are shown in fig. 16 (Ch~,lek, Lin, Eversole and Campillo [ 1991 ]). The spectrum in curve (a) was acquired from droplets without nigrosin (mi < 10-9, Qa > 7.0 • 108). MDRs with radial order number g = 1 - 4 are observable, and the spectral-integrated intensities of the MDRs are approximately equal. Curves (b) (c), and (d) correspond to increasing amounts of absorption (mi ,~ 1.6 • 10 -7, 8.2 • 10 -7, and 2.7 • 10-6, Qa ~ 4.3 • 106, 8.3 x l0 s, and 2.5 • l0 s, respectively). As the absorption increases, the high-Q0 modes (g - 1,2) disappear from the fluorescence spectrum because the absorption rate of light is faster than the leakage rate from the MDR (i.e., Qo 1 < Q~I, ~ is small and the fluorescence peak of high-Q0 MDRs is absorbed before it can leak out
1, w 5]
FLUORESCENCE AND LASING IN MICROSPHERES
6tk2
W A V E L E N G T H (nm) 600 5n,e
55
5,96
IU ,0 "7 W 0 (r W
i 1L L 11L LI ! L I ....
! ............. ~i .......................~ . . , , , ~
......
~ , + ~ ,
,,
I,
t
i
Fig. 16. Emission spectra of 15/2m-diameter rhodamine 6G-ethanol droplets for various droplet absorptions. Spectra (a) to (d) correspond to imaginary refractive indices of < 10-9, 1.6 x 10-7, 8.2 x 10-7, and 2.7 x 10-6, respectively. Mode assignments of the various features are shown with arrows up (down) for TE (TM) modes with mode number n and mode order g indicated by (n,g). Lower order modes are quenched by absorption more readily than higher order modes (Ch~,lek, Lin, Eversole and Campillo [1991]).
as the fluorescence circumvents the droplet rim). The g = 3 and g = 4 MDRs are relatively unaffected by the absorption because the leakage rate is still faster than the absorption rate (i.e., Qo 1 > Q~I, q~is near unity). The spectral-integrated intensities of the MDRs observed in figs. 16 (a)-(d) are well approximated by eq. (5.1) (Ch)lek, Lin, Eversole and Campillo [ 1991 ]). Rhodamine 6G fluoresces at wavelengths from less than 5 4 0 n m to greater than 600nm. However, the absorption band extends to wavelengths as long as 575 nm. Experiments in 20 ~tm-diameter droplets show that self-absorption by rhodamine 6G prevents g -- 1 (g = 2) MDRs from being observed at wavelengths less than 575 nm (566 nm) (Lin, Huston, Eversole, Campillo and Ch)lek [ 1992]). However, fluorescence spectra taken from similar microdroplets but containing 87 nm-diameter latex particles show g = 1 and 2 MDRs for wavelengths between 550 and 575 nm. The presence of small internal scatterers can actually enhance the detectable fluorescence emission (i.e. 0) from MDRs with high Q0. The latex particles provide a mechanism for light to scatter out of the microdroplet
56
NONLINEAR OPTICS IN MICROSPHERES
[1, w 5
in random directions. The scatterers cause the effective leakage rate for the high-Q0 modes to be faster than the absorption rate. The selective MDRs in fluorescence spectra from microspheres are enhanced when the pump radiation is resonant with an input MDR (Eversole, Lin and Campillo [1992]). At an input resonance with a particular g, fluorescence is most efficient for output MDRs with the same g (Owen, Chang and Barber [1982], Eversole, Lin and Campillo [1995]). 5.2. M I C R O S P H E R E L A S E R S
If a fluorophore is capable of sustaining a population inversion (i.e., laser dye, Er 3§ or Nd3+), a feedback-providing MDR can support laser action provided that the roundtrip gain is greater than the combined roundtrip leakage, scattering, absorption, and other perturbation losses. Because of cavity-QED enhancements to the Einstein-B coefficient and the small leakage losses on MDRs, substantially lower lasing threshold pump powers are expected for microspheres than, for example, the conventional jet-stream dye lasers. Here, we review the initial experiments involving lasing in both liquid and solid microspheres. Several possible applications of microsphere lasers are also discussed.
5.2.1. Development of microsphere lasers The threshold for lasing in microspheres is inferred from a distinct kink in the inelastic emission intensity from a single or collection of MDRs as the pump power is increased. The first observation of a sharp threshold in emission from a microsphere was reported in 1961 (Garret, Kaiser and Long [ 1961 ]). A threshold was observed for light emitted from the rim of a ~1 mm CaF2 sphere doped with Sm 2+ (cooled to 77 K). The rim emission was attributed to stimulated emission into whispering-gallery modes of the sphere. The emission intensity from the center of the sphere remained linearly proportional to the pump intensity and was determined to result from ordinary fluorescence. The polarization properties of the two emissions also indicated that the emission from the rim of the sphere was stimulated emission. The densely packed MDR peaks in the lasing spectrum could not be resolved because of the large size of the sphere. In addition, the sphere needed to be cooled to 77 K. Laser emission from liquid microspheres was confirmed to occur at wavelengths corresponding to MDRs in 1984 (Tzeng, Wall, Long and Chang [ 1984]). A monodispersed continuous stream of falling liquid ethanol microdroplets doped with rhodamine 6G (radius --30~tm) was illuminated with a focused
1, w 5]
FLUORESCENCEAND LASING IN MICROSPHERES ,
. , [
~
.
, ! [
57
.................. ,.-,. .....
=
-
RHO,i~AMtNE 6 G / E T H A N O L
Tc u
2,0
o
,I
A
A
4
-0
.,,II, q
-,-, ........ .., ........... -,,...1-.-g
9 ~-.-.~,,,~-~.~-.
4
O -4
I 0 -~
I
I 0 -2
L
I
~
10
|
r
z
r
I
puup POWER ( w l Fig. 17. The ratio of the spectrally integrated intensity from region A and region C as a function of the input pump-laser intensity. The displayed Ic/I A ratio deviates from a constant above 10-2 W, which is a measure of the laser threshold intensity for a single rhodamine 6G-ethanol droplet (Tzeng, Wall, Long and Chang [ 1984]).
cw argon-ion laser beam. Inelastic emission spectra were recorded in three wavelength intervals that corresponded to (A) a fluorescence region with some absorption (555-565 nm), (B) a transition region (590-600 nm), and (C) a lasing region with almost no absorption (600-610nm). The spectra in regions B and C showed regularly spaced MDRs. A plot of the integrated intensity in region C normalized to the fluorescence intensity in region A was a constant value with increasing pump intensity. The normalized intensity at approximately 0.01 W of pump power showed a sharp slope change, thus indicating the lasing threshold (fig. 17). Relaxation oscillations were observed in the inelastic emission from region C but not from region A, providing further evidence for laser emission from region C and normal fluorescence emission from region A. Soon after the confirmation of cw lasing in microdroplets, lasing pumped by Q-switched lasers was observed in dye-doped ethanol droplets (Qian, Snow, Tzeng and Chang [1986]) and dye-doped water droplets (Lin, Huston, Justus and Campillo [ 1986]). Images of the lasing microdroplets recorded in these experiments show a bright ring of emission highlighting the liquid-air interface. Potential applications of microsphere lasers in optical communications and optical computing require solid microsphere lasers that operate at room temperature. In 1987, cw laser oscillation at room temperature in a large sphere (5 mm diameter) formed from a single crystal of Nd:YAG was achieved (Baer [1987]).
58
NONLINEAR OPTICS IN MICROSPHERES
[1, w 5
The spherical laser typically operated multimode, but single-mode oscillation was attainable by careful control of the pump laser illumination geometry. The first polymer microsphere lasers (radius less than 200 ~tm) were dye-doped (Nile Red), polystyrene spheres (radii ranging 5-46 ~tm) (Kuwata-Gonokami, Takeda, Yasuda and Ema [1992]). The pump laser was a 520nm wavelength pulsed dye laser. Emission spectra from the dye-doped polystyrene spheres consisted of sharp peaks at wavelengths corresponding to MDRs when the pump power was above lasing threshold. Images of the illuminated spheres showed uniform emission below lasing threshold, but a bright rim above threshold. Laser action was further confirmed by measuring the temporal response of the inelastic emission above and below threshold. Above threshold the temporal response of the stimulated emission closely followed the temporal response of the pump laser (the pulse length was long compared with the MDR lifetime). Below threshold the temporal response of the emission had a slow decay typical of the spontaneous decay time of the dye molecule. Laser oscillation in Nd-doped solid microspheres has also been observed. In experiments on a 48.11 ~tm Nd:glass microsphere illuminated by a focused argon-ion laser beam (Wang, Lu, Li and Liu [1995]), the spontaneous and stimulated emission rates were observed to be enhanced by 103 at wavelengths corresponding to MDRs. A result of the cavity-QED enhanced Einstein A and B transition rates (comparable with the nonradiative rates) was anomalously high fluorescence intensity from the upper energy levels (740 and 810nm spectral bands). With sufficiently high pump intensity, multimode lasing near 860 nm was observed (in addition to lasing at the familiar 1060 nm band). The experiment had several drawbacks, however. The host glass was not highly transparent, which prevents very high-Q values (~ 10 l~ from being realized, and the microspheres had to rest on a glass plate. A technique to fabricate microspheres of radius a ~ 25-50~tm with very high-Q MDRs (Q > 109) entailed melting the end of a pure-silica communications fiber (20~tm diameter) with a CO2 laser (Collot, Lef+vreSeguin, Brune, Raimond and Haroche [1993]). The microsphere can be easily manipulated because it is attached to the end of the pure-silica fiber. Although the stem affects the modes near the pole of the sphere, the remarkably high-Q of modes near the microsphere equator are unaffected. Nd-doped silica microspheres are also fabricated by the same melting technique (Sandoghdar, Treussart, Hare, Lef+vre-Seguin, Raimond and Haroche [1996]). Fluorescence and lasing are pumped by a ~807 nm cw diode laser, which is current tuned to match an input-MDR frequency and efficiently coupled to the input-MDR by launching evanescent waves with a high-index prism. The inelastic emission is
1, w 5]
FLUORESCENCE AND LASING IN MICROSPHERES
59
coupled out of the microsphere by the same prism, sent to a monochromator with 0.02 nm resolution, and detected by a photodiode. The amount of pump power removed by the microsphere can also be measured by the decrease in the reflected intensity from the prism. To determine the MDRs that support lasing and to measure their cold-cavity Q values with gain narrowing, a second tunable diode laser operating near 1080 nm is scanned. With this apparatus, both single-mode and multimode lasing were observed, dependent on the pumping geometry of the Nd3+-doped silica microspheres on the fiber stem (Sandoghdar, Treussart, Hare, Lefrvre-Seguin, Raimond and Haroche [1996]). Very low thresholds (<200 nW of absorbed pump power) were reported for lasing modes with Q values greater than 2 x 108. The threshold for lasing was found to be inversely proportional to the cavity Q. For the case of multimode lasing, the equally spaced lasing MDRs all have the same (radial) g-mode order as the pump MDR, but differing (angular momentum) n-mode numbers. Furthermore, each of the narrow peaks of a particular n-mode number is resolved to be several closely spaced peaks, corresponding to lasing on several modes with different (azimuthal) m-mode numbers (where m ~ n). The (2n + 1) degeneracy of each n-mode is lifted because the fabricated microspheres are not perfectly spherical. Two-photon pumped lasing in microdroplets was observed by Kwok, Serpengiizel, Hsieh, Chang and Gillespie [1992]. Rhodamine 6G-and various coumarin-doped ethanol droplets illuminated with moderate intensity laser pulses (~ 1 GW/cm 2) exhibited lasing on MDRs at wavelengths to the blue of the pump laser.
5.2.2. Properties and applications of microsphere lasers Some of the properties and potential applications of microsphere lasers are reviewed. Although microsphere lasers have many characteristics in common with conventional lasers, several applications are tied to the unique properties of microsphere lasers. For lasing radiation to be detectable, the MDR must have the scattering efficiency r (see eq. (5.1)) to be of order unity, much like the case of fluorescence (Lin, Eversole and Campillo [1992a]). This condition is typically satisfied by many MDRs within the laser-gain profile. Thus, MDRs of both TE and TM polarization, several Gmode orders and multiple (usually consecutive) n-mode numbers are observable in the lasing spectrum. However, other 'hidden' MDRs are present with such high Q0 that they support lasing, but the light
60
NONLINEAR OPTICS IN MICROSPHERES
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is absorbed by the microsphere before an observable fraction can leak out, i.e., r << 1. The multimode nature of microsphere lasers is attributed to spatial-hole burning (Lin, Eversole and Campillo [1992a]). MDRs with the same n-mode number but different radial g-mode orders can lase simultaneously because their radial distributions are different (fig. 2), analogous to the case of different transverse modes of a linear-laser cavity. Similarly, MDRs of the same g-mode order but differing n-mode numbers do not share identical angular regions of gain within the microsphere, analogous to the case of longitudinal modes of a linear-laser cavity. MDRs of both polarizations can lase simultaneously for similar reasons. Although the lasing spectra are rich and complicated, the multiple lasing peaks can be useful for characterization of the microparticle (Chen, Mazumder, Chang, Swindal and Acker [ 1996]). In small microspheres (radius a < 20 ~tm), where the n-modes are well separated, accurate identification of the lasing MDR mode numbers, index of refraction, and size is possible (Eversole, Lin and Campillo [ 1992]). For larger microspheres, assignment is more difficult and less unique. However, the size of the microsphere can be determined using the approximate relation (Ch~lek, Kiehl and Ko [ 1978]) /~,2 tan-1 [m(r a~
-
2~a
_ 1 ]1/2
[re(to) z - 1]
'
(5.3)
where A~. is the wavelength separation between MDRs of identical g-mode order and polarization and with consecutive n-mode numbers (i.e., AA for n and n + 1). Equation (5.3) shows that the mode spacing is inversely proportional to the microsphere radius. Lasing spectra from dye-doped microspheres of known size confirmed this inverse relationship (Lin, Huston, Justus and Campillo [1986], Kuwata-Gonokami, Takeda, Yasuda and Ema [ 1992]). A consequence of eq. (5.3) is that a change in radius 6a results in a shift 62 in the wavelength of an MDR determined by a
6a = -~6~,
(5.4)
independent of the mode numbers g, n, and m. Spectroscopically, for an MDR with unknown g and n, a decrease of 6a in the radius is observed to cause a blue shift of the lasing peak. Size variations on the order of 6a/a ,~ 10-4 can be measured with the use of moderate-resolution spectrographs (~1 cm-1). This feature has been used to determine the evaporation rate of droplets in a
1, w5]
FLUORESCENCE AND LASING IN MICROSPHERES
61
continuous or segmented stream by simultaneously recording lasing spectra from multiple droplets in the stream and measuring the blue shifts of the lasing peaks from a series of downstream droplets (fig. 18) (Chen, Serpengtizel, Chang and Acker [ 1993]).
Fig. 18. The spectrally and spatially resolved CCD image of lasing droplets in a continuous droplet stream. The droplets are pumped by a single 10ns pulse from the second harmonic (532nm) of an Nd:YAG laser (Chen, Serpengiizel, Chang and Acker [1993]).
Lasing microspheres have been used to confirm predictions of the timeindependent perturbation theory (TIPM) described in w TIPM predicts that the (2n + 1) degeneracy of the azimuthal n-modes of an MDR with size parameter xn,e is lifted when the sphere is deformed. For quadrupolar deformations about the z-axis, the splitting has a quadratic dependence on azimuthal mode number m -- n cos 0 (0 is the angle between the normal to the orbit and the z-axis), yielding n + 1 discrete wavelengths (the +m a n d - m modes are still degenerate). By imaging the fight half of a magnified lasing microsphere onto the entrance slit of an image-preserving spectrometer and recording the spectra with a CCD, the dependence of the lasing wavelength along the half-tim of the droplet has been measured (Chen, Mazumder, Chemla, Serpengfizel, Chang and Hill [1993]). The
62
NONLINEAR OPTICS IN MICROSPHERES
[1, w 5
Fig. 19. (a) Schematic of a lasing oblate spheroidal droplet with the portion of the droplet (right half) that is imaged onto the entrance slit of a spectrograph and the wavelengths of the emission from the various regions along the droplet rim. (b) Same as (a), except that the droplet is prolate spheroidal. The lasing light that leaks from the entire rim of the droplet is spectrally resolved. The spectrally dispersed image shows a distinctive D-shape or c-shape because different m modes are spatially confined to different regions of the droplet rim. The orientation of the dispersed image indicates whether the droplet is oblate or prolate. (c) Spectrally resolved CCD image of a magnified lasing droplet (Chen, Mazumder, Chemla, Serpengiizel, Chang and Hill [1993]). w a v e l e n g t h from various points on the droplet rim is related to the vertical d i s p l a c e m e n t from the droplet equator along the slit o f the s p e c t r o m e t e r Az by
Z(Az) = ~0
[
e
1- ~ + ~
,
(5.5)
where A0 is the w a v e l e n g t h o f the degenerate M D R and e = (rp- re)/a is the droplet distortion parameter. F r o m eq. (5.5) both the m a g n i t u d e and sign o f e are determined. Figure 19 shows an i m a g e - p r e s e r v e d s p e c t r u m o f a lasing droplet.
l, w 5]
FLUORESCENCE AND LASING IN MICROSPHERES
63
Each of the D-shaped curves corresponds to lasing on a particular n-mode number MDR. By fitting a function of the form of eq. (5.5) to one D-shaped curve, the microsphere is determined to have e = -4 x 10.3 and X0 = 591 nm (Chen, Mazumder, Chemla, Serpengfizel, Chang and Hill [1993]). The damping time of quadrupolar shape deformation oscillations in falling droplets can be determined from the time-dependent oscillation amplitude of the wavelength shift of the lasing peaks. From the oscillation frequency and damping rate of the wavelength shifts, the surface tension and fluid viscosity are deduced, respectively (Tzeng, Long, Chang and Barber [1985]). For many applications it is desirable to control the laser emission from the microsphere cavity. Several methods that modify emission spectra from microsphere lasers, such as pumping geometry, injection seeding, particulate seeding, absorption modification, and Q-switching have been employed thus far. One method to alter the radial g-mode order of the MDRs that lase is by varying of the pumping geometry, and thus optimizing of the spatial overlap of the gain region with that of the MDR that provides the feedback (Knight, Driver and Robertson [1990], Lef~vre-Seguin and Haroche [1997], Lin, Eversole, Campillo and Barton [1998]). The high-Q (low-g) modes reside closest to the sphere surface (and penetrate the shortest distance outside the sphere), whereas the low-Q (higher g) modes have their peak intensity closer to the sphere center (and penetrate the furthest distance outside the sphere). With planewave illumination the gain is confined to the two interior hot lines, which have decreasing spatial overlap with MDR as the radial order number ~ decreases toward 1. With sufficiently high pump intensity the gain saturates uniformly within the sphere. Hence, many g-mode orders and n-mode numbers have access to the uniform gain region. By illuminating the microsphere with a loosely focused laser beam near the edge of the droplet, however, the gain region can have good spatial overlap with only low g-mode order MDRs. Further confinement of the gain region to a small volume by tightly focusing the pump laser beam near the microsphere surface can result in single-mode lasing (Baer [ 1987], Miuram, Tanaka and Hirao [ 1996]). Injection seeding is a common technique used to modify the output of conventional lasers (Erickson and Szabo [ 1971 ]). Initial experiments on injection seeding in dye-doped polystyrene microsphere lasers showed that by seeding an (g, n) MDR, lasing from modes with the same ~ and nearby consecutive n-modes (n' ~ n) is quenched (Kuwata-Gonokami, Ema and Takeda [1992]). The degree of quenching is greatest for adjacent n'-modes (In- n ~] = 1) and becomes smaller as the difference In- n ~] increases. Little or no quenching occurs for modes with g~ ~ g. Such quenching of other MDRs on seeding a particular MDR suggests
64
NONLINEAROPTICSIN MICROSPHERES
[1, w5
the necessity for good spatial overlap between the seeded mode and the quenched mode because gain depletion is most effective for MDRs with the same g but nearby n-modes. Intense elastic scattering of the seed laser prevented the intensity of the seeded lasing mode from being measured. This difficulty was solved by using a linearly polarized seed laser and a crossed analyzer in front of the spectrometer (Popp, Fields and Chang [ 1997a]). In this way, simultaneous detection of the enhanced emission from the seeded MDR and quenched emission from the other MDRs are possible. Figure 20 shows 9 'seeded' lasing spectra (as the seed is tuned through various MDRs) and a reference 'not-seeded' lasing spectrum. Enhancements to MDR lasing intensities as large as 40• the reference intensity were measured. In addition, hidden modes with low q~ become detectable by injection seeding (figs. 20f, i), thus indicating a significant enhancement to the internal intensity. Complete suppression of modes with I n - n ' l - 1 and g = g' is seen in fig. 20g. Again, the degree of suppression of the quenched modes is related to their spatial overlap with the seeded mode. In slightly deformed microdroplets, where the 2n + 1 degeneracy is lifted, injection seeding of the near-equatorial orbits (i.e., the m ~ n modes) further confirmed the importance of spatial overlap in the polar angles. The lasing from m t ~ n' ~ n modes was completely quenched, but the lasing of the near-polar orbit m' ,~ 0 modes was relatively unaffected by seeding (fig. 21). Scattering by small particle inclusions within a microsphere can significantly modify the lasing threshold, spectra, and emission spatial distribution. Investigations of dye-doped microspheres containing submicrometer-sized latex particles indicate that hidden MDRs with high Q0 become visible in the lasing spectra when the scatterer concentration is increased from zero because r is increased as the scattering Q/~ is degraded (eqs. (5.1) and (5.2)) (Armstrong, Xie, Ruekgauer, Gu and Pinnick [1992]). Lasing at all MDRs is suppressed when the concentration of the scatterers is high for two reasons: (1) dye molecules adsorb to the surface of the latex particles, thereby increasing the Frrsterassisted quenching of the laser-dye excited state; and (2) for each roundtrip the scattering-related leakage loss exceeds the gain. Introduction of a highly scattering fat emulsion into a dye-doped microdroplet also significantly modifies the lasing characteristics (Taniguchi, Tanosaki, Tsujita and Inaba [1996]). An optimal dye-intralipid mixture results in a lowering of the lasing threshold and an increase in emission intensity when compared with neat microdroplets. It is hoped that the lower thresholds and enhanced emission due to an optimal concentration of scatterers can be extended to other stimulated processes, such as stimulated Raman scattering, for detection of trace amounts of biocolloids
1, w 5]
FLUORESCENCE AND LASING IN MICROSPHERES
~
~
o~
f
ocml
j/
65
~TM21 209
o,
1"
(d)
1" TMIO 1
['~
~ 10} |TM935
l|
_~._~ ~ 626
628
(b) TM934 ~
_-,~ ....
rM~33 (a)
.-Ix_ _ ~
630 632 Wavelength (nm)
634
Fig. 20. Lasing spectra from ~ 21.7~tm dye-doped microdroplets. (a) 'Not-seeded' lasing spectrum. (b)-(j) 'Seeded' lasing spectra with the seed wavelength indicated by an T. All spectra have the same intensity scale. The intensity of the cutoff peaks are given by the numbers. The modes are identified and labeled as TM~n. Each spectrum is a 10 laser shot average. The spectra (f) and (g) show the major effects of seeding (Popp, Fields and Chang [ 1997a]).
66
NONLINEAROPTICSIN MICROSPHERES
[1, w5
Fig. 21. Seeding of m ~ n modes (mainly equatorial modes) in large droplets. The z-preserved spectra (a) for 'not-seeded' lasing in droplets and (b) for 'seeded' lasing droplets. The seed laser wavelength is indicated by the black arrow. Each spectrum is a 5 laser shot average (Popp, Fields and Chang [1997a]).
and biomolecules. The spatial distribution of light emitted from microdroplets containing relatively high concentrations of scatterers shows bright emission from the center of the microsphere, in contrast to the emission from neat and low scatterer concentration droplets which is predominantly from the rim (Taniguchi, Tanosaki, Tsujita and Inaba [ 1996]). This change in the lasing images is attributed to cessation of lasing supported by MDRs, which are confined to the rim. Multiple scattering by the dense loading of scatterers can cause mixing among ~, n and m modes. Lasing spectra recorded from these microspheres indicate a loss of MDR structure. Microsphere-laser wavelength tuning is also possible by control of cavity-Q values. Two mechanisms are responsible for the tunability by Q modification. First, as Q values are degraded, more gain is required for laser oscillation. In many dye-laser systems, laser action occurs at wavelengths to the red of the peak of the fluorescence or stimulated-emission, cross-section curve. To achieve threshold in the Q-degraded system, the lasing wavelengths shift to the blue to benefit from a higher gain coefficient. Second, as mentioned above, hidden modes with low (i.e., low mode order) can become observable with small amounts of Q-spoiling. Furthermore, low Q0 modes can become so degraded by the Q-spoiling that lasing is entirely quenched. Blue shifts in the lasing wavelengths by Q-spoiling have been achieved by introducing scattering particles (Taniguchi and Tomisawa [ 1994]) and by addition of the broadband absorber nigrosin (Mazumder, Chen, Chang and Gillespie [ 1995]) into the microsphere. In both investigations, blueshifts in wavelengths of up to 50 nm were observed by increasing the amount of Q-spoiling (by increasing the latex particle or nigrosin concentration). When the Q-spoiling is induced by
1, w 5]
FLUORESCENCE AND LASING IN MICROSPHERES
67
temperature-dependent absorption, the blueshifl can be used as a diagnostic for the temperature of the microsphere tim (Mazumder, Chen, Kindlmann, Chang and Gillespie [ 1995]). The effect of transient absorption on lasing in microspheres was studied by Kamada, Sasaki and Mauhara [1994]. Latex microspheres doped with both rhodamine B and 9, 10-diphenylanthracene (DPA) are illuminated with 355 nm pulses before pumping with the 532 nm pulse. The transient absorption of the excited state DPA causes the lasing of rhodamine B to be partially quenched. This technique has potential applications for using lasing microspheres as optical cavities for excited state absorption measurements in an analogous manner to cavity ring down spectroscopy. High-Q MDRs of microspheres provide effectively long path lengths (rc/m(oo)) in inexpensive and compact cavities, and their Q is extremely sensitive to small amounts of absorption. Introduction of a saturable absorber into a laser-dye droplet can enhance one lasing mode and partially suppress the intensities of other modes (Popp, Fields and Chang [1997b]). In these experiments, ethanol microdroplets containing DCM (a laser dye) and DQTCI (a saturable absorbed are illuminated with 7 ns pulses at 532 nm to pump lasing of DCM and with a 2 to 3 ns delayed pulse at ~643 nm to bleach the absorption of DQTCI. The intense bleaching pulse significantly reduces the absorption in the DCM lasing band by bleaching the absorption of DQTC! (a wide-opening saturable absorber). All lasing modes experience additional net gain because of the reduced absorption at all wavelengths. However, the lasing mode at the bleach-laser wavelength is selectively enhanced because it is also seeded by the intense bleaching pulse, whereas the emission from other MDRs is reduced due to cross saturation. Q-switching with a saturable absorber can therefore improve the contrast ratio between other lasing modes in microsphere lasers and the one mode that is coincident with the bleaching laser. Elastic scattering spectra recorded from these microdroplets for increasing incident intensities confirm that cavity Q values are increased as the absorption of the absorbing dye is progressively bleached (Popp, Fields and Chang [1997b]). The MDRs of a dielectric microsphere are also affected by the presence of other nearby dielectrics and metals because the field of the MDR extends outside the microsphere. The cases of a microsphere nearby another sphere (Fuller [ 1991]) and a flat dielectric (Chew, Wang and Kerker [ 1979]) have been treated theoretically. The nearby dielectric tends to shift the MDR wavelength and induce more leakage, i.e., degrade the MDR Q value. Experimentally, the presence of a microsphere within 100 nm of a lasing microsphere was shown to quench lasing on the dye-doped micropshere (Sasaki [1997]). Lasing spectra
68
NONLINEAR OPTICS IN MICROSPHERES
[1, w 5
from microspheres were also shown to be significantly modified by the presence of a flat dielectric (Sasaki, Fujiwara and Masuhara [1997]). Away from the glass plate, the lasing modes were of g = 2. When the microsphere was brought into contact with the glass plate, the lasing modes were identified to be of g = 1. The glass plate enhances the output coupling efficiency q~of the g = 1 modes so that they become observable (without the glass plate, r ,~ 0.12). Simultaneously, the output coupling of the g = 2 modes is so large that these modes can no longer provide sufficient feedback to support lasing. By monitoring the ratio of the lasing intensity from two peaks of different g-mode order as a lasing microsphere is brought near a surface, the gap between the microsphere and the surface can be monitored. By attaching a small microsphere (~ 1 ~tm) to the large lasing microsphere, the photon tunneling loss can be localized and the bi-sphere system used as a precise monitor of separation for near-field optical microscopes (Sasaki, Fujiwara and Masuhara [1997] ). 5.3. OTHER MICROCAVITY LASERS
Microstructures such as microcylinders, microdisks, microrings, and photonic wires are similar to microspheres in that they can strongly confine electromagnetic fields in 2-d or 1-d. Lasing can therefore be expected to occur on the resonances of these microstructures. MDRs have been observed in fluorescence spectra (Owen, Chang and Barber [1982]) and lasing spectra (Knight, Driver, Hutcheon and Robertson [1992], Pinnick, Fernandez, Xie, Ruekgauer, Gu and Armstrong [1992], Armstrong [1996]) from optical fibers coated with polystyrene with fluorescent dye, laser-dye liquid microcylinders, and laser-dye liquid in microcapillary tubes. For many applications in optical communications and optical computing, solid microstructures are more desirable. Photonic wire lasers (Zhang, Chu, Wu, Ho, Bi, Tu and Tiberio [ 1995]) have been fabricated with nearby resonant waveguides for more localized output coupling. Polymer microdisk and microring lasers (Kuwata-Gonokami, Jordan, Dodabalapur, Katz, Schilling, Slusher and Ozawa [1995]) have also been developed. Organic microsphere and microring lasers based on energy transfer among the molecules show promise in applications because of their low threshold. Electrically pumped organic microdisk LEDs were also demonstrated by Dodabalapur, Berggren, Slusher, Bao, Timko, Schiortino, Laskowski, Katz and Nalamasu [ 1998]. Multiple quantum-well semiconductor microdisk lasers with very low thresholds, of interest to the optical communications community, were fabricated by Slusher and Mohideen [1996], Ho, Chu, Zhang, Wu and Chin [1996]. These
1, w6]
NONLINEAROPTICALPROCESSES
69
lasers have radiation confinement around the disk but not perpendicular to the disk. Microdisk lasers can be made to operate single mode, but their emission is not directional. To achieve directional output, double-disk structures are used. One disk (at the lower deck) is the microcavity laser and the other disk (at the upper deck) is the output coupler. The output coupler has a small region (either a cutout or a tab) at which some of the circulating radiation has an angle of incidence less than the critical angle for total internal reflection and the Q is degraded. The coupling of laser emission from the lasing disk to the output coupler is controlled by the distance between the two disks (Chu, Chin, Bi, Hou, Tu and Ho [1994]). Quantum-cascade microdisk lasers were fabricated with lasing at 5 ~tm (Faist, Gmachl, Striccoli, Sirtori, Capasso, Sivco and Cho [1996]) and at 9.5 and 11.5 ~tm (Gmachl, Faist, Capasso, Sirtori, Sivco and Cho [1997]). When shaped like ARCs (w with quadrupolar shape deformations, the emission of the microlaser becomes more directional. Agreement between the measured farfield pattern and that predicted by ray-chaos theory is good (Gmachl, Capasso, Narimanov, N6ckel, Stone, Faist, Sivco and Cho [1998]). In addition to a decreasing lasing threshold, the maximum peak optical power emitted from these lasers increases as much as 1000x for a particular amount of quadrupolar deformation. The lasing mode is a stable orbit consisting of four bounces (referred to as a 'bow-tie' mode), with 0 only slightly greater than 0c.
w 6. Nonlinear Optical Processes Since 1984, a series of nonlinear optical effects have been observed in single micrometer-sized droplets. Currently, groups in about a dozen research laboratories are actively studying nonlinear optical phenomena in droplets that are flowing (as a single droplet in a linear stream or in a dense spray) or levitated (in air or a fluid). There is one main difference in detecting nonlinear signals from a liquid cell and from a droplet. For a cell the detector usually measures the signal generated along the propagation or along the coherent wave direction. For a droplet the detector measures the signal leaking tangentially from the droplet as the internal guided wave circulates around the droplet-air interface. Consequently, in the case of droplets, nonlinear waves are usually detectable in all directions. Furthermore, the detected intensity is not simply related to the internal intensity of that wave because less internal radiation leaks from the droplet when a nonlinear wave is on an MDR with large Q0, even though the internal intensity is sufficiently intense to generate other nonlinear waves.
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NONLINEAR OPTICS IN MICROSPHERES
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In this section we review the observation of spontaneous Raman scattering because it is a precursor to stimulated Raman scattering (SRS), third-order sum frequency generation (TSFG), coherent anti-Stokes Raman scattering (CARS), stimulated anti-Stokes Raman scattering, coherent Raman mixing (CRM), stimulated Brillouin scattering (SBS), and laser-induced radiation leakage from single microdroplets. Several other reviews that include discussions of nonlinear optics in microspheres are noteworthy (Chang [1990], Hill and Chang [1995], Armstrong [ 1996], Campillo, Eversole and Lin [ 1996], Mazumder, Chowdhury, Hill and Chang [ 1996], Cheung, Hartings and Chang [ 1997]). 6.1. SPONTANEOUS RAMAN SCATTERING
A theoretical description of Raman emitters within microdroplets has been formulated (Chew, Sculley, Kerker, McNulty and Cooke [1978], Kerker, McNulty, Sculley, Chew and Cooke [1978]). Raman spectra recorded from a glass fiber (Owen, Chang and Barber [1982]) and individual microspheres reveal numerous peaks superimposed on the bulk spontaneous Raman signal (Thurn and Kiefer [1984, 1985]). The sharp peaks observed in the microcylinder and microsphere Raman spectra are a demonstration of cavity-QED enhancements to the spontaneous Raman scattering cross section of the vibrational modes. These Raman peaks occur at wavelengths corresponding to MDRs of the particular microcavity (sphere, cylinder or disk). The Raman enhancement, as in the case of fluorescence, has the same spatial profile as that of MDRs and, hence, is confined to a radial region between a/m(~o) and a. Raman scattering from microparticles also depends on the spatial internalintensity distribution of the pump laser. Partial spatial overlap of the pump laser with the MDRs that are within the Raman band (Stokes and anti-Stokes shifted wavelength) is necessary for some peaks to be observed in the spontaneous Raman spectra. If the incident field is on an input resonance, the spatial overlap with an output resonance of the same radial mode order ~ within the Raman shifted band can be particularly large. The simultaneous coincidence of being on an input and output resonance is called 'double resonance'. For in situ Raman characterization of physical properties (e.g., radius and refractive index) and chemical composition, a microdroplet can be in a flowing stream or levitated either optically (Ashkin [ 1970]) or electrodynamically (Davis, Ravindran and Ray [ 1981 ]). Once the radial distribution of an MDR is known, to some extent, radial concentration profiling of layered flowing droplets is possible (Lin and Campillo [ 1995]). Input resonances can be determined from sharp increases in the spontaneous Raman signal (Schweiger [1990, 1991],
1, w6]
NONLINEAROPTICALPROCESSES
71
Schaschek, Popp and Kiefer [1993]). Raman spectroscopy has also determined (without container wall effects) the formation of metastable aqueous droplets at high supersaturation from hygroscopic levitated particles (Tang, Fung, Imre and Munkelwitz [ 1995]). 6.2. STIMULATEDRAMAN SCATTERING(SRS)
6.2.1. Cavity effects The morphology of the sample (spherical or cylindrical) is responsible for some unique features that give rise to the observation of SRS at moderate pump intensities. The threshold intensity for the excitation of first-order SRS at o91s = a,'L- (Dvib (where a,'L is the laser frequency and O)vib is the frequency of the Raman mode with the largest Raman gain) in microdroplets is much lower than that for SRS in an optical cell (Snow, Qian and Chang [1985a], Braunstein, Khazanov, Koganov and Shuker [1996]). SRS threshold in a droplet is reached when the round trip Raman gain (e.g. at o91s) exceeds the round trip loss of the Raman radiation around the circumference. Besides the usual loss mechanisms (leakage, absorption, scattering), intensity depletion processes (e.g., pumping other nonlinear processes) are now present. The Raman gain may be enhanced by cavity-QED effects (Campillo, Eversole and Lin [1991]). The SRS spectrum of substances with a broad Raman gain profile (e.g., water 400cm -1) consists of a series of equally spaced MDRs (Snow, Qian and Chang [1985b]). For microdroplets with a ~ 35~tm the mode spacing of the high-Q MDRs is 0.6cm -1. Comparison of observed SRS peaks with MDRs predicted by Mie theory indicates that the SRS MDRs have the same mode order g and consecutive mode numbers n (Lin, Eversole and Campillo [1990]). In droplets, pump depletion is not as significant as in bulk cells because of incomplete spatial overlap between pump and SRS waves. Hence, it is possible for the lower gain Raman modes and overtone/combination vibrational modes also to achieve threshold. Measured SRS thresholds of binary mixture droplets are lower than the threshold in optical cells, and the minimum detectable concentration (by SRS) of the minority species in droplets is lower than in bulk (Biswas, Armstrong and Pinnick [ 1990]). The SRS emission from droplets emerges uniformly along the droplet rim, in contrast to the SRS emission from an optical cell, which is only in the forward and backward directions. Comparison of the angular intensity distribution between SRS and nonresonant elastic light scattering from droplets showed that the SRS distribution was more isotropic (Pinnick, Biswas, Armstrong, Latifi,
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NONLINEAR OPTICS IN MICROSPHERES
[1, w 6
Creegan, Srivastava and Fernandez [1988]). Far-field measurements of the angular distribution in the equatorial plane of both elastic scattering and SRS indicate that the SRS exhibits a standing wave pattern, with ISRS o~ cos2(mr + Pm), which results from two counterpropagating waves having approximately equal amplitudes and an abitrary phase Pm (Chen, Acker, Chang and Hill [1991]). Surprisingly, the phase factor is stationary throughout the entire temporal growth and decay of the SRS pulse. The formation of a standing wave from forward and backward SRS waves implies, unsurprisingly, that the gain for both directions is similar. An efficient way to lower the SRS threshold intensity further is to tune the pump laser to an input resonance (Biswas, Latifi, Armstrong and Pinnick [ 1989], Lin, Eversole and Campillo [1990]). For the 'double-resonance' case, good spatial overlap exists between the MDR at the Raman wavelength and the MDR of the pump, and the Raman photons experience gain along the entire droplet rim. Continuous-wave SRS in CS2 microdroplets under this double-resonance condition was demonstrated by Lin, Eversole and Campillo [ 1992b] and Lin and Campillo [ 1994]. At present, a complete theoretical description of SRS within microspheres does not exist. However, an heuristic model for the growth and coupling of multi order Stokes SRS in droplets was presented by Serpengiizel, Chen, Chang and Hsieh [1992]. Multiorder SRS (Qian and Chang [1986]) occurs if the firstorder SRS (at O)ls -- O)L - - O ) v i b ) is sufficiently intense to serve as a pump source for another first-order SRS, thereby generating cascade second-order SRS (at O)2s -- ( / ) I s - (Dvib -- O ) L - 2tOvib). Because of the excellent spatial overlap among the various order SRS (i.e., the double-resonance effect), up to twentieth-order SRS has been observed (with a frequency at tOe0s = tOE-20t-Ovib) in CC14 droplets (Leach, Chang, Acker and Hill [1993]), with higher order SRS observation limited only by the red-sensitivity response of the detector. Multiorder SRS also may be generated by parametric mixing processes, which entail phasematching of the waves. The phase-matching concept had to be developed for spatial distributions among various MDRs (Serpeng'tizel, Chen, Chang and Hsieh [1992]). If the edge of the droplet is illuminated by a focused laser beam, the SRS appears to be supported by low-order (high-Q) MDRs (Chen, Acker, Chang and Hill [ 1991 ]). Similar results were obtained by the theoretical calculations of Barton, Alexander and Schaub [ 1989] and Khaled, Hill, Barber and Chowdhury [ 1992]. Theory suggests that the laser light tends to couple into low-order MDRs, even if the center of the focused Gaussian beam is outside the droplet and the laser wavelength is hundreds of linewidths off resonance.
1, w 6]
NONLINEAROPTICALPROCESSES
(a)
73
Laser ,
,
SRS
SBS
~
_ 5 nsec~.--
Time
5 nsec}-..--
Time
,, l 'er
I SBS
, _._.~
.....
5 nser ~,--
Time
Fig. 22. Single-shot time profiles of the input beam, SRS, and SBS. All three signals are simultaneously measured from the ethanol droplets (with a ~ 45 #m) that are illuminated with 1 GW/cm 2 (Zhang, Chen and Chang [1990]).
The SRS threshold also depends on the linewidth of the pump laser (Zhang, Chen and Chang [1990]). A multimode Q-switched laser (with a linewidth of 0.4 cm -1) pumps SRS directly, whereas a single-mode Q-switched laser (with a linewidth of 0.006 cm -1) pumps backward-stimulated Brillouin scattering (SBS), which then readily pumps the SRS because the double-resonance condition is satisfied. Simultaneous time-resolved measurements of SRS, SBS, and the laser signal indicates that the decay of SBS is correlated with the growth of SRS (fig. 22). The SRS threshold for multimode pumping (suppresses SBS) is three times higher than that for single-mode pumping (generates SBS). Several loss mechanisms limit the generation of SRS. They affect the total quality factor Qtotal of an MDR, which is given nearly identically to eq. (5.2): 1
0tota,
1
1
1
1
1
+ 0dep,'
(6.1)
with radiative leakage quality factor Q0, the internal absorption factor Qo, the
74
NONLINEAROPTICSINMICROSPHERES
[1, w6
scattering loss factor Q/~, surface or shape-distortion perturbation loss factor Qy and, a new term, the nonlinear optical depletion loss factor QdeplSRS requires Qtotal to be greater than 106-107 (Zhang, Leach and Chang [1988]). Internal scattering losses (Q~) caused by nanometer-sized latex particles leads to suppression of SRS (Xie, Ruekgauer, Armstrong and Pinnick [1993]). Because feedback needed for the SRS process is supported by high-Q MDRs, SRS suppression occurs at concentrations of the nanoparticles which are much lower than the concentration necessary for the suppression of elastic scattering peaks associated with much lower Q MDRs (~ 104-105) (Ngo and Pinnick [1992]). Perturbation losses Qy include laser-induced shape distortions and thermally induced surface tipples. However, an optimum distortion shape and amplitude exist for which input coupling of the external pump radiation is greatly increased while the roundtrip loss due to the localized shape distortion Qy does not spoil Qtotal significantly. A tightly focused, Q-switched mode-locked envelope of pump pulses directed at the droplet rim causes an accumulative localized shape distortion, which leads to an increased input-coupling rate (Cheung, Hartings and Chang [1995]). The enhanced input coupling results in a lowered SRS threshold for successive pump pulses because Qtotal of the supporting MDR is not significantly degraded by the localized distortion (Hartings, Pu, Cheung and Chang [ 1997]). Intensity depletion loss Qdepl resulting from pumping other nonlinear processes has been observed from temporally resolved SRS spectra (Hsieh, Zheng and Chang [1988], Pinnick, Biswas, Ch2~lek, Armstrong, Latifi, Creegan, Srivastava, Jarzembski and Fernandez [1988]). Both groups demonstrated that the growth of the second-order SRS is correlated with the decay of l st-order SRS (fig. 23). More generally, the growth of nth-order SRS is correlated with the decay of ( n - 1)th-order SRS. The SRS intensity depletion rate rSRlS(determined from the temporal measurements of the SRS) increases as the pump laser intensity is raised (Zhang, Leach and Chang [1988], Zhang, Hsieh, Chen and Chang [ 1989]). For a microdroplet the heuristic model for SRS of a single Raman-active vibrational mode qualitatively accounts for the measured time delay, growth and decay among first-, second-, and third-order SRS of nitrate ions in water (Serpengtizel, Chen, Chang and Hsieh [1992]). The model decomposes the standing wave of the MDR into two counterpropagating waves, circulating around the droplet equator. Another model that successfully describes multimode SRS in microdroplets treats the SRS field as a standing wave (Fields [1997]). The multimode model predicts that the intensity of first-order SRS from each
1, w 6]
NONLINEAR OPTICALPROCESSES .
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Fig. 23. Temporal profiles of the following experimentally observed pulses for aqueous NH4N03 solution: (a) input laser 1input(t), (b) first-order Stokes SRS Ils(t), (c) second-order Stokes SRS I2s(t), and (d) third-order Stokes SRS I3s(t). The multiorder Stokes SRS is associated with the Vl symmetrical vibrational mode of the nitrate ions; the time resolution is 0.4 ns (Hsieh, Zheng and Chang [1988], Serpengiizel, Chen, Chang and Hsieh [1992]).
mode is strongly affected by the generation of 'mixed' higher order SRS modes of both the strong- and weak-Raman-gain vibrational modes. Apart from pumping higher order SRS, intense SRS gives rise to third-order sum frequency generation, third harmonic generation, stimulated anti-Stokes Raman scattering, and stimulated low-frequency emission from anisotropic molecules (Hill and Chang [ 1995]). Stimulated Rayleigh wing scattering as well as four-wave parametric oscillation generated from interacting SRS waves were also observed by Lin and Campillo [1994]. 6.2.2. Enhancing SRS In an optical cell the SRS spectra are dominated by the vibrational modes with the strongest spontaneous intensity per unit frequency (i.e., Raman gain). The weaker gain modes cannot reach threshold because of pump depletion to the strongest gain modes. Because of incomplete spatial overlap between the SRS-supporting MDRs and the pump field, pump depletion effects in microdroplets are less significant than in optical cells. As the input laser intensity is increased, the growth of the Raman intensity can be divided into four regions: (1) low input intensity for which the Raman intensity grows linearly (spontaneous
76
NONLINEAROPTICSIN MICROSPHERES
[1, w6
Raman scattering); (2) moderate pump intensity for which, if the roundtrip loss is smaller than the roundtrip gain, SRS occurs and grows exponentially with the laser intensity (only in this region can the species concentration be related to the SRS intensity), (3) high pump intensities for which saturation of the SRS occurs, and (4) very high input intensities at which laser-induced breakdown (LIB) of the microdroplet occurs (Eickmans, Hsieh and Chang [1987], Hsieh, Zheng, Wood, Chu and Chang [1987], Biswas, Armstrong and Pinnick [1990], Chen, Mazumder, Chang, Swindal and Acker [ 1996]). These complications make SRS a more useful diagnostic for species identification (more of yes and no) and less useful for species concentration determination. In neat and multicomponent microdroplets, SRS is observable from both strong- and weak-gain Raman modes. Nevertheless, a minimum concentration is necessary for detectability of the weak-gain modes (typically from the minority species) (Biswas, Armstrong and Pinnick [1990]). In these investigations of the threshold behavior of SRS in binary-mixture microdroplets, the minimum concentration necessary for threshold of first-order SRS of the minority species was determined. The SRS of the weak-gain modes when pumped by the firstorder SRS of the strongest gain mode had the same concentration threshold as first-order SRS. However, SRS of the minority species is pumped more efficiently by the SRS of the majority species than by the pump beam (Mazumder, Schaschek, Chang and Gillespie [1995]). To illustrate this point, SRS spectra from ethanol droplets doped with 6% toluene are shown in fig. 24. SRS from the 2928 cm -~ C-H stretching mode of ethanol pumped by the input laser pulse (denoted 1E3 in fig. 24a) is intense because ethanol is the majority species. The toluene 1003 cm -l ring-breathing mode SRS generated by the input laser (denoted 1T 1 in fig. 24a) is weak and hardly distinguished from the background. However, the same toluene mode internally pumped by the 1E3 SRS mode is easily distinguished from the background (the combination mode is denoted 1E3+ 1T l in fig. 24b). Thus, minority species detection is facilitated by observing multiorder SRS lines. The efficiency of the internal SRS pumping was also confirmed in CC14 droplets, where the first-order SRS of the vl-mode of CC14 served as a more efficient pump for the SRS of the v4-mode (at o~ls - co~4) than the incident laser as a pump for the first-order SRS of the v4-mode (at a,'L -- tO~4) (Leach, Chang, Acker and Hill [ 1993]). The multimode SRS model is consistent with these observations (Fields [ 1997]). To use SRS as an optical identifier for microdroplet composition, it is desirable to enhance the detectability of the minority species. Several approaches that enhance SRS signals and reduce the minimum detectable concentration of
1, w 6]
NONLINEAROPTICALPROCESSES (a)
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6000
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.
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Stokes shift [cm"1] Fig. 24. SRS spectra of an ethanol-toluene mixture microdroplet. The droplet radius is a = 37.4 #m, and the laser wavelength is 486 nm. The concentration of toluene is 6% (Mazumder, Schaschek, Chang and Gillespie [1995]).
the minority species in a multicomponent microdroplet have been reviewed (Mazumder, Fields, Hartings, Pu, Kwok, Schaschek and Chang [1996]). Internal and external seeding at ( c o l - Wvib) can selectively enhance the SRS signal of the weak-gain mode. If the seed field intensity is greater than the spontaneous Raman scattering, the SRS builds predominantely from the seed photons and the observed SRS is the amplified seed. Internal seeding is achieved by addition of fluorescent dye molecules to the droplet, which fluoresce at the Raman-shifted frequency of the weak-gain mode. In this way the SRS signal can
78
NONLINEAROPTICSINMICROSPHERES
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be selectively enhanced. The dye additive has two effects. First, the fluorescence can efficiently seed the SRS because spectral and spatial overlap occurs between the MDR of the pumped fluorescence and the MDR of the SRS that is seeded. Second, 'extra' gain provided by inverted dye population may add to the Raman gain (Kwok and Chang [1992]). These two Raman-related effects also result in a partial suppression of lasing at high input intensities (Kwok and Chang [ 1993a]). Internal seeding of the minority species SRS can also be provided by SRS from the majority species (Pasternack, Fleming and Owrutsky [1996]). In these experiments one laser pumps SRS of the majority species (this SRS is the seed) and a second laser is tuned to generate SRS of the minority species at the same wavelength as a majority species SRS. By taking advantage of the preresonance and resonance Raman effects of the minority species, the strongest gain Raman mode of the minority species can also be selectively enhanced (Kwok and Chang [ 1993b]). Stimulated resonance Raman scattering of a rhodamine 6G mode from microdroplets with a low concentration of rhodamine 6G has been observed. However, the accompanying absorption band associated with the resonance Raman signal also degrades the Q values of the MDR. With external injection seeding at the Raman-shifted wavelength of the minority species, selective enhancement of the weak-gain mode SRS was observed by Fields, Popp and Chang [ 1996]. To prevent the elastic scattering of the external seed beam from overwhelming the SRS detection, two implementations were used. First, by using a broadband seed laser (~ 2 nm FWHM), the elastically scattered seed light is spectrally discriminated from the narrow-linewidth SRS. Second, by incorporating an analyzer in front of the detector that is crossed to the linear polarization of the seed laser, detector saturation is prevented. This technique has the advantage of not requiting any dye addition to the microdroplet and the weak-gain mode is selectively seeded and enhanced relative to other modes. In these experiments the enhancements of the weak-gain SRS by seeding were accompanied by a decrease in the intensity of the strong-gain SRS mode. Because pump depletion is assumed to be insignificant in microdroplets, this reduction of the strong-gain mode SRS is believed to be due to the competition among the second-order SRS modes. Recent modeling provided more evidence for this conclusion (Fields [ 1997]).
6.3. THIRD-ORDERSUM-FREQUENCYGENERATION Third-harmonic generation (THG) and, more generally, third-order sum-frequency
1, w6]
NONLINEAROPTICALPROCESSES
79
generation (TSFG) have been observed in liquid microdroplets (Acker, Leach and Chang [1989]). The nonlinear source polarization at the TSFG is pNLS(o)rSVG) = Z(3)E1(0)1)E2(0)1 )E3 (0)1),
(6.2)
where Z (3) is the third-order susceptibility, which is nonzero for centrosymmetrical media. E(0)1), E(0)2), and E(0)3) are the amplitudes of the electric fields at frequencies COl, 0)2 and 0)3, respectively. In microspheres the generating fields are at the frequency of the incident laser col and/or at frequencies of the j-order Stokes SRS 0)is. Energy conservation requires that 0)rsv~ = +0)1 4- (2)2 4- 0)3. Phase-matching conditions require that the resultant wave-vector for TSFG is kTSVG -- kl + k2 + k3. Acker, Leach and Chang [1989] reported the observation of many discrete peaks throughout the visible range, starting from the blue at the frequency 3a,L and extending to the red at frequencies at Orsv~(p) -- 30)L- P0)vib (where the frequency 0)rib corresponds to the Raman shift of the vibrational mode with the largest Raman gain, and p is an integer) in spectra taken from CC14 microdroplets pumped by a laser at the frequency COL.The peak at 3ok corresponds to THG of a~. The peaks at 0)rsv~ result from the generation of various TSFG frequencies of 0)c and SRS frequencies at 0)L- q0)vib, where q is an integer. In CC14 droplets a large number of TSFG peaks (up to p = 17) can be observed because the Raman shift of CC14 is small ( 0 ) v i b - - 459 cm -1) (Leach, Chang, Acker and Hill [1993]). At each droplet radius, however, only a few of the many possible TSFG peaks were detected. To simplify the TSFG spectra, experiments using D20 droplets were performed (Acker, Leach and Chang [1989]). Only first-order SRS is generated when D20 droplets are illuminated by an Nd:YAG laser (~i~ = 1064 nm) because the second-order SRS falls in the IR absorption band of D20 (fig. 25). By fine tuning the droplet radius, it was demonstrated that good spatial overlap of the generating wave and TSFG waves is an important but insufficient condition for the observation of TSFG in microdroplets. The TSFG intensity in droplets is much weaker than SRS, lasing, or SBS (e.g., it is four to six orders of magnitude weaker than multiorder SRS). However, the TSFG intensity per unit volume of the liquid is still orders of magnitudes greater in droplets than it is in an optical cell if the TSFG is on or near a droplet MDR. A theoretical steady-state model was developed to explain the experimental TSFG observations by Hill, Leach and Chang [1993] where the third-order polarization is generated by SRS or SRS in combination with the field at 0)L. The total TSFG power radiated (integrated over some frequency range, time interval, and solid angle) is proportional to the intensities of the generating waves, the
80
NONLINEAR OPTICS IN MICROSPHERES
"1
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t
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i
r
" ....
[1, w 6
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.
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i
. . . .
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Optical Frequency (cm"1) Fig. 25. Detected low-spectral-resolution TSFG spectra in the UV and visible range from D20 droplets (irradiatedby laser with/]'L = 1.064/tm and detected by a positive sensitive photomultiplier, Mepsicron). TSFG spectra are shown for five different droplet sizes (a)-(e), corresponding to five different droplet generator frequencies (fosc). The TSFG peaks are labeled according to their frequency shifts of pogvi b from the THG of the laser at 3COL(28195cm-1). For D20 droplets, (.ovib ,.~ 2450cm-l (Acker, Leach and Chang [1989]). third-order susceptibility X (3), the spatial overlap, and the frequency overlap KL. Analogous to phase matching of k-vectors for plane waves propagating through a medium of the length L, the phase-matching condition for a droplet is the spatial overlap over the droplet volume of the product of four complex MDR field distributions. The spatial integral of the products of four different MDRs is typically orders of magnitude smaller than the integral used in modeling lasing or SRS. Consequently, TSFG is expected to be much weaker than SRS or lasing, which is consistent with experimental results that the TSFG intensity is 10 -4 to 10 -6 times the SRS intensity (Acker, Leach and Chang [1989]). Another factor that accounts for the weakness of TSFG relative to SRS or lasing is that TSFG has an extremely narrow linewidth, which is approximated by the SRS frequency divided by the Q of the MDR that supports the SRS. These MDRs have Q values in the range of 106 to 108. Consequently only a small
1, w6]
NONLINEAROPTICALPROCESSES
81
probability exists of finding an MDR within even hundreds of linewidths of the TSFG frequency that has a good spatial overlap with the spatial distribution of the product of the three generating fields. An extension of nonlinear Mie theory was developed to calculate the angular dependence of THG (at 3 ~ ) from microdroplets illuminated with femtosecond laser pulses (at a,'L), and good agreement with experiments is achieved (Kasparian, Kr/imer, Dewitz, Vajda, Rairoux, Vezin, Boutou, Leisner, Hiibner, Wolf, W6ste and Bennemann [1997]). The most striking conclusions are that THG with femtosecond pulses is primarily radiated in the near-forward and near-backward directions, and that the intensity and angular distribution of THG does not depend strongly on the droplet size. Spectra recorded from these droplet also show a weak second harmonic peak (Kasparian, Kr~imer, Leisner, Rairoux, Boutou, Vezin and Wolf [1998]). 6.4. OTHERNONLINEARPROCESSES
6.4.1. Coherent anti-Stokes Raman scattering (CARS), stimulated anti-Stokes Raman scattering (SARS), and coherent Raman mixing (CRM) Nonlinear processes such as SRS and SBS are self-phase matched, whereas nonlinear processes such as CARS, SARS, and CRM in an extended medium require the phase mismatching among the various plane waves. As previously noted, spatial overlap of the product of the MDR waves is equivalent to phase matching with plane waves in an extended medium. CARS from microdroplets was observed by Snow, Qian and Chang [1985b] using a monochromatic pump laser beam (second harmonic output of a Q-switched Nd:YAG laser) and a broadband Stokes beam (provided by a multiwavelength dye laser). The dependence of CARS intensity on the relative input angle between the pump and the Stokes beams has been investigated for both water and ethanol droplets. When compared with the phase-matching angle required in an optical cell (with water or ethanol), that of a droplet is nearly the same except that the phasematching angle is broadened by the spherical interface. It is not understood why no MDR-related peaks were observed in the multiwavelength CARS spectra. Stimulated anti-Stokes Raman scattering (SARS) was also detected in CC14, water, and ethanol microdroplets (Leach, Chang and Acker [ 1992]). The second harmonic output of a mode-locked and Q-switched Nd:YAG laser was focused at the droplet equator. Both the pump field and the generated SRS propagate along the droplet circumference, resulting in a good spatial overlap over a long interaction length. Many higher order SARS peaks were observed from
82
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microdroplets because the low vibrational frequencies (vl at 459cm -1, v2 at 218 cm -1 and v4 at 314cm -1) allow the many orders of SARS and SRS to be covered by the spectrograph. The dependence of the SARS intensity on droplet size and the occasional lack of correlation between SRS and SARS spectra indicate that coincidence of MDRs with the output SARS frequencies may enhance a particular SARS peak. Coherent Raman mixing (CRM) was observed from water droplets illuminated by an IR beam of a Q-switched Nd'YAG laser at I~IR -- 1064 nm and the second harmonic output of a Q-switched Nd:YAG laser at /~L = 532 nm (Snow, Qian and Chang [1985b]). The CRM signal matched the SRS signal when I(tOiR) = 0. The fact that the coherent Raman intensity threshold with I(tOL) is lowered when I(tOiR) is increased indicates that the coherent Raman signal is generated with an extra gain provided by a phase-matched process among the E(tOlR), E(tOL), E(tOIR --toy), and E(tOL- toy) (Hanna, Yuratich and Cotter (1979), Snow, Qian and Chang [1985b]). The fact that spectral features and intensity of CRM are noted to be sensitive to tuning the droplet radius in order to satisfy the input resonance condition at tOE and (/)IR and to the output resonances at (o~ - toy) and (MR toy) strongly suggests the importance of spatial overlap among the MDRs in microdroplets, i.e., equivalent to phase matching in an extended medium. CC14
-
6.4.2. Stimulated Brillouin scattering (SBS)
Stimulated Brillouin scattering (SBS) is light scattering from coherent acoustic waves that are well within the Debye range, where tOAC = VskAC, vs is the sound velocity and ks is the acoustic wave vector. In microdroplets the SBS threshold is reached when the roundtrip leakage loss at the Brillouin-shifted wavelength is less than the roundtrip gain. The generation and suppression of SBS in microdroplets was reported by Zhang and Chang [1989]. The possibility of observing SBS from microdroplets was not initially obvious because, based on Lorenz-Mie theory, the averaged spectral spacing between all the MDRs for droplets with a radius of ~45 ~tm is ~ 0.5 cm-l(Hill and Benner [1986]), which is considerably wider than the maximum Brillouin shift (~ 0.02cm-l). However, the average spectral spacing of the MDRs is significantly smaller for falling droplets because of the lifting of the m-mode degeneracy ((2n+l) to m = -n .... , 0 , . . . ,n) for the slightly deformed droplet shape (Lai, Leung, Young, Barber and Hill [ 1990]). To observe SBS, ~45 ~tm radius droplets were illuminated by the second harmonic output of an injection-seeded, Q-switched Nd:YAG laser with a 10 ns pulse duration and linewidth of A ~ L ,~ 0 . 0 0 6 c m -1 (Zhang and Chang [ 1989]).
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(a) input
(b) water
(c)
8GW/cm 2
0.SGW/cm 2
CH30H
(d)
0.3GW/cm 2
CS2 (e) CS2
0.05GW/cm 2
Fig. 26. Single-laser-shot spectra of the SBS and the elastic scattering detected by an intensified linear photodiode array placed behind a Fabry-Perot interferometer with a free spectral range of 0.526 cm-1. The spectra shown are for three interferometric fringes. The single-mode input-laser beam is focused on the droplet rim, and the input intensity for each spectrum is indicated on the right. The elastic scattering spectrum (a) indicates the spectral resolution of the interferometer. The SBS spectra from water droplets (b) and from methanol droplets (c) indicate that only the Stokes component is detected (Zhang and Chang [1989]).
The SBS is distinguished from the laser scattering by use o f a Fabry-Perot interferometer. The obtained SBS spectra and/or the elastic scattering detected from various droplets (water, methanol, CS2) are shown in fig. 26. By focusing the pump beam at the droplet rim, the near-forward SBS circumnavigates with the same sense of circulation as the input light, whereas the near-backward scattered light circumnavigates with the opposite sense. The observed SBS shift with the pump beam focused at the center of the droplet is smaller than those with the pump beam focused at the edge (Zhang and Chang [1989]), because the center illumination corresponds to a scattering angle o f 90 ~ whereas edge illumination corresponds to a scattering angle o f 180 ~ the Brillouin gain being the largest at 180 ~ The measured temporal profile of the incident laser (single mode) and of the
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SBS and SRS generated in the droplets was shown in fig. 22 (Zhang, Chen and Chang [1990]). The decay of the first SBS pulse, which was pumped by a focused single-mode laser beam aimed at the droplet rim, was followed by the growth of the first SRS pulse, and the regrowth of the Q + 1)th SBS pulse occurred after the decay of thejth SRS pulse. According to the observed temporal correlation of the SBS and SRS pulses, the conclusion could be drawn that SRS within the droplet was pumped by the internal radiation of the SBS and not directly by the internal radiation of the single-mode laser pulse. Further supporting evidence for SBSpumped SRS was found when a multimode laser input was used. The incident multimode laser suppressed the generation of SBS (responding in the transient gain regime) and provided more gain for SRS. The SRS was directly pumped by the internal intensity of the multimode laser beam. The observed pump intensity threshold for SRS was 3 x higher for the multimode laser excitation (Zhang, Chen and Chang [1990]). Several treatments of the theory of SBS in microdroplets, which treated the waves as scalar (Chitanvis and Cantrell [1989]) and vector (Cantrell [1991a,b], Leung and Young [ 199 l a]) quantities, have appeared. These theories all consider the case for which the pump and SBS fields are both resonant with MDRs (the double-resonance condition). Hence, the SBS is predicted to leak tangentially from the microdroplet. Much like THG, good spatial overlap between the MDRs of the pump and SBS is important for SBS generation.
6. 4.3. Laser-induced radiation leakage At low input intensity the MDRs of a transparent spherical droplet have small amounts of light leakage (only through tunneling or diffraction loss because of the curved interface) and, thus, have high-Q values. The radiation lifetime r of the internally generated radiation of a frequency co near or on an MDR is equal to Q/co. Any inhomogeneous perturbations in the refractive index or any shape perturbation (globally or locally) can lower the Q value from that predicted by the Lorenz-Mie calculation for an ideal sphere (Hill and Benner [ 1988]). Other possible loss mechanisms are thermally induced surface tipples (Lai, Leung and Young [1990]) and unavoidable amounts of optical absorption associated with multivibrational absorption and residual absorption of dye-laser molecules. At high input intensity, depletion may result from the generation of nonlinear waves (Zhang, Leach and Chang [1988]). The inhomogeneous internal distribution of the incident field can introduce some inhomogeneously distributed perturbations. One of the perturbations is the laser-induced, intensity-dependent index of refraction change that can appear in several forms: (1) Am(co, I in)
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through the X~3) process; (2) Am(~o, AT) through the local temperature rise in the presence of some absorption; and (3) Am(~o, Ap) through the laser-induced electrostrictive effect that causes a density change Ap in the presence of high electric field gradients. Global shape distortion in highly transparent droplets, long after the laser pulse has ended, has been attributed to laser-induced electrostriction (Zhang and Chang [1988], Lai, Leung, Poon and Young [1989]). The observed temporal development of the shape distortion is remarkably well reproduced by calculations that include both the electrodynamic and hydrodynamic equations pertinent to the large spheres used in the experiment (Lai, Leung, Poon and Young [1989]). Electric field-induced quadrupolar shape distortions (one part in 105) were measured by using elastic scattering near an MDR by Arnold, Spock and Folan [ 1990]. Thermally induced droplet shape distortion by the absorption of a cw laser beam was used to determine the surface tension and viscocity of the liquid by Tzeng, Long, Chang and Barber [1985]. The laser-induced electrostrictive process can induce density gradients within the droplet volume. Subsequently, the density gradients cause Mie (elastic) scattering of the incident plane wave that is on an input MDR (Huston, Lin, Eversole and Campillo [ 1990], Lai, Leung, Ng and Young [ 1993]). During 150 ns the density gradient perturbation lowered the Q of the input MDR from 108 to 105 and increased the Mie scattering (hence, referred to as nonlinear Mie scattering). Illuminating a localized region at the droplet edge with a tightly focused beam consisting of a train of mode-locked pulses (100 ps pulses separated by 13.2 ns) also resulted in a delayed pulse-to-pulse increase of the elastically scattered light (Cheung, Hartings and Chang [ 1995]), just as in the above-mentioned nonlinear Mie experiment. However, in the latter study the Q-spoiling was attributed to small-scale electrostrictive distortion that was highly localized on the droplet surface. During the time between the successive pulses, the small distortion grows and enhances the input coupling efficiency of the externally focused input beam to the internal MDRs (Ng [ 1994]). Because of the increased input coupling, the internal intensity is increased and more light emerges from the nonperturbed parts of the droplet rim (Cheung, Hartings and Chang [ 1995]). The time-evolving rise of the measured scattered intensity mimics the time evolution of the internal intensity that is pumped by the external beam. Experimental indications show that, after passage of the intense pump beam, the laser-induced electrostrictive pressure (having compressed and rarefied sections) causes bubble formation (Bunkin and Karpov [1990]). In addition, because static gas pressure (usually N2) is applied to the vibrating orifice
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microdroplet generator, bubbles can form more readily in droplets. The gas bubbles that are produced, either by the laser beam and/or by the applied pressurized gas in the droplet reservoir, can cause additional elastic scattering of the radiation (Huston, Lin, Eversole and Campillo [1990]). Such bubbles effectively lower the Q values. At high input intensity other manifestations of the laser-induced electrostrictive mechanisms for extra light leakage from droplets can be seen. Ray optics and Lorenz-Mie calculations of the internal and surface intensity distributions showed that the input rays of the plane wave, after refraction by the spherical illuminated face, converge toward a ring on the droplet shadow face. This ring of ray concentration at the interface is referred to as the Descartes ring (Jarzembski and Srivastava [ 1989]). The calculations of the surface intensity at the Descartes ring (Jarzembski and Srivastava [ 1991 ]) and the initial experimental observation of increased SRS emerging from the Descartes ring motivated several additional experiments with droplets (Xie, Ruekgauer, Gu, Armstrong and Pinnick [ 1991 ], Xie, Ruekgauer, Gu, Armstrong, Pinnick and Pendleton [ 1991 ], Chen, Chowdhury, Chang and Hsieh [1993]) and liquid cylindrical jets (Pinnick, Fernandez, Xie, Ruekgauer, Gu and Armstrong [1992]). The observation of laser-induced leakage from the Descartes ring is significant because the emergence of internally generated light (e.g., of SRS or lasing) has never been accounted for by the standard treatment of leakage loss. Whereas the conventional tunneling or diffractive leakage is tangential to the spherical or cylindrical surface, the emission from the Descartes ring can be refractive if the internal angle is less than 0c. Figure 27 shows the contour plot of the internal intensity for an ethanol droplet with re(to) = 1.36 and a = 35.4~tm, together with a schematic of the formation of the Descartes ring. Experiments on such droplets indicate that extra light leakage does, in fact, occur at the Descartes ring (Chen, Chowdhury, Chang and Hsieh [ 1993]). However, the location of the Descartes ring and the maximum surface intensity strongly depends on the index of refraction and size of the droplet. In fig. 28, an internal-intensity contour plot for a CS2-ethanol droplet with re(to) = 1.54 and a = 25.5 ~tm is shown. The location of the surface bulges for the CS2 droplet was experimentally verified to occur near q~= 0 by observation of extra light leakage from that location (Chen, Chowdhury, Chang and Hsieh [1993]). A variety of possible mechanisms exists for extra leakage from the Descartes ring on the droplet surface. These mechanisms can be grouped according to their temporal responses, i.e., whether the perturbations are quasi-instantaneous or cumulative. Quasi-instantaneous response follows the laser pulse, and each pulse in the train of pulses is independent, without hysteresis. Cumulative perturbations
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Fig. 27. Contour plot of the internal intensity for an ethanol droplet illuminated with a nonresonant laser. The locations of the front-focal spot, back-focal spot, and maximum surface intensity (located at the Descartes angle CD) are shown. (b) The spatial distribution of the lasing or SRS field that is on MDRs is superimposed on the intensity contour plot. Possible locations are illustrated for extra elastic leakage because of surface bulge that is developing in time and for extra scatterings because of internal index-of-refraction perturbation Am(to). The amplitude of the surface bulges at r = OD is greatly exaggerated (Chen, Chowdhury, Chang and Hsieh [1993]).
d e p e n d on the time integral o f the input-laser pulse fluence, i.e., the pulse energy. Such perturbations can continue to develop even after the laser pulse has ended. A m o n g the possible m e c h a n i s m s are (1) A m ( o a , / i n ) = Ruekgauer, Gu, A r m s t r o n g and Pinnick [1991]); (2) Am(~o, A T ) =
nzlin(t) (Xie, (On/OT)AT,
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Fig. 28. Contour plot of the intemal intensity for a CS2-ethanol droplet illuminated with a nonresonant laser. The locations of the front-focal spot, back-focal, spot and maximum surface intensity (located at 0 = 0 and not at the Descartes angle 0D) are shown. (b) The spatial distribution of the lasing or SRS field that is on MDRs is superimposed on the contour plot. Possible locations are illustrated for extra leakage because of the surface bulge that is developing in time and for extra elastic scattering because of internal index refraction perturbation Am(to). The amplitude of the surface bulges at r = 0 is greatly exaggerated (Chen, Chowdhury, Chang and Hsieh [1993]).
where AT is the laser-induced temperature rise; (3) Am(to, Ap), where p is the local density change associated with the electrostrictive pressure and by the intense acoustic waves generated by the SBS field; (4) the Am caused by the plasma that is generated after laser-induced breakdown (LIB)
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(Xie, Ruekgauer, Gu, Armstrong and Pinnick [1991]); (5) localized surface bulging caused by the laser-induced electrostriction (Zhang and Chang [1988], Lai, Leung, Poon and Young [ 1989], Xie, Ruekgauer, Gu, Armstrong and Pinnick [ 1991 ], Xie, Ruekgauer, Gu, Armstrong, Pinnick and Pendleton [ 1991 ], Chen, Chowdhury, Chang and Hsieh [1993], Hartings, Pu, Cheung and Chang [1997]); (6) the lowering of the surface tension because of inhomogeneous temperature rise on the surface of an absorbing droplet, and (7) the localized acoustic waves generated by the SBS field. Future research will determine which of all these possible mechanisms is the dominant perturber.
w 7. Conclusion We have provided only a glimpse of the many facets of linear and nonlinear optics with spherical (circular) and deformed dielectric microparticles with large size parameters. While we were writing this chapter, new models, computations, and experiments appeared in journals from diverse fields. The intent of this chapter was to lay the foundation on which the reader could build his or her comprehension and appreciation of current and future articles in this burgeoning topic of dielectric microparticles and microcavities.
Acknowledgements The partial support of NSF (PHY-9612200) and ARO (DAAG55-91-1-0349) is gratefully acknowledged. Input from Dr. Seong-sik Chang was most helpful. The present address of MHF is MIT Lincoln Laboratory, 244 Wood Street, Lexington, Massachusetts, 02420-9185, and JP is Institut fiir Physikalische Chemic, Universitaet Wfirzberg, Am Hubland, 97074, Wfirzberg, Germany.
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E. WOLF, PROGRESS IN OPTICS 41 9 2000 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED
PRINCIPLES OF OPTICAL DISK DATA STORAGE
BY
JAMES CARRIERE, RAGHU NARAYAN1, WEI-HUNG YEH, CHUBING PENG, PRAMOD KHULBE, LIFENG LI, ROBERT ANDERSON2, JINHAN CHOI, MASUD MANSURIPUR Optical Sciences Center, Meinel Building, The University of Arizona, Tucson, AZ 85721, USA 1Calimetrics Corp., 815 Atlantic Ave., Suite 105, Alameda, CA 94501, USA 2Hoya Corp., Adva. Tech Lab, 3-3-1 Musashino, Akashima-Shi, Tokyo 196, Japan
97
CONTENTS
PAGE w 1.
INTRODUCTION
w 2.
W H A T IS I N A N O P T I C A L H E A D
w 3.
FOCUSING AND TRACKING
w 4.
MASTERING
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115
w 5.
SUBSTRATES . . . . . . . . . . . . . . . . . . . .
118
w 6.
MAGNETO-OPTICAL
127
w 7.
PHASE CHANGE
w 8.
DIFFRACTION
w 9.
FUTURE TRENDS IN OPTICAL DISKS AND DRIVES
REFERENCES
. . . . . . . . . . . . . . . . . . .
99
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101
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104
(MO) RECORDING . . . . . . . .
MEDIA . . . . . . . . . . . . . . .
147
FROM PERIODIC STRUCTURES . . . . .
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170 174
w I. Introduction Since the early 1940s, magnetic recording has dominated the data storage industry in the form of magnetic audio and videotapes and their respective recorders. However the advent of optical disk technology (Korpel [1978], Bouwhuis [1985]) on a commercial level in the fall of 1982 in the form of audio CDs, forever changed the face of the data storage industry. The high quality sound with little degradation over time from audio CDs far surpassed its competitors in the magnetic industry and the public demand grew, sparking advancements in the technology. The CD-ROM was introduced in 1985 as the standard for compact disk storage in the computer industry to take on the established hard and floppy disk storage systems, which dominate the industry. Today over half a billion CD audio players have been purchased by consumers and virtually every personal computer shipped contains an optical disk reader (Stinson [1997]). New uses for CD-ROMs were developed for interactive computer games and multimedia as well as the photo CD for still images, but the weakness of the CD-ROM was inherent in its name: R O M - Read Only Memory. A writeable disk was needed to reach the next level in optical data storage. Enter the CD-Recordable (CD-R) technology in 1990 for a one-time recording of information on optical disks the same size as a CD-ROM. The CD-R uses a layer of organic dye over the data area, which irreversibly changes its optical properties when the write laser pulse is applied to it. The CD-R was slow to catch on due to high prices, but a rapid decline over the last couple of years has resulted in a 200-300% increase in growth per year and the industry was on track to sell approximately 200 million CD-R drives in 1997 (Stinson [1997]). CD-R disks have now become common enough to be sold in any electronics store and the prices have dropped to a couple dollars per disk or less. The CD-R gave optical disks recording capabilities but were limited to a single recording. To truly compete with the magnetic storage industry, the CD-ReWritable (CD-RW) drive was introduced in 1997. A phase change material, which switches between crystalline and amorphous states upon application of a laser pulse, is used as the recording layer. The different reflectivities in the amorphous and crystalline states modulate the reflected laser 99
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PRINCIPLES OF OPTICAL DISK DATA STORAGE
[2, w 1
beam and create a data signal. Each disk can be written and rewritten on the order of 105 times during its lifetime at present, making it suitable for many storage applications. The overall reflectivity of the disk is unfortunately much lower than that of a CD-ROM or audio CD and as a result, only CD-RW drives and CD players that meet the new "multiread" specifications of the Optical Storage Technology Association can read these disks (Stinson [1997]). The CD-RW drive is much more expensive than a CD-R drive and the CD-RW disks themselves are more expensive than CD-R disks. Considering both disks hold the same amount of data, CD-R has a definite cost advantage for write-once applications but as the CD-R dropped in price with time, in all likelihood so will the CD-RW, to eventually replace the CD-R. The use of Magneto Optical (MO) disks is another method of solving the rewritability problem by using a magnetic field along with the laser to read and write data. Magneto Optical materials used to make these disks have the unique quality of rotating the polarization vector of the incident light. The disk's magnetization state determines the direction of rotation, so up-magnetized regions and down-magnetized regions rotate the polarization vector in opposite directions, resulting in the data signal. MO disks are currently available in both rewritable and Write Once Read Many (WORM) formats with multiple GB storage capacity. One major advantage of the MO disk is the increased lifetime of the disk over the CD-RW since it can be rewritten many millions of times compared to the 105 from the CD-RW. Unfortunately, since MO drives use a different reading/recording technique, they are unable to read the current CD technologies in existence. The MiniDisk also uses MO technology but is more compact than a standard MO disk (2.5-inch diameter). The erase/write arrangement of the MiniDisk has also been improved over a standard MO drive by modulating the magnetic field instead of the laser to record marks, allowing erasure and writing in only one pass (see w6.1.3). This reduces the cost of the MO system and conserves battery power to make it more attractive to the consumer. Currently 74 min. of digital recording can be stored on one MiniDisk, equivalent to a standard CD. The most recent optical disk technology to enter the market is the Digital Versatile Disk (DVD), which offers greater storage capacity than the CD. The individual marks on a DVD are smaller than those on a CD and are packed closer together, increasing the maximum storage capacity to 4.7 GB for a standard single-sided, single-layer DVD, which is seven times the maximum capacity of a CD-ROM. One standard DVD is capable of holding 133 min. of video complete with Dolby digital sound and room for three different language tracks. The single-sided, dual-layer and dual-sided, dual-layer versions of the DVD increase
2, w 2]
WHAT IS IN AN OPTICAL HEAD
101
the maximum capacity even further to 8.5 GB and 17 GB, respectively. DVD is available in three formats corresponding to those of CD, namely, DVD-Video, DVD-ROM, and DVD-RAM (or DVD-RW) (Usami, Inoue and Shigematsu [1997], Satoh, Ohara, Akahira and Takenaga [ 1998]). Recognizing the success of the CD, DVD has been made backward compatible with older CD technologies so a DVD drive can read a CD as well. Already the DVD-ROM drive is included in most mid- to upper-range computer systems in place of the CD-ROM drives of the past. The DVD-RAM (or DVD-RW) will eventually replace the CD-R and CD-RW drives. DVDRAM/RW also uses phase-change disks but to accommodate rewritability, the maximum capacity of the disk drops to 2.6GB per side. This is expected to increase to the standard 4.7 GB in the near future as the technology progresses. Currently DVD appears to be the definite wave of the future, and CDs will soon disappear just like the LP records of old but more importantly, so will VHS tapes become as obsolete as cassettes once the DVD-RAM/RW takes a firm hold in the industry.
w 2. What Is in an Optical Head
All of the optics in an optical disk system are contained within a small optical head which flies above the optical disk. Many books and papers such as Bouwhuis and Braat [1978], Braat [1985], and Mansuripur [1995] contain extensive descriptions of these optical heads. The main components of the optical head are a laser diode, a collimating lens, a polarizing beam splitter, a quarterwave plate, an objective lens, and the detection system. Figure 1 shows the basic format of these elements within the optical head, however the exact architecture of the optical head is different for each of the optical disk technologies. The laser diode, which can be of any standard wavelength (currently 780 nm for CD and 650nm for DVD), emits a beam that is linearly polarized in the X-direction as shown in fig. 1, and travelling in the Z-direction. The beam is diverging after leaving the laser cavity so a collimator is used to stop the divergence and collimate the beam. The beam passes through a polarizing beam splitter (PBS) which separates the incident and reflected beams. The beam then passes through a quarter-wave plate, which circularly polarizes the incident beam before being focused onto the disk by a substrate-corrected objective lens. If the substrate is ignored and the beam is focused through it, spherical aberration will result as the rays are bent towards the surface normal to focus behind the active layer, acting like a defocus in the system. To correct for this, the objective must
102
PRINCIPLES OF OPTICAL DISK DATASTORAGE
[2, w 2
Polarizing Quarter-Wave Beam Splitter Plate Laser ,* ~ - ~ 1 - i Collimator t t
X
t
Y
- - ~
Objective m
,Z
Detector System
O 9tical Disk
Fig. 1. A typical setup for an optical disk system. The X-polarized laser light is collimated by the collimator, then converted to circular polarization by the quarter-wave plate. Next, the objective focuses the light onto the disk where the data signal is read. The return beam is converted into Y-polarized light by passing through the quarter-wave plate once more so that all of the light is reflected from the polarizing beam-splitter towards the detector system.
be designed to have substrate correction to fully correct for spherical aberration and achieve the difffraction-limited spot that is necessary for a good signal. The numerical aperture of the objective lens (NA) and the wavelength of light (/l) used, determine the diameter of the diffraction-limited focused spot according to A D ~ NA"
(1)
Reducing the spot size by decreasing/l or increasing the NA of the objective allows for higher resolution and greater data density on the surface of the disk but both approaches have problems. For a long time, it has been difficult to manufacture short wavelength lasers that are small, sturdy, and inexpensive enough for use in these optical heads. Recent developments in the area of IIIV semiconductor diode lasers have created expectations of a large jump in storage capacity in the near future (Whipple [1998]). Increasing the NA of the objective is limited by the focus-error handling of the system since an increase in NA decreases the depth of focus according to: 6=
A NA 2"
(2)
Tolerances for disk tilt and inconsistencies in the thickness of the disk substrate are also limited by NA and are proportional to M(NA) 3 and ~/(NA) 4 respectively (see w5.3). As a result, the NA for objectives used in CD technology has been set to 0.45 and that for DVD to 0.6 to maximize storage capacity within tolerable limits.
2, w2]
WHATIS IN AN OPTICALHEAD
103
Photo Detector (Laser Monitor)
X
t
Y
Objective
Collimator
PBS Photo Detector 1
Differential ~
Output.......
I ~~ "'J
Optical Disk
Polarizing Beam Splitter
-~
Differential "~ I . . . .
Phase Plate
!
i
i Photo Detector 2
~ 1 Transimpedance Amplifier
Fig. 2. The optical head for an MO system is slightly different from that of the ROM or phase change. The polarizing beam splitter (PBS) has now been replaced by a leaky PBS which passes --80% and reflects the other 20% of the X-component while reflecting all of the Y-component. Interaction with the MO marks rotates the polarization vector of the incident beam, giving it a Y-component which is completely reflected by the leaky PBS. The phase plate removes any induced ellipticity from the beam. A PBS oriented at 45 ~ with respect to the incident polarization gives the optimum differential output signal.
The light reflected from the disk will be modulated by the k/4 deep pits (in the case of CD-ROM) which interfere destructively with the light reflected from the land and again passes through the objective lens which re-collimates the reflected beam. The quarter-wave plate now converts the circularly polarized reflected beam to a linearly polarized beam with the direction of polarization perpendicular to that of the incident beam (Y-direction as shown in fig. 1). This time when the beam strikes the PBS, the change in polarization ensures that none of the light is transmitted back to the laser diode to cause feedback and increase noise in the signal. After passing through the PBS, the reflected light containing the data signal is directed to the detector module. The RF data signal and the servo signals for automatic focusing and tracking are generated within this detector module. For MO detection, the required system is slightly more complicated as seen in fig. 2. The leaky polarizing beam splitter reflects a small portion of the incident
104
PRINCIPLES OF OPTICAL DISK DATA STORAGE
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beam to a photodetector used to monitor the laser power and transmits the rest to the disk. Without the quarter-wave plate, the light hitting the disk is completely polarized in the X-direction where the MO marks rotate the polarization to give it a Y-component, which is the MO signal. When the reflected light strikes the leaky PBS again, the entire Y-component and a fraction of the X-component are reflected through a phase plate (used to remove ellipticity from the beam), towards the detector module. A differential detection scheme is employed using a regular PBS at 45 ~ to the incident polarization, and two photodetectors, whose signals are amplified and sent to a differential amplifier. The result is the desired data signal. As previously mentioned, the return beam from the disk is used for more than just the data signal. Focusing and tracking can be achieved by several different methods using this return beam. The next section deals with the various methods available to produce the focusing and tracking servo signals.
w 3. Focusing and Tracking Manufacturing errors, thermal expansion of the substrate, disk tilt and eccentricity due to mounting errors cause an optical disk to wobble in and out of its intended position by + 100 ~tm in both the vertical and radial directions as it spins at several thousand rpm. According to eq. (2), the depth of focus is typically on the order of 1 ~tm for the optical disk system. The optical head must, therefore, have a mechanism to maintain the laser spot focused on the disk or within the range of depth of focus to ensure the smallest spot size for optimal recording and readout. In optical disks, the information is recorded along an area called a track. These tracks could be either concentric tings or a single spiral starting from the inner diameter of a disk and ending at the outer diameter of the disk. It is necessary for the optical head to be able to maintain the position of the focused spot on a given track to read or write information. An automatic tracking servo is required to achieve this purpose. To compensate the radial and vertical runout of the disk as it spins, the optical drive requires a closed-loop servo system to provide submicron focus and track-locking schemes. The servo system comprises optical position sensors coupled through complex feedback circuits to high-bandwidth actuators (Earman [ 1982], van Rosmalen [1985], Marchant [1990]. In practice, the objective lens is mounted in a voice coil actuator with a bandwidth of several kHz, and a feedback mechanism is used to drive the lens in both vertical and radial directions to maintain focus and tracking at all times.
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3.1. FOCUSING SCHEMES Many methods have been proposed to generate the focus error signal (FES), which serves as the feedback for the focus servo system. Cohen [1987] has thoroughly studied the various focus-error detection schemes. Most of the techniques that yield the FES use a field lens. After receiving the return beam from the objective lens, the field lens creates a secondary focused spot. Relying on the fact that the defocus of the disk correspondingly changes the secondary focused spot, it is possible to detect deviations from optimum focus on the disk by monitoring the shape, size or position of the secondary spot. A typical plot of focus-error signal versus disk defocus is illustrated in fig. 3. Often the FES is normalized by the total reflected signal to compensate for fluctuations in the laser power. Several criteria, such as defocus sensitivity, acquisition range, focus offset and linearity in the FES curve, are used to evaluate the efficiency of a given focus-servo technique. The sensitivity to defocus is related to the slope of the curve in the lock-on range, and the acquisition range represents the maximum distance from focus in which the system will be able to move in the correct direction and establish a lock-on focus. The focus offset is mainly due to system misalignment, wave aberrations, or residual aberration of the optical system. Another cause for concern in all practical schemes for focus-error and trackerror detection is crosstalk between these two servo channels. The servo signals FES
Offset y
Diskdefocus
Lock-onrange I
!
I
Acquisitionrange
!
Fig. 3. The typical focus error signal (FES) versus the disk defocus. Several criteria are used to evaluate a focus-servo technique, such as defocus sensitivity, acquisition range, focus offset, and linearity in the FES curve.
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for focusing and tracking are derived from the complex amplitude distribution in the exit pupil of the optical system. Hence, the presence of tracks that gives rise to the tracking signal will also affect the FES. This optical servo crosstalk, also known as feedthrough or pattern noise, is caused by the fluctuation of the optical intensity in the exit pupil of the system. These fluctuations may be caused by diffraction from the tracks in a pregrooved disk, aberrations, or substrate birefringence (see, e.g., Latta, Strand and Zavislan [1992], Bernacki and Mansuripur [1994]. Feedthrough, which creates a false FES, will pose a problem at the beginning and at the end of a seek process because of the limited bandwidth of the focus servo. Next, various optical techniques for sensing focus errors are reviewed and the corresponding optical systems are shown in fig. 4.
3. I.I. Pupil obscuration method This method closely resembles the Foucault knife-edge test used in astronomical applications. The obscuration technique positions an obstructing edge on the returning light path and places a split-detector exactly in focus (Braat [1985], Marchant [1990]. Figure 4a shows the optical layout for pupil obscuration detection. The detection scheme is called the knife-edge method if the obstructing edge is placed at the secondary focus and the split-detector is placed behind the obstructing edge (Bouwhuis and Braat [ 1983]). Geometrical optics illustrates that the normalized difference signal ( S l - S2)/(S1 + $2), of the split-detector results in an appropriate FES. If the disk is perfectly in focus, the illumination of the return beam on each of the detector halves is equal and the FES is zero. Any defocus of the disk, however, will shift the return beam entirely onto one or the other of the detector halves and results in a bipolar feedback signal. High sensitivity to disk defocus, long acquisition range, and less sensitivity to feedthrough are the advantages of using this difffraction-limited, focus-error detection technique; however, a critical alignment of the detector is demanded.
3.1.2. Bi-prism method Illustrated in fig. 4b, a bi-prism acts as a dual knife-edge (phase obscuration), focusing the light from each half of the aperture onto a separate split-detector (Braat [1985]). In contrast to the pupil obscuration detection scheme, the biprism technique allows the prism to be coarsely positioned at the aperture; however, it still requires the split-detector to be precisely positioned. Because it uses all the reflected light, the bi-prism technique is also more efficient compared to the obscuration method.
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Fig. 4. By the use of a field lens after the objective lens, the reflection from the disk forms a focused spot, whose size, shape, and location depend on the focus status of the disk. This provides feedback information for the focus-error detection schemes. Shown in the following figures is the reflection beam from the objective lens, the field lens (spherical lens, astigmatic lens, or ring-toric lens), and the detector set for various focus-error detection schemes.
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3.1.3. Astigmatic lens method
Devised by Bricot, Lehureau, Puech and Le Carvennec [1976], the astigmatic lens method is the most popular technique used in current optical drives. Shown in fig. 4c, the astigmatic sensor uses an astigmatic lens having two different focal lengths along orthogonal axes and a quad-detector positioned at the location of the circle of least confusion, the point intermediate between these two focal planes. Depending on whether the return beam is collimated (when the disk is in focus) or not (when the disk is out of focus), the astigmatic lens will form a circularly symmetric spot or an elongated line focus on the detector. The quad-detector oriented at 45 ~ to the image of the line-foci senses the change of the spot shape and yields the bipolar FES according to ((S1 + 5 3 ) - ($2 + 54))/(S1 + 82 + 53 + $4).
Cohen, Gee, Ludeke and Lewkowicz [1984] and Devore [1988] studied the effect of various system parameters on the FES for astigmatic lens method. Mansuripur [1987] and Bernacki and Mansuripur [1994] investigated the astigmatic system and the effect of feedthrough by the use of quasivector diffraction theory. Compared to other focus-error detection schemes, the astigmatic method requires many more adjustments and design considerations; however, it offers a compromise between sensitivity, acquisition range, and focus offset. Unfortunately, the astigmatic method operates in the regime of geometrical optics and is vulnerable to feedthrough.
3.1.4. Wax-wane method
Proposed by Arai, Hamada and Ogawa [1985], it is a simple system in which only a quad-detector is required. Figure 4d shows the optical system using the wax-wane focus-error detection scheme. The detector is positioned slightly out of focus and away from the optical axis. As the disk moves through the focal point, the spot projected on the detector will change its size correspondingly. By sensing the change of the spot size, the wax-wane technique detects the bipolar FES according to (a(S1 + $2)- ($3 + S4))/(S1 + $2 + $3 + $4), where a is an adjustable factor that makes the FES equal to zero when the disk is in focus. The wax-wane detection shows a large acquisition range but less sensitivity to disk defocus and less immunity to feedthrough. As investigated by Wang and Milster [ 1993], the insertion of an additional detector on the opposite side of the focal plane of the field lens improves the performance on the defocus sensitivity, acquisition range, and feedthrough.
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3.1.5. Ring-toric lens method Originated from McLeod [1954] and Goodell [1969], the ring-toric lens was presented by Mansuripur and Pons [1988] for using in the optical disk system. Figure 4e shows the optical layout of the focus error detection scheme implementing the ring-toric lens and a phi detector. The ring-toric method is a generalization of the bi-prism method; it is designed to bring a collimated beam to focus on a sharp, uniform ring. When the return beam is convergent, the ring-toric lens directs the rays towards the interior of the ring. A divergent beam, on the other hand, illuminates the exterior of the ring. Therefore, the bipolar FES is generated from the phi-detector according to ((S1 + S 2 ) - ( 8 3 + 84))/(S1 + $2 + $3 + $4).
The ring-toric lens shows superior performance over other methods of focuserror detection (Bernacki and Mansuripur [1992], Zambuto, Gerber, Erwin and Mansuripur [ 1994]). The circular symmetry of the ring-toric lens also simplifies alignment procedures. The ring-toric lens utilizes all the available focus-error information carried by the various rays of the beam and, consequently, is more sensitive to focus errors. The slope of the ring-toric FES is diffraction limited, whereas in most other methods (e.g., astigmatic method and wax-wane method) the slope is primarily limited by geometrical considerations. Placing the detector at the focal plane of the field lens also helps to reduce the feedthrough disturbances. Other approaches to the focus-error detection have been proposed including critical angle method (Ito, Musha and Kato [1983], defocusing fringes method (Braat and Bouwhuis [1978]), and spot-size detection (Elliott and Mickelson [ 1979], Yamamoto, Watabe and Ukita [ 1986]).
3.2. TRACKING SCHEMES
The feedback signal for controlling the position of the objective lens in order to maintain the position of the laser spot centered on any given track is provided by the interaction of the optical beam and the disk structure. Several auto-tracking schemes have been proposed to generate the tracking-error signal (TES). Devore [ 1988] and Marchant [ 1990] have provided thorough studies on various trackingerror detection schemes. Murakami [ 1984] characterized the performance of the tracking servo as a function of groove shape. Figure 5 illustrates the typical tracking-error signal versus the off-track distance. In this section, various optical techniques for sensing errors in the tracking position are reviewed.
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Track
i
I
'
I
I
TES
Fig. 5. The typical tracking error signal (TES) follows the grating profile when the focused spot crosses the track. At every center of the land or the groove, the TES is zero; any deviation from the center will result in the feedback signal.
3.2.1. Push-pull method The push-pull tracking mechanism (Bricot, Lehureau, Puech and Le Carvennec [1976]) relies on the presence of either grooves or a trackful of data on the medium. For a CD and CD-ROM, the prestamped marks along the track comprise a sort of discontinuous groove structure; however, the discontinuity is irrelevant to the operation of the tracking servo because it is at a much higher frequency than the tracking servo is designed to follow. For writable media such as CD-R/RW, MO, and DVD-RAM/RW, a tracking mechanism that is independent of the data pattern is required. In rewritable optical media, continuous land and groove areas are a popular form of preexisting tracks. Reflected from the periodic groove structure, multiple diffraction orders overlap with each other and form a baseball pattern at the exit pupil of the objective lens. The concept of diffraction from the grooved structure and the formation of the baseball pattern will be further reviewed in w8.1. Figure 6 demonstrates the basic principle of the push-pull method. If the focused beam is centered on a track, diffraction of light from adjacent grooves will be symmetric, and the image of the baseball pattern is then symmetric. On the other hand, if the focused spot moves away from the center of the track, an asymmetry develops in the diffracted light and consequently in the baseball pattern. A split-detector placed in the return path senses the asymmetry and results in a TES according to (S1-S2)/(S1 +$2). The groove depth is designed to be M8 to obtain the maximum TES (Marchant [1990]). However, compromise should be made if prestamped pits are used for tracking (M4 is the optimum depth of a pit) or if inter-track crosstalk is considered (Goodman
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FOCUSINGANDTRACKING
l
11
Fig. 6. Schematic diagram of the optical setup for the push-pull tracking detection scheme. Because of the interaction of the focused spot and the periodic structure of the disk, the multiple diffraction beams form the baseball pattem at the exit pupil of the objective lens. The split-detector senses the variation of the intensity distribution and generates the TES according to (S1 - $2)/($1 + $2). and Mansuripur [ 1996a] showed that Jl/6 is the optimum groove depth for landgroove recording). Devore [1986] used scalar diffraction theory, and Mansuripur [1987] used a quasi-vector diffraction program to model the TES for push-pull detection. Gerber and Mansuripur [1995] characterized the push-pull tracking signal for different optical disk, and studied the dependence of the tracking performance on the polarization state of the incident beam. Bartlett, Kay and Mansuripur [ 1997] analyzed the effects of disk tilt and lens tilt on push-pull tracking signal. High sensitivity to tracking error and its simplicity make the push-pull technique very attractive to the optical disk system. However, there are drawbacks behind the use of push-pull technique, for example, the region of linear response is narrow. Also, since this technique is based on the detection of the far-field pattern of the return beam, it is much more vulnerable to false TES created by defects or dirt on the disk or on the optical elements, substrate tilt, system misalignment, wave aberrations, or movement of the objective lens.
3.2.2. Three-beam method The three-beam method of tracking (Bouwhuis and Burgstede [ 1973]) has been extremely popular in read-only media, e.g., CD and CD-ROM, where continuous grooves are not present. Figure 7 shows the optical system for the three-beam detection scheme. By means of a diffraction grating in the incident path, a laser beam is divided into three beams, one of which forms the main spot and tries
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PRINCIPLES OF OPTICAL DISK DATA STORAGE
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Fig. 7. Schematic diagram of the optical setup for the three-beam tracking detection scheme. For simplicity, the two auxiliary beams are only shown in the process of focusing onto the disk. The TES is generated by comparing the signals between the auxiliarybeams. The center spot is used to read the data marks and to provide the focus-error signal. to follow the desired track while the other two are slightly off-axis and form auxiliary spots on either side of the main spot. Correspondingly, three detectors are then used to collect the reflected light from these three beams. Any movement of the central track away from its desired position will cause an imbalance in the signal level between the auxiliary detectors. A comparison of the two auxiliary signals provides sufficient information for the track-following servo. The three-beam technique is simple and relatively immune to disk tilt, lens shift, and disk defocus. However, because the auxiliary spots are only used to sense the tracking error, the laser power efficiency is low. The diversion of the laser energy into auxiliary spots imposes a serious limitation on the available power for writing marks.
3.2.3. Sample-seroo method Derived by Hazel and LaBudde [1982], the sample-servo technique appears mostly in WORM and phase change disks. In sampled-servo media as shown in fig. 8, a pair of wobble marks, either embossed into the disk during manufacturing or written during formatting procedures, defines the boundary of a given track. These discrete pairs of marks are placed on the media at regular intervals to maintain a lock on the track. Since these marks are slightly offset from the track center in opposite directions, the reflected light first indicates the arrival of one and then the other of these wobble marks. Depending on the position of the spot on the track, one of these two pulses of reflected light may be stronger than the other, thus indicating the direction of track error. The sample-servo tracking scheme is very simple to implement and does not require a pregrooved disk, thus tracking crosstalk can be avoided. However, it
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Fig. 8. Schematic diagram of the optical setup for the sample-servo tracking detection scheme. Since the pair of wobble marks is slightly offset from the track center in opposite directions, any deviation of the focused spot from the track center will result in a different signal level for each passing of the wobble mark. The TES is then generated by sensing the difference in the signal level between these two signals.
is much more sensitive to media defects and also uses precious media space for the overhead that can be used for data storage.
3.2.4. Differential phase-detection (DPD) method DPD tracking is currently used in the DVD system. The DPD signal relies on the diffraction of an off-track focused spot from the edges of the marks. As the focused spot crosses a mark edge, an asymmetric intensity distribution is generated at the exit pupil of the objective lens. The pattern of light intensity at the exit pupil rotates as the off-track spot moves from the leading to the trailing edge of the mark. Figure 9 illustrates the rotation of the intensity pattern at the exit pupil of the objective lens as the off-centered focused spot scans along a data mark. A quad-detector is used to capture the intensity pattern. Comparing signal (S1 + $3) and signal ($2 + $4), the two pairs of detectors receive equal amount of light when the focused spot is in the middle of the mark. At the leading edge less light reaches ($1 + $3) than ($2 + $4); at the trailing edge less light reaches ($2 + $4) than (S1 + $3). Therefore, the signal (S1 + $3) is not in
Fig. 9. In differential phase-detection (DPD) tracking scheme, a quad-detector is used to sense the rotation of the intensity pattern as an off-centered focused spot scanning along a data mark.
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Fig. 10. The corresponding response-distance delay between signal (Sl + S3) and signal ($2 + S4) when the focused spot scans over a mark. If the focused spot is off-centeredfrom the mark, the delay is generated as shown in (a) and (c). However, the delay is zero if the focused spot travels through the mark center as shown in (b). Converting the response-distance delay and the linear velocity of the track into a time difference generates the TES for DPD detection. The mark is 0.8 ~tm long, 0.35 ~tm wide, and M6 deep (4=0.65 ~tm). phase with signal ($2 + S4) if the focused spot deviates from the center of the track as shown in fig. 10. In terms of the delay in distance between these two signals, A~, and the linear velocity of the track, V, the bipolar tracking error signal is then decided by sensing the time difference A~/V. Peng, Yeh and Mansuripur [1998] studied the impacts of mark geometry, intertrack and intratrack crosstalk, system aberrations, and reflectance difference between marks and space on the DPD signal. Jeong, Kim and Kim [1997] analyzed the signal offset caused by optical asymmetry. The DPD signal is sensitive to the size and the shape of the mark, disk defocus and intertrack and intratrack crosstalk.
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w 4. Mastering 4.1. INTRODUCTION
Optical disks with static physical qualities such as the pits on a CD/DVD-ROM or the grooves on other disks are mass-produced by stamping out copies of an original disk called a "master". This master itself is created by a technique called "mastering" whereby the desired pattern is written into a photo-reactive material using a short wavelength (~400 nm) laser beam (Pasman [ 1985b], Wijn and Alink [1996], Block [1995]. Before raw data can be written onto a master, it must be properly encoded during pre-mastering. Pre-mastering encodes and formats the source materials by determining the sector address for each block of user data and its physical format such as header and error correction. Once the encoded data is verified and quality control is passed (Broers [1996]), mastering takes place. 4.2. MASTERING PROCESS
There are two different mastering processes, namely photo-resist and dyepolymer. The photo-resist process to make masters is based on well-known photo-lithographic techniques used in the semiconductor industry. Figure 11 shows a photo-resist mastering process. The photo-resist is spin-coated on a clean glass substrate. The Laser Beam Recorder (LBR) is then used to expose submicron-sized areas of this photo-resist layer. The exposed photo-resist is then developed which creates the pits. The glass master, now complete, is inspected and sent to electroforming. In this process, the (negative image) metal "father" is formed by plating a layer of metal such as nickel onto the glass master. The father can be used for molding in short production runs. For longer runs, a copy of the father is made by imprinting a positive "mother" from the father, then using the mother to create another negative "son" generally called a "stamper". Finally the stamper is checked for defects and signal quality before it is used for long-run replication. This photo-lithographic process provides the flexibility to obtain a large variety of groove and pit shapes. Another technique called dye-polymer or Direct Read After Write (DRAW]) (Wilkinson and Rilum [1997]) was developed in 1986 by Optical Disc Corporation (ODC). This mastering technique is based on a thermo-chemical decomposition process of polymer. This mastering process can be summarized as follows: First, the dye polymer is spin-coated onto the cleaned and polished glass substrate. A blue laser records pits directly without a developing process
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Fig. 11. Schematic diagram of a typical photoresist mastering process.
into the dye polymer, and the quality control red laser immediately provides real-time feedback to the mastering system. The glass master is then baked to eliminate residual moisture. After metallization, the final stamper is made. Since this DRAW mastering system has instant quality control and does not have a developing process, it can reduce the processing time. The major drawback of the dye polymer method is inter-symbol interference (ISI) due to heat diffusion between marks on adjacent tracks (Wijn and Alink [1996]). This ISI can damage the shape of the pits so it is difficult to obtain a wide variety of pit and groove shapes without complex write strategies.
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4.3. BASICPARAMETERSOF MASTERING
To have a good quality and high-density master, it is essential to determine the optimal mastering parameters such as pit geometry, beam spot size and light source. (Kaneda, Kubota, Yamatsu, Furuki, Kurokawa and Kashiwagi [1997], Kojima, Kitahara, Kasono, Katsumura and Wada [1997], Oka, Takeda, Kashiwagi, Yamamoto, Sakamoto, Kondo, Tatsuki and Kubota [ 1998]).The shape of the pits on an optical disk has a profound effect on the reliability during manufacture and playback. For easy replication, an ideal optical disk stamper would have no sharp cornered pits, and the pit edge slope would be as shallow as possible to make the injection molding process easier. The sharp corner and steep slope, however, cause more damage when removing the replica from the stamper. For improved playability, the pit should have a flat bottom and steep edges. Hence it is necessary to have independent control over the pit geometry parameters depth, width and angle of the edge. First, the pit depth can be controlled by the thickness of the coating, which is dependent upon the dilution of the photo-resist and the speed at which the glass substrate spins as the photo-resist is coated. Secondly, balancing the exposure intensity with the developing time can control the length, width and edge angle of a pit. One good approach in dealing with these conflicting pit geometry requirements is the deep groove method (DG) by Reactive Ion Etching (RIE) (Morita, Nishiyama and Furuta [ 1997], Morita and Nishiyama [1998]). In the case of rewritable optical recording systems such as the magneto-optical disk (MO) and the rewritable digital versatile disk (DVDRAM/RW), the deep groove method (DG) is employed to suppress the crosserase and the cross-talk, and to obtain good playability and higher storage density. The choice of the light source can increase pit density in order to satisfy today's industry specifications. Because pit density is dependent upon wavelength, the following considerations also must be taken into account (Wijn and Alink [1996]): If ~ > 400nm, standard optical components like AO-modulators, beam splitters and normal (objective) lenses can be used. A laser wavelength lower than 400 nm requires fused silica optical components. Another important parameter is the choice of objective and its numerical aperture (NA) to shrink the beam spot size. Once the disk structure has been created, ROM disks are coated with the protective substrate and packaged. Wfiteable disks on the other hand, require an additional optically active layer to store data. Two common materials used for these layers are Magneto Optical (MO) and Phase Change (PC) materials.
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w 5. Substrates 5.1. SUBSTRATEMATERIALS- REQUIREMENTS Just as paper holds the ink of handwritten words, so too must a substrate preserve the recording medium for many years. In this section, we review the attributes that make a good substrate for optical recording. Stability, overall disk flatness, roughness, optical properties, water absorption, resonant vibrational frequency, durability and practical issues such as cost, manufacturing ease, yield and groove/pit reproducibility must be considered when selecting a substrate. Traditionally, in order to protect the highly reactive rare-earth metals and other delicate film structures of the recording layer, through-substrate recording has been predominant. The optical properties of the substrate are therefore an important consideration. There are two main materials that are commonly used for substrates in optical recording: plastic and glass. 5.1.1. Plastic
The conventional substrate material of the optical recording industry is plastic. Plastic is inexpensive, durable and easily molded to reproduce even the most detailed sub-micron-sized patterns. However, plastic fails to have the high quality optical properties of amorphous glass. At higher rotational speeds plastics tend to vibrate and lack rigidity especially at elevated temperatures found inside disk drives. In the advanced storage magneto-optical specification (ASMO) the disk is thicker near the hub where it is 1.2 mm for stability and 0.6 mm near the edge, but at higher rotational speed the edges of the disk may still wobble. There are several kinds of plastics. Amorphous polyolefin (polyethylene) (APO) has low ductility so it is difficult to replicate the mold, but birefringence is low. Polymethyl methacrylate (PMMA) has a low glass transition temperature, which is good for flow, but it has high water absorption. It has an adverse effect on GdTbFe (Sato, Tsukane, Tokuhara and Toba [1985]) so an interfacial dielectric layer is needed before deposition of the magnetic layer. Polycarbonate is moderately good in all categories and remains the favorite of substrate makers. The molecular orientation in plastic substrates can influence its optical properties. Plastic is another name for polymer, which means many unit chains of monomers. The monomer units are joined together by breaking a carboncarbon double bond and inserting another monomer. As many as 1000 or more can be joined to form the long molecular chains found in plastic (polymers). Electrons are more easily polarized in the direction of the chain. This means
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119
light passing through a polymer chain will suffer from the collective motion of the electrons and the index of refraction is subsequently higher for light polarized in the direction of the chain. In the injection molding process the molten plastic is inserted into the center of the disk and flows radially outwards toward the edge. The long molecules favor lying in the radial direction. This molecular orientation is a source of birefringence in plastic substrates. Stress can also induce birefringence in substrates. Birefringence is proportional to stress and stress is proportional to the ratio of viscosity to cavity height, so it becomes harder to control birefringence in thinner substrates. Temperatureinduced stress can also cause a loss of signal contrast (Hsieh and Mansuripur [1997]). 5.1.2. Glass
Compared to plastic, glass has superior optical properties. However it is expensive and groove patterns are not easily reproduced. Formation of magnetic layers on substrates duplicated with photopolymerization is poor under humid conditions and a dielectric layer is needed between the substrate and magnetic layer similar to the case of PMMA plastic (Hartmann, Braat and Jacobs [ 1984]). The amorphous nature of glass reduces birefringence by usually more than an order of magnitude compared to plastic. 5.2. SUBSTRATEBIREFRINGENCE Substrate birefringence causes many difficulties in through-substrate optical recording such as beam astigmatism, elliptical polarization of the reflected light, mixing of TES, FES and data signals, loss of laser diode feedback isolation (Goodman and Mansuripur [1996b]) and a loss of common-mode signal rejection of the differential readout scheme. Pre-pits can tilt the index of refraction ellipsoid in the nearby data track causing noise (Sugaya and Mansuripur [ 1994]). Non-uniform birefringence from substrate stress or other in-homogeneity is another source of noise. Figure 12 shows the calculated effect of vertical birefringence on the magnetooptical (MO) signal. A larger NA objective lens and shorter wavelength reduces the output signal. The calculation includes the effects of diffraction through the objective lens and satisfies the boundary conditions imposed on the light at the surface of the substrate. In the calculation, the disk position is adjusted to obtain the smallest spot size and at high vertical birefringence, where the beam becomes astigmatic, the focus is chosen as the average between the two foci.
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PRINCIPLES OF OPTICALDISK DATASTORAGE I
-
'
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I
Birefringence 1.2 mm substrate '=--------e~_.~
NA=0.3x::633rrn
~-0
,,...
._~ -4
,
I
,
I
9
I
Vertical birefringence nt-nz (10 "4) Fig. 12. Influence of vertical birefringence on output signal level. Large NA objective and short wavelength enhances the birefringence loss of signal.
In the future, shorter wavelengths and larger NA objective lenses can be used to reduce the focused spot size. Consequently the birefringence An(A) in plastic increases, and the effect of the birefringence becomes larger through the wavelength relation of the phase difference between the polarization components
(3)
(2JrhAn) ar
Z
'
where AO is the phase difference, h is the substrate thickness, An is the birefringence, and ~, is the wavelength of light. A larger NA also increases the angle of the incident light so that the birefringence signal loss is worse. The current trend is towards thinner substrates to reduce the birefringence effect and aberrations. 5.2.1. Measurement methods
There are several methods to measure birefringence, which are shown in fig. 13. The birefringence can be determined by measuring the ellipticity imparted on a linearly polarized beam which is transmitted or reflected through the substrate (Fu, Yan and Mansuripur [ 1994]). An ellipsometer measures the ellipticity with a rotatable polarizer, a phase compensator plate and a detector (MacCrackin, Passaglia, Stromberg and Steinberg [1963]). First the ellipticity due to the in-plane birefringence Anll = n r - n t is measured with normally incident light.
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SUBSTRATES
O
Can measure at any point
X
Need ellipsometer
Calculation
A~=2nh/~, An[t
Can measure any point, same machine as I. Need ellipsometer A~=2nh/(~,cosO)x (-Anllcos20+An• sin20 )
121
Direct, simple, graphic Fourier transform gives radial variation
A simple lens setup
Can't measure single points on disk. Only Ani
Must first determine Anll using method I.
period=L/An•
sin20 = n2(An!i/An•
Fig. 13. Several methods for measuring birefringence in substrates. An[[ is determined from the ellipticity by the equation shown in column I of fig. 13. Then the incident angle of the light is made oblique so that the polarization has a z component and the vertical birefringence n • = n t - n z is determined from the previously determined Anil and ellipticity by the equation shown in column lI of fig. 13. In another method, light polarized 45 ~ with respect to the disk normal passes through the edge of the substrate. If the refractive index is different between the track and normal direction, the phase of each polarization component is shifted differently. As the light continues through the substrate, the polarization shifts from linear to elliptical, then circular, back to elliptical, then linear again and so on. Scattered dipole radiation from linear polarization perpendicular to the observation axis makes a bright stripe while that parallel to the observation axis makes a dark stripe. The period of the stripes is a measure of the vertical birefringence (Mansuripur and Hsieh [1994]). In the focusing lens method (Goodman and Mansuripur [1996b]) light bent from a lens passes through the substrate. Two dark spots appear that correspond to light passing along the two optical axes of the biaxially birefringent material, so there is no net phase retardation between n• and nil. The in-plane birefringence must be determined with another method, such as by ellipsometer, before the vertical birefringence is calculated. The orientation of the refractive index ellipsoid is quickly determined with this method by observing the direction of the line passing through the two dark spots. All methods for determining birefringence essentially involve the phase difference experienced by linearly polarized light incident between the two axes in question. In glass substrates for which the birefringence is usually low enough
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to not seriously affect the signals, a measurement of birefringence can also serve to indicate the stress remaining after the manufacturing process.
5.2.2. Lateral birefringence We now look at some measurements of birefringence in actual substrates. Inplane or lateral birefringence means the refractive index for light polarized in the radial direction (nr) is different than that for the track direction (nt). This birefringence increases the ellipticity of the reflected signal as it passes through the substrate. The signal loss increases with ellipticity. For a typical value of nil- 10-5 in a 1.2 mm thick plastic substrate (fig. 14), the phase shift is about 7 degrees at a wavelength of 650 nm or a 0.06 dB loss of MO signal. This signal loss is not as serious as the vertical birefringence signal loss. An average uniform lateral birefringence can be corrected with the addition of a phase retarder placed in the path of the reflected light. Also the incident polarization is usually chosen to lie along or perpendicular to the track (index of refraction ellipsoid) direction to minimize the perturbation of the incident light. Nonuniformity of the birefringence along the track direction can contribute to the substrate noise. Annealing reduces lateral birefringence in polycarbonate substrates. 1.4
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Light wavelength k (nm) Fig. 14. Lateral (in-plane) birefringence as a function of wavelength for polycarbonate and glass substrates.
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5.2.3. Vertical birefringence Vertical birefringence presents a more serious problem than lateral birefringence. Due to the asymmetric orientation of the polymer molecules in the substrate plane, the refractive index in the track direction (nt) is higher than that in the vertical direction (nz). A normally incident plane wave does not have a polarization component in the z direction, so the vertical birefringence has no effect on the plane wave. In the case of a focused beam, there is some nonzero z component of polarization due to the bending of the rays by the lens. The phase difference between the transmitted s and p polarization depends on the angle of incidence and the incident polarization direction. In the case that the vertical bireffingence is less than about 10-3 the phase difference between s and p polarization can be approximated by the equation in fig. 13, column II for oblique incidence. Typically polycarbonate has a vertical birefringence of about 5 • 10-4 (fig. 15). APO plastic has lower bireffingence than polycarbonate, about An• = 1.69• 10-4, and nil =2.45• -5. However, its low ductility makes reproduction of the injection mold pattern difficult. The glass bireffingence is lower by about an order of magnitude and is nearly independent of wavelength.
Fig. 15. Vertical birefringence as a function of wavelength for polycarbonate and glass substrates.
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5.3. TILT TOLERANCE One of the parameters that must be controlled to realize optical recording is substrate tilt. Under this category there are many factors that can cause the substrate to become tilted including warpage, thickness variations, hub mounting skew, vibration and shock. Disk tilt causes coma, tracking, and focusing problems (fig. 16). 5.3.1. Aberrations
As the focused light passes through the substrate, the phase of the light may become aberrated. Spherical aberration for a defined substrate thickness without tilt can be corrected by introducing a negative aberration in the objective lens. Aberration due to small deviations in the substrate thickness can sometimes be overcome by an automatic shift of the focus position by the focus servo. An actively tilted two-lens objective lens has been proposed to reduce aberrations (Nakamura, Senoh, Nagahama, Iwasa, Yamada, Matsushita, Sugimoto and Kiyoku [1997]) (Hendriks [1998]). Expressions for the aberration coefficients can be found in Braat [1997]. Disk tilt causes coma, which appears in the spot shape as a tail in the direction of tilt. The coma has a cubic dependence on the lens' NA (Braat [1997]). A typical tilt tolerance of + 5 mrad for a 1.2 mm thick disk at 650 nm wavelength through a 0.6NA objective lens produces about a quarter-wavelength of coma. A technique for correcting the tilt has been proposed by Gerber and Mansuripur [1996].
Fig. 16. Shift of spot position with substrate tilt.
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Astigmatism is one of the primary causes of mixing between the TES and FES. Astigmatism causes the wavefront to have two loci depending on the light polarization and the spot shape becomes symmetrically elongated. In the worst case, the astigmatic distortion is oriented 45 ~ to the track direction. Unlike spherical aberration, astigmatism is exacerbated by the automatic focus servo. In order to maintain a root-mean-square wavefront aberration below 0.07 ~, to satisfy the Marechal criterion, the tilt should be less than about 10~ Astigmatism also decreases the amplitude of the TES by as much as 1.5 dB for a 0.25 ~, aberration (Bernacki and Mansuripur [ 1993]). Astigmatism is mainly due to the laser, objective lens or substrate birefringence. The substrate tilt tolerance is limited more by crosstalk and jitter than by wavefront phase aberration.
5.3.2. Tracking 5.3.2.1. Intertrack crosstalk. We can understand the role of tilt on spot position and jitter by looking at the intensity distribution of light diffracting from an objective lens then passing through a tilted substrate. From the calculated intensity distribution like that shown in fig. 17 we can find the peak shift, decrease of the peak intensity and first-order diffraction peak shift. A tilt of 1~ and N A = 0 . 6 objective lens with 650nm wavelength light causes the zeroorder peak to shift by about 0.25 mm in the tilt direction. For a DVD disk
Fig. 17. Influence of substrate tilt on beam intensity distribution. With 10mrad of tilt the peak shifts about 0.25 ~tm and intensity is shifted from the main peak to the first order diffraction peak.
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this represents half of a track radial displacement, so a limit on the tilt is less than 4 mrad.
5.3.2.2. Edge shift or jitter Jitter caused by tilt in the track direction has an even larger effect than the intertrack crosstalk from radial tilt. The minimum DVD mark size is 0.41 ~tm so a zero-order peak shift below 0.04 ~tm is desirable. Jitter, or shifts in the timing of the marks, is usually the deciding factor for the maximum allowed tolerance of tilt between the substrate and objective lens. The first-order diffraction peak increases in intensity with tilt thus the step diverges from the ideal case. For a 10 mrad tilt, the ratio of first order to zero order peak intensity is about 0.18 or 15 dB and increases with shorter wavelength. The step response function develops an extra smaller step displaced from the main step. If the tilt does not change with time, then the response of the write and read filters can be tuned to account for this error. In the realistic case of nonuniform substrate thickness or wobble, the first-order peak intensity will fluctuate.
5.3.3. Focusing Substrate tilt also places demands on the focusing servo, especially in disks with larger radii because the actuator travel must be longer and the servo circuits need a higher dynamic range. For example, a 120 mm diameter DVD disk tilted by 4 mrad will wobble +240 ~tm near the outside edge of the disk. Smaller disks are easier to mount because the tilt does not affect defocus and, in addition, the vibrational resonance effect near the outer edge of the disk is reduced. 5.4. FUTURE TRENDS Much of the discussion above has been about the optical properties of substrates, especially the limitations of birefringence in the currently manufactured polycarbonate plastic. In order to circumvent this weakness of plastic with throughsubstrate recording, there is a growing interest to use air-incident (first surface) light recording. Issues such as birefringence and aberrations due to the substrate are eliminated. However, the corrosive rare-earth material must still be protected by application of some smooth overcoat to prevent oxidation. The overcoat material must be transparent and smooth. Typically sputtered SiN is used for this purpose. In the case of near-field recording, the surface roughness and flatness become important. The ideal near-field head travels only tens of nanometers above the
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surface of the disk, yet should never come into contact with it. The roughness of the surface and grooves determines the flow of air generated by the rotating disk. Roughness also contributes to the media noise, so a proper balance must be chosen to accomplish satisfactory head flying height without generating too much signal noise.
w 6. Magneto-Optical (MO) Recording 6.1. MO MEDIA
6.1.1. MO layers Amorphous Rare Earth-Transition Metal (RE-TM) alloy films are currently in use for magneto-optical data storage applications (Urner-Wille, Hansen and Witter [1980]). Continuous, uniform and smooth amorphous films of (TbyGdl-y)x (FezCOl-z)l-x (where x ~ 0.2, z ~ 0.9 and y ~ 0.9) can be deposited on large surface areas of polycarbonate substrates by sputtering from the alloy target or by co-sputtering from multiple element targets. Since the rare-earth elements are highly reactive to the atmospheric oxygen, a protective dielectric coating is required over these films. A typical disk structure scheme for magneto-optical (MO) data storage application is shown in fig. 18. The transition metal sub-network (FezCOl-z)l_x, which constitutes approximately 80% of the alloy, has a very small fraction of cobalt. Similarly, in the rare-earth sub-network (TbyGdl_y)x terbium is the main element. Due to the amorphous nature of these films, the chemical composition is not governed by stoichiometry and can be continuously varied without developing any nonuniformity or polycrystalline grains during deposition, until the desired properties for the data storage are achieved. RE-TM alloys have several interesting optical and magnetic properties, which are described in great detail by Mansuripur [ 1995] within the context of MO data storage.
Fig. 18. A typical magneto-optical disk structure.
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PRINCIPLES OF OPTICAL DISK DATA STORAGE
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MRERIITM TM[I
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The net magnetization of MO media is the difference of these two magnetizations which goes through zero at the compensation temperature.
The RE-TM alloys used in MO data storage are ferrimagnetic at room temperature. The magnetization of the TM sub-network is anti-parallel to the magnetization of the RE sub-network. The net magnetic moment is the vector sum of these two sub-network magnetizations. Figure 19 shows the temperature dependence of the magnetic moments of the RE and TM subnetworks. At very low temperatures, the RE magnetic moment is stronger than that of the TM magnetic moment. As the temperature increases, thermal disorder increasingly competes with the inter-atomic exchange interaction, thus decreasing the magnetic moments of both sub-networks. However, the magnetic moment of the RE sub-network decreases at a much faster rate than that of the TM sub-network and at some temperature, called the compensation temperature (Tcomp.), the net magnetization Ms becomes zero. Above Tcomp. the net magnetic moment is dominated by the TM sub-network. At the Curie temperature (Tc), thermal disorder overcomes the exchange coupling of magnetic dipoles and the material transforms to a magnetically disordered paramagnetic state. The composition of MO media is chosen such that Tcomp. stays near room temperature. During readout, laser light mainly interacts with the TM sub-network and gives a significant Kerr signal, even though the net magnetization is very small. This is because the magnetic electrons of Tb ions are 4f electrons, which are shielded by higher lying orbits and are therefore not accessible to the red laser beam.
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6.1.2. Magnetic anisotropy In a crystalline magnetic material there is a crystallographic direction along which magnetization is energetically preferred. This preferential axis is called the easy axis. For amorphous RE-TM thin films used in MO data storage the easy axis is perpendicular to the film surface and hence the film is said to possess perpendicular magnetic anisotropy. The magnetic anisotropy in these amorphous films is not well understood. It is generally believed that some kind of short-range atomic order and/or symmetry breaking at the film surface might be responsible for the observed preferential orientation of magnetic dipoles. This preferential magnetization is an advantage for MO data storage, as it allows magnetic dipoles of recorded marks on a storage layer to be oriented anti-parallel (up or down) with respect to the adjacent magnetic dipoles. This is an energetically favorable configuration because the dipoles in recorded domains remain in their naturally preferred direction, thus, providing high stability of recorded domains. Here it should be pointed out that the magnetic anisotropy (Ku) is a bulk parameter (the angle brackets represent spatial averaging). As stated earlier, MO films are considered to be homogeneous and uniform on a macroscopic scale. This uniformity and homogeneity, however, do not extend all the way down to the atomic scales. At spatial dimensions on the order of a few hundred nanometers, defects and spatial fluctuations in magnetic and/or structural parameters are believed to exist. These include spatial fluctuations in magnetic anisotropy about the bulk easy axis. This situation is called "random axis anisotropy", implying that the anisotropy axis of localized zones in nanometer scale are distributed randomly and independently about the bulk anisotropy axis. These random axes are oriented within a certain maximum solid angle O about the bulk easy axis, which is perpendicular to the film surface. This description of random axis anisotropy is helpful in understanding some aspects of coercivity, which is briefly described in w6.1.4. The strength of the bulk anisotropy constant may be determined by the application of an in-plane external field, say Hx along the x-axis (x-y plane being the film surface), to saturate the sample whose magnetization has been initially aligned along the z-axis. Now by monitoring the normal component of the magnetization (Ms), one may determine (Ku) from the curvature of the plot of (Ms) versus Hx (Mansuripur [1995]).
6.1.3. Thermomagnetic recording In thermomagnetic recording, a focused laser beam creates a localized hot spot and an external magnetic field reverses the direction of magnetization in that
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Fig. 20. A polarization optical micrograph of the array of magnetic domains written thermomagnetically for various combinations of magnetic field and laser pulse duration keeping the laser power fixed at 2.6 mW. The magnetic field varies as: 4.0, 3.0, 2.0, 1.0, 0.5 and 0.3 kOe from bottom to top and the pulse duration varies as: 800, 600, 400 and 200ns from left to right. The x - y dimensions are in ~tm.
localized area of micron dimensions. Erasure is similar to that of recording, except that the external magnetic field acts opposite to that of recording and brings the magnetization of the recorded mark back to its original state. Using this process the MO media can be written and erased several million times. Thermomagnetic recording is a complex process involving a nucleation of the initial domain, followed by its expansion and finally its stabilization as the heating is turned off (Giles and Mansuripur [ 1991 ], Hasegawa, Moroga, Okada, Okada and Hidaka [1991]). There is a combination of material characteristics and recording conditions that will be suitable for thermomagnetic recording. For example, if the material coercivity is low, the written domain will be unstable and could collapse before the end of the recording cycle. Similarly under a weak external field, a very short laser pulse or for a very thick MO film, the recorded domain may not be fully reverse magnetized or may produce an irregularly magnetized or noisy domain. A strong laser pulse may, however, burn the amorphous film or may locally crystallize it. The optimization of the recording process is an important aspect of media and system design. A picture of an array of recorded domains is shown in fig. 20 for various combinations of magnetic
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field and pulse duration at fixed laser power. From this picture one can see the variation in the size and contrast of written marks as the field and pulse duration are varied. One possible mechanism for erasure is that as the temperature rises and the coercivity drops, the forces of wall energy and external field overcome the forces of demagnetization and coercivity, thus forcing an inward collapse of the domain wall. Another possibility is that a new domain nucleates within the old one and expands until their walls meet. Both mechanisms may also work simultaneously. There are two schemes for thermomagnetic writing. In the first scheme known as laser power modulation (LPM), an information signal modulates the power of the write laser beam that creates a reverse magnetized domain on the MO disk as it moves under a constant magnetic field. In another method known as magnetic field modulation (MFM), laser power is kept constant, while the information signal modulates the direction of the magnetic field. The advantages of MFM are twofold: (i) direct overwrite is possible and (ii) domain wall positions are insensitive to defocus and laser power fluctuations, and can be accurately controlled by the timing of magnetic field switching. The disadvantage with MFM is that, the magnet should be small and must fly in a very close proximity to the surface of MO film. This creates a very tight mechanical tolerance requirement for flying the magnet.
6.1.4. Coercivity Thermomagnetic recording (and erasure) involves laser-assisted nucleation and growth of reverse magnetized domains under the support of an external magnetic field. Thus, the process of magnetization reversal is of considerable importance in optical data storage as successive data recording and erasure depend upon the repeated and reliable reversal of magnetization in micron-size areas on the medium. Coercivity describes the resistance of the magnetic medium to this magnetization reversal. A thin film (defined in x-y plane) that has a magnetic moment saturated in the positive z direction will need a minimum magnetic field to flip all its magnetic dipoles to the negative z direction. This minimum field is called the coercive field (Hc) or coercivity. The MO film for which the Kerr loop is shown in fig. 21 has a coercivity of +1.65 kOe. At ambient temperature MO films have coercivity on the order of several kOe. At elevated temperatures coercivity decreases, and at Curie point, the coercivity becomes zero. A typical Kerr hysteresis loop for an amorphous RE-TM thin film shown in fig. 21 shows several characteristics of MO media. The magnetization transitions from up to down (or from down to up) are very sharp. The reverse field does not affect the
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!
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~ -0.2 -0.4 -0.6
-6
. -4
.
-2
.
. 0 2 H (kOe)
4
6
Fig. 21. Polar Kerr signal versus magnetic field curve (from 6.0 ---, -2.0 kOe the magnetic domains are oriented upward giving rise to 0.4~ Kerr rotation and from -6.0 ~ 2.0kOe magnetic domains are oriented downwardgiving rise to -0.4~ Kerr rotation). The coercive field is also marked in the figure.
magnetization at which the coercive field is reached. At Hc the magnetization suddenly reverses the direction and saturates immediately. It further indicates that the remnant magnetization Mr and the saturation magnetization Ms are nearly equal (the value of ]0kl remains the same for all values of the field). Thus, once the sample has been saturated in an applied magnetic field, removing the field does not cause a reduction of the magnetic moment. This behavior is very important in MO storage, as it indicates that the recorded domain will remain fully saturated and give rise to a maximum Kerr signal during readout. Though useful in the phenomenology of bulk magnetization reversal, coercivity defined in the above manner is not a clear concept. Its relevance to the processes occurring at the spatial and temporal scales encountered during thermomagnetic recording raises many questions. First, there is a problem of distinguishing between nucleation and domain wall motion coercivities. Then there are questions regarding the speed and uniformity of domain wall motion, stability and erasability of the magnetic domain. All these issues are related to coercivity and the existing theories on magnetism are not fully capable of addressing them (Vonsovskii [ 1974], Coey and Ryan [ 1984]). The computation simulations by Giles and Mansuripur [ 1991 ] show that the field, that is required to initiate the reversal process in a truly uniform medium (i.e., the nucleation coercivity), is generally higher than that observed in practice. This indicates that voids, defects and random axis anisotropy have varying contributions to the bulk
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coercivity. It is observed that reverse magnetized seeds formed and stabilized in areas with large local anisotropy can substantially reduce the nucleation coercivity. On the other hand voids have an insignificant effect on the value of the nucleation field. Random spatial fluctuations and structural and/or magnetic defects also create barriers to domain wall motion. These barriers are only overcome if a sufficiently large magnetic field (in excess of the so-called wall motion coercivity) is applied. The wall coercivity in MO media is generally less than the corresponding nucleation coercivity. The strength of wall coercivity, of course, also depends on the type and size of spatial fluctuations and/or defects.
6.1.5. MO readout The information recorded on an MO media as described above can be read using the polar magneto-optical Kerr effect (Pershan [1967], Sprokel [1984]). When a linearly polarized light beam is normally incident on a perpendicularly magnetized medium, the plane of polarization of the reflected light undergoes a slight rotation. The polarization rotation direction is dependent on the state of magnetization of the reflecting medium. If the polarization vector rotates anti-clockwise from an up-magnetized region, then the same vector will rotate clockwise from a down-magnetized region. The MO media are characterized in terms of reflectivity R and Kerr rotation angle Ok (the angle at which the reflected light polarization is rotated with respect to polarization of incident light). Typical MO materials currently in use have about -t-0.5~ Kerr rotation angle, which provides sufficient signal-to-noise ratio for the reliable readout owing to very low media noise in these materials. As seen in fig. 21, the Kerr angle jumps between +0.4 ~ and -0.4 ~ with magnetization reversal. Thus, in MO readout, it is the sign of rotation angle that carries information about the state of magnetization of the medium, i.e., the recorded bit pattern. 6.2. METHODSOF MEDIA CHARACTERIZATION
6.2.1. Vibrating sample magnetometer Direct magnetization measurements in MO media are performed on a vibrating sample magnetometer (VSM). In the VSM a small piece of MO film is placed in between the pole pieces of an electromagnet. The sample is brought into mechanical vibration, which develops an oscillating magnetic field in the vicinity of the two pickup coils symmetrically located with respect to the sample. The e.m.f, generated on the pick-up coils is proportional to the oscillating magnetic
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field, which in turn is proportional to the magnetic moment of the sample. The constant of proportionality is established between the magnetic moment and the pick-up signal, using a standard (nickel) sample. Now the magnetization (magnetic moment per unit volume) of the sample can be traced as a function of the external magnetic field in a variety of ways. For example, one obtains a hysteresis loop when the field is perpendicular to the film surface. From the hysteresis loop, values of Hc and Ms can be evaluated. When the sample holder is rotated by 90 ~ the field is oriented along the direction of the film plane, and one measures the deviation of Ms from the easy axis, which allows determination of the constant of anisotropy (Ku). These measurements can be performed at various temperatures, yielding the plot of Hc(T), Ms(T) and (Ku)(T) for a given sample. From these results it is then possible to obtain the values of the Curie and compensation-point temperatures.
6.2.2. Magneto-optic loop tracer Another device routinely used for media characterization is the magnetooptic loop tracer. A schematic diagram of a loop tracer is shown in fig. 22a (Wolniansky, Chase, Rosenvold, Ruane and Mansuripur [1986], Hajjar, Zhou and Mansuripur [1990]). It has a large electromagnet capable of producing up to -t-20 kOe magnetic field in between the pole pieces. The rotating base of the electromagnet allows the field to be applied either parallel or perpendicular to the film surface. A linearly polarized 632.8 nm laser line from an He-Ne laser is used in normal incidence either in reflection (MO polar Kerr effect) or transmission mode (Faraday effect) to monitor the magnetization of the sample in the presence of a magnetic field. The reflected (or transmitted) laser light that carries the information on the state of magnetization in MO media is then directed to a differential detector assembly using a beam splitter. The output of the differential detector can be converted into Kerr angle, and a plot of Kerr angle (Ok) as a function of the applied magnetic field (B) can be obtained. The same differential detector can be used to measure the ellipticity of the reflected light by placing a ~,/4 plate in the path of the reflected beam. These measurements can be performed over a range of temperatures, yielding plots of 0k(T), ek(T) and Hc(T). If the field is applied in the plane of the sample, then the deviation of MTM from the easy axis as a function of the field may be used to determine (Ku). The loop tracer can also be used for Hall effect and magnetoresistance measurements. These measurements are carried out using the "four probe method" in three possible configurations as shown in fig. 22b. In perpendicular geometry the magnetic field (applied perpendicular to the film surface), direction
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3
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(b) Fig. 22. (a) Schematic diagram of MO loop tracer. The quarter-wave plate is inserted in the optical path when ellipticity measurements are desired. Inside the temperature-controlled stage, a four point probe is attached to the sample for measuring the magnetoresistance and Hall effect. (b) Three possible geometries for galvanomagnetic measurements.
of the current and the direction of the Hall voltage are all perpendicular to each other. Switching over to longitudinal geometry requires the field direction to be changed along the direction of current. The transverse geometry is the same as the longitudinal one, but for the interchange of current and voltage terminals. For measuring the exact values of electrical resistivity and Hall resistivity, one should apply van der Pauw formulas (van der Pauw [1958]) to the four-point probe data.
6.2.3. Polarized-light microscopy (static tester) Observing the magnetic domains and structure in MO films constitutes a useful approach to media characterization. Polarized-light optical microscopy has been
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Fig. 23. Schematicdiagramof modifiedpolarizationmicroscopeused for the observationof magnetic domains. used for studying the structure, dimension, jaggedness (or fractal dimension), nucleation, growth and collapse of magnetic domains in MO media (Takahashi, Niihara and Ohta [1988]). It also has been utilized in phase change media to study the crystalline marks on amorphous surfaces (or vice versa) by generating a reflectivity map of a section of surface area. The MO Kerr effect, which is used for the readout, also enables 2-D mapping of the variation in the state of magnetization in MO films. The diagram of a modified polarization microscope or "static tester", used for this purpose is shown in fig. 23. Here collimated output of a white light source is polarized and then focused on the sample using a microscope objective. The same microscope objective collects the light reflected from the sample. The change in the orientation of the polarization vector in the reflected light depends upon the local state of magnetization. This light then passes through a crossed analyzer to have the maximum image contrast and read by a TV camera, which is connected to a computer via a frame-grabber. In addition to these standard elements of a commercial microscope, the static tester contains a laser diode, an electromagnet, a piezo-driven x - y stage, a hot plate and a differential detector. The hot plate allows control of the temperature up to 120~ for high-temperature study of magnetic domains. The electromagnet applies a variable field in perpendicular up and down directions during writing, erasure, expansion and contraction of domains. The laser power can be varied for use in thermomagnetic reading, writing and scanning optical microscopy of magnetic domains. Apart from direct visualization of image in the TV monitor,
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a Kerr signal map of the section of the medium can be prepared by moving the base plate in horizontal x-y directions in very small step sizes and measuring the Kerr signal. This scheme produces high-contrast images of magnetic domain. The ultimate resolution of these images depends upon the wavelength and NA of the microscope objective. Typical images as seen through a polarization optical microscope are shown in figs. 20 and 24. The black-and-white regions correspond to up and down magnetized regions, respectively. Figure 20 shows a map of recorded marks, which were written using various combinations of magnetic field and laser pulse duration, while keeping the laser pulse power fixed at 2.6 mW. It is evident that the mark size decreases continuously with decreasing pulse duration from 800 to 200ns (left to right). The marks in the extreme fight are not fully reverse magnetized at their center. A similar behavior is observed while decreasing the field (from bottom to top), though the decrease in size is not as pronounced as observed with variation of laser pulse width. Figure 24b shows the effect of extemal magnetic field approximately equal to the coercive field in the absence of any laser irradiation on the magnetic domain shown in fig. 24a. Here one can notice growth in the size of the domain and appearance of several reversed magnetized branches at the bottom along with new nucleation centers. One such nucleation center is near the bottom right corner of the picture. Thus, the study of these pictures provides useful information to optimize the media characteristics during film deposition and to develop an optimized writing strategy.
6.2.4. Other high resolution imaging techniques The trend in MO data storage is towards higher recording densities and faster transfer rates. In order to progress in this direction, magnetic domains must be recorded whose dimensions are small and whose positions are precise. These domains must be stable in the presence of thermal gradients and stray external magnetic field. The magnetization within the domain should be uniform and the domain boundaries should be as sharp as possible to minimize the readout noise and jitter. Thus understanding the thin film magnetization dynamics within nanometer spatial resolution and nanosecond temporal resolution have become the focus of research activities. Magnetic force microscopy (van Kesteren, den Boef, Zeper, Spruit, Jacobs and Carcia [1991]) and Lorentz Electron Microscopy (LEM) (Chapman [1984]) are techniques capable of providing resolution below the wavelength of visible light. In a magnetic force microscopy instrument a sharp needle of cobalt or nickel is used to scan the surface of the magnetic film a few nanometers away. Through the deflection of the needle,
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Fig. 24. (a) Polarization micrograph of a magnetic domain taken using static tester. (b) Picture of the magnetic domain shown in (a) atter it was allowed to expand under a magnetic field equal to H c for a while in absence of any laser heating. Many branches appear on the main domain and a new nucleation center develops near the right bottom corner in the micrograph. which is proportional to the magnetic force exerted on its tip, magnetic force microscopy provides the picture of film's stray field pattern. LEM is a branch of phase contrast microscopy, based on the interaction between propagating electron waves and magnetic vector potential field. For a given electron trajectory, the interaction, known as the A h a r o n o v - B o h m effect, results in a phase delay directly proportional to the path integral of the vector potential. This 2-D phase delay may be mapped to provide the image o f the media. 6.3. EXCHANGE-COUPLED MAGNETIC MULTILAYERS MO media are generally used as high-density, low-cost rewritable information storage media. Especially, those having a recording layer composed o f an amorphous alloy o f a rare-earth metal (RE) and a transition metal (TM) exhibit a good balance between moderate carrier levels and very low noise. The exchange coupled magnetic multilayers proposed in 1981 (Kobayashi, Tsuji,
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Tsunashima and Uchiyama [ 1981]) have resulted in applications such as doublelayer films for high-carrier levels (Tsunashima, Kobayashi, Tsuji and Uchiyama [1981 ]), multilayer films for direct overwriting (DOW) by light intensity modulation (LIMDOW) (Saito, Sato, Matsumoto and Akasaka [1987]), multilayer films for magnetically induced super-resolution (MSR) (Aratani, Fukumoto, Ohta, Kaneno and Watanabe [ 1991 ]) and magnetic amplifying magneto-optical system (MAMMOS) for ultra-high density MO data storage (Awano, Ohnuki, Shirai, Ohta, Yamaguchi, Sumi and Torazawa [ 1996]). Combining LIMDOW and MSR, we can get media of both high-transfer rate and high density (LIMDOWMSR) (Hosokawa, Okamuro, Hashimoto and Miyata [1997]). 6.3.1. LIMDOW
LIMDOW media usually consist of several magnetic layers, such as readout layer, memory layer, intermediate layer, writing layer, switching layer and initialization layer. Figure 25 demonstrates the simplest LIMDOW disk structure and the concept of direct overwrite. In this configuration, one magnetic layer is the memory layer, the other is the writing layer. The Curie temperature of the memory layer, Tcl, is lower than that of the writing layer, Tc2. The recording scheme requires two magnets, one for initializing the writing layer (initializing field) and the other for writing (bias field). In every path under the initializing magnet, the writing layer is fully erased, but the memory layer, due to its high coercivity, retains its pattern of recorded domains. For readout, the laser power is set at a low level, P0, causing negligible heating of the magnetic layers. For writing, the laser is pulsed, either with moderate (P1) or high-peak power (P2), depending on whether the intended Laser Beam
Fig. 25. Schematic diagram of double-layer LIMDOW disk.
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domain's magnetization is up (T) or down (J.). The high-power pulses heat both magnetic layers above Tc2, and create ~. domains in both the writing layer and the memory layer by the + bias field. After the film temperature returns to the ambient temperature, the film passes under the initializing magnet, the domains in the writing layer are erased but those in the memory layer remain intact. The moderate power pulse P1 raises the temperature of the memory layer to about its Curie temperature, and erases magnetization of the memory layer but does not affect that of the writing layer. As the film cools down, the interlayer exchange coupling provides a field, which favors the formation of T domains in the memory layer. Thus laser pulses P1 and P2 create the desired T and I domains in the memory layer, regardless of any previously recorded data. This is the essence of direct overwrite. Initialization of the writing layer is a prerequisite for realizing LIMDOW. In fig. 25, the required field, Hini, for initialization may be expressed as follows: Hini > He2 + ~
O'MW
2Ms2t2
(4)
Here OMW represents interface wall energy between the memory layer and the writing layer, Hc2, Ms2, and t2 stand for coercivity, saturation magnetization, and thickness of the writing layer, respectively. Evidently, Hini decreases with decreasing ~rMW. The magnitude of OMW depends on the magnetic properties of the two layers, such as saturation magnetization, exchange stiffness, magnetic anisotropy, and thickness of both layers (Peng, Lee and Kim [ 1996]). An intermediate layer may be deposited between the memory layer and the writing layer to mediate the magnetic coupling between these two layers. Materials having very small effective perpendicular magnetic anisotropy can be used as the intermediate layer, which suppresses the magnetic coupling between the memory layer and the writing layer. Hence, as ~rMW is reduced, so is Hini. One of the advantages of using the intermediate layer is that the reduction of n i n i does not require special attention to the writing layer, such as its thickness or composition. Another advantage is that the intermediate layer improves the stability of a recorded bit in the memory layer. The magnetic coupling between the memory layer and the writing layer tends to align the orientation of the TM sublattice magnetization in the memory layer with that of the writing layer. This broadens the read power margin, and hence renders high reliability against ambient temperature. In order to miniaturize the MO driver, it is desirable to eliminate the initializing magnet. The elimination of the magnet leads to more complex exchange-coupled designs for direct overwrite. Figure 26 shows one such
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92~..~ \ P1 "~- P0
Time
(a) Recording
copy
Write field
+
(-300 Oe)
4,
initialization
r §
Initialization field (-3 kOe)
(b) after 92 irradiation
Memory layer ~ _ _ ~ ] Writing layer
_~
" I
~ ~
(c) after P1 irradiation Fig. 26. Principle of LIMDOW. (a) Laser output for direct overwrite; (b) magnetization status during cooling after light irradiation of P2 high power; (c) magnetization status during cooling after light irradiation of P1 moderate power.
scheme. In this scheme, a switching layer and an initializing layer (fig. 27) was added to the above double-layer structure (Fukami, Nakaki, Tokunaga, Taguchi and Tsutsumi [1989]). The initializing layer has the highest Curie temperature, and its magnetization is never reversed during reading/recording. The switch layer has a much lower Curie temperature than the initializing layer. The exchange coupling between the initializing layer and the writing layer exists only below the Curie temperature of the switching layer. In this way, the initializing layer plays the same role as the initialization magnet in the doublelayer structure. Except for the initialization, the essence of direct overwrite is the same as that in the double-layer structure. In fig. 26, the requirement for initialization becomes O'WS -- O'MW t2
> Ms2Hc2,
(5)
where aws is the interface wall energy between the writing layer and the
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Fig. 27. Schematic diagram of quadrilayer LIMDOW disk.
switching layer. The intermediate layer helps to reduce the magnitude of coupling between the writing layer and the memory layer, which makes eq. (5) easily satisfied (Fukami, Kawano, Tokunaga, Nakaki and Tsutsumi [1991], Muto, Shimouma, Nakaoki, Suzuki and Kaneko [ 1991 ]). Moreover, to enhance the readout, a readout layer can be deposited on the top of the memory layer (Kobayashi, Tsuji, Tsunashima and Uchiyama [ 1981 ], Hosokawa, Saito, Matsumoto, Iida, Okamuro, Kokai and Akasaka [ 1991 ]). This readout layer usually has a high Kerr rotation, such as GdFeCo and NdTbFeCo, but has a low coercivity. (The memory layer usually is TbFeCo, designed to have high coercivity at ambient temperature.) The domains in the readout layer are stabilized by the interlayer exchange coupling between the readout layer and the memory layer. 6.3.2. MSR
MSR makes use of the exchange-coupling force among the various layers of the multilayer stack. There are several types of MSR. Figure 28a shows the commonly used MSR structure. It consists mainly of three layers: a readout layer, an intermediate layer and a memory layer. As before, the intermediate layer is to make the magnetic transition between the other two layers smooth. It can be a magnetic layer like TbFe, or a dielectric layer like SiN, or it may even be absent. The memory layer, written onto at the time of writing, maintains a faithful copy of the recorded data at all times. The readout layer, on the other
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Fig. 28. The principle of magnetic super-resolution (MSR) in exchange-coupled triple-layer structures. (a) Disk configuration. (b) MSR by front-aperture detection. (c) MSR by rear-aperture detection.
hand, receives a copy of"selected" domains from the memory layer and presents this modified version to the readout beam. It is this selective presentation of the recorded domains to the read beam that achieves super-resolution (Wu and Yussof [1995]), since it removes the adjacent domains at the time of reading and essentially allows the read beam to "see" one domain at a time. Selective
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copying is activated by the rise in media temperature induced by the read beam itself. For MSR, there are slightly different ways for readout: front aperture detection, FAD, and rear aperture detection, RAD (Kaneko and Nakaoki [ 1996]). Figures 28b,c show schematic diagrams of the two detection schemes. In FAD, both the memory layer and the read layer normally contain identical copies of the recorded domains. Within a small area in the rear side of the focused spot, however, the rise of temperature weakens the coupling between the layers. At this point the magnetization in the rear side of the focused spot aligns itself with the applied field Hr, which is always in the same direction during readout. The beam thus sees the magnetized domains in the front aperture only. Superresolution is achieved by virtue of the fact that intersymbol interference from the domains in the rear aperture has been eliminated. Once the disk moves away and the temperatures return to normal, interlayer exchange regains its strength and the magnetization of the readout layer reverts to its original orientation. In RAD, as shown in fig. 28c, the read beam reads domains in the rear aperture of the focused spot while those in the front aperture are erased. Initially, the domains are recorded on both the memory layer and the readout layer, but the latter is erased prior to readout by the initialization field Hi. During readout, thermal effects of the read beam reduce the coercivity of the read layer within the rear aperture. As a result the underlying domains in the memory layer copy themselves onto the hot area of the read layer by the force of interlayer exchange. Thus super-resolution is achieved because the front aperture is erased and the only domain that is being read is within the rear aperture. When the disk moves away and the temperatures return to normal, the transferred domains persist in the read layer until such time as they are erased again by the initializing field Hi. From fig. 28c, it is evident that RAD can eliminate crosstalk from adjacent tracks. But a permanent magnet that can provide an initializing field of more than 3 kOe is required. If nonmagnetic materials are used as the intermediate layer, the initializing field can be reduced greatly (Matsumoto and Shono [ 1995], Kawano, Itoh, Yoshida and Kobayashi [ 1995]). GdFe can be also used for the intermediate layer to reduce the initializing field (Kaneko and Nakaoki [1996]). In both FAD and RAD, a carrier-to-noise ratio (CNR) of more than 45dB was obtained experimentally at the wavelength of ~ =690nm, and objective lens numerical aperture NA= 0.55 (Kaneko and Nakaoki [ 1996]). (The optical resolution limit for mark length is 2./(4 NA)= 310 nm.) A variation on the concept of FAD is central aperture detection, CAD. In CAD, the readout layer (e.g., GdFeCo) has in-plane magnetization at room temperature but perpendicular magnetization at high temperature (Yurakami,
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(b) i
i
|
|
.0~0--0
48 46
•.•44 42
/
/
o
40
.0
I
1.5
!
2.0
2'.5
3!.0
Read power (mW) Fig. 29. (a) Schematic mechanism of readout mechanism for central aperture detection (CAD) magnetic super-resolution (MSR). (b) Carrier-to-noise ratio (CNR) as functions of read power for a CAD-MSR disk. A disk track was recorded by a 7 MHz signal tone. The mark length is 0.5 ~tm.
Iketani, Nakajima, Takahashi, Ohta and Ishikawa [ 1993], Murakami, Takahashi and Terashima [1995]). Figure 29a shows the readout mechanism and fig. 29b shows the dependence of CNR as a function of read power. Data is recorded and stored in the memory layer in the form of magnetic domains. For readout, the thermal effects of the read beam raise the temperature in the region of the focused spot. The magnetization of the readout layer in the hot area becomes perpendicular to the film whereas the surrounding area remains in-plane. The orientation of the perpendicular magnetization is determined by the magnetic coupling between the readout layer and the memory layer. Since the in-plane magnetization does not contribute polar Kerr signal and the readout layer is thick (~50 nm), which is enough to block the light from reaching the memory layer, the read beam only "sees" those domains in the central region of the focused spot, achieving readout super-resolution. When the disk moves away and the temperatures return to normal, the magnetization becomes in-plane again. In this
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scheme, neither an initializing field nor a readout field is needed. Furthermore, cross-track crosstalk is suppressed to a large extent. In CAD-MSR using double-layer structures, due to the strong interlayer exchange coupling, the magnetization of the readout layer at the interface between the two layers is not in-plane. This reduces the performance for superresolution. To improve the readout, an intermediate layer can be deposited between the readout layer and the memory layer to mediate the magnetic coupling. If the intermediate layer also has in-plane magnetization at room temperature (e.g., GdFe), two masks, one at the front aperture and the other at the rear aperture, can be formed during readout. A CNR of 46 dB for 0.4 ~tm marks at A=680nm and NA =0.55 has been reported (Nishimura and Tsunashima [1996]). 6.3.3. MAMMOS
Magnetic domains expand or shrink under a certain external magnetic field. Based on this concept, a large MO signal can be obtained by dynamic domain expansion (Awano, Ohnuki, Shirai and Ohta [1997]). The disk structure for MAMMOS is similar to that for MSR, see fig. 28a. The readout layer has perpendicular magnetization and large polar Kerr signal at room temperature, such as GdFeCo; the intermediate layer can be dielectric, such as nitride, and the memory layer has good recording properties and high coercivity at room temperature, such as TbFeCo. The external field, H, for expansion and collapse of domains in the readout layer switches polarity at half of the clock for recording. Figure 30 schematically shows the readout mechanism for MAMMOS. At H < 0, there is no T domain in the readout layer because the
Fig. 30. Schematic diagram for magnetic domain expanding readout.
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external field is applied to the shrinking direction of the domain in the memory layer. At H = 0, the T domain in the memory layer makes a copy domain in the readout layer. Subsequently the external field (H > 0) expands the T domain in the readout layer rapidly. Thus the readout signal is increased drastically. As the external field returns to zero (H = 0), the expanded T domain in the readout layer shrinks to the same size as that in the memory layer. At this moment, the external field switches to be less than zero (H < 0), the copied T domain further shrinks and finally collapses. In summary, magnetic-coupled multilayers can be utilized to realize direct overwriting by light intensity modulation, achieve super-resolution beyond the cutoff frequency normally allowed by an objective lens and amplify readout signal by domain expansion. Of course, for the above readout mechanisms to be successful, the magnetic properties and thickness of all layers involved must be carefully selected. This presents a challenge for media manufacture.
w 7. Phase Change Media 7.1. INTRODUCTIONTO PHASE CHANGE RECORDING Phase change (PC) materials exist in two stable structural s t a t e s - amorphous and crystalline - and can be switched between these two states by the application of a high-power laser pulse (Feinlib, deNeufville, Moss and Ovshinsky [ 1971 ]). The amorphous to crystalline transformation is accompanied by large changes in the optical constants. A low-power laser beam is used to discern the amorphous state from the crystalline state by monitoring the reflected signal. In PC optical data storage, amorphous marks (information bits) that are as small as 0.4 mm are written on PC media. An amorphous mark is written by raising the temperature of the PC material above its melting point and cooling it rapidly below the crystallizing temperature. The amorphous state is also referred to as a glassy state. The glass formation process (Tauc [1974]) depends only on the suppression of crystallization during the cooling period. The amorphous mark can be erased (crystallized) by annealing the PC material at a temperature above the glass transition temperature but below its melting temperature. The annealing procedure allows the formation and growth of crystalline nuclei within the amorphous area. Thus information can be written, read and erased using a focused laser beam in a PC optical disk. A schematic diagram of the writing and erasing process is shown in fig. 31. The laser power is raised to a value, Pw, for a duration corresponding to the
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Time
! Track (crystalline state)
Written amorphous mark
Fig. 31. Schematic for laser power variation during writing and erasure process in a PC optical disk system.
length of an amorphous mark. A simple write strategy such as this produces a teardrop-shaped mark and so in practice, a multipulse write strategy is adopted to reduce the distortion of the mark shape (Akahira, Miyagawa, Nishiuchi, Sakaue and Ohno [ 1995]). The laser power is raised to a value, Pb, to facilitate fast crystallization of the amorphous mark. During the read process, the laser power is fixed at a sufficiently low value, Pr, such that there is no laser-induced crystallization. 7.2. MATERIALS FOR PC RECORDING
The alloys that belong to the family of chalcogenides are the most commonly used PC materials. A chalcogenide refers to an alloy comprised of at least one of the following elements, viz., selenium, tellurium, sulfur, etc. The chalcogenides can be tailored to satisfy the requirements of an optical data storage system. The material constraints imposed by the requirements of an optical data storage system are ease of amorphous mark formation, fast amorphous-crystalline (and vice versa) phase transformation, large signal to noise ratio between amorphous and crystalline states, stability of amorphous phase and re-cyclability. Recyclability refers to the ability to write information on the same track over a million times without significant degradation in the optical response of the PC disk.
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Historically, the material used in PC recording originates from the binary GeTe system which exhibits some of the essential qualities expected from a PC recording material (Libera and Chen [1990]). However, the crystallization speed depends strongly on the exact composition of the material. The stoichiometric compound in the GeTe binary system is GesoTes0. Depending on the atomic ratio between Ge and Te, the crystallizing speed varies over quite a wide range (ns to ms). The addition of antimony (Sb) eases this tight requirement of maintaining the exact atomic ratio between Ge and Te (Ovshinsky [1992]). This makes it possible for the commercial fabrication of Ge-Sb-Te (GST) PC material for data storage. The three stoichiometric phases in the GST system are Gel Sb4Te7, Gel Sb2Te4 and Ge2Sb2Tes, respectively. These compounds lie on the GeTeSb2Te3 pseudo-binary line and all three compounds exhibit high crystallization speeds. It was shown that the compositions that deviate from this pseudobinary line tended to exhibit longer crystallization times and higher crystallization temperatures (Yamada, Ohno, Nishiuchi and Akahira [ 1991 ]). The crystallization temperature and the crystallization speed impose constraints on the selection of the optimal PC media. For example, when the crystallization speed is very high the crystallization temperature is lower. The amorphous state is a metastable state and lower crystallization temperature implies that the written amorphous mark is unstable and hence makes the material unattractive for data storage purposes. Similarly, high crystallization temperatures mean longer crystallization times and this is also unacceptable in current optical data storage systems. The GST alloys, which fall along the GeTe-Sb2Te3 pseudobinary line, exhibit the proper balance of qualities necessary for an optical data storage system. Yamada [ 1997] studied the potential for high data-rate optical recording using the GST material. High data rate optical recording implies spinning the disk at velocities ~>20 m/s. It was shown that GST materials can be used to achieve data rates as high as 100 Mbps using the red laser and spinning the disk at 22 m/s. In this regime of operation, two key issues need to be addressed. (i) The dwell time of the laser on the media being less than 50 ns, the material should exhibit high crystallization speed while not compromising the stability of the amorphous state. (ii) The distortion of the written mark shape, caused by differences between the optical and thermal properties of amorphous and crystalline states must be controlled. The distortion of mark shape can be reduced by controlling the ratio of the absorbed energy in the amorphous state to the crystalline state. This issue has received wide attention from several researchers recently (Yamada [1997], Ide, Ohkubo and Okada [1996]).
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7.3. PC OPTICAL DISK STRUCTURE
Two major factors govern the design of a PC optical disk. The first is the optimization of reflectivity between the amorphous and crystalline states. This directly increases the magnitude of the signal and hence the signal-to-noise ratio (SNR). The second factor is related to the thermal design of the disk in order to optimize the rate of mark formation and erasure with the available laser power. The PC optical disk is typically a quadrilayer sample. The substrate is polycarbonate followed by an upper dielectric layer (ZnS:SiO2), the GST recording layer, the lower dielectric layer (ZnS:SiO2) and the aluminum layer. The aluminum layer is typically 100-150nm thick, the lower dielectric layer thickness is 20-60nm, the GST recording layer thickness is 20~ nm and the upper dielectric thickness is 150-200nm. These layers are deposited on the substrate by a sputtering process. In general there is an additional UV resin layer which is coated to protect the aluminum layer from damage during the handling of the disk. The UV layer plays no role in controlling either the optical or thermal properties of the disk. The disk structure is illustrated in fig. 32. This structure has also been referred to as the "rapid-cooled" disk structure and was shown by Ohta, Inoue, Uchida, Yoshioka, Akiyama, Furukawa, Nagata and Nakamura [1989] to have very high SNR and good overwrite characteristics. In addition, the rapid-cooled disk allowed a large tolerance to the laser power for erasure of amorphous marks. The upper dielectric serves the purpose of protecting the substrate from thermal damage. Apart from this protective function, the thickness of this layer
Fig. 32. Typical structure of a quadrilayer PC disk.
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can be used as a free parameter while optimizing the difference in reflectivity between the amorphous and crystalline states. The lower dielectric determines the cooling rate during amorphization since it acts as a thermal barrier between the recording layer and the aluminum layer. The thickness of the lower dielectric is determined by the thermal transport properties of the disk. Both the dielectric layers protect the recording layer from thermal damage by rapidly conducting the heat from the recording layer. During the melting process, the temperature in the recording layer can be as high as 800~ The dielectric layer also has a very low coefficient of thermal expansion thus preventing deformation of the disk structure at these extremely high temperatures (Ohta, Nagata, Satoh and Imanaka [1998]). Other approaches to prevent material flow and deformation of the disk have been attempted and can be found in recent literature (Kojima, Okabayashi, Kashihara, Horai, Matsunaga, Ohno, Yamada and Ohta [1998]). The aluminum layer is used to act as a heat sink thus protecting the recording layer from thermal stresses. The proximity of the aluminum layer to the recording layer determines the cooling rate, which controls the amorphization process. It has been shown that lower cooling rates (4~ produce amorphous marks with large crystalline grains at the periphery of the mark. These grains correspond to recrystallization from the mark edge during the cooling process. The rapidcooled disk structure with a high cooling rate (12~ was shown to exhibit no recrystallization during the cooling period (Ohta, Inoue, Uchida, Yoshioka, Akiyama, Furukawa, Nagata and Nakamura [ 1989]). If the aluminum layer is too close to the recording layer, then all the heat is dissipated quickly necessitating higher power from the laser. The aluminum layer is also used to tune the reflectivity of the disk. At a given wavelength of operation, the difference in the optical constants between the amorphous and crystalline states is fixed. This difference in the optical constants gives rise to a difference in reflectivity between the two states. A multilayer stack can further increase this difference in reflectivity using optical interference. The design of the multilayer stack has to be achieved within the constraints imposed by the thermal transport requirements of the disk structure. Commercial disks exhibiting values of SNR greater than 50 dB have been around since the commercial inception of PC data storage. 7.4. STATIC TESTER
The static tester that is used to study magnetic domains in MO materials is also useful for studying the crystallization and melting processes in PC materials. A schematic diagram of the static tester is shown in fig. 23 (refer to w6.2.3 for
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detailed description). The light from a laser diode (/l = 780 nm) is focused on the sample through a microscope objective. The NA of the objective can be chosen between 0.1 and 0.8. Depending on whether the laser beam is incident from the film side or through the substrate of the sample, the objective is corrected for spherical aberrations due to the substrate. The reflected light from the sample is directed through a polarizing beam splitter (PBS) to two separate photodetectors. The sum signal (S1 + $2) and the difference signal ( S 1 - $2) are amplified and displayed in an oscilloscope and stored in a computer. A single pulse is used to trigger the oscilloscope and the laser simultaneously. Hence the reflected signal is acquired in real time as the mark is being written. The sum signal is proportional to the total reflected optical power and is similar to the read signal in a phasechange drive. The measured reflected signal is converted to absolute reflectivity by calibrating the reflected signal using a silicon standard sample.
7.5. MEASUREMENTOF REFLECTIVITYPROFILES DURING CRYSTALLIZATIONAND MELTING USING A STATICTESTER The samples that have been studied are quadrilayer PC samples on a PMMA/ polycarbonate substrate. The PC layer composition is Ge2Sb2Te5 + 0.3 Sb. The geometric structure, layer thickness for the various samples measured and the optical constants (at 780 nm) for the various layers of the samples are listed in table 1. The reflectivity measured during crystallization (from as-deposited amorphous state) in sample Q 1 is shown in fig. 33. The characteristic feature easily seen in the figure is that there is a finite amount of time for the amorphous-crystalline transformation to start. Once the transformation is initiated (at a time, tonset, which is the knee of the reflectivity profile), it proceeds at a much faster rate until the area irradiated by the laser beam is completely crystallized. The reflectivity in the crystalline state being greater than in the amorphous state, Table 1 Listing of PC disk structure for samples discussed in the chapter No.
Thickness of layer (nm) ZnS:SiO2
GST
ZnS:SiO2
A1
Q1
68
100
Q2
77
25
25 50
100 100
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Fig. 33. Variationof reflectivity during crystallization(from as-deposited amorphous state) measured in the static tester. Notice that the onset of crystallization decreases with increasing laser power. the reflectivity profile increases after onset of crystallization. The saturation of the reflectivity profile is due to two effects. First, the temperature of the PC layer (at the crystalline-amorphous interface) is reduced to a level where the growth rates are very small. Secondly, the laser beam center is the hottest region and once crystallized the subsequent growth of this mark is quite slow, depending in a complicated manner on the thermal characteristics of the optical disk. The reflectivity profiles measured during the melting process in sample Q 1 are shown in fig. 34. Here the reflectivity starts at a value corresponding to the crystalline state. After the onset of melting, the reflectivity drops. The size of the molten pool determines the drop in the reflectivity value. In this sample there is a drop in the reflectivity even before the onset of melting (laser power less than 9.17 mW). This could be due to a variation in the optical constants or changes in the multilayer stack structure due to thermal deformation of the substrate. Note in table 1 that in this sample the upper dielectric layer is only 68 nm thick, and hence there is significant thermal communication between the hot recording layer and the plastic substrate. The drop in reflectivity could hence be due to deformation of the substrate. The reflectivity profile does behave in the manner shown in figs. 33 and 34 when the reflectivity difference between amorphous and crystalline states is large and the phase difference between the two states negligible. When the phase difference is nonzero, there will be some interesting diffraction effects,
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Fig. 34. Variation of reflectivity during melting (from crystalline state) measured in the static tester. In this sample there is a drop in reflectivity even when there is no melting (laser power less than 8.58 mW). This is attributed to deformation of the plastic substrate. This sample has a lower dielectric that is 70 nm thick and thermal simulations indicate that the substrate is raised to temperatures greater than 500~
which change the behavior of the reflectivity profile. The crystallization (from as-deposited amorphous state) measurements in sample Q2 (with a large phase difference between the amorphous and crystalline states) is shown in fig. 35. In this sample, the crystalline state reflectivity is larger than the amorphous state reflectivity. At first glance, the measurements in fig. 35 seem to contradict this fact. The behavior of the reflectivity profile can be predicted by considering diffraction at the phase boundaries. Figure 36 shows the result of a computer simulation performed using the program DIFFRACT TM. When the mark size is small compared to the focused spot, the reflectivity decreases due to light being diffracted out of the aperture of the objective lens. When the mark is large, the phase difference becomes inconsequential and the reflectivity increases back to the crystalline state reflectivity. Ide, Ohkubo and Okada [1996] have demonstrated using the phase difference between amorphous and crystalline states to read information in PC disks. This technique has the added advantage of increasing the crystalline state absorption to compensate for the differences in thermal behavior between crystalline and amorphous states.
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Fig. 35. Variation of reflectivity during crystallization (from as-deposited amorphous state) measured in the static tester. The drop in reflectivity can be explained only by considering diffraction due to the phase difference between the amorphous and crystalline states.
Fig. 36. Simulation of variation in reflectivity with mark size using a commercial program DIFFRACTTM. The simulation is performed for various values of phase difference between the amorphous and crystalline states.
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7.6. CRYSTALLIZATION/AMORPHIZATIONKINETICS Crystallization is a two-step process consisting of nucleation and growth (Tauc [1974]). Nucleation is the process where nuclei of the new crystalline phase are first formed from the parent amorphous phase. These crystalline nuclei subsequently grow rapidly to complete the crystallization of the area irradiated by the laser beam. The factors that control the rates of nucleation and growth constitute the subject of crystallization kinetics. The formation of the crystalline phase is initially a result of thermal agitation, which causes the molecules in the amorphous state to align themselves in a manner corresponding to the crystalline state. For the new phase to be stable, it has to satisfy certain thermodynamic conditions. The formation of stable nuclei is referred to as nucleation. The nucleation process may be homogeneous or heterogeneous in nature. The nucleation is considered homogeneous in pure systems. Nucleation as a result of impurities is referred to as heterogeneous nucleation. In general the nucleation rate has the following form:
Rn = Nav~ [-(AG*kT+AGa)]
.
(6)
In this expression, Na is the number of molecules per unit volume in the amorphous state, no is the atomic vibrational frequency (approximated by the Debye frequency), AG* is the activation free energy associated with nucleation, A Ga is the flee energy barrier between the amorphous and crystalline state, k is the Boltzmann constant, and T is the temperature. The activation free energy required for nucleation is different for homogeneous and heterogeneous nucleation. It is also a function of temperature and a detailed description of nucleation can be found in the following references: (Tauc [ 1974], Rao and Rao [ 1978], Christian [ 1965]). Following the nucleation of the crystalline phase, the crystallization process is completed by the growth of these nuclei. During the growth process, the molecules at the interface are transformed from the amorphous to crystalline state by a diffusion-like process. A rate equation describing the net movement of molecules is written to determine the growth rate, which is given by [-AG~a kT J
Rg =fAvoexp[
1 - exp ( VmAGv
Here, f refers to the fraction of available crystalline interface sites, A is the atomic jump distance during the growth of crystallites, A G~a is the interracial
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energy barrier associated with diffusion of atoms, Vm is the molar volume, and A Gv is the free energy difference between the crystalline and amorphous states. The factors f and AGv are functions of temperature and the reader is directed to the literature for more details (Tauc [1974], Rao and Rao [1978], Christian [1965]). The amorphization process is also referred to as the glass formation process. A liquid can be cooled to a glassy state by suppressing the process of crystallization. In PC media this is achieved by cooling the molten pool rapidly by employing the "rapid-cooled" structure described in w7.3. This essentially means that the temperature of the PC layer is reduced quickly to a value where there will be no nucleation and growth. Experimental measurements of crystallization in PC media have been used to indirectly determine the phase transformation kinetics parameters (Peng, Cheng and Mansuripur [1997]). The static tester was used to make extensive measurements of crystallization in PC media. A numerical temperature calculation for a multilayer stack irradiated by a laser beam along with the phase transformation model described is used to simulate the formation of a crystalline mark. The calculated reflectivity profile is compared with the experimental measurements. The parameters of the model are adjusted until a good fit between the measurement and calculation is obtained. Once the nucleation and growth model had been experimentally calibrated, the mark formation process in a spinning disk was simulated. The fundamental behavior of PC media such as crystallization, amorphization and re-crystallization of quenched amorphous state have been simulated. The results are shown in figs. 37-39. Figure 37 shows the simulated crystallization process caused by irradiation of 1.9 mW laser beam focused on the sample using a 0.4 NA microscope objective. The dark region represents the amorphous phase, while the bright regions are crystalline phase. The nuclei shown in the figure have been enlarged 25 times for easier visualization. Each frame is 2 ~tm • 2~tm. Figure 38 shows the formation of an amorphous mark by a 100 ns, 9.27 mW laser pulse starting at t = 250 ns (for t < 250ns the laser power is 4.37 mW). The laser spot moves from left to fight at 8.8 m/s. The figure is plotted in four gray levels from black to white, representing four different phases: molten pool, super-cooled liquid, amorphous and crystalline phases, respectively. Again, the nuclei shown in the figure have been enlarged 25 times for easier visualization. Each flame is 1.5 ~tm x 1 ~tm. Figure 39 shows the recrystallization of an amorphous mark. The laser power was 4 mW and the laser spot moves from left to right at 8.8 m/s. The dark regions represent the amorphous areas, while the white dots correspond to the crystalline areas. Each frame is 1.5 ~tm • 1 ~tm.
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Fig. 37. Formation of a crystalline mark using the crystallization kinetics model described in the chapter. The laser power was 1.9 mW. The measured reflectance profile using the static tester was used to calibrate the kinetics model. (a) 50ns exposure time; (b) 200ns exposure time; (c) 300ns exposure time; (d) 500ns exposure time.
Fig. 38. Formation of an amorphous mark in a spinning disk.
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Fig. 39. Erasure of a written amorphous mark. Notice that the edges of the amorphous mark remain amorphous after the laser moves away from the written mark. 7.7. FUTURE TRENDS IN HIGH-DENSITY PC RECORDING There are several avenues for progress in high-density data storage in PC media. The areal density can be increased by the arrival of the blue laser. With the shift towards blue wavelength and increase in the NA of the objective, areal densities of 10 GB/in 2 can be achieved. Near-field optical storage using the solid immersion lens (SIL) can further increase this value by at least an order of magnitude.
w 8. Diffraction from Periodic Structures The size and distance between data marks in optical disks are comparable to the wavelength of the laser light. In this regime of wavelength to characteristic lengths ratio the scalar approximation of diffraction, with which most optical engineers are familiar, is no longer valid. Indeed, using the scalar approximation, one cannot explain the experimentally observed polarization effects in the optical beams reflected from a disk, nor can one explain the existence of the surface wave excitations in a metallic or multilayer-coated dielectric disk (Gerber, Li and Mansuripur [1995], Yeh, Li and Mansuripur [1998]). To fully understand the interaction of a focused optical beam with an optical disk, a rigorous diffraction theory is necessary. Even in situations where the scalar modeling might be
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applicable, the rigorous theory should be used to check and define the domain of applicability of the scalar theory. Today there exist numerous electromagnetic scattering theories that in principle can be used to model optical disks. The one that we have chosen is derived from the grating theory. A grooved optical disk is in many ways just a diffraction grating. And the electromagnetic theory of gratings is probably the most complete and advanced among all electromagnetic scattering theories. To date, there are only three books published on diffraction gratings. The book edited by Petit [1980] is the only one that is devoted to the theory of gratings (but see also the book chapter by Maystre [1984]). The book by Hutley [1982] emphasizes the physics of gratings. Complementing the above two, the recently published book by Loewen and Popov [1997] focuses on the applications of gratings. In addition, a volume in SPIE's Milestone Series, edited by Maystre [1993] collects many important papers on all aspects of gratings up to 1992. The optical disk has three characteristics that are important to consider when one selects the optimum analysis method: (i) The disk grooves are shallow unlike many other applications of gratings. (ii) The disk materials may be metallic, anisotropic, and optically active (e.g., magneto-optic). (iii) The disks may include corrugated multilayer coatings. Among the more than a dozen grating modeling methods available today the coordinate transformation method of Chandezon, Maystre and Raoult [1980] can efficiently treat metallic and anisotropic media and it is most suitable for modeling multilayer gratings. (Useful references related to this technique are Chandezon, Dupuis, Comet and Maystre [ 1982], Li [ 1994, 1996], Li and Chandezon [ 1996], Harris, Preist and Sambles [ 1995], Hams, Preist, Wood and Sambles [ 1996], Inchaussandague and Depine [ 1996, 1997]. There are two ways to model a focused beam. In some numerical models the beam can be treated as a composite entity (Liu and Kowarz [1998], Marx and Psaltis [1997]. In order to use a conventional grating model, however, it is necessary to decompose the focused beam into a number of plane wave components. For each incident plane wave and each of the two independent polarizations, the grating model can be used to calculate the complex amplitudes of the diffraction orders. The total diffracted beam is then just the superposition of all the individual grating responses. The numerical results to be given in this section were obtained by this approach. The optical interaction in the disk region was calculated using a computer code based on the Chandezon method. The optical disk shows periodic structure owing to the presence of the continuous grooves or discontinuous data pattern. The interaction of the
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Fig. 40. The baseball pattern: the intensity distribution at the exit pupil of the objective lens after the focused light beam is reflected from a grooved disk. Usually, it is the interference pattern among zeroth and first diffraction orders, d+l and d-1 indicate the distance from the center of the first order to the center of the aperture.
focused spot with this periodic structure causes a periodic phase and amplitude modulation of the focused spot, and yields the desired signal for reading the data marks or providing the servo information. The fact that the period of the track for an optical disk is comparable to the optical wavelength makes it valuable to review the effects of diffraction in an optical disk system (see, e.g., Hopkins [ 1979], Pasman [ 1985a], Mansufipur [ 1989], Li [ 1994].
8.1. THE BASEBALL PATTERN The light reflected from the periodic structure (optical disk) gives rise to multiple diffraction orders that pass through the objective lens and form an image at the exit pupil of the lens. The image containing the interference pattern among multiple reflection orders is often called the baseball pattern. Figure 40 schematically shows the formation of the baseball pattern. In an optical disk drive, the pattern is often relayed through a field lens such as a spherical lens, an astigmatic lens or a ring-toric lens (refer to w 3) to generate the feedback signal for focusing and tracking servo and readout signal for data-reading. For a plane wave incident at an angle 0inc with respect to the surface normal
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of the periodic structure, the diffraction angle 0 m for mth order is given by the grating equation,
sin Om -- sin 0inc + m - , P
(8)
where/l is the optical wavelength and p is the period of the grating. Therefore, after reflection from the grooved structure, the various diffraction orders arrive at the exit pupil of the objective lens with different amounts of translation. For a focused beam used in an optical disk system, the diffraction orders appear as a cone of light at the exit pupil. If the objective lens satisfies Abbe's sine condition (i.e., it is an aplanatic lens), the center of the corresponding cone of light is then shifted from the center of the aperture by f sin 0 m --m)~f/p, where f is the focal length of the lens. As shown in fig. 40, for the mth-order diffracted beam the fractional shift of the center of the aperture relative to the aperture radius will be dm = sin Om/sin0 l e n s - - m M ( p . NA), where NA (= sin 0lens) is the numerical aperture of the objective lens. The amplitude and phase of the diffraction orders depend on the grating profile. In terms of the combination of the Fourier components, the complex reflection coefficient of the grating, r(x), may be written as
r(x) = ro + y ~ rme i2~mx/p,
(9)
m
where x is the coordinate across the groove, and m, the diffraction order, could be positive or negative. For a simple rectangular grating, the complex diffraction amplitude - the Fourier coefficients in eq. (9) - are then given by r0 = aroexp(iqgG) + (1 - a)rLexp(iqgL),
(10)
rm = asinc(mJra) [rLexp(i~L) -- rGexp(iqgG)] 9
Here, sinc(x)=sin(x)/x, a is the duty cycle of the grating, and rLexp(iqgL) and rGexp(iqg6) are the complex reflection coefficients for land and groove, respectively. The phase of the diffraction orders is also affected by the relative position between the focused beam and the grating. If the grating is shifted by Ax along the x-axis, its complex diffraction amplitude will be multiplied by a phase factor, exp(-i2JrmAx/p). It is this phase shift that results in the change of the
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Fig. 41. A schematic diagram of the experimental setup used to image the baseball pattern. Formed at the exit pupil of the objective lens, the baseball pattern is reimaged by a pair of relay lenses onto the CCD camera. intensity distribution on the baseball pattern and generates the tracking signal. With reference to fig. 6, the push-pull tracking error signal is derived from
I
( 7)1
TES cx $1 - $2 0( r0 + rlexp -i2r
ro + r-1 exp i2r P
After some straightforward algebra we obtain TES ~ a s i n c ( : r a ) r L r ~ s i n ( ~ L
-
~)sin
2~p
.
(11)
The TES of the push-pull tracking scheme is a sinusoidal curve (e.g., see Pasman [1985a], Marchant [1990]). As seen in eq. (11), to make the phase difference between land and groove, ~ L - ~6, equal to :r/2 and to obtain the maximum TES, the optimum groove depth is then ~ / 8 n for a reflection grating (here, n is the refractive index of the medium that fills the groove). Figure 41 illustrates the typical optical system used to examine the baseball pattern. Except for the relay lenses used to reimage the baseball pattern onto the CCD camera, it is similar to the setup of push-pull tracking detection shown in fig. 6. By using a dielectric grating with coated stacks of ZrO2-SiO2-ZrO2, Figure 42 shows experimental results of the baseball pattern for different optical wavelengths and objective lenses, and fig. 43 shows the corresponding computed results based on the rigorous vector theory of diffraction. For a metal grating
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Fig. 42. Measured baseball pattern for the incident beam with polarization parallel (top row) and perpendicular (bottom row) to the grooves. The sample is a dielectric grating with 0.6 ~tm period and a trilayer coating (ZrO2-SiO2-Zr02). (a,b) Laser light of 0.633 ~tm, 0.6NA objective lens. (c,d) Laser light of 0.544 ~tm, 0.6NA objective lens. (e,f) Laser light of 0.633 ~tm, 0.8NA objective lens. (g,h) Laser light of 0.544 ~tm, 0.8 NA objective lens. The anomaly line structure shown on the baseball pattern may arise from the coupling of waveguide modes.
Fig. 43. Simulated results corresponding to the experimental observations shown in fig. 42.
coated with a thick layer o f gold, fig. 44 shows the e x p e r i m e n t a l and c o m p u t e d baseball pattern. T h e line structure seen on the baseball pattern is due to the surface wave excitation and will be described in w 8.3. Braat [1985] and G e r b e r [1995] studied the effects o f wavefront aberrations
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Fig. 44. The baseball pattern for the incident beam with polarization parallel (top row) and perpendicular (bottom row) to the grooves. The optical wavelength is 0.633 ~tm, and the sample is a metal grating with 0.6 mm period and a thick layer of gold. (a,b) Experimental observation with a 0.6 NA objective lens. (c,d) Simulated results corresponding to (a,b). (e,f) Experimental observation with a 0.8 NA objective lens. (g,h) Simulated results corresponding to (e,f). The anomaly line structure shown on the baseball pattern may arise from the excitation of surface plasmon.
on the optical disk system. They both determined the influence of wavefront aberrations on various disk parameters and showed the degradation in signal power as a function of the spatial frequency of the data pattern on the optical disk. 8.2. DIFFRACTION FROM GROOVES THROUGH THE SOLID IMMERSION LENS (SIL)
The use of a solid immersion lens (Mansfield, Studenmund, Kino and Osato [1993], Terris, Mamin, Rugar, Studenmund and Kino [1994], Terris, Mamin and Rugar [ 1996]) in an optical disk system is another approach to increasing storage density and data rate. Figure 45 shows the optical disk system implementing an SIL. The hemispherical glass of refractive index n receives the rays of light at normal incidence to its surface. These rays come to focus at the center of the hemisphere and form a diffraction-limited spot. The spot size is n times smaller compared to the case without an SIL because the optical wavelength is reduced by n inside the hemisphere. To ensure that the smaller spot size does indeed increase the resolution of the system, the bottom of the SIL must either be in contact with the active layer of the disk or fly extremely close to it. The rays of light that are incident at large angles at the bottom of the SIL would have been reflected by total internal reflection except for the fact that light can tunnel through and cross the air gap which is small compared to
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Objective Lens
Solid Immersion Lens Air Gap Multilayer Coatings Substrate
Fig. 45. A schematic diagram of the experimental setup implemented with a solid immersion lens either flying above or sitting on the multilayer disk.
one wavelength. The tunneling mechanism or evanescent coupling is known as frustrated total internal reflection. The evanescent coupling strongly depends on the width of the air gap and is an important factor in the derived signal level. Mansfield [1992] analyzed the effects of wave aberrations and the system tolerance on the SIL. Mansuripur, Li and Yeh [1998] discussed the effect of the air gap on the depth of focus. They pointed out that the use of the SIL would not reduce the depth of focus if in the process of scanning, the SIL is moved with the sample. Based on the full vector theory of diffraction, Yeh, Li and Mansuripur [1998] studied the corresponding baseball pattern and the disk signal for the use of the SIL on the grooved structure. Figure 46 shows the experimental and computed results of the baseball pattern acquired by using an SIL for the dielectric grating and the metal grating mentioned earlier. The simulation results are based on vector diffraction calculations. 8.3. SURFACE WAVE EXCITATION
Another notable phenomenon in the interaction of the incident beam with the grooved structure is the excitation of surface wave (surface plasmons and waveguide modes) (see, e.g., Wood [1902], Dakss, Kuhn, Heidrich and Scott [ 1970]). The use of multilayer dielectric and metal coatings and high numerical aperture (NA) beams in modern optical disk technology inevitably entails the excitation of surface waves. The excitation of surface waves results in a drop in reflectance for certain angles of incidence; this may disturb the baseball pattern significantly.
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Fig. 46. The baseball pattern for the incident beam with polarization parallel (top row) and perpendicular (bottom row) to the grooves. The optical wavelength is 0.633 mm, the NA of the objective lens is 0.6, and a solid immersion lens with index n ~ 2 is used to increase the effective NA of the system. (a,b) Experimental observation for the dielectric grating. (c,d) Simulated results corresponding to (a,b). (e,f) Experimental observation for the metal grating. (g,h) Simulated results corresponding to (e,f). In a metal grating, the excitation of surface plasmons absorbs the energy of the incident wave and results in rapid and substantial variation in the reflected angular spectrum. An incident beam with perpendicular polarization (the polarization of incident beam is perpendicular to the groove) is required to excite the surface plasmons. Gerber, Li and Mansuripur [ 1995] demonstrated the surface plasmon excitation by observing the baseball pattern. As seen in fig. 44 for a metal grating, the line (or band) structure on the baseball pattern for perpendicular-polarization states may be due to the excitation of surface plasmons. For a grating with dielectric coatings, the coupling of incident beams into waveguide modes is responsible for the narrow dark or bright bands in the reflected angular spectrum. Both perpendicular- and parallel-polarization (the polarization of incident beam is parallel to the groove) states are able to generate the waveguide modes for dielectric gratings. As seen in figs. 42 and 43 for a dielectric grating, the line (or band) structure on the baseball pattern may be due to coupling into waveguide modes. With a solid immersion lens, the chance of coupling into surface waves is even greater because of the increased width of the angular spectrum. Gerber, Li and Mansuripur [ 1995] explored the coupling conditions to decide the location of the circular band structure on the baseball pattern for surface plasmon excitation. Typically, a medium with a large but negative dielectric
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constant e is a good host for surface plasmons. Tamir [ 1973] and Mansuripur and Li [ 1998] viewed the surface plasma as an inhomogeneous plane-wave solution to Maxwell's equations. 8.4. P O L A R I Z A T I O N
EFFECTS
The track pitch of current optical disks is comparable to the wavelength of the laser source. In this domain of pitch-to-wavelength ratio, incident beams with different polarization states will result in different complex diffraction amplitudes because of disparate response to the grooved multilayer. Furthermore, as mentioned in w 8.3, the use of multilayer dielectric and metallic coatings and high numerical aperture (NA) beams in modem optical disk technology inevitably entails the excitation of surface waves, which are polarizationdependent. For different polarization states the different intensity and phase distributions on the baseball pattern result in different data/servo signals. A good insight into the influence of the polarization effects on the various extracted signals is, therefore, important in the design and analysis of new generations of optical disks. In this section, a numerical model based on the rigorous vector theory of diffraction is used to analyze the interaction of the two independent polarization states with the optical disk. A V-shape grating, which is often used in an MO disk, is used in the simulation. The parameters of the grating structure and the optical elements are listed in table 2; the corresponding optical system with an SIL is shown in fig. 45 Table 2 The parameters of the optical disk system used to simulate the polarization effects on the disk signals System parameters
Specifications
System parameters
Specifications
A1 (n = 2, k = 7)
100 n m
V-shape grating grating period
1 ~tm
Substrate (n = 1.6, k = 0)
land width
0.7 ~tm
groove width
0.3 ~tm
Optical system
groove depth
0.1 ~tm
wavelength
650 n m
NA
0.6
solid immersion lens
n = 2
Coating structures SiN (n = 2, k - 0) T b F e C o (n = 3, k = 3.5)a SiN (n = 2, k = 0)
100 nm 20 n m
detector
50 n m
a T h e T b F e C o layer is treated as an isotropic material in the simulation.
split-detector (S1 a n d $2)
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Fig. 47. Results of vector diffraction calculations of SUM signal and differential signal for an incident beam with polarization parallel to the grooves. The parameters of the optical disk system are listed in table 1. The two dotted lines indicate the edge of the V-shape groove.
Distance (gm)
Distance (gm)
Fig. 48. Results of vector diffraction calculations of SUM signal and differential signal for an incident beam with polarization perpendicular to the grooves. Compared to parallel-polarized state (refer to fig. 9), perpendicular-polarized state shows lower resolution based on the SUM signal, but reveals stronger differential signals for the optical disk system listed in table 2.
and a split-detector is placed at the exit pupil of the objective lens. By inspecting the disk signal, SUM (S1 + $2) and differential signal (S1 - $2)/(S1 + $2), figs. 47 and 48 show the simulation results for different polarization states and different
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gap-width. The difference between parallel polarization (the incident polarization parallel to the tracks) and perpendicular polarization (the incident polarization perpendicular to the tracks) is found to be substantial. The perpendicular polarization shows stronger differential signal; however, the parallel polarization shows better resolution based on the SUM signal (the distance required to resolve a groove is shorter). For both polarization states, the differential signal decreases and the SUM signal increases as the air gap widens (most of the light is totally internally reflected back to the detector and less light is tunneling through the gap), and the response to various gap-width is different. This implies the necessity of considering the polarization effects in designing an optical disk system. Yeh, Li and Mansuripur [1998] showed the experimental results that verify the different behavior for different polarization states and demonstrated that the vector theory of diffraction is more suitable in predicting the disk signal for different polarization states than scalar theory.
w 9. Future Trends in Optical Disks and Drives
The two main goals in the optical data storage industry have always been greater storage capacity and faster data transfer rates. Increases in storage capacity can be accomplished by various means, the most common of which are decreasing the wavelength of light and increasing the numerical aperture of the objective lens. The data transfer rate can be increased by means such as parallel readout (Mansuripur [1995]) using multiple laser beams or by simply decreasing the mark size. Cost is another driving issue in the industry, making the systems and their components more efficient and less expensive decreases the overall cost of the system. Finally, decreasing the overall size of the system increases its uses in different applications, opening entirely new markets. 9.1. BLUE DIODE L A S E R S
In what is expected to have a major impact on the storage capacity of optical disks, Nichia Chemical of Japan has recently announced that they are capable of making a 30mW blue (~400nm) GaN-based diode laser for use in optical disk systems. The expected CW lifetime of these lasers is about 1000 hours at room temperature, making it sufficiently stable for most applications. To date, the main competitor of these blue diode lasers is Matsushita's second harmonic generator laser devices, which use a tunable distributed Bragg reflector and an 830 nm laser beam to achieve 415 nm. The effect of the decrease in wavelength
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would increase the storage capacity of a DVD-RAM from 2.6 GB per side to 15 GB per side (Whipple [1998]). There are still some problems with the lasers in their current form, namely that the Fabry-Perot laser cavity must be etched instead of the traditional cleaving due to the sapphire substrate used to grow the structure. Etching is much more difficult, inefficient and expensive so it inhibits mass production of the lasers. Other possible substrates could be used to allow the use of cleavage techniques but these are still being researched. While phase change materials would have no problem switching to a blue laser, the TbFeCo used in MO disks has a small Kerr rotation for blue light. This presents problems in readout (writing is still a thermal process) which must be resolved. Adding Nd or Pr to the TbFeCo greatly improves the performance at shorter wavelengths as shown by Gamino and McGuire [1985], Hansen, Raasch and Mergel [1994] and Peng, Kim, Cheong, Lee and Kim [1996]. Another possible solution is to make use of the separate reading and writing layers used in MSR disks (see w6.3.2) by optimizing the read layer for blue light using a GdFeCo layer which has a larger Kerr rotation at shorter wavelengths. During the writing process, the blue laser is only required to heat the sample so TbFeCo can still be used for the writing layer. Switching to blue light would also create backward compatibility problems with older disk technologies. As a result, two lasers would have to be installed in the optical head of any new blue laser systems to read all generations of disks. 9.2. MAMMOS AND DOMAIN WALL DISPLACEMENT (DWD)
Several techniques of increasing the MO signal have been presented. As shown in w6.3.3, MAMMOS increases the local read domain size. Larger spots result in a larger SNR. Recently, Shiratori, Fujii, Miyaoka and Hozumi [1998] proposed another technique called Domain Wall Displacement (DWD). DWD leaves the domain walls of each domain on the recording track unclosed at only the leading and trailing edges. As the readout beam approaches the domain wall, it is displaced in the higher temperature direction due to the temperature gradient of the beam spot. This domain wall displacement effectively expands the domain, resulting in a larger signal independent of the resolution of the optical system. 9.3. IMPROVED CYCLABILITY OF PHASE CHANGE MEDIA
The limitations on rewritability of phase change materials proposes another challenge. Kojima, Okabayashi, Kashihara, Horai, Matsunaga, Ohno, Yamada and Ohta [1998] have demonstrated that doping the GeSbTe recording layer
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used in phase change materials with a 2.7 % nitrogen concentration results in an increase of the maximum overwrite cycles to 8• 105. This order of magnitude increase in cyclability would make the lifetime of phase change disks comparable to that of MO disks. This technology will be applicable for optical disks with higher speeds and higher densities than those currently in use. Yamada, Otoba, Kawahara, Miyagawa, Ohta, Akahira and Matsunaga [1998] have also demonstrated another method of increasing cyclability. By adding interface layers of GeN around the GeSbTe active layer, the cyclability is increased to more than 5 • 105 overwrites and the crystallization process is accelerated. This accelerated crystallization rate allows higher revolution speeds for the disk of up to 12 m/s resulting in a data rate of 40 Mbps. The increased speed and cyclability of these methods will be a great improvement to phase change technology. 9.4. LIQUIDCRYSTALSERVOFOR ABERRATIONCORRECTION Coma caused by a tilt of the disk is a serious concern in optical head design. Recently, a liquid crystal servo was suggested (Murao, Iwasaki and Ohtaki [1996]) for correction of this tilt by imposing opposite coma on the beam. Applying different voltages to different areas of the transparent liquid crystal element will induce a variable phase across the aperture. By creating a servo system to modulate this phase, tilt-induced coma can be removed by the addition of negative coma. A similar technique can also be applied for reduction of spherical aberration. Reducing the sensitivity to these aberrations permits the use of a higher objective lens in the optical head for greater resolution. 9.5. SUBSTRATE/MASTERINGIMPROVEMENTS Deep, vertical walls and flat bottoms on grooves and pits result in the most desirable data signal. These features are difficult to manufacture with plastic substrates due to stresses caused in the mastering process. Morita and Nishiyama [ 1998] have shown that it is possible to reproduce steep walls and a narrow track pitch by injection molding processes. Improving the materials currently used as substrates will allow better manufacturing of grooves and pits, which results in lower noise and lower thermal and optical crosstalk (in the case of deep grooves). A decrease in noise and crosstalk effects could then be transferred to a higher density by decreasing the track pitch and mark sizes. 9.6. MULTILAYERROM DISKS Multilayer ROM disks can be created by removing the highly reflective A1 layer from the disk surface and mounting several data layers separated by a small air
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gap. The resultant disk will have a much lower reflectivity for each layer but the increase in storage density will be proportional to the number of data layers. Imaino, Rosen, Rubin, Strand and Best [1994] have successfully demonstrated a six layer ROM disk which operates within the specifications for CD drives. This idea would be a useful implementation for ROM disks and an extension to 10 layers or more is possible, but the more complex nature of rewritable disks would make multiple data layer disks challenging to produce. 9.7. NEAR-FIELDOPTICAL SYSTEMS Near-field optical systems are other promised devices that are under investigation for application in the optical head. The use of a near-field device is intended to increase the areal density of the optical disk. One method to fulfill this goal is to use an aperture probe, positioned closely to the disk (so named nearfield), to read or write the data pattern (Pohl, Denk and Lanz [1984], Betzig, Isaacson and Lewis [1987], Froehlich and Milster [1995]). An optical fiber or a waveguide is used to carry the near-field beam, the presence of the submicron aperture reduces the spot size on the disk, thus increasing the areal density. Betzig, Trautman, Wolfe, Gyorgy, Finn, Kryder and Chang [ 1992] have achieved an areal density of 45 Gbits/in 2 using a metallized, tapered optical fiber, however the disadvantages of this technique are a low optical efficiency and low data rate. Another approach to the near-field device is the use of a solid immersion lens (SIL) flying closely to the disk (refer to w5.3). Terris, Mamin and Rugar [ 1996] achieved an areal density of 2.5 Gbits/in 2 based on a SIL and an 830 nm light source. Unfortunately, the strict requirement on the thickness (about Z/10) of the air gap between the SIL and the disk is the main issue that must be overcome for commercial applications. Plastic surfaces may be too rough to fly the SIL sufficiently close to the disk. Developing glass substrates for use in SIL systems may ease these restrictions. 9.8. MINIATURIZATIONOF OPTICALHEAD COMPONENTS The increase in numerical aperture from CD to DVD systems creates a problem in backward compatibility between the two. If a DVD head is used to read a CD system, the increase in NA combined with the thicker substrate results in considerable spherical aberration that degrades the focused spot. Liu, Shieh, Ju, Tsai, Yang, Chang and Liu [ 1998] have shown that a dual-focus lens using a ring system with alternating DVD and CD aspherical surfaces is a low-cost method for allowing the optical pick-up head to read both disk types. They also showed that there is an improvement in spot quality, which will improve the readout
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[2
signal as well. The demand for compatibility between CD and DVD ensures that an optical head of this or a similar type is implemented. Holographic Optical Elements (HOEs) also appear to be gaining credibility in the optical disk industry. The advantage of a HOE is that it can combine multiple elements of the optical head together into one holographic element, thus decreasing the size of the system at perhaps a lower cost, but they are also far less efficient than the conventional components. Holographic beam splitters used in CD pick-up systems typically have an 8% round-trip efficiency, or using two photodiode arrays the efficiency can be increased to 16%. Compared to about 25% efficiency for conventional components, there is little incentive to use a HOE. Some of the difference comes from light losses due to different diffracted orders that are not used. Freeman, Shih, Chang, Wang, Chen, Chuang and Chang [ 1998] have suggested a high efficiency HOE which, using a three-beam grating built into the HOE, reclaims some of the lost light and results in an optical power increase of about 50%. The boost in power makes the round-trip efficiency of the HOE roughly equivalent to that of conventional components. With HOEs such as the one proposed, optical heads can be further reduced in size and possibly cost to improve the overall system. 9.9. FINAL REMARKS
Considering the incredible advances made in optical data storage within just the last five years, it is difficult to imagine where the industry will be 10 to 20 years from now. Many of the future trends listed above will shortly become a reality, changing the face of the industry yet again. Those trends that do not make it will most likely be improved upon and implemented at some later date. One thing is for certain, and that is optical data storage is experiencing an incredible boom with new ideas and technology emerging every day. It will be interesting to see what the future has in store for us all.
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E. WOLF, PROGRESS IN OPTICS 41 9 2000 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED
ELLIPSOMETRY OF THIN FILM SYSTEMS
BY
IVAN OHLiDAL AND DANIEL FRANTA
Department of Physical Electronics, Faculty of Science, Masaryk Unioersity, Kotl6~sk6 2, 611 37 Brno, Czech Republic
181
CONTENTS
PAGE w 1.
INTRODUCTION
w 2.
THEORETICAL
w 3.
PRINCIPLES OF ELLIPSOMETRY
w 4.
THEORY OF MEASUREMENTS
w 5.
ELLIPSOMETRIC SYSTEMS
w 6.
. . . . . . . . . . . . . . . . . . BACKGROUND
187
IN E L L I P S O M E T R Y . . . .
189
QUANTITIES OF IDEAL THIN FILM 196
QUANTITIES OF IMPERFECT THIN FILM
. . . . . . . . . . . . . . . . . . . . . .
w 7.
EXPERIMENTAL METHODS
w 8.
CONCLUSION
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211 235
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276
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278
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278
ACKNOWLEDGMENTS REFERENCES
183
. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
ELLIPSOMETRIC SYSTEMS
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183
182
w 1. Introduction Thin film systems are often encountered in practice. They are used in various branches of high technologies and industry. For example, these systems play an important role in nanotechnology, microelectronics, optoelectronics etc. Therefore it is necessary to have experimental methods that enable us to analyze thin film systems. Optical methods are useful for efficiently characterizing many thin film systems. This is why the optical methods suitable for analyzing the systems mentioned were developed extensively during the last three decades. Ellipsometry is one of the most important optical techniques used to analyze even very complicated thin film systems. Moreover, ellipsometric methods can be applied by using relatively simple experimental arrangements in comparison with the other techniques employed for characterizing complicated thin film systems (e.g., some electron beam techniques). These are the main reasons for the enormous development and progress of this experimental technique in the fields of fundamental research, applied research and practice. In this chapter, a review of both the important theoretical and experimental results concerning ellipsometry will be presented. It should be noted that in this chapter it is impossible to carry out a complete review of the results achieved in this field because of their enormous extent. We shall therefore limit the discussion to the most significant or representative results. Attention will be paid especially to the practical use of ellipsometry in obtaining both theoretical and experimental results.
w 2. Theoretical background In this section the theoretical background of ellipsometry will be presented. This means that the main theoretical considerations and formulae that make it possible to use ellipsometric methods for studying thin film systems will be introduced. 2.1. MODELOF THE THIN FILM SYSTEM We shall deal with the thin film systems corresponding to an ideal model. This ideal model is specified by means of the following assumptions: 183
184
ELLIPSOMETRYOF THINFILM SYSTEMS x
y
[3, w 2 x'
l ."z
I
s
,,o y t
I
eo Fig. 1. Schematic diagram of the incident wave on the system S and the emergent wave from this system. The symbols Ei and Eo denote the Jones vectors of the incident and emergent waves, respectively. Cartesian coordinate systems (x,y,z) and (xl,yl,z p) at the input and output of S are plotted as well.
(1) Optically the ambient of the system is formed by a non-absorbing, homogeneous, isotropic, material. (2) Materials forming the thin films and substrate of the system are optically homogeneous. In general it is assumed that these materials are optically anisotropic and absorbing. (3) The boundaries of the system are sharp (i.e., transition layers on the boundaries are not assumed). (4) The boundaries of the system are formed by parallel planes. In addition we shall deal with thin film systems that differ from this ideal model. Thus attention also will be paid to the systems that exhibit some defects. 2.2. JONES FORMALISM
Let us consider a polarized monochromatic plane wave incident on the thin film system. As a result of the interaction between the incident wave and the system, a modified plane wave emerges from this system in comparison with this incident wave. Figure 1 shows the schematic diagram of this situation. Two right-handed Cartesian coordinate systems (x,y,z) and (x',yt, z I) are associated with incident and the outgoing plane waves with the directions z and z t taken parallel to their wave vectors ki and ko, respectively (ki and ko need not be parallel). We shall assume that x and x' and/or y and y~ coordinate axes are parallel and/or perpendicular to the plane of incidence (the plane of incidence is given by the wave vector of the incident wave ki and normals to the boundaries of the system). Both the plane waves mentioned are described by the Jones vectors F,i (incident) and/~o (outgoing). These Jones vectors are two-dimensional, complex ones, i.e., Ei-/~iy
/~ix
)
and
Eo-- (
Eox
/~os
)
'
(2.1)
3, {} 2]
THEORETICAL BACKGROUND
185
where Ei~ and Eiy and/or Eox and Eoy denote complex amplitudes of the electric fields of the p- and s-polarized incident and/or outgoing monochromatic plane waves, respectively. The Jones vector completely characterizes the polarization state (polarization) of the polarized monochromatic plane wave, i.e., this vector expresses the absolute amplitude and absolute phase of the wave. In this chapter the components of the Jones vectors will be expressed by means of the Cartesian basis vectors (of course, one can use the other orthonormal basis vectors for describing these components). If the system is optically linear (i.e., if within the system nonlinear and depolarization effects are absent), one can write the following matrix relation:
(F-~ox
JxxJxy)(~,iy )
(2.2)
or, more concisely, /~o -- J/~i.
(2.3)
The 2 • 2 matrix ,] is called the Jones matrix of the optical system and, in general, its elements Jij are complex numbers. 2.3. REPRESENTATION OF POLARIZED LIGHT BY COMPLEX NUMBERS: THE POLARIZATION TRANSFER FUNCTION
Practically it is more useful to describe the polarization states of the light waves by means of the relative amplitudes and the differences of the phases corresponding to the p and s polarizations of these waves. It is thus suitable to define the following complex number )~"
L
2 - ^ 9
E~
(2.4)
The complex number )(o -- Eox/Eoy and/or Xi = E~/P~iy represents the polarization state of the incident wave on the thin film system and/or the polarization state of the outgoing wave from this system. Equations (2.2) and (2.4) imply that the relationship between Xo and ~'i is given by a bilinear transformation expressed by the following equation:
Jxx2i -k-Jxy ^ . 2~ = Jyx2i + Jyy
(2.5)
The relationship Xo = f(xi) in eq. (2.5) is called the polarization transfer (PTF) of the optical system. Note that the PTF is determined by
function
186
ELLIPSOMETRYOF THIN FILM SYSTEMS
[3, w 2
X
E~0 Fig. 2. A graphical representation of the azimuth 0 and ellipticity e.
the elements of the Jones matrix unambiguously but the inversion is not true. Therefore it is appropriate to define the normalized Jones matrix in the following way: 'In = (/51 t53 /512) '
(2.6)
where t51 = Jxx/Jyy, 152 = Jxy/J~ and P3 = Jyx/JAy. The PTF and the normalized Jones matrix are thus optically mutually equivalent. In this case the PTF is evidently given as follows: Zo = ,bl2i^^ + 152 P3Zi+ 1 "
(2.7)
Theory concerning the properties of polarized light is described in detail in the monograph of Azzam and Bashara [ 1977]. 2.4. DESCRIPTION OF POLARIZED LIGHT BY AZIMUTH AND ELLIPTICITY
In general, superposition of the linearly polarized waves represented by p and s polarizations gives the elliptically polarized wave. The electric vector E(t) of this elliptically polarized wave traces an ellipse at a fixed point in space (t is time). The form of this ellipse is determined by the angles 0 (azimuth) and e (ellipticity) plotted in fig. 2. The electric vector E(t) can be expressed in the following way:
E(t) = Re[(~'xX + Eyy) eit~
(2.8)
where x, y and to denote the unit vectors in the directions of the x-axis and y-axis and the angular frequency, respectively. The symbol Re denotes the real part of the corresponding complex quantity.
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PRINCIPLES OF ELLIPSOMETRY
187
It can be shown that the relations between the complex number ~' and the angles 0 and e can be derived in the following forms: 1 + i a tan 0 tan e 2' = tan 0 - i a tan e
(2.9)
and tan 20 =
2 Re(~) [212- 1
and
2 Im(~) sin 2e = o ~ 2 2 + 1'
(2.10)
where the symbol Im denotes the imaginary part of the corresponding complex quantity. It holds that o = 1 for right handed and a = -1 for left handed elliptically polarized light waves.
w 3. Principles of E l l i p s o m e t r y
Ellipsometry is a technique that enables us to measure the polarization state of the polarized light wave emerging from the system studied owing to the polarization state of the polarized light wave incident on this system. In principle it is possible to realize ellipsometric measurements in two basic ways. The first way concerns the ellipsometric measurements performed for the light waves specularly reflected or directly transmitted by the system. The application of the latter way is based on the ellipsometric measurements in waves scattered when the system is investigated. In practice the ellipsometric measurements are mostly carried out for the waves specularly reflected or directly transmitted by the system. Therefore we shall only deal with ellipsometry concerning the waves reflected or transmitted by the systems studied. If the ellipsometric measurements are performed in reflected light (reflection mode), the elements of normalized Jones matrix are expressed as follows: ^
^
^
rpp, f~2 = rps and /~3- rsp , (3.1) rss rss rss where ~.pp, ?~, ?p~ and ?~p are called the complex reflection coefficients of the f)l
-
system. In the case of ellipsometric measurements carried out in transmitted light (transmission mode), the elements of normalized Jones matrix are expressed as follows:
~31- ~pp
P2 = tps
and
133-
tsp
(3.2)
where tpp, ~, tps and t~p are called the complex transmission coefficients of the system.
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ELLIPSOMETRYOF THIN FILM SYSTEMS
[3, w 3
3.1. CONVENTIONAL ELLIPSOMETRY
If the optical properties of the thin film system imply that the Jones matrix is the diagonal one (Jxy = Jyx -- 0) it is possible to apply conventional ellipsometry. In this case the normalized Jones matrix exhibits the simpler form; i.e.,
1 )'
(3.3)
where ~ = ~P r,
or
r
~p. t~
(3.4)
The symbols ~p and/or ~ and tp and/or t~ denote the reflection Fresnel coefficient of the system for the p- and/or s-polarized wave and the transmission Fresnel coefficient of the system for the same waves, respectively (~'pp = ~'p, ~'~ = ~'~, rps -- rsp -- 0, l p p - - t p , t s s = t s and tp, = t,p = 0). Further, one can write t5 = tan tp e ia,
(3.5)
where tp and A represent the ellipsometric parameters of the system for the reflected or transmitted light waves (tp and/or A is called the azimuth and/or the phase change). The PTF is then expressed by means of the the following linear form: 20 = b 2i-
(3.6)
From the foregoing it is clear that within conventional ellipsometry, the PTF is uniquely determined by means of one pair of corresponding polarization states on the input )(i and the output Zo of the system under consideration. It should be noted that this conventional ellipsometry can be used to characterize the thin film systems formed by optically isotropic materials. Of course, in special cases of optically anisotropic systems, conventional ellipsometry can also be employed for their analysis. In these special cases anisotropic media forming the thin film systems have the principal axes parallel or perpendicular to the plane of incidence of light. 3.2. GENERALIZED ELLIPSOMETRY
If the Jones matrix of the thin film system is not the diagonal one (the PTF is given by the bilinear form), generalized ellipsometry must be applied. This
3, w 4]
THEORY OF MEASUREMENTS IN ELLIPSOMETRY
189
statement is true for optically anisotropic systems whose anisotropic materials exhibit the optical activity or whose principal axes are situated in general position with respect to the plane of incidence. Within generalized ellipsometry it is necessary to know at least three independent input polarization states (~1,)(i2,)(i3) and the corresponding three output polarization states (Zol,)(o2, Zo3) of the system studied. Both the normalized Jones matrix and the PTF are then determined unambiguously by these states in the following way: /51 = P2 =
2olXo2(Xil - 2i2) + )(o3)(o 1C~i3 - 2il ) + 2o22o3 (Xi2 -/~i3)
~
,
2o12o2(Xil -- 2i2)Xi3 + Xo3)(ol(Xi3 - 2il)/~i2 + )(o2Xo3(/~i2 - 2i3)2il
b /53 =
(3.7)
)(o3 (~il -- )(i2) + )(o2 (Xi3 -- )(il) + )(ol (Xi2 -- )(i3)
b
(3.8) ,
(3.9)
where b = 2o32i3 (/~il -- 2i2) + 2o22i2(/~i3 -- 2 i l ) + 2o12il Q~i2 -- )(i3)"
(3.10)
w 4. Theory of Measurements in Ellipsometry By means of ellipsometric measurements the PTF or the normalized Jones matrix of the optical system is determined. There are several kinds of arrangements (ellipsometers) that can be used to measure these quantities for the reflected or transmitted light waves by the systems under investigation. The schematic diagrams of the examples of the ellipsometers frequently utilized in practice are introduced in figs. 3 and 4. The ellipsometers corresponding to these schematic diagrams are known as the PCSA (polarizer-compensator-sample-analyzer) ellipsometers. These ellipsometers can operate in reflection (fig. 3) and transmission (fig. 4) modes. The ellipsometers PSCA and PSA also are often employed in practice. Here the principles of the ellipsometric measurements obtained by using the main types of the ellipsometers will be described briefly. 4.1. NULL ELLIPSOMETRY
Null ellipsometry is based on finding a set of azimuth angles for the polarizer, compensator and analyzer in such values that the light flux falling on the detector
190
ELLIPSOMETRYOF THIN FILM SYSTEMS
P
[3, w 4
A
C X
X~
S Fig. 3. Schematic diagram of the PCSA ellipsometer working in reflected light: L, P, C, S, A and D denote the light source, polarizer, compensator, sample, analyzer and detector, respectively.
P
C
A
Fig. 4. Schematic diagram of the PCSA ellipsometer working in transmitted light. All the symbols have the same meaning as in fig. 3.
of the ellipsometer is extinguished. If the polarizer, analyzer and compensator are optically ideal elements, the Jones vector of the wave incident on the sample/~i is given as (the PCSA ellipsometer with a quarter-wave compensator is considered) /~i -- l ~ ( - C ) ] c I~(C - P ) E p ,
(4.1)
where l~, Jc and/~e are expressed as follows: l~(a) = ( _c~
Jc =
0 -i
a cosSina)a ,
(4.2)
'
1
The matrix 1~ and/or ] c represents the transformation of the Jones vector under the effect of a coordinate rotation and/or the Jones matrix of the quarter-wave
3, w 4]
THEORY OF MEASUREMENTSIN ELLIPSOMETRY
191
compensator 1. The symbols P and C denote the azimuth angles of the polarizer and compensator, respectively. After inserting the expressions of the components of the Jones v e c t o r / ~ i into eq. (2.4), one obtains )~i = 1 + i tan C tan(P - C). tan C - i t a n ( P - C)
(4.5)
The Jones vector of the light wave falling onto the detector/~D is expressed as
/~D : ,IA I~(A)/~o,
(4.6)
where 1 0) oo
(4.7) 9
The matrix JA is the Jones matrix of the analyzer corresponding to its own coordinate system and/~o denotes the Jones vector of the light wave emerging from the sample. From eq. (4.6) one can see that /~o = (/~ox cosA +/~oy sinA ) 0
(4.8)
where A is the azimuth angle of the analyzer. If the zero light flux is recorded by the detector, i.e.,/~z) = 0, the following two conditions are simultaneously fulfilled: (1) The angles of the polarizer and compensator are set at the values corresponding to the linear polarization state of the wave emerging from the sample (the value of )(o = is a real number). (2) The transmission axis of the analyzer crosses the linear polarization of the light wave emerging from the sample.
Eox/Eoy
1 The matrix Jc expresses the influence of the compensator on the Jones vector of the light wave passing through compensator under the assumption that the x-axis of the coordinate system is oriented in the direction of its fast axis.
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ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 4
The polarization state of the light wave emerging from the sample is then given as follows: )(o = - tan A,
(4.9)
It should be noted that the angles P, C and A are measured from the plane of incidence and that the angle C is related to the faster axis of the compensator. The polarization states )(i and )(o expressed by eqs. (4.5) and (4.9) must then be substituted into eq. (3.6) and/or eqs. (3.7)-(3.10) if conventional and/or generalized ellipsometry is applied. In the special case of conventional ellipsometry, i.e., for C = ;r/4, the following simple equations are true =A =-A
and and
3Jr A=-~-ZP
for
Jr AC ( 0 , ~ ) ,
(4.10)
A=~-2P
for
AE(--~,0).
(4.11)
Details concerning null ellipsometry are presented in many works (e.g., in the monograph of Azzam and Bashara [1977] or in the paper by Merkt [ 1981 ]). 4.2. ROTATING-ANALYZER ELLIPSOMETRY
PSA ellipsometers are mostly employed within rotating-analyzer ellipsometry. The polarizer is fixed in a position corresponding to an azimuth angle P (P ~ 0,Jr/2,Jr, 3/21r) and the analyzer rotates, i.e., the azimuth angle A is a function of time 2. The Jones vector of the light wave falling onto the detector can be expressed by the following relation:
F.o(t) oc JA R(A(t)) Jn R(-P) E,e,
(4.12)
where ,], is the normalized Jones matrix of the system studied (see eq. 2.6). Hence light flux recorded by the detector is given as
I(t) cx ]/~ox(t)l2 o( 1 + Is sin 2A(t) + Ic cos 2A(t).
2 Rotating-polarizer ellipsometry is based on the same principle.
(4.13)
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193
If the function A(t) is linear, the coefficients Is and Ic are Fourier components of the harmonic light flux I(t). Within conventional ellipsometry these coefficients are expressed as follows: Is = 2Re(/5)cosP sinP ]/~]2 COS2 p + sin 2 p
and
Ic = ]/512c~ P - sin2 P 1/012cos 2 p + sin 2 p"
(4.14)
Equations (4.14) imply the following (see eq. 3.5): t
tan q t =
tanP~/{ +Ic
and
cosA=
-Ic
Is
(4.15)
v / l - / c 2"
It should be noted that within this ellipsometry, the sign of the phase change A is not determined unambiguously (i.e., it is impossible to determine the handedness of the polarization state of the light wave outgoing from the sample). A detailed discussion of the rotating-analyzer ellipsometry is performed, for example, in the works of Aspnes [1973, 1974], Azzam and Bashara [1977] and de Nijs, Holtslag, Hoekstra and van Silfhout [1988]. Expressions of the coefficients Is and Ic for generalized ellipsometry are presented in the paper by Schubert, Rheinlgnder, Woollam, Johs and Herzinger [ 1996]. In this case three or more azimuth settings P are needed for determining the normalized Jones matrix. However, the signs of imaginary parts of the complex elements of this matrix cannot be determined (this is the same situation as for conventional ellipsometry). 4.3. P H A S E - M O D U L A T E D ELLIPSOMETRY
In phase-modulated ellipsometry, PCSA or PSCA ellipsometers must be utilized. The polarizer, compensator and analyzer are fixed in certain positions. The phase retardation of the compensator 6 is a function of time. The Jones vector of the light wave falling onto the detector is expressed in the following way:
ED(t) oc JA (~(A)Jn (~(-C) Jc(t) R(C- P) Ee,
(4.16)
where ~
Jc(t)
=
(1
0 )
0 e ir(t)
(4.17)
is the Jones matrix of the compensator. Then the x-component of ED(t) is given as
/~9x(t) - (sinA sin C +/5 cosA cos C) cos(P - C) x eia(t)(sinA cos C - 15cosA sin C) sin(P - C).
(4.18)
If the azimuth angles of the elements of the ellipsometer used fulfill the relations P - C = +:v/4, C = 0, :v/2 and A = i:~/4, then eq. (4.18) implies the
194
ELLIPSOMETRYOF THINFILMSYSTEMS
[3, w4
following relation for the light flux I(t) recorded by the detector in this simple form:
I(t) cx 1 + lpl 2 -'l- 2Im(/5) sin 6(t) 9 2Re(/5)cos 6(t).
(4.19)
If the latter useful configuration of setting the azimuth angles of the elements expressed by relations P - C = • C = + : r / 4 and A = + : r / 4 is used, the relation for the light flux I(t) is given as follows:
I(t) cx 1 +
1/512i
2 I m p ) sin a(t) -4- (1 -1/512) cos 6(t).
(4.20)
Both the foregoing relations eqs. (4.19) and (4.20) can be written in one general expression
I(t) cx 1 + Is sin 6(t) + I~ cos 6(0.
(4.21)
The coefficients Is and Ic are called the associated ellipsometric parameters. For the configuration P - C = A = :r/4 and C = 0 and/or P - C = A = C = 3:/4 they are associated with the ellipsometric parameters tp and A in the following equations (see eqs. 3.5, 4.19 and 4.20): Is = sin 2 tp sin A
and
Ic = sin 2 tit cos A
(4.22)
and
Ic = cos 2 tp.
(4.23)
and/or Is = sin 2 tp sin A
Under the following assumption:
6(0 = .A sin tot
(4.24)
the functions sin 6(0 and cos 6(t) can be expressed by means of these series: oc
sin 6(t) = Z 2J2j + l(.A)sin[(2j + 1)tot], j=0
(4.25)
COS 6(t) = J0(A) + ~
(4.26)
ZJ2j(A) cos[2j~ot],
j=l
where Jn denotes the Bessel function of the n-th order, to is the angular frequency of the phase modulation of the compensator and .,4 is the amplitude of this modulation.
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After inserting eqs. (4.25) and (4.26) into eq. (4.21), one obtains the expression for the light flux I(t) in the form of the following series: I(t) o< 1 + 1(0 sin oot + 12(0cos 2~ot + . . . (higher harmonics).
(4.27)
Then the fundamental harmonic 1(0 and second harmonic 12(0 of the light flux are associated with the parameters Is and Ic through the following equations: Is =
J2(A)I~ J~ (A)[2Jz(.A) - J0(.A)I2o,]
(4.28)
and 4 =
2(0
2J2 (,A) - Jo(A)/2(0"
(4.29)
A detailed discussion of phase-modulated ellipsometry is carried out in the works of Azzam and Bashara [1977], Acher, Bigan and Dr~villon [1989] and Kim, Raccah and Garland [1992]. 4.4. OTHER ELLIPSOMETRIC TECHNIQUES
There are several other ellipsometric techniques that are also employed in practice. They are as follows: 9 Oscillating-analyzer ellipsometry: In this technique the PCSA ellipsometer is used. However, the analyzer of this ellipsometer is formed by a composed element consisting of an ac-driven optical rotator (Faraday cell) followed by the usual fixed analyzer. Theory of measuring within this ellipsometry is described by Azzam [ 1976]. 9 Rotating-compensator ellipsometry: The PCSA ellipsometers are used for this ellipsometry. The compensator rotates while both the polarizer and analyzer are fixed. Theory of this ellipsometric technique was presented by Hauge and Dill [1975]. A review of the other ellipsometric techniques with rotating elements is outlined in the paper by Aspnes and Hauge [ 1976]. 9 Rotating polarizer and analyzer ellipsometry: The PSA ellipsometer is used in this technique. The analyzer rotates with the polarizer synchronously. The ratio of frequencies of the analyzer and the polarizer is given by a rational number. Theoretical considerations concerning this ellipsometry are published in the paper by Chen and Lynch [ 1987].
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ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 5
9 Two-modulator generalized ellipsometry: The PCSCA ellipsometer is used in this case. The instrument consists of two photoelastic-modulators operating at different frequencies. With this technique it is possible to determine all six elements of the normalized Jones matrix from one set of positions of the elements of the ellipsometer unambiguously (see papers by Jellison and Modine [ 1997a,b]). 9 Return-path ellipsometry: This ellipsometry is realized by means of the PSA ellipsometers with one polarizer acting as both the analyzer and polarizer. The technique mentioned can be applied not only for an oblique incidence but also for the normal incidence of light onto the sample. This kind of ellipsometry is helpful for studying optically anisotropic materials. A theoretical background of this ellipsometry is described in papers by Azzam [ 1977, 1978]. In the literature the reader can find further special ellipsometric techniques that are not often utilized in practice. Therefore, they are not mentioned in this chapter.
w 5. Ellipsometric Quantifies of Ideal Thin Film Systems In this section an interaction of the light waves with the thin film systems corresponding to the ideal model defined in w 2.1 will be described by means of the optical quantities concerning ellipsometry. In principle it is sufficient to determine the values of the complex reflection and transmission coefficients ?'pp, t'ps, ~'sp, t'ss, tpp, lps, tsp and tss of the system studied. In cases where the system is formed by the isotropic materials, it is sufficient to determine the values of the Fresnel coefficients ~p, ~s, tp and ts. Using these coefficients one can determine the ellipsometric quantities; i.e., the PTF and elements of the normalized Jones matrix of the system in an unambiguous way (see w 3). In the following subsections formulae for the coefficients mentioned will be introduced for the different systems separately. It will be assumed that in all the materials forming the thin films and substrates of the systems considered below, currents caused by external electric fields and free electric charges will not take place. Thus the electromagnetic fields existing in these materials must satisfy the Maxwell equations in the following form: rotE rot[/-
oB 0t'
ob
0t'
(5.1) (5.2)
3, w 5]
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197
divD = 0,
(5.3)
divB = 0,
(5.4)
where/~ and//denote the electric and magnetic fields, respectively, and/) and/or represents the electric displacement vector and/or the magnetic induction vector. Further these quantities are mutually coupled by the material relations:
b = co ~ k,
(5.5)
k = ~ l~f/,
(5.6)
where e0 and ~ are the permittivity and permeability of vacuum, respectively, g: is the relative permittivity tensor and !~ is the relative permeability tensor. These tensors of the second order are generally the complex ones. Below we shall assume that ~ = 1 (from the practical point of view this equality is true for all materials within the region of frequencies of the electromagnetic waves in which ellipsometry is applied). For optically homogeneous materials the solutions of eqs. (5.1)-(5.6) will be assumed to be in the form of the monochromatic plane waves. This means that one can write /~(t, P) = A,b e i(c~
(5.7)
where .4,/~, m, t, k and ? are the amplitude of the electric field, the unit polarization vector, the angular frequency, time, the wave vector and the radius vector, respectively. From eqs. (5.1) and (5.6) it is implied that the magnetic field is expressed as follows:
k(t,~,) = -G-~(k^ • i,)
ei(~ot_/~)
(5.8)
5.1. OPTICALLY ISOTROPIC SYSTEMS
In the isotropic thin film systems the materials forming the individual media are characterized by scalar material quantities; i.e., ~2 = ~ ( 0 ) ) = ~2,
(5.9)
where h is the complex refractive index of the corresponding material. Every monochromatic plane wave propagating within the optically isotropic medium can be expressed by means of a superposition of two linearly polarized
198
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 5 ^
nj+l ^+
Aq,j+t
d
j-th boundary
(j+l)-th boundary
Fig. 5. Schematic diagram of the jth layer of the isotropic thin film system. waves. If these two linearly polarized waves are chosen so that one of them is polarized perpendicularly to the plane of incidence (s-polarization) and the latter one is polarized parallel (p-polarization) with this plane, it is possible to separate the solution of Maxwell equations corresponding to the system considered to two particular solutions belonging to these p- and s-polarized waves. We can then introduce the following 2 • 2 matrix formalism. The boundary conditions for the electromagnetic fields give the following matrix equation for the j-th boundary of the system: ^
,,~
^
Aq,j _ 1 "-" BqjAqj, where
Aqj
q = p, s,
(5.10)
denotes the vector whose components are formed by the complex A-+-
^ _
amplitudes Aqj a n d Aqj of the electric fields belonging to the fight-going and left-going waves inside of the j-th film, respectively (see fig. 5), i.e.,
^ AqJ=
Aqj A-qj
.
(5.11)
The vector Aq,j_ 1 has the same meaning for the ( / - l ) - t h film as the vector for j-th film. Thus matrix Bqj (refraction matrix) is defined as follows:
~qj=l(l~'qj)
9
hqj
(5.12)
This matrix expresses the binding conditions for the amplitudes mentioned at the j-th boundary. The symbols t'qj and tqj represent the Fresnel reflection and transmission coefficients, respectively, for the wave incident on the boundary
3, w 5]
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199
from the left side. They are given as follows (see, e.g., Vagi~ek [ 1960] and Knittl [1976]): ~pj __ l'lj COS Oj-1 -- nj-1 cos Oj
(5.13)
njcos0j_l +nj_lCOS0 j' ^
2 nj_l COS Oj'-1
(5.14)
tpj : ~'lj cos Oj._ l -l- ~'lj_ l COS Oj.' ~sj
~'lj-l COS Oj'- l -- l'ljcos Oj"
lCOSOj_+h cosg' tsj =
2 nj-1 COS 0j_ 1 nj_lCOSOj-_I +njcosOj.'
(5.15)
(5.16)
where the symbols by_ 1, hj, 0j-1 and t)j denote the complex refractive indices of both the (j-1)-th and j-th films and the complex angles of incidence and refraction of the j-th boundary, respectively. The complex angles Oj_ 1 and fulfill Shell's law; i.e., no sin Oo = hj-1 sin ~-1 = hj sin Oj,
(5.17)
where O0 is angle of incidence on the thin film system and no is the refractive index of the ambient (no is assumed to be a real quantity). The amplitudes of the waves inside of the j-th film corresponding to the j-th and (j + 1)-th boundaries are connected with the elements of the phase matrix "rj, i.e.,
O)
0 e-~
'
(5.18)
where
-- 4 os0j-
2Z hjdj cos 0j.
(5.19)
In the foregoing equation the symbol dj represents the thickness of the j-th film and Z denotes the wavelength of the wave incident on the system.
200
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 5
From the preceding considerations and equations, one can see that the following equation is fulfilled for the amplitudes of the incident, reflected and transmitted waves for a thin film system containing N films: Aqo -- ]3ql ~'lBq2~'2 . . .
BqN~rNBq,N+IAq-- I~r
(5.20)
where the vectors Aq0 and Slq exhibit the special form if the wave with the amplitude equal to unity falls on the system considered, i.e., 1 hqo-'-(~.q),
2q--( ~q
"
(5.21,
The symbols kq and ~q denote the Fresnel coefficients for reflected and transmitted waves by the system, respectively. From the foregoing it is apparent that they are determined as follows (see eq. 5.20): ? ' q - lQIq21 a n d /~/lq 11
~q_
1
(5.22)
]QIql I "
The matrix l~q is called the overall transfer matrix of the thin film system. The foregoing equations imply that the Fresnel coefficients Fq and tq must be understood as the functions of/l, no, 00 and the parameters characterizing the system. Thus one can write
l'q = ?'q(l~,Oo, no, dl, hl, d2 . . . . .
dN, hN, h)
(5.23)
tq = tq(/l, 00, no, dl, hi, d2, . . . , dN, nN, h),
(5.24)
and
where h - h N + l is the refractive index of the substrate. The ellipsometric parameters characterizing the isotropic thin film system can then be calculated by means of the following equations (see eqs. (3.4 and 3.5): tan ~ e iA - ~p
or
tan t/_/eia - ~tP.
~s
(5.25)
t~
5.2. OPTICALLY ANISOTROPIC SYSTEMS
In the anisotropic thin film systems the materials forming the individual media are characterized by permittivities that are mathematically described by the complex tensors of the second order from the general point of view.
3, w 5]
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201
From the Maxwell equations one can derive the following wave equation (see also eq. 5.7)"
k x (k x E')+ fi, E' = O,
(5.26)
where quantities &0 is given as ~ij -- (D2 ~l,of, o~ij -- k2 ~o.
(5.27)
Equation (5.26) can be rewritten into the following matrix form:
ayx
ayy - k2 - ~2
+ kxL
ayz
F_.~y
^ azz-kx
#z
=0
(5.28)
or, more concisely, lq/~ = 0. From the boundary conditions for the electric and magnetic fields, one can find that the x-components kx of the wave vectors must be identical in all media of the thin film system. These x-components of the wave vectors are thus real because they are given by the x-component of the wave vector of the wave incident on the system under consideration3 i.e. 2:r kx = ko sin 00 = -~--n0 sin 00.
(5.29)
In order to have nontrivial solutions, the determinant of the matrix lq in the eq. (5.28) must be equal to zero (detlq = 0). This gives the equation of the fourth order in lcz which yields four roots lczv (v = 1, 2, 3, 4) that correspond to two eigenwaves propagating in the positive direction of the z-axis and two eigenwaves propagating in the negative direction of the z-axis. If these roots are independent we can calculate the unit polarization vectors 4 of these waves by means of the solution of the following equations: l~lv/~v = 0
and
/~/~v = 1,
(5.30)
where 1~,, is matrix from eq. (5.28) if the substitution lcz = lczv is performed. The symbol asterisk denotes complex conjugation. Vectors/~ then correspond to inhomogeneous elliptically polarized plane wave in general.
3 Note that the Cartesian coordinate system is chosen in such a way that the z-axis is perpendicular to the boundaries of the system and the x-axis is parallel with the plane of incidence of light, i.e., /, = (k~,o, kz). 4 Normalization of the polarization vectors is not necessary.
202
[3, w 5
ELLIPSOMETRY OF THIN FILM SYSTEMS
If the solution of det/q = 0 contains two different roots and one double root fc=o, the unit polarization vectors corresponding to this double root cannot be determined by eq. (5.30) unambiguously. The unit polarization vectors of the waves belonging to the double root can then be chosen in the following way: Ps = (0, 1, O)
and
1
^
1
/~p : T-(ko x p~) = -r ko ko
^
O, kx),
(5.31)
where ko = ~/k~G + k}. The vector ibs and/or lop corresponds to transverse electric (TE) and/or transverse magnetic (TM) mode of inhomogeneous linearly polarized plane wave. In practice this situation sets in propagating the wave in the direction identical with one optical axis of the anisotropic materials. If the solution of det lq = 0 contains two double roots, the unit polarization vectors of the waves belonging to double roots can then be chosen in the following manner: 1
/~s+ = / ~ - = ( 0 , 1,0),
/~p = ~(k+ X/)s) k
1
and
pp ^-
=
-~(kk
Xbs),
(5.32)
where k+ and/or k- is wavevector of the right-going waves and/or left-going waves. In practice this situation corresponds to the propagation of the waves in the isotropic materials or the propagation of the waves along the optical axes of the anisotropic materials 5 The electric field inside the j-th thin film of the system considered is then expressed as follows: 4
F.j = ~ .4jr i~jv e i(t~
(5.33)
v=l
From eq. (5.8) it is implied that the magnetic field inside this film is given by the following relation: 4
flj ~ ~ .4jr ~ljve i('~
(5.34)
v=l
5 Note that the polarization vectors ibp = (+ cos 0, 0, sin 0) defined in eqs. (5.31) and (5.32) are not unit vectors, as defined in the definition of the Fresnel coefficients in eqs. (5.13)-(5.16).
3, w 5]
ELLIPSOMETRIC QUANTITIES OF IDEAL THIN FILM SYSTEMS
203
where ~j~, is the vector determining the direction of the magnetic field. This vector is expressed in this way
#j~ =/,j~ • bj~.
(5.35)
The continuity of L , Ey, & and/~y at the boundary between both (j-1)-th and j-th films imply the following equations for the amplitudes of the waves propagating inside the ambient, films and substrate of the system: 4
4
ftj _ 1,,.,~j -1, ,,,x = Z
Aj~, i~j ~ x,
(5.36)
.'~j - l ,v i~j - ~,v Y = ~-~ Ajv i~jv Y,
(5.37)
v=l
v=l
4
4
Z v=l
v=l
4
4 X
-
v=l
4
4
~-~Aj- l,v Oj- l,v Y = Z v=l
(5.3s
x,
v=l
Ajv gljv y,
(5.39)
v=l
where x and y denote the unit vectors in the directions of the x-axis and y-axis, respectively. These four equations can be rewritten in the form of the following matrix equation: (5.40) where the matrix l)j is given as follows:
{ i~jl X Pj 2X Pj 3X Pj 4 X ~
| i,j~ y
i,j2 y
bj3 y
l)J--kqjlx qjlY
qj2x qj2Y
qj3x qj4x ~IjZY qj4Y
ilj4 y "
(5.41)
If the j-th layer is formed by isotropic material, we can define the following identities" pjl - Ppj"+,i~j2 =- P~.," i~j3 - Psj"+and J~j4 ~ b ~ " The matrix l')j then exhibits the special form
iCzJ/~O -fCz~/~. o
O1 O1
o
" o
(5.42)
204
ELLIPSOMETRYOF THIN FILM SYSTEMS
[3, w 5
If the material of the ( j - 1 ) - t h medium of the system is also isotropic, the matrix 13s has evidently the following special form:
"J -- l); l l l)J -- ( " pj~) " sj~)) '
(5.43)
where flpj and ITIsjare the 2 • 2 matrices expressed in the foregoing section [see eq. (5.12)] and 0 denotes the zero matrix. It is then apparent that the eq. (5.40) corresponding to the 4 • 4 matrix formalism can be separated into two equations corresponding to 2 x 2 matrix formalism introduced in the foregoing section (see eq. 5.10). In general, the phase matrix of the j-th film can be written as follows:
e-'~J'~ 0 0 e-ik:J2dj 1"J=
o 0
o 0
0
0
e -i~j3dj
0
0
0
0
e -i~'j4dj
(5.44) "
It is further clear that the vectors of the amplitudes of the waves in the ambient and the substrate are associated by the following matrix equation: "~0 = B l ' r l m 2 ' r 2
--"
BNTNBN+I.4
= /~,
(5.45)
where M is the overall transfer matrix of the thin film system containing N films. If the p- and/or s-polarized wave with the unit amplitude falls on the system, the vectors ,40 and .31 exhibit these special forms, ^
f4o =
rpp 0
,4 =
(5.46)
,
r~p and/or
0 ~i0 =
~ps
?m 1
rss
,~ = '
0 tss
(5.47) "
o
Equation (5.45) then implies the following equations for the complex reflection and transmission coefficients of this system: ^
-
rpp --
M21M33 - M23~I31 1(/IlllQ[33 _ 1QI131QI31
,
(5.48)
3, w 5]
ELLIPSOMETRIC QUANTITIES OF IDEAL THIN FILM SYSTEMS
205
]QI41]QI33 - 1Q[43]QI31
?'sp -- IQIlll~/I33 _ I~/I13]QI31 ,
(5.49)
l~/I111Q[23 -- IQ[131Q[21
m
Fps -- ]~/IlllQ[33 _ ]QI13]~/I31
(5.50)
]QI111QI43 - ]Q[13IQI41
rss "
t~p
=
,
(5.51)
]QI111~i33 _ ]QI131~/i31 1~133 = ~11M33 - M13M3~' -IQI31
(5.52)
(5.53)
tsp --- 1QIll/QI33 _ ]QI13IQI31,
-lVI13
(5.54)
tps = /~/ii11~/I33 _ ]QI13]~/I31 , t s s --
]~/I11 ]QI1 llQI33 _ ]QI131~I31
(5.55)
Thus in general, it is evident that generalized ellipsometry should be used to analyze optically anisotropic thin film systems 6. In the following two sections we shall deal with two special cases of the anisotropic thin films systems; i.e., we shall deal with the systems containing media characterized with permittivities described by the symmetric tensors and the magneto-optical systems exhibiting the induced optical activity.
5.2.1. Systems containing media with symmetric permittivity tensors In this section we shall deal with the optically anisotropic thin film systems formed by the materials characterized by permittivities described by complex symmetric tensors g (~0 = ~ji). These anisotropic thin film systems can be formed, for example, absorbing anisotropic materials exhibiting a higher symmetry (see Born and Wolf [1999]). Moreover, these materials mentioned do not exhibit the optical activity. If the coordinate axes are identical with the principal axes of the crystal the permittivity tensor of this crystal can be written in the following simple way: ~2 =
(10 0) 0 ~2 0
,
(5.56)
0 0~3 6 The 4 x 4 matrix formalism presented in this section is called the Yeh's formalism in the literature (for details see Yeh [1980]).
206
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 5
where ~1, ~2 and ~3 a r e the principal permittivities. In this case one can show that the determinant det 1N (see eq. 5.28) obeys this relation:
o
~ - k~x - k ~
o
kxL
o
~ -k~x
: ~(~,-~)(~-k~x)
- k ~; k x ~3 ( a~~ - k ; - i ~ ) .
(5.57) The condition det lq = 0 implies that four monochromatic plane eigenwaves propagate within the crystal. The z-th components of the wavevectors o f these four eigenwaves exhibit the following values: kzl,2 = -+-V/&2-k 2 and kz3,4 = "["V/al(~3-k2)/t~3 9 These four values correspond to the TE (s-polarization) and TM (p-polarization) modes characterized with the following
kx).
polarization vectors:/31,2 = (0, 1,0) and ab3,4 -- (::[::V/~3(~3-k2)/al,O, It is evident that for the anisotropy described above, we can use the 2 • 2 matrix formalism introduced in w 5.1 7. The eigenwaves that are identical with linearly polarized TE and TM modes are also obtained where the tensor permittivity exhibits a more general form:
(
~xx 0 ~xz)
~=
0 (~
0
.
(5.58)
~xz 0 ~zz
The permittivity tensor expressed by the foregoing equation corresponds to the situation where the principal axes are oriented perpendicularly or parallel to the plane of incidence 8. If the principal axes are generally oriented owing to the plane of incidence, the permittivity tensor is given by the following unitary transformation: gZ= 1), gZ' 1~-~ ,
(5.59)
7 In this case we must substitute the complex refractive indices h in eqs. (5.13)-(5.16) and (5.19) by the following: hs2 = e2 for s-polarization and h2 = Jell3 + (e3-el) sin2 00]/e3 for p-polarization (see eqs. (5.9), (5.27) and (5.29)). One can thus use conventional ellipsometry to study the thin film systems exhibiting this anisotropy. Of course, the phase matrices of the p-polarization and s-polarization are mutually different. 8 In this case the refractive index of the TM mode depends on whether this wave propagates in the positive or negative direction within this anisotropic film.
hp
3, w 5]
ELLIPSOMETRIC QUANTITIESOF IDEAL THIN FILM SYSTEMS
207
where ~:' is the permittivity tensor expressed by eq. (5.56) and 1~ is the coordinate rotation matrix, which is given as follows (see, e.g., Goldstein [1980]):
-
1~ =
cos ~pcos O - sin ~pcos O sin 0 sin O cos 0 sin qs sin q~ - cos 0 cos ~psin q~
cos ~psin q~ - sin ~p sin q~ - sin 0 cos O + cos 0 sin ~pcos q~ + cos 0 cos ~pcos r sin 0 sin ~p
sin 0 cos ~p
'
(5.60)
cos 0
where ~p, 0 and 0 represent the Euler's angles. The tensor permittivity then exhibits the general symmetric form and the eigenwaves propagating in these media are elliptically polarized in general. One must thus use generalized ellipsometry for studying such anisotropic systems. Note that the foregoing formalism presented in this section can also be used to describe propagation of light in media formed by non-absorbing uniaxial and biaxial crystals whose permittivities are expressed with the real symmetric tensors (see the monograph of Born and Wolf [1999]). 5.2.2. Magneto-optical systems If a stationary magnetic field exists in a material medium, both the permittivity and permeability tensors g; and ~ become asymmetric ones. This stationary magnetic field corresponds to either an external magnetic field or an internal magnetic field existing in ferromagnetic materials. However, from a practical point of view, the tensor of relative permeability of the majority of the materials is equal to unity in the optical frequency range because the influence of the stationary magnetic field on this tensor is negligible 9. For the non-absorbing materials the tensor gz is Hermitian (~0 = ~]~, see Landau, Lifshitz and Pitaevski [1965]). In general one can express the influence of the constant magnetic field on the components of the tensor of permittivity of the magneto-optical materials by means of the following series (see Wettling [1976]): = tij +
+
GijkzM~Ml
+
9. . ,
(5.61)
where Mk is the k-th component of the magnetization vector ~/, kuk and/or Gijkt is the linear and/or quadratic magneto-optical tensor and e~f0) 0. denotes the
9 The nonnegligible influence of the magnetic field on the permeability tensor of the magneto-optical materials is discussed in the works of Sokolov [1967] and Krinchik and Chetkin [1959].
208
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 5
components of the permittivity tensor when M = 0. The effects connected with the linear and quadratic magneto-optical tensors are called the complex linear and quadratic magneto-optical effects in the literature (the effects of the higher orders are not studied in practice). The permittivity tensor g: must obey the general principle of the symmetry [i.e., Onsager relation: ~0-0l;/)= ~j/(-3;/)] and the point operations of the symmetry connected with crystallographic structure of the material under consideration. The Onsager relation implies the following conditions for the components of the linear and quadratic magneto-optical tensors:
g i i k = 0,
gij k = -gjik
quadand
aijk, = ajik, = ai.l'lk -- ajilk"
(5.62)
If the crystalline material does not exhibit any point operation of the symmetry (triclinic system) the linear and/or quadratic magneto-optical tensor has the 9 and/or 36 independent complex components. The existence of the point operations of the symmetry of the material causes the decrease in number of the independent components of the magneto-optical tensors mentioned. In the case of the highest syrmnetry of the crystallographic structure of the material (cubic system) these magneto-optical tensors only have the following four independent components: (5.63)
=
(5.64)
zzzz = & ,
Gxxyy = Gxxzz = Gyv= = G~xx = Gzzxx = (~zzyy= G2,
(5.65)
axyxy=axzxz=ayzyz--ayxxy--azxxz--azyyz=a3
(5.66)
9
The remaining components of these tensors are equal to zero. Relations between the components of the magneto-optical tensors for the different crystallographic structures, i.e., for triclinic, monoclinic, orthorhombic, tetragonal, hexagonal and cubic structures, were derived by Vi~fiovsk~, [ 1986a]. From the foregoing it is seen that the ellipsometric quantities of thin film systems comprising the magneto-optical materials can be calculated using the Yeh's formalism introduced in w 5.2 (see Vi~fiovsk~, [1986b]).
3, w 5]
ELLIPSOMETRIC QUANTITIESOF IDEAL THIN FILM SYSTEMS
209
For example, in the case of the cubic structure one can see that for the magnetization vector parallel to the z-axis the permittivity tensor is expressed as follows (see eq. 5.63):
gz =
(
~(o) _iOn(o) 0 ) ion(~ ~(o) 0 , 0 0 ~(o)
(5.67)
where iQ~ (~ = -KMz if the linear magneto-optical effect is taken into account (see Voigt [1898] or Lissberger and Parker [1971])10. For the wave propagating in the direction of the z-axis the determinant of the matrix detlq existing in eq. (5.28) is given in the following way: _ i,z
i&(~ 0
o
&(o)_/r
0
0
&(o)
= [(~(0) - k z"2) 2 - ((~(0) 0 ) 2] (~(0),
(5.68)
where &(0) = co2~Co@(O). The value of the foregoing determinant must be equal to zero. This fact implies that the quantity iCz can have four values" = + V/&(0) + &(0)O. The four monochromatic plane waves can thus propagate along the z-axis within the material considered (two waves propagate in the direction of the z-axis and two waves propagate in the opposite direction). The two waves propagating in the same direction along the z-axis are mutually different in the value of the wave vector and represent the left- and rightcircularly polarized eigenwaoes. These waves are described by the following polarization vectors (see eq. 5.30)" joy = (-+-l/x/~,i/x/~,0). In general, the superposition of these waves gives the elliptically polarized wave. Their state of polarization changes as the wave propagates. If we assume that the magnetization vector is perpendicular to the surface of the system studied and the normal incidence of light is realized, the Jones matrix ] of this system is invariant with respect to the transformation 1/ corresponding to the rotation of the coordinate system around the axis of the symmetry of the system considered [ l ~ ( - a ) ] l ~ - l ( a ) = ], where the matrix fl is expressed by eq. (4.2)]. From the foregoing matrix equation it is implied that the elements of the Jones matrix have to fulfil these equations: 3~ = -Jyy and
10 The quantity Q is called the linear magneto-optical constant or the Voigt magneto-optical parameter.
210
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 5
Jxy = Jyx. One can then write the following equation for the normalized Jones matrix: in= (~1Pl),
(5.69)
where/5 is the complex number in general. The PTF is then given as follows:
1
+k
(5.70)
It is evident that in this case one can characterize the magneto-optical system by two parameters. In practice this system is described by means of the polarization state of the outgoing wave )(o represented by the pair of the angles 0 and e (see eq. 2.10) corresponding to a chosen polarization state of the incident wave Xi. Note that in general (oblique incidence of light and general direction of the magnetization vector) it is necessary to use generalized ellipsometry for the unambiguous determination of the PTE However, in practice the angles 0 and are utilized for describing the magneto-optical systems even in the general case mentioned (of course, then the PTF of these systems cannot be evaluated). It should be noted that the magneto-optical effect corresponding to reflected and/or transmitted light is known as the Kerr and/or Faraday effect (both the Kerr and Faraday effects are usually associated with the linear magneto-optical ones). In practice the quantities called the polar Kerr rotation On and the polar Kerr ellipticity en are most often used. The quantity OK is defined as the difference between the azimuths of the outgoing and incident waves. The quantity en is the ellipticity of the outgoing wave if the incident wave is linearly polarized. The quadratic effects (usually observed in transmitted light) are called Voigt or Cotton-Mouton effects in the literature. Furthermore we can classify the magneto-optical effects by means of the directions of the magnetization vectors if/owing to the surface of the systems and the incidence plane of light. This means that we can distinguish the polar 11, longitudinal 12 and transversal 13
magnetization. A detailed review of the problems concerning magneto-optics and magnetooptical materials is presented in the monograph of Zvezdin and Kotov [ 1997].
1! M is perpendicular to the surface. 12 M is parallel to the surface and incidence plane. 13 M is perpendicular to the incidence plane.
3, w 6]
ELLIPSOMETRIC QUANTITIESOF IMPERFECTTHIN FILM SYSTEMS
211
w 6. Ellipsometric Quantities of Imperfect Thin Film Systems 6.1. ELLIPSOMETRIC QUANTITIES OF THE THIN FILM SYSTEMS WITH ROUGH BOUNDARIES
Roughness of the boundaries of thin films is a defect most frequently encountered in practice. Considerable attention has been paid to studying the influence of this defect on the optical properties of thin film systems (see e.g., the review by Ohlidal, Navrfitil and Ohlidal [1995]). The ellipsometric quantities of these systems are greatly influenced by roughness of the boundaries. The models of the rough boundaries can be divided into three basic groups, i.e., (i) periodically rough boundaries, (ii) randomly (statistically) rough boundaries and (iii) composed rough boundaries. In this chapter we shall only deal with the influence of random roughness of the boundaries on the ellipsometric quantities of the systems mentioned. Roughness of all the boundaries of the system will be represented by a stationary ergodic stochastic process, i.e., roughness will be homogeneous. Moreover, we shall assume that the materials forming the substrate and the thin films of the rough systems are optically homogeneous and isotropic. The heights of irregularities of the j-th boundary are characterized by the rms 14 value (standard deviation) ay defined as follows (see, Cram& [1946] and van Kampen [ 1981 ]): OO P
/ (z - ~j)Zwj(z) dz,
(6.1)
i t J
--0(3
where symbols z, ~9 and wj(z) denote a certain value, mean value and one-dimensional distribution of the probability density of the random function ~(x,y), respectively (x and y are the Cartesian coordinates in the mean plane of the rough boundary). In the following discussion it will be supposed that Zj -- 0. The quantity a is called the rms value of heights. The significant statistical quantity of the random function ~(x,y), i.e., the j-th rough boundary, is the autocorrelation function Gj. This function is defined as follows (see, Cram& [1946]): Oo
o<9
/ - o
Gj(Tx, Ty) = /
/Zl
z2 wj(z1,z2, Tx, Ty)dzldz2,
v~v
--OO --0(3
14 The abbreviation rms denotes root mean square.
(6.2)
212
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, {} 6
where the symbols Zl and/or Z2 and wj(zl,z2, rx, ry) denote the certain value of ~.(x,y) in the point [xl,yl] and/or the point [Xz,y2] and two-dimensional distribution of the probability density of ~.(x,y), respectively. The symbols rx and rv are given as follows: rx = x 2 - Xl and ry = y 2 - y l . The autocorrelation function is connected with the function of the spectral density of spatial frequencies of roughness of the j-th boundary Wj. This connection is given by the Fourier transform, i.e., OC
~(Xx, Kv) =
o<3
/ / Gj(T~.,Tv)e_i(K~r~+K,.r,.) dTxdTv . .'
~1 --
OC
--
(6.3)
(X)
where K~ and/or K,, denotes the x- and/or y-component of the wave vector of a harmonic component of the certain spatial frequency of roughness. If roughness of the j-th boundary is generated by an isotropic stationary ergodic stochastic process, the Fourier transform in eq. (6.3) is replaced by the Hankel transform in the following way:
l/
CX2
Wj(K) = ~
r Gj(r) J0(rK) d r,
(6.4)
0
where g 2 = K~ + K~2 and r 2 = r.~ + r~2,.The boundaries exhibiting this roughness are called homogeneously isotropically rough boundaries. For thin film system with the rough boundaries, the cross-correlation function of the j-th and k-th boundaries, i.e., G#, is important. This function is defined as follows: oc
oc
Gjk(Tr,Tv)=//ZlZ2Wjk(Zl,Z2, Tr,Tv)dZldZ2,
(6.5)
--0<3 --O(3
where the symbols Zl, z2 and w#(zl,z2, rx, ry) denote the certain value of ~(x,y) in the point [xl,yl], the certain value of ~k(x,y) in the point [Xz,y2] and two-dimensional distribution of the probability density of ~.(x,y) and r respectively. Of course, the cross-correlation function Gij is related to the function of the cross-spectral density of spatial frequencies by the Fourier or Hankel transforms (see eqs. 6.3 and 6.4). If we assume, for example, that the correlation function Gj(r) is given by the Gaussian function, i.e., when it holds that Gj(r) = %? exp(-r2/Tj2),
(6.6)
the quantity Tj (autocorrelation length) characterizes the linear dimension of the irregularities of the randomly rough boundary in its mean plane.
3, w 6]
ELLIPSOMETRIC QUANTITIES OF IMPERFECT THIN FILM SYSTEMS
213
The rms value of the derivative of ~j(x,y) of the j-th homogeneously isotropically rough boundary, i.e., tan/30j, is given as oo
tan2/30j
= / z '2 wj(z') dz',
(6.7)
--0<3
where z I and w' (z ~) denote a certain value of the derivative of the function ~.(x, y) in an arbitrary direction in the mean plane and one-dimensional distribution of the probability density of this function, respectively. It can be proved that oj tan/30j = v/2 Tj'
(6.8)
if the autocorrelation function Gj(r) is assumed to be in the form expressed by eq. (6.6) (see Levin [1960]). The quantity tan/30 is called the rms value of
slopes. Using the cross-correlation function Gjk(r) we can define the cross-correlation coefficient Cjk of the j-th and k-th boundaries, i.e.,
Cj.k_ G; ( ]_jk,Oj.
(6.9)
For the thin film systems encountered in practice, three kinds of models of the randomly rough boundary can be taken into account. They are as follows: (1) "Microrough" surface (MCRS): o << ~ and T << ~,. (2) Slightly rough surface (SRS): o << A and T ~ ~ (i.e., T < ~, T - ~, or
T>~). (3) Moderately rough surface (MRS): o < ~, and T >>/~. We shall deal with the ellipsometric quantities of the systems with the three kinds of rough boundaries separately.
6.1.1. Microrough systems If the boundaries of the thin film systems can be represented by the MCRS, one can use some of the formulae belonging to the theory of effective medium for expressing the ellipsometric quantities of these systems. In this case incident light does not "see" the structure of these boundaries. Within the theory mentioned one can replace the boundaries by fictitious films with suitable "effective" optical constants and thicknesses. The boundaries of these fictitious thin films are ideally smooth. The effective optical constants of the fictitious films are
214
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 6
dependent on the structure of the corresponding rough boundaries. In the first approximation, it is possible to assume that these fictitious films representing the rough boundaries are isotropic and homogeneous. Thus the values of the optical constants characterizing the fictitious films representing the boundaries can be calculated by means of formulae of the effective medium theory, i.e., by means of the Lorentz-Lorenz (LL), Maxwell Garnett (MG) and Bruggeman (BR) formulae. The LL, MG and BR formulae all have the same genetic form: 2h2
h~f + 2h 2 -P, h2 + 2h 2 +Pz h2 + 2h 2 ,
(6.10)
where her, hh, h l and n2 are the complex refractive indices of the effective medium, host medium and materials of types 1 and 2 in the host, respectively, and where pl and P2 represent volume fractions of materials of type 1 and 2 (Pl +p2 = 1). 9 In the LL formula, vacuum is considered as the host medium, i.e., hh = 1 (Aspnes, Theeten and Hottier [1979]). 9 In the MG formula (Maxwell Garnett [ 1904, 1906], the host medium is formed by one of two materials (hh = nl or he). 9 In the BR formula the host refractive index hh is replaced with hey in eq. (6.10) (hh = h~f), i.e., the effective medium itself acts as a host medium (Bruggeman [1935]). From the foregoing one can see that within the effective medium theory, the microrough boundary is characterized by the thickness of the fictitious film and volume fraction pl (or p2). This means that the microrough system consisting of N thin films generally contains (2N + 1) films because it contains (N + 1) rough boundaries replaced by the fictitious films. The Fresnel coefficients of this system, i.e., ~p, ~, ~p and t~, are calculated by using eqs. (5.20) and (5.22). Thus one can see that the Fresnel coefficients of the microrough system are generally functions of the following quantities:
dN, nN,, dN,N+I ,PN,N+I, n)
(6.11)
d01 ,P01, dr, h i , . . . , dN, nu,, dN,N+I ,PN,N+I, h),
(6.12)
rq -'- rq(/~, 00, no, d01 ,P01, dl, n l , . . . , lq -- tq(/~, 00, no,
where dj,j + 1 and/or pj,j +1 (j -- O, 1,..., N) denotes the thickness and/or volume fraction of the fictitious film representing the boundary between the j-th and (j + 1)-th media. The isotropic and homogeneous fictitious films do not represent an entirely satisfactory approximation of the boundaries corresponding to the MCRS
3, w6]
ELLIPSOMETRIC QUANTITIESOFIMPERFECTTHINFILMSYSTEMS
215
frequently. Physically it is reasonable to replace these rough boundaries by inhomogeneous fictitious film whose refractive index varies continuously along the direction perpendicular to the boundaries of the film (see, e.g., Nevot, Pardo and Corno [1988] and Szczyrbowski, Schmalzbauer and Hoffmann [1985]). The Fresnel coefficients of such the microrough systems must then be calculated by means of procedures that enable us to evaluate the optical quantities of the inhomogeneous thin films (see below). Note that within this theoretical approach the scattering of light by the boundaries is not respected. This means that the formulae for expressing the ellipsometric parameters of the thin film system with the boundaries corresponding to the MCRS presented here can only be employed where the total amount of the scattered light flux by this system is negligible in comparison with the light flux of the incident wave.
6.1.2. Slightly rough systems If the total flux of scattered light by the rough thin film system cannot be neglected, it is necessary to use some theoretical approach respecting the light scattering by the rough boundaries of this system. When the boundaries of the system can be approximated by the SRS, it is possible to use some of the perturbation theories. In this chapter we shall only demonstrate the utilization of the Rayleigh-Rice theory (RRT) (Rice [1951 ]) for deriving formulae that enable us to calculate the ellipsometric parameters of the system mentioned. Within the framework of this approach roughness of the boundaries is taken as a perturbation implying the existence of the perturbed electromagnetic fields. These perturbed fields together with the unperturbed electromagnetic fields ensure the validity of the boundary conditions at all the rough boundaries of the system (the unperturbed fields take place in the same system with the smooth boundaries). The perturbed fields allow us to calculate the scattered flux and corrections of the amplitudes of both coherently reflected and transmitted waves of the system (the directions of the coherent waves mentioned are the same as the directions of the specularly reflected and directly transmitted waves by the smooth system). The amplitudes of the coherently reflected waves from the system under consideration are then expressed as follows (Franta and Ohlidal [ 1998a]) 15:
~.q __ ~..~o)-k-A~.q,
q =p,s,
(6.13)
15 The amplitudes of the coherently transmitted waves by the system are given by the formulae similar to those presented above for the reflected waves [see eqs. (6.13) and (6.14)].
216
ELLIPSOMETRY OF THIN FILM SYSTEMS
where ?tq0) is the reflection Fresnel coefficient of the smooth system and the correction mentioned given as N+I N-el .,
=
A?q is
oc cx~
A~q=~ k~// /f/kq
[3, w 6
-- OO
(6.14)
-- OC
fjjkq
is a complicated complex function of the optical parameters of the system, wavelength and the angle of incidence of light falling on the system. Wj/ _= W/ denotes the spectral density of spatial frequencies of roughness of the j-th boundary (see eq. 6.3) and Wjk represents the cross-spectral density of spatial frequencies of the j-th and k-th boundaries 16. The concrete expressions of the functions are published for a general single layer in a paper by Franta and Ohlidal [ 1998a]. These functions for a special case of the single layer, i.e., for the layer containing the slightly rough upper boundary and smooth lower boundary that is placed onto the perfectly reflecting substrate, were presented by Krishen [ 1970]. Formulae expressing the belonging to the general rough system comprised of N films have not been presented so far. However, if the films of the rough systems are sufficiently thick, one can use very simple approximation for expressing the optical quantities of these systems (roughly speaking the rough films can be considered to be sufficiently thick if their thicknesses are greater than the values of the autocorrelation lengths of the boundaries). Within this approximation the optical quantities of the rough systems are expressed in the identical way as the optical ones corresponding to the same system with the smooth boundaries. Of course, the Fresnel coefficients of the smooth boundaries must be substituted by the Fresnel coefficients corrected for roughness of the boundaries in the formulae expressing the optical quantities mentioned. This means that the matrix Bqj taking place in the formalism of the smooth thin film systems has to be written as follows:
where
~kq(Kx,Ky)
functions~kq(Kx,Kv)
1_~( B q j = tqjR
1
--?'qjL
~'qjR
tqjRtqjL -- rqjRrqjL
)
(6.15) '
where the indices R and L distinguish the Fresnel coefficients corresponding to the waves incident on the boundary from the left side (fight-going wave)
16 It should be pointed out that the ellipsometric parameters of the thin film systems considered can be calculated by means of eqs. (6.13) and (6.14) only when the flux of scattered light by the system into the acceptance angle of the detector is negligible in comparison with the light flux corresponding to the coherent wave.
3, w 6]
ELLIPSOMETRIC QUANTITIES OF IMPERFECT THIN FILM SYSTEMS
217
and from the right side (left-going wave), respectively (i.e., from the ambient side and the substrate side, respectively). The concrete formulae for the Fresnel coefficients existing in the m a t r i x Bqj are presented, for example, in the paper by Schiffer [ 1987]. Further in the expressions of the elements of the phase matrix l"j the thickness of the smooth film must be replaced by the mean thickness of the rough film. It should be pointed out that the approximation discussed is supported with the theoretical results concerning the behavior of the electromagnetic fields at the neighborhood of the slightly rough boundary (for details see Franta and Ohlidal [1998b]). It should be noted that the other perturbation theory expressing the formulae for ellipsometric parameters of the single films with slightly rough boundaries has been presented in papers by Wind and Vlieger [1984, 1985]. Of course, the formulae published in these papers can only be employed for calculating the ellipsometric parameters of the rough single films exhibiting relatively small values of thicknesses (for details see the papers cited).
6.1.3. Moderately rough systems The interaction of light with thin film systems containing the rough boundaries corresponding to the MRS is frequently studied using diffraction theories. These diffraction approaches can be divided into two groups. The first group contains scalar diffraction theories and the second one comprises the oector diffraction
theories. The starting point of the scalar diffraction theory is the Helmholtz-Kirchhoff (HK) integral (see Born and Wolf [1999]); i.e.,
Eq(Fp)-- ~1
(rB) OG(rPon--rB) _ G(rp - rB)
On
dS,
(6.16)
where P,q(rB) and/or Eq(rp) is the local electric field at a point B in the rough upper boundary of the system and/or the electric field at a point P in the far zone. The symbol O/On represents a partial derivative with respect to normal in the rough upper boundary (the illuminated part of this upper boundary is denoted as S). The function (~ is expressed as G(rp - rB) = ~ 1
e-ik~Irp-rBI,
(6.17)
where k0 is the absolute value of the wave vector of the light waves propagating in the ambient and Irp- rBI denotes the distance of point P from a given point
218
ELLIPSOMETRYOF THINFILMSYSTEMS
B in the rough upper boundary
(rp
and
rB
[3, w 6
denote the radius vectors of the
corresponding points). For expressing the local field F.q(rB) in the HK integral the Kirchhoff approximation can be employed (see, e.g., Beckmann and Spizzichino [1963]). When the monochromatic plane wave is incident on the system and the ellipsometric parameters are studied in reflected light, the Kirchhoff approximation of the local electric field is expressed by means of the following equation:
Eq(rB)
= (1
+
?'~l))E, Oq(rB) ,
(6.18)
^r is the local reflection Fresnel coefficient of the system on the upper where rq boundary and Eoq(rB) is given as ^
^-k-
9
+
F_,oq(rB)= Aoq e-'ko r~,
(6.19)
where Aoq is the amplitude of the incident monochromatic plane waves 17 For expressing the ellipsometric parameters of the thin film systems the statistical mean values of the electric field at the point P in the far zone, i.e., (Eq(re)), are needed. The expression of (Eq(re)) is dependent on the model of the system containing the locally smooth boundaries. The reflection Fresnel coefficients of the rough thin film system containing the boundaries represented ^+
.
by the MRS can then be expressed as follows:
?q = (Eq(rp)).^+
(6.20)
Aoq Below we shall assume that the rough boundaries of the system are described by the normal (Gaussian) isotropic stationary ergodic stochastic processes.
17 It is evident that the local electric fields F.q(rB) can be correctly expressed by eqs. (6.18) and (6.19) only in the case where the rough boundaries of the system are approximatedby locallysmooth surfaces (the rough surface is locally smooth if the tangent plane to the surface in its arbitrary point deviates only slightly from this surface over the region the linear dimensions of which are much larger than the wavelength of incidence light, for details see the paper by Ohlidal, Navrfitil and Luke~ [1971] and monograph of Ogilvy [1991].
3, w 6]
ELLIPSOMETRIC QUANTITIESOF IMPERFECT THIN FILM SYSTEMS
219
First we shall deal with the identical multilayer system (IMS)18. The Fresnel coefficients ?q of this system are given by the following equation: oo
, , ? )(z~, z~) w , (z x) w , (zy) dzx, dz~,
~q __ e-2(okocos Oo)2
(6.21)
,11---CX3
where z~ and/or z~ denotes a certain value of ~(x,y) O~(x,y)/Ox and/or ~f(x,y) = O~(x,y)/Oy. The function w'(z~) and/or w'(z~) is the distribution of the probability density of ~(x, y) and/or ~f(x,y) (note that these distributions are =
~(l)l..t identical). The local Fresnel coefficients rq t~x,Z~) are expressed by eq. (5.22). In this equation one must substitute the angle of incidence 00 and the thicknesses dj- (j = 1,... ,N) by the local angle of incidence 0~0 and the local thicknesses 0, respectively. These local quantities obey the following equations:
cos 0~') = z~ sin 00 + cos 0o
and
d) 0 = dj cos/5,
(6.22)
COS/3
where
cos = V/1
+z7
(6.23)
The symbols 00 and dj denote the angle of incidence of light onto the mean plane of the upper boundary and the distances of the mean planes of the adjacent boundaries, respectively. After inserting eq. (6.21) into eqs. (3.4) and (3.5) it is evident that the ellipsometric parameters tit and A of the IMS are only dependent on one statistical quantity, i.e., on the rms value tan/50 of ~(x,y) and ~(x,y) (they are thus explicitly independent on the rms value c~). It should be pointed out that for the IMS with relatively small slopes of the boundaries (tan/3o < 0.1) eq. (6.21) can be simplified in this way (see Ohlidal and Luke~ [ 1972a]): ?q = e-2(okocos Oo)2(?~o~+ pq tan 2 fi0),
(6.24)
where Fq are the complicated complex functions of 00, dj, no, hj and h (see Ohlidal and Luke~ [ 1972b]).
18 This system is formed by identical thin films, i.e., the films whose boundaries are identical both statistically and geometrically.
220
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 6
It is clear that the formulae for the ellipsometric parameters of the IMS corresponding to waves transmitted will be similar to those belonging to the reflected waves from this system. Further we shall deal with the ellipsometric parameters of the general multilayer system (GMS). Within this system the rough boundaries are mutually correlated. The correlation of these boundaries is described by the crosscorrelation coefficients 19. We shall limit ourselves on the GMS exhibiting the rough boundaries whose slopes are very small so that the influence of these slopes can be neglected in deriving the formulae expressing the ellipsometric parameters of this system. The formulae enabling us to calculate the ellipsometric parameters of the GMS exhibiting the rough boundaries with greater slopes influencing these ellipsometric parameters have not been published so far. If the slopes of the boundaries of the GMS can be neglected, one obtains the following recursion formulae for the reflection Fresnel coefficients of this system: ~q __ l~q0 0 -k- Rql0_~ 1 + rqlRqll
and
q = p,s,
(6.25)
where ?ql is the reflection Fresnel coefficient of the upper boundary (see eqs. 5.13 and 5.15) and ^
" Rqjk =
(~,qjk + R q , j + 1, k ,, , 1 + t'q,j + 1 R q , j + 1,j+ 1
j = 1,... , N -
1,
k = O,... , j ,
(6.26)
^
k = 0,... ,N.
(6.27)
l~qoo = rql exp(-o 2ko cos 0o),
(6.28)
RqNk -- P'~tNk,
Further
(6.29) j
P'~ljk "-- t'q,j+' exp(-i Z v=k
j+l j+l
Jr)exp(- Z
Z
OuO~dud~Cu~)'
(6.30)
u=kv=k
1 9 Two special cases of the GMS are formed by the IMS (the cross-correlation coefficients are equal to unity and the values of o) and tan fi0j of all the boundaries are the same) and the uncorrelated multilayer system (UMS) (all the cross-correlation coefficients are equal to zero).
3, w 6]
ELLIPSOMETRIC QUANTITIESOF IMPERFECTTHIN FILM SYSTEMS
221
forj = 1,... ,N and k = 1,... ,j. The symbol Cu~represents the cross-correlation coefficient of the u-th and v-th boundaries (see eq. 6.9). The quantities J and fulfil the following equations: xv = 21cvd~cos O~,
(6.31)
0, = k, osO,,
(6.32)
and
Ou = kucoSbu-ku-1
c o s Ou-1,
/,/= k-k-
1,... ,j.
(6.33)
From the foregoing it is evident that within the approximation utilized the ellipsometric parameters of the GMS are dependent on the values of ~. and Cjk as for the statistical quantities (see eqs. 3.4, 3.5 and 6.25)20. Of course, if the slopes of the boundaries of the GMS are not negligible the ellipsometric parameters of this system are also dependent on the values of tan fioj. It should also be noted that the foregoing recursion formulae were derived on the basis of the recursion formulae presented for the normal incidence of light on the GMS in the papers by Ohlidal [1993] and Ohlidal, Vi~d'a and Ohlidal [1995]. It can be shown that the formulae for the ellipsometric parameters of the GMS corresponding to the transmitted waves are similar to those presented for the reflected waves above (see Ohlidal [1993]). The starting point of the vector diffraction theory is the Stratton-Chu-Silver (SCS) integral (see, e.g., Silver [1949]); i.e., 1
E(rp)- 4;rio9~/ ~
[(n x [-I)(A + k2)G + iwe(n x i?,) x ~7~] dS,
(6.34)
where n,/~ and if/are the local normal, local electric field and local magnetic field on the upper boundary of the system at the point kB, respectively, 0 is given by eq. (6.17) and V and/or A denotes the Hamilton and/or Laplace operator.
20 Note that the ellipsometric parameters of the IMS corresponding to this approximation are identical with those belonging to the multilayer system with ideally smooth boundaries.
222
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 6
If the IMS is taken into account, the SCS integral enables us to derive the following formula for the reflection Fresnel coefficients ?q of this system (see Ohlidal and Luke~ [1973]): ~q = e-2(ok0 cos 00)2
[[
• j j
- l 2 cos 00 e
(Pq+O
n ) L - (Pq+Od ) [ R p + (s-~ n ) ( s 0 n ) R p ] V
DF
--0(3
+
t)C.;0t)k+sDF~] ,q0I~F_ /
[~q+o~ ) ~ .)L + ~q+o, ) ~ n)r n)V -r
,J
• w'(Z'x) w ' (z~,) ' dz'~' dz~,'
(6.35) where
s~- k~
So- ko
ko '
ko '
"+ "(l). t Rq - 1 + r~ t,Zx,Z'v) , t=
s~ • n
V/1 - (s~ n) 2'
V = (s~ • So)n,
g(l)[..t t l~q -- (S 0 + S-~) + t q ~,~x,Zy)(SO -- S-~), -
d=s~ •
e=
so • n
V/1 - (s o n) 2'
f =s O •
L = (s~ n)(s o n ) - (s~ So),
V/, The local normal n has the components n = ( - ~ / cos /5, -z~,/ cos /5, 1/cos 13). For the relatively small slopes of the boundaries of IMS (tanfl0 eq. (6.24) can also be simplified (see Ohlidal and Luke~ [1973]), i.e.,
~'q = e-2'~176176176176~ +F'q tan 2 r 0 - (k~~ ?(p~ cot2 00].
< 0.1)
(6.36)
It is evident that the Fresnel coefficients of the IMS corresponding to transmitted light can be derived within the vector diffraction theory in the same way as in the case of the reflection Fresnel coefficients of this system. The Fresnel coefficients of the GMS exhibiting nonnegligible slopes for both the reflected and transmitted waves have not been derived so far.
3, w 6]
ELLIPSOMETRIC QUANTITIESOF IMPERFECT THIN FILM SYSTEMS
j
223
j+l
j-1
j+l
^
~j+l
o Fig. 6. Schematic diagram of the jth inhomogeneous layer of the thin film system.
6.2. OPTICAL QUANTITIES OF THE INHOMOGENEOUS THIN FILM SYSTEM
In practice one often encounters the defect of the thin films represented by profiles of the optical constants of these films along the direction perpendicular to their flat and smooth boundaries. This kind of optical inhomogeneity of thin film systems is characterized by the dependence of the refractive indices and extinction coefficients of the films on the coordinate of the z-axis (see fig. 6).
6.2.1. Exact solution From the Maxwell equations it is implied that the electric and magnetic fields, i.e., the vectors ~" and [-/, of the waves propagating in the inhomogeneous medium characterized by the optical constants depending on the Cartesian coordinates fulfill the following equations:
+ radl rad ] ,,
+ k0h2/~ = 0,
(6.37)
z ~ + 2 grad h --~ • rot f / + koh2H = 0.
(6.38)
n
n
If we assume that the wave vector of the wave incident on the system containing the inhomogeneous films lies in the coordinate plane (x,z) and that the complex refractive index of the medium only depends on the values of the z-th coordinate, i.e., h - h(z), it is possible to find two independent solutions of the foregoing
224
ELLIPSOMETRYOF THIN FILM SYSTEMS
[3, w 6
equations within the inhomogeneous films in the following forms (see, Jacobson [ 1964], Jacobson [ 1966] and Sheldon, Haggerty and Emslie [ 1982]): L = 0,
/~y = F ( z ) e -ik~
00,
/2/x = 0,
/2/~y = G ( z ) e -ik~ sin Oo,
(6.39)
/~z = 0
and (6.40)
/2/z = 0.
In eqs. (6.39) and (6.40) the symbols Ex, Ey and ~'z and/or/-/x,/2/y and/2/z represent the components of the electric field of the TE (s-polarized) wave and/or the components of the magnetic field of the TM (p-polarized) wave, respectively. The functions F(z) and (~(z) must then obey the following linear differential equations of the second order: d2ff(z)
dz 2
+ ~2(z)F(z) = 0
(6.41)
and d2G(z)
2 dh(z) d(~(z)
h2(z)G(z) = 0,
(6.42)
h(z) = ko 4h2(z) - sin 2 00 = ko h(z)cos O(z).
(6.43)
dz 2
n(z)
dz
dz
where
Each of the foregoing equations has two independent particular solutions corresponding to two independent boundary conditions. Let us choose the particular solutions ~ (z), ~(z), gl (z) and ~2(z) fulfilling the following boundary conditions"
(0) = gl (0) = 1,
d~dz(Z_____)) [ _ z=O
= 0
dgldz(Z)
(6.44)
z=O
and ~2(0) = ~2(0) = 0,
d~(z)
dz
z=0
d~2(z) iI = 1. dz ]z=0
(6.45)
Then the general solutions of differential equations under consideration, i.e., F(z) and (~(z), are given by the linear combinations of these particular solutions.
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ELLIPSOMETRIC QUANTITIES OF IMPERFECT THIN FILM SYSTEMS
225
Let us assume that the s-polarized wave falls onto the thin film system and that the j-th thin film is inhomogeneous as specified above. It is then suitable to connect the coordinate system with this film in the way presented in fig. 6. For the values of F(z) and its derivative at the j-th and (j + 1)-th boundaries the following matrix equation is fulfilled: (~jt~')~(~ljf2j)k f15 f2ff "t
( ^F j t +
1
(6.46)
'
or, more concisely, Fj -- XsjFj
(6.47)
+ 1,
where Fj- = F(dj), Fj+I = F(O), jT1j. = Z (dj), ~j = ~(dj) and ~:, F;+ i, J71) and . denote the derivatives of the corresponding functions with respect to z at the boundaries mentioned. If both the (j-1)-th and j-th films are inhomogeneous, one can write the following matrix equation for the vectors Pj_ 1 and Pj + 1" F j - 1 = Xs,j-1XsjFj
(6.48)
+ 1.
The foregoing matrix equation is evidently implied by the boundary conditions (the values of/~y and/~y are maintained at the boundaries). If the k-th film is homogeneous, it is possible to express the vector Pk as follows:
Pk =
(
1
1
ikohk cos 0k -ikohk cos 0k
A[k
649,
where the vector A~,k is defined in w 5.1. In a similar way we can introduce the same matrix formalism for the ppolarized waves. For these waves the boundary condition creates formally different equations (in this case the values of Hy and Hy/n 2 are maintained at the boundaries) and therefore the matrix equations equivalent to (6.46) exhibit the following form: " t "2 Gj/nfL
=
g2jnfR "t "2 gzjnfR/nfL "t "2 "2 glj/nfL
Gj+l
" t + 1/~.12R , Gj
(6.50)
or, more concisely,
~Tj = Xpj Gj + I ,
(6.51)
where hjt and/or hjR denotes the refractive index of the j-th inhomogeneous thin film at the j-th and/or (j + 1)-th boundary, i.e., hjL = h(dj) and hjR = h(O) (see fig. 6).
ELLIPSOMETRYOF THIN FILM SYSTEMS
226
[3, w 6
For the k-th homogeneous film it is evidently fulfilled that
Gk =
ik0 cos Ok ik0 cos Ok
Apk
For example, let us consider the three-layer system containing two inhomogeneous films and one homogeneous film (the homogeneous film is the middle film of the system). One can then write the following matrix equation expressing the relation between the amplitude vectors belonging to both the ambient and substrate surrounding this triple layer:
~e~q0 V~lqlXq 1V~lq2~r2~lql Xq3 Wqh q l~/lqhq, - -
=
(6.53)
~
where M q is the overall transfer matrix of this system. Of course, for the j-th inhomogeneous thin film we can also introduce the matrices Bqj and ~rqj in this way, ~
~
_|
~
Bqj = Wq,j_ 1,RWqjL,
,~qj = WqLjX-lqW j qRj~,~
~
(6.54)
where the matrix ~Vqj L and/or WqjRgiven by eqs. (6.49) and (6.52) belongs to the left and/or fight boundary inside of the j-th film. The matrix formalism concerning the thin film system with the inhomogeneous thin films is then identical with that introduced for the homogeneous thin film systems in w 5.121. From the foregoing it is evident that the Fresnel coefficients for both the reflected and transmitted waves by the thin film system containing the inhomogeneous films obey eqs. (5.22). In an analytical way one can find the exact solutions of eqs. (6.41) and (6.42) only in special cases of the profiles of the refractive index h(z) of the inhomogeneous thin films. This statement is true, for example, for both the linear and exponential profiles of the refractive index (see, e.g., Knittl [ 1976]). Numerically, it is possible to obtain the exact solution of eqs. (6.41) and (6.42) for every profile of h(z) of the inhomogeneous layer if a suitable numerical procedure is employed 22.
21 Note that for the inhomogeneous thin films the elements of the matrix I"~ are different for the s- and p-polarizations. 22 For this purpose the most suitable numerical procedure is the Runge-Kutta method (see, e.g., Sheldon, Haggerty and Emslie [ 1982] and Ralston and Wilf [1967]).
3, w 6]
ELLIPSOMETRIC QUANTITIES OF IMPERFECT THIN FILM SYSTEMS
227
6.2.2. Approximate solutions The solutions of eqs. (6.41) and (6.42) can be found by approximate procedures if the profiles of the refractive indices of the inhomogeneous films fulfill certain properties. If the gradient of h(z) of the film is relatively small, one can use the approximate procedure based on the geometrical optics for solving the equations mentioned. Note that this approximate procedure is also known as the WentzelKramers-Brillouin-Jeffries (WKBJ) method. Within this approximate procedure it is possible to express the two particular solutions of eq. (6.41) and/or (6.42) for the j-th inhomogeneous film in the following way (see, e.g., Knittl [1976]): ~+(z)-
1
e~:~(z) and/or
~,f(z)-
hi(z)e:~(z),
(6.55)
where ~-(z) = ~oz hj(~)d~.
(6.56)
The sign + and/or- denotes the particular solutions corresponding to the rightand/or left-going wave inside of the j-th inhomogeneous film. Within the WKBJ method it is also possible to neglect the terms containing the derivatives of hi(z) and hi(z) in the expressions of the derivatives of the functions~+(z) and ~ ( z ) . It is then possible to write the functions ~j(z), ~lj(z), f2j(z) and ~2j(z) defined by eqs. (6.44) and (6.45) as follows: (6.57)
'~lJ(Z) --
=
(6.58)
~ 1
(6.59)
1
_g.+(z) + g_(z)] '
(6.60)
where hi(O) = hjR and ttj(O) = kohjRcosOje. Thus within this WKBJ approximation the phase matrix of the inhomogeneous film generally given by
228
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 6
eq. (6.54) is expressed for both the s- and p-polarized waves in the following manner: 0
~e-~
'
(6.61)
where
~ = I h)2 - sin20o -sin
(6.62)
00'
From eq. (5.22) one can see that the ellipsometric parameters of the system containing the inhomogeneous films fulfilling the assumptions of the WKBJ method are independent on the constant ~. for both the reflected and transmitted light waves by this system 23. A further approximate approach for expressing the optical quantities of the thin film system with the inhomogeneous layers is based on developing the solutions of eqs. (6.41) and (6.42) into a power series in ko (see, e.g., Abel+s [ 1950] and Jacobson [ 1966]). The starting point of the mathematical procedure of this approach consists of the integration of eqs. (6.41) and (6.42) leading to the following formulae: F(z) = P ( 0 ) + z P ' ( 0 ) -
h2(~5)P(~e)d~5d~
(6.63)
and h2(O)
hZ(~)d~ -
h2(~)
h2(~)
(6.64) In the second point of this mathematical procedure the particular solutions of the foregoing equations corresponding to the j-th inhomogeneous film are developed into the power series mentioned, i.e., OC
OC
# v=0
tz ,
~ ( ) V'kZ ~!Y ( )
/ = 1,2.
(6.65)
v=0
23 It should be noted that this statement is also true for the reflectances of the system under consideration (only the transmissions of the system are dependent on this constant).
3, w 6]
ELLIPSOMETRIC QUANTITIESOF IMPERFECTTHIN FILM SYSTEMS
229
From eqs. (6.63) and (6.64) and the boundary conditions for the functions~j(z),
f2j(z), ~lj(z) and ~2j(z) (see eqs. 6.44 and 6.45) the following recursion formulae for the coefficients ~'(~) Jlj (Z), jr j~l(j~
g};)(Z) and g2j ^(v) (z) are implied: (6.66)
= 1, ~ 7 )(z) = z, 1 = 1,2
(6.67)
and ^(o)(Z)= 1 ^(o) 1 glj , gzj(Z) = njR ~n)(z) ~_~=_ 1
h2(~)
~0"Zh2(~)d~, h2(~)60
(6.68)
,~, ~j d~,
l = 1,2.
(6.69)
After inserting the values of the functions ~j(z), fzj(z), ~lj(Z) and ~2j(z) and values of their derivatives at the point z = dj expressed by the series found into eqs. (6.46) and (6.50), one obtains the elements of matrices Xsj and Xpj. In practice it is possible to take into account a finite number of terms in the series mentioned and therefore this theoretical approach is typically an approximate one. It is evident that for small values of k0 dj- one gets a good approximation by keeping only a small number of terms in eqs. (6.65) for calculating the ellipsometric parameters of the inhomogeneous thin film system. It should be noted that in practice it is sometimes useful to employ the approximation based on replacing the inhomogeneous layers by multilayers. The differences of the values of the refractive indices of the adjacent films in these multilayers or the values of the thicknesses of the films forming these multilayers must be sufficiently small. The optical quantities can then be calculated by means of the formulae corresponding to the thin film systems comprising the homogeneous layers (for details see Vagi~ek [1960], Heavens [1960], Berning [ 1963] and Jacobson [ 1966]). 6.3. TRANSITION LAYERS
In practice it is sometimes necessary to take into account very thin films taking place at the boundaries of the various thin film systems. The values of the thicknesses of these films known as the transition layers are usually equal to several nanometers. The layers mentioned originate in different ways. For
230
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 6
example, they originate by diffusion of materials forming the adjacent films of the system through the corresponding boundaries. A further example of the transition layers is represented by the thin films arising on the upper boundaries of the systems by miscellaneous chemical and physical processes (e.g., by oxidation and adsorption). From the point of view of the optical properties of the thin film systems studied in practice the transition layers mostly must be understood as the defect complicating the interpretation of experimental data achieved for these systems using various optical techniques (e.g., within ellipsometry). To be able to include the influence of the transition layers so as to correctly interpret the optical experimental data corresponding to the multilayer system studied, we have to create a suitable physical model of these layers. In the first approximation one can sometimes represent the transition layers by the homogeneous thin films (note that we shall only deal with the transition layers formed by the optically isotropic materials in this chapter). Thus the influence of these transition layers on the ellipsometric parameters and the other optical quantities of the thin film systems is expressed using the matrix formalism presented in w 5.1. However, the transition layers are often inhomogeneous; i.e., the refractive index of these layers is strongly dependent on the coordinate corresponding to the axis perpendicular to their boundaries. This means that in principle the approximation of the transition layers by the homogeneous very thin films is not always sufficient in practical problems. Thus these transition layers must then be represented by the inhomogeneous very thin films if their influence on the optical quantities of the thin film systems is accurately treated. The mathematical procedure that includes the influence of the very thin inhomogeneous films, i.e., the transition layers, on the ellipsometric parameters and the other optical quantities of the thin film system is relatively easy in comparison to the corresponding procedure that includes the influence of the general inhomogeneous thin film on these quantities of the system mentioned (see w 6.2). This is because we can use the roughest version of the approximation based on developing the particular solutions of eqs. (6.66)-(6.69) in the power series (see w 6.2.2). In these power series we can only restrict ourselves on the first terms. Thus within this approximation the functions ~ (z), ~ (z), ~1(z), f2/(z), ~l(z), ~2(z), g~l(z) and ~(z) can then be expressed as follows: ~ j ( z ) = 1,
(6.70)
f z j ( z ) = z,
=-kgf0 Z#(r
~.(z) = 1,
(6.71)
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ELLIPSOMETRIC QUANTITIES OF IMPERFECT THIN FILM SYSTEMS
~lj(Z) = 1,
1
~2j(z) = n~.2 R jr0 Z
/o z
h](~)d~,
cos
231
(6.72)
2.R ~
(6.73)
After inserting the foregoing equations into eqs. (6.46) and (6.50), one obtains the following forms of the elements of the matrices Xsj and Xpj representing the very thin inhomogeneous transition layer:
(
Xsj=
1
djk2(n~sin 2 0 0 - P ) 1
(6.74)
'
Xpj = djk2o(Ong sin 20o - 1) 1
'
(6.75)
where
1/0
p =~
hZ(z)dz,
1/0 hZ(z)dZ. 1
Q=~
(6.76)
For example, if we assume that the inhomogeneous transition layer takes place at the boundary between both the ambient and substrate, we can write the transfer matrix of this system in the following way:
l~/iq = ~r
f~q ~,~q.
(6.77)
Moreover if we assume that the refractive index of the transition layer mentioned continuously varies from the refractive index of the substrate to the refractive index of the ambient, eq. (5.22) gives the well-known relations for the Fresnel coefficients of this transition layer derived by Drude [ 1891 ] in another way (see also Vagi6ek [1960] and Knittl [1976]). This approximation is called the Drude
approximation. 6.4. VOLUME INHOMOGENEITIES
Volume inhomogeneities also belong to the defects of thin films, which play an important role in the optics of the thin films systems. In principle it is very difficult to include these inhomogeneities into the formulae expressing the optical quantities of the systems mentioned. So far only several kinds of the volume inhomogeneities have been taken into account in the expressions for the optical
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quantities of the thin films. The volume inhomogeneities exhibiting the linear dimensions substantially smaller than the wavelength of incident light 24 belong to the volume defects whose influence on the optical quantities of the thin film systems can be taken into account in a relatively easy way. For example, among the inhomogeneities of this type small voids inside of the materials forming the thin films can be included. To be able to include the existence of these voids in the thin films forming the systems, we can employ an effective medium theories approach for expressing the optical constants of these films. This means that one of the formulae for refractive index of twocomponent mixtures implied by the generic formula presented in w 6.1.1 (see eq. 6.10) must be used for this purpose. For example, the formula that expresses the refractive index of the j-th film of the system containing the small voids obeys the Bruggeman formula written in the following form: 1 - h)2,ef
#
PJ 1 + 2h~,ef +(1 - p j ) #^
- h2, ef +
2h2, ef = 0,
(6.78)
where pj is the relative volume of the film corresponding to the voids, hj that denotes the refractive index of the material of the film without the voids and hj, ef represents the refractive index of the entire film (it is assumed that the materials forming the thin films with the voids are optically isotropic). The further important defect of thin films is represented by their columnar structure. The materials forming many films prepared by various technologies are disposed in formations having the form of columns. These columns are aligned in the direction of growth of the film and long pores exist between them. The pores can be filled with water or the other liquids which depends on the atmosphere in which the film is placed. The optical properties of these films with the columnar structure can also be described by means of the formulae belonging to the effective medium theories. However, in general it is necessary to consider that there are several materials forming the columnar films because of the existence of the liquids within the pores (these pores can be saturated by the liquids or they can be partially filled with them). In this case, for example,
24 Of course, at the same time these linear dimensions of the volume inhomogeneities must be sufficiently large in order that one may use the phenomenological approach to describe the interaction of light with individual species of the inhomogeneous materials.
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233
the refractive index of the j-th columnar film containing M materials must obey the following Bruggeman formula: M
^
^
Zpjk n2k-n2ef k=l
n2k + 2n2, ef
M
"-0,
Zpjk = 1.
(6.79)
k=l
In practice the following intuitive formula is often used for expressing the refractive index of the columnar film: M nj, e f - -
ZPjk njk.
(6.80)
k=l
When the columnar films are placed in air, one can assume that three materials (M = 3), i.e., the material of the columns, water and air, mainly form the films mentioned. It should be emphasized that from the optical point of view the columnar films must appear as anisotropic films even when all the materials forming these films are optically isotropic (in this section we only deal with the columnar films consisting of the optically isotropic materials). This statement is especially true when the angles existing between the direction of the light wave propagating in the columnar film and the axes of the columns exhibit relatively large values. In this case the theoretical approach that allows us to express the interaction of the light waves with the columnar films must be based on other principles (see, e.g., Hodgkinson, Horowitz, Macleod, Sikkens and Wharton [ 1985], Hodgkinson and Wu [ 1993] and Wang [ 1995]). So far there has only been limited research on the interaction of light waves with the thin films that comprise larger volume inhomogeneities that cause the scattering of these waves.
6.5. THIN FILM SYSTEMS WITH OTHER DEFECTS
In practice we can also encounter other defects of thin films than those discussed in the preceding sections. Further defects relevant ellipsometry of thin film systems are as follows: 9 Non-uniformity o f thicknesses of the thin films. 9 Artificial optical anisotropy of materials of the films exhibiting continuous variations caused by the existence of stresses within these films.
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ELLIPSOMETRY OF THIN FILM SYSTEMS
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9 B l o c k or m o s a i c s t r u c t u r e of the films consisting of the anisotropic materials. Currently, the influence of the three foregoing defects on the ellipsometric parameters and the other optical quantities of the thin films systems has not been systematically investigated. Despite this fact it is still evident that these defects considerably influence the optical quantities of thin films. This statement is supported by the following reasons: It was shown that the non-uniformity of thicknesses had significant influence on the reflectances of the single layers. For example, by means of a special theoretical procedure the formula for the reflectance of the single layers with linear variations in thicknesses at normal incidence was derived (see, e.g., Pisarkiewicz [1994]). The numerical analysis of the reflectance formula mentioned shows that the values of the reflectances of the single layers are greatly affected even when the thickness variations are relatively small. The single crystal films formed by the optically isotropic materials to be produced by epitaxial growth on corresponding single crystal substrates can exhibit the artificial optical anisotropy caused by stresses originating inside these films in consequence of a misfit between the crystallographic structures of both the film and substrate. These stresses and the corresponding optical anisotropy continuously vary from the substrate to the ambient. Thus these films exhibit the optical properties influenced by a combination of both anisotropy and inhomogeneity (i.e., such films are optically inhomogeneous and anisotropic). Unfortunately, the influence of the defect mentioned on the optical quantities of thin films has not been studied sufficiently. Of course, it is clear that this problem can be solved by dividing the inhomogeneous anisotropic film into a multilayer system of homogeneous anisotropic layers with sufficiently small values of their thicknesses. One can then apply the combination of the mathematical procedures presented in w 5.2 and 6.2 to express corresponding formulae for the ellipsometric parameters and the other optical quantities of thin film systems containing these inhomogeneous anisotropic films. The third defect mentioned above, i.e., the block or mosaic structure of the films consisting of the anisotropic materials, is a particularly complicated one where the blocks of their crystallographic structure are oriented in various directions. One can see that this defect can be understood as a combination of two defects, i.e., as a combination of both the volume inhomogeneity and optical anisotropy. So far no substantial attention has been paid to solving the problem of deriving the formulae for the optical quantities of the systems containing such films even when many polycrystalline films exhibit this combined defect. Presently, the optical properties of the anisotropic films with block or mosaic structure are most often represented by the model of films containing an effective
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homogeneous anisotropic or isotropic material (see the paper by Collins, An, Nguyen, Li and Lu [1994]).
w 7. Experimental methods Experimental methods of ellipsometry can be divided into two basic groups. The first group consists of the ellipsometric methods applied for experimental studies of thin film systems after their preparation. They are usually utilized outside technological equipment. These methods are called ex-situ ones. The latter group is formed by the methods applied during the growth of the thin film systems. These methods are most often employed for checking the properties of the systems originating inside the corresponding technological equipment. They are called in-situ methods. Both the ex-situ and in-situ ellipsometric methods can be classified into two groups: 9 monochromatic methods 9 spectroscopic methods
The monochromatic methods and spectroscopic methods can further be divided into two other groups: - single angle o f incidence methods - multiple angle o f incidence methods
In practice one can thus encounter the following ellipsometric methods: the monochromatic single angle of incidence methods, monochromatic multiple angle of incidence methods, spectroscopic single angle of incidence methods and spectroscopic multiple angle of incidence methods. In principle all these methods consist of two modifications: 9 non-immersion modification 9 immersion modification
In the monochromatic and/or spectroscopic methods the values of the ellipsometric quantities of the thin film system are measured at a single wavelength and/or a family of wavelengths from a certain spectral region. Single and/or multiple angles of incidence ellipsometric methods are based on measuring these quantities at one and/or several angles of incidence of light on the systems. The non-immersion methods enable us to measure the ellipsometric quantities of the system in one ambient (mostly in air) whereas the immersion ones make it possibile to measure these quantities in various ambients (e.g., in different non-absorbing liquids). The foregoing methods can be applied in both conventional and generalized ellipsometry.
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ELLIPSOMETRYOF THIN FILM SYSTEMS
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Below we shall present some representative methods of conventional and generalized ellipsometries belonging to the methods introduced above. 7.1. MONOCHROMATIC SINGLE ANGLE OF INCIDENCE METHOD
This method can be called monochromatic single angle of incidence (MSAI) ellipsometry as well. In this method the ellipsometric parameters tp and A of the thin film systems are measured at one wavelength and one angle of incidence. The values of the two independent ellipsometric parameters ~ and A are thus measured for the system studied if this method is applied (the measurements of tp and A are usually performed in air)2s. MSAI ellipsometry was employed, for example, by Archer [1962] in ex-situ measurements of single layers of SiO2 placed onto the substrates formed with silicon single crystal wafers. This author measured the values of tp and A of the samples of this system Si/SiO2 for the angle of incidence of 70 ~ and the wavelength of 546.1 nm (the measurements were carried out by the PCSA ellipsometer). The individual samples of the system mentioned mutually differed in the values of the thickness of the SiO2 layer. The experimental data obtained were interpreted using the model of the ideal non-absorbing single layer, i.e., the following formula for calculating the values of the ellipsometric parameters of the system Si/SiO2 was used: tan tp
e ia =
rpl
+ ?'p2 e -i2k'
1 + ?pl ?'p2 e -i2k'
1 + rsl ?'s2 e -i2k'
(7.1)
rsl + ~'s2 e -i2k'
The formula above uses the general formulae for the ellipsometric parameters of the ideal isotropic thin film system (see w 5.1). From eq. (7.1) one can see that in principle every sample of the system Si/SiO2 is characterized by four optical parameters, i.e., by the refractive index n and extinction coefficient k of the silicon substrate and the refractive index nl and thickness dl of the SiO2 layer. Of course, it is impossible to determine the values of all the optical parameters mentioned characterizing a chosen sample of Si/SiO2 by means of the two values of the ellipsometric parameters. Thus the values of the optical constants of the silicon substrate had to be evaluated in an independent way. The value of the
25 It is interesting that in the last years MSAI ellipsometry is employed for studies of the thin film systems by means of a miniaturized ellipsometers formed by miniaturized optoelectronic components (see Stenkamp, Abraham, Amato, Zetterer and Pokrowsky [1995]). These ellipsometers connected with suitable optically active samples can serve as sensors for various purposes in industry.
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refractive index nl was then determined unambiguously, and the thickness dl, with an uncertainty corresponding to the periodicity of the fraction in eq. (7.1) due to this thickness (the SiO2 layers with the thicknesses laying within the interval 1.5-800 nm were studied). Note that the uncertainty in determining the thickness dl can be removed only when the other ellipsometric method is also applied to analyze every sample of the system Si/SiO2. MSAI ellipsometry was employed for ex-situ analyzing the single layers formed by miscellaneous materials (see, e.g., the papers by McCrackin, Passaglia, Stromberg and Steinberg [1963] and Luke~ [1972]). MSAI ellipsometry was also used to characterize the growth of various single layers. The principle of employing this ellipsometry for the in-situ studies is the same as for the ex-situ ones. For example, Leslie and Knorr [1974] employed MSAI ellipsometry for studying the growth of tantalum oxide single layers at the plasma oxidation of a tantalum substrate. This ellipsometry was often used for investigating the adsorption processes taking place on the surfaces of various solids (see, e.g., Meyer and Bootsma [1969] and Dorn, Ltith and Ibach [1974]) 26. If MSAI ellipsometry is applied for characterizing one sample of the thin film system, it is possible to determine the values of only two optical parameters of this system (the other parameters must be known or evaluated using independent methods). However, when several samples of the thin film system studied are simultaneously analyzed using MSAI ellipsometry (multisample modification of this method), one can determine the values of more parameters of this system if the samples mentioned mutually differ only in the values of one parameter. In this case it is possible to overdetermine the problem because one can obtain sufficient numbers of the measured values of t/.t and A making it possibile to evaluate the values of all the parameters characterizing all the samples of the system. For example, this multisample modification of MSAI ellipsometry was employed for the optical characterization of the absorbing molybdenum thin films and the absorbing molybdenum oxide (MOO3) films with different thicknesses in the paper by Lederich [1972] (the method was applied for the wavelength of 632.8 nm). Moreover, this modification of MSAI ellipsometry was employed by den Engelsen [ 1971 ] for analyzing optically anisotropic single Langmuir-Blodgett (LB) layers deposited on silicon wafers. These layers are
26 It should be pointed out that MSAI ellipsometry was also employed for both the in-situ and ex-situ studies of double layers and triple layers (see, e.g., Dinges [1981], Svitashov, Semenenko, Semenenko and Sokolov [1972] and Flamme [1980]).
238
ELLIPSOMETRY OF THIN FILM SYSTEMS
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formed by the uniaxial material whose optical axis is perpendicular to their boundaries. This means that the values of the ellipsometric parameters ~ and A were again measured for several layers with different thicknesses (the values of both the wavelength and angle of incidence were as follows: Z = 546.1 nm or 632.8nm and 00 = 67.8~ The family of all the values of t/t and A were then treated using the least-squares method (LSM). For example, by means of this procedure the following values of the optical constants of the LB layers of a 1:12 mixture of cyanin dye and arachidic acid were obtained for 3, = 546.1 nm: nlll = 1.511 -+- 0.004, nix = 1.532 + 0.004, kill = 0, kl• = 0.067 and d = (2.74 + 0.04)nm. The symbols nlll and kill and/or nix and kl• denote the refractive index and extinction coefficient of the layer corresponding to the wave polarized in the direction parallel and/or perpendicular to the optical axis. The symbol d represents the mean thickness of one monolayer of these LB layers. It should be noted that 10 layers were analyzed in this way (the thickest layer contained 80 monolayers with the thickness presented above). Further the LB layers of glycerol tripalmitate were also characterized by this method for ~, = 632.8 nm in this paper. Zhu, Lin and Wei [1992] used MSAI ellipsometry for characterizing the LB single layers of arachidic acid deposited on glass substrates in a modified way (~, = 632.8 nm and 00 = 70~ Kinosita and Nishibori [ 1969] employed MSAI ellipsometry for measuring the porosity of columnar MgF2 single films placed onto fused silica plates. First, they measured the mean refractive indices of these films in a specimen chamber filled in vacuum. The measurements of the same films then followed in a laboratory atmosphere and ethyl alcohol vapors. If the pore radii are sufficiently small, capillary condensation takes place and the pores will be filled to the brim with the condensed liquid (i.e., water or ethyl alcohol). They thus determined two values of the mean refractive index of the film studied corresponding to the pores filled in vacuum and the liquid. Then the intuitive formula for expressing the mean refractive index was employed (see eq. (6.80) for M = 2). Because the values of the refractive indices of the liquids mentioned are well known the authors could determine the values of refractive index of the columns nl and the packing density of pores (i.e., porosity) of the MgF2 films studied. It was found that the values of nl were identical with those corresponding to the bulk material of MgF2 and the values of the porosity were in the range 0.021-0.057. 7.2. M O N O C H R O M A T I C M U L T I P L E A N G L E OF I N C I D E N C E M E T H O D
In the following, this method will be called monochromatic multiple angle of incidence (MMAI) ellipsometry (in the literature this method is often called
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239
multiple angle of incidence (MAI) ellipsometry). In practice MMAI ellipsometry is frequently used for analyzing the thin film systems. In this method the angular dependences of the ellipsometric parameters of the system studied are measured for a chosen wavelength in air. The treatment of the angular dependences of tp and A is mostly performed by the LSM. Ibrahim and Bashara [1971] used MMAI ellipsometry to study the system formed by the silicon single crystal substrate and the native oxide layer. They measured the angular dependences of tp and A of the samples of this system for several wavelengths (296.7-435.8nm) but the treatment of the angular dependences was performed for each sample and each wavelength separately. They showed that the parameters characterizing the native oxide layer, i.e., the refractive index n l and thickness dl, could not reliable be determined by means of this method. However, the values of the optical constants of the silicon substrate were evaluated using the LSM with a satisfactory accuracy within this ellipsometry. It can be shown that in principle it is practically impossible to determine the values of the optical constants and thicknesses of the very thin films (i.e., the films with thicknesses equal to several nanometers or less) with sufficient accuracy (see, e.g., Loescher, Detry and Clauser [ 1971]). MMAI ellipsometry is mainly employed for the optical characterization of thicker single layers. For example, Pedinoff and Stafsudd [1982] reliably evaluated the values of both the refractive index and the thickness of single Si3N4 layers placed onto silicon single crystal substrate. (The thicknesses dl and the refractive index nl of a chosen Si3N4 layer were evaluated as follows: dl - 92.59nm and nl = 2). Moreover, the value of the complex refractive index h of the silicon substrate was determined in the value h = 3.866- 0.017i (A = 632.8 nm, 00 = 50-70~ It is necessary to point out that the values of the extinction coefficient of the silicon substrate strongly correlated with the optical parameters of the single layer and therefore its value cannot be determined with reasonable accuracy. MMAI ellipsometry was also employed for analyzing absorbing single layers. For example, the determination of the optical constants of chromium single layers deposited on glass substrates was evaluated by means of this method in the paper by L6schke [ 1981 ]. In this paper MMAI was applied for each of five wavelengths in the visible spectral range (00 = 35-75 ~ By treating the angular dependences of tit and A measured for every Cr film, the values of both the refractive index nl and extinction coefficient kl were determined for the selected wavelengths (the values of the thicknesses of the films were evaluated using the interferometric method applied with an interference microscope). The values of both the optical constants of the Cr films were evaluated with sufficient accuracy (e.g., for a
240
ELLIPSOMETRY OF THIN FILM SYSTEMS
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chosen Cr film the following values of the optical constants were obtained at the wavelength of 546.1 nm: nl = 2.418 4- 0.005 and kl = 2.131 4- 0.005). MMAI ellipsometry also can be employed for characterizing multilayer systems. This method was used to determine the values of the optical parameters of double and triple layers in particular. For example, the double layers formed by Si3N4 and SiO2 films deposited on silicon single crystal substrates were analyzed by MMAI ellipsometry in the paper by Paneva and Ohlidal [1986] (A = 632.8 nm and 00 = 45-75~ The values of the optical constants of the silicon substrate were taken from the literature and the values of the four optical parameters describing the double layers, i.e., the values of nl and dl (the upper film) and n2 and d2 (the lower film), were evaluated using the LSM. In the paper it was shown that the accuracy of determining the values of the parameters mentioned was strongly dependent on the values of these parameters (especially on the values of both the thicknesses). The following results for the chosen sample of the Si3N4/SiO2 and/or SiO2/Si3N4 were obtained: nl = 2.015 4-0.004, dl = (104.7 4- 0.2)nm, n2 = 1.439 + 0.002 and d2 = (202.1 4- 0.3)nm and/or nl = 1.441 + 0.009, dl = (224 + 2)nm, n2 = 1.981 i 0.004 and d2 = (94.0 + 0.2)nm. In this paper the three-layer system MgF2/Si3N4/SiO2 on the silicon substrate was analyzed as well. The following results were obtained for a chosen sample of this system: nl = 1.404 + 0.004, dl = (74.8 + 0.3)nm, n2 = 1.998 • 0.004, d2 = (196.3 + 0.6)nm, n3 = 1.457 + 0.003 and d3 = (109.1 + 0.3)nm. The authors also showed that MMAI ellipsometry was a very useful method for the complete optical analysis of the nonabsorbing double and triple layers if the thicknesses of the films forming the system mentioned are greater than about 50 nm. Gaillyovfi, Schmidt and Humli~ek [1985] analyzed the triple layers NOL/ p-Si/SiO2 placed on silicon substrates by MMAI ellipsometry as well (NOL and p-Si denote the native oxide layer and the film of polycrystalline silicon, respectively). They also showed that the ellipsometric method discussed was very helpful for the optical analysis of this system (A = 632.8 nm and 00 = 5080~ They obtained these results for a chosen sample: n2 = 3.9335 i 0.0008, k2 = 0.0405 + 0.0006, d2 = (119.88 + 0.05)nm and d3 = (302.19 4- 0.06)nm. Note that the values of the refractive indices of both the NOL and SiO2 film were fixed in the value of 1.46 and the complex refractive index of the silicon substrate was taken from the literature in the value of 3.881-0.018i (for details see the paper mentioned). MMAI ellipsometry can also be used for the optical analysis of anisotropic thin film systems. For example, Haitjema and Woerlee [1989] applied this method to study single films of tin dioxide deposited on quartz substrates
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separately at two wavelengths QI = 632.8 nm and/l = 1523 nm). They measured the angular dependences of the ellipsometric parameters within the following interval of the angles of incidence: 00 = 30-75 ~ For the wavelength of 1523 nm the SnO2 layers appeared to be the anisotropic (uniaxial) absorbing films. It was assumed that the optical axis was perpendicular to the boundaries of these films. The values of the following optical parameters characterizing the SnO2 layers studied were determined using this method by means of the LSM: nix, kl• nlll and kllJ. The values of the thicknesses dl and the refractive index of the quartz substrate were determined independently. The following results were obtained for a chosen sample of the SnO2 layer (dl = 196nm): nl• = 1.987 + 0.006, kl • = 0.087 + 0.021, n 111= 1.94 -t- 0.09 and kl II = 0.10 4- 0.15. From the literature it is evident that MMAI ellipsometry is a very useful method for the optical characterization of thin film systems exhibiting various defects. For example, this method was extensively utilized for characterizing many thin film systems with rough boundaries. The influence of roughness of the upper boundaries on the ellipsometric parameters was observed by Haitjema and Woerlee [1989] for the SnO2 layers at ~=632.8 nm (note that for ~= 1523 nm roughness of the upper boundaries was not considered, see above). For ~l=632.8 nm the layers of SnO2 could be considered to be nonabsorbing and isotropic films. It was shown that roughness of the upper boundaries of these films corresponded to the model of the microrough boundary. Therefore the treatment of the experimental data, the authors used the model of the nonabsorbing thin film with the upper rough boundary replaced with the homogeneous fictitious layer characterized with the refractive index nef expressed by the formula nZf = 0.5(n 2 + n2). They searched the values of the following parameters using the LSM: nl, dl and al (o1 ~ 0.67c, where c is the thickness of the fictitious film). For a chosen SnO2 layer, they determined these results: n = 1.83 + 0.04, dl = (581 • 20)nm and o"1 = (6 + 2)nm. Gaillyovfi [1987] studied the rough GaAs surfaces originated during thermal oxidation at a temperature of 450~ using MMAI ellipsometry after dissolution of the oxide films (X = 632.8 nm and 00 -- 50-80~ She again used the model of roughness represented by the fictitious film. For the treatment of the angular dependences of tp and A measured, the model consisting of the GaAs substrate covered by the double layer with the smooth boundaries formed by both the fictitious film and native oxide layer was used. The values of the refractive index nef and the thickness def of the fictitious film together with the refractive index nl of the native oxide film were determined. The thickness of the native oxide film was changed stepwise in the interval 2-5 nm. The extinction coefficient of the
242
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 7
fictitious film was fixed at certain values (see the paper cited) and the values of the optical constants of the GaAs substrates were taken from the literature. The results obtained for all the values of the thickness of the native layer were similar. For example, for d~ = 3.5 nm the following values of the parameters sought were found: nl -- 1.80 -+- 0.06nm, nef = 2.21 -+- 0.04 and def = (15.3 i 1.3)nm (roughness is represented by the values of nef and clef). By means of MMAI ellipsometry Ohlidal, Luke~ and Navrfitil [ 1974] analyzed the rough silicon single crystal surfaces covered with the native oxide layers (these surfaces were prepared by anodic oxidation of the smooth surfaces and successive dissolution of the resulting anodic oxide films). The authors applied the method for the wavelength of 546.1 nm within the interval 00 = 70-80 ~ For interpreting the experimental data they employed the formula expressing the ellipsometric parameters of the identical thin film derived using the scalar diffraction theory (see eq. (6.24) and the paper by Ohlidal and Luke~ [ 1972b]). The value of the refractive index nl of the native oxide layer was fixed at the value 1.46 and the values of the optical constants of silicon were taken from the literature. This means that the value of the thickness dl of the native oxide layer and the value of tan/30 were determined using the interpretation of the experimental data. The following values of the parameters mentioned were found for a chosen sample: dl = (5.5 + 0.5)nm and tan/30 = 0.100 + 0.005. In the same way Navrfitil, Ohlidal and Luke~ [1979] characterized the rough surfaces of GaAs single crystals originating during thermal oxidation at a temperature of 400-520~ (oxide films were again dissolved before ellipsometric measurements). For a chosen sample of this system (temperature was of 500~ the following results were obtained by the LSM: dl = (7.0 9 0.5)nm and tan/30 = 0.080 + 0.005 (the value of the refractive index nl of the native layer and the values of the optical constants GaAs were fixed in the values determined by independent ways). MMAI ellipsometry was also employed for the optical analysis of inhomogeneous isotropic thin films. Dutta, Candela, Chandler-Horowitz, Marchiando and Peckerar [1988] analyzed the inhomogeneous films that originated by highdose and high-energy oxygen ion implantation in silicon using this method (~, = 632.8 nm and 00 = 54.5-79.5 ~ after annealing in an inert ambient such as nitrogen or argon at above 1150~ to repair the damage introduced during implantation. These inhomogeneous films were approximated by multilayer systems (see w 6.2.2). The authors mentioned showed that the five-layer system could represent the inhomogeneous films studied from the point of view of the angular dependences of the ellipsometric parameters measured for these films (the LSM was used to treat the experimental data). The five-layer system
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mentioned was formed by a native oxide layer, a top silicon layer, an upper interlayer, a buried oxide film and a lower interlayer (the substrate was formed by the silicon single crystal). In the paper it was assumed that the top silicon layer and/or the buried oxide layer consisted of a single crystal of silicon and/or a perfect amorphous SiO2. The values of the thicknesses of the four films, i.e., the top silicon layer, the upper interlayer, the buried oxide film and the lower interlayer, together with the refractive indices of both the interlayers were thus sought by means of the LSM. The values of the optical constants of the native oxide layer, the top silicon layer, the buried oxide layer and the substrate were fixed in the values taken from the literature (the thickness of the native oxide layer was fixed in the value of 2 nm). For a chosen sample, the following results were obtained using the procedure described: d2 = (79.0 • 0.8)nm, d3 = (74 • 1)nm, d4 = (204 + 8)nm, d5 = (18 4-2)nm, n3 = 2.236 and n5 = 2.188 (the errors of the refractive indices were not presented). In the same way Ibrahim and Bashara [1972] investigated the inhomogeneous films formed by implantation of low-energy argon ions in the silicon single crystals. From the foregoing, it is thus evident that MMAI ellipsometry can be utilized for the characterization of complicated layered structures. The examples of MMAI ellipsometric studies of the layered systems introduced above are realized in reflected light (i.e., in reflection mode of MMAI ellipsometry). Of course, MMAI ellipsometric studies are sometimes performed in transmitted light as well (i.e., in transmission mode of MMAI ellipsometry). This statement can be illustrated by means of ellipsometric studies carried out in the paper by Ohlidal and Lukeg [1981]. These authors analyzed thin gold films deposited onto glass substrates by means of this method (several thin gold films with thicknesses in the range 25-45 nm were studied). For a chosen sample of the Au film they obtained the following results (A = 632.8nm, 00 = 4575~ nl = 0150 4-0.17, kl = 3 . 1 + 0 . 1 and dl = (45 4- 11) (the value of the refractive index of the glass substrate was determined independently). The authors mentioned showed that the values of the optical constants and thicknesses of the Au films investigated can be evaluated more precisely by the reflection mode than by the transmission one of this method (this statement is mainly valid for the refractive indices and thicknesses of the films under investigation). When MMAI ellipsometry is applied, one has to respect the influence of the back side of the glass substrate, i.e., the substrate side without the film, on the values ti t measured and interpreted within the transmission mode (for details see the paper cited). Note that this MMAI ellipsometry realized in the transmission mode was combined with MMAI ellipsometry applied in the reflection mode for analyzing the gold films mentioned as well (see w 7.6).
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It should also be noted that both non-immersion monochromatic ellipsometries, i.e., MSAI and MMAI ellipsometries, are often used for characterizing thin film systems not only in the visible region but also are frequently employed for this purpose in other spectral regions. These methods were utilized for the optical analysis of various thin film systems in the UV, IR and microwave regions. For example, in the IR region epitaxial layers of n-type silicon deposited on the silicon substrates were analyzed (see DeNicola, Saifi and Frazee [ 1972]). In this region single layers of ThF4 on ZnSe-substrates and single layers of ZnSe on KCl-substrates were analyzed as well (see Pedinoff, Braunstein and Stafsudd [1977]). In several papers metal films, i.e., Cr, A1, Cu films, were studied by ellipsometries mentioned in the microwave region (see, e.g., Konev, Kuleshov and Punko [ 1985]). In the UV region both MSAI and MMAI ellipsometries were utilized for characterizing thin film systems as well. For example, Johnson and Bashara [ 1971 ] used MMAI ellipsometry for analyzing very thin Ag tarnish films (dl < 10 nm) on thick Ag films for several wavelengths from both the UV and visible regions. 7.3. S P E C T R O S C O P I C SINGLE A N G L E OF I N C I D E N C E M E T H O D
During the last two decades methods of spectroscopic ellipsometry have been extensively employed for the optical analysis of thin film systems. The method of the optical analysis of the systems based on measuring the spectral dependences of tp and A of these systems at one angle of incidence can be called spectroscopic single angle of incidence (SSAI) ellipsometry. In the literature this method is often briefly called spectroscopic ellipsometry (SE). There are two procedures of interpreting the spectral dependences of tp and A. By means of the first procedure the values of the pair of tp and A are measured and treated for each wavelength separately. This means that in principle the treatment of the experimental data is the same as the treatment of those obtained by means of MSAI ellipsometry. Of course, in this way one can only determine the values of two parameters of the system studied for each wavelength (see w 7.1). For example, this procedure was employed for evaluating the spectral dependences of both the refractive index and extinction coefficient of ZnSe film prepared by molecular-beam epitaxy on GaAs substrate in the paper by Dahmani, Salamanca-Riba, Nguyen, Chandler-Horowitz and Jonker [ 1994]. The spectral dependences of the ellipsometric parameters of the system mentioned were measured in the region 250-830nm and interpreted using the ideal model of the substrate covered with the single film. In the spectral region of transparency of the ZnSe film the authors interpreted the data measured in
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the following way. The thickness of the film was fixed in the chosen values and the values of the optical constants of the ZnSe films were calculated for these values of the thickness at all the values of the wavelength of the spectral region. The spectral dependences of the optical constants obtained in this way exhibited oscillations in general. These oscillations were caused by the incorrect value of the thickness and deviations of the ZnSe film from the ideal model. The correct values of the thickness and optical constants corresponded to the smallest oscillations mentioned (this criterion was fulfilled for dl = 640nm). In the spectral region of absorption of the ZnSe film the system specified had to be replaced with the system formed by a semi-infinite substrate of ZnSe. In this region the spectral dependences of both the refractive index and extinction coefficient of the ZnSe film were determined using the formula that explicitly expressed both the optical constants of the semi-infinite substrate by means of its ellipsometric parameters that corresponded to a selected wavelength (for details see the paper mentioned). The latter procedure of treating the experimental data achieved within SSAI ellipsometry for the system formed with the substrate and the single film was used by Mori, Fujii, Xiong and Saitoh [1995]. They analyzed diamond-like carbon (DLC) films prepared by ionized deposition from methane on silicon substrate within the spectral region 300-850nm at the angle of incidence of 60 ~. They further assumed that the spectral dependences of the refractive index and the extinction coefficient were represented by the Cauchy formula and exponential formula, respectively: nl (Z) = Co + ~-2 + ~
and
kl (X) = A exp
~
- Eg
,
(7.2)
where Co, C1, (72, A, B and Eg are the material constants. The authors used the LSM for determining the values of the thickness and the material constants of the dispersion formulae. For a chosen sample of the DLC film they found the thickness in value of 614.5 nm. The spectral dependences of both the optical constants of this sample were presented in a graphic form. Note that in the latter procedure used to analyze the DLC films, the experimental data were treated simultaneously in contrast to the treatment of the data in the first procedure. The in-situ modification of SSAI ellipsometry is often employed for monitoring various kinetic processes concerning the thin film systems. For example, Logothetidis, Alexandrou and Papadopoulos [1995] used this modification for monitoring TiNx-thin films deposited by reactive sputtering on Ti and Si substrates. They measured the spectral dependences of ~ and A in the region 240-830nm (00 = 70.5~ The films mentioned were prepared under different
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ELLIPSOMETRY OF THIN FILM SYSTEMS
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conditions of depositing (the thicknesses of the films were several tens of nanometers). The authors replaced the system formed by the substrate and TiNx film with a fictitious boundary of an effective material characterized by a complex pseudodielectric function (e) expressed by means of the ellipsometric parameters measured using the following formula:
(~)=sin 200+
1+/~
tan 2 00 sin 2 00.
(7.3)
They investigated changes of the spectral dependences of this pseudodielectric function on the time of the deposition of the films mentioned. These spectral dependences were used for evaluating the values of plasma frequency tOp corresponding to the deposition times chosen (the Drude model of the dielectric function was used). Of course, for the short deposition times, i.e., for the very thin films, the values of tOp were misrepresented by the influence of the substrate because of penetrating light into the substrate. Moreover, for the long deposition times (i.e., for sufficiently great thicknesses of the films) these spectral dependences were utilized for the determination of the volume fractions of the components (i.e., voids, TiN and Ti) forming the materials of these films on the basis of the Bruggeman formula expressing the dielectric functions of these films. For short deposition times, the model of substrate covered with the film was used to interpret the experimental data, which allowed them to determine the spectral dependences of the dielectric function of the films. In the way described above the authors could then find the values of the volume fractions of the components of these very thin films. The studies presented enable the authors to reveal the influence of the deposition conditions on the basic properties of the TiNx-single films in real time. The in-situ modification of SSAI ellipsometry was used by many authors to monitor growth of miscellaneous thin films. For example, Picketing, Hope, Carline and Robbins [ 1995] utilized this method for studies of epitaxial growth of Sil_xGe~ single layers on silicon substrate (these studies mainly investigated the dependence of the thickness of the films on the deposition time). Basic applications of in-situ SSAI ellipsometry for monitoring some thin film systems used in the microelectronics industry are described in the reviews by Irene [ 1993] and Aspnes [ 1993]. SSAI ellipsometry is the suitable method for the optical analysis of the multilayer systems too. For this purpose the method was employed many times. Using SSAI ellipsometry Theeten, Aspnes, Simondet, Erman and Miirau [ 1981 ] analyzed the two-layer system formed by the silicon single-crystal substrate
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covered with the native oxide layer and Si3N4 film. These authors measured the spectral dependences of the ellipsometric parameters of the samples of the system under consideration in the range 210-830nm (00 = 67.08~ For interpreting the ellipsometric spectra measured, they used five models of the double layer specified above. Two models were represented by single layers. The first and/or second single layer was assumed to be formed by a single layer of Si3N4 and/or single layer consisting of a mixture of Si3N4 and SiO2. The other two models were formed by the double layers. The first of them contained the lower film of SiO2 with the fixed thickness in the value of 2 nm and the upper film of Si3N4 with the thickness sought. The latter model of the double layer consisted of the lower film formed by the mixture of voids and SiO2 and the upper one formed by the film containing the mixture of Si3N4 and SiO2. The fifth model of the system studied was formed by three films: by the lower one consisting of the mixture of voids and SiO2, the middle one formed by the mixture of amorphous Si and Si3N4 and the upper one formed by the film containing the mixture of Si3N4 and SiO2. The spectral dependences of the optical constants of SiO2, Si3N4, a-Si and c-Si were determined in independent ways and they were fixed at treating the experimental ellipsometric data using the LSM. The best fit of these ellipsometric data was achieved by means of the fifth model. The values of the parameters characterizing this model corresponding to the best fit are as follows: dl = (41.50 + 0.05)nm, pl(Si3N4) = 0.942 4- 0.003, d2 = (1.25 + 0.11)nm, pz(a-Si) = 0.993 + 0.037, d3 = (2.00 + 0.03)nm and p3(SiO2) = 0.942 + 0.003. The authors thus showed that SSAI ellipsometry was a very useful method for the optical analysis of the multilayer systems as well. The other example of the efficient application of SSAI ellipsometry for analyzing the multilayers is presented in the paper by Hulse, Heller and Rolfe [ 1994]. In this paper SSAI ellipsometry is used for the optical analysis of the two-layer system formed by the upper layer of Si and the lower layer of SixGel-x deposited by molecular beam epitaxy on the silicon single crystal substrate (the native oxide layer on the upper layer of Si was also taken into account). It is necessary to point out that in-situ SSAI ellipsometry was also employed for monitoring the multilayer systems. For example, Logothetidis, Alexandrou and Vouroutzis [ 1996] used this method to monitor Ti-Si multilayers deposited on the silicon single crystals by r. f. magnetron sputtering (Ti and Si layers exhibited the thicknesses in the values of 3.8 and 8 nm, respectively, and total thicknesses of the multilayer were about 93 nm). The ellipsometric spectra obtained were also utilized for the investigations of the influence of annealing on interdiffusion and formation of Ti-Si alloys and compounds at the boundaries of the multilayers mentioned. The other interesting example of the utilization of SSAI ellipsometry
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ELLIPSOMETRYOF THIN FILM SYSTEMS
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for in-situ studies of multilayers is presented in the paper by Heckens and Woollam [1995]. In this paper the multilayer system of SiC/TbFeCo/SiC deposited on the glass substrate by the magnetron sputtering was studied by this method in detail (these multilayers are employed as magneto-optical medium). SSAI ellipsometry often has been used to characterize the imperfect thin film systems. Considerable attention has been paid to the optical analysis of the inhomogeneous single films. For example, Kim and Vedam [1988] determined the spectral dependence of the refractive index of ThF4 deposited on vitreous silica substrates using SSAI ellipsometry in the region 250-900nm. These authors expressed the dispersion of the refractive index of these films by the formula which represents a first approximation of the well-known Sellmeier dispersion equation: B~ 2 n2(~,) = A +
(7.4)
They found that the spectral dependences of ~ and A measured for the samples of the ThF4 films could not be interpreted using the ideal model of the single layer. Therefore these authors tested the influence of inhomogeneity on the ellipsometric spectra measured. The best fit of the experimental data was obtained if they used a three-layer model for treating these data. Within the framework of this model the columnar structure was assumed for both the upper and the lower layers (the middle layer was assumed to be amorphous without voids). Therefore, eight parameters had to be sought (A, B, ~0, the thicknesses characterizing all the three layers and the volume fraction of the upper and the lower layers). Note that eq. (7.4) expresses the dispersion of the middle layer and the dispersion of the columns of both the upper and the lower layers. For a chosen sample of the ThFn-film they found by using the LSM the following values of the constants taking place in eq. (7.4): A = 1.665 + 0.005, B = 0.715 + 0.004 and 20 = (149 4- 1) nm. The values of the other parameters are as follows: dl = (0.2 4- 0.5)nm, Pl = 1.18 + 0.34, d2 = (139.7 • 0.5)nm, d3 = (39.1 9 0.4)nm and p3 = 0.080 q- 0.004. It should be noted that the values of the thickness and volume fraction of the upper film are peculiar. By means of the same model the authors interpreted the ellipsometric data measured for MgO films on the same substrates 27.
27 Chindaudom and Vedam [1994] presented several examples of the optical characterization of the inhomogeneous nonabsorbing single films on the nonabsorbing substrates by SSAI ellipsometry. These inhomogeneous films (HfO2, HfF4, A1F3, ThO2, ZrO2, etc.) were again replaced by the two-layer and three-layer systems.
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SSAI ellipsometry was also utilized for non-destructive depth profiling the inhomogeneous layers originated by implanting 28Si+ ions into the Si single crystals with { 100} orientation in the paper by Vedam, McMarr and Narayan [ 1985]. The inhomogeneous damaged layers that resulted from this method were again replaced by the multilayer systems. For example, for one of the samples studied it was shown that the best fit of the experimental data was achieved for a four-layer system. The layers forming this system are as follows: the native oxide layer, the upper layer of the mixture of a-Si and c-Si, the layer of crystalline silicon and the lower layer of the mixture of a-Si and c-Si. At the treatment of the experimental data, the LSM was used in such a way that the optical constants of both a-Si and c-Si were fixed in the values determined independently (the values of the refractive index of the native oxide layer were assumed to be equal to amorphous SiO2). This means that using the LSM the values of the thickness and the composition of the layers mentioned were sought (the composition of the mixture layers was expressed through the fraction volumes of the components). For the sample under consideration the following results were obtained: dl = (2.4 + 0.3)nm, d2 = (11.9 + 1.9)nm, d3 = (51.1 + 2.1)nm, d4 = (27 -+- 3)nm, pz(a-Si) = 0.18 • 0.03 and p4(a-Si) - 0.79 • 0.03. It should be noted that in the paper it was shown that the results achieved for the damaged layers by SSAI were in excellent agreement with the results obtained on the same samples using cross-section transmission electron microscopy (see the paper cited). SSAI ellipsometry was employed for detailed studies of rough surfaces covered by single-surface layers as well. Using this technique, Aspnes [1980] investigated the surfaces of InSb single crystals prepared by chemomechanical polishing and anodize-stripping. Ellipsometric spectra measured for the InSb surface in the spectral range of 210-830nm were utilized for evaluating the imaginary part ~2 of the pseudodielectric function. The spectral dependences of c2 were determined for different technological conditions. The spectrum with the largest values of e2 was considered as that which most closely approximated the spectrum of the imaginary part of the true dielectric function of bulk InSb. Thus, the dielectric function spectrum corresponding to the E2 spectrum with the largest values could be considered as that usable for studying the nature of the surface overlayers evidently existing on the InSb surfaces with smaller values of the e2 spectra. In the effective medium approximation, this dielectric function spectrum was employed for expressing the influence of surface roughness corresponding to the MCRS and overlayers on the ellipsometric spectra of the InSb surfaces, or on the dielectric function spectra of these surfaces determined from those ellipsometric spectra. It was shown that for a chosen sample of the InSb surface,
250
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 7
a good fit of the measured c2 spectrum was achieved for a void fraction of 0.36 4- 0.01 and thickness of the fictitious films of 12.1 4- 0.5 nm. The best fit was obtained when an overlayer of Sb on the fictitious film representing roughness was considered (its thickness dj was evaluated as follows: dl = (2.2 4- 0.6)nm). However, the improvement of fitting the experimental data by including the Sb film was not too remarkable. The thickness value of the fictitious film; i.e., the value of 12.1 nm, appears to be rather large because this thickness corresponds to the heights of the boundary irregularities, and the effective medium theory was employed for interpreting experimental data (very small roughness must be assumed within the effective-medium theory). Aspnes and Theeten [1979] employed SSAI ellipsometry for studying the optical properties of the interface between silicon single crystal and its thermally grown oxide film. Syton-polished surfaces of Si single-crystal wafers were oxidized at 1000~ in dry oxygen to an oxide thickness from the interval 100-300 nm. The spectral dependences of the ellipsometric parameters measured for the samples of the system mentioned were treated using several models by means of the LSM. It was shown that the best fit of the experimental data was achieved for a four-phase model (the substrate, transition interface layer, SiO2 layer and ambient). The transition layer was assumed to be formed by a mixture of Si and SiO2. The values of the optical constants of both Si and SiO2 were again assumed to be known. The values of the thicknesses of both the transition and SiO2 layers were thus sought, together with the value of the volume fraction of one of two components forming the mixture of the transition layer (this transition layer was considered to be optically homogeneous). For a chosen sample of the system studied the following values of the thickness and the volume fraction of SiO2 corresponding to the transition layers were determined: d2 = (0.7 4- 0.2)nm and p2(SiO2) = 0.2 4- 0.1. The authors stated that the results presented were incompatible with either microroughness of the boundary between Si and SiO2 or an abrupt transition from Si to SiO2, but rather supported a graded transition layer at the boundary mentioned. 7.4. SPECTROSCOPIC MULTIPLE A N G L E OF INCIDENCE M E T H O D
In this ellipsometric technique, the spectral dependences of the ellipsometric parameters of the thin film system under investigation are measured at several angles of incidence. This means that this technique can be called spectroscopic multiple angle of incidence (SMAI) ellipsometry. In the literature the technique mentioned is often called variable angle of incidence spectroscopic ellipsometry (VASE). The treatment of the experimental data achieved within SMAI ellipsometry can also be performed in two ways as in SSAI ellipsometry.
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Within the framework of the first way, the angular dependences of tp and A are separately treated by means of a suitable numerical procedure at each wavelength of the spectral range of interest. In the second one all the experimental data are interpreted simultaneously (i.e., the spectral dependences of tp and A measured for all the angles of incidence are treated together). Of course, in this case one must again assume concrete dispersion formulae expressing the spectral dependences of the optical constants of the materials forming the thin film system studied. It should be noted that during the last years, SMAI ellipsometry has been used extensively in the optical analysis of thin film systems. This is because SMAI ellipsometry enables us to obtain the most detailed information concerning the optical properties of the systems analyzed in comparison with the ellipsometric methods described above. In particular, SMAI ellipsometry was often employed for analyzing single layers. For example, Guo, Gustafsson, Hagel and Arwin [ 1996] used this method to determine the values of the thickness and the spectral dependences of the refractive index of thick transparent single films formed by benzocyclobutene (BCB) prepared by spin-coating on silicon wafers (after the BCB films underwent heat treatment in a nitrogen atmosphere). The thicknesses of the BCB films produced in this way were in the 200-9000 nm range. The ellipsometric parameters of these films were measured in the spectral region of 500-850 nm at three angles of incidence (00 = 65 ~ 70 ~ and 75~ The values of the thickness of the BCB films were estimated by means of measurements with a profilometer before the ellipsometric analysis. In this ellipsometric analysis the first method of treating the experimental data was utilized. The values of both the thickness and the refractive index of the BCB film investigated were thus determined using the following numerical procedure at each wavelength of the spectral range specified above. The thickness of the BCB film was changed in the vicinity of the value found by the profilometer. The value of refractive index of the film was evaluated for every wavelength using the corresponding angular dependence treated by the LSM. The value of the thickness and the spectral dependence of the refractive index corresponding to the minimum of the sum of squares Q, i.e.,
N
1
Q jZ= l 2 M - P
i=1
ao.,q,
~rij,a
1
(7.5) were considered as true ones for the film analyzed. In eq. (7.5) the symbol @,q, and/or ~j,a is the experimental standard deviation in ~/j,~p and/or Ao.,exp. P, M and N are the numbers of parameters sought, angles of incidence and wavelengths, respectively. In the paper the spectral dependences of the refractive
252
ELLIPSOMETRY OF THIN FILM SYSTEMS
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index of the BCB films with the thicknesses of 8982 nm and 673.2nm are introduced through the corresponding curves. The second modification of SMAI ellipsometry was used, for example, in the paper by Cho, Cho, Lee, Lee, Lee, Lee, Sun, Moon, Chung, Pang, Kim and Kim [1998]. These authors analyzed the silicon dioxide layers prepared by thermal oxidation on the crystalline silicon substrates. They measured the ellipsometric parameters of the samples of the system at 36 angles of incidence from 50 ~ to 85 ~ with a step of 1~ and 36 wavelength from 250 to 830nm. The optical constants of the silicon substrate were taken from the literature. In the first step of the interpretation of the experimental data the authors expressed the spectral dependence of the refractive index of the silicon dioxide layers by means of the Sellmeier formula (see eq. 7.4). They determined the values of the thicknesses and parameters taking place in the Sellmeier formula for the individual SiO2 layers using the LSM. It was found that the spectral dependences of the refractive index of the SiO2 layers were dependent on the values of the thickness of these layers (the thicknesses of the layers were in the region of 10-150nm). The values of the refractive index increased with the decreasing thickness of the layers studied. This fact was explained by the existence of the transition layers between the silicon substrates and the SiO2 layers. In the second step of the analysis of the samples the values of the volume fraction of the transition layers were determined on the basis of the fact that these transition layers were formed by the mixture of Si and SiO2 (the Bruggeman formula was employed for expressing the refractive index of the transition layers). The values of the thickness of the transition layers were fixed in the value of 1 nm and the values of the optical constants of SiO2 and Si were taken from the literature. McGahan, Johs and Woollam [1993] employed SMAI ellipsometry for the complete optical analysis of amorphous carbon hydrogenated films in the modification based on the simultaneous treatment of the experimental data corresponding to the films exhibiting the different values of the thickness in order to remove or reduce the correlation existing between the optical parameters characterizing these films (multisample modification of SMAI ellipsometry). It should be pointed out that the values of all the parameters of all the a-C:H films, i.e., the values of the thicknesses and the spectral dependences of both the refractive index and extinction coefficient of these films, were sought (of course, it was again assumed that the optical constants of all the films studied were identical). SMAI ellipsometry was used to characterize the multilayer systems as well. For example, Memarzadeh, Woollam and Belkind [1987] and/or Memarzadeh, Woollam and Belkind [1988] dealt with the characterization of the three-layer
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system formed by ZnO/Ag/ZnO and/or TiO2/Ag/TiO2 deposited onto the glass substrates using this method. The optical constants of ZnO and TiO2 determined for single-layer samples and the known optical constants of glass were used as fixed parameters, while the optical constants of Ag and the three-layer thicknesses were found by treating the angular dependences of t/.t and A of the three-layer sample studied for every wavelength using the LSM. For a chosen sample of TiOz/Ag/TiO2 the authors evaluated the following values of the thicknesses: dl - (33.1 + 0.3) nm, d2 = (21.1 -+- 0.4) nm andd3 = (28.9 i 1.1)nm (the spectral dependences of the optical constants of the Ag film of this sample are introduced in graphic form). SMAI ellipsometry was used to characterize special multilayer systems as well. By means of this method Anma, Yamaguchi, Okumura and Yoshida [ 1989], for example, determined the optical constants of (GaAs/A1As)N ultrathin-layered superlattices with N = 2,... ,20 prepared by MBE on GaAs substrates. The superlattices were regarded as homogeneous single layers. The total thicknesses of these superlattices were determined independently and values of their optical constants were evaluated by fitting the angular dependences of ~ and A measured in the wavelength region of interest (/~ = 350-850 nm and 00 - 40 ~ 60 ~ and 80~ The multisample modification of SMAI ellipsometry was used to analyze the system formed by the silicon single-crystal substrate and silicon dioxide film in the very interesting paper by Herzinger, Johs, McGahan, Woollam and Paulson [ 1998]. For this analysis six samples of the thermal SiO2 films with the different thicknesses (10-350nm) and one sample of the NOL were used. The spectral dependences of the ellipsometric parameters were measured in the spectral region 190-1650nm. The angles of incidence were selected in values of 40 ~ and 75 ~ for the thermal oxide films and of 70 ~ and 79 ~ for the NOL. In this paper several models of the system were examined. The most suitable model was formed by the Si substrate covered by the double layer consisting of the interface layer and SiO2 film. The spectral dependences of the refractive indices of both the SiO2 films and interface layers were expressed by the Sellmeier formula in the form: n2(~) = A + ~
B/~2 - C/~ 2.
(7.6)
The spectral dependences of the optical constants of the Si substrates were described by Kramers-Kronig consistent functions containing 58 parameters. Using the LSM the values of the parameters taking place in the spectral functions describing the optical constants of the substrate and SiO2 films together with the
254
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 7
values of thicknesses of both the interface layers and SiO2 films were sought. Parameters A, B, C and 20 of the Sellmeier formula of the interface layers were fixed (for details see the paper cited). In contrast to the other methods employed for determining the spectral dependences of the optical constants of silicon single crystals, this method described enabled us to evaluate these spectral dependences in the form of the analytical functions fulfilling the Kramers-Kronig relations. It is clear that in principle the modification of SMAI ellipsometry employing the simultaneous interpretation of the experimental data corresponding to several samples of a multilayer under investigation can also be applied for the characterization of all the samples of this multilayer. Of course, the samples used must mutually differ in the value of the thickness of at least one film under the assumption that the optical constants of the corresponding films of all the samples are the same. This modification is especially helpful if the multilayer with a strong correlation of the parameters sought is analyzed by SMAI ellipsometry. The ellipsometric method discussed was successfully used to characterize the thin film systems with defects. Below we shall deal with the utilization of SMAI ellipsometry for the optical analysis of the thin film systems exhibiting the following two defects: non-uniformity of the films in the thickness and roughness of the boundaries. The single films with the non-uniform thicknesses were studied in the paper by Pittal, Snyder and Ianno [1993]. By means of a suitable geometry of d. c., these authors produced magnetron sputtering ZnO-single films exhibiting linear non-uniformity in thickness (the substrates were formed by Si wafers). The spectral dependences of the ellipsometric parameters of such ZnO films were measured for two angles of incidence (00 = 70 ~ and 75 ~ in the spectral range 400-1000 nm. The ZnO films with linear thickness non-uniformity were modeled by dividing the ellipsometer-probed area across the film into smaller subareas, each with uniform film thickness. The reflected intensity, detected by the ellipsometer detector, is averaged over the probe area. The results of averaging the reflected intensity lead to averaging tan 2 ~ and tan ~ cos A for each thickness. The ellipsometric parameters measured are then given by the ellipsometric parameters corresponding to the above averaged quantities. The values of the optical constants of the ZnO films were determined independently. By fitting the experimental data, one then obtains the values of the central thickness dl and percentage non-uniformity/5. For a chosen sample of the ZnO film the authors found the following results: dl = 200.6 + 1.0 and /5 = 22.3 + 2.0%. Tarasenko, Jastrabik, Chvostovfi and Sobota [ 1998] characterized thin films of
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Ni, Mo and NixN films deposited on the Si-substrates. The thicknesses of these films were up to about 20nm. The spectral dependences of the ellipsometric parameters of the films studied were measured over 16 wavelengths in the spectral range of 348-806 nm. The values of the angle of incidence were varied between 45 ~ and 80 ~. Ten optical models were used to calculate the thicknesses of the layers and determine the quality of the films, i.e., their homogeneity and roughness. The optical constants of the materials Ni, Mo and NixN were determined using ellipsometric measurements carried out for relatively thick layers of these materials. The models were formed by various combinations of several layers representing roughness, native oxide layers and transition layers at the boundaries of the layers studied. The optical constants of the oxide films and Si-substrates were taken from the literature. It was shown that relatively thick layers exhibiting the thicknesses greater than 10nm had homogeneous compositions with the refractive indices close to bulk values and roughness less than 1 nm. The relatively thin metal and nickel nitride films exhibiting thicknesses less than 10nm were found to be probably discontinuous with inhomogeneous granular or island structure, which differs substantially from that of the relatively thick films. 7.5. IMMERSION MODIFICATIONS OF THE ELLIPSOMETRIC METHODS
In immersion modifications of the ellipsometric methods described above, the ellipsometric parameters of the thin film systems are measured in various non-absorbing ambients. These non-absorbing ambients are formed by air and miscellaneous non-absorbing liquids (benzene, pentane, toluene etc.). Thus in immersion ellipsometries the values of the ellipsometric parameters of these systems are also measured as functions of the refractive index of the ambient no. Of course, the dependences of the ellipsometric parameters on the refractive index no must be measured using suitable types of cells (see, e.g., the papers by Winterbottom [ 1946, 1955] and McCrackin, Passaglia, Stromberg and Steinberg [1963]). It should be emphasized that the immersion methods can only be used to characterize thin film systems that do not react with the immersion liquids, both physically and chemically. For example, non-absorbing liquids with a polar character of their molecules are not suitable for use in these methods (water is the typical representative of these liquids). The first immersion modification of ellipsometry can be called immersion monochromatic single angle of incidence (IMSAI) ellipsometry. Vedam, Rai, Luke~ and Srinivasan [1968] used this method to determine the values of both the refractive index nl and thickness dl of the single layers of amorphous SiO2
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[3, w 7
placed on the silicon single-crystal substrates. They measured the values of the ellipsometric parameters of this system for two non-absorbing ambients (air and toluene or benzene) at the angle of incidence 00 = 70 ~ (Z = 546.1 nm). By means of the measured values of these four ellipsometric parameters measured they could determined the values of nl and dl for every sample of the system mentioned (the individual samples of this system mutually differ in the values of dl of the SiO2 layers). The values of nl and dl characterizing all the samples were determined with a relatively high accuracy (the values of the thickness of the SiO2 single layers were in the interval of 8-27 nm). The authors mentioned found that the mean value of the refractive index of these single layers was greater than the value of the refractive index of bulk amorphous SiO2 at the wavelength introduced above. They found that nl = 1.484 instead of the value of about 1.46 (see, e.g., Landolt and Brrnstein [ 1962]). The method discussed was applied by Luke,, Knausenberger and Vedam [1969] to analyze the same system for the identical experimental conditions as in the foregoing paper cited (i.e., the ambients and the values of ~ and 00 were the same in both papers). The authors successfully employed this method to determine the values of nl and dl of three single layers of amorphous SiO2 exhibiting the thicknesses dl in values of 42, 50 and 90nm. In contrast to the paper by Vedam, Rai, Luke~ and Srinivasan [1968] they found that the nl values of these layers agreed very well with those belonging to bulk amorphous SiO2. It should be noted that in both papers mentioned the values of the optical constants of the silicon substrate were simultaneously determined using a special procedure (for details see the papers mentioned) 28. IMSAI ellipsometry was also utilized by Ohlidal and Libezn~, [1991] for the complete optical analysis of the native oxide layers taking place on GaAs single-crystal substrates. However, they employed the multisample modification of IMSAI ellipsometry based on measuring and treating the values of the ellipsometric parameters of several samples of the system mentioned with different values of the thicknesses of the layers. The ellipsometric parameters of the samples were measured for the wavelength of 632.8 nm at the angle of incidence of 70.25 ~ Air, acetone, toluene and carbon tetrachloride were used
28 In a similar way Jegorova, Ivanova, Potapov and Rakov [1974] and/or Jegorova, Potapov and Rakov [1976] analyzed relatively thin SiO2 films on the silicon single-crystal substrates (the thicknesses of these films were in the interval 2-50nm) and/or the single films of A1203 and SiO2 placed on aluminum substrates. Furthermore this method was also employedby Kao and Doremus [1994] to analyze of the thin SiO2 films on silicon single-crystal substrates whose thicknesses laid within a relatively extended interval (3-420nm).
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as the immersion media. Five samples of the system mentioned were measured under the experimental conditions described (the thicknesses of the native oxide layers of these samples were in the range 2-4 nm). The values of tit and A measured for these five samples were simultaneously treated by means of the LSM. The values of both the refractive index and thicknesses of the native oxide layers were determined with sufficient accuracy (e.g. n l - 1.7 • 0.1). Note, that by this method, these authors determined the values of both the refractive index and the extinction coefficient of the GaAs substrates with a high degree of accuracy (it should be pointed out that the values of the optical constants of the GaAs substrates, the refractive index and the thicknesses of the native oxide layers were simultaneously determined in this method). Lubinskaya, Mardezhov, Svitashov and Shvets [1986] used IMSAI ellipsometry (~ = 632.8 nm) to characterize thicker films of SiO2 (dl = 60-220nm), Si3N4 (dl = 107nm) and p-Si (dl = 10-120nm, p-Si denotes polycrystalline silicon that is slightly absorbing in the visible). Note that the sample studied was placed on the bottom of the beaker filled in the immersion liquids and the angle of incidence of light on the boundary between air and the liquid was of 70 ~ (the immersion liquids water, ethanol, toluene and acetone were used). The films analyzed were situated in the following system: SiO2/Si, Si3N4/Ge and p-Si/SiO2/Si. By means of this method they were able to determine not only the values of both the refractive index and thickness of the SiO2 and Si3Na-single films but they could also evaluate the values of both the refractive index and extinction coefficient of the substrates in contrast to non-immersion ellipsometries. The values of the optical constants of the SiO2 and Si3N4 films and the silicon and germanium substrates were found close to those presented for these materials in the literature. The values of the optical parameters of the p-Si films existing in the system p-Si/SiO2/Si were determined after evaluating the values of the optical parameters of the samples of the system SiO2/Si on which these p-Si films were deposited (i.e., at first the sample of the system SiO2/Si was prepared and analyzed and then the p-Si film was deposited on the sample mentioned and analyzed separately). An interesting modification of IMSAI ellipsometry was used to analyze surfaces of semiconductors covered with the native oxide layers in the paper by Moy [1981]. With this modification, a sample of the system mentioned was located under a glass lens in the form of a hemisphere. A layer of non-absorbing liquid was placed between this hemisphere and the sample. By mixing various non-absorbing liquids, one could obtain the mixture of the liquids exhibiting the refracting index whose value is practically identical to the value of the refractive index of the glass forming the lens. Because the values of the refractive indices of
258
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[3, w 7
the native oxide layers taking place on the semiconductor surfaces are very close to the refractive indices of the glasses employed for fabricating the lenses, it is possible to neglect the influence of the native oxide layers on the ellipsometric parameters measured for the specified system. This means practically speaking the ellipsometric parameters measured in this way correspond to those belonging to the semiconductor substrate free of the native layer. The values of both the refractive index and extinction coefficient of the semiconductor substrate investigated can then be determined using the values of ~ and A measured at one angle of incidence (the values of both the optical constants of Si, GaAs and InP substrates were determined by this procedure at the wavelength of/~ = 632.8). The immersion modifications of the other three ellipsometric methods, i.e., MMAI, SSAI and SMAI ellipsometries, are seldom employed in practice. For the multiple angle of incidence methods this is mainly due to the fact that their application is relatively experimentally difficult because special cells must be constructed. For example, Ohlidal and Libezn~, [1990] used the immersion modification of MMAI ellipsometry (IMMAI ellipsometry) for the optical analysis of the system formed by the silicon substrate covered with the native oxide layer. For measuring the ellipsometric parameters of the system they used a special cell containing four windows corresponding to two different angles of incidence, i.e., 00 = 64.93 ~ and 70.03 ~ These authors combined this method with MMAI ellipsometry to analyze the system mentioned (for details see w 7.6). The immersion modification of SMAI ellipsometry (ISMAI ellipsometry) was used to investigate the SiO2/Si interface (see paper by Yakovlev, Liu and Irene [ 1992]). It is evident that for the refractive index of the immersion liquid close to the refractive index of the SIO2, the spectral dependences of the ellipsometric parameters measured are very sensitive to the interface properties. The authors used this method to measure several samples of SiO2/Si in air and CC14 in the 310-520nm range at two angles of incidence of 72 ~ and 75 ~ From the measurement performed in air the thickness of the SiO2 overlayer was obtained. The structure of the interface between SiO2 and Si was studied using the spectral dependences of the ellipsometric parameters measured in CC14. The structure of the interface was described by the crystalline silicon protrusions forming hemispheres with an average radius R to be placed into a hexagonal network with an average distance between centers D of the protrusions (it was assumed that these protrusions are embedded into bulk SiO2). Further it was assumed that the protrusions and the region between them are covered by a layer of suboxide SiOx (0 < x < 2) with an average thickness L. This layer formed a chemical transition zone from Si to SiO2 (i.e., the "chemical" interface). Moreover, it was
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assumed that the whole interface was formed by one transition layer with an effective thickness L i n f -- R + L and an effective dielectric function Cinf, which represented a mixture of crystalline silicon, silicon oxide SiOx and SiO2 forming the overlayer. Then Cinf could be expressed using the Bruggeman formulae. The relative volume fraction of the transition layer was expressed from the assumed geometry. In order to model the evolution of the interface during annealing, they used a power law to describe the reduction of both the protrusions (physical interface) and the chemical transition layer (chemical interface with thickness L) (for details see the paper mentioned). After determining the values of the quantities characterizing the model of the interface studied, the authors found that two distinct stages of interface evolution took place. A fast initial stage with interface microroughness reduction followed by a slow decay of the interfacial suboxide. The interfacial suboxide reoxidation reaction dominated the interface evolution in the moderate temperatures region (Tan < 900~ and viscous flow became dominant at elevated temperatures (Tan > 950~ (Tan denotes annealing temperature). It should be noted that the immersion modification of SSAI ellipsometry (ISSAI ellipsometry) was also employed for studying the SiO2/Si interface in the papers by Yakovlev and Irene [1992] and Liu, Wall and Irene [1994]. In the paper by Zhao, Lefebvre and Irene [1998] this immersion modification was used to investigate both smoothing and roughening that occurred simultaneously at the SiO2/Si interface during the oxidation processes (the interface SiO2/Si was roughened artificially). In this paper it was found that smoothing dominated when the interface is too rough. Roughening was the dominant mechanism when the interface was very smooth (for details see this paper). 7.6. COMBINEDELLIPSOMETRICMETHODS In practice one can encounter methods based on a combination of the methods of monochromatic ellipsometry presented above. By combining these methods of monochromatic ellipsometry it is possible to obtain more extensive information about the thin film system studied. In other words the combination of the methods mentioned enable us to remove or reduce a correlation between the optical parameters of the system sought. In the papers by Ohlidal and Lukeg [198889] and Ohlidal and Libezn) [1990] the combination of MMAI ellipsometry with IMSAI ellipsometry was used to analyze the native oxide layers on the surfaces of single crystals of Si, GaAs and Ge. The angular dependences of ttt and A of the samples of these systems were measured for the wavelength of 632.8 nm in the range 450-80 ~ in air. Moreover, for the same wavelength
260
ELLIPSOMETRY OF THIN FILM SYSTEMS
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the values of the ellipsometric parameters of the same samples were measured for various immersion liquids at one angle of incidence. In the first paper cited the ellipsometric parameters were measured in distilled water and toluene at the angle of incidence of 70.03 ~. The angle of incidence of 70.25 ~ was employed for measuring the ellipsometric parameters in distilled water, toluene, acetone and carbon tetrachloride in the latter paper cited. In the combined method of MMAI and IMSAI ellipsometry the angular dependences of qs and A measured in air and the dependences of ~ and A measured at one angle of incidence in the immersion liquids were simultaneously treated by the LSM for every sample under investigation. Note that the values of the refractive indices of the immersion liquids were again measured independently. It was found that by using this combined method, one could not determine the values of all the optical parameters characterizing the samples analyzed, i.e., the values of both the refractive indices of the substrate and native oxide layer, the thickness of this layer and the extinction coefficient of the substrate (this was evidently caused by a correlation of the parameters mentioned). However, in the paper cited it was shown that it was possible to determine the values of both the refractive indices and the thickness of the layer of all the samples studied with sufficient accuracy if the values of the extinction coefficient of the substrates, i.e., Si, GaAs and Ge, had been considered as the known quantity taken from the literature. It was thus assumed that the values of the extinction coefficients of Si, GaAs and Ge laid within the intervals k = 0.02 -+- 0.01, k = 0.25 + 0.05 and k - 0.815 • 0.025, respectively. By means of this combined method the following results were achieved for chosen samples of the systems specified above: nl -- 1.53 + 0.02, dl = (3.3 • 0.1)nm and n = 3.879 + 0.003 (NOL/Si), nl = 1.63 i 0.07, dl = (3.8 + 0.5)nm and n = 3.862 + 0.007 (NOL/GaAs) and nl = 1.43 -4- 0.08, dl = (5 -+- 2)nm and n = 5.42 -+-0.08 (NOL/Ge). In the paper by Ohlidal and Libezn~, [ 1990] the optical analysis of the same samples of the systems mentioned; i.e., NOL/Si, NOL/GaAs and NOL/Ge, was also carried out using the combined method of MMAI and IMMAI ellipsometry (experimental data measured within the combined method of MMAI and IMSAI ellipsometry were completed by the values of tit and A measured for the samples mentioned in four immersion liquids specified above at an angle of incidence 00 = 64.93~ After the joint treatment of all the data the following results were obtained for the samples: nl = 1.53 + 0.01, dl = (3.15 -+- 0.09)nm and n = 3.871 + 0.003 (NOL/Si), nl = 1.65 + 0.06, dl = (3.36 -+- 0.06)nm and n = 3.859 -4- 0.005 (NOL/GaAs) and nl - 1.48 -+- 0.04, dl = (3.4 4- 0.4)nm and n = 5.42 -+- 0.04 (NOL/Ge). One can see that the values of nl, dl and
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n determined by both the combined method agree with each other within experimental accuracy. Moreover, it can be stated that by using the combined method of MMAI and IMSAI ellipsometry the accuracy in determining the value of dl is increased. In the paper cited the existence of the transition interface layer between the silicon single crystal and the native oxide layer mentioned above was taken into account. The values of the optical constants n2 and k2 and the thickness d2 of the interface layer were fixed at n2 = 3.29, k2 = 0.015 and d2 = 0.7 nm. These values of n2, k2 and d2 have been chosen on the basis of the results presented by Aspnes and Theeten [1979]. The values of n2 and k2 introduced above were namely calculated by means of Bruggeman's formula (see w 6.1.1) under the assumption that the composition of this layer was as presented in this paper (i.e., Si0.8(SiO2)0.2). In Bruggeman's formula, the following values of the optical constants were used: n(Si)=3.879, k(Si)=0.02 and n(SiO2)=1.46. After applying the procedure described above for the combined method of MMAI and IMSAI ellipsometry, the following values of nl, dl and n were obtained: nl = 1.48 + 0.02, dl = (2.79 -+- 0.09)nm and n = 3.868 + 0.003. In the paper by Ohlidal and Libezn~, [ 1991 ] the modification of the combined method of MMAI and IMSAI ellipsometry based on joint interpretation of several sample data was employed for the complete optical analysis of samples of the system NOL/GaAs (A = 632.8 nm, 00 - 45-80 ~ in air and 70.25 ~ in immersion liquids formed by carbon tetrachloride, acetone and toluene). The individual sample of the system mentioned exhibited different values of the thickness of the NOL (the differences in the values of the thickness were caused by the different times of exposure of the surfaces of GaAs single crystal to the laboratory atmosphere). The experimental data corresponding to the combined method mentioned to be measured for all the samples under investigation were treated together. The joint treatment of the data of all the samples caused the reduction of the correlation of the parameters sought so that it was possible to determine the values of all the parameters characterizing the samples of the system mentioned with sufficient accuracy. This fact is illustrated by the following values of the parameters determined and the values of their standard deviations: n = 3.8372 -+- 0.0015, k = 0.204 -+- 0.009, nl = 1.64 + 0.02, dl,1 = (2.27 + 0.10) nm, dl,2 = (2.66 + 0.10) nm, dl,3 -- (3.05 -+- 0.09) nm, dl,4 = (3.41 + 0.09)nm and dl,5 = (3.59-+-0.09)nm. Note that the symbols d1,1,... ,dl,5 denote the thicknesses of the NOL of the five different samples of the system under consideration. In figs. 7 and 8 the very good agreements between both the experimental and theoretical data are demonstrated for the sample with the thicknesses d1,1 and dl,5. These agreements imply that the present physical model of the system NOL/GaAs is satisfactory.
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ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 7
] 180 35 160 30
g--,
140
25
120
20
100 o,--, 80 o A o
10
W5 A1 A5
<1
60 40 20
I
I
I
I
45
50
55
60
0
I
I
I
65
70
75
,
I
0
80
Oo [o] Fig. 7. The angular dependences of the ellipsometric parameters ~ and A for two chosen samples of the system NOL/GaAs with thicknesses dl,l = 2.27 nm and dl,5 = 3.59nm. The points and/or curves denote the experimental and/or theoretical values of the quantity mentioned. !
|
i
!
|
|
11
160
10
140
9
120
8 E
100 g--,
7
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6 5 4
i I, o
3 2
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"~
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A5
20
I
I
I
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I
I
1
1.1
1.2
1.3
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no Fig. 8. The dependences of the ellipsometric parameters tp and A on the refractive index of ambient for two chosen samples of the system NOL/GaAs with thicknesses dl,1 = 2.27nm and dl, 5 -- 3.59nm. The points and/or curves denote the experimental and/or theoretical values of the quantity mentioned.
In a theoretical study Azzam, Elshazly-Zaghloul a procedure
of analyzing thin films by means of interpreting the ellipsometric
data obtained and Luke~
and Bashara [ 1975] suggested
in b o t h r e f l e c t e d a n d t r a n s m i t t e d
[1981]
used
a combined
l i g h t o f t h e s e films. O h l i d a l
ellipsometric
method
that employs
the
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EXPERIMENTAL METHODS
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simultaneous interpretation of the angular dependences of the ellipsometric parameters measured for light reflected and transmitted by the ambient side and the substrate side of thin films to realize the complete optical analysis of thin gold films deposited onto the substrate formed by planparallel glass plates. It should be emphasized that the ellipsometric parameters of the film in transmitted light related to light incident onto this film from the side of the ambient are identical with those related to light incident on this film from the side of the substrate if the corresponding angles of incidence fulfill Snell's law. In this combined method it is only necessary to measure one pair of the angular dependences of the ellipsometric parameters corresponding to the transmission mode (e.g., the angular dependences of the ellipsometric parameters measured in transmitted mode belonging to light falling on the film from the ambient side). The authors showed that in comparison with individual ellipsometries, i.e., external and internal reflection and transmission ellipsometries, and other standard optical techniques, one could obtain more precise values of the optical parameters characterizing these films using the combined method (external and/or internal reflection ellipsometry corresponding to light incident onto the film studied from the side of the ambient and/or the non-absorbing substrate). This conclusion was demonstrated by the values of the optical parameters obtained for several thin gold films with thicknesses in the range 25-45 nm at the wavelength of 632.8 nm (the value of the refractive index of the glass substrate was measured independently). For a chosen sample of the Au film the combined method yielded these results: n l = 0.212 + 0.015, kl = 3.337 • 0.018 and dl = (30.64 + 0.34). The thickness of this gold film which was determined by analyzing the Fizeau interference fringes (see Bennett and Bennett [ 1967]) was 30 + 2 nm which gives support for the correctness of the results achieved by means of the combined ellipsometric method discussed. It should be remembered that the measurements of the angular dependences of the ellipsometric parameters of all the gold films were performed in the ambient formed by air at the angles of incidence 00 in the range 45-80 ~. These angles of incidence correspond to light falling on the upper boundary of the film studied or on the back boundary of the glass plate without the film. The combined method of reflection and transmission MMAI ellipsometry was also employed for the complete optical analysis of absorbing double layers onto glass substrates (see Ohlidal and Luke~ [1984]). The samples of the following absorbing double layers were analyzed in this paper: MgFz/A1, A1/MgF2, TiO/Au and Au/Ni. It was found that this method was powerful for determining all the optical parameters characterizing the double layers mentioned above. This statement can be illustrated by the results achieved for a chosen
264
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 7
sample of the double layer MgF2/A1 (A = 632.8nm, 00 = 45-70 ~ in air): nl = 1.403 + 0.004, dl = (82.9 + 0.2)nm, n2 = 1.3 + 0.1, k2 = 7.23 + 0.05 and d2 = (26 -1- 3)nm. However, if the double layers containing two strongly absorbing thin films are analyzed using this combined method a lower precision is achieved in determining the values of their optical parameters. This statement is demonstrated by the results obtained for a chosen sample of the double layer Au/Ni: nl = 0.18 + 0.17, kl = 3.20 + 0.8, dl = (15 4- 4)nm, n2 = 2.12 + 0.03, k2 = 3.69 + 0.07 and d2 = (33 + 3)nm. To increase the precision in determining the values of the optical parameters of the double layers consisting of the two strongly absorbing films, one has to apply a modification of the combined ellipsometric method based again on a simultaneous treatment of the experimental data obtained for several samples of the same system as described in the paper by Ohlidal, Schmidt and Libezn2~ [1990]. These authors employ simultaneous interpretation of the ellipsometric parameters measured for light reflected and transmitted by the ambient side and the substrate side of the two or more samples of the double layer investigated. These samples must again have different thicknesses of both the strongly absorbing films (at least the thickness of one of the two films) and the same optical constants. In practice the two samples mentioned can often be prepared by depositing the two upper films differing in thicknesses onto the different parts of the lower film (for details see the paper cited). The authors demonstrated this procedure by characterizing the double layer Au/Cr placed on the substrate consisting of a glass plate. For a chosen sample of this double layer they obtained the following values of all the optical parameters (~ = 632.8nm, 00 = 50-80 ~ in air): n l - 0.25 + 0.03, kl = 3.27 + 0.01, dl,l = (17.5 + 0.4) nm, dl,2 = (26.2 + 0.5) nm, n2 = 3.42 + 0.06, k2 - 2.12 + 0.02 and d2 = (19.3 + 0.8)nm (dl,1 and dl,2 denote the different thicknesses of the upper film). From the foregoing it is apparent that the values of all the optical parameters characterizing the samples studied are determined with relatively high accuracy, a very important fact in establishing the practicality of the method (the values of the optical parameters of the samples of the Au/Cr-double layers with the other thicknesses were also determined with satisfactory accuracy). A correctness of results presented above was supported by means of measuring the values of both the thicknesses using the Fizeau method (see the paper cited). Practically the possibility of using the combined method of reflection and transmission MMAI ellipsometry for the reliable and precise analysis of the double layers is very significant. Namely, it is often necessary to characterize metal and semiconductor films protected against the influence of the surrounding atmosphere. These films must then be covered with protective layers and one has
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to carry out the optical analysis of an entire double layer in order to perform the characterization of the lower film investigated. The optical analysis of thin NiCr films deposited by the r. f. sputtering onto glass plates were analyzed in such a way in the paper by Ohlidal, Schmidt and Libezn~, [1989] (as the protective layers thin A1203 films served). Of course, the methods of spectroscopic ellipsometry presented above can also be combined. For example, in the paper by Bader, Ashrit, Girouard and Truong [1995] the method employing the combination of reflection and transmission SMAI ellipsometry was used to analyze gold films and indium tin oxide film deposited on glass substrates in the range 400-800nm (00 - 10-70~ Note that the authors employed the quantities corresponding to photoellipsometry for evaluating the experimental data (see the paper cited). 7.7. COMBINATION OF THE ELLIPSOMETRIC METHODS WITH THE OTHER OPTICAL METHODS
In practice a combination of the ellipsometric methods with the other optical ones appears to be very powerful. In several papers it was shown that this combination decreased the correlation between searched parameters of thin film systems under investigation and thus allowed to be analyzed the described systems with a greater number of parameters (e.g., the thin film systems exhibiting various defects). For example, in the paper by Elizalde, Frigerio and Rivory [1986] the amorphous metallic alloy films and gold films deposited onto silica substrate by d.c. sputtering at room temperature were analyzed using procedures based on interpreting the values of the reflectance R, transmittance T and ellipsometric parameters tp and A measured in the region 230-900nm (the reflectance, transmittance and ellipsometric parameters were measured at near-normal incidence, normal incidence and the 70 ~ angle of incidence, respectively). Two approaches were used to treat the data specified. In the first approach the measured values of the three quantities were used to evaluate the values of the thickness and optical constants at every wavelength of the spectral region of interest. The authors found that only the combinations [R,T,A] and [T,A, tp] yielded acceptable values of the optical parameters mentioned. In the second approach the measured values of all four optical quantities were used to determine the values of the optical parameters at the same wavelengths (in this case the LSM was employed for treating the data at every wavelength). The Au, CuzY, CuY, PdzY5 and PdY films were studied with thicknesses of several tens nanometers. The values of the optical constants of these films are
266
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 7
not introduced in the paper cited (only spectral dependences of the optical absorption corresponding to the sought optical parameters of the films mentioned are presented in a graphic form). Masetti, Montecchi and da Silva [ 1993] studied oxide overlayers on surfaces of polycrystalline zinc selenide originated by UV and ozone exposure of these surfaces. They used SSAI ellipsometry (00 = 75 ~ ~, = 500-840nm) and photothermal deflection spectroscopy (A - 500-840 and 10600nm). Surfaces of ZnSe were prepared in different ways. The values of the refractive index of the ZnSe-substrate were determined by means of spectral dependence of measured for the sample whose value of A corresponded to the overlayer with the minimum thickness (in their measurements this sample was characterized by the lowest value of A). The spectral dependence of the refractive index of the ZnSesubstrate determined in this way was used to evaluate the values of the refractive index and thickness of the overlayers at every wavelength (in the spectral region mentioned both the ZnSe-substrate and overlayers were considered nonabsorbing materials). For a chosen sample they determined the following values of the thickness and refractive index of the overlayer: d l = (26 4- 1)nm and nl = 2.74 + 0.01 for /~ = 500nm (in the paper the spectral dependences of n~ of the overlayers are presented through curves). Photothermal deflection spectroscopy was then employed to determine very small absorption of the overlayers. Ohlidal, Franta, Hora, Navrhtil, Weber and Janda [1998] combined SMAI ellipsometry and spectroscopic reflectometry to characterize slightly rough surfaces of single crystals of GaAs covered by native oxide layers. The surfaces of GaAs were prepared by thermal oxidation of GaAs wafers in air and by subsequent dissolution of the resulting oxide layers (for details see the paper cited). Spectral dependences of the ellipsometric parameters and/or spectral dependences of the near-normal reflectance of the GaAs surfaces with the NOL were measured in the spectral region 370-640nm and/or 220-820nm (ellipsometric spectra were measured for the angles of incidence laying in the range 64~176 All the spectral dependences; i.e., the spectral dependences of ~ and A corresponding to all the angles of incidence and the spectral dependence of the reflectance were treated by the LSM for every sample simultaneously. Several models of the rough surface of GaAs covered with the NOL were used. It was shown that the best fit of the experimental data of the sample studied were achieved for the model of the NOL formed by the ITF whose boundaries were described with the spectral density of spatial frequencies W consisting of a sum of two Gaussian functions with different values of cr and T (see w 6.1). In the paper mentioned the results of the
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EXPERIMENTAL METHODS
267
analysis of a chosen sample of the system under investigation are presented as follows: dl = (5.97 + 0.13)nm, Co = 1.64 + 0.02, C1 = (15700 4- 700)nm 2, Ol = (2.0 4- 0.1)nm, T1 = (12 4- 1)nm, o2 = (3.89 + 0.08)nm and T2 = (63 + 2)nm. Note that the constants Co and C1 correspond to the spectral dependence of the refractive index of the NOL expressed by the Cauchy formula (see eq. 7.2) and that the spectral dependences of the optical constants of the GaAs-substrate were fixed in the values taken from the paper by Aspnes and Studna [ 1983]. From the results presented one can see that the boundaries of the NOL on the rough GaAs-substrates exhibited composite roughness containing two components. The correctness of the results obtained in the paper cited was supported by the values of the quantities characterizing roughness of boundaries measured using atomic force microscopy (AFM). The combined method of SMAI ellipsometry and spectroscopic reflectometry was also employed for characterizing the NOL taking place on slightly rough surfaces of silicon single crystal in the paper by Ohlidal, Franta, Rezek and Ohlidal [ 1997]. However, this combined method was applied in the modification based on the simultaneous treatment of the experimental data obtained for all the samples under investigation (i.e., in the multisample modification). Both SMAI ellipsometry and spectroscopic reflectometry data were measured within the spectral range 240-830nm. The angles of incidence for measuring the ellipsometric and/or reflectometric data were 63 ~ 68 ~ 72 ~ 75 ~ and 77 ~ and/or 11 o. This means that by means of the LSM the experimental data corresponding to SMAI ellipsometry and spectroscopic reflectometry obtained for all the samples were treated together. The authors of the paper analyzed one smooth surface and the three differently rough surfaces of silicon wafers. It should be noted that the relative reflectances of the rough samples owing to the smooth sample at near-normal incidence were measured and interpreted. The rough silicon surfaces were prepared by anodic oxidation of the smooth surfaces followed by the dissolution of anodic oxide films originated produced by this anodic oxidation. Before the measurements the smooth Si surface was placed into the dissolving solution together with the rough Si surfaces so that one could assume that all the NOL existing on the smooth and slightly rough surfaces exhibited the same values of both the thickness dl and refractive index nl. The parameters of roughness of the boundaries of the individual NOL were mutually different (this fact was caused by different conditions of anodic oxidation). The interpretation of the experimental data was simplified by the fact that the spectral dependences of the optical constants of the silicon substrate and the spectral dependence of the refractive index nl of the NOL for all the samples were fixed in the dependences taken from the literature (it was again assumed
268
ELLIPSOMETRYOFTHINFILMSYSTEMS
[3, w7
35000 30000 --"'. N.X
!
\ \
25000 -~
20000
~:
15000
Ik
10000 "',,\
5000
9 "..... ,
0
.... 1
.......~..m____
i
0
0.02
_~
0.04 0.06 K[nm -1]
,
0.08
_~
0.1
Fig. 9. The power spectral density function W(K) of the rough Si surface covered with the NOL to be characterized with o4 = (9.03+0.01)nm and 7"4 -- (73.4+0.3)nm. The points denote the experimental values of W(K) determined by AFM and the curve represents the theoretical values of this quantities calculated using the values of o4 and T4 mentioned. that the refractive index of the NOL was identical with the refractive index of amorphous SiO2). For interpreting the experimental data of the rough samples the formulae corresponding to RRT were used. By means of the described procedure of interpreting the experimental data the following results were achieved: dl = (2.70 -F 0.04)nm, 02 = (2.15 + 0.06)nm, T2 = (29 i 2)rim, o3 = (6.16 + 0.02)nm, T3 = (47.6 + 0.3)nm, o4 = (9.03 + 0.01)nm and T4 = (73.4 + 0.3)nm. The symbols 02 and 1"2, o3 and T3 and o4 and T4 denote the rms values and autocorrelation lengths of the rough samples studied. The correctness of the results presented was supported by the agreement between the values of the roughest sample obtained by AFM and the combined optical method discussed. In the paper this agreement was mainly illustrated using the comparison of the values of the power spectral density of the spatial frequencies of roughness W determined by the optical method and AFM (see fig. 9). Note that in the optical method the spectral density W could easily be calculated by means of the values of 0 and T found because the Gaussian form of this quantity was assumed in the paper. The results obtained by the combined optical method were also supported by the agreement between the experimental and theoretical data corresponding to this method (see figs. 10 and 11). The multisample modification of the combined method of SMAI ellipsometry and spectroscopic reflectometry was successfully used to determine the spectral
3, w 7]
EXPERIMENTALMETHODS
269
Fig. 10. The spectral dependences of the relative reflectance of the rough Si samples studied (R1 is the reflectance of the smooth sample). The points and/or curves denote the experimental and/or theoretical values of the quantity mentioned.
Fig. 11. The spectral dependences of both the real and imaginary parts of the complex quantity /5 for smooth sample (/51) and the roughest one (/54) at the angle of incidence of 72~. The points and/or curves denote the experimental and/or theoretical values of the quantity mentioned. dependences of the optical constants of the silicon single crystal and silicon dioxide films in the paper by Ohlidal, Franta, Pinrik and Ohlidal [1999]. The method o f interpreting the experimental data employed in this paper is based on combining two procedures. The first of them utilizes the simultaneous interpretation of the data corresponding to all the samples at individual wavelengths and the latter one employs the simultaneous interpretation o f the
270
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 7
entire spectral dependences of all the optical data corresponding to all six samples of SiO2/Si analyzed. Using this method one can determine the spectral dependences of the optical constants of silicon without employing dispersion formulae (for details see the paper cited). 7.8. METHODSOF GENERALIZEDELLIPSOMETRY From the foregoing (see w 3.2 and 5.2) it is evident that only the thin film system formed by the anisotropic media with the generally oriented principal axes can be analyzed using methods of generalized ellipsometry. However, these methods of generalized ellipsometry are seldom employed in practice. The reason being that in the analysis of the optically anisotropic films using generalized ellipsometry one must determine relatively great numbers of the optical parameters that characterize these films. Thus the methods of generalized ellipsometry are very complicated. Moreover, for the majority of the anisotropic thin films analyzed it is possible to orientate the principal axes of the materials forming thin films in such the way that the methods of conventional ellipsometry can be used. In practice it is appropriate to use special ellipsometers to measure the values of the six independent elements of the normalized Jones matrix of the system studied. Jellison, Modine and Boatner [1997] employed the two-modulator generalized ellipsometer for investigating the surfaces of uniaxial rutile (TiO2) covered with isotropic thin films representing roughness of these surfaces. Three samples of the rutile surfaces were used in these experiments: two surfaces with the optics axis cut in the plane of the sample and one surface with the optics axis perpendicular to the plane. The authors measured the spectral dependences of the elements of the normalized Jones matrix in the spectral region 250-860 nm at the angles of incidence of 60 ~ and 65 ~ (i.e., generalized SMAI ellipsometry was thus used). To determine the thickness of the surface roughness layer, the fraction of rutile in this layer and the angles of the optics axis with respect to the plane of the incidence, they fitted the data corresponding to 400-860 nm in which rutile is non-absorbing (the spectral dependences of the refractive indices of rutile were taken from the literature). They then employed the values of these parameters for determining the spectral dependences of the optical constants of rutile (including the extinction coefficients) by treating the ellipsometric data corresponding to the entire spectral region 250-860nm. Note that for determining the spectral dependences of the rutile optical constants in the region mentioned, the authors did not use any dispersion formulae but they treated the experimental data for each wavelength separately. In an interesting paper by Lecourt, Blaudez and Turlet [1998a] the conventional spectroscopic ellipsometer with the rotating polarizer was used to
3, w7]
EXPERIMENTALMETHODS
271
Fig. 12. Schematic diagram of the principal axes al, a2 and a 3 of the LB film.
analyze the Langmuir-Blodgett (LB) films deposited onto silicon single-crystal substrates in generalized spectroscopic ellipsometry. The existence of the native oxide layers on the silicon surfaces was taken into account in the analysis of the LB films. The thicknesses of the native oxide layers were determined before preparing the LB films (the optical constants of silicon and the refractive index of the native oxide layers were taken from the literature). In the first stage of analyzing the LB films the authors assumed that these films were uniaxial. Further they assumed that the principal axis al of the films studied was perpendicular to their boundaries. This assumption enabled them to use conventional spectroscopic ellipsometry for determining the optical parameters describing these uniaxial films. Of course, in reality the LB films under investigation are biaxial ones. Anisotropy in the plane of the boundaries of these LB films is slight (this anisotropy is characterized by the difference An between the values of the principal refractive indices corresponding to this plane). However, it is further necessary to realize that the plane formed by the two principal axes a2 and a3 is not strictly identical with the plane of the boundaries and the third principal axis al is not strictly perpendicular to this plane (i.e., the principal axes al and a2 are rotated around axis a3 laying in the plane of the boundaries, see fig. 12). Thus there is a small angle 0 between the principal axis a l and the normal to the boundaries. The authors measured the light intensity detected by the detector of the ellipsometer as a function of the azimuth angle q~. The analyzer was in the positions corresponding to both p-
272
ELLIPSOMETRY
0.05
!
OF THIN
!
!
0.04
[3, w 7
SYSTEMS
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~ '"
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9
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9
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,~
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"b.~.. o i
0
50
i
100
i
150
i
200 q~[~
i
i
i
250
300
350
Fig. 13. Quantitiestip (squares) and fls (circles) of the LB film as functions of the sample azimuth cp at the selected wavelength Z = 280 nm. The dotted curves representthe values of flq calculatedusing eq. (7.8). and s-polarizations (the angle of incidence and/or the spectral region was of 70 ~ and/or 240-600 nm). In this way the authors obtained the values of the following quantities tip = 2Reff'psfi'pp) and fls = 2Re(~'spfi'ss). For a chosen sample of the LB film the dependences of these quantities on the azimuth angle q9 are plotted in fig. 13. Lecourt, Blaudez and Turlet [1998b] further showed that the Fourier components corresponding to the quantities flq(qg) were linearly dependent on 0 and An if these quantities 0 and An were sufficiently small, i.e. flq(l) = A(ql)(3.) 0
and
fiq(2) = A~2)(3.) An,
q = p,s,
(7.7)
where symbol/3q(1) and/or [3~2) denotes the first and/or second Fourier components defined by the following equation:
flq(qg) = g l ) sin(qg) + fl~2) sin(2qg).
(7.8)
The spectral dependences of these components determined for the same sample of the LB film are plotted in fig. 14. The spectral dependences of functions A~q~ (o = 1,2) needed to determine the values of 0 and An are given by the values of the parameters of the system evaluated in the first stage of the method. The values of 0 and An determined together with the values of the parameters evaluated using conventional ellipsometry characterize the LB films completely. Note that for the chosen sample of the LB films the authors evaluated the following values
3, w 7]
EXPERIMENTAL METHODS 0.08
,
,
,
,
273
,
,
0.06
F
|
~1~
o
~
o
0.04 0.02
O O
o
O
O
O
O
O
O
O
O
o -0.02 O
-0.04 -0.06
O
' 250
O
' 300
. 350
.
. 400
.
. 450
. . 500
550
600
~, Into] Fig. 14. The spectral dependences of Fourier components/3 (1),/3 (1),/3p(2) and/3} 2)
of
the LB film.
of total thickness d, 0 and An: d = 55.2 nm, 0 ~ 1.5 ~ and An = 0.01-0.02 (the spectral dependences of the principal refractive indices of the chosen LB film are introduced graphically in the cited paper). Other examples of the application of the generalized ellipsometry method for characterizing anisotropic films are introduced in the papers of Schubert [1998] and Elman, Greener, Herzinger and Johs [1998]. 7.9. E L L I P S O M E T R I C M E T H O D S IN M A G N E T O - O P T I C S
The ellipsometric methods applied in magneto-optics are based on measuring changes of the polarization states of the outgoing waves from the systems investigated corresponding to changes of the magnetizations of these systems. These polarization states are characterized by the angles c and 0 (see w 5.2.2). In practice the changes of the values of these angles are very small. Thus special ellipsometers must be employed for the magneto-optical measurements (e.g., Zvezdin and Kotov [1997]). Both the polar Kerr and Faraday first-order magneto-optical effects are employed for characterizing the magneto-optical materials most frequently. For example, in the paper by Vigfiovsk~,, N~vlt, Prosser, Lopugnik, Urban, FerrY, P~nissard, Renard and Krishnan [ 1995] the three-layer system Au/Co/Au deposited onto the glass substrate was studied using these effects. The spectral dependences of both the polar Kerr rotation and ellipticity and the polar Faraday rotation and ellipticity were measured for various values of the thickness of the Co layer in the spectral region 240-830 nm (the angle of incidence of light was
274
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 7
0.5
-0.5
_
9
ii-
I
I
I
I
-1000
-500
0 H [Oe]
500
I
1000
Fig. 15. Magnetic hysteresis loop observed for in-plane magnetic field along Fe [110] hard axis by Kerr rotation OK measurements(p, polarized light,/l = 632.8nm, 00 = 70~ and magnetic field H in the plane of incidence). of 5 ~ at studying the Kerr effect). These experimental data are compared with the theoretical ones calculated using the matrix formalism (the values of the optical parameters of the layers are assumed to be known). A relatively good agreement between the experimental and theoretical data was found. The linear magneto-optical effects can also be studied for the longitudinal magnetization. In general these longitudinal magneto-optical effects are weaker than the polar magneto-optical effects. However, in practice they are useful for studying the perpendicular magnetic anisotropy of the materials forming very thin films. These very thin films exhibit this anisotropy because of the behavior of the magnetization at their boundaries. In the paper by Vi~fiovsk~,, Kielar, N~vlt, Pafizek, Flevaris and Krishnan [1993] the polar and longitudinal magneto-optical Kerr effect is investigated for the multilayer systems formed by layers of Pt and Ni and layers of Pd and Ni. The multilayers formed by the P t ~ i layers and/or Pd/Ni layers were deposited onto glass and/or mica substrates. The individual thicknesses of the layers corresponded to several monoatomic layers. The multilayers studied were formed by tens of the individual layers. In the paper by Postava, Jaffres, Nguyen Van Dau, Goiran and Fert [1997] epitaxial Fe films (10-100 nm) deposited onto MgO-substrates were studied in reflected light (these Fe films were covered by Pt films with a thickness of 1.5 nm). The authors demonstrated the appearance of a strong asymmetrical hysteresis loop for p-polarized incident light (see fig. 15). They explained this behavior as the mixing of transverse magnetization contribution to the
3, w7]
EXPERIMENTALMETHODS
275
longitudinal magnetization measurements on the basis of quadratic magnetooptical effects. 7.10. DISCUSSIONOF THE ELLIPSOMETRICMETHODS The ellipsometric methods described above enable us to perform the optical analysis of the majority of the thin film systems taking place in practice. However, before the analysis of a concrete system one has to choose an adequate physical model of this system and to apply a sufficiently efficient method. In particular a choice of a true model of the thin film studied is a relatively difficult problem. This is because many thin film systems exhibit defects that are not evident at first glance. The incorrectness of the model may not be revealed if an insufficiently efficient ellipsometric method is used. From the foregoing sections one can see that the simplest ellipsometric methods such as the monochromatic methods or single angle of incidence methods are suitable for characterizing relatively simple thin film systems (e.g., single or double layers) fulfilling the ideal model without any defects. Further it is evident that the more complicated systems (e.g., the systems containing many films or the systems exhibiting defects) have to be analyzed using the more efficient methods (e.g., using SMAI ellipsometry or immersion modifications of ellipsometric methods). It should be noted that the combined methods are especially helpful for analyzing the complicated thin film systems. The multisample modifications of ellipsometries described belong among the very powerful methods as well. Within these combined methods and multisample modifications we can reduce or remove the correlation between the sought parameters of the system under consideration which enables us to improve both the accuracy and reliability of the results of the analysis of the system. Of course, the multisample modifications of ellipsometries can only be applied if it is possible to prepare the samples of the system studied which may differ in a small number of parameters (the difference in the values of only one parameter being optimum). This fact often represents a substantial limitation to using the multisample modifications in practice. It should be pointed out that before employing the ellipsometric method chosen for characterizing the concrete thin film system it is also reasonable to perform a prediction of estimations of errors of the parameter sought by a suitable numerical procedure. This prediction enables us to judge whether the method chosen is appropriate and sensitive for the characterization of the system under investigation. One of the numerical procedures suitable for this purpose was, for example, presented in the paper by Humli~ek [1985] (in this paper sensitivity extrema in MMAI ellipsometry of double layers was studied).
276
ELLIPSOMETRY OF THIN FILM SYSTEMS
[3, w 8
Of course, the procedure of this type can only be carried out if the values of the parameters sought can be estimated with sufficient accuracy and the precision of measuring the ellipsometric quantities is known. An analysis of the precision in measuring the ellipsometric quantities corresponding to the different experimental ellipsometric techniques mentioned in w 4 is relatively difficult. That is why many papers have been devoted to this problem so far. For example, the analysis of systematic and random errors of the measured quantities in the most employed rotating-analyzer ellipsometry is described in the paper by de Nijs and van Silfhout [ 1988]. The theoretical precision attainable from the types of ellipsometries summarized in w 4 is investigated in the paper by Aspnes [1975]. Of course, a detailed study of the problems of error analysis is also presented in the monograph of Azzam and Bashara [ 1977]. If the LSM is used to treat the experimental data, the choice of the quantity minimized has an influence on the correlation of the parameters sought. Thus this choice influences the quality of the results of the analysis carried out using the LSM. A theoretical analysis of this problem concerning special quantities minimized for selected thin film systems was carried out in the interesting paper by Kim and Vedam [ 1986]. These authors showed that in rotating-analyzer ellipsometry the most suitable quantity minimized was the sum of squares constructed from the Fourier components Is and Ic (see eqs. 4.14) which are directly measured using this ellipsometer. In general one may state that it is mostly appropriate to apply the LSM to quantities directly measured using the ellipsometer used. The ellipsometric methods mentioned are often employed together with the other physical methods of analyzing various thin film system. In this way one can obtain complementary information about properties of the system under investigation. For example, in the paper by Zajirkov~i, Ohlidal and Janra [ 1996] MMAI ellipsometry was employed together with spectroscopic reflectometry and X-ray photoelectron spectroscopy for characterizing thin films prepared by plasma-enhanced chemical vapor deposition from tetraetoxysilane (TEOS) and TEOS/methanol mixtures. One can expect that the trend of combining the ellipsometric methods and the other physical methods in the studies of thin film systems will be pronounced in the near future.
w 8. Conclusion In this chapter a brief discussion of the theoretical principles of ellipsometry has been presented. It has been shown that ellipsometric methods can be
3, w8]
CONCLUSION
277
classified into two basic groups that correspond to conventional ellipsometry and generalized ellipsometry. The theory of the measurements in ellipsometry has briefly been mentioned for the most employed ellipsometries in practice. Further the basic equations concerning the propagation of light in the ideal thin film systems have been introduced using the matrix formalism. Both the optically isotropic and anisotropic thin film systems have been taken into account in this part of the chapter. The special case of anisotropy caused by the magneto-optical effects in the materials has been included as well. Within the theoretical part the interaction of light with the thin film systems exhibiting defects also has been described. The main attention has been devoted to the influence of roughness of the boundaries, inhomogeneity in refractive index and transition layers on the optical properties of the film systems. All the substantial theoretical approaches expressing the interaction of light with these systems that allow us to calculate the ellipsometric quantities have been presented. In the experimental part of this chapter the review of the most important ellipsometric methods employed in practice has been introduced on the basis of the articles selected from the literature. It has been shown that these experimental methods of ellipsometry can be divided into four basic groups; i.e., into the groups formed by the methods of monochromatic single angle of incidence ellipsometry, monochromatic multiple angle of incidence ellipsometry, spectroscopic single angle of incidence ellipsometry and spectroscopic multiple angle of incidence ellipsometry. The typical examples of the ellipsometric methods playing a significant role in the fundamental and applied researches and various industrial branches have been presented. The individual sections of this chapter have been devoted to the immersion modifications of the ellipsometric methods specified above and the methods of generalized ellipsometry. The individual section has also been devoted to the ellipsometric methods in magneto-optics. The brief discussion of the ellipsometric method presented has been performed in the conclusion of the experimental part. The main emphasis has been on the discussion of the practical features of the ellipsometric methods described. Therefore the numerical results achieved by applying these ellipsometric methods to the concrete thin film system have also been presented. In the future one can expect that the ellipsometric methods will play an important role in investigating the thin films taking place within many fields of science and industry. This is implied because the ellipsometric methods can be applied using relatively simple experimental equipments in comparison with the other physical techniques suitable for studying the thin film systems and because the development of computer science allows one to improve the interpretation of the experimental data achieved within these methods. One can also expect that
278
ELLIPSOMETRYOF THINFILMSYSTEMS
[3
the combination o f the ellipsometric methods with other optical or non-optical ones will be useful in studies o f complicated thin film systems in particular.
Acknowledgments The authors are indebted to Prof. O. Litzman for his valuable discussion. Moreover, the authors wish to thank Prof. S. Vi~fiovsk~, and Dr. K. Postava for their help concerning the preparation o f the sections devoted to magneto-optics. This work was supported by the Grant A g e n c y o f Czech Republic and Grant A g e n c y o f Ministry o f Education o f Czech Republic under contracts no. 202/98/0988 and no. VS96084, respectively.
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ELLIPSOMETRYOF THIN FILM SYSTEMS
[3
Jegorova, G.A., E.V. Potapov and A.V. Rakov, 1976, Opt. Spektrosc. 41, 643. In Russian. Jellison Jr, G.E., and EA. Modine, 1997a, Appl. Opt. 36, 8184. Jellison Jr, G.E., and EA. Modine, 1997b, Appl. Opt. 36, 8190. Jellison Jr, G.E., EA. Modine and L.A. Boatner, 1997, Opt. Lett. 22, 1808. Johnson, J.A., and N.M. Bashara, 1971, J. Opt. Soc. Am. 61,457. Kao, S.-C., and R.H. Doremus, 1994, J. Electrochem. Soc. 141, 1832. Kim, C.C., P.M. Raccah and J.W. Garland, 1992, Rev. Sci. Instrum. 63, 2958. Kim, S.Y., and K. Vedam, 1986, Appl. Opt. 25, 2013. Kim, S.Y., and K. Vedam, 1988, Thin Solid Films 166, 325. Kinosita, K., and M. Nishibori, 1969, J. Vac. Sci. Technol. 6, 730. Knittl, Z., 1976, Optics of Thin Films (Wiley, London). Konev, V.A., E.M. Kuleshov and N.N. Punko, 1985, Radiowave Ellipsometry (Science and Technics, Minsk). In Russian. Krinchik, G.S., and M.V. Chetkin, 1959, Sov. Phys. JETP 36, 1368. Krishen, K., 1970, IEEE Trans. Antennas Propag. 18, 573. Landau, L.D., E.M. Lifshitz and L.P. Pitaevski, 1965, Electrodynamics of Continuous Media (Pergamon Press, New York). Landolt, H., and R. Brrnstein, 1962, Zahlenwerte und Funktionen, Vol. 7, Optische Konstanten (Springer, Berlin) pp. 2-415. Lecourt, B., D. Blaudez and J.M. Turlet, 1998a, Thin Solid Films 313-314, 790. Lecourt, B., D. Blaudez and J.M. Turlet, 1998b, J. Opt. Soc. Am. A 15, 2769. Lederich, R.J., 1972, J. Opt. Soc. Am. 62, 1524. Leslie, J.D., and K. Knorr, 1974, J. Electrochem. Soc. 121, 263. Levin, B.R., 1960, The Theory of Random Processes and Their Application to Radio Engineering (Sovyetskoe Radio, Moscow). In Russian. Lissberger, P.H., and M.R. Parker, 1971, Int. J. Magn. 1,209. Liu, Q., J.E Wall and E.A. Irene, 1994, J. Vac. Sci. Technol. A 12, 2625. Loescher, D.H., R.J. Detry and M.J. Clauser, 1971, J. Opt. Soc. Am. 61, 1230. Logothetidis, S., I. Alexandrou and A. Papadopoulos, 1995, J. Appl. Phys. 77, 1043. Logothetidis, S., I. Alexandrou and N. Vouroutzis, 1996, Thin Solid Films 275, 44. Lrschke, K., 1981, Cryst. Res. Technol. 16, K72. Lubinskaya, R.I., A.S. Mardezhov, K.K. Svitashov and V.A. Shvets, 1986, Surf. Sci. 177, 625. Luke,, E, 1972, Surf. Sci. 30, 91. Luke,, E, W.H. Knausenberger and K. Vedam, 1969, Surf. Sci. 16, 112. Masetti, E., M. Montecchi and M.P. da Silva, 1993, Thin Solid Films 234, 557. Maxwell Garnett, J.C., 1904, Philos. Trans. R. Soc. London 203, 385. Maxwell Garnett, J.C., 1906, Philos. Trans. R. Soc. London 205, 237. McCrackin, EL., E. Passaglia, R.R. Stromberg and H.L. Steinberg, 1963, J. Res. Nat. Bur. Stand. Sec. A 67, 363. McGahan, W.A., B. Johs and J.A. Woollam, 1993, Thin Solid Films 234, 443. Memarzadeh, K., J.A. Woollam and A. Belkind, 1987, Ellipsometry study of ZnO/Ag/ZnO optical coatings: Determination of layer thickness and optical constants, in: Optical Materials Technology for Energy Efficiency and Solar Energy Conversion VI, Proc. SPIE 823, 54-61. Memarzadeh, K., J.A. Woollam and A. Belkind, 1988, J. Appl. Phys. 64, 3407. Merkt, U., 1981, Appl. Opt. 20, 307. Meyer, E, and G.A. Bootsma, 1969, Surf. Sci. 16, 221. Moil, T., N. Fujii, Y.-M. Xiong and T. Saitoh, 1995, Thin Solid Films 270, 215. Moy, Y.-P., 1981, Appl. Opt. 20, 3821. Navrfitil, K., I. Ohlldal and E Luke,, 1979, Thin Solid Films 56, 163.
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E. WOLF, PROGRESS IN OPTICS 41 9 2000 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED
OPTICAL
TRUE-TIME
DELAY CONTROL
SYSTEMS
FOR WIDEBAND PHASED ARRAY ANTENNAS
BY
RAY T. CHEN AND ZHENHAI FU
Microelectronics Research Center, Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712, USA
283
CONTENTS
PAGE w 1.
INTRODUCTION TO PHASED ARRAY ANTENNAS
w 2.
PHOTONIC T E C H N O L O G Y IN PHASED ARRAY ANTENNAS
290
w 3.
BULK OPTICS AND ACOUSTO-OPTICS TRUE-TIME DELAY
297
w 4.
OPTICAL FIBER TRUE-TIME DELAY LINES . . . . . . .
304
w 5.
OPTICAL WAVEGUIDE-BASED TRUE-TIME DELAY LINES
324
w 6.
SUBSTRATE-GUIDED WAVE TRUE-TIME DELAY M O D U L E S
337
w 7.
WAVELENGTH-DIVISION MULTIPLEXED OPTICAL TRUE-TIME DELAY LINES . . . . . . . . . . . . . .
343
SUMMARY . . . . . . . . . . . . . . . . . . . . . .
354
w 8.
ACKNOWLEDGEMENTS REFERENCES
....
285
. . . . . . . . . . . . . . . . . .
355
. . . . . . . . . . . . . . . . . . . . . . .
355
284
w 1. Introduction to Phased Array Antennas 1.1. INTRODUCTIONTO PHASED ARRAYANTENNAS Military and commercial users have long sought an affordable means to transmit and receive real-time information directly or via satellite, to identify targets and to guide missiles. However, conventional, mechanically steered antennas are bulky, heavy and slow. Therefore, high-data-rate wireless communication has not been available to mobile platforms because the use of conventional, high-gain reflector antennas has not been physically practical. Usually, the radiation pattern of a single radiating element is relatively wide and the directivity is low. In many applications, antennas with very directional characteristics are needed and enlarging their electrical dimensions can do this. A way to do this without increasing the physical size of the individual element is to form an array. An antenna array is an assembly of radiating elements (of any kind: wires, apertures, micropatches, etc.) in an electrical and geometrical configuration. In most cases the elements are identical because this is convenient, simple and practical. In an array of identical elements, there are five controls that can be used to shape the radiation pattern of the array: the geometrical configuration of the overall array, the relative displacement between the elements, the excitation amplitude of the individual element, the excitation phase of the individual element and the relative pattern of the individual element. For a linear or planar array of identical elements with uniform amplitude and spacing and a progressive phase lead between adjacent elements (uniform array), the pattern of the array can be obtained by multiplying the array factor (which depends only on the geometrical and the electrical configuration) with the pattern of the individual elements. This method is called pattern multiplication. As illustrated in fig. 1, for linear array radiating elements with individual phase control, the far field pattern along the direction of q~ can be expressed as (Brookner [ 1991 ]) N E(OS, t) = ~ An exp (iOOmt) exp [i (~Pn + nkmAsin q~)], n=l
285
(1)
286
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
P h a s e shifter
T
Antenna
\(1)o
[4, w 1
Wavefront
element
1Xn power splitter
Fig. 1. Phased array antenna beam steering. where An is pattern of the individual element, (Dm is the microwave frequency, km--tOm/C is the wave vector, ~Pn is the phase shift and A is the distance between radiating elements. The dependence of the array factor on the relative phase shows that the orientation of the maximum radiation can be controlled by the phase excitation between the array elements. Therefore, by varying the progressive phase excitation, the beam can be oriented in any direction to give a scanning array. For example, to point the beam at an angle ~0, ~Pn is set to the following value
~p,, =-nkmAsin ~o.
(2)
For the scanning to be continuous, phase shifters are used to continuously vary the progressive phase. Phased arrays have made it possible to obtain better tapering of the illumination pattern to minimize sidelobes. Two basic methods are amplitude taper and space taper (Brookner [1991]). In the amplitude taper, the energy fed to each element is diminished toward the edges of the antenna to approximate the desired pattern. Lower sidelobe level arising from the use of a weighted illumination is obtained at the expense of a loss in directivity (taper efficiency). Some examples are cosine-squared-on-a-pedestal weighting, taperedTaylor weighting, Dolph-Chebyshev weighting. In the space tapering (array thinning) method, every active element carries full power, but instead of active elements being located at every radiating point, some are passive. Thus the active elements are less dense toward the outer edge of the antenna and the aperture gain is reduced by 3 dB. Array thinning allows for increased angle accuracy with minimum system cost increase. Building a thinned array also allows for future
4, w 1]
INTRODUCTIONTOPHASEDARRAYANTENNAS
287
growth by way of increased system sensitivity while still maintaining the same system beamwidth. These advantages are achieved at a price in reduced antenna gain, higher far out sidelobes and longer search time. Usually space taper is employed in the transmitter while amplitude taper is employed in the receiver. The phased array technology provides high bandwidth communication, from super-high-frequency (SHF) to extra-high-frequency (EHF), that is needed to move massive volumes of information. Thin; lightweight phased array antennas can steer beams electronically, permitting instantaneous connections between satellites, mobile platforms and stations. Phased array antenna systems also have the following advantages: simultaneous multimode operation (search, multitarget acquisition, multitarget track, multimissile guidance, automatic reacquisition of lost target, kill evaluation, passive detection, etc.), large power-aperture products, fast reaction time, automatic operation, electronic stabilization, reliability and maintainability. On the other hand, the phased array antenna has some disadvantages like complexity, high cost and difficulty in development. Phased array antennas have applications in both military and civilian systems. Military applications of phased array antenna include fire control radars, airborne radars, missile guidance, trajectory determination, and covert and mobile satellite communications. Civilian applications include air traffic and marine radar and smart antennas. For example, in commercial airlines, the phased array communication antenna system provides for the simultaneous communication of data and television signals. With a phased array communication antenna system on board, as well as a modified receiving system, passengers potentially will have access to the entire spectrum of commercial television channels and other services available from direct-broadcast satellites. For military customers, the communication antenna system will enable them to receive broad-band, highdata-rate Global Broadcast System (GBS) information by on-the-move platforms such as aircraft, surface ships, submarines, unmanned aerial vehicles and ground vehicles. The B-1B radar (APQ-164), developed by Westinghouse, was the first airborne system produced with a passive electronically scanned array antenna. 1.2. MODERNPHASED ARRAYANTENNA REQUIREMENTS The growing need for radar to provide increased range coverage, target identification, trajectory determination, faster data rate; to accommodate increased target densities; and to cope with high-speed jet propelled aircraft, ballistic missiles and satellites has imposed new technological and system design requirements on phased array antennas. These requirements are
288
TRUE-TIMEDELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 1
I n c r e a s e d c o v e r a g e range: The coverage range of phased array antenna is
(Brookner [ 1991 ]) R4= K~EoA2 s NF ~2' ~L
(3)
where E is transmission energy, a is radar cross section of target, A is array area, S/N is signal-to-noise ratio of single pulse required to obtain detection, L is loss, NF is noise factor and ~. is free space wavelength. Therefore, we need decreased noise power density, high transmission energy, large antenna aperture/high antenna gain, small beamwidth (~0.1~ high directive antenna gain (~58 dB) and minimized system loss. - I n c r e a s e d resolution: High range and angular resolution provide improved accuracy since accuracy is inversely proportional to signal bandwidth and directly to beamwidth. Higher accuracy leads directly to improved trajectory determination for locating launch and impact points of missile target and satellite orbit determination. Therefore, we need wide instantaneous bandwidth, narrow pencil beams, low peak, and average sidelobe level (typically -40 dB peak a n d - 5 5 dB average antenna sidelobe). - S h o r t e r r e a c t i o n time: Automatic operation and high-beam switching speed (typically 10-q0 ~ts) are required to provide high data rate and multifunction. -Physical r e q u i r e m e n t s : small size, low weight, easy to transport, high mobility. - E n v i r o n m e n t a l r e q u i r e m e n t s : wide operating temperature range, immunity to shock and vibration, resistivity to humidity, salt, fog, and fungus. - P r o d u c a b i l i t y , m a i n t a i n a b i l i t y , reliability, a n d low cost. 1.3. C H A L L E N G E S OF M O D E R N P H A S E D ARRAY A N T E N N A S
Future generations of satellites, aircraft, missiles, ships, and submarines will require phased array antennas and communication systems with strict performance requirements: bandwidths of an octave or more, dramatic reduction in size and weight, low observability, and greater isolation from electromagnetic interference and crosstalk among hundreds to thousands of modules, subarray feeds and elements. These requirements will drive innovative antenna elements and matching network designs, antenna feeds, and control interfaces, as well as novel backplane interface and signal distribution techniques. Conventional metallic waveguide and coaxial cable are heavy, bulky, loss, and susceptible to crosstalk. In addition, metallic cable- and waveguide-based distribution systems restrict the operating bandwidth and limit low observability.
4, w 1]
I N T R O D U C T I O N TO P H A S E D A R R A Y A N T E N N A S 0
,
,
/'
289
/ 9 f = 10 G H z
-5
f = 11 GHz
..... 9
-10 -15 "~ 9
-20
"~
-25
~
-30 -35 -40
38
40
42
44
46
48
50
52
Scanning direction
Fig. 2. Beam squint caused by frequency change in phase-shifter antenna array.
Phased array antenna systems combine the signals from as many as thousands of antenna elements to point a directive beam at some angle in space. The characteristics and angle of the beam are selected electronically across the array elements using analog or digital control of the amplitude and phase of excitation. Such an electronic phase control is accomplished through bulky, heavy coaxial waveguide feed. Furthermore, as higher frequency phased array operation is pursued, element spacing will become increasingly tight, making waveguide congestion and crosstalk at the array backplane serious concerns. In addition, conventional phase shift predetermined for a specific steering angle is de-coupled with scanning frequency, resulting in beam squinting when the frequency changes. Differentiating the above eq. (2), we have (Frank and Huting [ 1994])
A ~ : -tan 05o (Ao)m) (_Om 9
(4)
It is clear that for a fixed set of ~Pn's, if the microwave frequency is changed by an amount AOOm,the radiated beam will drift by an amount A 050. A simulated result is shown in fig. 2, where the number of antenna elements is 128, and the desired scanning direction is 45 ~. This phenomenon is the so-called "beam squint", which leads to an undesirable drop of the antenna gain in the ~0 direction. This effect increases dramatically as qJ0 increases.
290
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 2
w 2. Photonic Technology in Phased Array Antennas 2.1. O V E R V I E W
The use of photonics opens the possibility of unique, very high performance antenna systems while meeting stringent weight and size requirements. The photonic technology that can be used in phased array antenna systems includes fiber optics, optical waveguide, lasers, modulators, switches, detectors, acoustic optics, optical polymer, as well as other optically controlled components for the distribution and control of analog RF signals in antenna systems. Photonic links and systems can be used for array functions as simple as the transmission of digital signals to control the phase and amplitude weights for each element of an array. Once the RF signal is in the optical domain, one can use optical-based, space-time signal processing to perform beamforming and signal processing functions. This will ultimately allow some of the massive processing required for large digital beamforming phased array to be handled before the signal is digitized and put into the computer. The development of temperature-stable devices with broad bandwidth and low noise has been of critical importance to antenna and microwave applications. Fortunately, the search for high-speed interconnection for the optical digital telecommunications and the need to overcome the interconnect bottleneck in high-speed digital computing architectures have helped push the research and development of lasers, modulators, and detectors with high frequency characteristics. The optical waveguide material and optical fiber do not disturb the RF field since they are made of nonconducting dielectric. By using a lightwave as a carrier for the microwave signals that drive the radiation elements of the phased array, a delay network that is nondispersive over multibands of microwave frequencies can be realized by virtue of the small ratio of the signal bandwidth to the cartier frequency. Three optical alternatives to microwave beamforming network are (i) optical analog to RF beamforming network, where optical power splitter, phase-shifter and combiner replace their microwave counterparts; (ii) true-time delay (TTD) elements instead of phase-shifter; and (iii) a two-dimensional Fourier transform method. The first method is component-intensive. An M x N phased array system (M beams distributed to N antenna elements for simultaneous beam forming and steering) requires M 1 xN power splitters, M x N phase-shifters, N M xl power combiners, and N optical detectors/amplifiers feeding the N-element antenna array. It is doable, but complex. Furthermore, there is a beam
4, w 2]
PHOTONIC TECHNOLOGY IN PHASED ARRAY ANTENNAS
293
MM OPTICAL
BEAM-FORMING SYSTEM
flF
flF + fl
'11 I_S#MIGIvINA~ON [ Fig. 5. Fourier optical beamforming system in receive mode (9 1984 SPIE).
between signals from different directions. As shown in fig. 5, an array of RF mixers is required between the output of the processor and the antenna array. The frequency difference between the two light waves provides an intermediate frequency at the output of the RF mixers. The directional information of a received wave is contained in a phase ramp across the aperture of the array because of the difference in arrival time at the elements. At the output of the array of mixers, all the signals are coherent if a phase ramp of equal depth but opposite sign is generated in the optical processor. Signals that are of the same frequency band but from different directions have variable phases and suffer partial cancellations in the summation process. For further discrimination against sidelobes, the input mask of the processor can be selected to provide an amplitude weighting function. 2.3. OPTICAL RADIO FREQUENCY (RF) PHASE CONTROL
An optical phase-shifter can be realized by bulk optics, but it requires several hundreds of volts to operate it. Guided-wave light modulators have attracted wide attention because the drive power can be reduced remarkably. The principles of the bulk optical RF phase-shifter and guided-wave RF shifter are basically the same. The basic concept of the optical phase-shifter can be described by the architecture proposed by Matsumoto, Izutsu and Sueta [1991 ], which has been carried out using titanium diffused LiNbO3 optical waveguides. The schematic configuration of the proposed device is shown in fig. 6, which consists of an optical source, an optical interferometer, and an optical detector. The optical Mach-Zehnder interferometer fabricated on an electro-optic substrate contains an optical frequency-shifter and an optical phase-shifter. The incident CW light beam, which is given by Ea =Acosg2t, where #2 is the light frequency and A is the amplitude, is coupled to the interferometer, and is
294
TRUE-TIMEDELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 2
Input microwave signal Phase shifted
OFS~N~ I Laser I
mic~Voo~ rowave utput
OPS~
Ed~] Detect~ ]
M-Z lnterferometer T / V6~0 Control voltage OFS: Optical Frequency-Shifter
OPS: Optical Phase-Shifter
Fig. 6. Schematic configuration of microwave phase-shifter ( 9
IEEE).
then divided into two equal portions at the Y junction of the interferometer. In the first arm of the interferometer, the optical frequency-shifter operating at the microwave frequency to shifts the frequency of the light signal from s to s + to. The output of the optical OFS, Eb, is then given by A Eb = --~_cos [(s + o9) t]. x/2
(5)
In the second arm of the interferometer, the light beam is phase shifted so that the field Ec is expressed as follows: A Ec = ----~cos (f2t + Aq~), V2
(6)
where AO is the induced phase delay. At the output Y junction of the interferometer, two light signals are recombined, and the total light signal Ea becomes E d = - ~A
{cos [(g2 + 09) t] + cos (f2t + AO) }
.
(7)
The output is fed to a square-law detector, and an electrical beat signal is obtained, which is given by Vout -- const, x [1 + cos (o9t + Aq~)].
(8)
By using an ac-coupled detector to drop the dc component, the microwave signal fed to the frequency-shifter is retrieved with additional phase shift Aq~.
4, w 2]
PHOTONIC TECHNOLOGY IN PHASED ARRAY ANTENNAS
295
By changing the control voltage VA0 for the optical phase-shifter, the phase of the microwave output from the detector can be linearly shifted because Ar is proportional to the voltage VA0. 2.4. OPTICALTRUE-TIMEDELAYFEED FOR PHASED ARRAYANTENNAS In order to satisfy the ultrawide bandwidth operation of future phased array antennas, it is necessary to implement true-time delay (TTD) steering techniques such that the far-field pattern is independent of the microwave frequency. The underlying principle is that if the time shift is set according to a particular steering direction, the microwave phase shift at each antenna element, which is proportional to the microwave frequency fm, can follow the frequency scan to avoid beam squinting. In the TTD approach, the path difference between two radiators is compensated by lengthening the microwave feed to the radiating element with a shorter path to the microwave phase-from. A fixed set of delay lines compensates for the path differences corresponding to a particular steering angle at all frequencies. Specifically, the microwave exciting the (n + 1)th antenna element is made to propagate through an additional delay line of length Dn--nL(CI)o). The length of this delay line is designed to provide a time delay t. (q~o) =
(nAsin q~0)
(9)
for the (n + 1)th delay element. For all frequencies corn, ~Pn is given by ~0~ = -COmt~
(r
(10)
With such a delay setup, when the second phase term inside eq. (1) is changed due to frequency "hopping", the first term will change accordingly to compensate for the change such that the sum of the two remains unchanged. Thus, constructive interference can be obtained in the direction q~0 at all frequencies. Even if there is an instantaneous change in microwave frequency, the radiated beam will maintain its specific angular direction. A simulation result is shown in fig. 7, where the array contains 128 antenna elements and the desired scanning direction is 45 ~. These delay lines are usually accomplished by lossy and bulky metallic waveguide feeds, resulting in high cost and heavy weight. The progress in photonics technology in recent years has raised great interest in providing true-time delays using optical means. Compared to electronic phase-shifters,
296
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 2
0 -5
f= 20GHz
//
/\
f = 10GHz
-10
f = 15 GHz
;~ -15
.=
-2o -25
~
-:30 -:35 -40 ,
38
40
42
44
46
48
50
52
Scanning direction
Fig. 7. Far field pattern of true-time delay array antenna under different frequencies.
photonic TTDs offer wide bandwidth, compact size, reduced weight and very low RF interference. Notice that by setting the various time delays according to eq. (9), one can achieve "squint-free" beamforming only at one specific beam pointing direction q~0. In order to scan the beam into another angle ~/, a completely different reconfiguration of the delays has to be established. The total number of delay configurations is determined by the maximum steering angle and the minimum angular scan resolution. Obviously, this can be a very large number if continuous true-time delay is required. In practice, a total number of R = 2N discrete delay lines (called N-bit delay) is constructed as the time-delay unit. This approximation may introduce sidelobe errors. The set of discrete time delays selected for each steering angle represents a "quantized" approximation to a linear phase taper that dictates delay times of 0, At, 2At,..., RAt across the array. A higher degree of accuracy can be achieved with a smaller At, i.e., with more bits of resolution, then the antenna can be scanned at correspondingly smaller angular increments. Since practical PAA array elements number K can be as high as 10000, this results in a very large number (R• of total delay lines. Therefore, straightforward implementations of TTD result in large hardware complexities and are impractical. Architecture must compress the hardware with respect to the number of delays per PAA element. One approach to reduce the number of delays is to divide the radiating elements into several subarrays and use "time steering" for only the subarrays (Mailloux [ 1993]). Each subarray behaves as a smaller antenna that can be individually steered with phase-shifters over a reasonably
4, w 3]
BULK OPTICS AND ACOUSTO-OPTICS TRUE-TIME DELAY
'•f
297
LASER
SLM, (PBS,) SLM2 (PBS~)
SLMN(PBSN) (PDA)
Fig. 8. Bulk optics time delays for phased array antenna (9 1991 OSA). wide bandwidth. We may regard such a system as a hybrid system with both TTD and phase steering. Since the first demonstration of an optical true-time delay phased array antenna, many approaches for implementing optical true-time delay have been proposed and demonstrated. These approaches can be divided into several categories as will be described in the following sections. These approaches are bulk optics and acoustic optic TTDs, fiber-based TTDs, optical waveguide-based TTDs, substrate-guided wave TTDs, and wavelength division multiplexed TTDs.
w 3. Bulk Optics and Acousto-Optics True-Time Delay 3.1. BULKOPTICS TRUE-TIMEDELAYLINES Dolfi, Huignard and Baril [1989] proposed and experimentally demonstrated an implementation of optically controlled delays based on dual frequency laser beams, reflective 2-D spatial light modulator (SLM), and nonbinary cascaded free-space delay paths. Later, they extended the architecture into a binary nonreflective system based on polarization switching by N spatial light modulators that provides 2 N time delays to a phased array antenna with p x p radiating elements (Dolfi, Michel-Gabriel, Bann and Huignard [1991], Dolfi, Joffre, Antoine, Huignard, Philippet and Granger [1996]). The 0 to 27r phase of the microwave signals can be continuously controlled by a liquid-crystal spatial light modulator that operates in the birefringent mode. The microwave signal originates from the coherent detection of a dual-frequency laser beam obtained with an acousto-optic frequency shifter. The operating principle of the proposed binary nonreflective optical architecture is shown in fig. 8. A single-frequency laser beam (o~/27r) is focused through
298
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 3
an anisotropic acousto-optic Bragg cell (BC) excited by a continuous microwave signal 0c). The transmitted beam (~o/2Jr) and the diffracted beam (co/2;r +f) are cross-polarized and recombined on a beam combiner. When photodiode PD detects this dual frequency beam through a 45 ~ oriented polarizer (P), a heterodyned microwave beating signal at frequencyf is observed, which provides a phase reference to compare with the output delayed signals. The two beams at frequencies o)/2Jr and ~o/2;r + f travel along the same path, therefore their phase noises are correlated and the spectral width of the microwave signal is not affected by the laser linewidth. The dual-frequency beam then passes through a half-wave plate, where the light polarization is rotated by 45 ~ It is expanded and intercepts M0, a nematic liquid-crystal SLM of p• pixels that operates in the electrically controlled birefringent mode (parallel alignment), which provides an analog control of the microwave signal phase by changing the relative optical phase of the crosspolarized components of the dual-frequency beam. The polarization of the beam at frequency o)/2;r coincides with the orientation of the liquid-crystal molecules. So, according to the applied voltage Vk on each pixel, the refractive index n(Vk) experienced by this polarization continuously varies between ne and no, the extraordinary and ordinary refractive indexes of the liquid crystal, respectively. On the contrary, the beam at frequency ~o/2Jr + f experiences the constant refractive index no. The two polarizations are then recombined on a 45~ polarizer, and when a photodiode array detects this expanded beam, each photodetector delivers a beating signal of amplitude 1, ik(t) = ioCOS [2Jrfi + 2JreAn(Vk) /~
(11)
where ). is the laser wavelength, e is the liquid-crystal thickness of M0 and io=2(imi2~f+,o) 1/2, where im(i2:rf+,o) is the photocurrent delivered by a photodiode detecting the laser beam at frequency ~o/2Jr(o)/2~r +f). At the output of M0, the dual-frequency beam intercepts a set of electrically addressed SLMi, polarization beam splitters PBSi and prisms Pi. These components provide the parallel control of the time delays directed to the antenna as follows: each SLM consists of an array of p• pixels, and the beam polarization is rotated by 0~ or 90 ~ according to the applied voltage on each pixel. The electrically addressed Ferro-electric liquid-crystal cells SLMs are used because of their high extinction ratio and short switching time, which are well adapted to high-recurrence radars. Each PBSi consists of two polarizing beam splitters, joined side by side. When the polarization is horizontal ( T in fig. 9), PBSi is perfectly transparent and the light beam intercepts the next SLM, Mi +1. When the polarization is rotated by
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BULK OPTICSAND ACOUSTO-OPTICSTRUE-TIMEDELAY
299
Fig. 9. Operating principle of bulk optic time delays: (a) the position of Pi fixes the delay value; (b) the thickness of the glass plate 1 fixes the delay value (9 OSA).
90 ~ and becomes vertical (| in fig. 9), then the first polarizing beam splitter of PBSi is perfectly reflecting and deflects the beam toward a prism Pi. Pi acts as a comer cube, and the beam reflects off the second polarizing beam splitter of PBSi toward the next SLM. The collimated beam travels through all the PBSs and is focused by an array of microlenses (L) onto an array of p • high-speed photodiodes (PDA). For a given photodiode, the phase of the microwave beating signal is determined by the applied voltage on the corresponding pixel of M0 and by the choice of the PBSi on which the reflections occur. The positions of prisms Pi fix the delays, providing values according to a geometric progression: r, 2 r , . . . , 2 N- 1z-. The amplitude of the output beating signal of each photodiode is given by
ik(t) = i0cos 2Jrf t + 2Jrf Z
Ek,j 2 J - i t + --s
,
12
j=l
where, according to the polarization at the output of SLMj, Ek,j = 1 when PBSj reflects the portion of the collimated beam detected by the photodiode k (1 ~~/l. Riza [1991] modified Dolfi's interferometric architecture to form a noninterferometric time-delay-based optical beam former that operates in both the antenna transmit and receive modes. The system is incoherent, and uses intensity modulation, direct detection, and incoherent optical signal summation. The system is shown in fig. 10. A p-polarized laser L1 is intensity modulated by wideband microwave signal, which is transformed by the biasing/square root circuit and amplified by microwave amplifier. The modulatedp beam is expanded and collimated, and then passes through the PBS to illuminate a 2-D pixilated
300
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
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Fig. 10. Bulk optical beamforming system for phased array antenna that operates in both transmit/receive mode (9 1991 OSA). electrically addressed ferroelectric liquid crystal SLM1. Each pixel in SLM1 acts as a polarization rotator, which, depending on the applied pixel voltage, can rotate the beam polarization by 0~ or 90 ~ The n x n pixilated structure of SLM1 can generate n x n independent beams. These beams can then be independently time-delayed using cascaded N polarization rotating SLMs and N free-space delay units made from prisms and PBSs to give 2N values of time delay for n x n microwave signals driving an n x n array. To transmit a beam at a given direction, the electrical signals driving the SLMs are adjusted to give the required time delays for the n x n optical beams that generate the delayed microwave signals. The n x n independent optical beams at the output of the last delay unit can have either p or s polarization, depending on the path traveled by a particular beam. To detect all delayed optical beams while using the output PBS, additional optics need to be added because PBS is transparent to the p beams and reflects the s beams by 90 ~ One method is to place the polarization rotating components P, H and R before PBS. The p and s beams at the output of the last delay unit pass through the 45~ polarizer P that combines the p and s beams. The 45~ linear polarized beams are rotated by 45 ~ using the half-wave plate H, which causes the beams at the output of the half-wave plate to be p-polarized. This p-polarized light passes through the electrically controlled liquid crystal cell R, which rotates the light polarization by 0~ or 90 ~ In the transmit mode, the p-polarized light passes through R without rotation. This results in the p-polarized light passing through PBS and being focused onto the
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BULK OPTICS AND ACOUSTO-OPTICS TRUE-TIME DELAY
301
2-D detector array (DA) that generates the time-delayed microwave signals for driving the phased array. The delay configuration used in the system in fig. 10 to generate a beam in a specific direction remains unchanged when receiving in the same direction. In receive mode, the n x n microwave signals from the n x n antenna elements in the phased array are used to intensity modulate a 2-D array of n x n laser diodes. A lenslet array L is used to collimate the beams from the diode array. The s-polarized beams from the diode array are reflected by 90 ~ by the PBS to illuminate SLM1. The polarization rotation states of the pixels in SLM1 are reversed with respect to those in the transmit mode because in the receive mode the light from the diode array is s-polarized instead of p-polarized. R is turned on so that the p-polarized light incident upon R is rotated by 90 ~ to give s-polarized light that is reflected by 90 ~ by PBS, which directs the timedelayed n x n beams through a spherical lens S 1. The optical beams are delayed in the system so that for the particular antenna receive direction, all the beams at the output of PBS are aligned in time and the lens acts as an optical adder, summing the individual intensities from the beams on D1. A 2-D polarization rotating electrically controlled SLM labeled SLM-R can be used to replace P, H and R to achieve smaller optical loss. For a given setting of time delays, the polarization (p or s) of the n x n beams at the output of SLMN is known because the voltages applied to the SLMs controlling the delay units are known. In the transmit mode, by applying the appropriate voltages to the n xn pixels in SLM-R so that any incident s beams are changed to p beams while allowing the incident p beams to pass unaffected, all beams incident on PBS can travel through PBS and incident on the 2-D detector array. In the receive mode, the voltage settings are reversed on SLM-R, allowing all the s beams to pass unaffected while rotating the incident p beams to produce beams at the SLM-R output, all the beams incident on PBS are s-polarized and are directed to detector D1. The improved optical efficiency of this approach is acquired at the cost of an additional SLM plus some computer memory and control hardware. For receive mode operation the SLM1 and SLM-R switching times have to be compatible with the pulse repetition rates used by the radar. Riza also proposed a variety of 3-D polarization switched time-delay units architecture that uses imaging optics with free space and fiber paths (Riza [ 1992]) and experimentally demonstrated a high isolation birefringent mode nematic liquid crystal (BM-NLC) l x2 optical switch based 1-bit 3.33ns delay free space time delay unit with >30dB SNRs (Riza [1994a,b]). Fu, Schamschula and Caulfield [1995] described a modular solid optic design based on polarization switch and polarizing beam splitter with fast switching (ns-ms), arbitrary time
302
TRUE-TIMEDELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
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delay (ps-ns), and ruggedness for Riza's incoherent beamformer (Riza [ 1991 ]). Yao and Maleki [ 1994] extended the incoherent beamformer to femtoseconds of resolution and nanoseconds of total delay for ultrawide bandwidth applications. Fetterman, Chang, Scott, Forrest, Espiau, Wu, Plant, Kelly, Mather, Steier, Osgood, Haus and Simonis [1995] also demonstrated an optically controlled phased array radar receiver using the SLM/bulk optics approach at an optical wavelength of 1.3 ~tm that exhibited "squint flee" operation over the entire X-band (8-12GHz) based on the architecture proposed by Dolfi, but using NLC optical switches and external IO modulation of light from two distributed feedback (DFB) lasers. The use of computer processing techniques on the data collected has resulted in an angular accuracy of 1.4~ using only two antenna array elements and three SLM optical delay units. A configuration for the SLM receiver architecture was proposed and demonstrated that can increase the digital resolution from 15 sampled points to 64, improving the angular accuracy of the radar receiver without increasing the number of components. 3.2. A C O U S T O - O P T I C T R U E - T I M E DELAY LINES
Acousto-optic devices are used to provide time delay or used as a switch to select delay paths. Herczfeld's group (Jemison and Herczfeld [ 1993]) and Gesell, Feinleib, Lafuse and Turpin [ 1994] both demonstrated systems using AO devices to deflect the intensity modulated laser beam into desired delay paths to provide desired delays. Details of acousto-optic device control of phased array antennas can be found in Antenna Design with Fiber Optics by Kumar [ 1996]. Lin and Boughton [1989] proposed an acousto-optic multichannel programmable time-delay device using a multichannel Bragg cell as its key element. When a multichannel Bragg cell is applied with RF signals Si and is illuminated at the Bragg angle across the array of acoustic beams by a narrow optical beam, as shown in fig. 11, the diffracted beams will be intensity modulated by the acoustic waves in the narrow interaction regions. If the diffracted beams are monitored individually by an array of detectors, then each detector output will reproduce the RF waveform in that channel, but with a delay Tabs, given by the ratio of the distance of narrow optical beam to the transducer plane, li, and the acoustic velocity in the acousto-optic medium, V. rabs is absolute time delay taken by the signal Si to propagate over a distance li. If the optical beam is parallel to the transducer plane, the absolute delays are equal. When the incident beam is rotated by an angle as shown, however, absolute delays are made differently among the channels. In this case a differential delay Z'rel between the channels is created, where rrel =Ali/V. The absolute delay can be
4, w 3]
BULK OPTICS AND ACOUSTO-OPTICS TRUE-TIME DELAY 1 2 ,--,, ~ ..... ~ .... ~ f I
3 4 5 .-~ E~ ,,.,.,, __ ~.~.,'~
2
S
4
~ 5
O
t
303
TRANSMITrED BEAM PATTERN
I
T / t
%s = q _ . . .
_
_
T I t INPUT BEAM
7.reI = NARROW OPTICAL BEAM
Fig. 11. Multichannel acousto-optic programmable true-time delay line ( 9
SPIE).
varied by changing the distance li, while the differential time delay can be varied by changing the beam rotation angle 0. These two time delays can be varied independently. Also, since the aperture of the Bragg cell is completely accessible to the optical beam, both types of time delays can be changed continuously. The absolute delay can be obtained with a single-channel Bragg cell, but the differential delay is achieved due solely to the existence of multiple acoustic beams in the multichannel Bragg cell. When a narrow straight sheet beam is used, such as the one shown in fig. 11, the differential delays among the channels are a constant value. This is required for beamforming in a phased array antenna problem. Nonlinear beam cross sections would produce distributed delays across the channels. The key to realizing various implementations is to form the narrow sheet beam to illuminate the Bragg cell. These narrow incident beams can be formed by either inserting a slit opaque mask into a broad, collimated beam or a set of optical lenses. This heterodyne detection of multichannel AO-based timedelay concept was experimentally demonstrated using an in-line optical design and the rotation of a slit beam via a Dove prism. Toughlian and Zmuda [1990] proposed a phase-compensated heterodyne system to continuously vary the time delay. A deflection type AO modulator is used to spatially separate different frequency components. Lights of different frequencies then experience a different delay after incident on a rotating tilt mirror. Therefore the heterodyne-generated RF phase varies linearly with frequency, and in effect, provides true-time delay. Monsay, Baldwin and Caucuitto [1994] suggested a similar architecture but with a wider bandwidth and demonstrated it using a dichromatic (dual frequency) solid-state laser.
304
TRUE-TIMEDELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
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w 4. Optical Fiber True-Time Delay Lines The application of fiber optics in phased array antennas has been visualized for many years. Recently, the need to achieve large instantaneous bandwidths for wide aperture antennas further prompted the exploration of new concepts for broadband beam steering. Conceptually, a fiber-optic time delay system is extremely attractive because it is lightweight, compact, nondispersive over multiple microwave bands, and immune to electromagnetic interference, and has the potential to address phased array antennas that require extremely long delay. Optical fiber can support an information bandwidth of many GHz, and achieve delay of sub-nanoseconds. Multiple fibers can easily be used to feed a large number of radiation elements from a remote station. In 1979, Levine [1979] proposed the idea of using a passive multichannel fiber-optic delay lines and switchable transducer, such as optical to RF transducer (Photodetector), to select the desired time-delay signals. Since then, several approaches have been proposed, demonstrated and implemented using optical fiber to provide true-time delay signals for phased array antenna beamforming. These approaches can be divided into three major categories. The first one uses fiber bundles with different fiber lengths, which provide different delays. Desired delay signals are selected by different switching methods, like laser/detector switching and electro-optical switching. The second one also uses fiber bundles for each unit. However, the lengths of the fibers are the same in this approach. Different delays are produced by fibers with different dispersion effects and by using different working wavelengths. The third one uses a single fiber with embedded fiber gratings for each unit. Different delays are selected by changing the working wavelength. 4.1. F I B E R L E N G T H S W I T C H I N G T R U E - T I M E DELAY
4.1.1. Basic fiber switching time delay In 1984, Cardone [ 1985] reported the fabrication and partial testing of a receive mode three-beam electro-optical beamforming network for an eight-element linear array. An eight-horn array, each containing eight monolithic pre-amplifiers that drive the laser transmitters, produces amplitude modulation of the optical signals. An eight-fiber cable transmits the light signals to the remote (30 meters) optical beamforming network. Each light signal is power-divided into three signals and the signals for each beam are combined in three 8 x 8 star couplers. The fiber lengths between the optical dividers and combiners are cut to the proper
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OPTICAL FIBER TRUE-TIME DELAY LINES BIAS SWITCHING
305
8:1 FIBER-OPTIC
F ...... 1f 9
t
1
-'
~. 21
to o~
I
DC
I\
' r
RF ,..u.
OUTPUT
~;i
_~~"
"'4
PHOTO-
BIAS SWITCHING
Fig. 12. Three-bit fiber-optic time-shifternetworkusing laser switching (9
IEEE).
lengths and spliced to provide a summation of the detected RF signals in the desired beam direction. Multimode fiber, dividers and combiners were used for the broadband. The computed throughput loss for the breadboard hardware for each signal path from the laser input to the detector output was 52.38 dB. The measured loss averaged approximately 56 dB. A summation of the eight in-phase signals should produce a signal gain of 18 dB. Toughlian and Zmuda used the mirror deflection method to direct light beams into precut fibers of specific length to switch among different delays (Toughlian and Zmuda [1993]). Ng, Walston, Tangonan, Lee, Newberg and Bernstein [ 1991 ] proposed a fiberoptic delay network based on laser switching. A 3-bit time shifter is shown in fig. 12. It consists essentially of eight analog fiber-optic links. The lengths of the fibers in these links were cut to provide a pre-specified set of differential time delays determined by the antenna aperture and its maximum steering angle 0max. The eight delay lines provide 23 discrete incremental delays for achieving three bits of resolution in setting the microwave phase front. The microwave signal source is transmitted, after delay, to the output port by direct modulation of high-speed GaInAsP/InP buffed crescent lasers. During steering of the phased array, one delay line, as specified by the steering angle, is selected from each of such modules by biased switching of the eight laser diodes to provide time delay for the antenna subarray fed by the modules. The eight fibers in each module were spliced into a commercial star coupler whose optical output is coupled to a detector with a bandwidth of ~ 11 GHz. The set of discrete time-delay increments selected for each steering angle represents a quantized approximation to a linear phase taper that dictates delay times of r, 2 r , . . . , N r across the array, where r is given by (Asin Oo)/c. The smallest differential delay increment in each module corresponds to the least significant bit of the 3-bit approximation. The above ideas require N fibers and switch network to achieve N delay
306
TRUE-TIMEDELAYCONTROLSYSTEMSFOR WIDEBANDPHASEDARRAYANTENNAS [4, w 4
signals. For a large antenna array, the hardware requirement will be huge based on these architectures and is almost impossible. Some hardware-compressive architectures were proposed to solve this problem.
4.1.2. Square root cascaded delay line Soref [1984] proposed a hardware-compressive integrated and fiber optical design for programmable time-delay devices. Figure 13 shows such a voltagecontrolled 64-delay device consisting of two cascaded transmissive stages located on the same integrated optical wafer. The wafer contains two 1 • 8 electro-optical (EO) switching networks and two 8• 1 EO switching networks symbolized in the diagram by the branching arrays of 1 • elements. Two groups of fibers, eight strands in each, are doubly coupled to the wafer. Corresponding switch elements are addressed electrically so that one fiber delay path in the first stage is selected along with one fiber path in the second stage. Those fiber delays are additive, and N • different delays are available from N + M fibers. The length of the first-stage fibers are L l l - L 1 , L12-L1 +ALl, L13 =L1 + 2 A L l , etc., and the second-stage fibers have lengths L21 - L 2 , L22 - L 2 + AL2, L23 - L 2 + 2AL2, etc. The overall time delay tij is (Lli+L2j)/v, where v is the fiber's group velocity. The incremental delay Atij is (Lli-L1 +L2j-L2)/v. The shortest delay increments in the first and second stages are Atl =AL1/v and At2=AL2/v, respectively. If there are N fibers in each stage and AL2 = NzkL1, a set of N 2 equal delay increments, Atl,2Atl, 3Atl,... ,N2Atl, can be achieved. Since 2N fibers are needed to achieve N 2 delays, this structure is sometimes called the square root cascaded delay line (SRODEL). Lee, Loo, Livingston, Jones, Lewis, Yen, Tangonan and Wechsberg [1995] and Loo, Tangonan, Yen, Lee, Jones and Lewis [1996] developed cascaded
ZIL2=
single-mode ~
8Z~L~
fiber
electro-opt0cal 8x 1 switching network
:ptical
integrated
circui
Si-V-gr. ....
,
. ~
11
rflmicrowave electrical output
'
8xl -sw.
lxa-sw.
electrical
~modulation s.m.f. optical source
~
_
~
~
oo,,o.,
detector
s...!.
_ _
i
~
l
l
|
single-mode
.... ,.
isolator fiber lengthincrement =AL, 1
silicon V-groove array
electro-optical lx8 switching network
Fig. 13. Square root programmable fiber switching true-time delay line (9 1984 OSA).
fiber
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307
OPTICAL FIBER TRUE-TIME DELAY LINES
delcrv f i b r e . . , iine~J c o u p t e r Des sw~m ~aser ~ ~ I1 ~--.-4
to QP, te r~n Q
l
.
.
.
.
.
.
.
.
.
receive amp
.
-
.... i
to receiver
Fig. 14. Five-bit cascaded laser switching and bias switching time delay network (9
IEE).
switchable 5-bit photonic time-shiflers used in Hughes Research Laboratories phased array antenna based on laser and detector biased switch. The design of the 5-bit photonic time shift module is shown in fig. 14. Key components of the time-shifter include four pigtailed diode lasers, one 4 x 8 fiber coupler and two selector switches. In operation, the microwave signal modulates one of the four lasers through a 1:4 RF switch. The light is coupled into the 4 x8 fiber coupler and is incident on the detector array. By switching on one of the detectors in the selector switch, the signal is allowed to go through one of the 32 possible delay paths. A post-amplifier is then used to compensate for the conversion and fanout loss of the time-shifter. The 3-dB bandwidth of the device was 4.4 GHz. The measured RF insertion loss varied between 37 and 40 dB among the 32 delay paths. A 37-dB post-amplifier made the module almost transparent RF-wise. The measured noise figure of the module was 46 dB and the spurious-free dynamic range (SFDR) was determined to be 110 dB/Hz 2/3.
4.1.3. Binary fiber optic delay line Another way to compress the hardware requirement is the so-called binary fiber-optic delay line (BIFODEL). Figure 15 is the schematic drawing of the BIFODEL architecture proposed by Goutzoulis, Davies and Zomp [1989]. In this architecture, the microwave intensity-modulated optical signal from the laser DELAY
(MSB) b._~
CONTROL WORD (M B I T S ) (LSB) b~
b~
bo
2 M-IAT
Fig. 15. Programmable binary fiber-optic delay line (9
SPIE).
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TRUE-TIMEDELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
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diode is optionally routed through M graded-index multimode fiber segments whose lengths are precut so that the corresponding delays increase successively by a power of 2. The length of the shortest fiber has an associated delay of the desired delay resolution, AT. M 2 x2 optical switches are used to select the fiber segments through which the signal is routed. Each switch allows the signal to either connect (state bi = 1) or bypass (state bi--0) a specific fiber segment. By selecting the state bi of the switches, one may insert a delay T that can take any value, in increments of AT, up to the maximum value Tmax, given by Tmax "- (2 M -
1)AT.
(13)
The signal is converted back to an electrical signal through a photodetector after passing through the appropriate fiber segments. The output of the photodetector is buffered and further processed. The BIFODEL is one efficient programmable fiber optic delay line architectures because the number of fiber segments M required to achieve N delays is determined by a logarithmic relation, i.e., M=log2N. Consequently, the number of fibers needed ( 2 x M ) is also relatively small. Thus, this architecture is particularly suitable for applications in which a large number of delays are required and good signal quality at the output of the delay line is desired. In general, to avoid ambiguities as a result of signal splitting, the minimum programmability period Tpr, defined as the time difference between the setup of two consecutive programs occurring at times t - t i _ ~ and t = ti, must satisfy Tpr ~
Ts + Ti-1,
(14)
where Ts is the signal duration and T i - i is the total delay introduced at time t=ti_~.
Because of the relatively high crosstalk level (-20dB per switch) and the cost of the commercial available 2x2 integrated optical switches, electronic rather than optical switches were used in the demonstration. This requires the basic architecture of fig. 15 to be modified so that each fiber segment has a separate LD, detector and buffer. The electronic switches are commercially available, low cost, MESFET-based GaAs 1 x2 switches capable of operating at microwave frequencies (up to 6 GHz) with a switching time of ~50 ns and a crosstalk figure of < - 5 5 dB. Two such switches connected in a back-to-back configuration provide the 2 x2 switch operation and reduce the crosstalk figure to better than -65 dB. However, there are several other problems associated with the use of these switches. These problems include noise accumulation because of the cascaded active components, significant tipple in the frequency response
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OPTICAL FIBER TRUE-TIMEDELAYLINES
309
of the overall passband, significant increase in the third-order inter-modulation products (IMP), and increase in LD, detector, and associated hardware by a factor of ( M - 1). Such a demonstrated 7-stage system operated over the frequency band of 0.5 to 1.0 GHz and had a programmed delay of 0 to 5 ~ts with a resolution of 39ns. For 0 dBm input signals, the worst-case dynamic range (DR) was 36 dB and the IMP level w a s - 2 8 dBc.
4.1.4. M-ary fiber optic delay line Instead of just using binary delay segments, Ackerman, Wanuga, Kasemset, Minford, Thorsten and Watson [1992] demonstrated a 6-bit time-delay unit using three different switching architectures and compared their performance and complexity. Figure 16 shows three different architectures used to produce six bits of bi-directional time delay selectivity. The cascading of seven 2x2 switches is the simplest design, but in each path the signal incurs losses at 14 integrated-optic waveguide/optical fiber interfaces. Using three 8 x8 switch matrices minimizes the number of these waveguide/fiber transitions (six), but with the addition of complexity and many unused ports (12). Using four 4 x4 arrays of crossbar switches can reduce the complexity of the switch with only one more switch and less unused ports compared to the 8x8 switch matrices. The predicted overall optical insertion loss for the 4x4 crossbar
......
2x2~_ 8~
16~
32~
(a) T
.......
(b)
(e) Fig. 16. Alternative configurations of 6-bit time delay line employing integrated-optical switches and fiber delay lines: (a) seven 2x2 switches; (b) four 4x4 switches; (c) three 8x8 switches (9
IEEE).
310
TRUE-TIMEDELAY CONTROL SYSTEMS FOR WIDEBANDPHASED ARRAYANTENNAS
LiNbO 3 substrate
Optical waveguides
[4, w 4
v-Switch electrodes
R"~
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LiNbO 3 substrate -
,,
~
.
.
.
/
.
.
.
.
.
.
i
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.....
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. . . . .
~i
Fig. 17. Six-bit photonic time delay unit design, showing the four 4 • 4 crossbar switch arrays realized using two 3-inch LiNbO3 substrates (9 IEEE).
switches architecture was 15 dB. Figure 17 shows the demonstrated layout of sixteen crossbar switches in four cascaded 4 x4 matrix configurations and the 12 fiber lengths used in the unit to show how it can offer 64 different total delays (Or to 63r). In the demonstration, the delay step r was ~470ps and the required fiber length was 9.5 cm for this delay. The four switch arrays were designed to fit on two 3-inch Z-cut LiNbO3 substrates as shown in fig. 17. Each directional coupler switch had a full-time reversed A/3 electrode configuration to allow each switch to be operated by a single driver. Polarization-maintaining fiber is used for the delay lines to preserve the extinction ratio between the polarization-sensitive switches. The fiber delay lines were wound in a coil with a minimum diameter of 1.5 inches to minimize the size of the unit without inducing microbending losses. The fiber ends were placed in etched V-grooves and integrated with the switch arrays using UV-curing adhesive. The measured loss was less than 17 dB and the extinction ratio was greater than 14 dB. 4.2. WAVELENGTH-TUNABLE DISPERSIVE FIBER TRUE-TIME DELAY LINE
The optical dispersion-based true-time delay approach can provide continuously tuned delay signals and is made feasible by the widely tunable single polarization laser. Soref [ 1992] and Johns, Norton, Keefer, Erdmann and Soref [ 1993]) proposed and demonstrated a true-time delay network of the transmit-receive (T/R) system consisting of a group of identical microwave-to-optical-to-microwave fiber-optic data links and fig. 18 shows the proposed low-loss RF link. The wavelengthtunable laser diode is run in CW mode, and its emission is sent into a fast electro-
4, w 4]
OPTICAL FIBER TRUE-TIME DELAY LINES
CW wavelength tunable laser diode
Ultrafast electro-optic modulator (Intensity modulation)
IT~DI FEOMI ~~,
Electrical bias source for fast tuning of LD
311
Ultrafast photodiode
High dispersion single-mode fiber
Broadband microwave signal source
Fig. 18. Dispersive fiber variable-delay time delay line ( 9
Variably time-delayed microwave signal output
OSA).
optic waveguide modulator (EOM) that impresses a broadband microwave signal upon the light beam in an intensity-modulated mode. The microwave modulation frequency is the operating frequency of the antenna, and frequency hopping is feasible if desired. The modulated optical beam is then sent into a highly dispersive fiber-optic transmission line that has a total chromatic dispersion D(~.) of more than 50 ps/(nm km). This dispersion is approximately constant over the tuning range of the laser. A time-delayed microwave-modulated optical signal emerges from the fiber of length L, and the incremental delay r is proportional to the electrically induced wavelength shift A)~ of the laser, r = D(~)L(A~), which in turn is proportional to the DC electrical bias applied to the vertical-coupler filter section of the tunable laser diode. The delayed signal is sent to a fast photodiode (PD), which detects the modulated light beam, yielding a delayed microwave electrical signal that is then sent to a semiconductor transmitter module, where the signal is amplified and radiated by the antenna element. The tuning of the tunable laser diode is fast and therefore the electronically steered antenna radiation pattern can be reconfigured in a few nanoseconds. For an antenna array consisting of N radiating elements, 2N simple RF links shown in fig. 18 are needed. Esman, Monsma, Dexter and Cooper [ 1992] came up with the same idea and demonstrated a delay line consisting of o laser, Mach-Zehnder EO modulator, and up to 1.8 km of single mode fibers. Since long fiber is used, the attenuation and the temperature stability of the fiber are major concerns. Highly dispersive fiber can reduce the fiber length by ~70%. Based on this idea, Esman, Frankel, Dexter, Goldberg, Parent, Stilwell and Cooper [1993] demonstrated a time delay antenna, which utilized a wavelength-tunable laser and fibers with different dispersion characteristics in a standard externally modulated link configuration. The optical source used in the demonstration was an erbium-doped fiber ring laser, which provides stable single-polarization output and a 50 nm tuning range. A semiconductor laser can be used instead if the adjacent mode spacing is small so that effective continuous tuning may be obtained. To feed numerous array elements or subarrays, a true-
312
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND
~
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Fig. 19. Fiber-optic prism true-time delay antenna feed (9 1993 IEEE). time delay feed was formed by splitting the modulated single tunable laser source into a fiber-optic prism comprising numerous optical fiber links, each having the same nominal group delay but with slightly different net dispersion as shown in fig. 19. The fiber-optic prism was made by connecting varying amounts of highdispersion and nondispersion fiber as shown in fig. 19. Therefore, a change in the optical wavelength will result in different amounts of total delay change in each fiber link. At the center wavelength A0, all the time delays were matched by trimming the fibers and the main antenna beam was directed broadside. At wavelengths less (longer) than ~0, each of the prism fibers adds (subtracts) a time delay proportional to its dispersion, resulting in different time delays such that the main antenna beam is steered toward (away from) the nondispersion fiber side. A two-element antenna system comprised of a o laser source, an electrooptic microwave modulator, and a 1 )<2 50:50 fiber-optic beam splitter feeding two optical links was demonstrated. One of the optical links was the 108-meter section of high-dispersion fiber, and the other was a similar length section of standard nondispersion fiber. The fiber links were coupled to InGaAs p - i n photodiodes with a 3 dB bandwidth of 6 GHz. The photodiodes were attached through bias networks directly to the feeds of high-bandwidth horn antennas. The fiber-optic prism true-time delay feed exhibits several desirable features. The system bandwidth is expected to be limited only by the external modulators and photodiodes and other system parameters are projected to be 15-35 dB link loss, 34-41 dB noise figure, and 108 dB/Hz2/3dynamic range. Later, the same group (Frankel, Esman and Parent [ 1995]) demonstrated an eight-element antenna controlled by eight delay links with 0 to 1334m nominal lengths of high-dispersion fiber, working in transmit and receive modes. The combined beamformer RF insertion loss w a s - 4 0 dB. 4.3. WAVELENGTH-TUNABLEFIBER GRATINGTRUE-TIMEDELAYLINE In an optical true-time delay system, the required time delay is usually achieved by switching the optical signal through the appropriate length of optical fiber
4, w 4]
OPTICALFIBERTRUE-TIMEDELAYLINES
Fiber L....
,Modul.... ] //~Coupler' 'ILl__
I
GI"
G2
G3
313
G4
G5
G6
Digitizing Oscilloscope
.... Fig. 20. Programmable fiber Bragg grating delay line (9
IEEE).
or waveguide. However, alternate approaches based on wavelength switching can achieve a significant reduction in hardware and switch time. An alternative technique is to use multiple fiber Bragg reflection gratings (BRGs) distributed along the length of a single optical fiber. Different delays can be addressed by tuning the working wavelength. Ball, Glenn and Morey [ 1994] demonstrated a programmable fiber delay line, consisting of an externally modulated wavelength-tunable fiber laser with a small signal gain of 10 dB/m and a six-element wavelength-multiplexed fiber Bragg grating array as shown in fig. 20. The single-frequency wavelengthtunable Bragg grating fiber laser employed a master oscillator power amplifier designed to achieve 20 mW of low noise laser power. The RF signal was encoded onto the CW fiber laser output through an LiNbO3 Mach-Zehnder modulator. The modulator was fabricated by annealed proton exchange and had a 3 dB bandwidth of 600 MHz and an RF half-wave voltage of approximately 4 V. The modulator was pigtailed with polarization maintaining fiber on the input and nonpolarization maintaining fiber on the output. The optical delay line consisted of an optical fiber in which BRGs of various wavelengths had been written. The gratings were separated by 1 m to yield a differential round trip optical delay of approximately 10ns. The Bragg wavelengths of the six 50% reflecting grating were 1.5545-~tm, 1.5558 ~tm, 1.5561 gm, 1.5573 ~tm, 1.5579 ~tm and 1.5588 ~tm. A 3 dB four-port fiber coupler was used to connect the fiber laser to the fiber delay line. The delay line output was reflected back through the coupler and into a photodetector. An optical isolator following the fiber laser was used to eliminate destabilization due to feedback. The microwave delay line output was detected by a high-speed photodetector and, after amplification, sent to the antenna element. Molony, Edge and Bennion [1995] also demonstrated an optical fiber BRG based delay line, which can produce as small as a 31 ps time delay. In a basic 3-bit true-time delay unit based on a fiber BRG device as shown in fig. 21, the light from a tunable source is intensity modulated with the RF signal and is then coupled into the fiber BRG TTD element through a 3 dB four-port crossbar
314
TRUE-TIMEDELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS tunoble laser
GclAs optic~l m~dulator
diode
D
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--
control .
.
.
Bragg greting delay line ]
[
3~dBcouplerph~176176
.
speed photodetector high
~
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photodetector
Fig. 21. Setup for fiber Bragg grating delay line measurement (9 1995 IEE).
coupler or optical circulator. The fiber BRG TTD element is a single length of fiber with equally spaced, high reflectivity Bragg gratings of different central wavelengths distributed along its length. The reflected signal from the fiber grating TTD is selected by choosing the appropriate source wavelength. The reflected signal is separated from the incident light by the coupler or circulator and is detected by a high-speed photodetector. The 3 dB coupler introduces a 6 dB loss into the delay line, which is approximately equal to that suffered by a typical 4-bit optically switched TTD device, yet the fiber BRG approach could be expanded to more bits TTD with no increase in loss. An optical isolator could reduce the loss even further, to below 2 dB on each pass. The maximum number of discrete time delay elements that can be incorporated into a delay line is determined by the tuning range of the source and the optical bandwidth of the Bragg reflection. The optical bandwidth of a grating depends on the length of the grating, which in turn determines the minimum possible spacing between gratings in the delay line and hence the minimum achievable time delay. This can be summarized by the following expressions: Tmin -
6/l ~
2ndg 4nLg C ~2
~ ~,
(2nLg)' A~
g m a x ~'~
(36;~)'
C
(15) (16) (17)
where Tmin is the minimum achievable time delay, dg is the spacing between the gratings, Lg is the grating length, 6~, is the optical bandwidth of the Bragg reflection, Nmax is the maximum number of discrete time delay elements, AA, is the tuning range of the source, n is the refractive index of the fiber, and c is the free space speed of light. Cruz, Ortega, Andres, Gimeno, Pastor, Capmany and Dang [1997] proposed the use of chirped fiber BRGs as a true-time delay line for continuous steering of
4, w 4]
OPTICAL FIBER TRUE-TIME DELAY LINES
315
Modulating RF Signal Optical Carrier ~,,, /" .~
~
, ~
~ .~ ~ ~
t]!!!li!!!~I' i~
,
] I
i~l[illiil~T: ~I I i tll Chirped Fiber Grating ,.~..,
, , jI
~f~1I I ' ~ Le~(~) .
/
/
Optical Carrier ~1 <~ Phase Difference:
A~%=2nf~t(~)=4rff~Leff(~)n/c Fig. 22. Principle of fiber chirped grating delay line (9
IEE).
microwave-phased array antennas and demonstrated a chirped grating which can produce continuously tunable time delay by continuously tuning the wavelength of the optical carrier. Chirp grating is a highly dispersive reflector whose time delay depends strongly on the optical wavelength. As shown in fig. 22, assume that each wavelength is reflected from a single point of the grating, different wavelengths travel different distances Zeff(~,) and are then bounced back. The time delay is then given by 2Leff(~)n/c, where n is the average refractive index along the grating and c is the velocity of the lightwave. Hence, the fiber grating can produce a continuously varied time delay by changing the wavelength of the optical carrier. Recently, a good deal of attention has been given to the use of BRGs for microwave delay line applications. The concept introduced by Soref [1996] suggested that BRG's could be used to construct a fiber grating prism (FGP) beamformer architecture that has several distinct advantages. Zmuda, Soref, Payson, Johns and Toughlian [1997] experimentally implemented this idea and examined many of the related system-level issues. The system architecture for the FGP beamformer is shown in fig. 23. In the transmit mode, the tunable laser source (TLS) is externally modulated with an electro-optic modulator (EOM). This modulated light is split through an equal-path I:N power splitter and the N split beams are directed to the FGP by optical circulators to feed a group of N single-mode fibers (the receive EOM are not active in the transmit mode). Each fiber includes a spatially distributed array of BRGs, which collectively form the FGP. The different Bragg wavelengths /~1,... ,t~M of the various gratings are within the
316
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.
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Fig. 23. Transmit/receive optical beamforming system using fiber grating prism (9
IEEE).
tuning range of the laser A&. Reflected light is time delayed in accordance with the particular grating addressed and is directed to the circulator again. A second set of equal-length fibers transports the N circulator outputs to a set of wideband photodetectors (PD) located at the array which recover the delayed microwave signals that feed the antenna radiator elements. In the receive mode an unmodulated TLS (the transmit EOM is not active) feeds N optical channels as above. The received microwave signals are directed to a set of N EOMs. The modulated optical signals are then directed to the FGP via the optical circulators. These optical signals are summed together and sent to a PD for recovery of the collective microwave signal. Wavelength shift in the TLS selects a desired outgoing and incoming beam direction. In essence, the FGP receive system acts as a matched filter. Figure 23 also illustrates the spatial layout of the fiber prism's Bragg gratings for an example of N =4. Here, the main lobe of the phased-array antenna can point in any one of five discrete directions (M= 5) as selected by the laser wavelength. The dashed lines show the progressive spatial separation of neighboring gratings, within a fiber, and from fiber to fiber. The major loss mechanisms of the architecture are the couplers (6-dB roundtrip), the 1:N splitter/summer (10 logN dB), the circulator (1 dB per direction). The FGP introduces a loss 1 - R , where R is the reflectance of the selected grating. The reflectance of the BRGs can be greater than 99%, while the M - 1 unselected gratings per channel do not introduce appreciable optical loss, typically less than 0.4 dB. Generally, an EOM biased at quadrature contributes a total insertion loss of approximately 6-8 dB. Splices, connectors, isolators,
4, w4]
OPTICALFIBERTRUE-TIMEDELAYLINES
(a)
IT..~, ~.~L JOE I I~
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e--
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~.--- L/cf------[
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_
317
.
~r
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U
L 0 Pulse Length ~" in units oftime Cf: Speedof light in a fiber
Fig. 24. Multiwavelength serially fed optical time delay beamforming network: (a) timing unit; (b) distribution network (@1996 IEEE). polarization control and the like will add another few dB of excess loss to the system. Since the system is fiber-based, optical amplifiers can be incorporated into the design to compensate for the insertion losses in the system. Each leg of the FGP had a typical noise figure of 20 dB on average over the 3.5 GHz band. The spurious free dynamic range (SFDR) is 42 dB/Hz. Cohen, Chang, Levi, Fetterman and Newberg [ 1996], Chang, Tsap, Fetterman, Cohen, Levi and Newberg [ 1997] and Tsap, Chang, Fetterman, Levi, Cohen and Newberg [1998] proposed a photonic true-time delay transmitter/receiver system suitable for RF phased array antenna. This system uses a serial-feed concept that represents a major simplification in both optical and microwave components compared with conventional parallel systems. The system consists of a single tunable laser, optical modulator, and fiber grating time delay element. The transmit function can be divided into two parts, the timing unit and the tapped optical delay line feed, as shown in fig. 24. The fast and broadband tunable laser can be electronically tuned in less than a nanosecond and its wavelength can change by at least 20nm using proper current adjustment. Light from the laser is amplitude modulated at the desired microwave operating frequency using a high-speed optical modulator and is gated at the radiated pulse width. The modulated light then passes through an optical circulator and enters a fiber with distributed fiber Bragg reflection gratings. The timing information
318
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 4
is obtained via the fiber grating which yields a wavelength-selective propagation delay for each gated pulse. The number of different time delays is determined by the number of available wavelengths. A laser which can be continuously tuned over a 20 nm range and fiber grating reflectors with a 0.3 nm FWHM linewidth (corresponding to a 53 GHz bandwidth) allow access to at least 60 different time delays with time-delay accuracy around 1 ps. Each serially fed optical pulse with unique wavelength has a unique time delay relative to the RF pulse gate timing. The number of pulses in the fully loaded line corresponds to the number of radiating elements in the array (or subarray). Reflected light from the grating returns to the circulator and is directed onto a tapped optical delay line. The tapped delay line consists of a fiber having equally spaced taps with each tap connected to an optical detector. The optical propagation time between these taps corresponds to the length of the longest radiated pulse. From the circulator, each gated pulse will arrive simultaneously at the intended tap on the delay line with the additional delay for beam steering imposed by the fiber grating. Essentially, the delay line "stores" the pulses until it is fully loaded with the correct pulse adjacent to each tap. And then, the microwave signals from the optical detectors, located at each tap in the fiber manifold feed, are switched to the antenna element and all pulses on the line are radiated simultaneously. Each antenna element's microwave pulse has the correct time delay, as set by the fiber Bragg grating reflector, to form a radiating beam in a desired direction. To change the radiating direction, another set of different wavelengths with a different wavelength separation will be chosen. The number of radiating directions is constrained by the number of available laser wavelengths, the tunable range of the laser and associated Bragg grating reflectors. After the signal is radiated, the detector outputs are simultaneously switched away from the antenna elements and the fiber manifold feed can be reloaded. In the receive mode, only the RF phase is required at each antenna element for a given direction of observation; there is no optical or microwave pulse gating. Using the same timing and tapped delay lines as in the transmit mode, local oscillator (LO) signals, in exact reverse phase as used in transmit mode, are supplied to mixers located at each antenna element. The reflected light from the time delay unit goes through the circulator and enters the distribution network which supplies each mixer with the LO signals for mixing with the received microwave signals. The signals, mixed to baseband, from each antenna element are then added coherently in a simple summation signal processor to form the received antenna beam signal. Due to the power splitting along the tapped delay line, each photodetector receives 1/N of the useful optical power, where N is the number of radiating
4, w 4]
OPTICAL FIBER TRUE-TIME DELAY LINES
319
elements. This corresponds to a 10 log N dB of optical power loss for each element. For a practical size (N ~ 100) of subarray or array, this loss can be compensated by a single optical amplifier (+20 dB or more) if necessary. For quasi-CW transmit or receive mode systems, a modification of the basic system permits each optical pulse to be accessed in turn by each element so that there is no 1/N loss. 4.4. OTHER FIBER TRUE-TIME DELAY APPROACHES
In polarization-based systems, the changes in the desired state of polarization (SOP) will lead to unwanted effects such as signal fading, loss in optical heterodyne efficiency in interferometric systems, increase in unwanted noise sources, and overall higher loss in the system. Systems that use fibers must use PM fibers to retain the high degree of the desired SOP. However, any changes in the PM fibers' optical birefringence caused by the change in the environment or even the optical wavelength change can degrade the desired SOP of the travelling light beam. Any major changes in SOP caused by the fiber segments, or for that matter, any other components cannot be tolerated, especially for multichannel delay line application, where many fiber segments with a cascaded architecture are needed. The cost of PM fibers is another important issue. Riza [1995] proposed fiber birefringence compensated N-bit M channel switched fiber delay line as shown in fig. 25 for example, where N = 3 bits and M-- 16 channels. Here 16 linear polarized and collimated beams enter the system and pass through a 4 ) 4 pixelated array of a polarization-mode spatial light modulator (SLM). Each SLM pixel can rotate the input beam linear polarization by 90 ~ when turned on and 0~ when turned off. The SLM combined with the PBS forms a multichannel optical switch where, depending on the SLM setting, TOP VIEW: Fiber Lens
TDU TDU TDU Bit 1 Bit 2 Bit 3 r,,:-,:,.\\\\ ~'-.\~,',",,\\~.~-,,.x~Mirror
Fiber___.~l IT! /LT'I i~-I F~,,,d~yP,ot~tor Laser IN
SLM ~ l l ~ II ~ l l PBS \ ~ ill ~ Ill ~ i l l /
"_2 ~
Photo~or ~
j X/4 plate
16 delayed
~'\\\\\\~,x\\xxx-\\-~ P Mirror
Fig. 25. Fiber birefringence compensated N-bit M channel switched fiber delay line ( 9
SPIE).
320
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 4
beams are directed toward the fiber paths or straight through to the next SLM. For a horizontally polarized input beam with the SLM pixel on, the vertically polarized beam exited from the SLM is deflected 90 ~ by the PBS into a fiber coupling lens that is connected to a single mode non-PM fiber of a certain desired length. The vertically polarized input light into the fiber exits the fiber via another fiber lens. Depending on the fiber birefringence the travelling light has suffered (this depends on the individual fiber and the external conditions), the fiber exited light is no longer vertically polarized and the SOP lies somewhere different on the Poincar6 sphere. This perturbed light passes through a Faraday rotator to strike a mirror that reflects the light back through the Faraday rotator and the birefringent fiber. This beam retracing operation coupled with the Faraday rotator and mirror induces symmetry properties on the Poincar6 sphere rotations, causing the fiber entrance and exit polarization states to be always orthogonal points on the Poincar6 sphere. Thus, if vertically polarized light enters the non-PM fiber, then horizontally polarized light exits the fiber regardless of the birefringence efforts in the fiber and the fiber-coupling optics. This preservation of the linear SOP is critical for operation of the proposed switched delay line. The horizontally polarized light exiting the fiber after the mirror reflection travels through the PBS to be retraced through a quarter-wave plate-mirror arrangement. The light returning from this mirror is again vertically polarized and is reflected by 90 degrees by the PBS to enter the next SLM that forms the next bit in the delay line. Gesell, Turpin and Rubin [ 1995] used fixed length fiber array with BRG fiber cavity of different lengths on each fiber to provide variable delays for different frequency components of the modulated input optical signal. Each spectral component of the signal to be delayed is phase shifted by an amount proportional to the frequency of that spectral component and proportional to the desired time delay. These phase shifted spectrum components are then summed to obtain the desired delay signal. In this approach, a continuously tuned delay can be generated and high dynamic range can be obtained. Figure 26 shows an example of the optical fiber cavity system for generating the time delay of a signal. The light from a single mode CW laser is coupled into a single mode PM fiber and is split into two paths: the signal path and the reference path. Within each of these two paths, the light is further split into a number of channels with one reference channel corresponding to one of the signal path channels. In the signal channelizer leg, the light is modulated by the signal to be delayed in an electro-optic modulator. The modulation of the laser light actually broadens the spectral line of the laser light to the bandwidth of the signal. The modulated light then passes through an optical isolator to
4, w 4]
OPTICAL FIBER TRUE-TIMEDELAYLINES fibers with cavities on substrate
321
optical amplification
optical isolation "
. . . . . . .
U
II1,,-
:,__1
~1111111
!/!i"
r
s(t-~)
9
"
photodiodes optical fiber
amplitude, and I phase shift control
'~ EO amplitude* and phase modulators
* Implementationof amplitudecontrol is dependent on the intended application. Fig. 26. Fiber optical cavity time delay line (@1995 SPIE).
eliminate optical signals reflecting back from the channelizer portion of the signal channelizer leg. After passing through the optical isolator, the light enters the channelizer portion of the signal channelizer leg and is further split into a number of beams, each entering a fiber with a resonant cavity of a different length. The mirrors defining the ends of these cavities are BRGs. BRGs with reflectance as high as 99.9% can be written into the fiber. Only light that is in the pass band of each of the cavities will enter the cavity and pass on through. The other portion of the light will be reflected back by the first Bragg grating mirror. A multiport optical circulator is used to sequentially pipe the reflected light from one cavity into the next to avoid the optical loss resulted from a parallel distribution of the light into the fiber cavities. As indicated in the figure, the length of each of the cavities is different so that each cavity passes a different spectral component of the signal. The spectral component out of each of the resonators is coupled to a photodiode reference for heterodyne detection with
322
TRUE-TIMEDELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS [4, w 4 optical isolation
s •L)
IImoclmatorl F9." [ - -]l ~ "-~ I
I
k,,,
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k~
w[ control2~
~,.
"
control3~
~-
~ control n ~
' Fibe'rChannelizer- ~ Optical Amp . H. EO phase shifters 1 ~-
~
EO phase shifters 2 ~-
/ L
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_ I
[
I
EO phase shiflers n ]
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,
~ l ) -
~ -- ~ - -
IX,._| ~ | 9
@ 9
~ , w k9
(~-~t_~N)
)
S(t_tl:3) 9
photodiodes Fig. 27. Fiber optical cavity architecture for transmit array (9 1995 SPIE).
a phase-shifted reference. In the reference path, the light is split into the same number of channels as that in the signal path. The optical reference signal in each of these channels is amplitude controlled and phase shifted in the electro-optic modulator by the relative amounts that are required to obtain the desired signal time delay out of the device. Channel-dependent amplitude control can be used to null strong narrowband interference when used in the receive mode. These phase-shifted references are used in the process of heterodyne detection to phase shift the spectral components of the signal being delayed. Heterodyne detection between each of the signal path channels and a corresponding reference path channel takes place on a wideband photodiode. Use of electro-optic modulators allows for precise and rapid change of time delays for rapid beam steering. Beams can be switched in tens of nanoseconds or faster. Suitable technologies for fabrication of these devices include LiNbO3 and electro-optic polymers. With the ability to finely adjust the phases of reference signals over a continuum, a continuum of time delays can be obtained. Figure 27 shows the architecture for simultaneously generating N time delayed signals for steering a linear RF array with N elements or subarrays. To simultaneously generate the different time delays, each frequency component of the spectrum of the signal to be delayed, as obtained from the fiber optic channelizer, is phase shifted by a different relative amount. For the ideal case of delaying a signal without introducing frequency dependent distortion, the phase shift module can be constructed such that one electrical control signal will maintain the proper linear relationship between the phases. Spectrally dependent adaptive beamforming requires independent control of the relative phase and amplitude of each spectral channel, i.e., two control signals for each channel.
4, w 4]
OPTICAL FIBER TRUE-TIME DELAY LINES
323
p,sl(t) [_Delaymodule ~ Control1 ..
Is2(t)
E Delaym~ Control2--
{
~ ~ - ~ "
s(t)
s(t-
' !Sn(t)
l Delaymodule ] Controln
s(t'~2)
~is3(t)
|Delay module~ .. Control 3-
s(t-'~l).
s(t-%)
~l,
Fig. 28. Fiber optical cavity receiving array architecture (9 1995 SPIE).
I H
fiber
Fig. 29. Fiber optic time delay device with PZT stretcher (9
IEEE).
Figure 28 shows the architecture to implement true-time delay beamforming for a receiving array. In this figure each delay module block corresponds to the signal delay architecture that was illustrated in fig. 26. The signal, sk(t) from the kth array element or subarray modulates the light in the kth delay module. The sum of the delayed signals corresponds to the signal arriving at the RF array from the desired direction. Herczfeld, Daryoush, Kieli, Siegel and Soref [1987] proposed and demonstrated two other methods for achieving time delay using optical fiber. One method uses the physical stretching of the fiber. It is accomplished by wrapping the fiber around a piezoelectric (PZT) ring, as shown in fig. 29, which expands upon the application of an electric field and thus stretches the fiber. The time
324
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 5
i rnicrobending fore pig-tailed laser J-..aL.t ' ---1"
mecl~ splice I ' 9 ., , ....
-
i~ ~ s t e p - i n d e x fiber
r
pig-tailed PIN
.
mech. splice 11
~-:-~._,
."7-
J /
,'
,
conne~or
Fig. 30. Mode-switching-inducedtime delay (@1987 IEEE). delay is given by r =(AL) n/c=nLs/c, where L is the fiber length and s is the piezoelectric compliance ratio. Another approach they proposed is to use different group velocities of guided modes of different orders to obtain different delays. As we know, in a step index multimode fiber, the different modes travel with different velocities. Lower-order modes take a shorter time to transit over the fiber optical link than higher-order modes. If a narrow group of lower order modes are launched in a multimode fiber and due to external perturbation, like microbending force, they are redistributed to higher order modes, then a time delay is observed. The proposed device architecture is shown in fig. 30.
w 5. Optical Waveguide-Based True-Time Delay Lines The working frequency of phased array radar can range from UHF to the X band for ground-based or shipboard antennas, and up to a millimeter wave for airborne applications. With the increase in operation frequency and angular scan resolution, the delay length accuracy down to micrometers is required, while the largest delay length can still be tens of centimeters for even a moderate number of array elements. Conventional fiber-delay lines are trimmed mechanically to pre-designed lengths that are determined by the aperture size and steering angle of the phased array. It is difficult for the optical fiber switch architecture to meet such high demands on the delay resolutions. In addition, the optical fiber/switch interfaces in such a system contribute to the system optical insertion loss and final module cost. Because most of the fast electro-optic switches are polarization sensitive while standard communication fibers are not, maintaining the polarization can increase the technical difficulty and the cost. Integrated waveguide-based delay line technology offers several advantages over its fiber-based counterparts. First, the waveguides are typically fabricated
4, w 5]
OPTICALWAVEGUIDE-BASEDTRUE-TIMEDELAYLINES
325
Table 1 Waveguide material properties Material
Refractive index (n)
Attenuation (dB/cm)
Stability (dn/dt)
GaAs/A1GaAs (@1550nm)
3.45
0.15 (straight); ~1 (curve)
5x10 -4
LiNbO3 (@632 nm)
2.20
0.2-0.5
Ge-doped silica
~ 1.5
~0.1 (curve)
10-5
BK-7 Glass (@632 nm)
1.515
0.019
5•
Polymers (@632 nm)
1.5-1.6
<0.1
10-4
(@1550nm) -6
by photolithographic processes with dimensional precision of micrometers. This enables better differential delay accuracy, especially for the fine bits of the timeshifter. Second, the time-delay module can be packaged more compactly if the waveguide wafer board can be adopted as a monolithic or hybrid integration platform for the active components. A more compact, lightweight unit that is more suitable for airborne or space borne applications can be made. Third, the possible of integration of the detectors to the optical waveguides eliminates a delicate fiber-device interface between the delay lines and optoelectronic switches that control the delay times. The all-integrated photonic delay line modules on a single wafer can greatly improve the optical insertion loss performance since waveguide-fiber interfaces can be totally avoided. Finally, since guided lightwave maintains the same polarization, all the waveguide devices including modulators and switches can be designed for any convenient polarization without the need of polarization-maintaining fiber pigtail alignment. There are several waveguide materials employed in true-time delay control of phased array antenna, like GaAs, LiNbO3, BK-7 glass or silica, and polymers. Their properties are listed in table 1 (Hartman, Corey, Kenan and Belcher [1991]). The ideal material should exhibit a large refractive index, low optical attenuation, and a small temperature stability coefficient of the refractive index. However, there is some tradeoff between the refractive index and the attenuation of these materials as shown in table 1. GaAs exhibits a large refractive index, but it has a large attenuation and a large dn/dt. Glass, on the other hand, exhibits a significantly lower attenuation and temperature dependence of refractive index and, unfortunately, a much lower refractive index. The low-cost silica technology also requires thermal optic switches that are slow and may consume more
326
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 5
electrical power. Furthermore, the use of glass or silica materials implies a hybrid structure since electronic components such as detectors cannot be monolithically integrated, although the development in VLSI fabrication technology makes the hybrid device feasible with small compact units. LiNbO3 waveguide is usually used as the switching or modulation elements. Electro-optical polymer materials can be used as the waveguide as well as the active switching and modulating elements while easing the fabrication process, therefore they are attracting more and more attention.
5.1. GaAsBASEDOPTICAL WAVEGUIDETRUE-TIMEDELAY GaAs waveguide true-time delay lines were demonstrated by several research groups. The concept proposed by Sullivan, Mukherjee, Hibbs-Brenner, Gopinath, Kalweit, Marta, Goldberg and Waiterson [ 1992] is shown in fig. 31. It consists of cascaded pairs of low-loss optical waveguide delay lines and 2 x2 EO waveguide switches. The fundamental building block in this approach is the switched time-delay element (STDE), a 1-bit time-delay generator consisting of two EO switches connected by two delay lines of different lengths. One of the delay lines is the reference delay path, which has the same length for all bits. The nonreference delay path provides a relative time shift ranging in integral multiples of 2 -1 from the most significant bit (MSB) to the least significant bit (LSB). The fiber delay line can be integrated with the waveguide delay line to provide long delay.
Fig. 31. Optical time delay unit using GaAs photonic integrated circuit (9 1992 SPIE).
4, w 5]
OPTICAL WAVEGUIDE-BASEDTRUE-TIMEDELAYLINES
Fig. 32. GaAs optical waveguide design (9
327
SPIE).
The waveguide material was grown by organometallic vapor-phase epitaxy at atmosphere pressure using the double heterostructure design as shown in fig. 32: a 1 ~tm-thick A10.04Ga0.96As upper clad, a 2 ~tm-thick GaAs core, and a 4 g m thick A10.04Ga0.96As lower clad. All layers were unintentionally doped and typically had a net carrier concentration less than 1 x 1015 p type. Very strong lateral confinement was obtained by deeply etching through the waveguide core using a Cl-based plasma process. The single-mode attenuation for a straight waveguide using this design and fabrication approach was typically about 0.6 dB/cm at 1.3 ~tm. The best-case optical loss achieved for 180~ curved waveguide with ROC ~1 mm was 1.5 dB and the waveguide corner bend loss was 0.7-0.9 dB at 1.3 ~tm. The EO switches used were two-mode interference (TMI) devices (i.e., zerogap directional couplers) because of the considerable fabrication simplicity and potential benefit of reduced voltage-length product compared with directional couplers. The total switch loss was calculated to be about 1 dB due to mode mismatch at the switch input and output channels and the losses along the switch. The voltage-length product was estimated as 7 v.cm at 1.3 ~tm. The measured mean total insertion loss was around 15 dB for the 2-bit time-delay generator, which included unoptimized losses of over 3dB per TMI switch. Optimization of the 2 x2 switch design and fabrication technologies was expected to yield a 7-bit time-delay unit with total losses of approximately 20 dB. Utilization of fiber-delay lines, in conjunction with GaAs integrated delay lines, will provide very long delays with losses dominated mostly by the fiber-
328
TRUE-TIMEDELAYCONTROL SYSTEMS FOR WIDEBANDPHASED ARRAYANTENNAS
RF MODULATED j OPTICAL --INPUT (Z = 1.3 l~m)
WAVEGUIDE-COUPLED
[4, w 5
M S M D E T E C T O R ARRAY
Lo
I1
1 x 4 SPLITTE GaAIAs/GaAs RIB W A V E G U I D E L 1 = Lo
!
L2 = L o + AL L 3 = L o + 2zSL L4 = L o + 3~L
Fig. 33. Layout for monolithic 2-bit time-delay network integrating GaAs waveguide and MSM detectors (9 1994 IEEE).
chip interface. Again, the losses can be overcome with low added noise by using rare-earth doped fiber amplifiers. Ng, Yap, Narayanan, Hayes and Walston [1994] and Ng, Yap, Narayanan and Walston [1994] demonstrated monolithic waveguide TTD integrated with MSM detectors on GaAs substrate. Figure 33 shows the layout of such a 2-bit time-delay network integrating rib-waveguides and waveguide-coupled MSM detectors on GaAs substrates. The network consists of four delay lines whose lengths were designed to be L0, L0+AL, L0 +2AL, L0 + 3AL. After going through the 1 x4 splitter, which consists of two cascaded 1 x2 Y-branches, the RF modulated optical signal is delayed by the propagation time to, to + r, to + 2 r, to + 3 r in the four rib-waveguides. The detector array coupled to the waveguides serves as an optoelectronic demodulator/switch that selects one of the four RF delay signals for the radiating element it addresses. For example, by turning on the bias of the detector integrated to the waveguide with length L0 + 2AL, we "switch" out the RF signal delayed by to + 2r. Because the photocurrents of the MSM detectors are always clamped to a null at zero bias, on/off ratios of better than 40 dB for the RF output from the detector array can be easily achieved. The monolithic time-delay network was fabricated from epitaxial layers grown by MOVPE in a single step. The overall loss of these S-bends, including contributions from the propagation loss, bending loss and transition loss between the curved and straight segments in the S-bend, was estimated to be ~1 dB/cm for S-bends with radii of curvature greater than 2 mm. The step size of 0.338 cm provided a delay step r of ~40ps.
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OPTICAL WAVEGUIDE-BASED TRUE-TIME DELAY LINES
329
5.2. SILICA BASED OPTICAL WAVEGUIDETRUE-TIME DELAY Because silica waveguides typically demonstrate a propagation loss of less than 0.1 dB/cm, they are more attractive for steering phased array antennas with large aperture sizes and wide scan angles. Figure 34 shows the 4-bit silica-waveguide time-shift network demonstrated by Ng, Loo, Jones, Lewis, Livingston and Lee [1995]. The delay times in the time-shift network are varied by means of optoelectronic switching. The delay path of the RF-modulated optical carrier is defined by selectively turning on a laser and detector located at the beginning and the end of each analog option, from L0=L] +L5 to L = L 0 + 15AL (i.e., four bits of resolution). The time-delay networks were designed to control the subarrays in a 4 • element conformal array whose aperture size is ~1 m • The radiating elements of the conformal array were arranged to lie on an arc with a ~3.05 m radius. Inside the conformal array, eight photonic time-shifters were used to steer eight subarrays, each consisting of 4 • radiating elements. The large aperture size and wide scan-angle requirement (of +60 ~) for this particular array imposed a step-size r of ~0.25 ns for the quantized delay times. Using a time delay of ~50 psec/cm for silica-waveguide, the estimated AL is 5 cm in fig. 34. As shown in fig. 34, the delay lines were integrated on two silica-waveguide chips (I and II) that were coupled to the input and output ends of a 4 • 4 silica star-coupler (~2 cm long). The dimensions of chips I and II were 6 cm• 7 cm and 5 cm• 6 cm, respectively. By using a "loop-type" geometry with a bend radius of curvature larger than 6.7 mm, delay lines with lengths of 7.2 cm, 27.2 cm, 47.2 cm and 67.2 cm were integrated on chip I. Similarly, waveguide delay lines with lengths of 6 cm, 11 cm, 16 cm and 21 cm were integrated on chip II. Four lasers were coupled to chip II. When one of the four lasers was biased to LASER BIAS SWITCH
.r__j
DIODE / SILICA SILICA ~' / WAVEGUIDE WAVEGUIDE T'I" / c~p. . c,,,J -J.-L..~ r--- .... "~ L I + ~ . \ r ....... "~ F~"--'~_ , ,.s , _ _ -~ L, ! r-=~= r,~puR.... / I .T-v"J"-T"------'~ L1 § r~ x 9 ~T .." ~ :,', I~L.__. ', L s * ~ ,~ "~'! "~n 'i Ii /
INPUT- --'[ L,
LASER
~.....
: "~ :,-s
: L..~2AL '
+~"
.
~\ ~ :
~
U
.
I
'
4x4 INTEGRATED OPTIC STAR-COUPLER
~ i ,,-,,
; : , '. . . .
I1
',m
/
t rm: J
DETECTOR ARRAY & BIAS CONTROL CIRCUIT
(~L = 5 I:m FOR A R P A / R O M E C O N F O R M A L A R R A Y )
Fig. 34. Schematic of 4-bit silica-waveguideoptical time delay network (9
SPIE).
330
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 5
approximately three times above threshold, the RF input to the network was modulated onto its optical output. By biasing up the appropriate laser, light was selectively coupled into one of the four shorter delay lines on chip II (located at the input side of the star-coupler): Ls, L6 (= L5 + AL), L7 (= L5 + 2AL), L8 (=L5 + 3AL). The optical cartier was split into four by the star-coupler and then coupled into delay lines L1, L2 (=L1 +4AL), L3 (=L1 +8AL) and L4 (=LI + 12AL). One of these four optical branches was selectively detected by "switching" on the photodiode pigtailed to the appropriate delay line. In practice, this was accomplished by turning off the voltage applied to the gate of a GaAs MESFET ("normally on") connected in series with the photodiode. Therefore, laser switching and detector switching controlled the two finer bits and the two higher bits of the 4-bit time delays, respectively. In the silica-based approach, the waveguides and detectors are no longer integrated on the same substrate. This requires the MSM detector array to be fabricated from an epitaxial system that would minimize the detector dark current. Here, the MSM detectors were fabricated from a wafer of In0.s3Ga0.n7As (~1 ~tm thick) grown on InP. A layer of In0.52A10.a7As (~300 A thick) was used to enhance the Schottky barrier heights of the finger electrodes. The optical insertion losses between the input and output ports consisted of a fan-out loss of ~8 dB (at the 4 x4 star-coupler), interface coupling losses (of 0.5 dB/interface) between silica chips, and propagation loss in the silica waveguides, which was measured to be less than 0.1 dB/cm. Paquet, Chenard, Jakubczyk, Belanger, Tetu and Belisle [1995] proposed a different switching scheme for high-silica (SiO2/Si) waveguide TTD lines. Thermo-optic interferometric switches are used for selecting between a short reference path and a longer delayed path. Time-delay values are generated by the path difference between the reference and delayed path. Six switches are used for a binary cascaded 5-bit device, the last one being used as a combiner. Therefore, 32 different delay times can be selected with proper switching configurations. The device was fabricated on SiO2/Si. The total insertion loss of the device was measured to be 20dB. The measured extinction ratio of 13 dB can be improved to 20 dB by better control of key fabrication parameters, which have been determined through modeling and simulation. While the compactness of the integrated optic device is attractive, the power requirement in a thermo-optic switch, 500 mW, is high for some applications. For systems that require a large number of delays, the large insertion losses associated with the optical switching network that selects the appropriate delay line are a serious limitation. Yegnanarayanan, Trinh and Jalali [1996] proposed and demonstrated a photonic filter that uses the optical carrier wavelength to select the desired time delay.
4, w 5]
OPTICALWAVEGUIDE-BASEDTRUE-TIMEDELAYLINES
331
~j
c~upl~S 1 .4
/
(b)
Fig. 35. (a) Recirculating photonic filter wavelength selective TTD (@1996 OSA); (b) schematic of arrayed-waveguide grating demultiplexer (9 OSA).
Figure 35a shows the proposed integrated waveguide implementation of the wavelength-selective time-delay function. The concept uses an arrayed waveguide grating in a symmetric feedback (recirculating) configuration. The modulated optical cartier is steered by the waveguide grating to the appropriate integrated delay line, depending on the cartier wavelength. The signal is delayed and is then fed back into the symmetric input port. The grating focuses the delayed beam into the common output port. This TTD can be considered topologically equivalent to two phased arrayed-waveguide gratings (AWGs) in series with a delay element in between as shown in fig. 35b. Three sequential operations are performed by the single device: wavelength demultiplexing, time delay, and wavelength multiplexing. The advantages of the recirculating structure compared with the two devices in a series are twofold: the structure occupies roughly one-third of a chip area and, more importantly, this configuration ensures that the filters that perform the multiplexing and demultiplexing have identical spectral responses (the same filter does both). Because of inevitable processrelated variations, any two devices will have slight variations in their spectral response, rendering the series-connected approach inoperative. The recirculating photonic filter has no 1/N loss. All the power at a given wavelength is diffracted into the output port. Furthermore, high resolution (6-8 bits) can be obtained in a compact integrated device. A prototype recirculating photonic filter TTD device was realized by use of an eight-channel arrayed-waveguide grating demultiplexer with 0.8 nm channel spacing (free spectral range 6.4 nm). The device was based on silica waveguide technology and operated in the 1.5 gm wavelength range. Light from an external cavity tunable laser was directly RF modulated at 10-40 MHz and was coupled into the arrayed-waveguide grating chip. The output was detected by an InGaAs
332
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 5
Table 2 Key design parameters for silica- and GaAs-based integrated time-delaynetworks Properties
Silica
GaAs
Substrate
8" ( 12" max )
6"
Waveguide material
Ge-doped silica (buried)
GaA1As/GaAs(Rib)
Loss (dB/cm) (for curved guides)
~0.1
~1
Refractive index (n)
~1.5
~3.45
Loss per ns of delay (dB)
~2
-8.7
Minimum bend radius (mm)
--5 (An/n ,~ 0.65%)
-3 (An/n ,~ 2%)
Fiber coupling loss (dB)
~-0.5
~3
Device integration
No (hybrid platform)
Detector switch
p - i - n photodiode. The total optical insertion loss for each wavelength channel was ~6 dB. The passband spectrum revealed slightly nonuniform insertion loss and inter-channel crosstalk. The crosstalk for the arrayed-waveguide grating filters used was - 2 6 dB; however, values below - 3 0 dB have been reported. Ng, Yap, Narayanan, Liu and Hayes [1994] compared the Silica based TTD and GaAs based TTD in detail. Table 2 lists the key design parameters for silicaand GaAs-based integrated time-delay networks. The silica waveguides are usually composed of a buffed Ge-doped SiO2 embedded inside SiO2. These core and cladding layers are typically deposited by flame hydrolysis on Si substrates approximately four inches in diameter. On the other hand, the waveguides for the GaAs-based technology are of the rib-type. The epitaxial layers from which these rib-waveguides are fabricated are grown by MOVPE on GaAs substrates that are typically two inches in diameter. Partially because they are buffed waveguides, the propagation loss of straight silica waveguides is as low as 0.04dB/cm. This figure is to be compared with typical losses of 0.5 dB/cm for straight GaA1As/GaAs rib-waveguides. Because the refractive index difference between cladding and core ( A n / n ~ 0.65%) is smaller for the silica waveguides than for their GaAs counterparts ( A n / n ,~ 2%), the minimum bend radius of silica guides is o n l y - 5 mm. The fact that silica waveguides cannot be "bent as tightly" as GaAs waveguides is compensated by the availability of larger Si-substrates. The typical losses of silica (high An type) and GaAs curved waveguides with radii of curvature approaching their bend minimums are, respectively, -0.1 dB/cm a n d - 1 dB/cm. However, a more meaningful figure of merit for comparing their losses (in the context of a time-delay network) is the loss incurred for generating
4, w 5]
OPTICAL WAVEGUIDE-BASEDTRUE-TIME DELAYLINES
333
unit time delay. Because GaAs has a higher refractive index of 3.45 (vs n = 1.5 for silica), it can generate more time delay (At= nL/c) per unit length (L) than silica. Taking into account the loss per unit length and the time-delay per unit length, we obtain a propagation loss of N2 dB/nsec for silica waveguides vs. ~9 dB/nsec for GaAs rib-waveguides. Finally, silica waveguides demonstrate a lower coupling loss (of only 0.45 dB, i.e., ~10%) to single mode fibers than GaAs waveguides, because the core/cladding of the single-mode silica waveguides is very similar in refractive indices and physical dimensions to optical fibers. This, in turn, reduces the mode-mismatch between the optical fiber and silica waveguide. However, a GaAs-based technology allows the detector switches to be integrated monolithically with the waveguide delay lines. This not only enables the time-delay network to be packaged more compactly, but also eliminates an extra coupling interface between the delay lines and optoelectronic switches. Finally, the optical input to a GaAs integrated time-shift network is distributed to its delay branches by cascaded Y-branch (1 xN) splatters. In contrast, N x N starcouplers based on a Fourier optics design are used in a silica-based network to distribute the optical inputs. The latter design would enable the implementation of higher bits of resolution without incurring excess splitting loss.
5.3. POLYMER-BASED OPTICAL WAVEGUIDE TRUE-TIME DELAY
Polymer NLO materials have unique features that are particularly attractive for the photonic on-wafer delay line applications. They can be easily spin-cast on a large piece of wafer, say a 6" silicon wafer. Long waveguide delay lines can be achieved on a single wafer, and these delay lines can be integrated with waveguide EO switches and broadband EO modulators, all made of NLO polymers. The high resolution of the delay lines can be guaranteed by the advanced microelectronics fabrication technology. By carefully designing the waveguide layout and using waveguide crossing, large delays over tens of centimeters can be obtained. Fast switching to nanoseconds can be achieved with low electrical power consumption with a lumped circuit structure. Using epitaxial lift-off technology, III-V active devices, such as semiconductor lasers and photodetectors, can be integrated on the delay line unit based on lowcost Si wafers. Furthermore, combined with advanced microwave monolithic integrated circuit (MMIC) technology, a highly functioning chip that combines supporting electronics circuitry and optical delay lines is achievable. With the advances in material synthesis and device fabrication, polymer on-wafer photonic delay line chips are increasingly realistic. It provides a unique opportunity to
334
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
~z--'n~2~021cS
~
~
[4, w 5
"~ V-groove At 1----nAR]011c
V2
$2 V V-groove
pc
M-Z modulat~ m
RF in
SO I Vo --
~
,%--'n,~R0%/c
RF out
Fig. 36. Three-bit integrated polymer waveguide delay line integrating with EO modulator and control circuit (9 SPIE). develop a unit that is highly integrated, easy to control, broadband, low optical insertion loss at a low cost. A delay line based on fiber optics, polymer waveguide and broadband EO modulators and switches proposed by Wang, Shi, Lin and Bechtel [ 1996] is shown in fig. 36 to illustrate the concept. The modulator was a Mach-Zehnder type modulator fabricated using PUR-DR19 polymer. Polymer EO modulators with bandwidths as high as 60 GHz have been demonstrated. Two waveguide routing methods can be used to form a delay unit. For small delay differences, the simplest way is to branch the guided lightwave to different waveguides of desired lengths using 2 • 2 switches as shown in fig. 36. The directional coupler type switch was designed and fabricated using a thermal crosslinked electrooptic polymer, PUR-DR19. The calculated switching voltage was ~33 V. The path length difference can be adjusted by different radii of the curvature. In an alternating routing scheme, one output of a switch is connected to the input of the same waveguide in the same switch through a waveguide loop. Using the other waveguide for the unit input and output, a switch cross-state will provide delay while a bar-state gives no delay. This configuration is more suitable for longer delays because only one delay waveguide is required. In both cases, a built-in constant phase delay exists which can be compensated for by all the elements in the final assembling stage. For delays over tens of centimeters long, waveguide crosses may be needed for the two-dimensional structure. The total system optical insertion loss will be determined by switch loss, waveguide bend and cross loss, and the input/output loss at the fiber pigtails. The loss of the PUR-DR19 polymer waveguide including bending loss is 0.5 dB/cm. The total optical insertion loss was estimated to be 20 dB for an 11-bit device feeding 64 elements phased array antenna working at 60 GHz with 30 ~ scan angle and 1o scan resolution. The switching speed is limited by the driving circuit to ~ 1 ns.
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OPTICAL WAVEGUIDE-BASED TRUE-TIME DELAY LINES
335
Fig. 37. Schematic diagram of twin-beam optical TTD line based on photonic polymerwaveguide: (a) top view; (b) side view. A new type of optical true-time delay lines based on photonic polymeric waveguide circuits in conjunction with electrically switched high-speed photodetectors shown in fig. 37 was proposed and demonstrated by Tang, Wu, Fu, An, Han and Chen [1999]. The device is capable of providing true-time delays from 1 ps to 50ns for wideband multiple communication links in a compact miniaturized scheme. The system uses (i) an innovative ultralong photonic polymeric channel waveguide in conjunction with (ii) an array of surface-normal fanout gratings, (iii) two laser diodes for generating optical RF cartier based on optical heterodyne technique, and (iv) an inexpensive photodetector array. This system eliminates the need for fast wavelength tunable laser diodes, long bulky bundles of fibers and/or expensive optical 2 x2 waveguide switches. Unlike any conventional approach where one TTD line can provide only one delay signal at a time, the proposed true-time delay module is capable of generating all required optical true-time delay signals simultaneously to all antenna elements. As a result, a large number of true-time delay combinations can be provided for the phased array antenna simultaneously by electronic switching the photodetector array fabricated under the polymeric waveguide as shown in the figure. Such a monolithic integration not only reduces the cost associated with optoelectronic packaging, but also reduces the system payload with an improved reliability. A significant reduction of cost, weight and power consumption is expected. Unlike expensive electro-optic switches and wavelength tunable laser diodes, high performance photodetectors are inexpensive and can be cost-effectively
336
TRUE-TIMEDELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 5
Fig. 38. Electrical switching diagram of a detector-switched optical TTD line.
fabricated into a large array based on the technologies originally developed for optical imaging and fiber-optic communications. High-speed PIN photodetectors have a typical bias voltage of 3-10 volts with a bandwidth up to 100 GHz. The electric diagram of the detector-switched optical true-time delay module is shown in fig. 38. Such a hybrid integration of detectors with the optical waveguides eliminates the most difficult optoelectronic packaging problem associated with the delicate fiber-detector interface and/or fiber-switch interface. It not only reduces the cost associated with optoelectronic packaging, but also reduces the system payload with an improved reliability. A significant reduction of cost, weight and power consumption is expected. The unique optical amplification feature of photonic polymers allows us to fabricate a large number of fanout gratings over an ultralong optical channel waveguide. The optical propagation loss and fanout loss is compensated by the optical gain throughout the waveguide delay line. As a result, a large number of time delays can be obtained by using a single laser diode for advanced photonic radar systems that often have 103 to 105 antenna elements. The optical gain is provided within the photonic polymeric waveguide doped with rare-earth ions and pumped by a third laser (~.3) from another end of the waveguide. In order to obtain uniform fanouts, the optical gain in the waveguide section between two fanout gratings has to be engineered to exactly compensate the sum of the waveguide propagation loss and optical fanout loss. The delay at each detector is equal to the time of flight along the waveguide circuit to the selected waveguide grating coupler. Because the length of waveguides is defined by photolithography, the proposed TTD module can provide a 0.1 ps true-time delay resolution over a 50 ns range. The thin-film nature of polymers allows us to fabricate the proposed TTD module (made out of waveguide circuits and waveguide gratings) on any
4,, w 6]
SUBSTRATE-GUIDED WAVETRUE-TIME DELAY MODULES
337
substrate of interest, using standard VLSI technologies originally developed for the microelectronics industries.
w 6. Substrate-Guided Wave True-Time Delay Modules Fiber-delay lines are not as compact as waveguide-delay lines and are not precise enough for very high frequency. Waveguide-delay lines are compact and precise, but they suffer from some disadvantages like high waveguide loss, complexity in fabrication, and high cost, especially the high-speed switches and modulators. Substrate-guided wave true-time delay lines proposed and demonstrated by Chen and Li [1996] and Li and Chen [1997] take advantage of both the fiber-delay lines and the waveguide-delay lines and don't have some of their disadvantages. The delay lines are provided by both optical fibers (for longer delay) and substrate-guided wave delay lines (for precise short delay). One single-delay module can simultaneously generate all the required delay signals for the whole antenna array. Therefore the device is very hardware-compressive. The fiber and the substrate-guided wave delay lines can also serve as dispersion elements to provide continuously tuned delay signals by changing the optical carrier frequency and therefore can control the antenna array for continuous scanning instead of discrete scanning. The device is easy to fabricate, able to achieve different delay ranges and different delay steps, and is very cost effective. 6.1. SINGLE WAVELENGTH SUBSTRATE-GUIDED WAVE TRUE-TIME DELAY
Figure 39 illustrates the basic system architecture of this 2-D substrate-guided wave optical true-time delay line. A 1 xN fiber beam splitter with pre-determined output fiber lengths is used to provide N delay signals, each with an Mr delay increment. Each of the N delay signals from the 1 xN fiber beam splitter is coupled into the substrate surface - normally with a specific substrate bouncing angle through a volume holographic grating coupler and then zigzagged within the substrate through total internal reflection (TIR) with ~100% reflection efficiency. Portions of the substrate-guided waves are sequentially extracted surface- normally through a holographic grating output coupler array. Figure 40 illustrates the structure of one of the N TTD sub-units with delay paths provided by cascaded substrate-guided optical fanouts. The input holographic grating coupler is designed to couple the surface normal incoming light into a substrateguided mode having a fixed bouncing angle of greater than the total internal reflection, say 45 ~. The output holographic grating couplers extract an array of
338
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 6
Fig. 39. Structure of substrate-guided wave true-time delay line.
Fig. 40. One-dimensional optical delay line based on substrate guided mode and holographic grating coupler. substrate-guided beams into a free space 1-D array having M surface normal fanout beams. Different optical delays are obtained at subsequent fanouts due to the extra bouncing distance light propagated within the substrate. The time delay between every two bouncings is designed to r, i.e., the delay step is r. Thus, N x M delay lines are achieved. The fanout optical delay signals can either be detected by high-speed photodetector array and then sent to antenna transmitters by programmed switching or they can be switched in optical domain and then sent to the detectors behind each antenna element or subarray using optical fibers. Fiber optical amplifiers can be put in these fibers or the 1 x N fiber splitter or both to compensate for the optical insertion loss of 1 x N splitter and 1 x M even fanouts. Based on this architecture, up to 7-bit TTDs have been demonstrated (Fu, Li and Chen [ 1998a,b], Fu and Chen[ 1998b]). The delay steps were easily changed by changing either the thickness of the substrate or the bouncing angle
4, w 6]
SUBSTRATE-GUIDED WAVE TRUE-TIME DELAY MODULES
339
Ele or (
I!1 ... III 0
Phase ~hifler
D
Delay Signal
~o
,-,
Inddent wavefront
~
Wavefront ~lt of subarray
Fig. 41. Single substrate-guidedwave TTD unit controls one antenna array. inside the substrate, and delay steps from 15 to 100 ps were demonstrated (Li, Fu and Chen [1997], Fu, Li and Chen [1998a], Fu and Chen [1998b]). Since all the delay signals are achieved simultaneously, each of such true-time delay modules can be used not only to provide true-time-delay signals to one antenna element but also to provide delay signals to several antenna elements simultaneously for multibeam steering (Fu and Chen[ 1998a]). For a 7-bit TTD unit, an example is shown in fig. 41, where a 7-bit module is used to control 9 antenna subarrays (one of these 9 subarrays is provided with zero delay signal). Each subarray has several antenna elements, which share the same time delay signals. A phase-shifter is behind each antenna element to fine-tune the phase delay and therefore to scan with a small angle increment. The possible scanning angles and the corresponding delay signals needed by each subarray are shown in table 3, where r is the minimum delay needed. As shown in fig. 41 and table 3, to stare in one direction, say-45 ~ the microwave switches behind the subarrays connect the desired delay signals of 0, 16r, 32r, 48r, 64r, 80r, 96r, 112r and 128r to subarray 1, 2, 3 4, 5, 6, 7, 8 and 9, respectively. To stare into another direction, say +32 ~ the switches controlled by the control signals send delay signals of 0, 12r, 24r, 36r, 48r, 60r, 72r, 84r and 96r to subarray 9, 8, 7, 6, 5, 4, 3, 2, 1 and 0, respectively. In such an arrangement, the antenna array can be easily controlled by one true-time delay module and electronic or optical switches. Therefore, complexity of the system hardware is dramatically reduced. The input to the 1 • fiber beam splitter can be microwave-modulated optical signals or the optical heterodyning of two tunable lasers with slightly
340
TRUE-TIMEDELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 6
Table 3 Configuration of the delay signals for 7-bit TTD devices to control 9 subarrays (0-45 ~ Scanning
Delay step between subarrays (r)
1
2
3
4
5
6
7
8
9
-45
0
16
32
48
64
80
96
112
128
-41.5
0
15
30
45
60
75
90
105
120
15
-38.2
0
14
28
42
56
70
84
98
112
14
-35.1
0
13
26
39
52
65
78
91
104
13
-32
0
12
24
36
48
60
72
84
96
12
-29.1
0
11
22
33
44
55
66
77
88
11
-26.2
0
10
20
30
40
50
60
70
80
10
angle (o)
16
-23.4
0
9
18
27
36
45
54
63
72
9
-20.7
0
8
16
24
32
40
48
56
64
8
-18.0
0
7
14
21
28
35
42
49
56
7
-15.4
0
6
12
18
24
30
36
42
48
6
-12.8
0
5
10
15
20
25
30
35
40
5
-10.2
0
4
8
12
16
20
24
28
32
4
-7.6
0
3
6
9
12
15
18
21
24
3
-5.1
0
2
4
6
8
10
12
14
16
2
-2.5
0
1
2
3
4
5
6
7
8
1
0
0
0
0
0
0
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0
different wavelengths. Up to 50GHz heterodyned microwave signals were generated and detected by the two tunable lasers working at 1550nm. The volume holographic coupling gratings were created by two-beam interference recording method on photopolymer film, which can be easily dry-processed and has very good stability. Diffraction efficiencies as high a s - 9 9 % have been achieved. The substrate used can be BK-7 glass or quartz, which has very low absorption loss, high temperature stability and low dispersion. A 3 dB bandwidth of 2.4 THz of such a 5-bit TTD module was experimentally confirmed by optical auto-correlation and Fourier transform. The uniformity of the fanout beams can be within +5% (Liu, Zhao, Lee and Chen [1997]), which makes it easy to use amplitude tapering to decrease the sidelobes. The surface-normal fanout architecture facilitates the surface-normal integration of the device with photodetector array, and therefore eliminates the delicate interface between
4, w 6]
SUBSTRATE-GUIDED WAVE TRUE-TIME DELAY MODULES
341
the optical delay lines and detecting/switching elements. Due to the collinear multiplexibility of the delay lines, a high packing density can be achieved (10 delay lines/cm 2 has been demonstrated) (Fu and Chen [1998b]). The total optical insertion loss of the device from the fiber beam splitter input to each individual delayed output is less than 3k + 2 dB for a k-bit delay line. To decrease this loss, a laser array can be used to provide N modulated signals instead of the l x N beam splitter. And again, erbium-doped fiber amplifier (EDFA) with a low-noise figure can be used to effectively compensate for the loss. 6.2. WAVELENGTHFINE-TUNINGSUBSTRATE-GUIDEDWAVETRUE-TIMEDELAY The volume holographic grating couplers for input and output coupling exhibit dispersion effect. The angular deviation A0 of the diffraction angle with respect to a perfect phase-matching diffraction angle was deduced and experimentally confirmed to be (Zhou, Fu and Chen [1998]) A~ A0 = --z-tan 0. A
(18)
Therefore, if the wavelength of the beam incident to the structure shown in fig. 40 changes slightly, the corresponding diffraction angle within the substrate will follow this change as shown in fig. 42. As a result, the length of the optical delay path between every two bouncings will change slightly, which causes the signal delay to change in a small range (Fu and Chen [1998b], Fu, Zhou and Chen [1999]). For the central wavelength, the delay between two successive bouncings is given by r =
2hn
(19)
COS0 X C'
where h is the height of the substrate, n is the refractive index of substrate, which is 1.528 at 1550nm, c is the velocity of light in free space, and 0 is
Fig. 42. Wavelengthfine tunable TTD line.
342
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 6
the bouncing angle in the substrate. For 0 around 45 ~ the delay between two successive bounces will change with a working wavelength by an amount of Ar =
2hnsin 0 2hnsin 0 AX A0 = ~tan cos 2 0 x c cos 2 0 x c 2,
0 ~
2hn
cos0xc
A~,
A/~
- r--~-.
(20)
Therefore, the delay step can be continuously tuned by tuning the working wavelength instead of the quantized value. By replacing the N output fibers from the 1 • fiber beam splitter in fig. 39 with a highly dispersive fiber which has the same dispersion characteristics as that of the substrate-guided wave devices and employing a tunable laser source, continuous delay tuning around N • discrete delay signals can be achieved and only N • detectors are needed to detect all the delay signals from the true-time delay module (Fu, Zhou and Chen [ 1999]). The fanouts under different working wavelengths need to be focused onto the small sensitive areas of the high-speed photodetectors. As described in w 6.1, one 7-bit module can be used to steer the beam of a phased array antenna with 9 antenna subarrays into 33 discrete directions among +45 ~ range under one wavelength. Once the configuration of the delay signals is set to steer the radiated beam to one discrete direction, a continuous scanning in a small range around this discrete direction can be achieved by continuous tuning of the working wavelength around the central wavelength. From table 3, suppose the delay step between two antenna subarrays for mth scanning angle is rm, the smallest delay step under center wavelength is r, if the working wavelength is changed by A~,, rm will change to r m = mr
~
+ 1 .
(21)
To achieve continuous scanning, i.e., continuous delay step, the following equation should be satisfied: A Tm
T m --
--
Tm - 1 +
A Tm-
(22)
1,
i.e., A),
A~
mr--~-(m - 1)r + (m - 1)r 2, .
mr-
(23)
Therefore, the required minimum wavelength change is given by A~ 2,
1 -
2m- 1
.
(24)
In this way, the delay line is pseudo-analog and continuous scanning can be achieved. The simulated far field patterns of an antenna array with 128 elements
4, w 7]
WAVELENGTH-DIVISION MULTIPLEXED OPTICAL TRUE-TIME DELAY LINES
343
Fig. 43. Simulated result of pseudo-analog TTD line for continuous beam steering. controlled by such pseudo-analog TTD lines under 5 different wavelengths are shown in fig. 43. Under the central wavelength, the scanning direction is 45 ~ This scanning direction will deviate by ~0.75 ~ with the working wavelength change of 20nm.
w 7. Wavelength-Division Multiplexed Optical True-Time Delay Lines The straightforward implementation of fiber-optic TTD for large PAAs results in very large amounts of hardware, which makes the TTD concept impractical in this form. The hardware complexity is proportional to the product of the number of PAA elements (K) and the number of PAA steering angles (R), and K and R are in the 102-104 and 102-103 ranges, respectively. Hardware-compressive TTD architectures that reduce the overall hardware with respect to both K and R must be used. Wavelength-division multiplexed technology can dramatically reduce the hardware requirement for the true-time delay lines.
7.1. PHOTONIC TRUE-TIME DELAY BASED ON WAVELENGTH-DIVISION MULTIPLEXING Freitag and Forrest [ 1993] proposed architecture based on a coherent, multiwavelength transmission network as shown in fig. 44. A microwave signal A~sinco~t modulates the intensity of the output of a wavelength-tunable laser array of n lasers, each with an optical frequency of COok.The modulated optical signals are delayed by different amounts of kAt, where k = 1, . . . , n. These n delayed signals given by A~sin[CO~(t + kAt)] sin COoktare then combined onto the singletransmission fiber with optical amplifier. The optical fiber amplifier compensates
344
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 7
S
L_ I I~,tLaser /Uray
-~ J
I~
LO.LaserArray
II P,etltnmmLas~ Fig. 44. Schematic diagram of coherent phased array transmitter antenna (9
IEEE).
for signal losses and prevents degradation of S/N. Either erbium-doped fiber amplifiers (EDFAs) or semiconductor optical amplifiers (SOAs) can be used in this application. Advantages of the EDFA are a higher S/N than for SOAs, whereas SOAs offer a higher degree of device integration. After transmission over the fiber, the signals are distributed to n coherent receivers using a 1 • splitter. Again, optical amplifiers compensate for splitter loss and prevent undue S/N degradation. Each of the n optical signals are incident on n receivers, along with n local oscillator (LO) signals that are tuned to provide receiver 1 with frequency tOol, receiver 2 with tOo2, etc. Given that the kth LO is at Bksin(tOokt + #k), then the signal at the output of the kth receiver is proportional to: A~tBkGkcosCksin[tO~(t+kAt)]/2, where Gk is the product of all the gains and losses in the link. This signal is then delivered to the kth antenna element at the RF frequency (typically 20-60 GHz) by frequency shifting. To sweep the beam, the input laser array (or, alternatively, the LO array) frequencies are changed to create a new correspondence between a delay and an antenna element. The output signal of the detector is proportional to the LO strength B,. Typically, this can be as high as 10mW at A = 1.55 ~tm, thus providing considerable gain at the antenna end. Both microwave amplifier and LO gain can be tuned at each antenna element to arbitrarily "shape" the beam. To achieve a high S/N, combiners/splitters with EDFAs are implemented as follows: The n signals are combined by m (n/m + 1)x 1 passive couplers followed by m EDFAs, with n being the number of delays. The use of multiple, low-gain coupler stages minimizes spontaneous emission noise common in systems which combine high loss and high gain in only a few amplifier stages. Each combiner merges n/m signals along with a pump signal for an EDFA. After amplification,
4, w 7]
WAVELENGTH-DIVISION MULTIPLEXED OPTICAL TRUE-TIME DELAY LINES
345
Fig. 45. TTD phased arrayantenna feed systemwith four optical channels and two coherentreceivers. Inset: optical heterodyne AM receiver (9 1996 IEEE). the signals were combined by an m x 1 passive coupler and transmitted along the fiber transmission line. An EDFA is also used on the transmission line to further improve signal level. Similar to the combiner, the signals are split by a 1 x(m + 1) passive splitter, amplified by m EDFAs, and then further split by m 1 x (n/m) passive splatters. For a system with a 128-channel link using EDFAs with an optical gain of~22 dB and a noise figure 4 dB, the overall SNR is ~40 dB and the system optical insertion loss is -3 dB. Xu, Taylor and Forrest [1996] demonstrated a true-time delay phased array antenna feed system based on multi-channel optical heterodyne detection techniques. Figure 45 shows a four-optical channel, two-antenna element demonstration system. The four /l = 1.55 ~tm DFB laser transmitters are each thermally tuned to have slightly different central optical frequencies. The light from the four transmitter lasers is amplitude-modulated (AM) by microwave using LiNbO3 Mach-Zehnder (M-Z) modulators. The modulated optical signals are then transmitted through different fiber lengths before being combined by a 4x 1 combiner composed of a series of 2x 1 combiners. The signals then pass through a single transmission fiber before being routed to two optical receivers through a series of splitters. Next, these signals are separately mixed with two optical local oscillator (LO) signals at the input of InGaAs p-i-n detectors with 70% quantum efficiency at ~= 1.55 Bm. By tuning the LO DFB lasers, each receiver is able to detect and demodulate the microwave signal from any of the four transmitter laser channels. The laser frequencies can be adjusted by independent current and temperature controllers with a temperature stability of +0.02~ Since the modulated signals arrive at the receivers through
346
TRUE-TIME DELAY CONTROL
SYSTEMS FOR WIDEBAND
PHASED ARRAY ANTENNAS
[4, w 7
PAA
RF in
Demultipixer
r
P rogrammable-Dispersio n
Matrix Monolithic Mode-Locked
Laser (as multi-~, source)
~
,2r
i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' 2X2 ' OpUcal , Switch
i i i
"Dispersive" Elements
I '
J
Fig. 46. MWOCPAAarchitecture using multiwavelength laser and programmabledispersion matrix (9 IEEE).
different fiber lengths, true-time delays are introduced among the four channels. Additional optical channels can provide more delay combinations, while more splitting before the optical receivers can be used to accommodate additional antenna elements. The demodulation scheme used is a wideband filter rectifier-narrow band filter structure (WIRNA, shown in fig. 45 inset), which employs a squarelaw device (SLD) to demodulate the AM signal. This receiver architecture is relatively insensitive to laser linewidth, and the dynamic range is comparable to that of optical direct detection schemes. In the demonstration, the intermediate frequency (IF) passband is from 2 to 8 GHz, and the narrow band amplifier operates from 0.8 to 1.5 GHz. The SLD used can be either a frequency doubler or an FET mixer. For optically coherent detection, the light polarization can be controlled using polarization maintaining (PM) fibers and PM fiber couplers throughout the system. In the demonstration, single-mode non-PM fibers and couplers were used along with manual polarization rotators. These components did not appear to significantly degrade the system performance or stability. The total optical insertion loss of the demonstrated system was 20 dB, with 12 dB loss from the 3 dB couplers, and 8 dB from the Mach-Zehnder modulator. The system SNR w a s - 5 0 dB and the dynamic range was about 54 dB. A multiwavelength optical-controlled phased array antenna (MWOCPAA) based on dispersion optical elements proposed by Tong and Wu [ 1996] is shown in fig. 46. It consists of a monolithic mode-locked laser (MLL) which functions as the multiwavelength source, an external electro-optic (EO) modulator, a
4, w 7]
WAVELENGTH-DIVISION MULTIPLEXEDOPTICAL TRUE-TIMEDELAYLINES
347
programmable dispersion matrix (PDM) and a wavelength-division-multiplexed (WDM) demultiplexer. The multiple wavelengths are simultaneously modulated by the external EO modulator and then sent through the PDM for true-time delay processing. The PDM with n-bit resolution comprises n "dispersive" elements with an exponentially increasing dispersion, Do, 2D0, . . . , 2 n- 1D0 (ps/nm), and (n + 1) 2• optical switches to program the total dispersion, as illustrated in the inset of fig. 46. The dispersive elements can be made of any wavelengthdependent time-delay units such as dispersive fibers, grating fibers, or WDM delay lines. The total dispersion of the PDM is then
DxoT = ~ 2 i- 1DoSi,
(25)
i=1
where Si--O o r 1 is the state of the ith optical switch. By programming the optical switches, the DvoT can vary from 0 to (2 n - 1)D0 ps/nm in increments of Do ps/nm. The resulting time delay between adjacent optical channels is AT =DToT AZ, where A~ is the channel spacing. If the PDM is implemented using dispersive fibers, uniform wavelength spacing is required. This can be achieved by using the mode-locked supermode as the multiwavelength source in which the wavelength spacing is precisely equal to the mode-locking frequency. The WDM demultiplexer directs /~i to the ith element of the array, generating a relative linear time shift of {0, AT, 2 A T , . . . , (m - 1)AT} across the array elements. The steering angle 0 is then given by 0 - sin-l(cAT/A) for all RF frequencies, where A is the distance between array elements, and c is the velocity of light in fleespace. For an n-bit PDM, the total optical insertion loss from the input of PDM to the output of the WDM demultiplexer is n + 4 dB. 7.2. PHOTONIC TRUE-TIME DELAY BASED ON ARRAY PARTITION AND WAVELENGTHDIVISION MULTIPLEXING
Goutzoulis and Davies [ 1990, 1992] and Goutzoulis, Davies, Zomp, Hrycak and Johnson [1994] employed BIFODELs (as described in w 4.1) in conjunction with optical WDM to compress the hardware. Hardware compression with respect to the number of steering angles R is most efficiently accomplished by the use of BIFODELs. In a BIFODEL, the optical signal is optionally routed through M fiber segments whose lengths increase successively by a power of 2. The various segments are addressed by a set of M 2 • 2 optical switches. Because each switch permits the signal either to connect or bypass a fiber segment, a delay T
348
TRUE-TIMEDELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 7
INPUT MICROWAVE RADAR SIGNAL TO BE TRANSMITTED
T
IREFERENCE
.
.
.
.
.
i
EFERENCE
REFERENCE
~,~oo~ L ... ~~_~,
~ ,,,,:o,,e,
.
1
.....
OPTICAL MUX
,, i
ri:
- ~ :
~,
E-CHANNEL OPTICAL SPLITTER
:
~
I ,,,
.... . . -
IN-CHANNELI IN-CHANNELI IN-CHANNEL] L OErtUX J
1~1h22~3 h~ ,~lh2A3 !
OUTPUT SIGNAL DELAYS SET- I
OUTPUT SIGNAL DELAYS SET-2
*Xm ~ I A 2 ~ 3 OUTPUT SIGNAL DELAYS SET-3
hlw ~1~zh3
~
OUTPUT SIGNAL DELAYS SET-4
/~I
[N-CHANNEL l / OEMUX I 'hsh~t~3 OUTPUT SIGNAL DELAYS SET-E _
~N
!
TO PHASED ARRAY RADAR ANTENNA
Fig. 47. BIFODEL-based compressive delay line architecture for transmit mode ( 9
OSA).
may be inserted in increments of AT up to the maximum value of (2 M - 1)AT, where AT is the delay that corresponds to the shortest fiber segment. Therefore the complexity of the BIFODEL grows slowly with increasing R, in fact as log2R, and so only a small amount of hardware is needed even for large R. Hardware compression with respect to the number of PAA elements K is accomplished by the partitioning of the PAA. In a K-element partitioned linear one-dimensional PAA, there are E subarrays, each having N elements, such that K = N E . In this case, the delay required by the ith element of the jth subarray is equal to the delay of the ith element of the first or reference subarray (RS) plus a bias delay. This bias delay depends only on j and is common to all the elements of a given set. Therefore, if partitioning is used, the total number of different types of BIFODEL is reduced to N + E (i.e., N for the RS plus E for the bias delays) compared with K for the noncompressed implementation. Using wavelength division multiplexing, this compression can be implemented very efficiently. In the transmit mode as shown in fig. 47, N RS BIFODELs with outputs at wavelengths /~1, /~2, - . . , /~N are driven in parallel by the microwave radar signal. The N BIFODELs outputs are multiplexed via an N-channel multiplexer (MUX), the output of which is divided into E channels via an
4, w 7]
WAVELENGTH-DIVISION MULTIPLEXEDOPTICAL TRUE-TIME DELAYLINES RECEIVING P H A S E D A R R A Y A N T E N N A I
,
.......... SET- I
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ELEMENTS
] . . . . . . . . . . . . SET-2 SET-E
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IOPTIC2L.uxl
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" (E-2)TNmin
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l
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349
, ,; .........
t DELAY / FOR 1 THIS
F
E-CHANNEL
4~ " N - C H A N N E L .... OPTICAL D E M U X
I
II ' ":- [ ~
Tlmln
9
I
....
VECTOR SUM OUTPUT
Fig. 48. BIFODEL-based compressive delay line architecture for receive mode (9
OSA).
optical 1 • channel splitter. All but one of the splitter outputs independently drive a bias BIFODEL, each of which is followed by an optical N channel demultiplexer (DEMUX). The splitter output channel that does not drive a bias BIFODEL is also demultiplexed. Since the optical inputs to each bias BIFODEL contain N optical wavelengths with different delays, the DEMUX output will also contain N wavelengths A1, ~,2, 999 AN, each with different delay required for the RS signals. In addition, the outputs of each of the DEMUXs following bias BIFODELs contain an additional common bias delay with respect to the RS delay signals. The antenna array is then properly fed by using similar wavelength outputs to drive similar location elements in each subarray. The resolution of the reference BIFODEL is Tmin and that of the bias BIFODEL is NTmin. In the receive mode shown in fig. 48, the same architecture is used but in reverse. The output of each PAA element in a subarray drives a laser diode (LD) of a different wavelength. Elements with similar locations in different subarrays drive LDs of the same wavelength. For each PAA subarray, the LD outputs are multiplexed and drive a bias BIFODEL with the RS output now driving the largest bias BIFODEL and with the subarray E output requiring no bias. At the output of the bias BIFODELs, all bias delays have been removed, and the
350
TRUE-TIMEDELAYCONTROLSYSTEMS FOR WIDEBANDPHASEDARRAYANTENNAS [4, w 7
wavefront appears like a sawtooth having a period of a subarray. Then the outputs of all the bias BIFODELs are combined via an E-channel optical combiner, the output of which is subsequently demultiplexed. Each of the DEMUX outputs drives one RS BIFODEL, which eliminates the in-subarray delays. The outputs of the reference BIFODELs are added up via a combiner to provide the desired vector SUM. The same architecture can be used for both transmit and receive modes, resulting in systems of reduced cost, size, and power, because the BIFODELs can be configured for two-way optical propagation using bi-directional fiber transmission together with reversible splitters/combiners and MUX/DEMUX devices. The desirable properties of hardware compression and transmit/receive reversibility of the system do not significantly affect other performance characteristics such as bandwidth, simultaneous radar beams at different frequencies and reconfiguration time. However, there were some additional DR losses due to the crosstalk of twice as many switches per path (compared to a single BIFODEL), other crosstalk-type losses due to output DEMUXs, losses due to increased link losses arising mainly as a result of the 1 • optical splitter, and the various extra insertion losses due to the increased number of components in the signal path. Based on Goutzoulis' architecture, several different ways to provide the intra-subarray delays were proposed. Wavelength partitioning of the array has been shown to be a powerful technique for achieving hardware compression in true-time delay beamforming networks. In order to obtain a larger hardware compression in the required number of delay line interconnects in the beamformer, Minasian, Alameh and Fourikis [ 1995] presented a wavelength division multiplexed (WDM) delay unit that efficiently provides the reference delay (intrasubarray delays) functions in the partitioned array architecture. This concept exploits the wavelength independence of the optical carriers to maximize the utilization of the available optical fiber delay lines. A 16-wavelength example of the proposed structure is shown in fig. 49. The 16 wavelengths are grouped into 4 subgroups, each experiencing different delay over fibers with different lengths. The outputs of the fiber with different wavelengths are combined together and then regrouped into 4 subgroups with each of the four wavelengths in the same subgroup in the first delay path distributed to a different subgroup. The wavelengths in four subgroups then experience another set of different delays with 4r differential delay as shown in the figure. In this way, each unique wavelength experiences a unique extra delay ranging from 0 to 15 r, i.e., ~1 is delayed Tmin, ~ 2 _ i s delayed Tmin + T, . . . , ~16 is delayed Tmin + 15r. This gives a linear delay taper that can be used to provide the reference delay function
4, w 7]
WAVELENGTH-DIVISION MULTIPLEXED OPTICAL TRUE-TIME DELAY LINES
~u
-. - - ~ -'e
~
Delay 0
~.,, ;~, ~,
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o~= ~,,~...~,.
Delay 4'r :Combiner Coupler ! ~ ' ~kao, ~,,,, ~,,=
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Fig. 49. Wavelengthpartitioning WDM delay line (9 1995 John Wiley & Sons, Inc.).
X
k,. ~
k,, 9"
x,
--J
L,
~. k,. ~.,,. ~.,=
<
~,_o
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Fig. 50. Wavelengthpath control unit layout (9
John Wiley & Sons, Inc.).
in the partitioned beamforming network. The delay unit can be upgraded to any number of wavelengths by straightforward modification. This delay unit utilizes all the delay lines simultaneously, by routing different wavelengths along paths, some of which are shared by several wavelengths. A WDM delay unit that uses P - 2 r wavelengths requires 2 l o g z P - 2 r different delay lines. For example, a 256-wavelength unit requires only 16 delay lines. This represents large hardware compression. The wavelength path controller (WPC) units in fig. 49 are essentially wavelength demultiplexers/routers. They guide several wavelengths into same outputs. The layout of the WPC unit is shown in fig. 50. The WPC units can be realized in either fiber form or by using an integrated optic structure. The complete architecture for an array of N - 2 1 antenna elements (divided into E subarrays) and M beams using this WDM arrangement with P - U
352
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
Set#1 --
.
..
.
Set #2 --
.
.
Set #E --
-
[4, w 7
ArdeN'~
P:M
Fibre Bias Delay Lines E:I
1
!
~"" ~ " " ~"
9
Reference Delay Unit
Delays
J Delay [ Unit Optical
Receiver Output for individual
beam
directions
Fig. 51. Wavelength multiplexed beamforming architecture with WDM delay units (9 Wiley & Sons, Inc.).
John
wavelengths is shown in fig. 51. This utilizes Goutzoulis' bias delay line structure, but introduces the new WDM delay traits to provide the reference delays in the network. This is a highly efficient structure that combines the WDM delay unit concept with the partitioning technique to give large hardware compression especially for large arrays. The total number of delay lines is given by
K = M(E + 2r) = M ( ~--7+ 2r ) .
(26)
The optimum value of the integer r that minimizes K can hence be obtained as r o p t = l o g 2 ( n l 2 (2)) = l - 2 .
(27)
Then the minimum number of delay lines required in the beamformer for an N-element array with M beams is Kmin
--
2Mlog2N.
(28)
Riza and Madamopoulos [1996] and Russell, Marinilli, Green, Preiss and Jain [ 1996] proposed a fully reversible transmit/receive mode TTD phased array
4, w 7]
WAVELENGTH-DIVISION
Controller
MULTIPLEXED
OPTICAL
TRUE-TIME
DELAY
353
LINES
Site To Past Pttxcs~g
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Receive Mode
',
,,~,v lvx~A
.......................
:
* This Single Channel P T D L can also be a grating fiber-based switched delay line
, '
,
!
A2 , " A
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/
,i .
i ,
;
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Sub-Anay
" \ ,
i~r-:
i
t
,
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(K -I)T
Fig. 52. CREOL WDM reversible photonic control system for PAA (9
SPIE).
antenna architecture based on array partition that uses dispersive optical elements (similar to the PDM proposed by Tong and Wu [1996] in w 7.1) to provide the intra-subarray delay control and uses a multiple switched delay channel for intersubarray bias delay control. The system proposed by Riza and Madamopoulos is shown in fig. 52. This wavelength multiplexing technique is also based on partitioning the PAA into subarrays, where there are a total of N subarrays, with M antenna elements per subarray. The ruth element of the (n + l)th subarray requires a time delay that is equal to the time delay of the ruth element of the nth subarray plus a bias delay. Thus, the photonic control system was designed as a cascade of two photonic subsystems. The first switched photonic delay line (PTDL) sub-system will give the required time delays that control the M antenna elements in each subarray. The second switched PTDL subsystem will give the additional required bias delays for controlling each of the N subarrays. In the transmit mode, M high-power diode lasers (H-LD) that operate at different wavelengths, Xl, %2, . . . , AM, are modulated by the transmit signal. An M:I optical wavelength multiplexer is used to couple the signals at
354
TRUE-TIME DELAY CONTROL SYSTEMS FOR WIDEBAND PHASED ARRAY ANTENNAS
[4, w 8
different wavelengths into a single channel dispersive fiber-based F-bit switched PTDL system. The different wavelengths travel in the dispersive fiber with different velocities, obtaining small time delay differences with respect to each other. The longest time delay achieved is rV(&M). A 1:N optical splitter is used to split the M delayed signals into N sets of different wavelength signals. These N sets of ~,1, A2, . . . , AM signals then pass through the G-bit N-channel switched bias time delay system, obtaining the appropriate bias delays with a differential delay of T-rv(AM). Finally, N 1:M optical wavelength demultiplexers are used to separate the different M signals that drive the elements of each subarray. The required time delayed signals are detected by the photodetectors at the antenna element optoelectronic transmit/receive module. In receive mode, essentially the same hardware is used for both the transmit and receive modes of the antenna. Each antenna element is connected to a low noise laser diode (LD). The M outputs of each antenna subarray modulate the M LDs, which operate at the M different wavelengths. The wavelength of the LD used for the ruth element for each of the N subarrays is the same. Light propagates in the opposite direction in the system when compared to the transmit mode. For each subarray, the wavelengths are multiplexed and pass through the bias PTDL. After the bias PTDL, the bias time delays have been canceled. An N:I optical combiner is used to couple all the signals to the single channel dispersive fiber PTDL system, where now the shorter time delays corresponding to the individual elements of each subarray are canceled. The received optical signals are directed to the detectors of the controller optoelectronic n-transmit/receive modules, after which all the signals are electrically summed, and sent to the electronic receiver for post-processing. Russell, Marinilli, Green, Preiss and Jain [ 1996] used a similar structure shown in fig. 53. Multiple dispersive fiber-delay channels were used to parallelly provide the intra-subarray delay signals to the bias delay paths instead of using one dispersion channel and 1 • splitter in Riza's architecture. Riza and Madamopoulos [1997] extended this concept to 2-D beamforming and describe various optical time-delay unit designs including both dispersive and grating-based fibers.
w 8. Summary We have reviewed the basic principles of phased array antenna and optical technology for phased array antenna. Then a brief review is given to Fourier optics beamforming, optical RF phase-shifter, and optical true-time delay for
4]
REFERENCES
355
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wideband phased array antenna. Different technologic options and the status of the photonic true-time delay devices demonstrated or reported to date have been described in detail, including bulk optics true-time delay, fiber optics true-time delay, optical waveguide true-time delay, substrate-guided wave true-time delay, and wavelength-division multiplexed true-time delay. The number of contributors to optical true-time delay control of phased array antennas is large, and we apologize for any omission in reference to their work.
Acknowledgements The authors wish to express their gratitude to Dr. Suning Tang from Radiant Research, Inc, Austin, Texas, for his careful reading of this manuscript and his very useful comments. This work was supported by Office of Naval Research, Air Force Research Lab, Department of Advanced Research Projects Agency, 3M Foundation, and the Advanced Technology Program of the State of Texas.
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E. WOLF, PROGRESS IN OPTICS 41 9 2000 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
BY
JAN PE~dNA JR
Joint Laboratory of Optics, Palack)) University and Institute of Physics of the Academy of Sciences of the Czech Republic, 17. listopadu 50, 772 07 Olomouc, Czech Republic; E-mail: [email protected]
AND
JAN PE~NA
Department of Optics and Joint Laboratory of Optics, Palack)) University, 17. listopadu 50, 772 07 Olomouc, Czech Republic; E-mail: [email protected]
361
CONTENTS
PAGE w 1.
INTRODUCTION
. . . . . . . . . . . . . . . . . . .
w 2.
Q U A N T U M E V O L U T I O N OF I N T E R A C T I N G O P T I C A L F I E L D S 366
w 3.
Q U A N T U M STATISTICAL P R O P E R T I E S OF I N T E R A C T I N G OPTICAL FIELDS . . . . . . . . . . . . . . . . . . .
w 4.
370
C O U P L E R S B A S E D ON S E C O N D - H A R M O N I C A N D SUBHARMONIC GENERATION
w 5.
363
. . . . . . . . . . . .
375
COUPLERS BASED ON NONDEGENERATE OPTICAL PARAMETRIC PROCESSES
. . . . . . . . . . . . . .
392
w 6.
COUPLERS BASED ON THE KERR EFFECT
w 7.
COUPLERS BASED ON R A M A N AND BRILLOUIN SCATTERING . . . . . . . . . . . . . . . . . . . .
401
w 8.
MISCELLANEOUS COUPLERS
. . . . . . . . . . . .
412
w 9.
CONCLUSIONS . . . . . . . . . . . . . . . . . . . .
414
w 10. A C K N O W L E D G M E N T S REFERENCES
. . . . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
362
395
416 416
w 1. Introduction
Optical couplers employing evanescent waves play an important role in optics, optoelectronics and photonics because they may be conveniently used in the switching of light beams. They also provide a tool for controlling light by light. Amplitude and intensity behaviour of linear couplers has been investigated extensively (Yariv and Yeh [1984], Solimeno, Crosignani and Di Porto [1986], Saleh and Teich [1991]). Only recently quantum statistical properties of linear couplers have been studied (Janszky, Sibilia and Bertolotti [1991], Lai, Bu~ek and Knight [1991], Pefinov~, Luke, Kfepelka, Sibilia and Bertolotti [1991], Janszky, Adam, Bertolotti and Sibilia [1992], Bandyopadhyay and Rai [1997]). Such investigations were able to examine complete quantum statistics of these devices applying methods and results obtained earlier in quantum optics of "pure" optical processes, including "transfer of nonclassical properties" among different channels of the coupler. Linear couplers also provide ways to generate number-phase intelligent states. Substantial progress in controlling light beams has been achieved when nonlinear waveguides with both linear and nonlinear coupling have been taken into account (Finlayson, Banyai, Wright, Seaton, Stegeman, Cullen and Ironside [1988], Townsend, Baker, Jackel, Shelburne III and Etemad [1989], Leutheuser, Langbein and Lederer [1990], Weinert-Raczka and Lederer [1993], Assanto, Laureti-Palma, Sibilia and Bertolotti [1994], Hatami-Hanza and Chu [1995], Hansen, Kloch, Aakjer and Rasmussen [1995], Weinert-Raczka [ 1996]). This gave new possibilities of fast all-optical switching, including digital switching, and reduction of switching power. New ways to control optical beams in nonlinear couplers have been discovered. The most important property of these devices is that they may generate, "transfer", and control nonclassical light that exhibits squeezing of vacuum fluctuations and/or have sub-Poissonian photon statistics. Coupled-mode analysis is usually applied in traditional treatment to the theory of nonlinear couplers (Yariv and Yeh [1984], Yasumoto [1991], Yasumoto, Maekawa and Maeda [1994], Yasumoto [1996]). This approach is suitable even for more complex couplers, for instance those composed of three nonlinear waveguides (u Mitsunaga and Maeda [1996]). Phase mismatches in nonlinear optical processes (Shen [1984], Schubert and Wilhelmi [1986], 363
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QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 1
Boyd [1992], Miranowicz and Kielich [1994], Agrawal [1995], Reinisch, Blau, Vitrant, Primeau and Baltog [ 1996]), quasi-phase-matching, and cascaded nonlinear processes as well as Kerr-type processes lead to interesting effects in nonlinear couplers. Bifurcation analysis is useful in some types of couplers (Ankiewicz and Akhmediev [ 1996]), especially in those with fluctuating coupling parameters (Mostofi, Malomed and Chu [1997]). All-optical switching can be achieved (Fazio, Sibilia, Senesi and Bertolotti [1996]). Besides classical nonlinear crystals nematic liquid crystals also can be enployed (Karpierz and Wolifiski [ 1995]). Also double-core optical fibres are potentially very interesting (Malomed, Peng and Chu [1996]). Nonlinear optical couplers become very important for nondestructive measurements of coupling constants (Brooks, Ruschin and Scarlat [1996]), and optical sensors (Luff, Harris, Wilkinson, Wilson and Schiffrin [ 1996]) as well as realization of logic operations (HatamiHanza and Chu [1996]). They have also been used in experiments with multibeam interferences (Soldano and Pennings [1995]). Phase mismatches and cascading of nonlinear processes play an important role in the behaviour of nonlinear optical couplers (Assanto and Torelli [1995], Hache, Zrboulon, Gallot and Gale [1995], Cerullo, De Silvestri, Monguzzi, Segala and Magni [ 1995], Picciau, Leo and Assanto [1996], White, Mlynek and Schiller [1996], Kobyakov and Lederer [1996], Saltiel, Tanev and Boardman [1997], Koynov and Saltiel [1998]). Also feedback can be used to control the behaviour of beams in nonlinear optical couplers (Artigas, Dios and Canal [1997]). Interesting effects can arise from the competition of degenerate and nondegenerate optical parametric processes (White, Lam, Taubman, Marte, Schiller, McClelland and Bachor [1997]) and quasi-phase-matching (Fejer, Magel, Jundt and Byer [ 1992], Myers, Eckardt, Fejer, Byer, Bosenberg and Pierce [1995], Myers, Eckardt, Fejer, Byer and Bosenberg [1996]). Nonlinear waveguide materials used in composing nonlinear couplers provide new possibilities in constructing switching and memory elements for all-optical devices. These elements are necessary for further development of optical processing and computing. Classically, all-optical devices are analyzed from the viewpoint of their amplitude or intensity dependences. However, they can be treated fully in quantum theory. Noise of light beams in nonlinear couplers is naturally included in this quantum treatment. A general theory of quantum statistical properties of "pure" nonlinear optical processes has already been developed (see, e.g., Chap. 10 in Pefina [ 1991 ], some new results may be found in Chizhov, Haus and Yeong [ 1995], Yeong and Haus [1996], Chizhov, Haus and Yeong [1997], Kalmykov and Veisman [1998], Driihl and Windenberger [1998]). This theory has been recently applied to nonlinear couplers composed of various
5, w1]
INTRODUCTION
365
linear and nonlinear waveguides (Pefina [1995a] and references therein). The theory is particularly useful for the description of generation and "transmission" of nonclassical light. It can also include effects stemming from both linear and nonlinear phase mismatches. In order to determine quantum statistical properties of light beams, closedform solutions of nonlinear operator equations have to be found first. One can apply short-length approximation. This approximation can even be extended to higher powers of coupling constants when symbolic computation is used (Pefina [1995b], Pefina and Bajer [1995], Mi~ta Jr and Pe~ina [1997]). Or pump modes can be assumed to be in strong coherent states. This leads to linearization of the operator equations and their solution is then easily found (Pefina and Pe~ina Jr [1995a, 1996], Korolkova and Pe~ina [1997b]). Rotatingwave approximation together with the existence of some integrals of motion are sufficient to obtain analytical solutions in Kerr nonlinear couplers (Korolkova and Pe~ina [ 1997a,c]) including contradirectional propagation (Ariunbold and Pefina [2000]). Numerical analysis may be adopted for any nonlinear system (Chefles and Barnett [1996], Ibrahim, Umarov and Wahiddin [2000]). The quantum statistical description of optical fields includes photon-number distribution, its factorial moments, quadrature and intensity variances, various squeeze variances, phase distributions, etc. All these quantities depend on the position in a coupler. Schrrdinger-cat states (Buick and Knight [1995]) can be transmitted through nonlinear couplers (Janszky, Petak, Sibilia, Bertolotti and Adam [1995]). Entangled superpositions of macroscopically distinguishable states can be generated in a nonlinear directional coupler with second- and third-order nonlinear materials (Mogilevtsev, Korolkova and Pe~ina [1996]). There are solid-state applications leading to the so-called bandgap quantum coupler (Mogilevtsev, Korolkova and Pe~ina [1997]). The model of coupled harmonic oscillators is useful for the description of some nonlinear interactions (Abdalla [ 1997], Abdalla, Ahmend and A1-Homidan [1998], Abdalla and Bashir [ 1998]). Squeezed light can be generated in couplers with cascaded second-order nonlinearities (Berzanskis, Feller and Stabinis [1995]). Quasi-phase-matched materials are conveniently used (Noirie, Vidakovi6 and Levenson [1997], Chickarmane and Agarwal [1998]). Instabilities in cascaded processes can be controlled (Kheruntsyan, Kryuchkyan, Mouradyan and Petrosyan [1998]). The most useful nonlinear optical processes for the generation of light with nonclassical properties are optical parametric processes (Janszky, Sibilia, Bertolotti, Adam and Petak [1995], Korolkova and Pe~ina [1997d]), Kerr effect (Chefles and Barnett [ 1996], Korolkova and Pe~ina [ 1997a,c]), Raman scattering,
366
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 2
and Brillouin scattering (Olivik and Pefina [1995], Pe~ina Jr and Pefina [1997], Fiurfi~ek and Pefina [1998, 1999b, 2000b]). Nonclassical properties of light beams in nonlinear codirectional and contradirectional couplers based on nonlinear materials with the above mentioned processes are reviewed in this contribution. Both linear and nonlinear (Abdalla, E1-Orany and Pefina [2000]) coupling of modes between waveguides is assumed. Couplers with contrapropagating beams are described by a quantum consistent method (Pefina and Pefina Jr [ 1995b,c]). This method even provides Hamilton formulation of the description of such couplers (Luis and Pefina [1996], l~ehfirek, Mi~ta Jr and Pefina [1999], Fiurfi~ek and Pefina [2000c]). Losses stemming from the interaction of light beams with the surroundings are included. Phase mismatches inside nonlinear waveguides and between them are also taken into account. The coherent-state technique represents a fundamental tool of our treatment. It permits us to describe coherence and fluctuations of optical beams as boson cooperative phenomena (Klauder and Skagerstam [1985], Perelomov [1986], Zhang, Feng and Gilmore [ 1990], Kim and Noz [ 1991 ], Pefina, Hradil and Jurro [1994], Walls and Milburn [ 1994], Vogel and Welsch [ 1994], Mandel and Wolf [ 1995], Barnett and Radmore [ 1997], Scully and Zubairy [ 1997]). We restrict ourselves only to spatial behaviour of interacting optical fields in our treatment. Spatio-temporal descriptions in analogy to soliton physics (Chu, Malomed, Peng and Skinner [1994], Skinner, Peng, Malomed and Chu [1995], Malomed, Skinner and Tasgal [1997], Lakoba, Kaup and Malomed [1997], Mostofi, Malomed and Chu [1998], Peng, Malomed and Chu [1998]) can open new frontiers in this field.
w 2. Quantum Evolution of Interacting Optical Fields The dynamics of mutually interacting optical fields in a nonlinear coupler can be described conveniently using effective Hamiltonians or momentum operators (Sczaniecki [1983], Graham [1968], Graham and Haken [1968], Pe~ina Jr [ 1993]). In this case, degrees of freedom of a material system are not explicitly taken into account and the evolution of mutually interacting optical fields is described by a nonlinear interaction Hamiltonian or momentum operator. Such a description keeps the most important features in the evolution of interacting optical fields. Also damping of optical fields in a material system can be consistently included using damping constants and fluctuating forces. We can follow time or spatial development of optical fields using Hamiltonian or momentum operator and these descriptions are equivalent if dispersion in waveguides is omitted. We formulate the models of nonlinear couplers in spatial
5, w 2]
QUANTUM EVOLUTION OF INTERACTING OPTICAL FIELDS
367
domain using momentum operators, because dispersion of optical fields in materials can be described in this way. An arbitrary operator ~i(z) of the optical field obeys the following Heisenberg equation in the interaction picture: dA(z)
_
dz
i [~int(Z),A(Z)]
h
(2.1)
where Gint(Z) denotes the momentum operator characterizing nonlinear interaction, h is the reduced Planck constant, and the symbol [, ] denotes a commutator. Using interaction pictures throughout the chapter we eliminate fast free-field oscillations from the description. In general the nonlinear operator equations arising from eq. (2.1) cannot usually be solved directly and various approximations have to be employed. Equation (2.1) can be formally solved in the form
~i(z) = exp._
[~oZdzt~int(Z')l zi(z = 0),
(2.2)
Gint stands for a superoperator defined a s ~ i n t ( Z ) . . . = -(i/~)[Gint(Z),...] and the symbol exp+__ means an exponential superoperator with increasing arguments ordered to the left. In short-length approximation the solution is given by the first terms in the expansion of the exponential function in eq. (2.2). This approximation is valid when nonlinear interaction is weak (smaller values of nonlinear interaction constants, shorter interaction regions). Usually secondorder expansion is sufficient and gives at least the right tendency of the evolution. In any case this is a good description for single-passage propagation. We may also linearize eq. (2.2) around its stationary point; i.e., we may assume the operator A(z) in the form A(z) = /stat nt- t~2(Z), where Astat is the amplitude of a coherent state characterizing the stationary point and 6A(z) describes linear operator corrections to the stationary point. The set 6,~ = {6A1,6A~,&i2,6A~,...,6A~,6A~} T (T means the transposition) of where
annihilation (&ii) and creation (6A~) operators of the linear operator corrections of the interacting fields then obeys the system of linear differential equations: d&~(z) dz
- M(z)bJ,(z);
(2.3)
the matrix M(z) characterizes the interaction around the stationary point. Parametric approximation is based on the assumption that some of the interacting fields (pump fields) being at the beginning of the interaction in
368
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, {} 2
coherent states are strong in comparison with the other fields and thus are influenced only slightly by nonlinear interaction. They can be then approximated by classical deterministic functions and are usually included into effective coupling constants. This leads to a system of linear differential equations similar to that in eq. (2.3) in many cases. Quantum features of interacting fields are kept in this formulation; however, some nonlinear features like saturation, bifurcations, etc., are lost. Equations (2.2) can be in special cases solved exactly, e.g., when integrals of motion are known (for instance in Kerr media). However, in general they can be solved only numerically. In this case the nonlinear operator equations are written in some basis and the resulting system of linear differential equations is solved. When describing statistical properties of real nonlinear couplers, inclusion of the influence from "the surroundings" of interacting optical fields (reservoir) is usually important. This influence leads to damping of optical fields and also to the occurrence of noise (fluctuation-dissipation theorem). The HeisenbergLangevin equations can be applied in this case: dA(z) _ M(z)A(z) + t,(z), dz ^t
(2.4)
^t
^t
where A = {.,~,,A1,A2,Az,...,,,~,,,A,,} r, f, = {L,,L~,Lz,L~,...,L,,,Lt} T and the operators of the Langevin fluctuating forces Zi,Z~ are assumed to have the following properties:
(Lj(z))
:
0,
(i.j(z)Zk(z')) : (Lt (z)Z~(z')) : O, (Zt (z)Zk(Z')) = 2yj(nrj)bjk6(Z - z'), (Zj(z)Z~(z')) = 2yj((nrj) + 1)6jk6(z - z').
(2.5)
Brackets in eqs. (2.5) denote averaging over reservoir; }9 stands for a damping constant of the jth mode, and (n,.j) means a mean number of reservoir photons or phonons in the jth mode which is determined in thermal equilibrium at the temperature T by the expression (nrj} -- [exp(h~./KT)1]-1, ~- being the frequency of the reservoir mode j and K is the Boltzmann constant. The Kronecker symbol is denoted as 6jk and 6 is the Dirac delta function. The above relations can be derived in the framework of the Wigner-Weisskopf approximation if broad reservoir spectra are assumed. A general solution of eq. (2.4) for the operators A(z) can be written in terms of the operators A(0) at z = 0" A(z) = P(z, 0)A(0) +
/o z dz'P(z,z')L(z'),
(2.6)
5, w 2]
QUANTUM EVOLUTION OF INTERACTING OPTICAL FIELDS
369
where the evolution matrix P(z,z') is a solution of the equation dP(z, z')
dz
= M(z)P(z,z');
P(z',z') = E;
(2.7)
the symbol E stands for identity matrix. If the matrix M is z-independent, eq. (2.6) simplifies to .~(z) = P(z).~(0) +
dz'P(z - z') L(z')
(2.8)
and P(z)= exp(Mz). The expression for A(z) in eq. (2.8) can be recast into the following form: J,(z) =
(2.9)
+ /=1
The matrices U(z), V(z) and F(z) containing the elements Ujl(Z), Vjt(z) and Fj(z) can be expressed in terms of eigenvalues Ak and the corresponding eigenvectors Yk (MYk = AkYk) of the matrix M. We get in terms of the matrix Y (composed of the eigenvectors Yk) and its inverse matrix y (y = y-I): Ujk(z) =
~-~exp(~lZ)Y2j-l,lYl,2k-1, l=1
Vj/~(z) =
~exp(~lz)Y2j-l,lYl,2k. l=1
(2.10) Using the relations in eq. (2.5), the correlation functions of the stochastic forces ~(z) can be written as follows:
2n 2n exp([X,~ + I],m,]Z) - 1 qm,m t (P; (Z)pk(Z)) - - Z Z Y2;'-l,mY2k-l'm' m= 1 m, = 1
~ * -+- ~m'
2n 2n exp([Xm + Xm']Z)(FJ(Z)~"k= Z Z Y2j-l,mY2k-l,m, ~m 4- I~m'
(2.11) 1 qm,m I ,
m=lm'=l
where
qm,m' = ~ Yl [Y;,2t-lYm',2'-I(nr,) 4-Y;,21Ym',21 ((Ylrl)-Jr-1)], /=1
Om,rn' -- ~
(2.12)
]r [Ym,2lYm',21-1(nrl) q-Ym,2l-lYm',2l ((nrl) + 1 ) ] .
/=1
If ~ + ~m' = 0 o r ,~m nt- ~rn' - 0, the expressions in eq. (2.11) have to be replaced by their limit for (~m*+ '~m') ---+0 or (~m + '~m') ~ 0.
370
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
The standard boson commutation relations among the operators
[5, w 3
.4j(z) and
AJ(z) for j
= 1,..., n must be fulfilled for every z as a consequence of unitary evolution. This property provides identities among the coefficients Uj.k and Vjk in the solution, which are useful for checking numerical calculations. Sometimes they can be used for simplification of the solution. Equations of motion for (propagating along the -z axis) are classically derived by the replacement by in the equations which would describe their propagation along the +z axis. The same method can also be applied to quantum fields. The obtained equations cannot be derived directly from a Hamiltonian, but a Hamiltonian associated with the system comprising contrapropagating fields exists. It is obtained if we take the Hamiltonian describing a system with all fields propagating in one direction
contrapropagatingfields d/dz -d/dz
(along the +z axis) and make the substitution .4k ~ A~ for the fields propagating along the -z axis. Such a Hamiltonian provides equations of motion and the equations describing correctly the system with contrapropagating fields are then obtained by returning to the original variables by means of the substitution -4~ +--'Ak for the fields propagating along the -z axis in the final solution (Toren and Ben-Aryeh [1994], Pefina and Pefina Jr [1995b,c]). The standard bosonic commutation relations are assumed to be fulfilled among field operators from the set composed of the operators at z = 0 for fields propagating along the +z axis and the operators at z = L (L being the length of the crystal) for fields propagating along the -z axis. It can be shown that bosonic commutation relations are then also obeyed among the operators at z - L for fields propagating along the +z axis and the operators at z = 0 for fields propagating along the -z axis if quadratic Hamiltonians are considered (Luis and Pefina [ 1996]).
w 3. Quantum Statistical Properties of Interacting Optical Fields Nonclassical properties of interacting optical fields like squeezing of vacuum fluctuations, sub-Poissonian photon-number statistics and antibunching of photons may be conveniently described in the framework of generalized superposition of coherent fields and quantum noise. A normal characteristic function CH({fij},z) defined as
CH({[3j},z)=Tr{exp[~.~fij,~j(z)lexp[_~[3j.,ftj(z)]~(O)}=l j=l
(3.1)
represents a useful tool for the description of quantum statistical properties of optical fields. The symbol/5(0) means the statistical operator at z = 0 and
5, w 3]
QUANTUMSTATISTICALPROPERTIES OF INTERACTING OPTICAL FIELDS
371
Tr stands for a trace. The normal characteristic function Cx({/~},z) has a Gaussian form for generalized superposition of coherent fields and quantum noise appropriate for the description of linearized problems (Pefina and I~epelka [1991]): C x ( { ~ } , z, =
exp{~I-Bj(z)l[3jl2+(~Cj.(z)~.*2+c.c.)+(~.*(z)-c.c.)] j=l
j = 1 k = ly'
(3.2) where c.c. denotes complex conjugate terms. The complex amplitude ~(z) of the jth mode is given by
~j(z) =
[Uj-k(z)~k(0) + Vjk(z) ~ (0)1,
(3.3)
k=l
where ~j.(0) are the amplitudes at z = 0 and the matrices U(z) and V(z) are given in eq. (2.10). The coefficients Bj(z), Cj.(z), Djk(z) and Djk(z) occurring in eq. (3.2) are determined as follows (Pefina and Kfepelka [ 1991]):
k=l
k=l
9
= <(~ij(z)) 2) k=l
k=l
Djk(z) = <~j(z)~k(z)> -- <~.(z)/%'k(Z)> + ~ { UjI('Z)Ukl(Z)CIA+ %I(Z)Vkl(Z)C;A+ [Ujl(Z)Vkl(Z)+ Vjl(Z)Ukl(Z)]BIA } /=1
- ~ r)l(z)Vkl(Z), j ~ k, /=1
bj(z):
--
-
-
-
~ { [Uj*l(Z)Ukl(Z)+ I'~(Z)Vk/(Z)] 01.4 + ~*l(Z)Ukl(Z)ClA+ Uj*l(Z)Vkl(Z)C;.A } /=1
+ ~ U)*l(Z)Ukl(Z),
j ~ k,
/=1
(3.4)
372
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 3
and AAj(z)= Aj(z)- (Aj(z)), AAJ(z)= A J ( z ) - (AJ(z)). The quantities BjA and Cj.a appropriate for z = 0 are related to antinormal ordering of field operators [Bj(0) = B j . a - 1, Cj(0) = Q.a] in order to describe nonclassical initial states. They are assumed in the form
BjA = cosh 2 (rj) + (noj),
Cj.A = -~I exp (i0j) sinh (2rj).
(3.5)
The symbol rj denotes the squeeze parameter of the jth mode, 0j- stands for the squeeze phase of the jth mode, and (n0j) describes external noise at z = 0 in the jth mode. Statistical independence of the interacting fields at z = 0 is assumed (i.e., Djk(O) = Djk(O) = 0). The normal characteristic function CAr({/~},z) describes statistical properties of the interacting fields completely and various statistical quantities like photonnumber distribution, moments of integrated intensity or quadrature variances can be derived in terms of the parameters occurring in CAc({~-},z).
3.1. S I N G L E - M O D E CASE
A photon-number distribution p(n,z) is defined in terms of the density matrix r as (nl[o(z)ln), where In) stands for the Fock state with n photons, and can be determined according to the Mandel photodetection formula (e.g., Pe~ina [1991])
p(n,z) = fO cx~PN(W,z)~.W n e x p ( - W ) d W ,
(3.6)
where PH(W,z) denotes a distribution of the integrated intensity W related to normal operator ordering,
P v(W,z) = / ~H(a,z)b(W-lal2)d2a,
(3.7)
where the integration is taken over the whole complex plane of the field amplitude a. The Glauber-Sudarshan quasidistribution ~ ( a , z ) may be determined as the Fourier transform of the normal characteristic function Cx([3,z). The photonnumber distribution p(nj, z) of mode j can be expressed in terms of the Laguerre
5, w 3]
QUANTUM STATISTICAL PROPERTIES OF INTERACTING OPTICAL FIELDS
373
polynomials Lk 1/2 defined by Morse and Feshbach [1953] (K~trsk~t and Pefina [1990]): 1
p(nj,z) = (EjFj)I/2
( 1 ) 1- ~
nj exp
(AuAzj) Ej Fj
nj 1 • k~O = F(k+~ 1 ) l-7(n _ k + I)
1- ~ 1 I
(3.8)
XLk,/2(_ Ej(EjAlj- 1) )1_1/2 ( A2j ) *"nj-k \-Fj(Fj - l) " The symbol F denotes the gamma function,
Ej(z) = Bj(z)+ 1-ICj(z)[,
Fj.(z) = Bj(z)+ 1 + [Cj(z)l
(3.9)
are the quantum noise components, and
1[
1
(3.10)
A1,2j(z) = ~ I~j-(Z)l2 ::~ 21Cj(z)l (~2(z)CJ*(Z)+ c.c.)
denote the coherent signal components. Normal moments of the integrated intensity IVy of mode j are given by (Kfirskfi and Pefina [ 1990])
(Wf) =
PA/'(W,z)W k dW 0~00~176 nj~
(nj- k)~
=k!(Fj-llkl~or(l+ ~= •
Ej - 1
1
l)F(k-l+ A~
I)
Fj.
,)
(3.11)
1
)
where hj = A~Aj and Ej(z), Fj(z) and A 1,ej(z) are given in eqs. (3.9) and (3.10). Reduced moments of integrated intensity (Wjk)/(Wj.)k - 1 for k = 2,3,... are convenient for the description of nonclassical light because they are classically greater than or equal (in a coherent state) to zero. If they are less than zero, the classical inequality (W k) ~> (W) k is violated. If (W 2) ~ (W) 2, the field has sub-Poissonian photon-number statistics.
374
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
Variances of the quadrature components qj expressed as (Pefina and K~epelka [ 1991 ])
-- Aj +
AJ and Pj ----
((A~)j(z)) 2) = 1 + 2{Bj(z)+ Re[Q(z)]}, ((A/~j(z)) 2) = 1 + 2{Bj(z)- Re[Cj(z)]},
[5, w 3
-i(Aj -AJ) are (3.12)
where Re denotes a real part. Maximum amount of squeezing of vacuum fluctuations is indicated by the principal squeeze variance Aj (Lukg, Pefinov~ and Pefina [ 1988]):
~,j(z) = 1 + 2[Bj(z)-
ICj(z)l].
(3.13)
Squeezing occurs if ((A~j)2), ((A/3j)2) or & is less than one. Uncertainty in mode j can be characterized by the quantity uj defined as:
uj(z) = ((A~j(z))2) ((A/3j(z))2).
(3.14)
3.2. COUPLED-MODE CASE
An optical field is assumed to be composed of two modes in this case. The distribution PA;(W,z) of the integrated intensity W of the field composed of modes j and k is determined as follows:
PA/'(W,z) = f OAr(aj, ak,Z)b(W- ]~.12 -]akl2)d2ajdZak
(3.15)
and ChAr(aj, ak,Z) denotes the two-mode Glauber-Sudarshan quasidistribution. The expressions for the photon-number distribution p(n,z) and the moments of integrated intensity (Wj~(z)) are more complex and the corresponding procedures of their determination can be found in papers by Kfirskfi and Pefina [1990], Pefinovfi [ 1981 ], Pefinovfi and Pefina [ 1981 ]. Variances of the quadrature components qjk = qj + qk and Pjk = Pj + Pk and the principal squeeze variance Ajk are determined as follows: ((A~jk(z)) 2) = 2 { 1 + Bj(z) + Bk(z)- 2Re[[)jk(Z)] +Re[G(z) + Ck(z) + 2Dj.k(Z)]}, ((A/3jk(z)) 2) = 2 { 1 + Bj(z) + Bk(z)- 2Re[bj.k(z)] -Re[Cj(z) + C~(z) + 2Djk(Z)]}, ~jk(Z) = 2 { 1 + Bj(z) + Bk(z)- 2Re[bjk(Z)] -[Cj(z) + Ck(z) + 2Djk(z)[}.
(3.16)
Squeezing is reached if ((A~tjk)2), ((A/3jk)2) or ~,jk is less than two. Uncertainty in the two-mode field is characterized by the quantity ujk (ujk = ((Aqjk) 2) ((A/3jk)2)).
5, w 4]
COUPLERSBASED ON SECOND-HARMONICAND SUBHARMONICGENERATION
375
Local oscillators for each mode in the scheme of homodyne detection can be used and then the optimization of local-oscillator phases and intensities leads to a generalized squeeze variance ~,c (for details, see Fiurfigek and Pefina [2000a]). Optical fields can also be superimposed in nonlinear media and then homodyne detection of an outcoming beam can be applied. Sum or difference squeeze variance describes statistics of the field in this case (Hillery [1987, 1992]). Principal, generalized and sum squeeze variances can be defined also for fields composed of, in general, N optical beams (Fiufftgek and Pefina [2000a]). 3.3. PHASE PROPERTIES
Quantum phase properties may be conveniently studied using phase-space methods (Tanag, Miranowicz and Gantsog [1996]). Phase distribution is then determined as a marginal distribution of the corresponding quasidistribution. We further consider the quasidistribution cI)A(a,z) related to antinormal ordering of field operators with respect to its positive semidefiniteness (see, e.g., Pefina [1991]). The phase distribution PA(r is determined as follows:
PA(r
=
oA(a,z)lal dial. f0 ~176
(3.17)
Statistical uncertainty of the phase distribution Pjt(O,z) can be conveniently described by the phase dispersion a~(z) defined as (Bandilla and Paul [1969]) og(z) -- 1-(cos(q~))~- (sin(q~))~ = 1 - I (exp(iq~))q~[2.
(3.~8)
The problem of quantum phase can also be treated by defining a suitable phase operator or by using an operational approach to the definition of quantum phase (for recent reviews, see Pefinovfi, Lukg and Pefina [ 1998], Luis and Sfinchez-Soto
[20001).
w 4. Couplers Based on Second-Harmonic and Subharmonic Generation 4.1. CODIRECTIONAL COUPLER WITH X(2) AND Z( 0 MEDIA
Second-subharmonic mode ( b l ) and pump mode (b2) nonlinearly interact in a waveguide with Z (2) medium. The second-subharmonic mode bl also interacts linearly with mode a in a linear waveguide. The corresponding momentum
376
QUANTUMSTATISTICSOF NONLINEAROPTICALCOUPLERS
[5, w 4
operator in interaction pictures is written in the form (Pefina and Pefina Jr [1995a]): Gint =
-hlCab,2a2g, -
hFbA2b,2g2exp(iAkbZ) + h.c.,
(4.1)
l(ab , denotes the linear coupling constant of modes a and bl and Fb is the nonlinear coupling constant between modes bl and b2. The nonlinear phase mismatch Akb is defined as Akb = 2kb~ -kb2 and kb~ (kb2) means the wavevector of mode bl (b2). The symbols -4a, Abe, a n d Ab2 stand for optical-field operators of modes a, bl, and b2 in interaction pictures; h.c. means Hermitian conjugate. The momentum operator Gint in eq. (4.1) provides the following Heisenberg equations: where
dAa dz dab, dz dab2 dz The
9
,
^
- - ltCabIAbe,
(4.2)
itCab,Aa - 2iF6*A~,Ab2 exp(-iAkbz),
iF6AZb, exp(iAkbz).
conservation
2f(Z)2a(Z)
+
-
const
is
fulfilled by the solution of eqs. (4.2). 4.1.1. Short-length approximation
Assuming incident coherent states in all three modes (/~a ~ ~a, Abl -"+ ~bl, ,4b2 ---* ~b2 for z - 0) and using second-order solution, squeezed and subPoissonian light for small values of [Fb[Z can be generated in single mode bl and in compound modes (a, bl) and (bl,b2) if completely stimulated (~bl r 0 and ~b2 ~ 0) or partially stimulated (~b, ~ 0 and ~b2 = 0) nonlinear process occurs (Pefina [ 1995a]). Principal squeeze variances and variances of integrated intensity then take the form ~+,
- 1 + 2(4[Fb~b212z 2 --[2iF6*~b2z + [Fbl2~2 z 2 + F~Akb~b2z2[),
Zab, = 1 + 4]Fb~b2[az2 - [2iFb*~b2Z+ [F612~2 z 2 + 2F~G*b,~62z2 + F~,Akb~e2za],
Zblb2
=
1 + 4[Cb~bal2Z 2 - [2iF~ ~b2z
+ [Cbl2~glZ2 -k- 4lCbl2~bl ~b2Z2 -k- Fff Akb~bzZ2[, (4.3)
5, w 4]
COUPLERS BASED ON SECOND-HARMONIC AND SUBHARMONIC GENERATION ((AWbl)
2)
=
377
(2iFb~2 ~b*z + c.c.) + 21Fb 2z2(21~5~12 + 121~b,~b~[2 -
[~bl ]4) +
4(1-'blfabl ~a~bl ~b2Z2 + C.C.)
- ( r ~ a k ~ , ~ ; z : + ~.r
((AWabl) 2 )
=
(2iFb~21
~b2z 71-2Fb~ab 1~a~bl ~b2Z2 -- FbAkb~21 ~b2Z2
(4.4)
+ c.c.) + 2l/'bl2z2[2l~b~ 12 + 12[~b~b~[ 2 - [~b, 14],
((AWb, b2 2 )
=
(2iFb~21 ~b29 Z @ 41-'bKabl~a~bl ~b29 Z2 -- FbAkb~21~b*2Z2 + c.c.) + 2[Fb12z212[~b212 + 81~b~~b212 -
[~b114]
9
According to eqs. (4.3) and (4.4), phase mismatch may support the generation of light with nonclassical properties if phases of the incident fields are suitably chosen. Second-order, shorth-length approximation indicates the conservation of coherence in modes a, b2 and (a, b2). Using an iterative solution up to (Fbz) lz, it has been shown by Pefina and Bajer [ 1995] that coherence is conserved for relatively long distances. Squeezed light as well as sub-Poissonian light can be obtained also in modes a, be, and (a, b2).
4.1.2. Parametric approximation The assumption of a strong coherent field in mode b2 with the amplitude ~b2 leads to linearization of the operator equations (4.2) (Pefina and Pefina Jr [1995a]): dz
-
itCab~eib~ '
deib-------5'- -itCableia - 2iFb*ei~ ~b2 exp(-iAkbz). dz
(4.5)
The solution of eq. (4.5) can be reached by finding eigenvalues and eigenvectors of the corresponding matrix, as discussed in w 2. The obtained expressions are complex and can be found in Pefina and Pefina Jr [1995a]. The analysis of the dependence of eigenvalues on parameters enables us to determine qualitatively the behaviour of mode intensities; they can oscillate or exponentially increase. In our case, there are two degenerate eigenvalues (Akb = 0 is assumed): ~1,2 -- ]I-'b~b21i (IFb~b212 --I~ab112)1/2.
(4.6)
According to eq. (4.6), if the nonlinear coupling characterized by ]Fb~b2[ is stronger than the linear one, IFb~b2[ > ]tCab~I, the eigenvalues are real. In the opposite case, ]Fb~b2[ < ]tCab~l, the eigenvalues are complex and oscillations occur in the spatial development of quantities characterizing the fields. We divide the discussion of the behaviour of the coupler into three parts as follows.
Incident coherent states and [/'b~b2[ > [tCab~[. If the phase condition 2 arg(~bl)- arg(~b2) = 3:/2 is fulfilled, sub-Poissonian light in mode bl can be
378
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
< ~ :
!
Fig. 1. P h o t o n - n u m b e r distribution p ( n , z )
i!r
[5, w 4
,
for m o d e b I shows oscillations in n and parity interchange;
F b = 1, tCab ~ = 1, ~a = 1, ~eb, = exp(iar/4), ~b2 = 10, r a = rb~ = 0, and ( n o a ) = (nOb 1 ) = 0 (after
Pefina and Pefina Jr [ 1995a]).
generated. The photon-number distribution p(n,z) of mode bl develops from Poissonian distribution of the incident coherent state through sub-Poissonian distribution for small z to super-Poissonian distribution for longer z (see fig. 1). Oscillations occur in p(n,z) (Schleich and Wheeler [1987], Pefina and Bajer [1990]) and parity interchange also can be observed. If the above mentioned phase condition is not obeyed, a weakly sub-Poissonian light can be observed in mode a owing to the transfer of light from the nonlinear part of the coupler. Nonzero nonlinear phase mismatch Akb leads to oscillations in the reduced factorial moments of the photon-number distribution p(n,z) (they equal to the reduced moments of integrated intensity W) of mode bl (see fig. 2) (Pefina and Pefina Jr [1995c]). The period of oscillations is inversely proportional to Akb. Negative reduced moments of integrated intensity (further only moments) of mode bl occur also for longer z. Non-zero linear phase mismatch Akab, (Akab, = ka - kb,) also introduces oscillations in z and may provide negative moments in modes a and (a, bl). Higher values of Akb and Akab, lead to a frequent repetition of areas with sub-Poissonian statistics, but they suppress oscillations in photon number n in the photon-number distribution p(n,z). Squeezed light can be generated in all three modes a, bl and (a, bl). In general, phase mismatches diminish values of squeezing and lead to oscillations.
Incident coherent states and JFb~b21 < II~abll 9 Squeezed light and light with negative moments (sub-Poissonian light) can be obtained in the same modes as
5, w 4]
COUPLERSBASED ON SECOND-HARMONIC AND SUBHARMONIC GENERATION
379
2.0 r==q
I
1.5 1.o
J ~
~
0.5 0.0
0 . ~
~
f
~
0.0
~
I
~
~
0.5
~
~
~
~
J
1.0
~
I
~
~
1.5
~
~
I
J
~
2.0
~
~
I
r
2.5
J
~
~
4
3.0
z
Fig. 2. The second reduced moment ( W 2 ) / ( W ) 2 - 1 ( , ) and mean integrated intensity (W) (full curve without symbols) for mode bl oscillate in z; F b = 1, ~abl = 1, ~a = 1, ~bl = exp(i~/4), ~b2 = 2, Akb = 100, Akab~ = O, ra = rb~ = 0, and (noa) = (n0b~) = 0 (after Pef'ina and Pef'ina Jr [1995c]). 0.08
]
0.06 w~4
0.04
0.02 0.00 -0.02 -0.04
, , ,
0.0
i , , ,
0.1
f
0.2
r
]
~
I ' ' '
0.3 Z
I ' ' '
0.4
I ' '
0.5
0.6
Fig. 3. Reduced moments o f integrated intensity ( W k ) / ( W ) k - 1 for k = 2 (,), 3 (o), 4 (A), and 5 (<)) for mode a; F b = 1, l~ab 1 = 2, ~a = 1, ~b 1 = exp(br/4), ~b2 = 2, r a = rbl = 0, and (noa) = (nOb 1 ) = 0 (after Pef'ina and Pef-ina Jr [1995a]).
for ]Fb~b: ] > [Ifabl I" However, a strong linear coupling provides better conditions for the occurrence of light with sub-Poissonian statistics in mode a (for negative moments of mode a, see fig. 3). Usually light with sub-Poissonian statistics occurs only in one of the modes a and bl for a given z, but as z increases the modes interchange their properties periodically. In some cases light with sub-Poissonian statistics can be reached in both modes simultaneously. The greater the linear coupling constant ItCab, ], the easier the "transfer of nonclassical properties of light" to mode a. For non-zero nonlinear phase mismatch Akb the modes can return periodically to states which are close to a pure state, in contradiction to the case Akab, :/: O.
380
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 4
Incident squeezed light and noise. Nonclassical properties of incident beams are smoothed out during their propagation along z in general, but light with nonclassical properties may again occur having its origin in the nonlinear process. Incident noise rules out any nonclassical behaviour of optical fields. For great values of ItCab, [ high amount of noise interchanges between modes a and bl. Quantum zero and anti-zero effects were demonstrated in more complex systems by l~ehfirek, Pefina, Facchi, Pascazio and Mi~ta Jr [2000]. 4.2. CONTRADIRECTIONAL COUPLER WITH
X(2) AND
X (1) MEDIA
We consider the same coupler as in w 4.1 and assume that mode a propagates against modes bl and be. The operator equations are derived from eqs. (4.2) by adding the sign - in front of the derivative dAa/dz (Akb = 0 is assumed):
- itCab,Ab,
dz
dJb, '
dz
.k
-
ilCab,,4a - 2iFb*A~ Ab: ' '
dJb2 dz
,,2
iFbAb~. (4.7)
The solution of eqs. (4.7) obeys the conservation law
--2~(Z)2a(Z ) + 2~,(Z)2b,(Z ) + 2A~2(Z)2bz(Z ) = const; 2~(O)2a(O)+2~,(L)2b,(L)+ 2A~: (L)Ab: (L)
i.e.,
: A~(L)Aa(L)+ ~t~, (O)/lb, (0)+ 2A~2(O)Ab2(0). Eqs. (4.7) can be derived in quantum theory. The annihilation (creation) operator 2 0 ( 2 ~ ) o f the contrapropagating field a must be replaced by a creation A't ' ^'t t ' (annihilation) operator A a (Aa); i.e., 2 a -'-'+A a and A, ~ Aa" This scheme is based on the decomposition of the electromagnetic-field vector-potential operator into harmonic plane waves and takes into account the opposite direction of propagation of mode a. The momentum o p e r a t o r ai'nt then has the form A' t A t
A'
A2
(4.8)
Gint = -htfablAa Ab, - hff bAblA 2 + h.c. and the Heisenberg equations are written as follows: t
dAa dz
m
itCab,.~ "
dAb, dz
~
i~Cab,Aa_2iF~,j~jb2 ' AIt
dAb2 dz
iFb.,~2b~ " (4.9)
' At A't When we return to the original operators (Aa ~ Aa and A a ~ Aa) in eqs. (4.9), we arrive at equations (4.7) derived "classically". We note that despite the fact
5, w 4]
COUPLERSBASED ON SECOND-HARMONICAND SUBHARMONICGENERATION
381
that the operator solution is obtained for z E <0, L>, the interacting fieldsare determined consistently by quantum theory only for z = 0 and z = L (Luis and Pefina [1996]).
4.2.1. Short-length approximation The solution of eqs. (4.9) up to Z 2 (L 2) is formally the same as for the copropagating fields, only the operator Aa(0) at the beginning of the coupler has to be replaced by the operator.4a(L) appropriate for the end of the coupler (Pefina and Pefina Jr [1995b]). Consequently statistical properties of the interacting fields are the same as those discussed in w 4.1.1. Higher order corrections cannot be obtained in this way because the higher-order iterative solution does not provide a unique way for expressing Aa(0) in terms of.4a(L).
4.2.2. Parametric approximation Assuming a strong coherent field in mode b2, the eqs. (4.7) are linear and thus the operators
,4a(L), A~(L), .4b,(L),
and A~I(L) can be expressed as linear
combinations of the operators Aa(0), A~(0),-4bl (0), and A~ (0). The solution is finally reached when the operators Aa(0),-4ta(0), Ab~(L), and . ~ (L) are expressed in terms of the operators Aa(L), A~(L), ,4b~(0), and 3g~ (0). The commutation relations for incident beams,
[.4a(L),.,~(L)] =
[Ab~(0),2~1 (0)] -- 1,
[A.(L),Ab,(o)] = [.4a(L),.~ (0)] = 0, (4.10)
determine the validity of the commutation relations for outgoing fields: [2a(0),A~(0)] = [Ab] (L), A~I (L)]
=
1,
[Aa(0),Abl(L)]
=
[.4,(0),.4~,
(L)]
= O.
(4.11) Eigenvalues of the matrix of the eqs. (4.7) are degenerate and real: ~1,2 =
[1-'b~b2[ -~" (I/'b~b2 [2 + Ii~ab112)1/2.
(4.12)
The behaviour of optical fields (Pefina and Pefina Jr [1995b,c]) is similar to that for the codirectional coupler discussed in w4.1; i.e., sub-Poissonian light and squeezed light can be obtained in all three modes a, bl and (a, bl) under suitably chosen conditions. Moreover light with sub-Poissonian statistics can be reached
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
382
[5,
w 4
o o
-0.2
--
!
'--
'
-
0.6
. . . . . . 0.8
,
-
. . . . . 9
1.0
-
-+
- +~-
.......................... 0.0 O.ie 0.4
Fig. 4. R e d u c e d m o m e n t s o f integrated intensity ( W k ) / ( W )
-+
....
0.6
9---
+-~
0.8
-9----~
1.0
k - 1 for k = 2 ( , ) , 3 (o), 4 ( A ) , and 5
( 0 ) for m o d e a are negative; 1"6 = 1, ~Cab1 = 20, ~a - 1, ~b~ = exp(i~r/4), ~b2 = 2, ra = rb 1 = 0, and (noa) = (nOb I } = 0 (after Pefina and Pefina Jr [1995b]).
in modes a and (a, bl) and also for asymptotically long couplers (see fig. 4) if the process of second-subharmonic generation as well as the linear coupling are sufficiently strong. This is important from the viewpoint of stability of light leaving the coupler, because the optical fields are highly stable in the region of longer L. "Transmission of nonclassical light" from mode bl into mode a is better than in the case of a codirectional coupler. Non-zero values of the phase mismatches Akb and Akabl may support the occurrence of negative moments in modes a and (a, bl) in long couplers under certain conditions (Pefina and Pefina Jr [ 1995c]). Noise of the incident field in mode a can be suppressed and light with subPoissonian statistics can be generated for longer L (see fig. 5). Sub-Poissonian light is not reached when higher values of incident noise in mode a occur, but the noise in mode a can still be substantially reduced. Even smaller values of noise in mode b~ destroy sub-Poissonian statistics of mode a and lead to lower values of squeezing in mode a. Greater values of the phase mismatches Akb and Akab, suppress sub-Poissonian statistics in mode a for longer L, but noise can still be reduced. If incident squeezed light in mode a has super-Poissonian statistics with oscillations in photon number n, the oscillations are reduced in the region with small L by the interaction. However, they can occur for longer L again having their origin in the nonlinear process (see fig. 6). Phase distributions of both modes a and bl (Mi~ta Jr, l~ehfi~ek and Pefina [ 1998a,b]) splits into two identical components (see fig. 7) during the evolution along z if 2arg(~b2)- arg(~a) = ;r/2 and if ]Fb~b:] > ]tCab,[. For longer L they become zc-periodic. On the other side for 2arg(~b:)- arg(~:a) = 3~/2 the phase distributions maintain one-peak structure and phase dispersion ~r~
5, w 4 ]
COUPLERS BASED ON SECOND-HARMONIC AND SUBHARMONIC GENERATION
383
4 -
I
3
-]
1
0.0
p t J l d , l l l l , l ~ 0.2 0.4 0.6
, p ~ , , i 0,8
].0
L Fig. 5. Reduced moments of integrated intensity ( W k ) / ( W ) k - 1 for k = 2 (.), 3 (o), and 4 (/k) and mean integrated intensity (W) (full curve without symbols) for mode a; F b = 1, rabl = 10, ~a = 1, ~b~ = exp(iJr/4), ~b2 = 2, r a = rbl = O, (noa) = 1, a n d ( n 0 b 1 ) = 0 (after Pefina and Pei-ina Jr [1995b]).
~-<5~ 7 ~ ~ "~0 ~2.~ ~
~'~
Fig. 6. Photon-number distributionp(n,L) for mode a and incident squeezed light; F b = 1, tCab~ = 10, ~a = 1, ~bl =
exp(iJr/4),
~b2 = 2, r a = rbl = 1,
and ( n o a )
= (nObl) = 0
(after Pefina and Pe~ina Jr
[1995b]). defined in eq. (3.18) is minimal. Phase squeezed states, i.e., states with a lower value o f phase dispersion than that for a coherent state with the same m e a n n u m b e r o f photons, can be generated if the linear coupling is sufficiently strong (ll-'b~b2 [ < ItCab~ [). Incident noise in m o d e a means higher values o f incident phase dispersion, but values o f the phase dispersion can be substantially reduced during the interaction in the coupler. The phase distributions o f modes a and bl are strongly correlated and two
384
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 4
Fig. 7. Phase distribution PA(q)) for mode a gets a two-peak structure for longer L; Fb = 1, tfab I
=
1,
~a = 1, -~b, = exp(i:r/4), -~b2 = 2, ra = rb~ = 0, and (noa) = (nOb,) = 0 (after Mi~ta Jr, l~ehfi6ek and Pefina [ 1998a]).
Fig. 8. Phase distribution P.A(Oa) of mode a at L = 1 in dependence on the incident phase Cb~ of mode bl; Fb = 1, tCab, = 1, ~a = 1, I~b, I = 1, ~b2 = 2, r a = rb, : O, and (nOa) = (nOb I ) = 0 (after Mi~ta Jr, l~ehfi~,ek and Pe~ina [ 1998a]).
possible phase distributions in mode a mutually shifted by Jr for longer L may be reached in the dependence on the value of the incident phase of mode bl (see fig. 8). 4.3. C O D I R E C T I O N A L C O U P L E R W I T H ONE X (2) A N D T W O X (1) M E D I A
We can add a new mode into the coupler discussed in w 4.1 being linearly connected with second-harmonic mode b2. This configuration provides a broader variety of regimes suitable for the generation of nonclassical light (Pefina and Bajer [ 1995], Pe~ina [1995b]). The momentum operator Gint f o r such a coupler is in the form Gint
= _hl(ablJaj[l
_
^ " t + hFbAbl ^2 ~t~2 + h.c.; htCcb2AcAb2
(4.13)
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COUPLERSBASED ON SECOND-HARMONICAND SUBHARMONIC GENERATION
385
l~ab 1 and ~cb2 a r e the linear coupling constants and Fe stands for the nonlinear constant characterizing second-harmonic and subharmonic generation. The solution of the corresponding Heisenberg equations, dAa
dz
dAbl _
ilCab,Aa + 2iFb*J~,Je 2,
dz
(4.14)
d~ie2 _ dz dz
--
iK. e2~i~ + iFe~ize~ ' --
9 ,,~..A ~ ^ 1 "- 2 A I - 2 , ' ~ ' c b D
the q u a n t i t y A~(z)Aa(z ) + A~I(Z)Abl(Z) "-[- 2A~2(z)Ab2(Z ) "q- 2A~(z)Ac(z) conserved during the propagation. Second-order perturbation approximation leads to the conclusion that squeezed light can be obtained in modes bl, (a, bl), (c, bl) and (bl, b2):
keeps
&bl -- 1 -212iFff~b2z-[Fbl2~2 z 2 + ~cb2Fff ~z2[, &ab, - 1 -12iFff~b2z- [Fb[2~2 z 2 + l~cb2I'~cz 2 + 2~Cab~l-'ff~b2z2l,
t~cb1 = 1 - [ 2 i / ' f f ~ b 2
Z-
[/-'b12~21Z 2 +
l~cb2l-'~cZ2l,
(4.15)
Zb~b~ = 1 --12iYb*~b~z- IYbl2ff2 z 2 + lCcb2F;~cZ 2 - 4lFel2~e~e2 z2 . If the process is completely spontaneous (~a = ~bl = ~b2 -- ~ = 0), all modes conserve coherence in second-order approximation. However, if mode c is stimulated ( ~ ;~ 0), coupling between the linear and nonlinear parts of the coupler leads to squeezed light: &b~ : 1 - 2 I t b*K:~<~
Iz2,
&abl : ~r
= &bib2 -- 1 - IFb~cb2 ~c Iz2-
(4.16)
According to the expressions in eqs. (4.15), sub-Poissonian statistics can be reached, e.g., in mode bl, and linear coupling can support the generation of light with sub-Poissonian statistics in mode bl provided that the phases of incident fields are suitably chosen (for details, see Pefina and Bajer [ 1995]). These results are also applicable to the case when modes a and c propagate against modes bl and b2 (Pefina [1995b]). Higher-order approximations can be obtained using symbolic computer calculations (Bajer and Lison~k [1991], Bajer and Pef'ina [1991]) which manipulate the objects according to boson commutation relations. They use the
386
I
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
(a)
[5, w 4
(b)
&
1
b
a
0.9
0.o5
b
12
12
0.f
O.7
0.85
0.6 0"50
0.81 0.1
0.2
0.3
0.4
0"75O
f~ 011
['bZ
0;2
013
0.4
['bZ
Fig. 9. (a) Principal squeeze variances of single modes [a] Ac, [b] i~,b2, [c] /~.a and [d] '~'bl, and (b) principal squeeze variances of compound modes [a] Acb2, [b] ~'ac, [c] ~'ab2, [d] '~'blb2, [e] ~abl and [t] '~.cb] indicate the generation of squeezed light; lr 1 - lr 2 =-3/-'b, ~a = ~c -- 1, ~bl -- ~b2 - - 0 (after Pe~'ina and Bajer [1995]).
fact that the operator solution/i(z) of eq. (2.1) can be expressed in the Taylor expansion:
(4.17)
k!'
k=O
where the kth-order commutator/)k is determined as follows: Dk = [(~int,bk-11,
Do = A(0).
(4.18)
The inclusion of higher-order terms (up to the 12th order) in the expansions showed that squeezed light can also be generated in modes a, b2, c, (c, b2), (a, c) and (a, b2) (see fig. 9). Sub-Poissonian light may occur in single modes b2 and c and in compound mode (a, b2). 4.4. CODIRECTIONAL COUPLER WITH TWO X (2) AND ONE X (1) MEDIA
We consider a coupler composed of three parts; the process of secondsubharmonic generation occurs in two parts and a mode in the third part of the coupler serves to connect the second-subharmonic modes linearly. The corresponding momentum operator Gint is in the form: Gint =
,,2
^t
^2 ^t
^t
^t ^
hI"aAa,Aa2 + hFcAc]Ac2 + hl(abAalAb q- htCbc,Ac~Ab + h.c.
(4.19)
5, w 4]
COUPLERS BASED ON SECOND-HARMONIC AND SUBHARMONIC GENERATION
387
The momentum operator Gint in eq. (4.19) gives the Heisenberg equations
dAal - i t C a b A b + 2i/-'*A~lAa 2
dz
dzia2 _ i / - ' a ~ l
dz dzZib
. , ^
. , ^
(4.20)
dz - ltfabAal + ltf['cAcl'
dz
_ irbcl& +
, ^t ^ Ac
2irc Ac
'
dAc2 _ i~czi~ 1
dz
"
The solution of eqs. (4.20) conserves the quantity ~tal (Z)2al (Z) -4- 22~2(Z)Aa2(Z ) -4~i~(z).,ib(z) + ft~1 (z)~,iCl(z) + 2~i~2(z)~ic2(Z). A generalization of eqs. (4.20) to the case of non-zero linear and nonlinear phase mismatches was obtained by Mi~ta Jr and Pefina [ 1997]. The short-length solution for coherent incident fields (Mi~ta Jr and Pefina [1997]) showed that squeezed light occurs in modes al, Cl, (al, b), (b, cl), (al,a2), (al,cl), (al,c2), (a2,cl) and (cl,c2). Sub-Poissonian light may be obtained in modes al and Cl.
4.5. CODIRECTIONALCOUPLERWITH TWO
X (2)
WAVEGUIDES
Couplers composed of two waveguides (Pefina and Pefina Jr [1996]) in which the process of second-subharmonic generation occurs are considered. The fundamental mode in the first (second) waveguide is assumed to be in a strong coherent state with the amplitude ~a2 exp[i(ka2 + kb2)z/2] (~b2 exp[i(ka2 + kb2)z/2]). The coupler is then described by the following momentum operator (~ in the Heisenberg picture: = hkal~l~al~lal --I-
hkb~^tabl abl ^
-Jr [hl-'a~i2al~a2* exp[-i(ka2 + kb2)z/2]
(4.21)
+hFbgt~, ~b*2exp[-i(ka2 + kb2)z/2] + htcabhal ab~ ^ t + h . c .]. The wavevectors in the second-subharmonic (pump) modes are denoted a s kal and kbl (ka2 and kb2), Fa (Fb) is the nonlinear coupling constant in the first
388
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, {} 4
(second) waveguide, and the constant tCab describes linear coupling between the second-subharmonic modes. Substituting ha, = Aai e x p
we arrive at the following set of operator equations with constant coefficients: dAa I
. Aka
- AI ^
-- l ~ A a l
9 . ^ I + 2iFa~a2f~ ' + llr
dz 2 dAb, . Akb + AI ^ - l ~ A b , + ilfabAa, + 2iFff ~b2A~, ; dz 2
(4.23)
Aka = 2ka, - ka 2, Akb - 2kb, - kb2, and AI = (kb: - k~:)/2. Assuming phase matching, arg(Fa~a2) - arg(Fb~b2) = 0 or ~, and ]Fa~a21--]Fb~b21, one can find a simple analytical solution of eqs. (4.23) (Korolkova and Pefina [ 1997d]) and it has an oscillating character for [tCab[ > 2[Fa~a2 [,
whereas it behaves exponentially for Irabl < 21ra~a~l. The condition I~:abl 21/'a~a~ I is the most suitable for the generation of single-mode squeezed light. Discussion of statistical properties of the coupler can be conveniently split as follows. I n c i d e n t c o h e r e n t states a n d p h a s e matching. Statistical properties of the optical
fields strongly depend on the phase differences 0 a -- 2 arg(~a, ) - arg(~a2)- arg(F~) and Cb = 2 a r g ( ~ b , ) - arg(~b2)- arg(Fb). If q~a (r is equal to -at/2, subPoissonian statistics for smaller values of z in mode al (bl) occur. If O a and q~b equal-Jr/2, Ira~a~l ~ Irb~b=l, and the linear coupling is weak, sub-Poissonian statistics are obtained in modes al, bl, and (al,bl) for smaller values of z. For larger values of z, the photon-number distributions are superPoissonian with oscillations in photon number n and also parity interchange is observed. Squeezing is reached in modes al, bl and (al, bl) for longer values of z. Greater values of ItCabl suppress oscillations in photon-number distributions and they lead to lower values of squeezing. If ]tCabl is large enough, oscillations in z occur. They cause the occurrence of negative moments for larger values of z, but absolute values of the negative moments decrease with increasing ItCabl. If the waveguides are not in balance, i.e., [Fa~a21 4:]Fb~b2 l, the difference between the intensities of modes al and bl decreases with increasing ItCab[ and the coupler behaves similarly as in the balanced case. If q)a = -~r/2 and Ob = at/2, light with negative moments can be generated only in mode al for weaker linear coupling. Larger values of I~abl introduce
5, w 4]
COUPLERSBASED ON SECOND-HARMONICAND SUBHARMONIC GENERATION
389
Fig. 10. Mean integrated intensity (W) (solid curve without symbols) and negative reduced moments of integrated intensity (Wk)/(W) k - 1 for k = 2 (,), 3 (o), 4 (A), and 5 (0) for mode ( a l , b l ) ; I'a~a2 = exp(iJr/2), I'b~b2 - 0.999exp(-iJr/2), lr = 1, ~)ka - -100, 6kb = Akab = 100,
~al - ~b~ = 1, ra~ = rbl = 0, and (n0al) = (nOb~)= 0 (after Pefina and Pefina Jr [1996]).
oscillations in z and negative moments can occur in modes al and bl. Also squeezed light may be reached only in single modes al and bl. If Oa = q~b = :r/2 and ItCabl is so large that oscillations in z occur, negative moments in all three modes al, bl and (al, bl) also are reached for larger values of z in the balanced case. Negative moments for larger values of z are lost if the coupler is not in balance. Squeezed light can also be generated in all three modes. Incident coherent states and nonzero p h a s e mismatches. It is convenient to introduce the nonlinear phase mismatches 6ka = 2kal - (ka2 + kb2)/2 and 6kb = 2kbl - (ka2 + kb~)/2 with respect to the phase oscillations of the strong fundamental fields in modes a2 and b2. Non-zero nonlinear mismatches 6ka and 6kb cause oscillations in z and lead to regions with negative moments in all three modes. These regions repeat periodically along z. Non-zero values of 6ka and 6kb suppress oscillations in photon-number distributions. The oscillations in z occurring for larger values of [l~ab [ a r e superimposed on the oscillations given by 6ka and 6kb. Sub-Poissonian light and squeezed light can be achieved in
modes al, bl and (al, bl) also for larger values of z under suitably chosen values of parameters. Non-zero phase mismatches 6ka, 6kb and Akab (Akab = kb~ - kay) may stimulate the generation of light with larger absolute values of negative moments (see fig. 10). Incident squeezed states and noise. Incident squeezing is gradually suppressed
by the dynamics of the nonlinear process, but squeezed light having its origin
390
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 4
in nonlinear dynamics can be observed for larger values of z. Incident noise smoothes out oscillations in n in photon-number distributions. It contributes to the values of mean intensities, moments, and squeeze variances by approximately the same amounts for all z and suppresses nonclassical properties of light. Karpati, Adam, Janszky, Bertolotti and Sibilia [2000] discussed all-optical switching in this system. 4.6. C O N T R A D I R E C T I O N A L C O U P L E R W I T H T W O X (2) M E D I A
We assume that modes bl and b2 in the coupler discussed in w 4.5 propagate against modes a l and a2 (Pefina and Pefina Jr [1996]). Equations governing the dynamics of the coupler can be derived from the momentum operator in eq. (4.21) and the operator equations in (4.23): dAa, dz
- i
A k a - A l Aa, 2
" * "
+ ll(abAb'
. + 2ira
"t
~a2Aa" (4.24)
d,4b~ dz
. Akb + AI ,
-- l ~ A b , 2
--iICabAa, -- 2iF~b22~ .
If arg(Fa~a2) = arg(Fb~b2) + Jr = 0 or Jr, [Fa~a21 = IFb~b21, and phase matching is assumed, the solution of eqs. (4.24) has oscillating character (Korolkova and Pe~ina [1997d]). A substituting scheme for these devices was suggested by Fiurfi~ek and Pefina [2000c]. I n c i d e n t c o h e r e n t s t a t e s a n d p h a s e m a t c h i n g . If the linear coupling constant
is small, the contradirectional coupler behaves similarly as the codirectional one discussed in w 4.5.1. If Irabl is large, Oa = ~6 = - r and Ir~a~l ~ Irb~l, the optical fields have a tendency to return to coherent states for larger values of L in all three modes al, bl and (al, bl) (see fig. 11). States closed to coherent states are obtained for smaller values of L for larger values of I~:abl. When the coupler is unbalanced, negative moments occur in all three modes only for small values of L, but negative moments are kept also for large values of L in a single mode with smaller pumping (given by I r . ~ l , Irbffb~l). If ~a = - : r / 2 , Ob -- :r/2 and [tCab[ is large, negative moments in mode bl (in general in a mode with incident super-Poissonian statistics) occur if L is large (see fig. 12 where a typical spatial development of the second reduced moment of integrated intensity for several values of l(ab is shown). If Ca = 0b = r and [l(ab ] is large, negative moments can be obtained only in single modes a~ and bl for smaller values of L provided that the coupler is balanced. If the coupler is unbalanced, negative moments, even
5, w 4]
COUPLERS BASED ON SECOND-HARMONIC AND SUBHARMONIC GENERATION
391
2.0
~J >~
.5 1.0 0.5 o.o -o.5 -1.0 0.0
O.
0.4
0.6
O.
L Fig. 11. Mean integrated intensity (W) (solid curve without symbols) and reduced moments of integrated intenisty ( W k ) / ( W ) I ' - 1 for k = 2 (.), 3 (o), 4 (/k), and 5 (<)) for mode al develop to values appropriate for a coherent state for longer L; Fa~a2 = Fb~b2 = exp(ier/2), ICab = 10, 6ka = 8kb = Akab = O, ~al = ~bl = 1, ral = rb~ = O, and (noah) = (nOb1) = 0 (after Pefina and Pefina Jr [ 1996]). 0.4
~-~
I
-
0.2
0.0 J ~
-0.2
-0.4
F
0.0
0.5
~
1o0
l
1.5
Fig. 12. The second reduced moment of integrated intensity <W2)/(W)2- 1 for mode bl for different values of tCab: tCab = 0 (*), tcab = 2.5 (o), tcab = 3 ( A ) , tcab = 5 ( 0 ) , tcab = 10 ([2]), and tCab = 2 0 (e); Fb~b2 = 0.1 e x p ( - i J r / 2 ) and values of the other parameters are the same as in fig. 11 (after Pefina and Pefina Jr [ 1996]).
for great values of L, occur only in a mode with stronger pumping. In general, squeezed light can be generated in modes al, bl and (al,bl), but increasing values of ]tCabl lead to lower values of squeezing.
Incident coherent states and non-zero phase mismatches. Oscillations along z caused by non-zero phase mismatches 6ka, 6kb and Akab occur only for smaller values of L. They are smoothed out by nonlinear dynamics for larger values of L. The greater the linear coupling constant tc~b is, the more rapidly the oscillations are suppressed.
392
QUANTUMSTATISTICSOF NONLINEAROPTICALCOUPLERS
[5, w 5
Incident squeezed states and noise. Greater values of moments of incident squeezed light may be replaced by negative moments (even for great values of L) if the light is injected into a mode which provides negative moments for great values of L with incident coherent states. Incident squeezed light in a mode suppresses the occurrence of negative moments in the other mode. Incident squeezing is in general smoothed out for longer L. Incident noise is suppressed if it is injected into a mode having negative moments for great values of L with incident coherent states. Otherwise the noise survives in the coupler and contributes to the statistical quantities by approximately the same amounts independently of L. w 5. Couplers Based on Nondegenerate Optical Parametric Processes 5.1. CODIRECTIONAL COUPLER WITH TWO X(2) MEDIA
A coupler composed of two nonlinear waveguides with nondegenerate optical parametric processes is investigated (Janszky, Sibilia, Bertolotti, Adam and Petak [1995], Herec [1999], Mi~ta Jr [1999], Abdalla, E1-Orany and Pefina [1999], Mi~ta Jr, Herec, Jelinek, l~eh~rek and Pefina [2000]). Pump (ae, bp), signal (as, bs), and the idler (al, bi) modes in one waveguide are assumed to interact linearly with their counterparts in the other waveguide. The coupler is described by the following momentum operator:
=h
Z ki~l:~li-l-h [l"a~lapr i=ap,as,al,bp,bs,bl
+ [,babpabsab,^ ^t ^t + h.c.]
+ IhtCp~lap(l~bp+ hlCs~las(l~s+ hlcI(lal~l~l-Jr-h.c.]
(5.1) ,
where Fa (Fb) is the nonlinear coupling constant in waveguide a (b) and top, tCs and tr stand for the linear coupling constants between the pump, the signal and the idler modes. The symbol ki denotes the wavevector of the ith mode along z-axis. The coupler has two symmetries. The first one is based on the exchange of subscripts a and b together with substitutions tCp -~ tc~, lCs -~ tc~, and tCl ~ Ic[. The second symmetry arises when the subscripts S and I are exchanged. Taking into account these symmetries, the description of the coupler may be considerably shortened. Substituting
hje = Ajp exp(ikpz),
hjs =/~js exp(iksz),
hj, = [tj, exp(iklz),
for j = a, b, (5.2) where ke = (kap + kb,~)/2, ks = (kas + kbs)/2, and kI = (kai + kb,)/2, we arrive at the following set of equations:
5, w 5]
COUPLERSBASED ON NONDEGENERATEOPTICAL PARAMETRICPROCESSES
d3
--- ap _
dz
Akp iAap + lICfiAbp"* " 2
+ lI-' a " * "AasAal"
dAa s .AksA . .^ d z - 1 - - - ~ as + ltCsAbs +
exp(iAkz),
^ ^t iFaAa~Aa, exp(-iAkz),
dAal - i A-~~Aal + 1lr *Ab," + il-'aAapAas"
dz
393
(5.3)
^t exp(-iAkz),
where Akp = kap - kbp, Aks = kas - kbs, Aki = ka, - kbz, and Ak = ks + ki - kp. 5.1.1. Short-length approximation
Iterative solution of eqs. (5.3) up to the second order together with the assumption of incident coherent states leads to the conclusion that squeezed light can be obtained only in compound modes (ae, as), (as, ai), and (as, bl) (Herec [1999]): i~aeas = 1 -Jr-IFal21~a~l(l~a~l-
]~a~])z2,
l~asai -- 1 - - [ F a ( 2 i ~ a ~ Z -- ( k a ~ - k b , - kbl)~apZ 2 - 1C;~bp Z2)
-IFal2~as~a, Z2f + 2lFa[21~ae[2Z 2, Zasb, = 1 +
(IFa~ap[ 2 +
IVbffb~12-
IFatCi~ap +
(5.4)
Fb~cy~b~[)z2.
According to eqs. (5.4), squeezing in mode (ap, as) ((ap, ai)) occurs if the amplitudes of the incident coherent states satisfy the condition I~a~l > I~a~l (l~al[ > [~ap 1).
Short-length approximation provides the following expressions for singlemode variances of integrated intensities: ( ( A m a e ) 2) -- 0,
(5.5)
((AWas) 2 ) -- 2lFa~ae~asl2z 2, ( ( A m a / ) 2 ) = 2]Fa~ae~all2Z 2,
i.e., mode ap remains coherent and modes as and ai are super-Poissonian. 5.1.2. Parametric approximation
Assuming strong coherent fields in pump modes ae and bp (hap --~ ~ap, abp --~ ~bp) and phase matching, equations (5.3) can be simplified (Mi~ta Jr [ 1999]) to d~ t dmas 9 ," dz - 1K'sAbs +
iI-'a~aP2~l,
dza,
--
ilCi2~1 -- ll""*~'* a ~ap Aas
(5.6)
394
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 5
(equations for ,4bs and A[, are obtained from symmetry). Three different regimes occur in spatial evolution of the coupler according to the character of eigenvalues of the system of linear differential equations (5.6): (I) Ifa < -2b (a = [tCs[2 + [K'I[ 2 --[l"a~ae[ 2 --[l"b~bp[ 2, b = [l(sl( I -- F*I'b~ap~bpI), the eigenvalues are real,
l(
'~,1,2,3,4 -" nt-~
)
v / - a - 2b + v/-a + 2b ,
(5.7)
and the amplification of the interacting fields occurs. (II) I f - 2 b < a < 2b, the eigenvalues acquire the form: ~,1,2,3,4 = ~
1 (x/-a +
2b=lzi~)
(5.8)
and damped oscillations are obtained. (III) If a > 2b, the purely imaginary eigenvalues,
i(
'~,1,2,3,4 -- -'l-~
)
v / a - 2b :t: x/'a + 2b ,
(5.9)
determine oscillatory behaviour of the coupler. If the incident fields are assumed to be unsqueezed, squeezed light cannot be obtained in single modes (Mi~ta Jr [ 1999]). However, squeezed light may occur in compound modes. Assuming the phase condition arg(F~*~ase~a,e~p) = -~/2 (optimal for the generation of nonclassical light) in waveguide a and spontaneous process in waveguide b, squeezed light and also light with sub-Poissonian statistics can be generated only in mode (as, at) provided that the linear coupling constants Irsl and Irll are small. Greater values of Itcsl and IrlI introduce oscillations along the z-axis and lead to squeezed light in modes (as, bi) and (bs, ai) and sub-Poissonian light in modes (as, b1), (bs, ai) and (bs, bi). If stimulated processes occur in both waveguides and phase conditions optimum for nonclassical-light generation in both waveguides are fulfilled, squeezed light and sub-Poissonian light is observed in modes (as, a1) and (bs, bi) even for weak linear coupling. Stronger linear coupling stimulates the occurrence of squeezed light and light with sub-Poissonian statistics in modes (as, bD and (bs, ai). If it holds that arg(Fi*~is~i,~p) ~ -Jr/2 for i = a,b, squeezed light is reached in modes (as, at), (bs, bl) and (as, bt), (bs, aI) (only for larger linear coupling). For larger values of ]tCsl and Itr and under the phase condition arg(Fa ~as~bl~ap) = arg(Fb*~as~b,~p) -- 0 mode (as, bi) provides sub-Poissonian light (see fig. 13).
5, w 6]
COUPLERS BASED ON THE KERR EFFECT
395
10 O6
I ~ x
O6 o4 02 O0
~
0 . ~
' ~
0.0
'
I
0.4
r
,
r
I
~
~
'
0.8
I -~=-'
1.2
r
I
~
~
1.6
Z
Fig. 13. R e d u c e d m o m e n t s o f integrated intensity (Wk)/(W)k- 1 for k = 2 (,), 3 (o), 4 (A), and 5 (<)) for m o d e (as, bi) m a y take on negative values; Fa = Fb = 1, tr s - 8, trI = 0, top = 0, and ~ap = ~bp = ~as = ~bs = ~ai = ~bl 1 (after Migta Jr [1999]). =
The coupler can be conveniently used for the transfer of energy between modes without introducing additional noise (Janszky, Sibilia, Bertolotti, Adam and Petak [ 1995]). Energy transfer is controlled by the pump-field intensity and this property may be used for all-optical switching. A coupler with the same configuration of modes but with one nonlinear waveguide (supporting parametric process) and one linear waveguide has been investigated by Herec and Migta Jr [1999]. They have shown that from the viewpoint of nonclassical-light generation this coupler behaves similar to the coupler discussed above.
w 6. Couplers Based on the Kerr Effect
Propagation of light in media with Kerr nonlinearities has been studied extensively mainly in optical fibers (for a recent review, see Sizmann and Leuchs [ 1999]). Optical fibers as well as certain organic polymers with high third-order nonlinearities may be used for the construction of couplers based on Kerr effect. A coupler with two nonlinear waveguides (denoted as a and b) with Kerr nonlinearities is quantally described by the momentum operator aint (ka = kb is assumed for the wavevectors) (Pefina, Horfik, Hradil, Sibilia and Bertolotti [ 1989], Bertolotti, Sibilia, Horfik, Pefina and Janszky [ 1990], Chefies and Barnett [1996], Korolkova and Pefina [ 1997a], Fiurfigek, I~epelka and Pefina [1999a,b], Ibrahim, Umarov and Wahiddin [2000], Ariunbold and Pefina [2000]): Gint = n g A a Aa + n g A b A b + h g a b A a A a A b A b +
hKabAaA
+ h.c..
(6.1)
396
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 6
We assume that the Kerr nonlinear coupling coefficients in both waveguides are the same and are denoted as g. The real constant gab describes nonlinear coupling of modes and a real constant tCab characterizes linear coupling of modes. The corresponding Heisenberg equations have the form
diia _ llfabA b + 2ig(ii~Aa)lia + igab(Zi~iib)Aa,
dz
(6.2)
dab _ ilCaAa +
dz
9
At ^ ^
2ig(A~Ab),4b+ lgab(AaAa)Ab.
The operator AaA ^t^ a + 2~Ab is a constant of motion of eqs. (6.2). If we substitute the operators Aa and iib by the operators B, and B6,
Aa "-- --~l [Ba exp(itcabz) + Bb exp(-itcabz)] (6.3)
,4b : ---~l [Ba exp(itCabz) - Bb exp(-itCat, Z)] and neglect fast-oscillating terms, we arrive at a soluble set of nonlinear operator equations:
d& --
ig ((1 +
]/ab)B~aBa-Jr-2B[Bb)/ ha,
\ dz dBb - i g (2B~h \ - - a 4;-(l dz
-k- Yab)B~Bb)
(6.4)
hb;
Yah = gaj(2g). The approximation based on the omission of fast-oscillating terms is valid only for smaller values of the product of nonlinear coupling constants and variable z (Fiur~ek, K~epelka and Pefina [1999a]). Taking into account the conservation of the operators /~a~Ba and B~/~b along z (B~a(Z)Ba(Z) -- / ~ a ~ ( 0 ) h a ( 0 ) , B~(z)Bb(z ) -- /~(0)Bb(0)), the solution of eqs. (6.4) can be found (Korolkova and Pefina [1997a]): /~a(Z) = exp
[igz ((1+ yab)B~(O)Ba(O)+ 2/)~(0)/~b(0))] Ba(0),
Bb(z) = exp
[igz
(6.5) (2Ba~(0)ba(0)--I-(1-k-
Yab)B~b(O)bb(O))] Bb(O).
Using the transformation given in eq. (6.3) in the solution (6.5), we get the solution of the original equations (6.2). We note that the obtained solution is exact for Yah -- 1. Analogous results can be found for the contradirectional Kerr coupler (Ariunbold and Pefina [2000]).
5, w 6]
COUPLEnS BASED ON THE KERR EFFECT
397
An exact solution of eqs. (6.2) can be found numerically by solving the corresponding Schrrdinger equation in the Fock basis [na,nb) (Chefles and Barnett [ 1996], Fiur/t~ek, Kfepelka and Pefina [ 1999a,b]) provided that incident pure states are assumed. The eigenstate [~(z)) is then expressed as follows: O<3
ItIl(z)) - E
O0
E
Cna'nb(Z)[na'nb)'
Cna,nb(Z) =
(na, nbl tI-t(z)).
(6.6)
na = 0 nb= 0
Determination of the coefficients C,,a,,,b(Z) is possible even for greater values of n~ and rib, because the Hilbert space of all states splits into subspaces invariant with respect to the momentum operator Gint in eq. (6.1) as a consequence of the relation [2~2 a -+-2~2b, a i n t ] - - 0. The phase distribution P.a(r of mode a defined in eq. (3.17) may be expressed in terms of the density matrix elements Pma,na = ~mb:0 Cma,mbCn*,mb as follows (Fiurfi~ek, K_fepelka and Pefina [1999a,b]): o<3
PA(r
= -~ E
f(1 + [m + n]/2) exp[i(n-m)r Pm,n(Z)
(6.7)
m~n=O
where F denotes the Gamma function. The procedure for obtaining a singlemode phase distribution P.A(Oa,Z) c a n be easily generalized for two-mode fields and a two-mode joint phase distribution P.A,ab(~Oa,Ob,Z) may be obtained (Fiur~ek, I~epelka and Pefina [1999a,b]). A distribution P A,a_b(AOab,Z) of the phase difference ACab = ~ a - Cb can then be determined: oo
P'A'a-b(A~ab'Z) : 2-~ E
o<3
E
Pmamb'na(mt'+ma-na)(Z)
ma,na=O mb=max(O,na--ma)
• F(1 + [ma + na]/2)F(1 + [2mb + ma - na]/2) v/ma!na!mb!(mb + ma -- na)! • exp[i(na - ma)Aq~ab], where
(6.8)
Pmamb,nanb --- Cma,mb Cna,nb"
N-fold symmetries of phase distributions are clearly indicated by higher values of the Fourier coefficients F(n,z) defined as follows:
F(n,z) = ](exp(inr
I.
(6.9)
The Kerr coupler behaves periodically under special conditions. Namely, if there is no linear coupling between the waveguides (tcab = 0) and gab/g
398
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 6
Fig. 14. Phase probability distribution PA(~a,z) of mode a shows periodicity in z as well as N-fold symmetry in 0a; tCab= 1, g = 0.1, gab = 0.2, ~a = 2, and ~b = 0 (after Fiunigek, I~epelka and Pefina [ 1999a]).
is expressed as a fraction of integers. In this case the state of the field is given as a discrete superposition of a finite number of coherent states for lengths z = Mzr/N, where zr describes the period of spatial oscillations, and M and N are primes. The single-mode phase distributions PA(r as well as the quasidistributions ~A(a) have N-fold symmetry. Also if 2g = gab, ICab ~ 0, and the ratio g/l(ab is equal to a quotient of two integers (real Kab is assumed), periodic behaviour occurs (see fig. 14 for a typical behaviour of the phase distribution PA(r The overall period of oscillations is determined both by the period z l = zr/g arising from nonlinear phase modulation and by the period z2 = :r/lCab having its origin in linear coupling. If g/l(ab is not equal to a fraction of two integers, beats between the two oscillations are observed and the incident state is never fully reconstructed. We note that the period of energy exchange between two waveguides is given by z2. Also the evolution of the phase distribution PA,a_b(A~ab, Z) is periodic with the period z2 and the phase distribution PA,a-b behaves in the same way as if the coupler was linear and the modes were in coherent states at z = 0 (Fiur~i~ek, I~epelka and Pefina [ 1999a,b]). Such a behaviour reflects that the nonlinear phase modulation described by g is fully compensated by the nonlinear coupling described by gab. In general, the behaviour of the coupler will be dependent on the parameter ~1 = 12g--gab[(l~a[ 2 + [~bl2)/l(ab (Chefles and Barnett [1996]). Assuming mode a being in a coherent state with the amplitude ~ ~ 0 and mode b in vacuum state (~b = 0), there is a threshold at the value r/= 4. Oscillations and their collapses and revivals occur below the threshold. At the threshold the incident energy is asymptotically uniformly distributed between the modes. The incident energy
5, w 6]
399
COUPLERS BASED ON THE KERR EFFECT 4
'
' eo o
9
62
0
0
20
40 z
60
80
Fig. 15. Mean photon number of mode a (solid curve) and mode b (dotted curve) for the coupling function tCab(Z) with hyperbolic-secant profile (for details, see Korolkova and Pef'ina [1997c]); g = 0.1, gab = 0, ~a = 2 and ~b = 0.
is trapped into mode a above the threshold. The spatial evolutions of the mean photon numbers in modes a and b and the phase-difference distribution P.A,a-b are closely related. An optimum energy exchange between modes a and b occurs if the difference of the phases of the complex amplitudes of modes a and b equals :r/2 o r - : r / 2 . The phase-difference distribution Pjt,a-b has a peak around n'/2 or -r in this case. The mean photon numbers of modes a and b then oscillate in z. There may appear a collapse of energy exchange between the waveguides w h e n (A~Aa) ~'~ (A~Ab) for a quite long interval o f z (see fig. 15). The evolution of the mean photon numbers is characterized by damped oscillations and then a collapse occurs. The envelope of damped oscillations is given by the nonlinear parameters g and gab whereas the period of oscillations is determined by the linear coupling constant tCab (Korolkova and Pefina [1997a]). This collapse is reflected in the shape of the phase-difference distribution PA,a-b which undergoes a bifurcation and gets two-fold symmetry. A revival of energy exchange occurs after every collapse, but the revival is not complete. This is illustrated in fig. 16. At z - 0 the distribution PA,a-b is constant as a consequence of incident vacuum state in mode b. The distribution P,a,a-b has one peak in the region of periodic exchange of energy, namely at AOab = 3ZC/2 for z = :r/4 and at AOab = Jr/2 for z = 3zc/4. The distribution P.A,a-b has two-fold symmetry for z = 10n: when a collapse of energy exchange occurs; it has two peaks at Ad/)ab = zc/2 and 3:r/2. The point z = 20:r is already in the area of revival and that is why the distribution P.A,a-b has only one peak. However, the distribution P.A,a-b is flatter compared
400
06]
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 6
i
o5J
i
t ~c I
~o
3
, /
1
0
t
0
1
2
I
~
3
I
4
I
I
5
i
I
6
ACab Fig. 16. Phase-difference distribution P.A,a_b(Aq)ab) for z = 0 (solid curve w i t h o u t symbols), z = : r / 4 (o), z = 3 J r / 4 (V]), z = 10:r ( A ) , and z = 20~r ( 0 ) reflects collapses a n d revivals o f m e a n p h o t o n numbers: tCab = 1, g = O. 1, gab = 0, ~a = 2, and ~b = 0 (after Fiunigek, K~epelka and Pe~'ina [ 1999a]).
to that at z = 0, because the revival is only partial. Collapses and revivals do not occur in the region near the threshold. Squeezed light can be generated in mode a which is in a coherent state with ~a ~ 0 at z = 0. Non-zero values of the nonlinear coupling constant gab lead to the occurrence of squeezed light also in mode b being in vacuum state at z = 0. The degree of principal squeezing depends strongly on the intensities of modes a and b, because they lead to nonlinear phase shifts which are the basis for quantum effects. The principal squeeze variances/~a and Zb are greater than or equal to 1/2. Also the compound mode (a, b) can be squeezed, but squeezing is weak in this case. Photon-number statistics in modes a and b are sub-Poissonian when values of the parameters are suitably chosen. The character of the evolution of mean photon numbers in the regions of revivals can be controlled by the z-dependent linear coupling constant tCob(Z) (Korolkova and Pe~ina [1997c]). A z-dependent linear coupling constant can be
5, w 7]
COUPLERS BASED ON RAMAN AND BRILLOUIN SCATTERING
401
obtained by adjusting the coupler geometry. Switching of energy between the waveguides is achieved in this way (see fig. 15); the switching characteristics are determined by the profile of the coupling function tCab(Z) (for details, see Korolkova and Pefina [1997b]). The coupling function tCab(Z) may also be designed in order to preserve squeezing in a given waveguide. An interesting analogy of Kerr coupling can also occur in Bose-Einstein condensates (Kuang and Quyang [2000]). Chaos in coupled Kerr oscillators was discussed by Grygiel and Szlachetka [2000]. w 7. Couplers Based on Raman and Brillouin Scattering A great deal of attention has been paid to the couplers which are based on Raman and Brillouin scattering (Pefina Jr and Pefina [ 1997], Fiurfigek and Pefina [ 1998, 1999a, 2000a,b]). The momentum operator G,
j aj, + j=
,
A ht + h~jshjlhJvht + h.c.
I=L,S,A,V
A At A At h.c.] + V[h trsaasabs + ]'llfAaaAabA -k_~ (7.1) is appropriate for a coupler composed of two waveguides. The constant ~j~ (~j~) describes the Stokes (anti-Stokes) nonlinear coupling in waveguidej, whereas the linear coupling constants tCs and tea have their origin in the coupling of Stokes and anti-Stokes modes in different waveguides through evanescent waves. The momentum operator 0 in eq. (7.1) reflects a symmetry of the coupler being based on the exchange of suffices (aL, as, aA,av) and (bL, bs, bA,bv) and the replacement of the constants tea and tCs by their complex conjugate ones. Assuming strong coherent states in pump modes aL and bL (hj~ ~ ~jL exp(ikjLz) for j = a,b), the momentum operator 0 in eq. (7.1) provides the following equations for operators in interaction picture Cij, = hj~ exp(-ikj~z)): Aas
dz
alia A
dz
dziav
dz
At
- igas exp(iAkasz)Aa~ + itc~ exp(-iAksz)~ib~, (7.2)
-igaA exp(iAkaAz)~iav + itc~ exp(-iAkAz)~ibA, -
YavAav + igaA* exp(-iAkaAZ)~iaA + igas exp(iAkasz)~s
+ Lay
gjs = gjs ~L and gjA = gjA~j~. Vectors characterizing phase mismatches are defined as follows: Akj~ = ~L - kj.V - k/s , AkjA = kj.~ + kjv - kj~, Aks = kas -- kbs,
402
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 7
and AkA = kaA- kb,. We have introduced damping of the vibration mode av; a damping term (--ga~ta~) and the operator Lay a Langevin force have of
occurred in the third equation in (7.2) (for details, see Pefina [1991]). If we omit damping of the vibration modes (~v = 0 for j = a,b), a conservation law of the number of photons in the coupler may be derived from eqs. (7.2): Y~'~j= a,b(A;v (z)Aj,, (z) + ~tj.,(z)Aj~ (z) - ~tjs (z)AJs (z)) = coast. 7.1. SHORT-LENGTH APPROXIMATION
A short-length solution of eqs. (7.2) for Brillouin scattering (vibration modes are in coherent states at z = 0) and for incident coherent states in Stokes and antiStokes modes leads to the conclusion that sub-Poissonian light and squeezed light can be generated only in compound modes (as, aA), (as, av) and (as, bv) if the phases of incident fields are suitably chosen. Variances of integrated intensities have the form ((AWasa4) 2) = 2lgasl2l~asl2Z 2 - (gaAgas~a*s~a g 2 + C.C.), ((AWasa,,) 2) = 2[(igas~as~av z + c . c . ) + ]gas[2(1 + 3]~:as [2 + 3[~av]2)z 2 + (gas ga49 ~a, ~" f:* f:* z2 nt- C.C.)], -as* Z2 -+- C.C.) -+- (gasXSbbs~av
((AWash,,) 2) = 2[gasl2[~asl2Z 2 + 2lgbslZl~b~,12z2 -(gbsK~as~,v z2 + C.C.).
(7.3) Linear coupling between the Stokes modes as and bs enables the generation of sub-Poissonian light in mode (as, by). It can also support the occurrence of negative moments in mode (as, av). The principal squeeze variances are determined as follows:
l~asaA 211 + [gasl([gas[- Iga, {)z2], l~asa,. 211 - 21g~slz + Igas[2z2], =
=
(7.4)
l~asbv 211 + ([gas[2 + [gbsl2)z 2 -[gbsrSlz2]. --
Considering Raman scattering (vibration modes are in chaotic states at z = 0 with mean photon numbers nay and nb~), the short-length solution indicates negative moments of integrated intensity only in mode (as, aA): ((AWasaA) 2) = 2lga~[2(nav + 1)l~a, 12Z2 + 2lgaA[2nav[Gl2z 2
- [gaAgas(2nav + 1)~a*~g*~Z2 + C.C.].
(7.5)
5, w 7]
COUPLERS BASED ON RAMAN AND BRILLOUIN SCATTERING
403
We note that second-order, short-length approximation cannot reveal the role of Stokes and anti-Stokes linear couplings in this case. On the other hand squeezed light can be obtained in modes (as, aa), (as, av), (aA, av), (as, bv) and (aA, bv):
l~asaA
=
211 + Igas 12(nav + 1)z 2 + IgaA[enavz2
-IgaAgasl(1 + 2nay)Z2], '~'asav
=
211 + Flay -- 2[gas [(nay + 1)z + 2[gas [2(nay +
1)z 2
- IgaA12nazZ2], XaAav -- 211 +nav--(ig~,~navZ+C.C.)+]gasl2(nav + 1)Z2],
(7.6)
2asb~ = 211 + nb~ + Igasl2(nav + 1)z 2 + Igb, 12(nbv + 1)z 2 -[gbAl2nb~z e --[gb~lCsl(nbv + 1)Z2], ZaAb~ -- 211 + nb~ + Igb~ [2(nb~ + 1)z 2 + Iga~ 12na~zz -
[gb~ 12nb~z e - (gb*~tCAnb~z2 + C.C.)].
Expressions in eqs. (7.6) show that Stokes coupling can generate squeezed light in mode (as, by) and anti-Stokes coupling can provide squeezed light in mode (aA, by). 7.2. A N A L Y T I C A L A N D N U M E R I C A L R E S U L T S
If we consider that the nonlinear process occurs only in waveguide
a (gbs =
gb~ = 0), omit damping of the vibration mode av, and assume phase matching of all modes, eigenvalues of the system of linear differential equations written in (7.2) can be found (Fiurfigek and Pefina [1999b]): 1 /~ 1,2,3,4 -- -l-
~ / - p + V/p 2 - 4 q ,
p = [tcslZ + ltCAIZ--(Igas[Z-- [gaAl2),
/15 = 0,
(7.7)
q= ItfStCA]Z+ ltCsgaAIZ-- [tCAgasl2.
(7.8) According to the character of eigenvalues in eqs. (7.7) we can divide the space of parameters into four regions: (I) q < 0: two real and two purely imaginary eigenvalues exist, (II) q > 0, p < 0, p2 > 4q: four real eigenvalues occur, (III) q > 0, p > 0, p2 > 4q: there are four purely imaginary eigenvalues, (IV) q > 0, p2 < 4q: four eigenvalues are complex in general. These regions are schematically shown in the Ires[2 - [rAlZ-diagram in fig. 17. Exponential amplification of the solution occurs in regions I, II and IV. These
404
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
i
I~al2
,,
I
(a)
I,~AI2
/!I I I I ~
(b)
'
I
[5, w 7
III
/ tKsl2
0
Fig. 17. [1r [tCA[2-diagram of the coupler with nonlinear process in mode a for (a) Iga~l2 -IgaAI2 > 0 and (b) Iga~l2 -IgaAI2 < 0. Dashed lines given by the formula I~:sI2 = Iga~l2 indicate the asymptotes of hyperbolae (after Fiurfi~ekand Pefina [1999b]). regions are not suitable for nonclassical-light generation because the noise increases strongly. The purely oscillating character of the solution in region III is caused by a strong Stokes linear coupling which prevents amplification and is suitable for nonclassical-light generation. We note that we can move between different regions by changing the intensity of the pump fields. If damping of the vibration mode av is taken into account, division of the space of parameters into the above defined four regions is approximately valid and exponential amplification occurs also in region III except in the cases defined by the condition ]tCs[ = I~:AI and Iga~l2 - [goAla <~ O. Numerical calculations also revealed that phase mismatches support exponential amplification. When nonlinear processes occur in both waveguides and losses in the vibration modes are included, exponential amplification is present for almost all values of parameters. A special case optimum for squeezed-light generation has been studied analytically by Pefina Jr and Pefina [1997] and Fiur~t~ek and Pefina [ 1999b]. It is given by the conditions I~sl = I~AI,
Igb~f = Clga~l,
Igb~f = Clgo~I,
arg( tcs tcAgasg~g~,sg~, ~) = zr, Igor[ 2 - l g a , I 2 < 0 , [gb~ [2 [gbA[2 < 0 ,
(7.9)
where c/> 0 is an arbitrary real number. Squeezed light and light with sub-Poissonian statistics cannot be generated in single modes if incident coherent states or coherent states with superimposed noise are considered. This property has its origin in the form of the momentum operator (~ (Pefina Jr and Pefina [ 1997]). That is why we further pay attention only to two-mode fields. The conditions ]gas ] < [gaA I and [gbs [ < IgbA I suitable for nonclassical-light generation are also assumed. We further discuss the behaviour of Brillouin and Raman couplers separately.
5, w 7]
COUPLERS BASED ON RAMAN AND BRILLOUIN SCATTERING
405
0.05 i 0.00
, oo ! x j
t/
--o.lo - 0 . 15 -0,20
o
1
2 z
B
4
Fig. 18. R e d u c e d m o m e n t s o f integrated intensity { W k ) / ( W ) ~ - 1 for k = 2 (,), k = 3 (o), k = 4 ( A ) , and k = 5) ( 0 ) for m o d e (as, aA) are negative in some regions o f z ; gas = 1, gaA = 2, tCS = - 1 0 , ~as = 2, ~av = 1, ~bs = 2, and values o f the other parameters are zero (after Pef-ina Jr and Pe~ina [1997]).
7.2.1. Coupler based on Brillouin scattering Influence of Its on Brillouin scattering in one waveguide. If stimulated Stokes and spontaneous anti-Stokes processes (gas ~ O, g~A ~ O, ~as ~ O, ~av ~ O, ~a~ = O) Occur in waveguide a, sub-Poissonian light can be obtained in mode (as, av). Squeezed light occurs in modes (as, aA) and (as, av). Spatial development is characterized by oscillations with the period 1/v/Iga~l 2 - ]gas[2. If the antiStokes process is also stimulated (~a~ ~ 0), sub-Poissonian light also can be generated in mode (as, aA) provided that the phases are suitably chosen (arg(~av~asgas) = -Jr~2, arg(~av~aAgaA) = --a:/2 (see Pieczonkovfi and Pef-ina [1981]). Stokes linear coupling (trs ~ 0) supports negative moments of integrated intensity in mode (as, aA) (see fig. 18). It also leads to squeezed light and light with sub-Poissonian statistics in modes (bE, av) and (bE, aA) which are composed of single modes in different waveguides. Non-zero values of the Stokes linear coupling constant tCs shorten the period of spatial oscillations and they lead to a tendency to conserve incident statistics. Influence of irA on Brillouin scattering in one waoeguide. If a spontaneous Stokes process and stimulated anti-Stokes process (gas ~ O, gaA ~ O, ~as = O, ~av ~ 0, ~aA ~ 0) in waveguide a occur, sub-Poissonian statistics can be reached in mode (as, aA) provided that arg(~aA)= i ~ / 2 . Squeezing is obtained in modes (as, aA) and (as, av). Negative moments of integrated intensity and squeezing in mode (as, bA) for shorter z can be reached owing to anti-Stokes linear coupling (teA ~ 0). Anti-
406
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 7
Stokes coupling does not support the generation of nonclassical light and it gradually destroys squeezing in modes (as, aA) and (as, av). Non-zero values of the anti-Stokes linear coupling constant ~cA cause a fast increase of mean intensities, their moments and principal variances, but there may also occur periods with noise reduction.
Behaviour of the nonlinear coupler. Stimulated Stokes and anti-Stokes processes in both waveguides are assumed. Then squeezed light and light with subPoissonian statistics can be reached in modes (as, aA), (as, av), (bs, bA) and (bs, by) (see fig. 19) if the phases are suitably chosen. Non-zero Stokes coupling (Its ;~ 0) preserves nonclassical properties of light in these modes and it also induces sub-Poissonian light and squeezed light in compound modes composed of single modes in different waveguides; namely in modes (bs, aA), (bs, av), (as, bA) and (as, by). A typical behaviour of a compound mode is shown in fig. 19: regions with slightly negative moments of integrated intensity are followed by short regions where a high increase of noise occurs. High increase of noise is connected with a decrease of intensities. Nonclassical light cannot be generated in modes (as, bs), (aA,bA) and (av, bv), but these modes have a strong tendency to return to coherent states. Higher values of the anti-Stokes coupling constant Ir lead to higher values of moments of integrated intensity. This means that negative moments of integrated intensity may occur only for small values of z or may not occur at all. Non-zero
? 7
1
] 0
2
1
3
z Fig. 19. Mean integrated intensity (W) (solid curve without symbols) and reduced moments of integrated intensity ( W k ) / ( W ) k - 1 for k = 2 ( , ) and k = 3 (o) for mode ( a s , a A ) ; gas = 1, gaA = 2, lCS = 6i, gbs = 1, gbA = 2, ~as = - - 2 i , ~aA = 2i, ~av -- 1, ~bs = --2i, ~b~ = 2i, ~bv = 1, and values o f the other parameters are zero (after Pefina Jr and Pefina [ 1997]).
5, w 7]
COUPLERS BASED ON RAMAN AND BRILLOUIN SCATTERING
407
values of teA usually cause higher values of principal squeeze variances, but they can support squeezed-light generation in some modes (e.g., in mode (as, aA)).
7.2.2. Coupler based on Raman scattering The occurrence of regimes with nonclassical effects is similar as in the case of Brillouin scattering. However, the role of various phase relations is smaller owing to the chaotic statistics of phonon modes.
Influence of tCs on Raman scattering in one waoeguide. Assuming a stimulated Stokes process and spontaneous anti-Stokes process, squeezed light is present in modes (as, aA) and (as, av). A stimulated anti-Stokes process moreover creates light with sub-Poissonian statistics in mode (as, aA). Figure 20a shows a typical
Fig. 20. (a) Photon-number distribution p(n,z) and (b) reduced moments of integrated intensity (Wk)/(W) k - 1 for k = 2 (.), k = 3 (o), k = 4 (A), and k = 5 (<5) for mode (as, an); gas = 1, gaA = 2, trS = 0.5, ~as = --2i, ~aA = 2i, ~bs = i, (hOar) = 0.1, and values of the other parameters are zero (after Pef-ina Jr and Pef'ina [1997]). spatial evolution of the photon-number distribution: it evolves periodically from Poissonian statistics through sub-Poissonian statistics to super-Poissonian statistics (compare moments of integrated intensity in fig. 20b). Light with negative moments of integrated intensity cannot be obtained in mode (as, av) owing to the chaotic statistics of the vibration mode. Non-zero Stokes coupling supports the generation of sub-Poissonian light in modes (as, aA), (as, av), (bs, aA) and (bs, av) and squeezed light in modes (bs, aA) and (bs, av). Greater values of the mean number of phonons nav destroy subPoissonian statistics in all modes.
Influence of tea on Raman scattering in one waoeguide. Squeezed light can be obtained in modes (as, aA) and (as, av) if a stimulated anti-Stokes process is assumed. Negative moments of integrated intensity can be reached in mode
(as, aA) provided that nav is small. Squeezed light and sub-Poissonian light in
408
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 7
mode (as, bA) may occur if the anti-Stokes coupling constant Ir is non-zero. However, greater values of K'A lead to a degradation of nonclassical properties of light.
Behaviour of the nonlinear coupler. Nonclassical effects occur in the same compound modes as in Brillouin coupler for non-zero Stokes coupling, i.e., light with sub-Poissonian statistics and squeezed light can be obtained in modes (as, aA), (as, av), (bs, bA), (bs, bv), (bs, aA), (bs, av), (as, bA) and (as, by). Reduction of the incident noise in compound modes (including phonon modes a v and by) can be obtained for greater values of the Stokes coupling constant I :sl. Greater values of the anti-Stokes coupling constant IleAl result in the loss of negative values of the moments of integrated intensity. They also usually lead to greater values of principal squeeze variances, but squeezed light may still occur in some modes (e.g., in mode (as, aA)). Greater values of the mean phonon numbers nay and nbv conserve negative moments of integrated intensity and squeezing in compound modes excluding the phonon modes. They suppress nonclassical properties of light in compound modes involving the phonon modes av and by. The influence of damping and noise in all modes both in Brillouin and Raman couplers has been studied by Fiurfi~ek and Pefina [1999a] in detail. It has been shown that greater values of damping constants as well as greater values of the mean numbers of chaotic phonons and photons completely destroy the nonclassical properties of light. A gradual loss of negative values of moments of integrated intensity with an increasing mean number of chaotic phonons (nrav) is demonstrated in fig. 21. Effects caused by linear and nonlinear phase mismatches have been described by Fiurfi~ek and Pefina [ 1998]. According to the value of any phase mismatch, three different regimes can be distinguished. If the value of a phase mismatch is small, the spatial development of the quantities discussed throughout the chapter is affected only slightly. Greater values of the phase mismatch introduce oscillations into the spatial development of the studied quantities and strongly change their behaviour. Great values of the phase mismatch effectively suppress the influence of the process described by the appropriate linear or nonlinear coupling constant and the system behaves as if the process does not occur. Nonzero phase mismatches can support the generation of nonclassical light in two cases. The first case occurs if the value of a phase mismatch is great enough and eliminates the influence of a coupling constant which destroys nonclassicallight generation (see fig. 22). Non-zero phase mismatch can also change values of some phases to the values suitable for nonclassical-light generation in some
5, w 7]
COUPLERS BASED ON RAMAN AND BRILLOUIN SCATTERING
409
1.0
o.8_o.6 / /!,,,~' / ,'"~:/ '", ,/"/"'~"'',//', I
-.1~ O.4 0.2
;,,.,,,,"/~ '", ;, ; ,,.,,'1it,';'',,,,~,, //~\ , ,,, , ,, /,~ I ~~//~
", ~///
" / " '/ ~ '"
-o.2 lj --0.4
ill 0.0
iii
liB iT1 iI l l ~ l l
0.5
1.0
1.5
ilrlll
2.0
2.5
iI~i
lIT11~
3.0
3.5
ll~ 4 0
Fig. 21. Second reduced moments of integrated intensity (W2)/(W) 2 1 of mode (aA,bs) lose their negative values with increasing values of the mean number of phonons (nra v); gs] ---- gas, (nrav) = 0 (solid curve), (nrav) = 0.1 (dashed curve), and (nrav) = 1 (dotted curve); gas = 1, -
gaA = 2 , 1(S = --10, ya V -- 0.25, ~a S = 0.5, ~a A = 2 , ~b S --~ i,
and values o f the other parameters are
zero (after Fiur~i~ek and Pe~ina [ 1999a]). 0.4
0"3 t
~L 0.2
0.1 v
0.0 -0.1
-0.2 0.0
1.0
2.0
3.0
4.0
5.0
Z
Fig. 22. Second reduced moments of integrated intensity (W2)/(W) 2 1 of mode (as,aA) are negative if the value of AkA is sufficiently great; AkA = 0 (solid curve with o) and AkA = 150 (solid curve without symbols); gas = 1, gaA = 2, trS = - - 1 0 , K"A = 10, ~as = 1, ~av = 1, ~bs = 1, and values of the other parameters are zero (after Fiur~t~ekand Pe~ina [1998]). -
regions of z. In particular, great values o f the linear Stokes phase mismatch Aks suppress the influence of tCs and destroy nonclassical properties o f the generated light in this way. On the other hand, great values o f the linear anti-Stokes phase mismatch AkA "switch off" the influence o f teA and thus improve the conditions for nonclassical-light generation. Non-zero values o f the nonlinear Stokes phase mismatch Akas do not create conditions suitable for the generation of light with sub-Poissonian statistics, but they support the generation o f squeezed light in
410
QUANTUM STATISTICSOF NONLINEAROPTICALCOUPLERS
[5, w 7
modes (as, aA) and (bs, aA). The nonlinear anti-Stokes phase mismatch AkaA can partially compensate for the effects caused by the nonlinear Stokes phase mismatch Akas. Phase properties of this coupler are discussed by Fiurfi~ek and Pefina [2000b]. 7.3. RAMAN COUPLER WITH BROAD PHONON SPECTRA
Phonon systems with broad spectra are described by multimode boson fields and these fields can be eliminated from the description of the coupler using Wigner-Weisskopf approximation. Assuming strong pump beams in coherent states, the dynamics of the Stokes and anti-Stokes fields is governed by the following equations (Pe~ina [ 1981 ], Fiurfi~ek and Pe~ina [2000a]): dAas - )tasAas + )tasaA exp[i(Aka~ +
dz
+ itr exp(-iAksz)Abs dftaA _ dz
AkaA)Z]2[~
+ Zas,
(7.10)
YaAZia.,- Ya~a, exp[i(Akas + Aka,)z]A~s "
"
+ i ~ exp(-iAkAz~bA + Za~, and similar equations f o r Abs and AbA-The constants )'as, YaA and )tasaA in eqs. (7.10) are determined according to the relations ~9, =
a~lgJ,~~12pjv ,
YjA = arlgj.~.~ I2p j ~ ,
rj~j~ = a r g j s g j ~ 2 ~ p j ~ ,
(7.11) where pj,, means the density of phonon modes in field jz (j = a, b). The symbol Zas (Za.4) in eqs. (7.10) stands for an operator of the Langevin force in mode as
(aA). Exponential amplification of photon-field amplitudes occurs for most of the values of parameters. The amplification does not appear under the conditions I r s l - I~AI,
[YjA [ > [Yj~ [, IYb~l = clYa~l,
zxks + akA = 0, A kj~ + AkjA = 0, j = a, b
(7.12)
IY~I = ClYaAI,
= arg(tcs I(A )tasa , ]f~s bA ) = Y'C, where c ~> 0 is a constant. The conditions described in eqs. (7.12) are optimum for the generation of squeezed light.
5, w 7]
COUPLERS BASED ON RAMAN AND BRILLOUIN SCATTERING
411
Fig. 23. (a) Principal squeeze variance 3, and (b) generalized squeeze variance/~G of mode (a S, aA) at z = 2.4 as functions of the linear coupling constants trS and trA show squeezing; 7as = 0.5, YaA = 1, ~as = ~aA = ~bs = ~bA = 0, and values of the other parameters are zero (after Fiurfi~ekand Pefina [2000a]). Principal squeeze variances, generalized squeeze variances, and sum squeeze variances of the fields in Raman coupler have been analyzed by Fiurfigek and Pefina [2000a]. It has been shown that generalized squeezing occurs if the Glauber-Sudarshan quasidistribution does not exist as an ordinary function. If a field exhibits generalized squeezing, it also shows principal squeezing and sum squeezing. Squeezed light can be generated only in compound modes, namely in modes (as, aA), (bs, bA), (as, hA) and (bs,aA). Principal squeezing occurs only if IYjsl < [Yj~l (J = a,b) and values of the phase q) defined in eq. (7.12) have to lie near Jr. Generalized squeezing appears in mode (js,jA) ( j -- a, b) regardless of the values of the nonlinear coupling constants, i.e., light exhibiting generalized squeezing may be obtained even if exponential amplification takes place. Both principal squeeze variances and generalized squeeze variances reach their lowest values if I~:sl = I~:AI (see fig. 23). The role of phase mismatches and mean numbers of reservoir phonons is similar to that discussed in the previous subsection. 7.4. LINEAR OPERATOR CORRECTIONS Quantum properties of the optical fields in Brillouin and Raman couplers can be described by linear operator corrections to a classical solution if the pump fields are weak and cannot be treated classically. Parametric approximation is not valid in this case. The stationary point of a Brillouin coupler with all modes (jc, i s , jA, j r , j = a, b) being pumped by classical external fields has been found by Fiurfigek and Pe~ina [1999a]. The stationary point exists only above a certain threshold given by the strengths of external fields. If only modes aL and by a r e externally
412
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 8
pumped, the stationary point is unstable. If also Stokes modes (as, bs) or antiStokes modes (aA, bA) are externally pumped, the stationary point becomes stable and the method of linear operator corrections can be applied. If the stationary point is unstable, another method can be used. An operator A of an optical field can be expressed as A(z) = ~(z) + AA(z), where ~(z) = (A(z)) and (AA(z)) = 0. Then a system of equations for the mean values ~- and second-order moments of the linear operator corrections (coefficients B, C, D, and b in eq. (3.2)) can be derived invoking approximations (Olivik and Pefina [ 1995]). Numerical results show that light with sub-Poissonian statistics cannot be generated, but squeezed light can be obtained in laser modes aL and bL (Fiurfi~ek and Pefina [ 1999a]).
w 8. Miscellaneous Couplers 8.1. B A N D G A P C O U P L E R
The coupler consists of one central waveguide (a) which interacts linearly with a greater number of mutually noninteracting waveguides (bj) in its surroundings (Mogilevtsev, Korolkova and Pefina [1997]). It is described by the following momentum operator (~: N
N
0-- hka~l~as -k-h Z kbj~i~s~lbj+ h Z gabj((lbs~lt a -}-s j=l
(8.1)
j=l
where ka (kbs) is the wavevector of mode a (bj), gabs is the linear coupling constant between modes a and bj, and N denotes the number of surrounding waveguides. The corresponding Heisenberg equations are linear, N
dha - ikaha + i Z dz
gabsgtbs'
j=l
(8.2)
dhbs
dz - ikbsgtbs + igabsgta'
and their solution can easily be found. The behaviour of the coupler for the incident coherent state in mode a and vacuum states in modes bj has been investigated by Mogilevtsev, Korolkova and Pe~ina [ 1997]. If the spectrum of coupling constants gabs is described by a smooth Gaussian function, the mean number of photons in mode a shows that revivals
5, w 8]
MISCELLANEOUS COUPLERS
413
1
~o,
~s5
~ 0.6
~25
-..
. .
> <
-
~.
\
o2
5
10
i 15
9
J "k.r~ 20
,
-
/
.
. .. "*,.
,
,
25
3O
. . - ....................... "" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5
.:'" 10
15
z
Z
Mean number of photons in mode a for linear (a) and nonlinear (b) central waveguides in case of the Gaussian spectrum of coupling constants (dotted curve) and for the spectrum with a gap (solid curve); ~a = 1, ~a2 = 0 0 5 , a n d )' = 0 . 4 6 (for details, see Mogilevtsev, Korolkova and Pefina
Fig. 24
[1997]).
and decays between the revivals are exponential (see fig. 24a). The revivals are suppressed in the limit of dense spectrum. If the spectrum has a gap and the wavevector ka lies in the gap, decay of the mean number of photons in mode a is suppressed and a "steady state" is formed until a revival occurs (see fig. 24a). The suppression of the decay is caused by a partially coherent response of the reservoir modes bj to the changes in mode a. The coupler may serve for the generation of N identical replicas of the incident state in mode a, because a replica of the incident state in mode a occurs at each mode bj for z greater than a certain value z0. If the central waveguide a contains a second-order nonlinear medium, the momentum operator Gn has the form: h
Gn = G + -~ [~a2 exp(2ikaz)ata 2 + h.c.],
(8.3)
where (~ is given in eq. (8.1) and ~a2 stands for the amplitude of the pump field in the central waveguide. Equations corresponding to the momentum operator Gn have been solved and the behaviour of the coupler has been analyzed by Mogilevtsev, Korolkova and Pefina [ 1997]. The parametric process in waveguide a is inhibited if the spectrum of the coupling constants gabj is Gaussian and I~a2] ~ ~, where ), characterizes interaction with the reservoir modes bj (see fig. 24b). On the other hand, the spectrum with a gap suppresses the parametric process only for z < z0 (see fig. 24b). For z > z0 the parametric process develops and light with sub-Poissonian statistics and squeezed light are generated. If ]~a2 ] > Y, the parametric process occurs in waveguide a and the mean number of photons in mode a increases exponentially with the rate 2(1~a21- )') for a
414
QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
[5, w 8
Gaussian spectrum, whereas the rate equals 2[~a2 ] for the spectrum with a gap and only for z > zo. 8.2. COUPLER COMBINING 2'(2) AND X (3) MEDIA
We consider a coupler with second-subharmonic generation in waveguide a. The phase of the second-subharmonic mode a is influenced by mode b in the second waveguide with Kerr medium. The interaction momentum operator aint describing the coupler has the form (Mogilevtsev, Korolkova and Pefina [ 1996]): aint =
~t2 ~2
^t ^
^t ^
hga(A~ 2 + Zl2a)+ ngtu% At, + hgabAaAaAbAb,
(8.4)
where g~ describes the process of second-subharmonic generation and includes the coherent pump amplitude, gb stands for the Kerr constant, and gab means the nonlinear coupling constant between modes a and b. The corresponding Heisenberg equations, dA o
dz - 2igaA~ + igAa]Vb, dAb
dz
(8.5)
^t ^ ^
- igabAaAaAb + 2igbNbAb,
]Vb = A~Ab, can be solved owing to the fact that the number of photons in mode b is conserved during the evolution along z (A[b(Z) = Arb(0)). Assuming incident Fock state with Nb photons in mode b, the solution for the operator Aa(z) oscillates in z for 2ga < Nbgab, whereas it has an exponential character for 2ga > Nbgab. Assuming incident coherent state with the amplitude ~b in mode b and vacuum state in mode a, three regimes in the evolution can be distinguished. If 2ga >> [~b[2gab, mode b has no influence on the mode a and the mean number of photons in mode a increases exponentially with z. If 2ga ~ I~bl2gab, both oscillating and exponential terms contribute significantly to the solution. Entanglement between modes a and b develops and the overall state of the compound mode (a,b) is such that Schr6dinger-cat states of mode b are associated with every Fock state of mode a. The Schr6dinger-cat states evolve from two-component states to multi-component states with increasing z. If 2ga << I bl=ga6, the solution has an oscillating character. The process of secondsubharmonic generation is strongly suppressed and the mean number of photons in mode a remains small. Squeezed light can be obtained in mode a in all regimes.
5, w9]
CONCLUSIONS
415
w 9. Conclusions Quantum statistical properties of optical fields in codirectional and contradirectional nonlinear couplers composed of both linear and nonlinear waveguides have been described. Nonlinear waveguides have been assumed to be composed of materials with second- and third-order nonlinearities, in which parametric processes, Kerr effect, and Raman and Brillouin scattering can occur. Spatial evolution of quantum fields has been described by the Heisenberg equations. The Heisenberg equations include terms describing both the influence of linear and nonlinear phase mismatches and effects originating in damping of optical fields. The equations have been solved either exactly in the operator form or under suitable approximations (short-length approximation, parametric approximation, linear operator corrections to "classical solution"). Quantum statistical properties of optical fields have been conveniently described in the framework of quantum generalized superposition of signal and noise. Photon-number distribution, reduced moments of integrated intensity and various squeeze variances have then been determined both for single and compound optical fields. Nonlinear processes in waveguides provide nonclassical light like squeezed light and light with sub-Poissonian statistics under certain conditions. Linear and nonlinear coupling of modes between waveguides leads either to stimulation of nonclassical-light generation of nonclassical light or to suppression of the nonclassical character of the generated light. Coupling between modes can lead to "transfer of nonclassical light" from one mode to the other mode. Properties of a mode can be controlled by parameters of other modes being coupled with the first one. For example, all-optical switching can be reached in certain types of couplers. Phase mismatches play an important role in the interaction among modes. They change the phases of interacting fields locally in space and can stimulate the occurrence of light with nonclassical properties in some spatial regions. Greater values of a phase mismatch strongly suppress the effect of coupling among the corresponding modes and this results either in stimulation or suppression of nonclassical properties of optical fields. External reservoir noise and damping of optical fields lead to a degradation of the nonclassical character of optical fields. On the other hand, noise present in incident fields can be reduced in certain types of couplers. Long contradirectional couplers provide a stable source of nonclassical light under certain conditions. Also highly nonclassical states like states with oscillations in photon-number distributions or Schr6dinger-cat states can be generated in nonlinear couplers. Damping of a field in a central waveguide can be partially inhibited if the field is coupled to many neighbouring waveguides.
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QUANTUMSTATISTICSOF NONLINEAROPTICALCOUPLERS
[5
We mainly discussed how nonclassical properties of optical fields in nonlinear couplers can find their application in many areas of applied optics, including optical communications, precise optical measurements, and optical computing. The authors hope that the review will contribute to the stimulation of experimental activities as well as applied research in this field.
w 10. Acknowledgments This work was supported by the Czech Ministry of Education under the Complex Grant No. VS96028 and the Research Project CEZ: J14/98:153100009 "Wave and Particle Optics". Support from Grant No. 19982003012 of the Czech Home Department is also acknowledged. The authors thank J. K~epelka, J. l~eh~6ek, J. Herec, L. Mi~ta Jr, and J. Fiurfi~ek for their help with preparation of the article.
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E. WOLF, PROGRESS IN OPTICS 41 9 2000 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED
QUANTUM PHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
BY
A. L u I s AND L . L . SANCHEZ-SOT0
Departamento de Optica, Facultad de Ciencias Fisicas, Universidad Complutense, 28040 Madrid, Spain
421
CONTENTS
PAGE w 1.
INTRODUCTION
. . . . . . . . . . . . . . . . .
w 2.
STOKES OPERATORS AND PHASE DIFFERENCE . . . . .
425
w 3.
QUANTUM RELATIVE PHASE FORMALISMS
9 .
434
w 4.
P H A S E - S H I F T D E T E C T I O N IN SU(2) I N T E R F E R O M E T E R S .
454
w 5.
F R O M T W O - M O D E P H A S E TO O N E - M O D E P H A S E A N D
.
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BACK . . . . . . . . . . . . . . . . . . . . . . . .
465
w 6.
CONCLUSIONS
473
w 7.
ACKNOWLEDGEMENT
REFERENCES
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
473
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w 1. Introduction The subject of quantum phase has experienced a renewed interest in recent times. The rapidly increasing effort devoted to this topic reveals a latent expectation ready to trigger a considerable amount of work. Several reasons can explain this situation. It is indisputable that phase is a peerless tool for solving and understanding classical optical phenomena. The failure of first and eminent attempts to translate this variable into the quantum domain thus gives the impression that something relevant is missing in our understanding of the quantum world, especially in quantum optics. This alone would justify the devotion of a significant degree of effort to fill the blank. Phase should be subject to quantization and for a sufficiently small number of particles quantized phase effects should be accessible for experimentation. Leaving aside such abstract motives, some other more practical reasons might also be invoked. The ultimate performance of the measurement of very small phase shifts is always of great interest because this is the basis of the most sensitive devices, such as gravitational wave antennas and laser gyroscopes, for example. These devices already can operate at the shot-noise limit, so that the uncertainty in the measured phase-shifts is due solely to the quantum fluctuations of the light injected into the interferometer. Also, efficient communication near the ultimate quantum limit would require not only engineering states coding information with minimum error, but also estimating phase accurately from single measurements. The list of quantum phase applications easily could be enlarged. One might argue that quantum optics can subsist without quantum phase. In fact its role has been occupied by the better behaved field quadrature operators. However, the question arises as to what extent a suitable solution of the phase problem would boost the realm of quantum optics by introducing a tool inheriting the significance of its classical counterpart. Naturally, the primary objective in this context has been the quantum description of the phase of a single-mode field. The progress made is manifest and the work in this subject and the goals achieved have already been well reviewed. 423
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QUANTUM PHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 1
As knowledge about quantum phase improves, a rapidly growing amount of theoretical as well as experimental work is focusing directly on the phase difference as the relevant variable in this context. Strictly speaking, only relative variables are of interest in physics. Most, if not all, methods of practical phase measurement are two-mode arrangements measuring phase difference as a comparison of phases in two different modes. Classic interferometers are a suitable example. Finally, any phase value must always be defined relative to a phase origin that sooner or later has to be explicitly described. We speak of single-mode phase when such origin is assumed to be classical and well defined. It might be foreseen that the study of phase difference would not add anything not already included in the current knowledge about single-mode phase. In other words, relative phase should be constructed merely as the difference of phases. Maybe surprisingly, experience demonstrates that this is not the case since there are theoretical as well as experimental results which cannot be accounted for by using the difference of phases. As an example we have the discrete character of phase difference for small photon numbers. This means that it makes sense to study the quantum description of phase difference without any previous assumption about single-mode phases. Many authors have followed this possibility with the hope of circumventing the difficulties that quantum phase has encountered from the beginning of the quantum theory. The purpose of this work is to review the main solutions proposed so far. Concerning the statistics of two-mode fields and their phase properties, the Stokes parameters are a relevant tool. Their definition explicitly involves phase difference. Moreover, they are measurable quantities in the quantum as well as in the classical domains. In particular, lossless passive interferometers measure the Stokes operators. These variables are at the heart of most of the theoretical and experimental work on the subject so they provide a suitable guiding thread. Stokes operators are formally equivalent to an angular momentum and this allows us to describe at a time a great variety of different physical situations, even beyond quantum optics. This is the case with atomic interferometry and spectroscopy, for example. Then, the issue of defining and measuring phase difference is connected with topics like the improvement of frequency standards, for instance. We devote w2 to presenting basic definitions. The different approaches to quantum relative phase are presented in w3. Quantum noise in spectroscopy and interferometry is a subject of fundamental and, to an increasing degree, of practical interest. The improvement of interferometric measurements is a relevant stimulus impelling the study of quantum phase. We consider the problem of precise phase-shift detection in w4.
6, w 2]
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In this context, linear, lossless and passive interferometers (which include classic devices) are the simplest arrangements where phase difference manifests itself in an observable way as an intrinsic two-mode variable. Finally in w5 it is shown how the main proposals for the quantum single-mode phase emerge from two-mode formalisms when one of the modes is in a suitably chosen reference state. It is important to stress that the converse is not true, and no one of the phase-difference approaches considered in w3 can be derived from the same one-mode formalisms they define. Throughout we have tried to minimize abstract mathematical discussions which might be regarded as void of meaning until the development of the subject leads to a more firm connection with experimental and observable consequences. Single-mode phase marks the boundary of this work. Whenever we enter the subject of one-mode phase the reader is referred to the excellent reviews and books already available. Among them we can quote as especially useful the following ones: Carruthers and Nieto [ 1968], Bergou and Englert [ 1991 ], Shapiro and Shepard [1991], Physica Scripta T 48 [1993], Vogel and Welsch [1994], Luk~ and Pefinov~i [ 1994], Pefina, Hradil and JurSo [ 1994], Lynch [ 1995], Tana~, Miranowicz and Gantsog [1996], Pegg and Barnett [1997] and Pefinov/t, Luk~ and Pefina [ 1998]. w 2. Stokes Operators and Phase Difference
In this section we recall basic definitions taken from classical and quantum optics that will be necessary later. We also discuss why there is no straightforward quantum translation of phase and phase difference. In all these points the quantum treatment follows the classical description of the problem as closely as possible. 2.1. STOKES PARAMETERS AND STOKES OPERATORS
In classical optics, the Stokes parameters offer a very convenient tool for describing of a two-mode field (Jauch and Rohrlich [1959], Azzam and Bashara [ 1987]). They are measurable quantities fully equivalent to the coherence matrix (Born and Wolf [1980], Mandel and Wolf [1995]). Denoting by al, a2 the classical complex amplitudes of two field modes of the same frequency, the following time-independent Stokes variables can be defined
So = a~a, + a~a2, Sy = i ( a ~ a l - a l a 2 ),
S x -- a~a 2 + a~al,
Sz =ala, -a~a2,
(2.1)
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QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 2
where t denotes complex conjugation. If a l, a2 fluctuate, these four variables (So,S) will fluctuate as well. The Stokes parameters (so, s) are defined as the averages of eq. (2.1), sj = (Sj), for j = 0, x, y, z. Frequently, these definitions are applied to two copropagating field modes with orthogonal polarizations. However, they can be applied as well to any pair of independent field modes, for instance propagating along different directions and having the same polarization state. For copropagating modes the Stokes parameters determine the intensity and the (average) polarization state, while for field modes with the same polarization they contain the intensity and the visibility of the interference fringes where the two modes overlap. It can be seen that S 2 = S2, so only three of the four variables in eq. (2.1) are independent. Because of this, it is customary to represent the Stokes variables as points S/So on a sphere of unit radius, the Poincar6 sphere (Bom and Wolf [1980]). Pure monochromatic fields are represented just by a single point on the surface of the sphere, which represents a definite polarization ellipse. Fluctuating fields will be represented by a probability distribution on the surface of the sphere. The phase difference q) between mode a l and mode a2 can be simply defined by the relation q) = a r g ( a l a ~ ) : arg(Sx- iSv).
(2.2)
This is to say that the phase difference is the azimuthal angle on the Poincar6 sphere. The associated polar angle 0 can be defined as cos 0 = Sz/So. The definition in eq. (2.2) applies provided that Sx * 0 and/or Sy ~: O. The case S~ - S,, = 0 corresponds to the north or south poles (0 = 0, Jr), which are completely specified by their value of the polar angle 0. At the poles q) is not defined, but classically this is not a serious problem since it only occurs at two isolated points. From this classical viewpoint So, q) and 0 are three parameters or coordinates representing a possible realization of a two-mode field. Their values or their probability distribution for any field state can be fully determined by simultaneously measuring (So,S). This is conceivable because the signal mode amplitudes al, a2 can be split as many times as required. Nothing prevents such copies from being accurate replicas of the incident amplitudes, so that different Stokes variables can be measured at once. In a straightforward quantification of the problem, the classical complex amplitudes are replaced by operators satisfying the commutation relations [al,a~] - [a2,a~] = 1, [al,a~] = [al,a2] = 0, and from now on 1- will denote
6, w 2]
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427
Hermitian conjugation. The Stokes variables (eq. 2.1) become Stokes operators verifying the commutation relations
[Sx, Sy] = 2iSz (and cyclic permutations), [S, S0] = 0.
(2.3)
These commutation relations allow the identification of S as an angular momentum or spin J by J = S/2 (in units h = 1). As a matter of fact eq. (2.1) is the boson representation of angular momentum operators first proposed by Schwinger [1965]. On the other hand So is the operator representing the total number of photons in both modes. Because [S, So] = 0, the total Hilbert space '1-~1 @ "1-[2 of the two-mode field can be split as a direct sum of subspaces ~n invariant under the action of the Stokes operators oo
(2.4)
~"/1 @ ~-'/2 = ( ~ ~"/n, n=O
where 7-/, is the subspace with fixed total photon number n. Denoting photon number states in both modes as [nl, n2), we have that ~"/n is spanned by the basis [nl,n- nl) with nl = 0, 1,... ,n, so all ~'/n are finite-dimensional spaces with dimension n + 1. Each subspace 7-/, can be regarded as describing an abstract angular momentum j = n/2 with the correspondence [nl,n2) =
[j,m),
(2.5)
where [j, m) denotes the simultaneous eigenvector of j 2 and
Jz with eigenvalues
j(j + 1) and m respectively, and j
nl nt-n2 -
-
-
,
2
nl - n 2 m
-
-
-
.
2
(2.6)
There are some hallmarks distinguishing quantum from classical Stokes variables. For example, in the quantum case no field state can have definite values of the three Stokes operators S because they do not commute. There is no state with a well-defined polarization ellipse, exactly analogous to the fact that quantum particles do not follow single classical trajectories. Equivalently, any exact simultaneous measurement of S, or any pair of its components, is precluded. The quantum S unavoidably fluctuates and the points on the Poincar6 sphere lose their meaning and no longer can represent field states with definite values of the Stokes variables.
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QUANTUMPHASE DIFFERENCE,PHASE MEASUREMENTSAND STOKESOPERATORS
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The quantification of the problem replaces the classical relation between numbers S 2 = S 2 with the quantum one between operators S 2 = So(So + 2). This, along with [(S)l ~ (So), implies that (z~) 2 = (A~x) 2 + ( A & ) 2 + (ASz) 2 > 2(So),
(2.7)
where (z~A) 2 -- (A 2) - (A) 2. This shows that the quantum fluctuations of S have a lower bound. As in classical optics, in quantum optics a degree of polarization P assessing the fluctuations of S can be defined. As discussed by Alodjants and Arakelian [1999], depending on how P is defined it may or may not reflect the quantum uncertainty of the polarization ellipse. If it is defined as usual
p_ v/~ 2 (So) '
(2.8)
there are states with P = 1, as shown for example by Tanag and Kielich [1990]. This definition does not reflect the quantum fluctuations of the polarization ellipse. However, as proposed by Alodjants and Arakelian [ 1999], the polarization degree might also be defined as P'
-
g~2
(X/-~
~-~2
-
< 1,
(2.9)
v/(So(So + 2))
and no field state is fully polarized in the sense of having P' = 1. Both definitions have the same classical limit (So) ~ ~ , but they differ significantly in the quantum regime (So) ~ 0. In particular, field states arbitrarily close to the twomode vacuum can have P = 1 while P' ~ 0. In this context we can mention the visibility operator introduced by Gennaro, Leonardi, Lillo, Vaglica and Vetri [1994] that is able to describe and evaluate quantum phase correlations between two field modes. The unavoidable quantum uncertainties lead one to inquire about states with minimum fluctuations. For instance, we can consider the states satisfying the equality (AS) 2 = 2(S0). These states can be defined within each subspace 7-/~ and their expression in the number basis is
1n, 0,r
1
= (1 + 1ff12)~
~-~ ( n ) 89 nl
~"']nl,n-nl),
(2.10)
F/l=0
having the mean values (Sx) = n sin 0 cos r
(Sy) = - n sin 0 sin q~,
(Sz) = - n cos 0,
(2.11)
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429
where ~, 0, q~ are related by ~--tan(0/2)e i0. They can be expressed also as In, 0, q~> = e ra~ a2-r*a2t al l0 ' n>,
(2.12)
where r = (0/2)e i0. These states are known as SU(2) coherent states or atomic coherent states (Atkins and Dobson [ 1971 ], Radcliffe [ 1971 ], Arecchi, Courtens, Gilmore and Thomas [1972], Perelomov [1986], Fonda, Manko~-Borgtnik and Rosina [1988], Mandel and Wolf [1995]). They are closely related to the ordinary quadrature coherent states (Glauber [1963]) ]a> = e-Ial2/2 n__ ~0 an In>"
(2.13)
It has been shown by Atkins and Dobson [1971] that the two-mode coherent states [al, a2) are a Poissonian superposition of SU(2) coherent states [al, 0/2> = e -[fi[2/2 n ~= 0 fin
(2.14)
~
where
- v/lal 12 + la212
a2
la21'
e i0~tan
0 0/1 . 2 a2
(2.15)
A very relevant property of the Stokes operators is that they are measurable quantities which can be determined from simple experiments. Intensity or photon-number measurements at the outputs of a two-beam linear lossless and passive device are the measurement of So and a linear combination of S for the input fields. This is any time-independent quadratic combination of the input complex amplitudes. Two-beam energy-conserving linear devices are made of beam splitters, mirrors, and phase plates, and include classic instruments like Mach-Zehnder, Michelson, and Fabry-Perot interferometers. Linearity and energy conservation imposes that the input-output relations caused by these devices are rotations of S, the total photon number So being a constant. This implies that there is a tight connection between linear lossless passive devices and the SU(2) group, locally isomorphic to the rotation group O(3), whose infinitesimal generators are precisely the Stokes operators. This has been shown in a general, elegant, and sound analysis by Yurke, McCall and Klauder [ 1986]. It permits us to fruitfully understand the action of basic optical devices as angular momentum rotations (Yurke, McCall and Klauder [ 1986], Campos, Saleh and Teich [1989], Leonhardt [1993]). For example, we have from eq. (2.12) that
430
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 2
A r
BS~ a~ Fig. 1. Outline of a four-port homodyne detector simultaneously measuring So and a linear combination of and S v. It consists of a 50:50 beam splitter and a phase shifter.
Sx
SU(2) coherent states are produced whenever one of the input ports of a beam splitter is in vacuum. Linear lossless and passive interferometers will be referred to as SU(2) interferometers below. The direct measurement of the photon number in each mode gives the simultaneous measurement of So and Sz. The measurement of the phasedependent quantities Sx, Sy requires the use of an interferometric arrangement, like the (four-port) homodyne detector schematized in fig. 1 (Noh, Fougbres and Mandel [1991, 1992a,b], Vogel and Welsch [1994]). For copropagating modes the arrangement in fig. 1 easily can be implemented by using phase plates and polarizing beam splitters. The two signal modes al, a2 are mixed at a 50:50 beam splitter BS that we will assume is described by real reflection and transmission coefficients with a Jr phase change in the upperside reflection. Before impinging on the beam splitter the phase of the mode a~ can be shifted by some amount ~. Then, the photonnumber detection at the outputs a]al, a~a2 corresponds to the measurement of the following operators for the input fields: So -- a~ a l q- a~ a2 = So,
Sz -'- a[ a l - a~ a2 - c o s
~)Sx.-bsin ~)Sy.
(2.16)
We notice that the measurement of Sx or Sv requires different arrangements, that is, a different phase shift (r - 0 and ~ - :r/2, respectively), while So can be measured together with any of them. This is reminiscent of the commutation relations in eq. (2.3). The statistics of the measurement are given by the projection of the input signal state on the simultaneous eigenvectors [k) of So and Sr = cos r + sin ~Sy S0]k) = (nl + n2)lk), So[k ) = (nl - n2)lk),
(2.17)
where n l, n2 are the corresponding photon numbers recorded at the outputs of the beam splitter. These vectors Ik) are given by [k) - v/~1
v/ni'n2 '1
(e_iCa~
+a~) n, (e_iCa~ _a~) n2 10, 0),
where 10, 0) is the two-mode vacuum and n = nl + n2.
(2.18)
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431
This simple interferometric arrangement is the basis of what seemingly was the only measurement of quantum phase performed until very recently. This is the measurement of phase fluctuations carried out by Gerhardt, Btichler and Litfin [1974] and the related one by Matthys and Jaynes [1980]. A radiation field of few photons was amplified and later mixed with a reference beam at a beam splitter. The data so obtained have been compared with the predictions of most of the theoretical descriptions of quantum phase, including the phase-difference approaches to be examined in this work. The formal identification of the Stokes operators as an angular momentum permits us to draw parallels with a great variety of diverse physical systems, in particular collections of two-level atoms (Dicke [1954], Arecchi, Courtens, Gilmore and Thomas [1972]). An assembly of n two-level atoms can be conveniently described by the collective angular momentum J = ~
ak,
(2.19)
k=l
where the sum runs over all atoms in the sample and ak are vectors with components ~rk~ - [ek)(gk] + Igk)(ek[, ~rk,y = i([gk)(e~ I -lek)(g~[),
(2.20)
Ok,z = [ek)(ek[- [gk)(gk[, where le~) and [gk) are the excited and ground energy levels of the kth atom. It is easily seen from eq. (2.12) that the SU(2) or atomic coherent states are always of the form In, 0,q~) = H~__all,
0, q~)k,
(2.21)
where 0 0 [1, 0, q~)k = cos ~[gk)+e ir sin ~[ek).
(2.22)
This is the product of n independent uncorrelated atoms, each one in the same coherent state I1, 0,q~)k (Kitagawa and Ueda [1993]). 2.2. QUANTUM AND CLASSICAL PHASE DIFFERENCE
The impossibility of any exact noiseless simultaneous measurement of Sx and Sy seemingly implies that the classical route to the phase difference represented by
432
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 2
eq. (2.2) is blocked. The phase difference apparently faces the same difficulties as the single-mode phase. We will see in the next section some procedures to get around this obstacle. Before examining them, we show here that the lack of a straightforward quantum translation of phase and phase difference can be traced back to classical physics. Such analysis might also serve to illustrate the advantages and peculiarities that quantum relative phase offers. Classically, every dynamical variable plays different roles. They are coordinates or functions in phase space, assigning to each system state the value of the magnitude they represent. But they can also be regarded as infinitesimal generators of transformations (Saletan and Cromer [1971]). For example, the Stokes variables generate rotations on the Poincar6 sphere. In particular, Sz generates rotations around the z axis, shifting the phase difference or azimuthal angle ~ which is its conjugate variable. As a coordinate, the only minor particularity associated with q~ is its lack of definition at the poles. Concerning its role as generator of transformations the difficulties are far more serious. According to classical Poisson brackets between conjugate variables (Carruthers and Nieto [ 1968], Saletan and Cromer [1971 ]) the transformation generated by q~would shift cos 0, ~ being constant. The points on the sphere would move towards one of the poles along the meridian they occupy. Such a transformation is precluded classically for several reasons. In the first place, a whole parallel would be mapped on a single point, the corresponding pole. Also, there would be no image for points closer to the pole than such a parallel. The opposite occurs at the other pole. Moreover, phase-space orbits would cross at the poles. Single-mode phase behaves similarly replacing poles with the origin of the complex amplitude plane (Pefinovfi, Luk~ and Pefina [1998]). Thus r fails to be a well-behaved generator of global classical transformations on the sphere. Nevertheless, we might still consider local transformations involving limited regions far enough from the poles. For the one-mode phase this possibility has been examined in the quantum case by Bialynicki-Birula and Bialynicka-Birula [ 1976] and Cibils, Cuche, Marvulle and Wreszinski [ 1991 ]. The standard translation of classical variables into Hermitian (self-adjoint) operators simultaneously accounts for their role as coordinates and as generators of transformations, which merge in a single object, the operator. The coordinate function is represented by its eigenvalues and eigenvectors, which provide the set of admissible values and their probabilities. The operator generates unitary transformations through its complex exponential. According to the preceding analysis, we might foresee the lack of an operator for the phase difference because of the impossibility of the transformations
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433
it should generate. Such transformations would be precluded by the bounded character of Sz/So (for the single-mode phase, we have the bounded-from-below nature of the number operator). Nevertheless, it must be taken into account that not all conclusions of the classical analysis can be straightforwardly translated into the quantum case, in particular due to the lack of precise meaning of points on the Poincar6 sphere. We will return to this point in w3.5. In principle, it might be thought that to renounce the idea of an operator for the phase difference is to abandon completely the possibility of a quantum representation for this variable. However, it has been shown that it is possible to translate into the quantum domain the coordinate role of classical variables, thus splitting this role from their function as generators of transformations, which can be then dismissed. A probability distribution P(O) for any phase-angle q~ (or any other variable) can be assigned by means of a family of operators A(O) parametrized by r (positive operator measure) in the form P(r = tr [pA(r
(2.23)
where p is the density matrix of the system (Helstrom [1976], Yuen [1982], Peres [1993]). The statistical nature of P(r leads to the following conditions on A(q~) guaranteeing that P(O) is real, positive and normalized A t (r = A(q~),
A(q~)>~ 0,
~ ~ de A(q~) = I,
(2.24)
where I is the identity. In addition to these statistical conditions, some other desirable requirements may be imposed. Concerning phase-angle variables and following Leonhardt, Vaccaro, Brhmer and Paul [ 1995], Busch, Grabowski and Lahti [ 1995] and Pegg and Barnett [ 1997], a desirable property is the phase-shift condition
eiO's'-/ZA(~fl) e-iO'sz/2 = A(O + Or),
(2.25)
expressing that phase-angle and Sz are canonically conjugate variables. This condition is equivalent to say that
A((/)) = eir
-iOSS2.
(2.26)
Another natural requirement can be the commutator [S0,A(O)] = 0, expressing that the phase difference is invariant under equal phase shifts in both modes ei0'S~
e -i0's~ = A(r
(2.27)
This commutator also means that the phase difference and the total photon number (or total intensity) are compatible observables and can be measured simultaneously.
434
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 3
In general, the operators A(r will not be projectors on orthogonal states, that is tr[A(r162 ~ 6(r r and in such a case they are referred to as a nonorthogonal positive operator measure. This implies that the field property described by A(O) cannot be represented by any Hermitian operator acting on the system Hilbert space, since in such a case it would be an orthogonal positive operator measure. In other words, there can be phase probability distributions but no Hermitian operator representing the phase observable. Nevertheless, every nonorthogonal positive operator measure can be defined by a Hermitian operator but acting in an enlarged space including auxiliary degrees of freedom in a fixed state. Then A(~) arises after tracing over these additional degrees of freedom. Such a Hermitian operator and enlarged space are referred to as a Naimark extension or also as a generalized measurement (Helstrom [1976], Yuen [1982], Peres [1993]). There are some relevant features distinguishing this formulation of quantum variables from the more standard one in terms of Hermitian operators. In the general case it is possible that no state has a definite nonfluctuating value of ~. In other words, it can occur that there is no quantum state for which P(r has a non-zero width (Grabowski [1989], Shapiro and Shepard [1991]). This can be regarded as A(r provides an intrinsically noisy or fuzzy representation of r (Hall [1991], Hall [1993]). This conforms well with the fact that its measurement unavoidably involves auxiliary quantum fluctuating degrees of freedom, as implied by the Naimark extensions.
w 3. Quantum Relative Phase Formalisms In this section we present a brief overview of different proposals concerning how the relative phase or phase difference should be described in quantum terms. These formalisms regard phase difference as a dynamical variable. This means that they assume that it is legitimate to ask about the value of the relative phase as an intrinsic two-mode field property when the field is in an arbitrary state. The solutions proposed are very different and include abstract operators defined from first principles as well as positive operator measures and operational definitions based on feasible measurements. Despite their differences, most of them are directly related with the Stokes operators. It is worth pointing out that in all these approaches the relative phase they define cannot be expressed as the difference of phases. Also, most of them predict a discrete character for this variable. The formalisms included in this section are not the only ones dealing with phase for a two-mode field. Here we focus on those treating both modes on an
6, w 3]
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435
equal footing. There are other two-mode formalisms mainly aimed to describe the phase of a single mode, where the other mode acts as an auxiliary system prepared in a fixed and known state. We will consider them in w5 when examining how relative phase serves to define one-mode phase. 3.1. PHASE DIFFERENCE FROM STOKES PARAMETERS
Although the commutation relations preclude inserting into eq. (2.2) the outcomes of a (noiseless) simultaneous measurement of Sx and Sw, some information about phase difference can be obtained if Stokes variables are replaced by Stokes parameters r = arg(sx- iSy)-- arg((a,a~)).
(3.1)
This does not attempt to define an operator or a probability distribution since this is a relation between numbers and not between field variables or operators. It can be regarded as providing a mean, average or preferred value (Luk~ and Pefinovfi [1991], Opatrn~, [1994], Pe~inovfi, Luk~ and Pefina [1998]). Knowledge about phase difference fluctuations may be obtained by propagating the corresponding uncertainties of the Stokes operators. This approach has been applied and compared with other possibilities by Tana~ and Kielich [1990], Tana~ and Gantsog [1992a,b] and Luis, Sfinchez-Soto and Tana~ [1995]. It has been used also for the definition of a phase standard for Bose-Einstein condensates by Dunnigham and Burnett [1999]. The relation in eq. (3.1) suggests the definition of (noncommuting) cosine and sine of the phase difference as operators proportional to Sx and Sy respectively, in analogy with the so-called measured phase operators introduced by Barnett and Pegg [1986]
C = KSx,
S = gSy,
(3.2)
where K is a state-dependent number (Noh, Foug&es and Mandel [1991, 1992a,b], Shumovsky [1997]). In comparison with more rigorous definitions this has the advantage of being more closely related to feasible experiments as well as simplifying calculations, as pointed out by Lynch [1987]. 3.2. EIGHT-PORT HOMODYNE DETECTION
While quantum theory precludes the simultaneous noiseless measurement of Sx and Sy, nothing prevents measuring them at one time with less than perfect accuracy.
436
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
BS2.N]lra2~ a2-- !~ g2 ~ a2o
t
a3 L/
BS3~ a4
a,
[6, w 3
a6~ BS5 ~as I ~ )./4 . a,
a,o~ "BS,
7 Fig. 2. Outline of an eight-port homodyne detector made of four 50:50 beam splitters and a quarterwave plate.
In classical optics the joint noiseless measurement of Sx and Sy is possible and it can be achieved by properly splitting the input signal fields and measuring a different Stokes variable at each copy. A viable way to define and measure quantum phase difference is to directly transfer such classical arrangements to the quantum domain, leaving aside any other mathematical criteria. These have been termed operational definitions. A suitable example is many-port homodyne detection. Although such interferometric arrangements have been discussed before (Walker and Carroll [ 1984, 1986], Walker [ 1987]), the emphasis has not been on the problem of extracting the quantum phase difference until recently. The eight-port homodyne detector is the basis of the remarkable operational definition introduced and developed by Noh, Foug+res and Mandel [1991, 1992a,b, 1993a] and Mandel and Wolf [1995]. It has been successfully carried out experimentally for coherent fields, with special insight into the quantum limit of small photon numbers, partially coherent fields (Foug+res, Noh, Grayson and Mandel [1994], Fougbres, Torgerson and Mandel [ 1994]) and nonclassical downconverted fields (Foug+res, Monken and Mandel [1994]). This proposal has already been well examined and discussed before so we will merely focus on relevant two-mode features, especially its connection with the Stokes variables. The definition of the input, internal and output complex amplitudes is shown in fig. 2. The role of the input beam splitters BSI, BS2 is to provide two accurate copies of the input signal fields al, a2. To this end modes al0, a20 are always in vacuum. We will assume that the beam splitters are 50:50 and described by real reflection and transmission coefficients with a Jr phase change in the upperside reflections. After shifting by Jr/2 one of the copies of al, the intensities 13,14,15,
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QUANTUMRELATIVEPHASEFORMALISMS
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16 leaving the output beam splitters BS3, BS5 are measured. Classically we have al0 = a20 - 0 so that 14-13 ~ Sx, 16-15 ~ Sy and this provides the classical phase difference using eq. (2.2). This scheme also gives the measurement of the total intensity in signal modes as So = 13 + 14 + 15 + 16. In the quantum case intensity can be replaced by the photon number so we have the natural identification of phase difference as (3.3)
q~(k) = arg [n4 - n3 - i(n6 - ns)],
where k = {n3,n4,ns,n6} represents the photon-number readouts at the corresponding output ports. Nevertheless a different definition of q~ in terms of k has been found and examined by l~ehfi~ek, Hradil, Zawisky, Pascazio, Rauch and Pefina [1999]. We will return to this point in w3.3. Taking into account explicitly the vacuum state in modes al0, a20 the statistics P(k) of the measurement can be written in the form P(k) = tr(plk)
a~)/'/6 lO, O)
[k) = 2~ v/n3 !n4 !n5 !n6! (3.4) n5 + n6. These states are eigenstates of So with eigenvalue n, Solk) = nlk), reflecting the fact that the measurement of the total photon number in signal modes is, as classically, exact or noiseless. This also implies that this definition of phase difference is compatible with the total photon number. The vectors [k) are not orthogonal nor linearly independent. This is clearly seen because their number within each subspace ~n exceeds its dimension n + 1. Then Ik)(kl is a nonorthogonal positive operator measure and this scheme, regarded as a measurement performed on the signal modes alone, cannot be represented by a Hermitian operator (Englert and Wddkiewicz [1995]). Thus, the four-mode measurement, including modes al0, ae0, is a Naimark extension of the two-mode measurement described by the positive operator measure [k)(kl. Since the Stokes operators cannot be measured simultaneously, there must be relevant differences between the classical and quantum perspectives. The main difference is that the quantum vacuum is a fluctuating field that will affect the fields leaving the input beam splitters. No accurate copies of the signal field state can be obtained, and the classical reasoning fails. This is a particular example of the general result stating that arbitrary quantum states cannot be copied or cloned (Wootters and Zurek [1982], D'Ariano and Yuen [1996], Lindblad [1999]). This with n = n3 +
n4 +
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QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 3
implies that the classical relationship between output intensities and input Stokes operators is broken and the commuting measured observables
~Sx = 2(at4a4-a~a3) ,
~Sy = 2(a~a6-a[as),
(3.5)
are no longer proportional to Sx and Sv, respectively. Nevertheless, it is always possible to interpret the vacuum fluctuations as noise of quantum origin so that Sx, S,. would represent a noisy, fuzzy or unsharp simultaneous measurement of Sx, Sv. In fact we have the mean values (5'x) = (S~) and (Sv) = (Sv)- The effect of the vacuum fluctuations first appears when computing the uncertainties (Luis and Pefina [1996a]) ( A L ) 2 -- (AS~) 2 + (30),
( A ~ , ) 2 - - ( A S v ) 2 -t-(So),
(3.6)
where some extra noise proportional to the total intensity of the signal fields adds to the intrinsic fluctuations of S,- and Sv. The relative importance of this noise becomes negligible in the classical limit (So) ~ oo since in such a case ASx/(S0) ~ ASx/(So). Accordingly, eq. (3.3) should be regarded as a noisy, fuzzy or unsharp measurement of the phase difference. This agrees with the one-mode limit, which will be discussed in w5. The basis and motivation of this scheme and related ones are deeply rooted in classical optics, where vacuum means a null complex amplitude. This entails that arrangements which are equivalent classically can differ significantly in the quantum case because of vacuum fluctuations, leading to different statistics for the same input state. This has been regarded as a lack of consistency of these operational definitions which are then regarded as phase-dependent measurements rather than representing the phase itself (Barnett and Pegg [ 1993], Pegg and Barnett [1997]). On the other hand, it has been argued that this demonstrates that there is no unique quantum description of the phase difference (Noh, Fougbres and Mandel [1993b]). Another point of discussion is related to the proper interpretation and the way to handle the outcomes with n3 = n 4 and n 6 = n 5 , which are the readouts leading to S~ = S~, = 0. These are the poles of the Poincar6 sphere, where phase difference is not defined. Classically this is not a problem because all variables have continuous ranges of variation and the poles are two isolated points of null measure. Things are different in the quantum case since Sx, Sy take discrete values and the outcomes 5"x = 5"v = 0 have always a finite probability, generally nonnegligible. This is especially relevant as the field state tends to be the twomode vacuum since in such a case Sx = Sy = 0 becomes the most probable event.
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Several proposals concerning how to deal with this problem have been suggested. Some authors have proposed to discard such outcomes, renormalizing the remaining probability distribution (Noh, Fougbres and Mandel [1992a, 1993e], Torgerson and Mandel [1997], Hakio~lu [1999]). This has been criticized as being tantamount to altering the input state, effectively fabricating a new one (Hradil [1993a], Hradil and Bajer [1993]). Other authors have proposed to interpret such phase-inconclusive outcomes as being equivalent to a completely random readout for the phase difference, in agreement with a maximum-likelihood analysis of phase measurement (Raymer, Cooper and Beck [1993], Hradil [1993a], Richter [1997]). Moreover, zero-counts in homodyne detection can in principle include complete information about the field statistics (Wallentowitz and Vogel [1996], Banaszek and W6dkiewicz [1996], Mogilevtsev, Hradil and Pefina [1998]). Another noticeable feature of this scheme is that the possible phase-difference values in eq. (3.3) form a discrete numerable set. This discreteness precludes the fulfillment of the phase-shift property (2.25) since the phase shift r can take any value. This can be regarded as a discrepancy between different facets of a single concept. To solve such discrepancy a probability distribution defined on a continuous range can be obtained by repeating the measurement after controlled phase shifts q~j of the input fields (Noh, Fougbres and Mandel [ 1993c,d]). After each Oj this arrangement can be regarded as a noisy simultaneous measurement of the rotated Stokes operators
Sx(cp/) = cos CjSx + sin CjSy, Sy(Oj) = cos cpjSy - sin $jS~.
(3.7)
Each shift Cj can be interpreted as a displacement of the allowed values for ~, so that the output k is assigned the phase difference r ~j. After a large enough number of phase shifts, the possible values for 0 will tend to cover a continuous range with probabilities P(r = tr (peir
(3.8)
where
,4(0) = ~1 Z
e-iO(k)Sz/2 k)(k [eiC/)(k)Sz/2.
(3.9)
k
This P(q~) satisfies the shifting property (2.25) by construction. As discussed by Fontenelle, Braunstein and Schleich [1996], this procedure may circumvent the difficulties linked with the ambiguous phase data Sx = Sy = 0
440
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
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because of the continuous character of the final probability distribution, as in classical optics. The relative importance of these points depends on a direct linking of the measurement with some prescribed field variables, namely Stokes parameters and phase difference. From a different perspective, it can be interpreted that the measurement provides information about the signal fields, without prejudging an immediate connection with any definite field property. Instead, we can examine for which signal field observables it is possible to (indirectly) determine their statistics once the whole statistics P(k) of the measurement has been obtained. Among other examples, it has been shown that it is possible to obtain the statistics associated with all phase-difference proposals compatible with the total photon number (Luis and Pefina [1996b]). As a particular example, Richter [ 1997] has shown that P(r in eq. (3.8) can be obtained from the data recorded without shifting the input fields. 3.3. MANY-PORT H O M O D Y N E D E T E C T I O N
In the eight-port scheme each input signal mode is split once by mixing it with vacuum at a beam splitter. Such a technique can be generalized by considering an arbitrary number of splittings, which increase the number of measurements that can be simultaneously performed on the same input state (Walker [ 1987]). For instance, the eight-port detector can be enlarged by inserting two additional beam splitters in the input path of both signal modes al, a2. This leads to two new output ports, as well as the implication of two new auxiliary field modes in vacuum. With the help of these two new outputs it is possible to measure (unsharply) all Stokes operators (So,S) simultaneously. This has been studied for copropagating modes (Hakio(glu, Shumovsky and Aytfir [ 1994], Hakio(glu [ 1999], Alodjants and Arakelian [ 1999]) as well as for interferometric arrangements (Luis and Pefina [1996a]). A very interesting many-port interferometer has been studied by Raymer, Cooper and Beck [1993] following the semiclassical approach of Walkup and Goodman [1973]. It has been experimentally carried out for the study of phase correlation effects in stimulated Raman scattering by Smithey, Belsley, Wedding and Raymer [1991 ] and Belsley, Smithey, Wedding and Raymer [ 1993]. A closely related approach was developed by Mertz [1984, 1988]. This method is the spatial sampling of interference fringes. In a possible realization, illustrated in fig. 3, the directions of propagation of the interfering signal modes al, a2 are not orthogonal and after mixing them at a 50:50 beam splitter (BS) two spatial distributions of interference fringes appear. The fringes are recorded by
6, w 3]
QUANTUM RELATIVEPHASE FORMALISMS
441
Z,q,XgZhth4tA~
ale
,~
a2g~ Fig. 3. Outline of a many-port homodyne detector. Two fields are mixed at a 50:50 beam splitter. The outgoing fields are detected by two photodetector arrays.
two orthogonal photodetector arrays (assumed to be of unit quantum efficiency) at 45 ~ from the faces of the beam splitter. Each array comprises N identical detectors (pixels), much smaller than the fringe period. The relative phase is determined by fitting a sinusoidal function to the data after each exposure. By making many measurements on similarly prepared systems the statistical distribution of the relative phase is obtained. As in other similar situations, the best performance is obtained by considering the difference of photon numbers Anj, j - 1 , 2 , . . . , N , between equivalent pixels on different arrays. If the pixel width is small enough, such differences of photonnumber operators become Anj oc Z
t e i2a~W-(e-e')j + H . c . alea2e,
(3.10)
g,g'
where ale, azg, are the complex amplitude operators corresponding to the input modes at each face (1 or 2) of the beam splitter. The exponential factor depending on g, g' comes from the transverse propagation constants. The signal modes are formed by one mode ale and another mode azg,, say g = go, gt = 0. The rest of the modes are in vacuum. In order for the field to spatially resolve each pixel the number of modes effectively involved is N per beam. It is useful to examine these expressions from the point of view of classical optics. In such a case, the complex amplitudes of all vacuum modes can be replaced by zeros, so we have Anj c< cos ~j Sx + sin CjSy,
2Jr g . q~j = -~- 0J,
(3.11)
and each pair of pixels measures a rotated pair of Stokes variables. This classical expression coincides with the quantum mean value of eq. (3.10) so we can regard this scheme as a noisy simultaneous measurement of the rotated Stokes operators (3.11), all of them at once.
442
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 3
The phase difference between the signal fields can be determined from the expected harmonic dependence of the registered photon-number differences Anj on q~j, as it is expressed in eq. (3.11). Assuming that each array embraces an integer number of periods of the spatial fringe, a simple estimator r of the phase difference after each measurement is given by the discrete Fourier transform (3.12)
o
This fringe estimator arises purely from intuitive reasoning rather than from any particular optimality criterion. The use of joint maximum-likelihood estimation has been examined by Walkup and Goodman [1973] and Raymer, Cooper and Beck [1993]. In this context, l~eh~i6ek, Hradil, Zawisky, Pascazio, Rauch and Pefina [1999] and l~eh~i6ek, Hradil, Du~ek, Haderka and Hendrych [2000] have shown that eq. (3.12) is a valid maximum-likelihood estimator only for continuous Gaussian signals with phase-independent noise. In the general case the optimum relation between the inferred phase and the outcomes Anj is no longer given by eq. (3.12). This has been experimentally demonstrated in neutron interferometry. These results apply as well to the eight-port scheme in w3.2, as discussed by the same authors. Like the preceding example, this scheme is also a generalized measurement, so it does not define an operator for the quantum phase difference, but a nonorthogonal positive operator measure. The probability of each outcome k = {nits} is given by projection of the two-mode field state on the signal modes (now denoted by a l, a2) on the nonorthogonal family of unnormalized vectors
,
Ik) - (2N)~/-/ff IHi2~=l ~
,
( e-iCja~-(-1)Ua~ ) "j'' 10' 0),
(3.13)
where njt, is the number of registered photons at the pixel j of the array /t, and n - ~J,F~nJFe is the total number of recorded photons. These vectors are eigenstates of the total photon-number operator for the signal modes S0[k) - nlk ), so also in this case phase difference and total number are compatible observables. Leaving aside different constant phase relationships, this scheme can be viewed as a generalization of the eight-port detector and, accordingly, they share some characteristics. For example, there are outcomes for which the discrete Fourier transform in eq. (3.12) vanishes and no phase can be inferred. There are also some significant differences. For a given total photon number n, the number of possible outcomes (and then the number of phase-difference
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QUANTUM RELATIVE PHASE FORMALISMS
443
values) is much larger than in w3.2, due to the larger number of output channels. For example, if go = 1 and we have a one-photon state (n = 1), there are at least N possible phase-difference values, which can be a very large number in comparison to the four allowed values in the case of the eight-port homodyne detector. Many-port homodyning can approach a quasicontinuous range of variation for the phase difference even in the case of very weak fields. All the rotated Stokes operators (3.11) are simultaneously measured in every single run, while in order to achieve a similar goal the eight-port scheme requires the repetition of the measurement after an accordingly large number of phase shiftings of the input fields. We have quoted some examples of devices that provide the same answer for classical phase difference but differ substantially in the quantum regime. They not only predict different probability distributions, but even different sets of allowed values for r The mutual comparison of these arrangements raises interesting questions concerning the amount of quantum noise that effectively influences the measurement, its dependence on the number of ports and the way they are arranged, as well as whether there is an optimal number of ports (Raymer, Cooper and Beck [ 1993]).
3.4. RADIATION PHASE
The preceding examples show how quantum phase difference can be operationally defined by admissible measurements. Another approach to the problem is to consider that phase might be defined as well operationally by practical generation processes. If the vacuum has uniform phase distribution, any field state must acquire its phase properties in the process of generation. In particular, Shumovsky [ 1997] and Shumovsky and Mfistecaplio~lu [ 1997] have argued that the conservation of total angular momentum is responsible for the transmission of quantum phase properties from the source to the radiated field. Focusing on the Jaynes-Cummings model for an electric-dipole transition, five generalized Stokes operators for a three-mode field have been defined. The relevant ones concerning phase difference ,~x, Sy are defined as the operators which complement the atomic phase operators with respect to integrals of motion, being
Sx = a~al q-a ia 0 q-a~a2 q- H. c., (3.14)
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QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
~v=i (a~al +a~ao + a ~ a 2 -
[6, w 3
H. c . ) ,
where al, a2 are circularly polarized signal modes with opposite helicities, while a0 specifies the linearly polarized longitudinal component which in the far zone can be assumed to be in the vacuum state (Shumovsky and Miistecaplio~lu [ 1998a,b,c]). The original Stokes operators would be recovered provided a0 = 0, so that Sx, S~, are quantum noisy counterparts of Sx, Sv. These two operators commute [Sx, Sy] = 0, so they serve to define the phase difference via eq. (3.2) or via eq. (2.2) and looking for the eigenvectors and eigenvalues of Sx - iSy (Shumovsky [1998, 1999]). Moreover, the commutator [L,~v] = 0 also implies that L , Sy can be simultaneously measured. In fact, these same three-mode operators in eq. (3.14) are measured in a six-port homodyne detector, as shown by Paris, Chizhov and Steuernagel [ 1997]. Finally we can note that also this definition predicts a discrete character for the phase difference as well as compatibility with the total photon number in signal modes. 3.5. P O L A R D E C O M P O S I T I O N OF STOKES OPERATORS
The preceding formalisms interpret the expression (2.2) as a relation between numbers. Another approach is to regard eq. (2.2) as a relation between operators defining a phase-difference operator. In such a case there is no need to assume that the measurement of the phase-difference operator r should rely on the measurement of Sx and Sv. To avoid any problem linked with the natural periodicity of r it is preferable to work with the complex exponential E = e ir so that eq. (2.2) can be arranged as
Sx- iS,, = ala~ = Ev/a~ala2a ~ = v/a~a2ala~E,
(3.15)
where E is a unitary operator exponential of the phase difference which must be found. There is no ordering problem in these equations since the operators in the square roots are fixed by the assumed unitarity for E. In contrast to the similar polar decomposition for a one-mode field, this one has unitary solutions (Luis and S~nchez-Soto [1993], S~nchez-Soto and Luis [ 1994], Yu [ 1997a,b], Yu and Zhang [1998], Wu and Zhang [2000]). Since after eq. (2.2) we regard q~ as a function of Sx and Sy, the operator q~ should commute with So. Then, it is natural to look for solutions satisfying [S0,E] = 0 so that
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QUANTUMRELATIVEPHASEFORMALISMS
445
the phase-difference operator and the total number of photons commute. This allows us to solve eq. (3.15) on each subspace ~n of fixed total photon number n. Then, the solution for E can be written as
oo
E = Z E(n)'
(3.16)
n=O where E (n) is the solution of eq. (3.15) acting on 7-r The general relation between Stokes operators and angular momentum implies that within each Hn the problem is equivalent to the polar decomposition defining the azimuthal angle of an angular momentum j = n/2. This problem has been well studied by L6vyLeblond [1973], Santhanam [1976], Goldhirsch [1980], Vourdas [1990], Barnett and Pegg [1990a], Ellinas [1991a], Nienhuis and Van Enk [1993] and De La Torre and Iguain [1998]. The action of E ~n) on the number basis is
E(n)lnl,n- nl) E(n)lo, n)
= In1 - 1 , n - n 1
+ 1),
ifnl * 0 ,
(3.17)
In, 0),
= e i(n+l)r176
where q~n) are arbitrary constant phases. Some restrictions can be imposed on q~"), but it seems that they do not fix them completely (Luis, Sfinchez-Soto and Tanag [1995]). The simultaneous eigenvectors of E and So are
1
Ir
-- 4/~vt -JI- i
~
einl~(n) r In1, n - nl),
(3.18)
nl=0 with Elq}~n)) = eiO2n)Iq~n)>, Solqff )) = nlqff )) and r
= q~n) +
2r n+l'
r = 0, 1,
. n. "" '
(3 19)
The joint probability distribution for total number and phase difference is
P(n, r
_ tr (plr162
9
(3.20)
This operator E was introduced and confronted with early experimental results on phase measurements by L6vy-Leblond [ 1976, 1977]. It has been applied to the study of phase difference between Bose-Einstein condensates (Javanainen and Wilkens [1997]), to analyze the performance of quantum clocks (Peres [1980],
446
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 3
Bu~ek, Derka and Massar [ 1999]) and it also appears when considering the problem of accurate phase-shift detection (Luis and Pefina [1996c], Derka, Bu~ek and Ekert [ 1998], D'Ariano, Macchiavello and Sacchi [ 1998]). It has been used also to examine the mechanisms which enforce complementarity in two-path interferometers (Luis and Sfinchez-Soto [1998], Bj6rk and S6derholm [1999]). The allowed values for the phase difference form a discrete set, everywhere dense but not continuous. For a total photon number n there are n + 1 possible values uniformly distributed on the circle so their spacing is 2 ~ / ( n + 1). This spacing accounts for the minimum detectable phase change (Heisenberg limit) which is known to scale as 1/n (Ou [1997]), as we shall see in w4. Among the striking points of this operator we have that the two-mode vacuum 10, 0) is an eigenvector of E. Also, any choice for the constants r n) singles out a set of n + 1 states and phase values among a continuous set of largely equivalent states and phases. In other words, the vectors [r n)) are no longer eigenstates of E after a phase shift (unless the phase shift is 2r + 1), for integer k, as discussed by Hakio~lu [1998]). Here again discreteness prevents the fulfillment of the phase-shift property (eq. 2.25). These ambiguities are alleviated precisely by the existence of the Heisenberg limit, since it implies that phase difference cannot be determined with precision better than l/n, irrespective of the method used. As it occurs in the preceding formalisms, here discreteness also prevents the writing of E as the product of phase operators for each mode. This in turn implies that E lacks the usual mathematical properties of a difference, that is El3 ~ E12E23, where E O. is the exponential of the phase difference between modes ai, a]. This point, along with the proposal of a quantum addition rule for phasedifference operators, has been discussed in detail by Yu [ 1997a]. This question has been examined also by Kar and Bhaumik [1995] for the case in which the Hilbert spaces of modes al, a2 were finite-dimensional, as it is usually considered within the Pegg-Barnett formalism for the phase (Barnett and Pegg [ 1989], Pegg and Barnett [ 1989]). There is no simple commutation relation between E and Sz. The most simple expressions are obtained on the Weyl form within each subspace 7-/, and involving only discrete phase shifts (Santhanam and Tekumalla [1976], Ellinas [1991b]). Nevertheless, the conjugate relation between S~ and E can be examined easily from the action of E on the basis of eigenvectors of Sz shown in eq. (3.17). There we see that E shifts S. by -1, except for the eigenstate with minimum eigenvalue S~ = -n, which is transformed into the state with maximum eigenvalue Sz = n. Such transformation is cyclic and (E(")) n is proportional to the identity on 7-/, (Santhanam [1976, 1977]).
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QUANTUMRELATIVEPHASEFORMALISMS
447
In w2 we argued that classical phase difference cannot be properly regarded as a generator of transformations because of the orbit crossing at the poles and the absence of image/preimage at the arctic/antarctic areas. However, in the quantum case there is a unitary operator E representing Sz displacements as far as possible. In the quantum case the orbit crossing obstacle is removed because of the quantum lack of precise meaning of the points on the Poincar6 sphere. The removal of this difficulty allows us to map the region close to one pole (classically without image) on the regicn close to the other pole (classically without preimage) (Ellinas [ 1991 a]). This might explain why the two-mode polar decomposition has unitary solutions Where the single-mode one fails. It might be said that the quantum phase difference is better behaved than both classical phase difference and singlemode phases. This quantum solution cannot be easily transferred to one-mode phase. In such a case, the phase space is the plane instead of the sphere and classical difficulties arise at a single point, the origin. The problem of orbit crossing will be again removed by quantum mechanics, but the absence of images/preimages is difficult to solve unless resorting to some phase-space cutoff far away from the origin (Santhanam [1977], Pegg and Barnett [1988], Barnett and Pegg [1989]), or introducing a field state representing an infinite number of photons (Vaccaro [ 1995]). Both possibilities behave as if they were introducing an effective remote pole which together with the origin play the same function as the two poles of the sphere. Concerning the practical measurement of this operator, it has been shown by Bj6rk and S6derholm [1999] that the transformation of photon-number measurements into phase measurements would require extremely nonlinear field couplings, which are rather difficult to achieve experimentally for arbitrary input signal fields (Sanders, Milburn and Zhang [1997] have proposed to use quantum computers to transform photon counting into phase measurements). Nevertheless, for small photon numbers (up to n = 2) a clever experimental implementation has been found and carried out by Trifonov, Tsegaye, Bj6rk, S6derholm and Goobar [1999] and Trifonov, Tsegaye, Bj6rk, S6derholm, Goobar, Atatfire and Sergienko [2000] using only linear components. This is possible because a suitable unbalanced beam splitter transforms the number state ]nl = 1,n2 = 1) into one of the phase states (3.18). Then the probability distribution P(n = 2, q~2)) is measured by recording photo-detector coincidence counts between the two output channels of the beam splitter after suitably phase shifting the input signal field. Concerning indirect measurements, it has been shown that it is possible to
448
Q U A N T U M P H A S E D I F F E R E N C E , P H A S E M E A S U R E M E N T S A N D STOKES OPERATORS
[6, w 3
determine the whole probability distribution P(n, q~n)) for any input state from the statistics of the eight-port homodyne detector examined in w3.2 (Luis and Pe~ina [ 1996b]). Marburger III and Das [1999] have shown that if two boson modes are in a phase difference state (3.18) the visibility of interference fringes is less than unity, while the SU(2) coherent states In, 0,r in eq. (2.10) with 0 = :r/2 have unit visibility. This is because such observation of interference is given by the measurement of the Stokes operators, whose fluctuations depend on amplitude as well as on phase fluctuations of the interfering fields. Phase states have two much weight in states with very different number occupations, and visibility is reduced when amplitudes are unequal. 3.6. RELATIVE-PHASEOPERATOR As we have mentioned above, the difficulties in defining the quantum phase for a single-mode are usually ascribed to the bounded from below spectrum of the number operator, which is the quantum counterpart of the nonnegative character of the radial coordinate in classical phase space. The situation changes when dealing with a two-mode field. In such a case, the relative phase is expected to be canonically conjugate to the number difference Sz, which is not bounded from below. This means that the unitary shifting of the number difference is possible, as demonstrated by Rocca and Sirugue [1973] and Ban [1991a, 1992a, 1993a]. Such a unitary operator (denoted by D and referred to as exponential of the relative phase) is defined by the following action on the number basis
Dim, n))
= [m- 1,n)),
(3.21)
where [m,n)) are number states Im, n)) - [nl,n2) with m = n l - n2 and n = min(nl, n2). This parametrization and the action of D in the number basis is illustrated in fig. 4. The eigenvectors of D are
In,O)) - ~
1
Z
eimr n)),
(3.22)
//1 ------2r
with Din, O)) = eiOln,r This is valid for any ~, so this operator has a continuous spectrum. The probability distribution for the relative phase is given by the orthogonal positive operator measure O(3
A(r = ~ n=0
In, q~))((n, r
(3.23)
6, w 3]
QUANTUM RELATIVEPHASE FORMALISMS n,--O
n=l
n=2
449
m=l m-o
,," nl
./
,i"
2
0
m=-I
" ; ,,if" n=2
1
2
3
n2
Fig. 4. Illustration of the parametrization m = nl - n 2 , n = min(nl, n2) of the number states In 1,n2). The arrows show the action of the operator D exponential of the relative phase.
Here again there seems to be no factorization of D as product of one-mode operators. The operator D transforms properly under phase-difference shifts
eir
-iO'sz = e-i0'D.
(3.24)
Equations (3.21) and (3.24) show that the mutual relation between D and Sz is the expected one for complementary observables. However, as discussed by Ban [1992b], it is not clear whether D actually represents phase difference. In this respect it should be taken into account that the shifting relations [eqs. (3.21), (3.24)] are not the unique properties that should be accomplished by the quantum phase difference. Some further relations with other observables also isolate it among other similar variables. For instance, as discussed in w2.2, commutation with the total number So is a desirable property satisfied by all the approaches examined so far. However in this case we have [D, S0] ;~ 0. This lack of commutation allows a continuous range of variation for q~, in contrast with all the other formalisms. In any case, the shifting properties (3.21) and (3.24) establish a strong relation between D and quantum phase. Interesting connections with other phase approaches are revealed by the one-mode limit examined in w5. This operator has been applied to the investigation of Josephson junctions (Ban [ 199 l a,b]), to the definition of time in quantum mechanics (Ban [ 199 l b,c, 1993b]) and to the study of phase relaxation processes (Ban [1991d]). 3.7. DIFFERENCE OF PHASES
So far we have examined the definition of phase difference or relative phase without resorting to the individual phases of the field modes involved. Actually, in this review we follow the opposite route. As a matter of fact we have noticed that no approach factorizes as difference of phases, while it is always possible
450
QUANTUM PHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, {} 3
to use a two-mode formalism to define the quantum phase for a single mode, as will be discussed in w5. Despite this, it is worth considering the result of defining r as difference r = q~l - r of one-mode phases q~l, r (Pegg and Barnett [1989, 1997], Barnett and Pegg [1990b], Gantsog and Tana~ [1991a], Pegg and Vaccaro [1995], Luk~ and Pefinov~i [1996]). In the most general case the variables q~l, 0z will be described by positive operator measures Aj(Oj), j = 1,2, satisfying the statistical conditions (2.24). The positive operator measure for the phase difference is simply defined by the natural relation '4(0) = Ji,'r dq}2 Al(r + r162
(3.25)
It is possible to go further if Aj(dpj),j = 1,2 satisfy the phase-shift property 9
t
t
9
t
t
e,O a, aj Aj(~j le -'0 a/aj = Aj((/)j + (/)1),
(3.26)
which is fulfilled by a broad class of phase approaches (Leonhardt, Vaccaro, Brhmer and Paul [1995], Busch, Grabowski and Lahti [1995]). In such a case it can be easily seen that A(r satisfies the shifting property (2.25) for the phase difference and also commutes with the total photon number [So, A(O)] = 0. This commutator allows us to split A(q~) as a sum of independent contributions A(n, r on each subspace 7-/, oc
A(O) = Z A(n, 0),
(3.27)
n=O
which serves to define a joint probability distribution for total number and phase difference as P(n, r = tr[pA(n, q~)]. These properties have further consequences. The commutation with So along with eq. (2.26) imply that any P(n,r is a periodic function of q~ with no more than 2n + 1 Fourier components. Then P(n, ~0) can be completely fixed by knowing its value P(n, q~rn)) at 2n + 1 points, like r = 2arr/(2n + 1) with r = - n , - n + 1 , . . . , n, for instance (Luis and S~nchez-Soto [ 1995, 1996]). Then, the information provided by a continuous q~is redundant and a discrete numerable set of phase-difference values is informationally complete. We find in this way an implicit discrete character which was explicit in the preceding approaches. These conclusions are valid for any phase formalism satisfying eq. (3.26). Two of them have received especial attention, the formalism based on the SusskindGlogower phase states and the one derived from the Q function.
6, w3]
QUANTUMRELATIVEPHASEFORMALISMS
451
By a variety of different arguments, the Susskind-Glogower phase states (Susskind and Glogower [1964]) O(3
0 ein0ln) , , / g1y ~ __
I0)-
(3.28)
are regarded as providing the most accurate description of the phase of a single mode by means of the nonorthogonal positive operator measure A(q~)= Iq~)(q~l (L6vy-Leblond [ 1976], Helstrom [ 1976], Holevo [ 1982], Shapiro, Shepard and Wong [ 1989], Hall [1991 ], Shapiro and Shepard [ 1991 ], Lukg and Pefinovfi [1991, 1993], Leonhardt, Vaccaro, B6hmer and Paul [1995]). These vectors are the eigenvectors of the Susskind-Glogower phase operator EIq~) = eiq~10), where OG
E = ~
In - 1)(n[
(3.29)
n=l
is the nonunitary solution of the single-mode polar decomposition a = E x/~a = x/-a--asE (Carruthers and Nieto [ 1968]). Going to the two-mode case we have O0
Iq~l)lq~2)- V / ~
.
n~Oelnr
(3.30)
where I,,, o) -
1
~einl4)lnl,n_nl)
(3.31)
nl =0
and q~= ~ 1 - q}2. Then we have
A(n, O) = In, O)(n, r
(3.32)
The phase-difference states In, q~) are not orthogonal, so A(n, ~) is a nonorthogonal positive operator measure. Given the general relation between Stokes operators and angular momentum, this formalism can be translated immediately into the quantum description of the azimuthal angle of an angular momentum j = n/2. This has been studied in detail by Grabowski [1989] and Sanders and Milburn [1995]. Leaving aside trivial equivalences, it can be seen that this positive operator measure is the only one having the shifting property (2.25) and being made of
452
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 3
pure states. Then, the In, r basis serves to define the representation in which Sz is a purely differential operator Sz = 2i0/0r as shown by Sanders and Milburn [1995]. This positive operator measure is also distinguished by its optimal properties concerning efficient measuring strategies for phase estimation and quantum decision theory (Helstrom [1976], Holevo [1982], Hall and Fuss [1991], Sanders and Milburn [1995], Sanders, Milburn and Zhang [1997], D'Ariano, Macchiavello and Sacchi [1998]). It has been used also to examine the enforcement of complementarity in two-path interferometers (Luis and SfinchezSoto [1998]). It has been compared to the experimental results of Gerhardt, Biichler and Litfin [ 1974] by Lynch [ 1990], Gerry and Urbansky [ 1990] and Tsui and Reid [ 1992]. This approach has been compared to the experimental results obtained in eight-port homodyne detection in most of the contributions quoted in w3.2. Furthermore, this formalism has been applied to the study of several processes like the correlated-emission laser (Lakshmi and Swain [1990]), the phase difference fluctuations of the quantum beat laser (Orszag and Saavedra [ 1991 ]), the propagation of the phase difference in nonlinear media (Tana~ and Gantsog [ 1991 ], Tana~, Miranowicz and Gantsog [ 1996]) and to the examination of the phase properties of pair coherent states (Gantsog and Tana~ [ 199 l b]). Concerning the measurement of this probability distribution, the analysis made in w3.5 could be transferred here due to the strong similarity between this approach and the phase-difference operator defined there. We could add here the potential measurement of the phase probability distribution proposed by Barnett and Pegg [ 1996] and Pegg, Barnett and Phillips [ 1997]. The signal state is mixed at a 50:50 symmetric beam splitter with a so-called reciprocal-binomial state (Moussa and Baseia [1998]). The outcomes with no photons at one of the outputs provide the probability distribution defined by the phase states (3.28) after suitably shifting the input state. Despite the similarity of this approach with the polar decomposition in w3.5, significant differences can be found. In the first place, we have discrete versus continuous character. This can be relevant when computing averages. Moreover, while the probability P(n,r ")) is contained in P(n,~), there is no equivalent relation between the marginal probability distributions for ~ because the spectrum q~) of the operator is different in each subspace 7-/~. Furthermore, P(n, ~) contains more information than P(n, 0~")) because P(n, ~) is fixed by its value at 2n + 1 0-points while there are only n + 1 allowed values for ~ ) . The continuous range of variation for ~ has the advantage that there is no set of privileged phase values and phase vectors. But it must be taken into account that this does not imply arbitrary phase (or angle) resolution for fixed n. It has been
6, w3]
QUANTUMRELATIVEPHASEFORMALISMS
453
demonstrated that there are no states with fixed n and arbitrarily well-defined phase difference or azimuthal angle (Grabowski [1989]). Within this same framework can be placed one of the earliest descriptions of phase difference, that introduced by Carruthers and Nieto [1968]. They defined the cosine C and sine S of the phase difference as C=~ I(E1Et2+E~E2)=
Ldr
cos q~za(O), (3.33)
S=~ where Ej, j = 1,2 are the corresponding Susskind-Glogower operators for each mode. These operators do not commute, [C,S] ~ O, and this prevents the unitarity of C + iS -- E1Et2 (Vaglica and Vetri [1984], Santamaura, Vaglica and Vetri [ 1987]). It must be pointed out that in the original proposal the operators C, S instead of A(O) were the basic quantities describing phase difference. The predictions based on these operators have been compared to results of experiments by Nieto [1977] and Rauch, Summhammer, Zawisky and Jericha [1990]. They have been applied to study the dynamics of Josephson junctions (Nieto [ 1968, 1969], Tsui [1993]) as well as used in a phase-difference model of interaction on lattices (Bogoliubov and Nassar [ 1997], Bogoliubov, Izergin and Kitanine [1997]). Another notable approach uses the coherent states (eq. 2.13). A probability distribution for the phase can be obtained from the radial integration of the Q function Q(ot) = (atpla>/n. This corresponds to defining the positive operator measure (Paul [ 1974])
A(r = -~l f 0 ~ drrlot = reiO>(ot = rei0].
(3.34)
Among other relevant features which make this approach very attractive is that it can be experimentally measured, for example in double homodyne or heterodyne detection as we shall see in w5 (Shapiro and Wagner [1984], Lai and Haus [1989], Leonhardt and Paul [1993a,b], Freyberger and Schleich [1993], Freyberger, Vogel and Schleich [1993a,b]). It easily can be seen that this formalism fulfills the phase-shift property (3.26). After eq. (2.14), a two-mode Q function leads directly to a probability distribution on the Poincar~ sphere defined by the SU(2) coherent states In, 0, q~) in eq. (2.10). In fact, we have that in this formalism A(n, q~) - n + 1 43r
dO sin Oln , 0, q~>(n, 0, 0[.
(3.35)
454
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 4
The result of this integration is not a projector on a pure state, so this positive operator measure might be regarded as a noisy version of eq. (3.32) in agreement with the work of Leonhardt, Vaccaro, B6hmer and Paul [1995]. This approach has been compared to results of experiments by Bandilla [ 1991] and Tsui and Reid [ 1992]. As before, eq. (3.35) provides immediately the definition of the azimuthal angle of an angular momentum or spin withj = n/2, as discussed by Agarwal and Singh [ 1996]. SU(2) coherent states with 0 = r have been regarded as phasedifference states and used to describe the relative phase of two Bose-Einstein condensates (Leggett and Sols [1991], Castin and Dalibard [1997], Sinatra and Castin [ 1998]).
w 4. Phase-Shift Detection in SU(2) Interferometers The approaches in w3 focus on the quantum translation of the relative phase as a dynamical variable. As mentioned in w 1, a relevant motive impelling the interest in quantum phase is that the detection of phase shifts is one of the most accurate procedures to monitor very small changes of any variable of interest. From a pragmatic perspective, the primary purpose of the quantum phase topic would be to accurately detect phase changes more than to define abstract phase operators or probability distributions. A phase change is an unknown nonrandom classical parameter. In principle, any phase-sensitive measurement will serve to detect a phase change. In fact, it has been argued that simple arrangements can do as well as, or even better, than sophisticated phase concepts (Hradil [1995], Hradil, My~ka, Opatrn~ and Bajer [ 1996]). In fig. 5 we have schematized the structure of a general two-mode arrangement aimed to detect phase shifts. Block G represents the preparation of a suitable two-mode field state that undergoes a phase shift r Block M represents the measurement to be performed. From this standpoint, the quantum phase problem becomes how to optimally estimate q~after one of the possible outcomes k of M has occurred. This is to look for the optimal input state, measurement
Fig. 5. General scheme for phase-shift detection in a two-modefield arrangement. Block G represents the preparation of the field state [~) undergoing the phase shift r Block M represents the phasedependent measurementperformed and k is the outcome.
6, w 4]
PHASE-SHIFT DETECTION IN SU(2) INTERFEROMETERS
455
and mathematical data treatment (Braunstein, Lane and Caves [ 1992], Braunstein [ 1992]). This implies that, while the statistics of measured variables is always a linear function of the density matrix, the statistics of inferred variables can be a nonlinear one, especially when accumulating data from multiple measurements (Hradil, My~ka, Opatrn~, and Bajer [1996], Hradil and My~ka [1996]). All the available information about the phase shift is contained in the statistics of the measured events P(k[r this is the probability of measuring k when the actual phase shift is r We shall call r the estimate of the true but unknown r According to a Bayesian formulation of the problem, P(k[r serves to define a posterior probability distribution for r of the form P(r ec P(k[r = r where we have assumed no prior knowledge about r (Helstrom [1976], Cousins [1994]). From this point two main routes can be followed. The posterior probability P(r can be used to get a definite and deterministic relation r such that the outcome k is interpreted as the detection of the phase shift r Such relation r will depend in general on the input state as well as on the measuring arrangement. It can be established in as many ways as a distinguished r value can be extracted from a whole probability distribution P(r One of the most studied is maximum likelihood which picks the r value maximizing P(r this is the phase shift that makes the occurrence of the actual outcome k the most probable event (Helstrom [1976], Shapiro, Shepard and Wong [1989], Shapiro and Shepard [1991], Braunstein [1992], Braunstein, Lane and Caves [1992], Lane, Braunstein and Caves [1993]). Since k is random, the estimate r is also a random variable obeying a probability distribution of the form P(r162 o< P(k(r 0). In a slightly different approach, P(q)lk) is regarded as a true probability distribution (Hradil [1995]). As argued by Zawisky, Hasegawa, Rauch, Hradil, Mygka and Pe~ina [ 1998] this is more in accordance with the probabilistic nature of quantum theory. In principle, detecting k is not exactly the same as measuring phase, so in quantum mechanics it is not granted that a one-to-one relationship +-+ k should exist. The global performance of a detection scheme followed by this data strategy is provided by the quantum average of posterior probabilities P(r162 = E k
P(r162
(4.1)
This formalism has been applied to neutron interferometry operating at the quantum limit by Hradil, My~ka, Pefina, Zawisky, Hasegawa and Rauch [1996] and Zawisky, Hasegawa, Rauch, Hradil, My{ka and Pe~ina [ 1998] showing that this strategy works well even in the limit of very few particles. In such a limit this data analysis works better than maximum likelihood, while they coincide in
456
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 4
the regime of large particle numbers (l~eh~i6ek, Hradil, Zawisky, Pascazio, Rauch and Pefina [ 1999]). Irrespective of which of the two preceding routes is followed, the performance of a detection procedure, or phase resolution, has to be estimated from the probability distribution P(~Ir in terms of its width as a function of ~. This can be done in many different ways, leading to different conclusions. As examples of performance measure we have reciprocal peak likelihood, variance, dispersion, entropy, confidence intervals, and so on (L6vy-Leblond [1976, 1985], Luk~ and Pefinov~i [1991], Hradil [1992a], Jones [1993], Bialynicki-Birula, Freyberger and Schleich [ 1993], Hall [ 1993], D'Ariano and Paris [ 1994], Opatrn3~ [ 1994], Hillery, Freyberger and Schleich [1995], Hradil, My~ka, Opatrn~" and Bajer [ 1996], Sanders, Milburn and Zhang [ 1997]). Whatever the choice is, the result will depend in general on the true phase shift r This is because most of the practical phase-dependent measurements are not shift invariant, that is P(k[O) ~ P(r r for any suitable set of constants ~k. The global performance of a detection procedure regardless of the actual r can be measured by introducing a cost function C(~, ~) assessing the cost of errors. Averaging overall possible phase shifts ~ we obtain the average cost
(4.2) where we have assumed that all r are equally probable. Once a particular cost function is chosen, it is natural to ask which observable should be measured and which should be the input state experiencing the phase shift in order to obtain the best possible sensitivity (minimum cost). Depending on the criterion used, this can lead to some of the abstract approaches in w3 (Holevo [ 1982], Shapiro and Shepard [ 1991 ], Luis and Pefina [ 1996c], D'Ariano, Macchiavello and Sacchi [ 1998]). However, optimization of estimation problems is difficult, and even for some simpler cases no solutions are known. Moreover, even if the solution is found, usually there is no known way to actually perform the optimal measurement and/or there is no current experimental procedure generating the optimal input states. This has led many authors to focus directly on accessible measurements and practical input states from the very beginning. The weakness of such approaches lies in that they leave no assurance that the ultimate sensitivity is achieved. In order to resolve very small phase shifts, classic devices like Michelson, Mach-Zehnder or Fabry-Perot interferometers have proven their usefulness. These linear passive devices are examples of SU(2) interferometers as shown by Yurke, McCall and Klauder [ 1986]. Currently, it is possible to reach experimental
6, w 4]
PHASE-SHIFT DETECTIONtN SU(2) INTERFEROMETERS
457
A .-/~
v
j
Fig. 6. Mach-Zehnder interferometer for the detection of small phase shifts.
conditions where the effect of classical or technical noise sources is almost completely removed so the fluctuations of the measured observables are quantum mechanical in origin. The performance of these interferometers is only limited by the quantum fluctuations of the Stokes operators. Some examples have been provided by Moss, Miller and Forward [1971], Schoemaker, Schilling, Schnupp, Winkler, Maischberger and Rfidiger [1988], Stevenson, Gray, Bachor and McClelland [1993] and Bachor [1998], to mention only a few examples. A relevant method to perform phase-dependent measurements is the MachZehnder interferometer schematized in fig. 6. The first beam splitter BS1 acts as the source block G in fig. 5 preparing the two-mode field state [~p) undergoing the phase shift. The output beam splitter BS2 serves to transform the photon-number measurement at the output ports into the simultaneous measurement of So and another Stokes operator on [~p). Without loss of generality we will consider that Sy is such a Stokes operator. This is a particular example of the block M in fig. 5. For very small q~ a linearization of the phase-dependent quantities is often sufficient. In two-path SU(2) interferometers a phase shift r will change the mean value of the measured operator from (Sy) to cos q~(Sy)+ sin r ~- {Sy)+~(Sx) provided that r << 1. In these conditions q~ can be detected only if q~(Sx) is larger than the fluctuations ASy of the measured observable. Then, the minimum detectable phase change 6r is (provided (S~) ~ 0)
ASy
60-I<Sx>l ~
1
(4.3)
aSz'
where we have used the uncertainty relation ASzASy >~ [(Sx)[ implied by the commutation relations (2.3). The behavior of the Fabry-Perot interferometer (or any other multipath interferometer) is slightly different, although these same tools can be applied leading to similar conclusions, as shown by Yurke, McCall and Klauder [1986] and D'Ariano and Paris [1997].
458
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 4
Hillery and Mlodinow [ 1993] have shown that this result is quite general and can be applied to any phase-dependent measurement M. If the phase shift is implemented by the unitary operator exp(-iCN), then we have AM
-
=
60 Id(M)/dOl
AM
1 I([N,M])I >~ 2AN'
(4.4)
where the uncertainty relation AMAN ~> [([N,M])[/2 has been taken into account. This relation can be used even when M is a unitary operator, for instance by computing AM as the dispersion (AM) 2 = 1 - [ ( M ) I 2 (L6vy-Leblond [ 1976], Luk~ and Pefinovfi [ 1991 ], Hradil [ 1992a], Opatrn~, [ 1994]). If moreover [(IN, M ] ) / = I(M)I, eq. (4.4) gives (60) 2 -
1
[(M)l:
1,
(4.5)
which is another known measure of phase uncertainty (Pefinovfi, Luk~ and Pe~ina [1998]). These expressions (4.3) and (4.4) have the form of an uncertainty relation. However, notice that in this context they are definitions of 6r Nevertheless, it is possible to arrive at similar relations after a previous independent definition of 6~ (Hilgevoord and Uffmk [ 1983, 1985], Uffmk and Hilgevoord [ 1984], Uffink [1985, 1993], Anandan and Aharonov [1990], Vaidman [1992], Horesh and Mann [1998], S6derholm, Bj6rk, Tsegaye and Trifonov [1999]). Since ~ is a parameter, these kinds of uncertainty products are often called parameter-based uncertainty relations. If in eq. (4.3) we use the general relation AS~ <~ (So), we conclude that 6~ ~
1
(So)"
(4.6)
This means that there is a fundamental quantum limit in these precision phase measurements, which depends on the total intensity of the field state I~) undergoing the phase shift. This is known as the Heisenberg limit. Although the situation considered here is a particular case, the general validity of this limit has been confirmed by very different analyses. For example, a series of arguments has been provided by Ou [ 1996, 1997], including very fundamental proofs based on the complementarity principle of quantum mechanics. This applies to single measurements. According to Braunstein, Lane and Caves [1992], the quantum limits to the sensitivity for multiple measurements are not yet well known.
6, w 4]
PHASE-SHIFT DETECTION IN SU(2) INTERFEROMETERS
459
The Heisenberg limit allows us to throw more light on the formalisms considered in w3. Since no arbitrary precision is allowed for a finite total number of photons n, either the spectrum of the phase difference is discrete (with an spacing scaling as 1/n), or it is an intrinsically fuzzy or unsharp description of the relative phase. As discussed in w2, the angular-momentum picture of a two-mode field allows us to translate these results into precision atomic spectroscopy. For example, population spectroscopy using the Ramsey method (Ramsey [ 1963]) is formally equivalent to a Mach-Zehnder interferometer, being q~proportional to the product of frequency and time. A lower limit to the uncertainty 6m in frequency determination is then simply obtained by replacing 6r by t6oo, where t is the total measurement time. The maximum precision achievable by using n atoms is 6o2 >/ 1/(tn) (Wineland, Bollinger, Itano and Heinzen [1994]). When [~p) is a coherent state, eq. (4.3) gives an optimum value 6q~ = 1/v/n, where n = (So) is the mean total number of photons or atoms. This equality is obtained when the coherent state has 0 = zc/2. This is the standard photon noise or projection noise in population spectroscopy (Itano, Berquist, Bollinger, Gilligan, Heinzen, Moore, Raizen and Wineland [ 1993]). For example, for a laser source of 1 mW the photon noise represents a phase uncertainty 6q~ _~ 10-7 rad (Bachor and Manson [ 1990]). The same phase uncertainty is obtained when repeating the measurement n times on the same one-photon state (Braunstein, Lane and Caves [ 1992]). This conclusion is more clear if we refer to population spectroscopy where SU(2) coherent states are made of independent uncorrelated atoms in the same internal state, as shown in eq. (2.21) (Kitagawa and Ueda [1993]). This also has been demonstrated in atomic interferometry and spin polarization measurements (Scully and Dowling [1993], Sorensen, Hald and Polzik [ 1998]). This phase resolution 1/v/~ is far from the bound 1/n established by the Heisenberg limit, especially when n is large. The improvement of the phase sensitivity beyond 1/x/~ is a subject of fundamental as well as practical interest. This is especially true for spectroscopy on trapped atoms where the number of particles is fixed and kept small. A dramatic example has been put forward by Wineland, Bollinger, Itano and Heinzen [1994] showing that an atomic clock using 10 l~ atoms operating at the Heisenberg limit would yield the same precision in 1 s as an atomic clock using the same number of uncorrelated atoms over a period of 300 years. In contrast to more involved statistical data analysis, the simplicity of the derivation of eq. (4.6) allows us to find easily the states I~p) that optimize the sensitivity. A necessary condition to reach the Heisenberg limit in eq. (4.3) is
460
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 4
that A~ASv -I<Sx>l. Since S commutes with So, the optimal states I~p) can be found within each subspace 7-/, as solutions of the eigenvalue equation
(cosh
OSv + i sinh tgS~)I~P) = 2tuliP),
where 0 is a real parameter, m = -n/2,-n/2 eigenvalue eq. (4.7) can be written also as
(e~
+ e-OSt_) I~p) = 4ml~p) ,
(4.7)
+ 1,...
,n/2, and S0[~P) = n[~p). The
(4.8)
where S~ = Sy + iSz (Aragone, Guerri, Salam6 and Tani [1974], Aragone, Chalbaud and Salam6 [1976], Ruschin and Ben-Aryeh [1976], Rashid [1978], Trifonov [ 1994a,b, 1997], Brif [1997], Campos and Gerry [ 1999], Pef'inov/t, Luk~ and K~epelka [2000]). A phase space representation of these states on the Poincar6 sphere has been computed by Dowling, Agarwal and Schleich [1994]. A generation mechanism based on the state reduction associated with quantum measurement has been presented by Luis and Pef'ina [ 1996d]. These states achieve the equality of the uncertainty relation ASzASy ~ I(Sx)], so they might be called minimum uncertainty states. However, since [Sy, Sz] is not a number but an operator, the mere equality does not imply that ASzASy takes its minimum value, and they are often called intelligent instead of minimum. In our context, the relevant property of these states is that they can lead to better phase resolution than coherent states. This is also referred to as spectroscopic squeezing and occurs provided that the variance of an angular momentum component normal to the mean angular momentum vector (J) is smaller than j v / ~ (Wineland, Bollinger, Itano, Moore and Heinzen [1992], Kitagawa and Ueda [1993], Agarwal and Puri [1994]). There are other definitions of spin squeezing (Walls and Zoller [ 1981], W6dkiewicz and Eberly [1985], Barnett and Dupertuis [1987], Hillery [1989], Dupertuis and Kireev [1990], Alodjants and Arakelian [1999]), but this is the one relevant in this context. The fulfillment of the eigenvalue eq. (4.7) is not sufficient to improve the coherent state resolution. The problem has been studied by Hillery and Mlodinow [1993], Brif and Mann [1996a] and Agarwal and Puff [1994]. In particular Brif and Mann [ 1996a] have found exact analytical expressions for these vectors and the phase resolution. The best resolution is reached when t9 ---, 0 and m ~ 0, leading to 6r ~ v/2/v/n(n + 2), which scales as x/2/n for n >> 1. In w2 we have shown that two-mode quadrature coherent states are SU(2) coherent states. Concerning squeezing, it can be asked whether two-mode
6, w4]
PHASE-SHIFTDETECTIONIN SU(2)INTERFEROMETERS
461
quadrature-squeezed states will lead to a superposition of states fulfilling eqs. (4.7) and (4.8) within each subspace ~n. According to Bandyopadhyay and Rai [1995] it appears that this is not the case. However, despite the lack of an exact relation, it has been shown that quadrature-squeezed states display the SU(2) or spectroscopic squeezing relevant here (Yurke, McCall and Klauder [1986], Hillery and Mlodinow [1993], Bandyopadhyay and Rai [1995], Brif and Mann [1996a]). The performance improvement of phase-shift detection by using squeezed states is well known (Caves [1981], Bondurant and Shapiro [1984], GeaBanacloche and Leuchs [1987]) and has been experimentally demonstrated in quantum-noise-limited interferometers by Xiao, Wu and Kimble [1987], Grangier, Slusher, Yurke and LaPorta [1987] and Sorensen, Hald and Polzik [1997, 1998] (see also Slusher, Hollberg, Yurke, Mertz and Valley [1985], Shelby, Levenson, Perlmutter, DeVoe and Walls [1986] and Wu, Kimble, Hall and Wu [1986]). The effect of detection efficiency has been examined by Paris [1995]. Since quadrature squeezing has revealed to be determinant in precise phase detection, many approaches focus on interferometric arrangements embodying nonlinear media, where it is known that quadrature squeezing can be produced (Pefina [ 1991 ]). The analysis of these active nonlinear interferometers would lead us far away both from the Stokes operators (that no longer describe the field observables measured in such arrangements) and from the definition of phasedifference as an intrinsic two-mode variable. More details can be found in the works of Yurke, McCall and Klauder [1986], Hillery and Mlodinow [1993], Brif and Mann [ 1996a,b] and Brif and Ben-Aryeh [1996], for example. The search for field states ]~p) surpassing the sensitivity achievable with coherent states need not necessarily focus on minimum or intelligent states, and some other possibilities can be found in the literature. This is because we can be interested in spectroscopic squeezing without attempting to reach the absolute minimum. Also, there are interesting situations where the simplified analysis leading to eq. (4.3) is not valid. For example, we have the combination of two eigenstates of Sy with eigenvalues close to zero, as proposed by Yurke [ 1986] and Yurke, McCall and Klauder [1986], ]~P) = ~
1
(l J, O)y + ]j, 1)y),
(4.9)
where we have used the notation Sy]j,m)y = 2mlj, m)y, So[j,m)y = 2jlj, m)y and j has been assumed an integer. It has been shown that in this case the sensitivity
462
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 4
scales as 60 -~ 2/n when n >> 1 (Yurke [1986], Yurke, McCall and Klauder [ 1986], Brif and Mann [ 1996a]). In the case of Bose-Einstein condensates, these states might be generated after an atomic detection performed on two interfering condensates, as shown by Wong, Collet and Walls [ 1996], Castin and Dalibard [ 1997] and Dowling [ 1998]. The squeezed states (4.7) are optimal in the limit t9 ~ 0. According to Brif and Mann [1996a], this limit should be treated with care since the case t9 - 0 does not preserve basic properties of the intelligent states with 0 ;~ 0. When # = 0 we have that [~p) is an eigenstate of Sv. Specifically, the case m = 0 has been examined in detail by Holland and Burnett [1993]. Such state [j,0)3, would be produced by combining two identical number states In, n) at a 50:50 symmetrical beam splitter. Holland and Bumett [1993] argue that if two classical fields of the same nonfluctuating intensities are combined at a 50:50 beam splitter, the output fields have a well-defined classical phase difference, irrespective of the input phase noise. The input state [n,n) accomplishes the requirement of identical nonfluctuating amplitudes in the quantum regime. These arguments have been studied in detail by Hillery, Zou and Buick [1996] obtaining slightly different conclusions but confirming that beam splitters transform well-defined difference of amplitudes into well-defined phase difference. Concerning the quantum analysis for the state [j, 0)y we have, (Sx) = (ASy) = O, so the performance of this proposal cannot be determined by using eq. (4.3), making it necessary to resort to a more complete statistical analysis. After computing the width of the posterior probability distribution P(~[k) using the variance, it has been argued by Holland and Bumett [1993] that this scheme will operate close to the Heisenberg limit (see also Hradil and My~ka [1996] and Dunningham and Bumett [2000]). Kim, Shin, Ha, Kim, Park, Noh and Hong [ 1998], Kim, Pfister, Holland, Noh and Hall [ 1998a, b] and Kim, Ha, Shin, Kim, Park, Kim, Noh and Hong [1999] arrive at the same conclusion using fourthorder correlations. These authors also have examined the effect of experimental imperfections. As discussed by Sanders and Milbum [ 1995] and Ou [1997], the requirement of identical number states at the two input ports can be accomplished using the two-mode squeezed vacuum generated in spontaneous parametric downconversion. In fact, this possibility has been already carried out experimentally by Rarity, Tapster, Jakeman, Larchuk, Campos, Teich and Saleh [ 1990], Ou, Zou, Wang and Mandel [1990] and Ou, Rhee and Wang [1999] (see also the closely related measurement performed by Kuzmich and Mandel [ 1998]). In such a twomode squeezed vacuum, the photon number n has a thermal distribution so that
6, w 4]
PHASE-SHIFT DETECTION IN SU(2) INTERFEROMETERS
463
low values for n have always the highest probabilities. Nevertheless, Kim, Pfister, Holland, Noh and Hall [1998a,b] have shown that the details of the statistics of n are irrelevant and the final performance only depends on the mean value of the total photon number. The proposal of Holland and Burnett [1993] has been translated to atomic spectroscopy in Ramsey-type interferometers for Bose-Einstein condensates by Bouyer and Kasevich [ 1997]. A quantum nondemolition intracavity measurement very similar to this one has been proposed and analyzed by Braginsky, Gorodetsky and Kalili [1998]. Nonclassical polarization states of light along with quantum nondemolition measurements of the Stokes operators have been introduced and applied to phase measurements by Alodjants, Arakelian and Chirkin [ 1997, 1998] and Alodjants and Arakelian [ 1999]. Going back to the atomic context, the improvement of the sensitivity requires the use of samples of correlated atoms. Several procedures have been presented, involving nonlinear couplings of trapped ions via their internal and external degrees of freedom as well as methods involving the interaction with squeezed light (Walls and Zoller [ 1981 ], Barnett and Dupermis [1987], Agarwal and Purl [ 1988, 1990], Wineland, Bollinger, Itano, Moore and Heinzen [ 1992], Wineland, Bollinger, Itano and Heinzen [1994], Kuzmich, Molmer and Polzik [1997], Saito and Ueda [1997], Molmer [1999], Sorensen and Molmer [1999]). Their generation after interaction with coherent states of light has been demonstrated by Ueda, Wakabayashi and Kuwata-Gonokami [1996] and Saito and Ueda [1999]. An entangled ensemble of 107 cold atoms has been generated and observed by Hald, Sorensen, Schori and Polzik [1999]. Kuzmich, Bigelow and Mandel [1998], Kuzmich, Mandel, Janis, Young, Ejnisman and Bigelow [1999] and Kuzmich, Mandel and Bigelow [2000] have demonstrated that it is possible to prepare spin-squeezed states by using the quantum state reduction associated with measurement after coupling the interfering system with an auxiliary angular momentum. On the other hand, Kitagawa and Ueda [1991, 1993] have proposed nonlinear MachZehnder interferometers generating squeezed fermion states approaching a phase sensitivity 6q~ _~ n -5/6. Scully [1985] has suggested improving phase-shift sensitivity by using correlated spontaneous-emission lasers. Another way to approach the Heisenberg limit in spectroscopy has been put forward by Bollinger, Itano, Wineland and Heinzen [ 1996]. Concerning the state ]~p) experiencing the phase shift, they propose to use I~p) = ~
1
([j, -j) + [j,j)),
(4.1 O)
464
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 4
where IJ, i j ) are the corresponding simultaneous eigenstates of Sz and So. These states have been termed maximally correlated states in the sense that the measurement of the energy of any atom determines the state of all the others. After a phase shift r the state in eq. (4.10) becomes l/p) = ~
1
([J'-j) + eiejOlj,j)) ,
(4.11)
so that the phase shift is amplified (so to speak) by a factor of 2j. These states are intelligent states of the parameter-based uncertainty relations mentioned above (Horesh and Mann [1998], Srderholm, Bjrrk, Tsegaye and Trifonov [1999]). Phase amplification also has been studied by D'Ariano, Macchiavello, Sterpi and Yuen [ 1996]. For the states in eq. (4.11) we have (J) = 0 for any r so the J are not phase-dependent observables. Bollinger, Itano, Wineland and Heinzen [1996] propose to measure M = II2J=lOk,x on [lp), which is the product of population inversions ~rk; after applying a Jr/2 pulse on [~p) (this is to measure the product of population inversions on ei;rJ'/2l~p)). This is a phase-dependent observable with (M) - cos(n0) and AM = [sin(n0)[. The application of formula (4.4) gives a phase uncertainty 6~ = 1/n which is the Heisenberg limit. It must be taken into account that two phase shifts differing by 2Jr/n are indistinguishable, which leads to an ambiguity (free spectral range) of the same order of 6r Methods to prepare the states (4.10) have been discussed by Cirac and Zoller [ 1994], Gerry [ 1996], Bollinger, Itano, Wineland and Heinzen [ 1996], Zheng and Guo [ 1997a,b], Molmer and Sorensen [ 1999], Jaksch, Briegel, Cirac, Gardiner and Zoller [1999], Steinbach and Gerry [1998], Gerry [2000], Rauschenbeutel, Nogues, Osnaghi, Bertet, Brune, Raimond and Haroche [2000] and SackeR, Kielpinski, King, Langer, Meyer, Myatt, Rowe, Turchette, Itano, Wineland and Monroe [2000]. Weak force detection with Bose-Einstein condensates prepared in the state (4.10) has been analyzed by Corney, Milburn and Zhang [ 1999]. The actual precision achievable with the states (4.10) has been examined by Huelga, Macchiavello, Pellizzari, Ekert, Plenio and Cirac [1997] after including decoherence effects. They argue that these states do not provide much better resolution than coherent states. Nevertheless, according to the same authors, some other partially entangled preparations can improve the sensitivity beyond the shot-noise limit of uncorrelated particles. The counterpart of the states (4.10) for field modes is I~,> - ~
1
(In, O) + 10, n>),
(4.12)
6, w 5]
FROM TWO-MODE PHASE TO ONE-MODE PHASE AND BACK
465
where Inl,n2) is again the usual photon-number basis. Their performance in the detection of phase shifts has been studied by Bj6rk, Trifonov, Tsegaye and S6derholm [1998] and S6derholm, Bj6rk, Tsegaye and Trifonov [1999]. For n = 1 this is a one-photon state divided between two modes and it can be generated by using a standard 50:50 beam splitter. For n > 1 it might be said that eq. (4.12) is also a split state where n particles behave as a single indivisible entity. The effective de Broglie wavelength of the compound is the individual wavelength divided by the number of particles, with the corresponding enhancement of sensitivity. For n > 2 an ordinary linear beam splitter would not serve to produce such a state. It would be necessary to use nonlinear collective beam splitters so that all the photons are transmitted or all are reflected. This leads to the so-called de Broglie interferometers which have been examined in detail by Jacobson, Bj6rk, Chuang and Yamamoto [1995] and Barnett, Imoto and Huttner [1998]. They have been successfully carried out experimentally by Rarity, Tapster, Jakeman, Larchuk, Campos, Teich and Saleh [ 1990] and Fonseca, Monken and Pgtdua [1999].
w 5. From Two-Mode Phase to One-Mode Phase and Back
Any description of relative phase naturally embodies a formalism (or even more than one) for the phase of a single mode. This occurs because one of the modes can act as a phase reference. Such a reference mode must be prepared in a suitable state, fixed and known in advance. In fact, most quantum descriptions of phase implicitly or explicitly involve auxiliary degrees of freedom. This is particularly true for nonorthogonal positive operator measures. In order to be implemented, the system space has to be embedded in a larger space (Naimark extension). The purpose of this section is to show how most of the relevant quantum phase formalisms emerge from the two-mode approaches considered in w3. Once such a goal is achieved, we realize that every relative-phase approach offers two ways to deal with phase difference. We have the direct definition and also an indirect one via the difference of the single-mode phases that the same approach defines. It has been demonstrated theoretically as well as experimentally that these two procedures do not give the same result. This proves that phase difference is a more fundamental variable than single-mode phase. Phase difference naturally reproduces one-mode phase while the opposite is not true. On the other hand, this is stressing that the quantum relative phase approaches examined in w3 are not expressible as difference of phases and,
466
QUANTUMPHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 5
accordingly, do not obey the mathematical properties of a difference. In other words, the addition rule for quantum phase difference diverges from that of classical phase differences (Yu [1997a]). 5.1. O P E R A T I O N A L D E F I N I T I O N S
The operational defmitions considered in w167 3.2, 3.3 and 3.4 lead to a suitable one-mode phase description when the reference mode, say ae, is in a coherent state of very large amplitude. In order to see how this occurs, it can be advantageous to consider first the same limit for the homodyne measurement of the rotated Stokes operator S~ = a~a2e -ir + ala~e i~ examined in w2.1. It is known that if mode a2 is in a coherent state ]a) with ]a I ~ c~ the measurement of SO becomes the measurement of the quadrature X~ of the field mode al 1
~S~ ~ Xr = a~e -ir + ale ir
(5.1)
where without loss of generality we have assumed a to be real (Lai and Haus [1989], Vogel and Welsch [1994], Mandel and Wolf [1995]). Now we can focus on the eight-port homodyne detector with a strong enough coherent state la) in the input mode a2. Then, the output modes a2, a20 of the beam splitter BS2 are also in strong coherent states and the photon-number measurements after the output beam splitters BS3, BS5 become the measurement of one-mode quadratures of modes ~l and hi0 respectively
(5.2)
where a has been assumed to be real and
l--i(al-al), and similarly for the field mode al0. It can be verified that [.~, ~'] - 0 and this scheme can also be described as being the measurement of the operator .~" + i~" = 2 (al + aI0 ) .
(5.4)
When mode al0 is in the vacuum state classically we would have 2a10 = X10 + iY10 = 0 and this arrangement would provide the measurement of
6, w 5]
FROM TWO-MODE PHASE TO ONE-MODE PHASE AND BACK
467
the complex amplitude al via the simultaneous measurement of the two quadratures X = X1, Y = Y1. The phase of al would be measured as r = arg(A" + i~')= arg(X1 + iY1). In the quantum case the noiseless simultaneous measurement of X1 and Y1 is precluded by the commutation relation [X1, Y1] = 2i. In other words, .~ ~ X1, Y ~ Y1 because al0 is an unavoidably fluctuating complex variable. Then, this scheme is a noisy, fuzzy or unsharp simultaneous measurement of X1 and Y1 (Lai and Haus [1989], Stenholm [1992], Freyberger, Heni and Schleich [1995]). This same one-mode limit is obtained by using other measuring arrangements, like heterodyne detection (Yuen and Shapiro [1980], Yuen [1982], Shapiro and Wagner [ 1984], Hradil [ 1992b], D'Ariano and Paris [1994]). Also the multimode scheme considered in w3.3 has this same one-mode limit (Walker [1987], Raymer, Cooper and Beck [ 1993]). In fact, the fringe sampling might be regarded as the counterpart of heterodyne detection in the spatial domain. The relationship between these schemes has been studied by Leonhardt and Paul [1993a,b]. There is another very interesting two-mode scheme introduced by Luke, PeHnovfi and I~epelka [1994] leading to this same limit but via a different route. Very often the problem of quantum phase and rotation angles are treated alike. But quantum rotation angles do not face the difficulties that quantum phase encounters. For instance, the instantaneous angle that the electric field of a plane wave traveling along the z axis makes with the x axis can be determined by simultaneously measuring the components Cx and ~y of the electric field. This is possible in the classical as well as in the quantum domain since [C~, gy] = 0. Classically, for a completely circularly polarized field (right-handed for example), phase and angle are proportional, so an angle measurement is equivalent to a phase measurement. If we express the measured observables C~, Cy in terms of the complex amplitude operators a+ for circularly polarized modes, we have C~c+ iEy cx a+ + at_,
(5.5)
and classically a right-handed field is characterized by a_ = 0. In the quantum case the complex amplitude a_ is always a fluctuating variable, even if mode a_ is in the vacuum state. The standard accurate measurement of gx + igy becomes fuzzy or noisy when regarded as a measurement of a+ alone (PeHnovfi, Lukg and Pe~ina [ 1998]). The statistics of these different schemes can be described in a unified way denoting the measured observable in eqs. (5.4) and (5.5) as a + b t, where a and b are the complex-amplitude operators for the signal (a = al, a+) and auxiliary modes (b = al0,a_) respectively.
468
QUANTUM PHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 5
Concerning phase, the operator a + b t, unlike a, admits a polar decomposition, being possible to define the unitary operator
R=
~/a + bt at+b'
(5.6)
as shown by Hradil [1992b, 1993a], Luke, Pefinov~i and K~epelka [1994] and Fan and Xiao [1996a]. Very interesting relations between this operator R and the relative-phase operator D in w3.6 have been found by Hradil [ 1993b], Ban [1994] and Luke, Pefinovfi and I~epelka [1994]. They establish an attractive connection between ideal and feasible phase descriptions. The statistics P(~) of the measurement of a + b t is given by the projection of the field state in modes a, b on the vector 1~) defined by the eigenvalue equation (a + bt) I~) = ~[be),
(5.7)
where ~ is any complex number. These are also the eigenvectors of R. For each outcome ~ the accordingly measured phase is q~= arg ~. This eigenvalue equation has been solved in a number of different ways (Helstrom [1976], Shapiro and Wagner [1984], Hong-Yi and Klauder [1994], D'Ariano and Sacchi [1995], Freyberger, Heni and Schleich [1995], Fan and Xiao [1996a,b]). The solution can be written in the form 1~) = O(~)l~ = 0),
1
1~ - 0) - - ~
~
Z(-1
)n
In, n),
(5.8)
n=0
where In, n) are number states in modes a, b and D(~) is the displacement operator D(~) = exp(~a t - ~*a) (Pefina [1991]). In our context, the total density matrix always factorizes as a product of density matrices for the signal Pa and auxiliary Pb modes p - Pa @ Pb, where Pb will be assumed to be in a fixed state (usually the vacuum). Taking into account this factorization, we have that this scheme can be regarded as a generalized measurement on the signal mode described by the positive operator measure
P(~)
= tT a L O a A ( ~ ) ] ,
A(~) : 1D(~)/~Dt(~),
(5.9)
where CX3
p -- : r t r b 6obl
=
(-1)"+"'(nlPbln')ln') (n[,
-- 01) -
(5.10)
n,t/t=0
is a density matrix in mode a (Hall and Fuss [1991], Hradil [1992b]). The statistics P(~) arise from the comparison or overlap of Pa with a reference state
6, w 5]
FROM TWO-MODE PHASE TO ONE-MODE PHASE AND BACK
469
/5 displaced by ~ (Aharonov, Albert and Au [1981], O'Connell and Rajagopal [ 1982], W6dkiewicz [ 1984, 1986], Royer [ 1985, 1989], Walker [ 1987]). If, as usual, Pb is the vacuum, then/5 is the vacuum, A(~) is the projector on a coherent state and P(~) is the Q function of the state in mode a. This implies that the measured phase probability distribution is the radial integration of the Q function quoted in w3.7 (Lai and Haus [1989], Stenholm [1992], Leonhardt and Paul [1993a], Freyberger and Schleich [1993], Freyberger, Vogel and Schleich [1993a,b]). After this limit, these operational schemes offer two possibilities to determine the phase difference. It can be measured directly as in w167 3.2 and 3.3, or indirectly by measuring the phase of each mode in turn against the field of a strong reference field and then applying the procedures of w3.7. These two possibilities lead to different probability distributions, especially for weak fields. This fact was already noticed by Nieto [ 1977], L6vy-Leblond [ 1977] and Pegg and Vaccaro [1995] and later shown explicitly in this operational context by Torgerson and Mandel [1996] and Fontenelle, Braunstein and Schleich [1996]. 5.2. P O L A R D E C O M P O S I T I O N OF STOKES OPERATORS
A suitable phase description can be obtained provided that the field state in the mode acting as reference has a large number of photons. In such a limit only projectors In, qff))(n, r with large n are of interest. The eigenvalues r approach a continuous range of variation and A(n, qff)) can be safely replaced by A(n, r in eq. (3.32). The conclusion is that, in the limit of high photon number for the reference field, the polar decomposition of Stokes operators tends to the phase-difference formalism derived from the Susskind-Glogower phase states examined in w3.7. If the density matrix factorizes p = Pl | P2, we have from eq. (3.25) that P(q~) : trl [Pl A(~)],
(5.11)
being A(q}) = f2ar dq~'P2(q~')Iq~+ q}')(q}+ q~'l,
(5.12)
where P2(r (r being Iq~) the Susskind-Glogower phase states (3.28). We can see that the field state in mode a2 acts as an imprecise or fuzzy origin for the phase in mode al. Such a fuzziness can be reduced by choosing the state P2 of the reference mode such that P2(r is as narrow as possible. A suitable choice is
470
QUANTUM PHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 5
a coherent state of large enough amplitude. In this context a useful representation of strong fields has been found by Klimov and Chumakov [ 1997] which allows us to deal with periodic Gaussian functions on the circle. This one-mode limit was considered by Luis and Sfinchez-Soto [1993] and later analyzed in detail by Davidovi6 and Lalovi6 [1999]. A similar conclusion was obtained by Ellinas [1991a] directly from the polar decomposition by performing a group contraction limit. Some numerical results for particular examples can be found in Luis, Sfinchez-Soto and Tana~ [1995]. After this one-mode limit we could also consider the indirect definition of relative phase as difference of phases, as suggested by Pegg and Vaccaro [ 1995]. This leads to the formalism examined in w3.7 instead of the starting point in w3.5, with the similarities and differences already discussed there. 5.3. RELATIVE-PHASE OPERATOR
The relative-phase operator discussed in w3.6 leads to the same one-mode phase description of the preceding subsection w5.2, but via a different strategy. Contrary to the previous examples, in this case a suitable choice for the state in the reference mode is the vacuum. The reason for this choice can be envisaged from fig. 4. If mode a2 is in the vacuum state [0)2, then the operator D is equivalent to the Susskind-Glogower exponential of phase operator (3.29). Equivalently, from eq. (3.22) we have 2(01n,r = 6,,0[q~) where [O) are the Susskind-Glogower phase states (3.28) in mode al (Ban [1993b]). More precisely, the restriction of D to the subspace n = 0 reads oo
Ol.=0 = ~
I n - 1 , 0 ) ) ( ( m , 0[ = E , I0)22(01+E2t10),l(01.
(5.13)
m-------oo
This restriction can be regarded as equivalent to the operator proposed by Newton [ 1980] and Barnett and Pegg [ 1986] after doubling the one-mode Hilbert space by introducing fictitious number states with negative n. A different relation between these same operators and D has been found by Yu [1997b]. The operator (5.13) also has been examined by Shapiro and Shepard [1991] and D'Ariano, Macchiavello and Sacchi [1998] from the point of view of quantum estimation theory as a Naimark extension of the Susskind-Glogower positive operator measure for a single-mode field. Shapiro [1993] has studied the communication capabilities associated with the measurement of the operator (5.13) introducing a scheme for zero-error probability phase-conjugate quantum communication at a finite average photon number.
6, {} 5]
FROM TWO-MODE PHASE TO ONE-MODE PHASE AND BACK
471
Finally we can point out that Din= 0 is formally equivalent to the angle operator for a plane rotor, such as a bead on a circular wire, after identifying the states ]m, n = 0)) with the eigenstates of the angular momentum component Lz (Carruthers and Nieto [1968], Lukg, Pefinovfi and K~epelka [1994]). 5.4. H O M O D Y N E D E T E C T I O N W I T H N O N T R I V I A L AUXILIARY M O D E
In all the preceding approaches, the role played by the reference mode is merely auxiliar and to some extent its explicit appearance might be regarded as an undesired consequence of quantum physics providing no contribution other than noise. However, some two-mode approaches have been introduced recently where the potential usefulness of the auxiliary degree of freedom is exploited. The homodyne detector with an intense enough reference field (local oscillator) provides one of the most efficient arrangements for phase scan. It has been used to define the so-called measured phase operators as the one-mode limit of eq. (3.2). Cosine and sine of phase are defined to be proportional to the corresponding quadratures of the field through a suitable state-dependent constant of proportionality (the amplitude of the field) that has to be measured independently (Barnett and Pegg [1986]). This definition was compared to the experimental results of Gerhardt, Biichler and Litfin [ 1974] by Lynch [ 1987]. In this context, a six-port homodyne scheme has been proposed by Ritze [1992] where an additional beam splitter allows a (noisy) simultaneous measurement of the amplitude and the desired quadrature. Standard homodyne detection also serves to define an observable phase distribution by considering only the null results, as suggested by Vogel and Schleich [ 1991 ]. Leaving aside these examples, four-port homodyning is not suitable for measuring completely unknown probability distributions for the phase of arbitrary field states. In order to be fully useful as phase detector there must be some previous knowledge about the phase of the measured state. The local oscillator must be in quadrature with the phase of the system, the phase fluctuations must be small and the amplitude of the signal field must be known. A modification of four-port homodyne detection with a strong local oscillator has been introduced by Wiseman [ 1995] and developed by Wiseman and Killip [1997, 1998], which exploits its intrinsic two-mode character. According to Wiseman and Killip [1997] heterodyne detection (or equivalently multi-port detection) does not require any previous knowledge about the field state because all field quadratures are sampled equally, at the expense of introducing additional noise. The proposal of Wiseman [1995] falls somewhere in between homodyne
472
QUANTUM PHASE DIFFERENCE, PHASE MEASUREMENTS AND STOKES OPERATORS
[6, w 5
and heterodyne detection. The phase of the local oscillator is continuously adjusted by a feedback loop over the course of every single measurement to be always in quadrature with the phase being detected. Such phase is inferred from the photocurrent record so far. The final performance of the measurement depends on how this inference is carried out. With a suitable strategy these authors have shown that this arrangement approaches the direct measurement of the probability distribution defined by the Susskind-Glogower phase states (3.28). Hradil and My~ka [ 1996] have suggested a relation of this scheme to the proposal of Vogel and Schleich [ 1991 ]. A closely related scheme has been introduced by Chizhov, De Renzi and Paris [ 1998] who consider a measurement of a + b t performed in two steps. In the first one, the measurement is carried out with mode b in vacuum. In the second step, the measurement is repeated but with mode b in a squeezed vacuum state whose phase is matched to the system phase extracted from the first step. This leads to a phase sensitivity scaling 1/n, where n is the total number of photons. Feedbackassisted homodyne and double-homodyne detection of phase shifts also have been addressed by D'Ariano, Paris and Seno [1996]. Recently, another interesting proposal dealing with the eight-port homodyne arrangement as a detector of phase shifts has been introduced by D'Ariano and Sacchi [1995]. Before entering the measurement device, the input state [~p) in modes a, b undergoes a phase shift to be determined from the result of the measurement of the operator a + b t. In order to use the full potential of the arrangement, the input state [~p) is taken to be as close as possible to one of the eigenstates [~e) of a + b t in eq. (5.8). The generation of the states [~) can be approximated by the output state It/) produced in spontaneous parametric downconversion (Pefina [ 1991 ]) oo
It/) = V / 1 - Ir/[2 ~
rfln, n),
(5.14)
n=O
so that It/) ~ I~ = 0) provided that r/ ~ -1. The displacement D(~) of Ir/) can be achieved by mixing one of the down-converted modes with a strong coherent field at a beam splitter of high transmittance (Busch, Grabowski and Lahti [1994]). Some features of this scheme recall the proposal of Holland and Burnett [1993] in w4. Both consider as input states the eigenstates of the measured observable. Also, after Sanders and Milburn [1995] and Ou [1997], both can be implemented using the same state [~7).
6]
REFERENCES
473
w 6. Conclusions In this article we have reviewed the existing approaches to the quantum description of the phase difference. Close examination shows that most of them admit the Stokes operators as a common denominator. The Stokes operators allow the establishment of a fruitful connection between the quantum phase problem and a great variety of fundamental and applied domains, such as interferometry and spectroscopy. Furthermore, the Stokes operators are easily measurable quantities. All this can speed up the conversion of phase into a valuable tool in quantum optics and related areas. Despite their differences, the formalisms proposed so far demonstrate theoretically and experimentally that phase difference is a more fundamental variable than single-mode phase. Phase difference allows us to circumvent some of the difficulties that phase has encountered from the very beginning of quantum theory. Well-behaved operators can be defined and accessible measurements are possible. Moreover, this variable embraces one-mode phase as a limiting case while revealing quantum features (like discreteness, for example) that the mere difference of phases cannot account for. Finally, we think it is worth stressing that suitable two-mode arrangements approach in practice phase measurements that up to recently were regarded as far from any implementation. We have tried to present a reference list as complete as possible but, given the rapidly growing volume of contributions on the diverse topics within the scope of this article, some important omissions are inevitable. We regret such omissions and hope that the interested reader will encounter these works in due course.
w 7. Acknowledgement We would like to thank Professors S.M. Barnett, G. Bj6rk, D.T. Pegg and J. Pefina for a careful reading of the manuscript and valuable suggestions.
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E. WOLE PROGRESS IN OPTICS 41 9 2000 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED
OPTICAL WITH
SOLITONS
A QUADRATIC
IN MEDIA
NONLINEARITY
BY
CHRISTOPH ETRICH 1, FALK LEDERER 1, BORIS A. MALOMED2, THOMAS PESCHEL 1 AND ULF PESCHEL 1
l Institute of Solid State Theory and Theoretical Optics, Friedrich-Schiller-Universit6t Jena, Max-Wien-Platz 1, D-07743 Jena, Germany; 2Department of Interdisciplinary Studies, Faculty of Engineering, Tel At)it) University, Tel At)iv 69978, Israel
483
CONTENTS
PAGE
w 1.
INTRODUCTION
w 2.
DERIVATION OF T H E BASIC E Q U A T I O N S . . . . . . . .
w 3.
M O D U L A T I O N A L I N S T A B I L I T Y OF C O N T I N U O U S P L A N E
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WAVE S O L U T I O N S . . . . . . . . . . . . . . . . . .
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S O L I T O N S IN P L A N A R W A V E G U I D E S . . . . . . . . . .
510
S O L I T O N S IN P E R I O D I C W A V E G U I D E S T R U C T U R E S BRAGG SOLITONS . . . . . . . . . . . . . . . . . .
w 6.
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S O L I T O N S A N D T H E I R B I F U R C A T I O N S IN N O N L I N E A R COUPLERS
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D I S C R E T E S O L I T O N S IN W A V E G U I D E ARRAYS
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MULTIDIMENSIONAL SOLITONS . . . . . . . . . . . .
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CONCLUSIONS . . . . . . . . . . . . . . . . . . . .
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ACKNOWLEDGMENTS REFERENCES
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542 547
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w I. Introduction In 1961, shortly after reliable laser sources were available, the experimental observation of frequency doubling [second harmonic generation (SHG)] of red light was reported by Franken, Hill, Peters and Weinreich [1961]. This virtually first experiment of nonlinear optics employed the second-order (or Z ~2)) nonlinear response of a quartz crystal. Since this pioneering discovery, frequency conversion has steadily been a subject of active research. Sum and difference frequency generation are regarded as versatile tools to extend the available frequency domains of laser sources which are usually limited by the spectral response of the gain material. Recent advances in the fabrication of nonlinear materials with large quadratic (Z ~2)) nonlinear coefficients (for an overview see Fejer [1998]) and the mastery of the periodic poling technology (quasi-phase matched materials), originally suggested in the seminal paper by Armstrong, Bloembergen, Ducuing and Pershan [1962], by groups around the world (for details see Fejer [1998]) have allowed for a considerable increase in conversion efficiencies. Highly reliable and efficient frequency converters are now commercially available. In most cases, traveling-wave geometries driven by short, intense, highly focused pulses are used, considerable pump depletion taking place. Obviously, a simple approximation neglecting the reciprocal action of the generated radiation on the pump field is no longer valid in this case. The diffraction of the beam focused onto the nonlinear crystal as well as the action of dispersion on the pulse shape start to play a major role too. Hence, an appropriate description of the spatio-temporal evolution of the fields involved becomes nontrivial and theories beyond stationary plane wave models have to be developed. The lack in the theoretical understanding of these complex phenomena has stimulated a strong research activity during the past several years. It has been found that a great deal of the physical processes involved can be understood in terms of the formation of spatial, temporal or spatio-temporal localized structures or solitons. Spatio-temporal solitons are often referred to as light bullets (LB). It was observed that fundamental harmonic (FH) and second harmonic (SH) waves tend to form a symbiotic state as nondiffracting beams in the spatial domain (Torruellas, Wang, Hagan, Van Stryland, Stegeman, Torner and Menyuk [1995], Schiek, Baek and Stegeman [1996]), stable pulses 485
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OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 1
in the temporal domain (Di Trapani, Valiulis, Chinaglia and Andreoni [ 1998]) or even light bullets in the spatio-temporal domain (Liu, Qian and Wise [1999]). These self-organized, localized objects are commonly referred to as solitons, although they lack some important properties of genuine solitons in integrable systems in a stringent mathematical definition. But, nevertheless, solitons in media with a quadratic nonlinearity also exhibit a remarkable robustness. Like for cubic nonlinear media, traditionally considered as the natural arena for optical soliton formation, quadratic soliton formation relies on a balance between linear diffractive or dispersive effects and a nonlinearily evoked phase modulation. Although true stationary states are usually not attained in the conversion process (e.g., only one input frequency) the solitons mark attractors of the nonlinear evolution. The fields either spread into linear radiation, or eventually approach a close vicinity of a soliton state and oscillate around it (Torner, Menyuk and Stegeman [1994], Etrich, Peschel, Lederer, Malomed and Kivshar [1996]). A basic understanding of soliton formation in this particular nonlinear environment is necessary to estimate the efficiency as well as length and time scales of the conversion process. Another driving source for the renewed interest in frequency conversion processes originates from a completely different approach, viz., the requirement to implement efficient all-optical processing schemes. Previously the most active research in this area of nonlinear optics relied on cubic nonlinearities. In that case, a mere phase modulation is the primary effect to be exploited but no appreciable generation of a new (triple) frequency is observed. In fact, quasiinstantaneous Kerr nonlinearities occur in many materials, including direct semiconductors in the transparent wavelength domain and in silica optical fibers. In these fibers, the cubic nonlinearity allows for a perfect compensation of dispersive effects, leading to the formation of stable solitons at mW power levels. Solitons can be successfully employed to increase the transmission capacity of long-haul fiber-optical communication lines (for a review see Hasegawa and Kodama [1995]). In contrast to this continuous progress, all-optical data processing still requires unrealistic high power levels. The available fast cubic nonresonant nonlinearities are basically too weak to allow for logical operations within chip dimensions. Therefore, more efficient schemes have been sought. A step forward in this direction was performed by realizing that only a powerdependent phase shift is required, which may be caused by any nonlinear effect. In particular, the fundamental harmonic accumulates such a phase shift during consecutive up- and down-frequency conversions shown experimentally by DeSalvo, Hagan, Sheik-Bahae, Stegeman, Van Stryland and Vanherzeele [ 1992] and calculated analytically by Kobyakov, Peschel and Lederer [ 1996] and
7, w 1]
INTRODUCTION
487
Kobyakov and Lederer [1996]. Provided that the overall conversion rate is small, this cascading mechanism (subsequent up- and down-conversion), resembles the action of some effective cubic nonlinearity (cascading or local limit). The general validity of this concept was successfully proven experimentally. Different basic photonic components employing the cascading mechanism, such as directional couplers (Baek, Schiek and Stegeman [ 1996]) and Mach-Zehnder interferometers (Back, Schiek and Stegeman [1995]), have been shown to operate at moderate power levels. Also, the formation of 1D spatial solitons in a waveguide, similar to those generated by the cubic nonlinearities, was demonstrated (Schiek, Baek and Stegeman [1996]). In the cascading limit, the strength of the effective cubic nonlinearity is proportional to the inverse of the phase mismatch between the fundamental and second-harmonic waves involved in the conversion process. Its further increase requires a reduced mismatch, resulting in considerable pump depletion. It turned out that the properties of solitons generated in this case considerably depart from the traditional ones governed by the nonlinear Schr6dinger (NLS) equation. One fundamental difference is that the quadratic nonlinearity allows for stable multidimensional solitons and does not lead to a wave collapse in any physical dimension. This fundamental property was discovered theoretically back in 1981 (Kanashov and Rubenchik [1981]), while, as mentioned above, stable self-guided, two-dimensional beams in quadratically nonlinear materials were observed experimentally by Torruellas, Wang, Hagan, Van Stryland, Stegeman, Torner and Menyuk [1995]. These effects, offering a new subject of fundamental research and having a great potential for the applications mentioned above, have stimulated a great deal of theoretical and experimental work during the last decade. Most effects observed for the continuous-wave case, i.e., in the absence of diffraction and dispersion may be neglected, can be readily understood on the basis of the seminal paper of Armstrong, Bloembergen, Ducuing and Pershan [1962] except for the explicit form of the phase modulation of the individual waves involved (see Kobyakov, Peschel and Lederer [1996] and Kobyakov and Lederer [ 1996]). But just this phase modulation plays a key role for the implementation of alloptical effects and for soliton formation in quadratic media. However, most theoretical activities were concentrated on the new aspects of the frequency conversion process in conjunction with diffraction and/or dispersion, i.e., the investigation of solitons supported by the quadratic nonlinearity. In fact, 25 years ago Karamzin and Sukhorukov [1974] had predicted that the fundamental and second-harmonic waves may propagate as a mutually trapped localized state. But, for a long time, these solitons were regarded as a certain peculiarity, and the
488
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 1
above-mentioned original paper predicting them was almost forgotten. Although the solitons supported by the quadratic nonlinearity were rediscovered several times (e.g., by Hayata and Koshiba [ 1993] and Werner and Drummond [ 1993]), a systematic investigation of their properties has been carried out only recently. In particular, in the 1D case (waveguide) it was found that the system of equations describing the interaction of FH and SH waves in the presence of diffraction or dispersion is always nonintegrable. However, neglecting these terms but taking into account a group-velocity mismatch (temporal walk-off) between the harmonics, the system is integrable by means of the inverse scattering transform (Kaup [1978]). Moreover, it was found earlier that a more general model of the nondispersive three-wave interaction (one wave is a pump, which may decay into an idler and a signal wave), with walk-offs between all the waves, is also integrable, not only in the one-dimensional, but in the multidimensional case as well (Kaup [1976]). Note that SHG is a particular (degenerate) case of the three-wave interaction, when signal and idler waves are identical to each other. The lack of a complete analytical description has stimulated the development and introduction of new theoretical concepts to describe nonintegrable systems. While first studies were mainly based on a numerical solution of the propagation equations (see, e.g., Tomer, Menyuk and Stegeman [ 1994, 1995]) more sophisticated analytical and semianalytical tools have been developed since. Families of stationary soliton solutions were found numerically (Buryak and Kivshar [1995a]) and their stability was investigated in detail (Pelinovsky, Buryak and Kivshar [ 1995]). It has been realized that the lack of integrability is not only a drawback, but may also give rise to new interesting dynamics. The solitary waves generated in nonintegrable models with quadratic nonlinearity may be sometimes fairly robust (Etrich, Peschel, Lederer, Malomed and Kivshar [1996]), while in other cases they may be unstable and decay in various ways (Pelinovsky, Buryak and Kivshar [ 1995]). These solitons may form unstable bound states (Wemer and Drummond [ 1993], Boardman, Xie and Sangarpaul [ 1995], Mihalache, Lederer, Mazilu and Crasovan [1996]), or may fuse as a consequence of their mutual interaction (Etrich, Peschel, Lederer and Malomed [ 1995], Baboiu, Stegeman and Torner [1995], Constantini, De Angelis, Barthelemy, Bourliaguet and Kermene [1998]). Collisions or other strong perturbations can readily excite extremely long-lived intemal oscillations of the solitons (Torner, Menyuk and Stegeman [1994], Etrich, Peschel, Lederer, Malomed and Kivshar [1996]), which are related to the existence of an internal mode of the soliton (Etrich, Peschel, Lederer, Malomed and Kivshar [1996]). The latter property is not known in integrable soliton models.
7, w 1]
INTRODUCTION
489
Because solitons in a quadratically nonlinear environment are built from two or three waves, they may be generated in different ways. Usually, the launch of a finite beam, i.e., at FH frequency, is sufficient to eventually generate a stable soliton. If the power of the beam at the FH exceeds a certain threshold, a soliton is formed, excess power being shed off in the form of quasilinear radiation (Torner, Menyuk and Stegeman [ 1994], Torruellas, Wang, Hagan, Van Stryland, Stegeman, Torner and Menyuk [1995], Schiek, Baek and Stegeman [1996], Fuerst, Canva, Baboiu and Stegeman [ 1997], Schiek, Baek, Stegeman and Sohler [ 1999]). Further increase of the power of the beam may lead to the onset of its modulational instability (Fuerst, Baboiu, Lawrence, Torruellas, Stegeman, Trillo and Wabnitz [1997]). The unstable initial beam breaks up into a number of different peaks, each giving rise to an individual soliton. Modulational instability is of fundamental interest by itself. It has been studied in detail analytically for cw states by Trillo and Ferro [1995] and He, Drummond and Malomed [1996]. For the down-conversion process (i.e., a SH input), soliton formation is initiated by a quantum-noise seed of the FH component. A very weak seed triggers the soliton formation, being considerably amplified (Canva, Fuerst, Baboiu and Stegeman [ 1997], Di Trapani, Valiulis, Chinaglia and Andreoni [ 1998]). In the context of quadratically nonlinear systems, certain linear properties of the material, which have no physical relevance in the one-component case, i.e., the NLSE, start to play an important role. The sign and magnitude of the propagation constant difference between the FH and SH waves (phase mismatch) critically affect the nature of the quadratic interaction. Different combinations of dark and bright solitons are found according to the sign of the mismatch between the harmonics, as well as the sign of their relative dispersion (diffraction) coefficient (Buryak and Kivshar [1995a]). The temporal (group-velocity difference between the waves) as well as the spatial walk-off between the waves in birefringent media, also influence the soliton formation (Torner, Mazilu and Mihalache [ 1996], Etrich, Peschel, Lederer and Malomed [1997]). It has been experimentally shown that they may also give rise to new all-optical steering and dragging (Torruellas, Wang, Torner and Stegeman [1995]) or switching operations (Torruellas, Assanto, Lawrence, Fuerst and Stegeman [ 1996]). Recently it was shown that the spatial and temporal walk-offs may compensate each other to a certain extent, while allowing for the formation of temporal solitons (Di Trapani, Caironi, Valiulis, Dubietis, Danielus and Piskarskas [ 1998]). Although a deep understanding of the fundamental processes involved in the conversion process has been achieved in the course of recent years, there still remain many unsolved problems. Many interesting phenomena
490
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 1
that have been theoretically predicted to exist are waiting for experimental observation. New types of solitons are expected to exist in such systems as quadratically nonlinear waveguides equipped with a Bragg grating (T.Peschel, Peschel, Lederer and Malomed [1997], Conti, Trillo and Assanto [1997], He and Drummond [1997]), in quadratically nonlinear directional couplers (Mak, Malomed and Chu [1997, 1998c]) and in evanescently coupled waveguide arrays (T. Peschel, Peschel and Lederer [1998], Darmanyan, Kobyakov and Lederer [1998], Kobyakov, Darmanyan, Pertsch and Lederer [1999]). It is also worthy to mention solitons predicted in various three-wave models that do not reduce to the standard SHG ones (Tran [ 1995], U. Peschel, Etrich, Lederer and Malomed [1997], Mak, Malomed and Chu [1998b]), solitons in media with competing quadratic and QPM-induced (Clausen, Bang and Kivshar [1997]) or material cubic nonlinearities (Karpierz [1995], Trillo, Buryak and Kivshar [1996]) and optical shock waves (Darmanyan, Kamchatnov and Lederer [1998]). Despite a considerable theoretical progress in the study of all these advanced types of solitons and the potential that many of them have for applications in all-optical schemes, experimental evidence for their existence has not yet been obtained. Only very recently, direct links between nonlinear optics in quadratically nonlinear media and quantum description have been established. In particular, strong quantum correlations between the fields generated in the down-conversion process and the formation of quantum gap solitons (He and Drummond [ 1997]) have been predicted. Although the formation and dynamics of solitons in quadratically nonlinear media is still a subject of active research, a certain level of completeness has been achieved within the past several years. Therefore, it is worthwhile and quite useful to summarize the results obtained in this field. The aim of this article is to give the reader an introduction into this rapidly evolving field and to provide an overview of the current research activities. In order to place the subject into the proper context, we primarily focus on theoretical concepts and predictions and only concisely review the experimental papers. As already mentioned the experimental confirmation of numerous effects studied theoretically or numerically is still a challenging task. In this respect a major task of this contribution is to attract attention of a broader audience to this rapidly growing and exciting field of nonlinear optics. We first dwell on the derivation of the basic equations describing the field propagation in various configurations exhibiting a quadratic nonlinearity. This concerns bulk media, film and stripe waveguides; two-core waveguide couplers and arrays and waveguides with a Bragg grating. Here, we exclusively focus on conservative (Hamiltonian) systems, because most of the experimentally accessible geometries are traveling-wave configurations in relatively small
7, w 2]
DERIVATION OF THE BASIC EQUATIONS
491
samples where losses are negligible. Consideration of soliton formation in quadratically nonlinear dissipative and/or amplifying media (Torner [1998a], Peng, Malomed and Chu [1998], Crasovan, Malomed, Mihalache and Lederer [1999]) or the formation of localized structures in externally driven cavities (Etrich, Peschel and Lederer [1997], Staliunas and Sanchez-Morcillo [1997], Trillo and Haelterman [1998], Jian, Torruellas, Haelterman, Trillo, Peschel and Lederer [1999]) are beyond the scope of this review.
w 2. Derivation of the Basic Equations Because most papers about solitons in quadratic media usually start from a reduced or normalized set of equations, the goal of this section is to fill the gap between experimental settings and normalized nonlinear partial differential equations. Thus the aim is a consistent derivation of the basic equations describing the spatio-temporal evolution of the field in a linearly birefringent and quadratically nonlinear medium. In addition to the nonlinear polarization we also take into account linear polarization terms, which can stem from adjacent waveguides (two-core coupler, waveguide array) or periodic variations of geometric or dielectric properties along the guide (Bragg waveguide). The basic configurations we are interested in are bulk media, film and stripe waveguides. Our concept is based on the dynamic evolution of normal modes of the unperturbed system under the influence of perturbations. Because soliton formation relies essentially on the balance of nonlinear phase modulation and dispersion and/or diffraction, these effects have to be reasonably taken into account. In order to do this it is natural to use the normal modes in the Fourier space with respect to time and space. The dynamics of these modes upon propagation, driven by the perturbations, are described by ordinary differential equations. A final inverse Fourier transform yields the partial differential equation for the spatio-temporal evolution. In order to keep prominent effects in birefringent materials, such as the temporal and spatial walk-off, it would be most appropriate to depart from the normal modes in a linear anisotropic medium. Unfortunately, this approach is too involved and does not straightforwardly yield the desired results. Because of this, we take advantage of the normal modes of an isotropic medium, i.e., stationary plane waves or guided modes with a propagation constant not depending on the direction of propagation. Then we account for the anisotropy as the leading perturbation in order to derive new normal modes. The dynamics of these modes are then governed by the remaining perturbations.
492
OPTICAL SOLITONSIN MEDIAWITH A QUADRATICNONLINEARITY
[7, w 2
2.1. BULK MEDIUM
We first consider the linear problem for the propagation of a spatio-temporal pulse at mean frequency too in an anisotropic medium. The real-valued dielectric tensor which can be arbitrarily oriented in the laboratory system (x,y,z) is introduced as
eO(oa) = e(a))60 + Ao.(oo),
Aij = aji,
(2.1)
i,j = x , y , z ,
where IA0.1 << lel is assumed. The wave equation in frequency space is then derived as 0) 2
0) 2
V2E(r' ~
V [VE(r, ~
[e(og,I + A] E,(r, to, = -E0r - p n l ( r , ~o) '
+7
(2.2) with I = (60), A = (a/j) and ~nl the nonlinear polarization. The tilde denotes quantities in Fourier space. We assume that the z axis coincides with the principal propagation direction, which implies that only small spatial frequencies k2, k 2 ( ( k 2 with k2(co) = (~02/c2)e(09) will contribute to the formation of the beam for widths exceeding 10~/x/e. Performing the spatial Fourier transform (with respect to x and y), introducing the slowly varying amplitudes Ai as El(Z; kx, ky, co) = Ai(z; kx, ky, oo) exp (ikoz) + c.c., where ko = k(w0) and c.c. denotes the complex conjugate, and omitting ~nl, we obtain from eq. (2.2) for the first two components
0A1 2iko ~ - - + (k2 - k2)A1 + k2A1 + kxkyA2 + kxkoA3 - i k • OA3 & 0) 2
~
03 2
.....
03 2
+ --c-TAxxA1+ --C--~-AxyA2+ --~AxzA 3 = 0,
(2.3) 0A2 2 i k 0 ~ + (k 2 - k2)712 + k,,ky74, + k27-12 + kykoA3 - i k y OA30z (.0 2
+
.v
0) 2
.v
0) 2
-~A,,yA, + --~-AyyA2 + --~AyzA3 = 0,
where kz2 = k2(co)- k2 - k y2 (kz = ko at kx = ky = 0 and co = o90). The amplitude A3 in eqs.(2.3) can be eliminated using the fact that the dielectric displacement
7, w2]
DERIVATIONOFTHEBASICEQUATIONS
493 N
vector is divergence free, or using the third of eqs. (2.2) for A3. In lowest order with respect to the small quantities kx, ky, co-COo and Aij this leads to A3 = -
kx N
ky-
]
gA1 + gA2 + Axz(000)A1 + Ayz(000)A2 9
(2.4)
We now substitute eqs. (2.4) into eqs. (2.3) and take into account leading terms up to second order with respect to kx, ky, co-COo and A O. only. Then the Fourier backtransformation in space and time is performed as aj(r, t) = exp(i000t)/f/dkxdkyd00 Atz(kx, ky, 00; z) exp [i (kxx + kyy - cot)] ,
which yields the following expressions for the FH components:
i
( 0~~ + k0--~ Oal " " 'Oa2 + Zlxz(,000)-~y" " 'Oa2 t0al ) + i2kZc 032 2 I2Axz(00o) __0. x - + Zlyz{,000)--0~ k~t 02al 20t 2
i
+~01 ( o2al--~-+ -~---T-02al)+ 2k0c2002[Axx(000)a 1 + Axy(00~ ] =0 Oa2 Oa, Oa, tOa2 + iZkzc 0)2 2 2Ayz(000)_~__ + Axz(000)--0~ + Ayz(000)--~--
+ k6--~
k~t 02a2
1 (02a2 02a2~ 002 - - ~ + OY2 J + 2koC ~ 2 0 t 2 t- ~0
[Axy(00~ + Ayy(00~
-- O,
(2.5) where k~ = (0k/000)1~o= ~ and k~' = (02k/0002)1~o=~. Note that corrections to k~ leading to different inverse group velocities have been neglected. Similar equations can be derived at the frequency 2000 for the SH waves (a3 and a4). Sorting out the nonlinear coupling between the four waves via
l ~ y ( r , 00) = ~1 j / d 001 d ~ )(jkl (2)(--(001 h- 002); 001,0)2) 130 (2.6) x Ek(r, 001)El(r, 002) 6(o01 + 092 - to),
9
,
I~
~
+
+
4-
I
,
~
I"~,~+I
l
i
-I-
-t-
~
I .~
-t~....
" ~
~ ~I~
-, _~, S
"
~
E
I
...
i'..,.}
-F
l
,
~
~:
~
,
~
I
-" - -~- ~ .
"
~
,
~
+
+
4-
l
~
+
-t-
~" ~ ~I~ ~I~
l
,
l
o~
~
~.
-*
~
I
l
~
~"
~
"
~
~
~
+
4-
~
+
-I-
I
~.~~
l
,.
~9
-*
~
l
>~
I
~..-,..'~
~ . . ~
"
~
~
+
4-
I
~
+
-t-
I
I
~
o
~ ~
o
>
,o
z
7, w2]
DERIVATIONOFTHEBASICEQUATIONS
495
A few remarks about these equations should be made. Obviously the system of eqs. (2.7) describes the evolution of four fields, two at each frequency. A restriction to a reduced set of equations with three (vectorial interaction) or even two (scalar interaction) fields has to be based on physical arguments, as, e.g., vanishing nonlinear coefficients (cf. Butcher and Cotter [ 1990]) or missing phase matching conditions for one kind of interaction. The two polarization components at one frequency are mutually coupled by the components of the dielectric tensor. This concerns linear (no derivatives) and walk-off (first derivatives with respect to x and y) terms. The linear coupling between a l, a2 and a3, a4 can be transformed away by rotating the respective polarization vectors. This transformation can be derived from the eigenvalue problem
0)2 (Axx Axy) (al,3) =~,(al,3) Axy Ayy a2,4 a2,4
2kc 2
(2.8)
'
with the eigenvalues
V1,2 -- 4k,c~176 oc2 [Axx+ Ayy _~_g/(Axx _ Ayy)2 _k_4Axy1 (2.9)
k2~ 20)2 [Axx4-Ayy q-v/(Axx - Ayy)2-+-4Axy] .
,V3,4
Eliminating the linear coupling leads to the final set of equations for type II or vectorial interaction:
( Oa~
10a~l
i \ Oz + Ogl Ot
6xl
Oa~ --~ -
6yl
Oa~l) --~
132102a~1 1 ( Oea~l 02a~l) 2 0 t 2 + ~1o --~ + - ~
2- - y-l a '* a 3' exp [-i (kla~ + k~a~- k~)a~)z] i
(~_
+
10a~2
Vg, Ot
6x2 Oa~2- 6y2 Oa~2)
--~
--~
[3210 2a~ 1 ( 0 2a~ 0 2a~ ) 2 0 t 2 + 2-~10 ---~T+ ~ -
= -yZdl* a 3' exp [-i (kl~ + k ~ - k~a~)z] i
( ~_~ -
+
10a~3 Og20t
6x3
Oa~ Oa~3) ~ -- Oy3--~
[32202a~3 1 ( OZa~ 02a~3) 20t 2 + ~ ~T- + - ~
2y3a~la2' exp [i (kla~ + k~oan_ k~a~)z] ,
(2.10) where 1/Vgj= k~j, fi2j - k~, kl~ - klo + Vl, k2a~ = k,o + v2, and ko~ = k20 + v3. The coefficients 6x,yj and the effective nonlinear coefficients ~. can be determined from the transformation. Basically in eqs. (2.10) only terms with the same phase mismatch Ak = kta~ + k2a~ - koa~ are kept. The coupling of the fields due to
496
OPTICAL SOLITONS IN MEDIA WITH A QUADRATICNONLINEARITY
[7, w 2
spatial walk-off is neglected because the mismatch (expressions exp[+i(vl - v2)], exp[+i(v3- v4)]) between the two polarization components in an anisotropic crystal are too large in a usual configuration (birefringent phase matching) to allow for a coherent energy exchange. Eqs. (2.10) provide the most general description of the spatio-temporal field propagation in an arbitrarily oriented biaxial crystal with a quadratic nonlinearity provided that the paraxial approximation holds and that the index difference between two corresponding polarizations is large. This equation should hold down to pulse lengths of about 100 fs and beam widths of 10/tm. The pertinent effects are described by the parameters Vg (group velocity), 6 (spatial walk-off), 132 [group velocity dispersion (GVD)] and 1/ko (diffraction coefficient). In the scalar case the first two equations of (2.10) become identical, the mismatch is A k = 2kla~- k~ and the factor '2' disappears on the fight-hand side of the third equation. We note in this context another peculiarity of systems with quadratic nonlinearity; namely the change of the sign of nonlinearity can be easily compensated for by a change of the sign of the SH amplitude. This is in contrast to the situation encountered in cubic media. For further use it is necessary to reduce the number of parameters in eqs. (2.10) to the minimum. This can be achieved by transforming the equations into a reference frame moving with the pulsed beam a 'l, i.e. X ~ X + bxlZ , y --~ y + (~ylZ, t ~ t - Z/Og 1 and by introducing normalized quantities as X =x
t
y=Y
W
W
Z=Z-dd, r = v % l & l l , U2 = L d ~ a ~ ,
U 1 = Ld ~ a / l ,
/3=AkLd, Ld=kl0W2,
U3 = L d ~ a ~
(2.11)
exp (-i/3Z),
where w is an arbitrary parameter which can be related to the beam width and Ld is the diffraction length. The basic normalized equations which describe
the spatio-temporal evolution of the field envelope in a birefringent nonlinear quadratic bulk medium are then OU1 1 [-sgn (/321)02 Ul 02Ul 02UI ] i-0-f+2 t. - - ~ + - ~ + - ~ J + U~ U3 =O' i Ik-~ +Ot2T-a2X-~-Ot2Y--~
j +5
-sgnCfl21)--0-~+ ~
+--O~J
+ ufcr3-o, i k--~
+CtT-~
-- ~3X - ~ - -- O t 3 Y - ~
+ ~ L-O'--~ +P
~
+
-flU 3+2U 1u2 =0,
(2.12)
7, w 2]
DERIVATIONOF THE BASIC EQUATIONS
497
where we have used the abbreviations ~Ld aT =
~--~
(1 Og2
/322
o = sgn(~2) ~ (
1 ) , Ugl
ajX,Y-- Ld ((~xyj W '
~xyl) '
j=2,3,
klo P - k20"
(2.13) Eqs. (2.12) and (2.13) are the point of departure for our further studies of temporal, spatial and spatio-temporal localization effects in quadratic bulk media where these general equations are specified and simplified accordingly. 2.2. WAVEGUIDES
Now we derive the evolution equations in a film waveguide where diffraction is confined to one dimension and the particular properties of waveguide modes have to be accounted for. To get the evolution equations in stripe waveguides these results can be straightforwardly simplified using the corresponding modes and dropping diffraction effects altogether. The derivation is similar to the bulk case. Starting points are the normal modes of the isotropic film waveguide, characterized by the real-valued dielectric function e(x; co). They can be separated into TE and TM modes, which for propagation in z-direction are ETE(X; CO)= ~TE(X;ky = 0, cO)exp(ikTE(to)Z) = (0, 0TEy, 0) exp(ikTEZ), ETM(X; tO) = ~TM(X;ky = 0, tO) exp(ikTM(tO)z) = (0TMx, 0, OTMz)exp(ikTMz), (2.14) where kTE and kTM are the propagation constants of the modes. Any perturbation will again affect the z-dependence of the mode amplitudes. Starting from Maxwell's equations, rewriting them for the transversal components of the electric (Et) and magnetic (Ht) fields and taking advantage of the reciprocity theorem, one arrives at the evolution equation for two arbitrary guided fields (see, e.g., B6rner, Mtiller, Schiek and Trommer [1990]): oo
OO
0 dx Ozz
.v 9 2t • I-Ilt
+
"~* • H2t Elt
= itO
dx E]~P
(2.15)
z --OO
--oo
where the superscripts TE and TM are omitted for the moment. Now we identify field '1' as a mode of the unperturbed, ideal waveguide but expressed for small inclinations in the transverse direction (y):
El (x,z; ky, to) = e.~(x'~ ky, tO) exp [i (kyy -k- kzz)] , Hl(X,Z;ky, tO) - hla(x;ky, tO)ex p [i ( k y y + kzz)] ,
498
OPTICAL SOLITONS IN MEDIA WITH A QUADRATICNONLINEARITY
[7, w 2
where the inclined field structures in case of the electric field are for TE polarization
~TE(X, kv, tO) =
0, kT----E'kT----E OTEy(X;tO),
(2.16)
and for TM polarization
~TM(X,k,,,O~)=(1 ' 0,0)OTMx(X;~O)+ (0, kTM' ky kTM kz) where kz =
eTM z(X; tO),
(2.17)
k ~ - k2. Field '2' is a superposition of all forward (A~) and
backward (Bl,) propagating modes assuming that the field profiles are not changing (B6mer, Miiller, Schiek and Trommer [ 1990]):
It
(2.18) where in contrast to the bulk case the amplitudes A and B contain fast variations in z. Using the orthogonality of the guided modes, we end up with the evolution equations 9OA~,(z; ICy, ~o) 1
&
+
(o~) - AF,(z;ky, oJ) = -~-~,
~;,(x; ky, ~o)~(x,z; ky, oJ), --OG OG
--1
OqZ
+
(fO) --
~
e/t(X, ky, og)P(x, z, ky, (.o), --DG
(2.19) without any further approximation. Here Pl, is the power per unit width carried by the mode g, defined as P!, = Re[fdx (~F, x h/,)~]. Obviously the evolution equations for forward and backward propagating waves only differ by the sign in front of the z-derivative. Thus in what follows we restrict ourselves to the forward propagating amplitudes. It is also assumed that in the vicinity of tOo the propagation constants satisfy kTE(tO) "~ kTM(tO) = k(tO). We proceed now similarly to the bulk case considering first the anisotropic part of the polarization P. Substituting the modes from eq. (2.16) and eq. (2.17) into the first of
7, w 2]
DERIVATIONOF THE BASICEQUATIONS
499
eqs. (2.19), expanding with respect to the small quantities ky, t o - too and A,y and performing the Fourier backtransformation
a~,O',z,t) = exp [-i (koz -
mot)]//dkydto~l,(z;
ky, co)exp [i (kyy -
cot)] ,
we finally get for the slowly varying amplitudes a~,
{Oa; 10a~) i\ Oz + -Og- - ~
f1202a~ 1 02a~ ( O) + 0)22 + ~ /@v--i6.v~ av 20t 2 2ko v = TE,TM oo
_
(koz
COo exp [-i 4P~
--
co0t)] f a x ~/~Ir ~_.~nl --OO
(2.20) where ko = k(too), 1/Og = k~ and/32 = k~' as before. The coupling coefficients are oo
K'TETE
]eTE yl 2
toOEO j d x
-
-
4PTE
Ayy, --OO (30
K'TETM --
fdxe y fax
4PTE
(eTMxaxy q-eTMzAyz) ,
--OO
oo
K'TMTE -- ~o~o
eTEy (e~MxAxy + e~MzAyz) ,
4PTM
--OO
oo
K'TMTM
EITMx,aAxx§
4PTM --CX3
oo
t~TETE
to0E0 /
-
-
2koPTE
dx [eTEy
[2 Ayz ,
--OO oo
~ E VM
4koPTEtoOeO /dx
eTE~*y [eTMxAxz + eTMz (Ayy + Azz)] ,
--0(3 oo
_ co0e0 /
0TMWE -- 4k0PTM
dx eTEy (e~MxAxz -t- e~MzAyy) ,
500
OPTICAL SOLITONS IN MEDIA WITH A QUADRATICNONLINEARITY
[7, w 2
oo
+c ) xy +2 I T zl
6VMTM = 4k0PTM --OO
Equations (2.20) describe the spatio-temporal evolution of the envelopes of a guided field in a birefringent film waveguide under the influence of a nonlinear polarization. Temporal and spatial walk-off, 1D diffraction and pulse spreading induced by the group velocity dispersion are included. For our purpose another situation is interesting as well, viz., stripe waveguides (O/Oy = 0) where the coupling of forward and backward waves (Bragg waveguide) or the coupling between adjacent waveguides (directional coupler or coupled waveguide array) is considered. This can be described by adding additional polarization terms to pnl (see the following subsections). For the film waveguide with merely a nonlinear polarization we can proceed as in the bulk case. At the frequency 2to0 (SH) equations similar to eqs. (2.20) hold. Transforming away the linear coupling (via/c,v), sorting out the nonlinear coupling and using a scaling similar to eq. (2.11), we finally get for vectorial interaction
OUl 1 1--~- + -~ 9
i i
( (0 3 0U2
[
--sgn (s~l)
0U2
~-a2v~j+~
~
02UI O2UI] + + OT2 Oy 2
,
[ 02 2 02 2] OT2 Oy 2 0 3) +-~1( - a O02T z3 + p o02y 23)
1
~+av~-a3v~
U2* U 3 = 0
- - s g n (/~21)
+
-4- gl* U3 - 0
,
(2.21) These are the same as eqs. (2.12) except that spatial walk-off and diffraction are limited to one transverse direction. Remember that around too and 2to0 we have assumed kxE1 --~ kvM1 and kvE2 --~ kvM2, respectively. The nonlinear coefficients yj- contain now overlap integrals between the different waveguide modes. This is due to the integral on the r.h.s of eqs. (2.20). Finally, for stripe waveguides the derivation follows identical lines. The modes ~F,(x,y; to) of the ideal waveguide are now hybrid and depend on both transverse coordinates whereas the propagation vector k(to) has only a z-component. Spatial walk-off and diffraction are absent and the overlap integrals in the nonlinear coefficients now have to be taken over x, y.
2.2.1. Bragg waoeguide A stripe Bragg waveguide is characterized by a periodic modulation of the cross section or the refractive index along the propagation direction. The main purpose
7, w 2]
DERIVATION OF THE BASIC EQUATIONS
501
of taking into account the anisotropy of the medium in the previous subsections was to demonstrate the origin of spatial walk-off. In stripe waveguide this is absent, and in the scaled evolutions equations the anisotropy is only apparent in the phase mismatch terms as far as our approximation is concerned (small anisotropy). Thus for the sake of simplicity the anisotropy is neglegted in what follows. Also, to keep the number of equations and parameters reasonably small we consider only the scalar interaction. Thus, accounting for the nonlinearity and the peculiarities of stripe waveguides mentioned above and taking into account the corresponding equations for the backward propagating waves, we get from eqs. (2.20) (3o co i
+ ----~-Vgl
+ Y l a l a 2 e x p ( - i A k z ) = - - ~ l exp[-i(kl~176
dx dY e~Pg ' -co -co
,0..)
i \ Oz + --Og2 - - ~
.o
+ Y2a2 exp(izlkz) = - ~
exp [-i (k20 z - 2mot)l
co co jj
dx dy e~Pg ,
-oQ -co
i
co co
(.., ,0.1) Ylb~b2exp(iAkz)=--~l.oexp[-i(-kl~176 - - - ~ - + ----&--Vgl
j:
+
dxdye~bPg '
-co -0(3 (x) 0(3
i --~z + ---& Og2
+ Y2b2 exp(-iAkz)= - 2--~2 090 exp [-i (-k20z - 2~Oot)]
dx dy e~b Pg ,
-oG -co
(2.22) where now subscripts '1' and '2' refer to FH and SH waves, Pj denotes the power of the guided modes, and A k = 2k10- k20. Moreover, we have dropped the GVD term (/~ ~ 0) because GVD stemming from the grating-induced coupling exceeds by far this term. The perturbation polarization Pg, evoked, e.g., by an arbitrary grating of depth d ruled at the top face of the waveguide, can be written as
2 Pg(x, y,z, t) = e0 Z [aJ(z' t)ej (x' y;jo~ exp (ikj 0z) + bj(z, t)~jb(x, y;jmo) exp (-ikj0z)] j=l
x exp(-ijmot ) [eguide(J~%)-1] ~
fm exp im---~z
,
m=-oo
(2.23) where A is the grating period and fm depends on the specific grating profile. If 237 = 21~2 w i t h I~1 , ~21 ~ 2 J ~ / / e , we now assume that 2k~0 2~ 5 _ 261 and 2k20 - 2-3-
502
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 2
insert eq. (2.23) into eq. (2.22) and take only terms with a small mismatch into account, we obtain i
+
i
, Oa,)
~
i -~
. 1
+ 'gla a2 exp(-iAkz) + K'glbl e x p ( - 2 i b l z ) = 0
Og1 Ot
+ vg20t
lO ,)
+~ ~ Ogl
i -~
+
GOt
+ y2a~ exp(iAkz) + ~gzb2 e x p ( - 2 i ~ z ) = 0 ,
,
(2.24)
,
+ ylb I b2 exp(iAkz) + tcglal exp(2i61z) = 0 + y2b 2 exp(-iAkz) + tcg2a2 exp(2i62z) = 0
Og2 Ot
with the linear coupling coefficients
j(-o060 [6guide(j(_o0)
_
1]s
f / d x dy e;ejb. gr
The wavevector mismatches 61 and 62 can be related to frequency detunings from the Bragg condition. We may assume that at a frequency a,~ the Bragg condition 2kl0(toB) 2;r a - 0 holds. For small deviations we obtain then 61 - - ( O h 3 - 0913)/O81 and 62 as a function of 61. In the literature two different but equivalent variants are used, viz., 62 = 261 - Ak or 62 = 2(09o - toB)/Ug2 nt- A k B with AkB - k20(2a,~)- 2zr/A. Without loss o f generality we set X'gl - ]X'gl ]. N o w a normalization o f eq. (2.24) can easily be performed using
U1 = ~ 1 v~l y2al exp (iblz) ,
U2 = ~ 1 Yx/--~bl exp (-i61z) ,
V1 = ~ 1 yla2 exp (i62z) ,
V2 = ~
Z = I/r Iz,
T = Vgl I/r
t,
1
~tl b2 exp (-i62z) ,
]~ = A k / I K ' g l ]
,
q = AkB/[K'gl I 9
Thus we arrive at the normalized equations
OU1
i
OU1 )
-0-Z +--0-~
i
--~+c--~-j
+ff2UI+U~VI-q-U2--O, +Y2V1 + U 2 + x V 2 = 0 , (2.25)
i \---~i
+ --0-T
+..QU2+USV2wU1--O,
(or2 or2) ---0-Z + c-0-~-- +I2V2+U2+
r*
VI=0,
where c = Ogl/Og2, ~ = ( ( D o - (DB)/(IK'gl IOgl), K" = tCgZ/ltCgl I and g2 = 2 s s = q + 2cg2.
or
7, w 2]
DERIVATIONOF THE BASICEQUATIONS
503
2.2.2. Waveguide coupler We assume that two identical stripe waveguides (symmetric couplers) are located in close proximity and thus perturbing each other by the mutual field overlap. As before, we restrict ourselves to the scalar case. We can take advantage of the first two of eqs. (2.22) but now for two different guides, and we include GVD. We denote the FH and SH envelopes by al~ and a2,, respectively, where the subscript # - 1,2 labels the guide number. This leads to
Oallt i--~-z4
10all~)
[321 02allz +
2
Vgl 0t
0t 2
Ylal,a2, exp(-iAkz) *
OO o(3 -
0)0
4P~
_
exp[_i(kl0z
(_,o0t)]
f f dxdyel~tP "~* --OO --CX3
( Oa=. 10a=.) i \ ~ -t
15=02a2~ + 72a~ exp(iAkz) 2
Vg20t
-
(J)0 e x p [ _ i ( k 2 0
2P2
9
(2.26)
Ot 2
(2O(X) z _ 2 0 ) 0 / ) ] f fdxdy,~.c e2~tP --OO --OO
where pc accounts for the perturbation polarization due to the respective other guide. Although the guides were assumed to be identical (same mode profile and same propagation constants) we have to distinguish between the modes in the different guides because of their different locations. Moreover these modes are nonorthogonal. Nevertheless in a very reasonable approximation we can neglect all terms proportional to f f d x dy ej~ej3-~t. The linear polarization pc essentially leads to two effects, a change of the propagation constant of the mode in guide/t, evoked by the index distribution in guide 3 - / t , and a mutual periodic power exchange (for details see, e.g., Chuang [ 1987]). This polarization can be written as
2
2
pC(x,y,z, t) = eo ~ Z
[Aej~ajn(z,t)~jn(x,y;jmo) exp (ikj-0z)] exp(-ijmot) ,
#=lj=l
(2.27) with Aej.n = ej.(x,y;jcOo)- ej,(x,y;jcOo) where ~(x,y;jo)o) and ejn(x,y;jcOo) are the dielectric functions of the two-core configuration and the isolated guide/~, respectively. Inserting now eq. (2.27) into eq. (2.26) we obtain
504
OPTICAL SOLITONS IN MEDIA WITH A QUADRATICNONLINEARITY
i
Oall 10all ) ~ + Og1 Ot i
i
Ot
f12102a12 2
O~2
9 F ]/la12a22
(Oa2210a22) f12202a22 ~ Ot q
exp(-iAkz)
= --K'llall
f122 02a21 2 0/2 F y2a21 exp(iAkz)=-K21a21
Og20t
Ogl
, ~- y l a l l a 2 1
Ot2
2
--~-Oa211Oa21)
( -Oa12 10a12 ) - ~ -~ i
&l 02all
[7, w 2
--"
082
2
0/`2
exp(-iAkz)
K'22aa2,
= --K'llal2
-- K ' 1 2 a l l
--/('21
-- K22a21
2 exp(iAkz )
~- ]/2a12
-- K ' 1 2 a l 2
=
a22
,
(2.28) where the coupling coefficients ~F, are defined as (DO
1CJ'l--
OO
(2O
4Pj
O(3
4Pj --00
OO
--(30 --(X)
--0~
OO
Cx3
~ 2 - 4Pj
OO
(2.29)
dy A6y'2ejl ej2 -- 4~ --OO --OO
--O(3 - - O O
Finally, eqs. (2.28) can be normalized by introducing a retarded time as t ~ t - Z/Vgl, the pulse length To, the dispersion length L 3 and normalized quantities as (see also eq. 2.11) U1 =
V1 = L 3)tl a21 exp [-i (Ak + 2 K ' l l ) Z-"
U2 = L3 x/'Yl ) ' 2 a 1 2 exp(-itqlZ),
Lfiv~)'2all exp(--iK'llZ),
L3 '
T -t
To
Lfi'
z] ,
V1 = L 3 Yl a 2 2 exp [-i (Ak + 2X'll) z] ,
r~ [fi21 [ "
(2.30) Ultimately we get the equations 1 0 2U1 ).sgn (~l) - - ~ - + Ul*Vl +K1U2 = 0'
OU1 i OZ
i (OVI_o_Z+ aT fiT -OV1)
002V12 OT2 -flVI+U?+K2V2--O
OU2 1 02 U1 i~ - ~ sgn (l~l) OT 2 + U~ V2 + K1 U1 -- O,
i
(OV2
OV2) 002V2 ~V2_f_U2+K2V1= 0
--~q-OlT--~
20T 2
(2.31)
7, w 2]
DERIVATION OF THE BASIC EQUATIONS
505
where we have used L/3(1 aT = -~0
[3 = (Ak
1)
Ug2
Ugl
'
+ 2K'll - K'21)L/3,
/322
a = sgn q~2)
Kj =/r
9
Equations (2.31) describe the field evolution in a symmetric directional coupler with a quadratic nonlinearity where temporal walk-off and GVD are taken into account. 2.2.3. Waveguide arrays
We conclude this subsection by describing an array of N identical waveguides that exhibit nearest-neighbor coupling. Here we are only concerned with the stationary case, i.e., where all time derivatives can be set equal to zero. Thus we can generalize and simplify eq. (2.28) to 9daln , 1---~-- + ylalnazn exp(-iAkz) = -tcllaln - K'lZ(aln+l + aln-1),
(2.32)
. dazn 1--~z + yzaZn exp(iAkz) = --K'21azn -- K'22(a2n+l + a z n - 1 ) .
The coupling coefficients (2.29) are now l~j"1 --
jmoeo f f d x dy ~
I jnl 2 ,
~2-
j mo eo / / d x 4Pj
"* " dy AEjn+ l ejnejn+ l '
NN
NN
where 'NN' indicates that only nearest-neighbor interaction is taken into account. We basically use the same normalization as in the two-core case, but replace the dispersion length L/3 by an arbitrary scaling length L0, which can be chosen such that either one of the coupling constants or the effective phase mismatch are scaled to unity. Similarly to eqs. (2.30) we have Un = L o x / ~ ]I2aln exp(-itcllz),
Vn = Lo Y1a2n exp [-i (Ak + 2tell) z] ,
Z-
z
Lo ' (2.33)
which leads to
idUn d Z + 2 U~, Vn + Cu ( U, +I + U,-1) = O, dVn i - d z - [3v, + u 2 + Cv (v,+l + Vn-1) = O,
(2.34)
506
OPTICALSOLITONSIN MEDIAWITHA QUADRATICNONLINEARITY
[7, w 3
m
where now the effective phase mismatch/3 and the coupling coefficients Cu,v are given by D
3 = (Ak
+ 2Kll
-- K'21)L0,
Cu = K'I2L0,
Cv = K22L0.
(2.35)
w 3. Modulational Instability of Continuous Plane Wave Solutions Continuous plane wave (cw) solutions are the simplest exact solutions of the nonlinear wave equations, with an amplitude which does not depend on the transverse temporal or spatial variables. The analysis of the modulational instability (MI) of the cw solutions against small perturbations that break the uniformity of the solutions is a necessary step in the consideration of more intricate properties of nonlinear media, first of all, because MI can transform cw solutions into soliton arrays (see a discussion of the physical purport of MI in the book by Agrawal [1995]). The modulational instability of the cw solutions of the basic (scalar, or Type I, see the previous section) SHG model was independently investigated by Trillo and Ferro [1995] and by He, Drummond and Malomed [ 1996]. This section is mainly based on the latter work, where the analysis was more general, including both spatial- and temporal-domain versions of the basic model (some particular results for MI in the temporal domain were also obtained by Buryak and Kivshar [1995b]). For scalar interaction the general equations (2.21) simplify to 0 2 U1
0 2 U11
OU1
1
ON2
1 ( - (Y0 2 U 2+ 02U2) -[~U2 + U? = 0
i--~- + ~
--sgn (1~1)
i--~- + -~
OT 2
OT 2
+
Oy 2
+ U ; U2 -- 0
,
(3.1)
p Oy 2
where the SH is now denoted by U2 and the walk-off terms are neglected (for the possibility of transforming these terms away see w4). Note that the factor in front of the nonlinearity in the second of the above equations is scaled out. If we consider either diffraction in a film or dispersion in a channel waveguide (such that the equations have only two independent variables instead of three) eqs. (3.1) simplify further. Introducing the transformation U1 = Ul eitcZ ,
U2 = u2 e2itcZ
(3.2)
7, w 3]
MODULATIONAL INSTABILITY OF CONTINUOUS PLANE WAVE SOLUTIONS
507
leads to 9Oul
r
02Ul
1-0-~- -~ 2 0 r 2 OU 2
x'ul + u 1u2 = 0,
(3.3)
S 02U2
i --~-~ ~ 2 0 " g 2
q u 2 + u21 = O,
where r = +1, s = p in the spatial (r = Y) domain, r = -sgn(/~l), s - - a in the temporal domain (r = T) and q = fi + 2tr As will be demonstrated in the following sections, the case of interest for the formation of bright solitons in the temporal domain is the one if, at least, the FH dispersion is anomalous (r = + 1). Given the transformation of eq. (3.2) the cw solutions correspond to constant solutions of eqs. (3.3). They can be assumed as real and are given by u20 = x'q,
U20 =
x'.
(3.4)
This solution exists only for tcq > 0, i.e., tr and q are both either positive or negative. For q = 0 eqs. (3.3) have also a degenerate solution corresponding to a pump wave at SH with no FH present, i.e., Ul0 - 0 with u20 being arbitrary. To analyze the modulational instability of the cw solutions eqs. (3.3) are linearized with respect to small deviations from these solutions. Seeking solutions proportional to exp[i(AZ- g2r)] of the linear system, where f2 is an arbitrary real wavenumber or frequency, leads to an eigenvalue problem for A. For the degenerate solution the analysis is, as a matter of fact, trivial. In this case the FH and SH perturbations evolve independently. It is straightforward to obtain the corresponding stability condition u20 < K"2. For the nondegenerate cw solutions the associated characteristic equation for A yields A2 = 1 ( R •
2
(3.5)
4A)
where
R = ~1 s
+ s 2) + ~,-22(sq + rtr + 4tcq + q2,
1 2 (89 A = ~g2
+ 2 r s t c + q ) ( 1 ~sg-24 + qg22 - 4 r t c q )
(3.6) .
As usual, the stability condition amounts to Im A(f2) > 0 which must hold for all g2. Since R and A are real the eigenvalues A may appear in purely imaginary pairs, A = +iAi, in real pairs, A = -+-&, or in quartets, A = + ~ 4- iAi, where the coefficients '~,r,i a r e real, and the two alternate signs 4- in the last expression are
508
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 3
mutually independent. From this it follows that (neutral) stability is possible only if all the eigenvalues are purely real. Equation (3.5) then implies R > 0, A > 0 and R 2 - 4 A > 0 for all values of Y2. The condition R > 0 is always fulfilled. From A > 0 it is straightforward to get for r=+l
:
r=-l"
s<0,
q<-2tcs,
s>0,
q>2tcs.
(3.7)
Note that rs < 0 is required for stability. The condition R 2 - 4A > 0 is more difficult to exploit. Following He, Drummond and Malomed [ 1996] we look for the intersection points of the straight lines q = ks and R 2 - 4A = 0 in the (s, q)plane for fixed Y22. Substituting q = ks into R 2 - 4A = 0 and solving for s the intersection points are given by s = (.(22 +12k) 2 { - 8 x ' k + T + 4 V/lck[-rf2 4 - 2(/r + rk)g2 2 + 4tck - T] } s = (f22 +12k) 2 {-8tck - T + 4 V/tCk[-rg2 4 - 2(x" + rk)g2 2 + 4tck + T] } T = v/g22[g2 6 + 4 ( r r + k)(2 4 + 4k2g2 2 - 1 6 r k ( 2 r + rk)].
(3.8) This is now expanded for large g22 which yields for r=+l:
s = 1 + 2(tr
1
k)~-22 -t- 4 v / - 2 t c k ~ - / + O(1/g24),
1
s = -1 - 2(tr - k)~--/+ O( 1/~'-24), (3.9) r=-l:
s = 1 - 2(tr + k)~-52 4- O(1/g-24), s = -1 + 2(tr + k) ~---~-+ 4 ~ - - k ~---~ + 0(1/g24).
Thus R2lines from
for g22 ~ ~ the lines s = + 1 are approached. Substituting this into 4A = 0 and solving for g22 yields a complex solution for s 2, i.e., the s = 4-1 are never crossed varying g22. Hence the conditions for stability eqs. (3.9) are for
r-+l" r--1
s>-l :
s> 1
(/c,q>0), (/c,q>0),
s<-l s< 1
(x',q<0),
(3.10)
(x',q<0).
For arguments that there are no other domains of stability as far as the condition R 2 - 4A > 0 is concerned or that the respective complementary domains
7, w 3]
MODULATIONAL INSTABILITY OF CONTINUOUS PLANE WAVE SOLUTIONS
509
are covered with intersection points we refer to He, Drummond and Malomed [1996]. For x',q > 0 eqs. (3.7) and eqs. (3.10) are summarized in fig. 3.1. Thus, cw states in the temporal domain cannot be modulationally stable unless the dispersion coefficients have opposite signs at FH and SH (which is quite possible in available Z (2) materials). In contrast, the cw state in the spatial domain is never modulationally stable, as the effective diffraction coefficients are positive for both harmonics. Note that a lack of modulational stability of cw states makes it unlikely to have stable dark solitons, for which the cw states serve as a background. However, a detailed consideration of dark solitons is beyond the framework of this review.
Fig. 3.1. Domains of modulational stabilityand instabilityof the cw solutions of the one-dimensional scalar model in the (s, q)-plane for tr = 1.
The MI analysis can be extended into various relevant directions. The most natural generalization is to consider SHG models with walk-off or a larger number of components. Very recently, He, Arraf, De Sterke, Drummond and Malomed [1999] considered MI of cw states in four-wave models of a Bragg grating equipped with a quadratic nonlinearity. In this model, the MI analysis proves to be extremely involved. Another generalization is MI in the simple scalar (Type I) model for the multidimensional case, which is determined by an interplay of the temporal dispersions at the two harmonics and spatial diffraction (Musslimani and Malomed [1998]). In some cases where the cw states are modulationally unstable, they decay into chains of bright solitons. A detailed analysis of the bright soliton states in various forms will be presented in detail in the following sections of the review.
510
OPTICAL SOLITONS IN MEDIA WITH A QUADRATICNONLINEARITY
[7, w 4
w 4. Solitons in Planar Waveguides In this section we deal with one-dimensional solitons in quadratically nonlinear media, as they can be found in waveguides. Here the transverse dimension is either space (film waveguide) or time (channel waveguide). The simplest case is scalar or Type I interaction neglecting walk-off. Since this is a particular case of vectorial or Type II second harmonic generation without walk-off the first subsection treats this in detail, discussing the two-parameter family of resting soliton solutions (for a more general case see Boardman, Bontemps and Xie [1998]). Of special interest is the Schr6dinger limit since as pointed out in the introduction the evolution equations are not integrable if a transverse dimension is included. As a consequence the dynamical response of the system is richer than in the case of a cubic nonlinearity. In particular the solitons can become unstable. The last point in this subsection is devoted to the variational approach which is shown to be a convenient tool and reproduces the basic soliton dynamics. In the next subsection walk-off is included leading to moving soliton solutions. Here the discussion is limited to scalar interaction. The last two subsections deal with special aspects of quadratic solitons. Particular soliton solutions which do not exist in the cubic case are multiple hump solitons, which are always unstable but show an interesting decay behavior. The final subject addressed is the persistent oscillations of solitons which may be the consequence, e.g., of an imperfect excitation process. 4.1. VECTORIAL INTERACTION WITHOUT WALK-OFF
In this subsection we consider one-dimensional solitons in scalar and vectorial second harmonic generation without walk-off. Starting point are eqs. (2.21) without the terms referring to walk-off. Considering only one transverse coordinate, i.e., spatial or temporal propagation, leads obviously to the same evolution equations. For the sake of generality we start with the vectorial case. With above assumptions the evolution equations for the two fundamentals UI, U2 and the second harmonic U3 are 9 oqU1 1 0 2 U1 1 - - - ~ q- ~ Oy-----T+U~U3=O,
.OU2
1 02U2
1--ff2- + -~ or-----T+ u ? u3 = o,
. OU3
p 02U3
1--~-- + 2 0 Y 2
[JU3+2U1U2=O'
(4.1)
7, w 4]
SOLITONS IN PLANAR WAVEGUIDES
511
where/3 is the (rescaled) phase mismatch between the two FHs and the SH. Eqs. (4.1) are written for the spatial case which is mainly dealt with in this subsection. Thus p is the ratio of the wavenumbers of the FHs (assumed to be equal) and SH. Writing eqs. (4.1) for the temporal case we would introduce ~ - o with o > 0. Thus all results hold for the temporal case as well. In the spatial case a good approximation is p = 1/2. The analysis can be limited to/3 = + 1. Solutions for other values of/3 can be derived via a simple scaling transformation (see for example, Torner [ 1998b]). Setting U1 = U2 the equations for scalar interaction are recovered. The above equations yield two conservation laws which play a central role in the analysis, the total energy Q and the imbalance C:
Q = f d r (Iu, I~ + IN212+ Iu312),
(4.2)
c =/dr(IU, l~- IU212).
(4.3)
Apart from these two conservation laws there are two other ones, the Hamiltonian and the momentum.
4.1.1. Hamiltonian system Here we demonstrate that eqs. (4.1) can be considered as a Hamiltonian system and that the conservation laws of eqs. (4.2, 4.3) can be derived from translational and a phase invariance. Thus the Hamiltonian itself is also a conserved quantity. The Hamiltonian H[U1, U{, U2, U~, U3, U~] as a functional of the fields is
l O U~--~ 1 2 1 0 U+-~--~ 22 H = / [ dY
2 [3 2 -k-p -ON3 ~ + ~lg3l -
* 1
gl g 2 g ~ - g ~ g ~ u3
(4.4) The evolution equations can then be obtained via functional derivatives of H: 9OU1 1 - -
6H -
OU~ OZ
i--
. OU2 1 - -
-
oz
~ u~ '
6H 6Ul'
oz
i
6H
i OU3
~ u~ '
2 oz
--
6H --
OU~ OZ
6H 6U2'
(4.5)
~u; '
i OU; 2 OZ
6H 6U3'
i.e., Un and U~*, n = 1,2, U3/v/-2 and U~/v/2 are conjugate to each other.
(4.6)
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The Hamiltonian and thus eqs. (4.1) are translationally invariant with respect to Y and Z or replacing the fields by U1 exp(i01), U2 exp(i02), U3 exp[i(q~l + r In the latter case an infinitesimal transformation 6r yields H[U1, UI* , W2, W2*, U3, U3*]
= H[UI(1 + i6q~1), Ul*(1 -i6q~l), U2, U2*, U3(1 + i6r
U3*(1 - i6q~l)]
= H[U1, W~, W2, W;, W3, W3*] f[.6H 6H +
6r
dr
(4.7)
6H .6H ] l~-~ll W1 --i~-~l. Ul* + i~--U-~3u 3 - l ~ 3 . W; ~ 1 .
Using eq. (4.5), eq. (4.6) and the fact that H is invariant results for arbitrary in 0 f dY []U1 12 + ~lU31 1 2] = 0. OZ
(4.8)
Similarly for infinitesimal transformations 6~2 one obtains
o az
f
1 2] =0. dY[IU2i +~lu31
(4.9)
Linear combinations of eq. (4.8) and eq. (4.9) then give the total energy and the imbalance. The translational invariance with respect to Y (infinitesimally U,,(Y + 6Y) = U,,(Y) + (OU,,/OY)6Y) leads in a similar way to the conservation of the momentum (definition below), while the translational invariance with respect to Z means the conservation of H.
4.1.2. Soliton solutions A one-parameter family of resting soliton solutions was identified in Buryak and Kivshar [ 1995a] (scalar case) and the two-parameter family in Buryak, Kivshar and Trillo [ 1996] and U. Peschel, Etrich, Lederer and Malomed [ 1997] (vectorial case). To determine the soliton solutions first the following transformation is performed: UI = Ul eirm
z
,
" , U2 -" u2 elK'2Z
" , U3 = u3 el(tcm+x2)Z
(4.10)
where K'I, K"2 are the parameters of the family of solutions. The conservation laws are not affected by this transformation. The transformation of eq. (4.10) leads to 9 0Ul 1 02Ul 1--ff~ + 2 0 Y 2 K'lUl + UzU3 = O, 9 Ou2 1 02u2 1-ff~ + ~ Oy----T
p GQZu3 .0/,/3 1--0-~- -t 2 0 Y 2
-
K'2u2 + u~u3 = 0,
(fl + K'I + K'2)u3 + 2UlU2 = 0.
(4.11)
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l Uno12 4 s
i
-5
[
0
)
Y
5
Fig. 4.1. Typical soliton solution for tq = 1, tr -- 1.5 and/3 = -1. Solid and dashed lines correspond to the two fundamental (upper solid line first FH) and second harmonics, respectively. The above equations could be rescaled further using the parameters tr = K'2/K"1 and a = (/3 + tr + tCz)/tcl. Apart from reducing the number of parameters this scale has the advantage that there is no discontinuity caused by different signs of/3. For various reasons this scaling will be used for the case with walk-off. In this case using eqs. (4.11) turns out to be more natural. Solitons are now determined as localized stationary solutions u,o(y) of eqs. (4.11), i.e. equating the Z-derivative to zero. We are interested in bright solitary waves on a zero background. The stability of the zero background or trivial homogeneous solution of eqs. (4.11) is determined substituting un = 6u, exp(i~Z)exp(ikY), uT, = 6u, exp(i~Z)exp(ikY) and linearizing with respect to 6u~, 6u, which yields the following solutions of the corresponding characteristic equation: ~,5,6= • ~lpk2 +/3 + K'I + K'2). (4.12) These are also dispersion relations for linear plane waves, i.e. solving eqs. (4.11) without the nonlinear terms. Since the solutions given in eqs. (4.12) are real, the background is always modulationally stable. From eqs. (4.12) the values of the family parameters for which solitons may exist can be determined. In order to allow asymptotically for evanescent solutions there should be a gap between ~1,2 , A3,4 and ~s,6. This yields tr tr > 0 and for negative phase mismatch 1,2 =
-~"(1 k 2 -k-K"1 ) ,
~3,4 = -k-(1 k 2 + K'2),
K'I +K" 2 > _ / 3 .
Up to above phase transformation the soliton solutions are real-valued (see fig. 4.1 for an example). Examples of branches of soliton solutions in terms of the energy Q are displayed in fig. 4.2 where the dynamical variable uo = f d Y (IU10] 2 -k-IU20[ 2 - I U 3 o ] Z ) / Q is used, which is the relative energy
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OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
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C=O
/ i
C = 0.02
/ i
C=0.1
s 0
10
20
O
Fig. 4.2. Relative energy difference u0 between the two fundamentals and the second harmonic versus the total energy Q of exact numerical soliton solutions for different imbalances C for/3 = 1 (upper branches) and/3 = -1 (lower branches). Solid lines refer to stability and dashed lines to instability.
difference between the two FHs and the SH. As stated above, it is sufficient to restrict t o / 3 = 4-1. F o r / 3 = 0 there would be just a straight line in fig. 4.2 which is approached by the branches for/3 - + 1 for large Q. For vanishing imbalance (C = 0) the scalar case is recovered. If/3 < 0 there is a limit point, i.e., a point connecting two branches o f solutions which are due to the coexistence o f two different solutions with the same energy (fig. 4.2). The lower branch approaches u0 = - 1 for Q ~ c~, i.e., all the energy is in the
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0.10
0.05
0.00
I
0
4
'
Q
I
8
Fig. 4.3. Loci of limit points in the (Q, C)-plane referring to soliton solutions from the variational approach (solid line) and the exact numerical analysis (dashed line) for/3 = - 1 .
SH with the width diverging. This is not possible for finite imbalance. If the imbalance C is small, there is a second limit point. The branch emanating from this limit point is extremely small and seems to terminate. In this case there is no energy in one of the FHs, in which one depends on whether C > 0 or C < 0, and most of the energy in the SH. For a finite imbalance it is not possible that all the energy is in the SH since it is a conserved quantity. For larger [C I the limit points vanish (cf. fig. 4.2 and fig. 4.3). Again, at the termination point of the remaining branch there is no energy in one of the FHs. If/3 > 0 the energy can approach zero. In this case all the energy is in the FHs (u0 = 1). For p = 1/2 eqs. (4.1) are invariant under a Galilei-like transformation (compare below when walk-off is included). In this case we have a three-parameter family of solutions (tel, tr and a velocity). It should be mentioned that for tq = tc2 = to0 = / 3 / ( p - 2) one member of the two-parameter family of soliton solutions is available analytically (Karamzin and Sukhorukov [ 1974]):
Ul0 = u20 -
3to0 f p 1 2--V2cosh2(~x/~0/2x),
u30 -
3to0 1 2 coshZ(v/~x)
"
(4.13) For other analytical types of solitary wave solutions see for example, Karpierz and Sypek [1994].
4.1.3. Schr6dinger limit In this limit it is assumed that the phase mismatch is large, i.e., that propagation and diffraction (dispersion) effects in the third of eqs. (4.1) can be neglected
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OPTICAL SOLITONSIN MEDIA WITH A QUADRATICNONLINEARITY
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(cascading or local approximation). This leads to cubic nonlinearities in other two of eqs. (4.1). Some fundamental experiments were carried out in Schrrdinger limit where only a small fraction of the total energy is in the (Schiek, Back and Stegeman [1996]). Thus substituting U3 = 2U1Uz/fi into first two of eqs. (4.1) gives 9O U 1
1 02 UI
'--~--+2 or 2 .0U2
1 02U2
2
[2
2
12
+fi]U2 U,=O,
~-b-z ~ 2 or2 ~ l g ~
the the SH the
(4.14)
g2=o.
From the above equations we have that in the vectorial case the Schrrdinger limit leads to cross-phase modulation only. The scalar case gives the nonlinear Schrrdinger equation. Assuming U1 = U2 = U we have
OU 1 02U 2 i-b--Z+ ~-0-~ + ~ fuI2 u = 0,
(4.15)
which has the well-known stationary one-parameter family (to) of single soliton solutions U =
X/~ eiXZ, cosh(x/~Y)
(4.16)
with a small SH part U3 = 2U2/fl. Thus to have bright soliton solutions/3 must be positive, which means the nonlinearity is effectively focusing. To mimic a cubic nonlinearity via quadratic ones is the very idea of the so-called cascading. It is obvious that the method outlined above fails if the SH plays a role or if the phase mismatch is negative. Also if the solitons are interacting, the above description fails since the nonlinear Schr6dinger equation is integrable whereas eqs. (4.1) are not. In numerical collision experiments Schrrdinger solitons pass through each other. Soliton solutions of eqs. (4.1) merge and create a new state if they approach each other with sufficiently small velocities (for p - 1/2, cf. Werner and Drummond [ 1993] and Etrich, Peschel, Lederer and Malomed [1995]). Note that performing the limit using eqs. (4.11)/3 is replaced by 2tr +/3.
4.1.4. Stability of solitons As mentioned above, the system of eqs. (4.1) is not integrable. Thus the stability of the soliton or actually solitary wave solutions is an important issue. As
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517
displayed in fig. 4.2 for negative phase mismatch, branches of solitons have at least one limit point where solutions usually change their stability behavior. In what follows we are going to determine the boundary in parameter space which separates stable and unstable domains. We follow the analysis in Etrich, Peschel, Lederer and Malomed [1997]. For the vectorial case the stability behavior was discussed in Buryak, Kivshar and Trillo [1996] and U. Peschel, Etrich, Lederer and Malomed [1997]. To determine the stability eqs. (4.11) are linearized around a soliton solution UnO. Substituting the ansatz u, = Uno q- 6Une i~'Z, u n = UnO -t- -~UneiZZ into eqs. (4.11), linearizing with respect to 6Un, 6Un and introducing the variables X+ = (6//1 + 6Ul, (~U2+ 6//2, 61/3 + (~U3)T , X_ = (6Ul -- 6Ul, 6 t / 2
--
6u2, 6u3
-
6u3)
(4.17)
T ,
we arrive at the following eigenvalue problem for Z: L+x+ = Zx_,
L_x_ = Zx+,
(4.18)
with 1 02 -~ Oy--~ - 1r
+u30
u20
1 02
L+ =
+u30
20Y 2
2u2o
2ulo
tr
ulo
9
(4.19)
p 02 ~ Oy--5 -([3 + tq + tr
The eigenvalue problem can be decoupled to give L_L+x+
= Z2x+,
L+L_x_
=
Z2X_,
(4.20)
The linear problem has three lOcalized or bound states with corresponding eigenvalue & = 0: Oxo X+t --
OY
'
x-t = 0,
X+p = 0 ,
X_p = (ulO, 0 , / / 3 0 ) T ,
X+q = 0 ,
X_q = (0,/,/20, u30) T ,
(4.21)
which correspond to translational and a phase invariance of eqs. (4.11). Here the three-component vector x0 = (Ul0, u20, u30)T was defined and the superscript T denotes the transposed. Apart from the above three bound states, we identified
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OPTICAL SOLITONS IN MEDIA WITH A QUADRATICNONLINEARITY
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numerically a fourth nontrivial one X+b , X-b with real squared eigenvalue ~2. At a critical tc2 = tCZc for the other soliton parameter tq fixed (or vice versa) the squared eigenvalue changes sign and the solitons become unstable. To derive the manifold in parameter space that separates stable and unstable domains we assume that at the critical point the bound state is a linear combination of zero eigenmodes, i.e., X+b = fX+p + g X + q = 0 , X_b ---fX_p + gX_q. Here we made use of the fact that L• are even operators and the nontrivial bound state is even. We assume deviations tc2 = tC2c + ~2~c22, tr = -+-1 and introduce the following expansion: =Zle+~2 s , _(0) _(1) _(2)..2 X+b =X+b + X + b ~ + X + b t +...
(4.22)
,
X_ b __ X(__~)+ X{__lb)~_+_X(?)~2 + . . . ,
with X+b-(~= 0 and X(~) =fX_p + gX_q (at K2c). Above expansion is substituted into the linear problem of eqs. (4.18) to get, up to order c 2" O(1)-
L(~
o(~)-
(1) = L(+O)X+b (2) = L(+0) X+b
o(e2) 9
(~ = 0 ,
L(~ (~) = 0 ' - "~-b
~lX(___~) '
~2x(O) _ + ~1 X_ o ,
(4.23)
L(0) (2)
0X~--~} = A1 _(1) - X-b -- K22L(-0) 0K2 X+b '
where L~ ) are the operators L+ taken at tC2c and _(0) X+b = 0 was used. The first of eqs. (4.23) is trivially fulfilled since xt~ is a linear combination of the zero modes with respect to L_. In the fourth of eqs. (4.23) the relation ~o (L(O)x(~) _ _ = 0 was used. The inhomogeneous equations at order e are solved by X(1) = ~ 1 / j 0 x ~
+u
0x~
+g0-k-;
'
XOb) = 0.
(4.24)
Now the fourth of eqs. (4.23) leads to a solvability condition, which is due to a nontrivial null space of the adjoint operator L* of L_" L*_x0 = 0, L~(x_p- X_q) - 0. Calculating the scalar products between x0, X_p - X_q and the fourth of eqs. (4.23) yields (x0,_(1) X+b ) -- 0 ,
(X_p -- X_q, X+b _(1)\] = 0 ,
(4.25)
where the scalar product is defined as (Xl,X2) = f_~dyx~T 9 x2 with the dot denoting matrix multiplication. Equations (4.25) establish a system of linear
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1.0 e
O.5
",,,,~ble'~
no soliton solution
0.0
'
0.0
I
'
N
0.5
1.0 K"1
Fig. 4.4. Domains of stability and instability in the (tq, tc2)-plane for/3 =-1.
equations for the constants f and g. The requirement of a nontrivial solution yields
OQ oc
OQ oc
0tel 0tr
0tr 0tel
- 0,
(4.26)
where Q and C have to be calculated with the soliton solutions u,0. Note that the same procedure applied to the second and third of eqs. (4.23) leads to a trivial result. The curve separating stable and unstable domains in parameter space as calculated from eq. (4.26) is displayed in fig. 4.4. Note that the unstable domain is extremely narrow. The coefficients Ai arise from solvability conditions for higher orders. Thus the soliton solutions become unstable if the vector function (Q(K'I, K'2), C(K'I, K'2)) is not locally invertible. The above analysis relies on the fact that there is not another bound state which has for instance always ~2 < 0. In this case the soliton would be always unstable regardless of the transition described by eq. (4.26). For single hump solutions the above approach can be applied since there is only one nontrivial bound state. But in the case of double hump solutions the situation is different (see below). In the scalar case the condition of eq. (4.26) corresponds to OQ/Otr = O, i.e., Q as a function of tel has a local extremum (Vakhitov and Kolokolov [ 1973]). The instability occurs for negative phase mismatch only. In fig. 4.2 the limit points correspond to eq. (4.26). The soliton solutions destabilize and stabilize at these points. The curve separating stable and unstable domains in parameter space translates in the (Q, C)-plane to fig. 4.3. The imbalance between the two FH waves has a stabilizing effect (cf. fig. 4.3). Finally the stability of the soliton solutions can be confirmed by means of numerical experiments (see for instance U. Peschel, Etrich, Lederer and Malomed [1997]).
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OPTICALSOLITONSIN MEDIAWITHA QUADRATICNONLINEARITY
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4.1.5. Variationalapproach In this subsection we employ the variational or Lagrangian approach to approximate the soliton solutions and to describe their stability behavior. The approximation of the soliton solutions is by suitable trial functions with z-dependent parameters. The variational approach reduces the infinite number of degrees of freedom of the system described by eqs. (4.1) to a finite number leaving a system of ordinary differential equations for the z-dependent parameters (cf. also {}8). The Lagrangian from which eqs. (4.1) can be derived is given by
=
If
[i( 2 0 0 l
d r" d Z
U r -b-Z
OO~)
- u1--ff2-
-
21 2OU1 _~
.
i( ON2 OS~ 1 OU212 + -~ U;-~- - U2 0Z / -- -2 --~ i(
OU3
OU;)
+ -~ V;-~- - V3--~ fllU312 +
2
(4.27)
p OU3 2
- -4 - ~
U~ U~ U3 + UIU2U; ]
It is convenient to use Gaussians as trial functions:
U1 --
02 --
1
1
(2,7"g')1/4
1
1 (2~) 1/4 1
v/Q(1 + u) + 2 c e ivy'v/-~ e-rt2V2eiaV2, 9
CQ(1 + u) - 2C e 1q92~
03 -- (2~)1/4 v/Q(1 - u) e i~c3~
y2 iay2
e -r/2 e
,
(4.28)
e -q2 Y2e2iay2,
where the variable u = f d Y (I Vl ]2 + [g2 ]2 _ ]U3 [2)/Q, t he phases q0,, n = 1,2, 3, the inverse width r/and the wave from curvature a are functions of Z. Q and C are the conserved quantities energy and imbalance. For simplicity we introduced a common width for the three fields. Note that u is defined in the interval [-1 + 2]C]/O, 1] and that [C] < O. For u = 1 the SH vanishes while for u = -1 + 2]C]/Q one of the FHs vanishes depending on whether C is positive or negative. Substituting the ansatz of eqs. (4.28) into the Lagrangian, integrating over Y and varying the resulting effective Lagrangian with respect to the functions u(Z),
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Cgn(Z), r/(Z) and a(Z) yields the following set of evolution equations for these quantities: du dZ - 2Q'v/-~v/(1 - u)[(1 + u) 2 - C '2] sin(cg), d99 _ f i - ~ 3 r/2 + Q ' x / ~ (1 + u)(1 - 3 u ) + C '2 cos(C), dZ v/(1 - u)[(1 + u) 2 - C '2 ] dr/ dZ da dZ
_
2at/, a1 ( 5
nt- 3 U ) r / 4
- 2a2 - g1 Q, r/2 x/~v/(1 - u)[(1 + u) 2 - C '2] cos(qg),
(4.29) where q9 = qgl + cg2 - q93, Q' = (2~) -1/4 x/2Q/3 and C' = 2C/Q. Here we restricted to p = 1/2. Eqs. (4.29) can be written in terms of a Hamiltonian system with Hamiltonian a2
H = -flu+ 2~-~ + J(5 + 3u)t/2 - 2 p ' x / ~ v / ( 1 - u)[(1 + u) 2 - C'21 cos(q)) (4.30) and conjugated variables u, q9 and a, 1/02. Thus H from eq. (4.30) is a conserved quantity. Since we are interested in stationary solitary wave solutions approximated by eqs. (4.28) we seek steady state solutions of the system of ordinary differential equations. Equating the z-derivatives in eqs. (4.29) to zero gives immediately 990 = 0 for the phase difference and a0 = 0 for the wave front curvature. The second and fourth of eqs. (4.29) can then be solved numerically for u0 and r/0. It turns out that there is no solution with ~0 = Jr which also solves the first of eqs. (4.29). Figure 4.5 displays some examples for various imbalances C and fi = + 1. Comparing this with fig. 4.2 for the exact numerical soliton solutions there is very good agreement. Thus the variational approach gives a good approximation of the solutions under consideration. Also the stability behavior is recovered. A linear stability analysis of the steady-state solutions of eqs. (4.29) shows that they destabilize and stabilize at the limit points. Again the stable branch emanating from the second limit point for C ~ 0 is very small. It terminates where the square root in eqs. (4.28) becomes zero, i.e., there is no energy in one of the FHs and most of the energy is in the SH. A simpler variational approach than the one presented above yields analytical expressions for the different quantities, but not the stability behavior. In particular, it is easy to allow for individual widths of the two FHs and the SH (U. Peschel, Etrich, Lederer and Malomed [1997]). For ways to take walk-off into account see Agranovich, Darmanyan, Kamchamov, Leskova and Boardman [1997].
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OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
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C=O
C = 0.02
-
0
~
C =0.1
10
20
Fig. 4.5. Relative energy difference u0 between the two fundamentals and the second harmonics versus the total energy Q of soliton solutions from the variational approach for different imbalances C for fl = 1 (upper branches) and/3 = -1 (lower branches). Solid lines refer to stability and dashed lines to instability. 4.2. SCALAR INTERACTION WITH WALK-OFF The soliton solutions dealt with in the previous subsections can be considered as real-valued. They are basically described in terms o f their family parameters. Introducing walk-off (spatial or temporal) in general yields an additional family parameter and the soliton solutions b e c o m e complex-valued functions (Torner, Mazilu and Mihalache [1996], Etrich, Peschel, Lederer and M a l o m e d [1997]).
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Thus for simplicity we consider only scalar interaction with walk-off (for vectorial interaction with walk-off see Mihalache, Mazilu, Crasovan and Torner [1997]). To avoid dealing with too many parameters a different scaling is used here to point out the number of family parameters. For scalar interaction including walk-off the version of the evolution equations corresponding to eqs. (4.1) is
.OU1 1--~
OU2"~
002
i
~-+ar--O-~j
1 0201 q- -~ O T-----~ -t- U ~ U2 = O, (4.31)
O 02 U2
~ 20T
2 fiU2+U2=O"
where now U1 denotes the fundamental and U2 the second harmonic. Note that the factor in front of the nonlinearity in the second of the above equations is scaled out. Eqs. (4.31) are also written for the more appropriate temporal case, since p = 1/2 in eqs. (4.1) (or a -- 1/2 here) is a very special case which becomes clear below. The temporal walk-off or group velocity mismatch is denoted by at. Apart from a Hamiltonian eqs. (4.31) have two other conserved quantities, the energy Q and the momentum P:
o = fdr
(lull 2 + Iu212),
1 / d T [U I ~T-Og~ OWl 1 ( og~ og2"~] - U ; - ~ + -~ U: - S T - - U ; --fff - I .
(4.32)
P = ~
The solutions of eqs. (4.31) that we are interested in are solitons with bright shapes. Because of the group velocity difference FH and SH components of a solution tend to separate from each other. On the other hand, for a solitary wave to exist both constituents must move with a common velocity. The question is whether the nonlinear interaction is able to prevent this separation. It is expected that the complex term introduced by the walk-off in general leads to a chirped solution. Thus another parameter of the solutions is the average frequency (or momentum). We are now looking for a two-parameter family of solutions with parameters x" (wavenumber) and v (velocity). To this end we introduce the following transformation of eqs. (4.31):
t= V / r-~-o~ (r - vz) , Ul =
~
(~ __. (2o-1)o+av V/K- v~2 ,
~ = (K-- ~) z U1,
u2 = -7--j-v2 ~ 1 ~-2i(tc-v2)Z,.,-2ivT~ I(- T a = fi+21r176 o-~ 2 , g:
U2 ,
(4.33)
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OPTICALSOLITONSIN MEDIAWITHA QUADRATICNONLINEARITY
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giving . Oul
1
02/,/1
1---&- -~ 2 0 l
2
U l + U ~ U 2 -- O ,
(4.34) i
( O/d2 ~+6
_~)
O" 02u2 + 20l 2
au2+u2=O"
Thus the solutions are described by the parameters 6 and a. As above bright solitons u,0 are now calculated as stationary solutions of eqs. (4.34). The solitons calculated in this way can be considered as moving ones of eqs. (4.31) with propagation constant K" and velocity v, thus establishing a two-parameter family of solutions. Note from eqs. (4.33) the interplay among the velocity v, the walkoff ar and the dispersion coefficient a in the expression for the rescaled walkoff 6. This indicates that the group velocity mismatch and the velocity of a soliton solution act similarly. This fact can be conveniently exploited, since a shift of the velocity v -~ v + v0 (v0 = - a r / ( 2 a - 1)) and of the propagation constant tc --~ tc + (v + v0/2)v0 allows the restriction to aT = 0. Provided that a r 1/2 and the phase mismatch is renormalized as fl ~ fl + a 2 / ( 2 a - 1). For 6 = 0 the real-valued families of soliton solutions are recovered. The oneparameter family (parameter Ic) corresponds to a r 1/2, i.e., v = 0 for aT = 0 or v - - a T / ( 2 a - 1) otherwise. It is just a limiting case within a broader class of solutions. The two-parameter family (parameters Ic and v) is obtained in the case a = 1/2, i.e., a r = 0. As can be seen from the transformation of eqs. (4.33), the moving solutions of this family can be generated directly from the resting ones. The soliton solutions obtained for nonvanishing 6 are complex-valued with a nontrivial phase, i.e., they have a chirp (see fig. 4.6). There is no limitation with respect to the rescaled walk-off 6 for solitons to exist. The lower boundary of the rescaled phase mismatch a depends on b and can be obtained from the linear dispersion relations as above. Introducing the frequency s instead of the wavenumber k they are ~,1,2-4-i
)
--~-+ 1
, (4.35)
'~3,4 -- + i
I2/ ) b Q -4- ~
2 + a-
b2 ~-~
]
from which the limit of the continuous spectrum is min{1, a - 62/(2a)}. For a = 62/(2a) the gap of the continuous spectrum vanishes. This also marks the
7, w 4]
SOLITONS IN PLANAR WAVEGUIDES
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3 2 =
I . - .t/~
0 0.0 %
~
~,- ~ . .
-0.5 .
-1.0
-
'
-8
~
.
I
'
0
8
t
Fig. 4.6. Intensities and time-derivatives of the phases of a chirped soliton solution for 6 = 0.6, a = 0.5 and o" = 0.8. Solid and dashed lines correspond to the fundamental and second harmonics, respectively.
t
stable
u
__./J.""
"nsta~' no soliton solution
__
o
I
1
'
~
2
Fig. 4.7. Domains of stability and instability of soliton solutions in the (6, a)-plane for o = 0.8.
limit of existence for bright solitons (see dashed line in fig. 4.7), which is due to the requirement that these solutions have evanescent tails. The boundary in parameter space separating stable and unstable domains is obtained from an expression similar to eq. (4.26). In terms of the family parameters tr and o it is
OQ OP OQ OP 0to 0o
0o 0to
- 0,
(4.36)
where Q and P have to be calculated with the soliton solutions U10, U20 (Etrich, Peschel, Lederer and Malomed [1997]). Equation (4.36) would be much more complicated in terms of 6 and a, whereas the manifold separating stable and unstable domains is much simpler using these parameters (fig. 4.7).
526
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 4
4.3. EFFECTIVE POTENTIAL AND MULTIPLE HUMP SOLUTIONS
As mentioned above, only a few analytical soliton solutions for particular values of the family parameters are available. An alternative way of examining the different types of stationary localized solutions is to consider them as homoclinic trajectories of classical motion in a potential (He, Werner and Drummond [ 1996]). To demonstrate this method we focus here on the simplest case (scalar, no walk-off). It is most convenient to start with the stationary version of eq. (4.34): 1 d2ul - - ~ --//1 + U l U 2 -- 0 , 2 dt 2 O d2u2 2 dt 2 otu2 + u21 = O,
(4.37)
where real-valued solutions are assumed. In fact this assumption is confirmed by all numerical investigations. If real-valued solutions exist no solitons with a nontrivial phase dynamic are found. Substituting Ul = x/-O--~a, u2 - b into eq. (4.37) gives d2a
- 2 a + 2 a b = O,
dt2 d2b
2a
(4.38)
dt 2
o
b+a 2
=0,
which describes the motion of a particle in the two-dimensional potential V = a 2b - a 2 - a b2 ' tY
(4.39)
where the spatial coordinates correspond to the field amplitudes a and b. Since we are looking for localized solutions the particle should start at the origin (a = b = 0 for t = - o c ) and finally return to it (t = +e~). Due to energy conservation the particle is not able to cross the line where the potential has the same value as at the origin, i.e., V - 0. Approaching this line the particle is reflected and moves back. No such line exists for a negative SH (b < 0) and the particle does not return to the origin. Thus solitons always have a positive SH. In this case two lines V -- 0 exist with a < 0 and a > 0, respectively (cf. fig. 4.8a). In the case of a single hump solution the particle is reflected once at one of these lines and returns on the same trajectory (see line in fig. 4.8a), as
7, w 4]
SOUTONSIN PLANARWAVEGUIDES
527
Fig. 4.8. Contour plots of the effective potential (eq. 4.39) together with soliton solutions (represented by bold lines in the corresponding potential) for (a) a = 0.5, (b) a -- 0.1 and (c) a = 0.3 (~r = 0.5). The line V = 0 is marked correspondingly. its incidence is along the normal to the line. Thus in this case the soliton is symmetric and the turning point corresponds to the e x t r e m u m o f the single h u m p solution (see fig. 4.8b). If the initial trajectory deviates slightly from the previous one, the particle approaches the line V = 0 with an oblique angle. N o w it cannot return on the same trajectory. In most cases it will be bounced back and forth between the two lines V = 0 forever without returning to the origin. But there are exceptional cases where the particle returns to the origin after some reflections. Every reflection corresponds to an additional hump o f the soliton solution. In the case o f a double-hump solution the particle starts at the origin, approaches the first line V = 0, turns back and crosses the axis u - 0 with an angle o f :r/2.
528
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
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For a < 0 it performs the mirror-imaged trajectory and returns to the origin (see line in fig. 4.8b,c). Thus the two humps of the FH of the resulting solution have opposite signs (see fig. 4.8b,c). In general for a given potential of eq. (4.39) an arbitrary number of reflections may occur. The intensity of the resulting multiple-hump solution is always symmetric but the sign of the humps of the FH alternates (He, Werner and Drummond [ 1996], Mihalache, Lederer, Mazilu and Crasovan [ 1996]). Therefore we may conclude that in contrast to integrable systems even for a fixed propagation constant, many stationary localized solutions exist. The physical interpretation of this phenomenon is that the multiple hump solutions are basically bound states of the lowest order (single hump) solitons where the FHs are out of phase (phase difference Jr). They attract each other through the inphase SHs. In contrast the alternating sign of the FH components results in a repulsion and prevents a fusion. An essential prerequisite for the existence of these bound states seems to be the dominant role of the SH providing the binding force. Double-hump solutions exist for negative phase mismatch only. In this case the SH dominates. Near the lower limit of existence of localized solutions the maxima of a double-hump solution are very close to each other (see fig. 4.8c). Increasing the propagation constant results in a separation of the humps. Now the binding mechanism is restricted to the interaction of the soliton tails. Consequently no bound solutions exist as soon as the tail of the SH is more evanescent than that of the FHs. This yields an upper limit of existence (Etrich, Peschel, Lederer, Mihalache and Mazilu [1998]). The method presented above works for more complicated systems as well. Although the dimension of the effective potential is higher in the case of the vectorial interaction (three interacting fields) or for nonvanishing walk-off (four real-valued components) we still find multiple-hump solutions (for details see Etrich, Peschel, Lederer, Mihalache and Mazilu [1998]). Multiple-hump solutions seem to be always unstable. For double-hump solutions the linear stability analysis yields more than one nontrivial bound state of the corresponding eigenvalue problem (fig. 4.9, Etrich, Peschel, Lederer, Mihalache and Mazilu [1998]). One of them always has a negative 72 and the solution is unstable. Propagating such a solution, the syrmnetry is spontaneously broken, i.e., the solution decays asymmetrically, no matter whether walkoff is present or not (Haelterman, Trillo and Ferro [1997], Etrich, Peschel, Lederer, Mihalache and Mazilu [1998]). The decay is either into an oscillating single-hump solution or into two single-hump solutions moving away from each
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SOLITONSIN PLANARWAVEGUIDES
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0.1
0.0 2
-0.1
-0.2
-0.3
i
0.0
'
12
0.
0.4
Fig. 4.9. Squared eigenvalues ~2 of the nontrivial discrete states of the linear problem versus a. The bold solid line marks the limit to the continuum.
Fig. 4.10. Evolution of the amplitude of the fundamental (left) and second harmonics (fight) displaying spontaneous symmetry breaking of chirped double hump solitary waves. (a) annihilation of one hump for a = 0.4 and (b) break-up of the humps for a = 0.7 (6 = 0.5).
o t h e r (fig. 4.10). This c a n be e x p l a i n e d in t e r m s o f the b o u n d states o f the linearized problem. 4.4. PERSISTENT OSCILLATIONS The excitation o f a soliton (Artigas,
Torner and Akhmediev
[1999])
in a
530
OPTICAL SOLITONSIN MEDIAWITHA QUADRATICNONLINEARITY
[7, w4
quadratically nonlinear medium generally leads to a perturbed state which displays persistent periodic oscillations. Experimentally such a state is due to the fact that usually a soliton is generated from the FH. Oscillating perturbed states are also obtained (numerically) if two moving solitons collide and merge (Etrich, Peschel, Lederer and Malomed [ 1995]) or if an unstable soliton decays to a stable one (Pelinovsky, Buryak and Kivshar [1995]). The aim of this subsection is to understand the nature of these oscillations. Their amplitude is not a new soliton parameter. But in numerical simulations these oscillations appear to be very persistent (Etrich, Peschel, Lederer, Malomed and Kivshar [ 1996]). This is somehow in contrast to integrable systems where perturbations decay very fast. Here we use eqs. (4.34) without walk-off, i.e., 6 = 0:
9Oul 1 02U2 1---~- + 2 0 y 2
Ul + ul u2 = O,
(4.40) On 2
O 02U2
i--~- + 2 0 y 2
au2 + u~ = 0,
with o = 1/2 which is the appropriate choice for the spatial case. To examine the oscillations a soliton is perturbed and propagated over a certain distance. To this end a particular shape of the initial perturbation is chosen, such that the energy of the initial wave is not changed and the radiation minimized:
2 02 ] 89 Uno(O) U2o(y) u.(z= o)= IU2.o(y)+ O~U2o~)l. Iy-- 0 ~2 ]
(4.41)
where un0, n = 1,2, denotes a stationary soliton solution of eqs. (4.40) and is the amplitude of the perturbations. A typical example of the persistent oscillations of a perturbed solitary wave is displayed in fig. 4.11. The oscillations are quite regular. It is essentially the widths of the solitons that oscillate. There is hardly any energy exchange between the FH and SH and the peak amplitudes are oscillating in phase. The idea is now that, if the amplitudes of the oscillations are not too large, they may be compared with the solutions of the linearized eqs. (4.40). Linearizing eqs. (4.40) around a stationary solution as in w4.1.4, i.e., un = u~0 + 6u~ exp(blz), uT, - u,0 + 6u, exp(iJlz), we arrive at the following eigenvalue problem for the propagation constant of the perturbation 2.: L+x+ = ~x_,
L_x_ = ~x+,
(4.42)
7, w4]
SOLITONSIN PLANARWAVEGUIDES
531
Fig. 4.11. Typicalexample of persistent internal oscillationsof a soliton solutionexcited with ~ = 0.4 for a = 0.5. Displayed are the intensities of the (a) fundamental and (b) second harmonics. with L+ =
1 02 2 0y 2
1 + u20 2u10
Ul0 02 ~ Oy-5 - ot
,
(4.43)
and the definition of x+ from eq. (4.17). Similar to w4.1.4 the linear problem has two trivial localized states with zero eigenvalues, corresponding to translational invariance and the invariance due to an arbitrary phase of the two fields, and a nontrivial one with real A2 ~ 0. This discrete eigenvalue must obey the inequality A2 < min{ 1, a2}, which marks the boundary of the continuous spectrum. The eigenvalue Ab as a function of a is displayed in fig. 4.12a. It should be noted that the eigenvalue almost touches the continuum (corresponding to the SH) at a _ 0.4. For larger a the distance to the continuum increases again until it approaches the continuum (corresponding to the FH). Recently it was suggested by Pelinovsky, Sipe and Yang [1999] that this eigenvalue disappears close to the Schr6dinger limit. At a _~ 0.106 the solitons destabilize and A2 changes its sign (fig. 4.12b). Comparing the numerically evaluated frequencies of the internal oscillations of the perturbed solitary wave with Ab, there is good agreement even for stronger perturbations [~ ~_ 0.15 in eq. (4.41)]. Thus the assumption that the oscillations are adequately described by the linearized system seems to be correct. The reason for the persistence of the oscillations is that a bound or localized state is discrete, i.e., it is in the gap of the spectrum of the linear waves (linear dispersion relation). Only higher harmonics generated by the nonlinearity radiate. The damping is extremely nonlinear. Weak oscillations are practically undamped
532
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY 1.5
(a)
[7, w 4
/ / /
1.0
'~b 0.5
0.0
'
I
'
1
I
'
2
3
0.8 (b)
0.4 2
0.0
0.0
0.5
O~
1.0
Fig. 4.12. (a) Eigenvalues '~b and (b) squared eigenvalues A2 of the nontrivial discrete eigenstate versus a. The straight dashed lines in (a) mark the limit of the continuous spectrum.
whereas strong oscillations decay quickly until they have reached a moderate amplitude. There is a maximum amplitude which hardly can be exceeded. The amplitude is strongly dependent on the separation of the discrete state from the continuum (cf. fig. 4.13). Thus for larger a the oscillations disappear, no matter whether the discrete mode still exists or not. Near the critical point (a < 0.2) a beating is observed. There is an additional fast oscillation which can be shown to be damped (Etrich, Peschel, Lederer, Malomed and Kivshar [1996]). The reason is a mode that is bound with respect to the FH and not the SH. The dominating SH in the equation for the FH can be considered as an effective potential. The quasi-bound mode is in the gap of the spectrum of linear waves of the FH but not the SH. Such resonances (quasibound modes) in the continuum are well known in quantum mechanics as socalled Fano resonances (Fano [ 1961 ]).
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SOLITONS IN PERIODIC WAVEGUIDESTRUCTURES- BRAGG SOLITONS
533
0.2 or
2~,b (or
1), 1 -Xt, (a,> 1)
oscillation am plitude
0.1
0.0
'
I
1
9
i
o~
I
2
'
3
Fig. 4.13. Gap between the continuous spectrum and the discrete eigenvalue ~.b (solid) and final oscillation amplitude of the intensity (fundamental, dashed) of the internal oscillations versus a for ~ = 0.15.
w 5. Solitons in Periodic Waveguide Structures - Bragg Solitons
Soliton formation in a quadratically nonlinear waveguide with a periodic modulation of the cross section or the dielectric properties (Bragg grating) is based on the interaction of FH and SH forward and backward propagating waves. The Bragg grating provides for both the coupling of these waves and the large dispersion required for soliton formation. Thus, it is reasonable to term these solitons Bragg grating or Bragg solitons. Since bright Bragg solitons emerge inside the gap of the linear transmission spectrum or the so-called stop band of the grating, they are also frequently referred to as gap-solitons (see, e.g., Christodoulides and Joseph [1989] and De Sterke and Sipe [1994] for the case of a cubic nonlinearity). Bragg solitons in quadratic media differ from their cubic counterparts considerably. Novel effects include forbidden domains inside the stop band, coexistence of in-phase and anti-phase solitons and stable multi-hump solutions (T. Peschel, Peschel, Lederer and Malomed [1997]). Numerical investigations predict interesting phenomena like soliton trapping (Conti, Trillo and Assanto [ 1998b,c]) or all-optical switching and memory effects (Conti, Trillo and Assanto [ 1998b] and Conti, Assanto and Trillo [ 1998]). The solitons under consideration here involve colinear FH and SH fields. A different situation, where the FH waves propagate perpendicular to the grating pitch whereas the SH field evolves along the pitch, was dealt with by Mak, Malomed and Chu [1998c].
534
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 5
5.1. BASIC CONCEPTS We start from the basic eqs. (2.25) derived in w2,
(o< o<)
i \ OZ + OT J + ff2U1w U~ V1w U2 -O' OV1 OV1 "~ -i --~+c--~j+K2VI+U2+IcV2=O, (5.1)
(or2 ov2'
i \--O--z + OV J +QU2+U:~V2+U1 = 0 , or2
ova)
i --~-~+c--ff~-j
-
+(2V2+UzZ+
,~.
V, = 0 .
The parameters tc and c denote the ratios of the coupling constants and the group velocities of the FH and the SH, respectively, g2 is the detuning of the FH from the Bragg resonance and s is related to it by g2 = 2 s - 2 c ~ + q with fi and q being the phase mismatch and the detuning of the SH wave [for details see eq. (2.25)]. We note again that tr is complex-valued. Its argument cannot be removed by a phase transformation of the field amplitudes (He and Drummond [1998]). In contrast to the cubic scenario, where analytical solitary wave solutions exist (Aceves and Wabnitz [1989], Feng and Kneubiihl [1993]), exact solutions of eqs. (5.1) are only known for isolated points in parameter space (see Conti, Trillo and Assanto [1998a]). These exact solutions correspond to particular cases of resting solitons with a real-valued coupling constant for the SH. They can be derived if the condition 2g2 + g2 + x" = 2(1 +c)s + q + tr = 0
(5.2)
holds and they have a real-valued SH. The above condition implies that in a particular geometry there is only a single solution with a fixed frequency defined by c, q and to. As a consequence of a real-valued SH only the sum of detuning (q) and coupling constant (to) enters the condition (5.2), i.e., the effects of SH coupling and detuning are identical for this particular type of solitons. The shapes of some of these exact solutions are depicted in fig. 5.1. If the frequency is close to the gap edge, the shape resembles that of Bragg solitons in cubic media. Closer to the gap center double-humb structures evolve. At negative frequencies both FH amplitudes are in-phase while they are anti-phase
7, w 5]
SOLITONS IN PERIODIC WAVEGUIDE STRUCTURES - BRAGG SOLITONS Frequency: -0.9
0.6
Frequency: -0.5
i
/\
0.5 0.4
0.4 . . . . . . . . . . . . . 02 "0
0.3 0.2
..........
--,
"", ", i\. . . .
0.1 0
-0.4 -0.6
-0.2 -0.3
~. . . . . . . -5
1
0 5 Z Frequency: -0.2
0.5
-.
0 -0.5
-1.5
~.... !
-
-2 -10
-5
0 Z
-10
t
-
FH Phase
.....
SH Amplitude
......
i
,': ,.i,"
i
-5
0 5 Z Frequency: -0.0001
10
0
,.,. . . . . . . . . . . . . . .
i,,,,i/" _
-
10
FH Amplitude
": 'i' ", / . . . . . . . . . . . . . . .
-0.8 -1 /
i -10
535
-0.5
I
5
-1
-1.5
10
,.
k
-10
~,', ,
, '
ilJ ,
.
-5
0 Z
5
.
.
.
10
Fig. 5.1. E x a c t s o l i t o n s o l u t i o n s for d i f f e r e n t v a l u e s o f t h e F H f r e q u e n c y d e t u n i n g E2.
at positive frequencies. For the rather involved analytical expressions describing these solutions we refer to Conti, Trillo and Assanto [1998a]. If eq. (5.2) does not hold, eq. (5.1) has to be solved numerically. But here we focus on approximate solutions to get some physical insight into the formation of quadratic Bragg solitons. A comparison with numerical solutions will be performed if appropriate. There are two different approximate methods to solve eqs. (5.1) for arbitrary parameters. The first one relies on the multiplescale approach which, in the limit of a weak nonlinearity, reduces a Braggtype system of equations to the respective second-order partial differential equation(s), i.e., the scalar limit of eqs. (2.21) (Conti, Trillo and Assanto [ 1997]). This corresponds to the transformation of the Bragg equations to the nonlinear Schr6dinger equation in the cubic case (De Sterke and Sipe [ 1994], Conti, Trillo and Assanto [1997]). The multiple-scale method allows for an arbitrary choice of the system parameters (detuning and coupling) but is limited to low power levels. The other common approximation, used to solve problems involving a quadratic nonlinearity, is usually referred to as cascading limit. The underlying assumption for this approach is that for a large detuning the SH is locally downconverted. Thus propagation terms in the SH equations of eqs. (5.1) may be
536
OPTICAL SOLITONS IN MEDIA WITH A QUADRATICNONLINEARITY
[7, w 5
neglected and the equations describing the FH evolution now contain an effective cubic nonlinearity. In, e.g., a quadratic film waveguide without corrugation, this local approximation leads to the nonlinear Schrrdinger equation (see, e.g., Schiek, Baek and Stegeman [ 1996]). That is why sometimes this concept is also called the Schrrdinger limit (cf. w4.1.3). For Bragg gratings with a quadratic nonlinearity there is another option to arrive at the local limit of the multicomponent system (5.1). If we neglect propagation effects for the SH field, the respective equations from (5.1) are v, = -kU~ + a u ? ,
v, - -k* u? + a u ~ ,
(5.3)
where we have introduced the effective wavevector mismatch A = ~'~---/([K'I2 - ~ 2 ) and the effective coupling constant k = tc/([tc[ 2 - ~2). Inserting eqs. (5.3) into (5.1) we obtain
(o o) (olot o) i
~
+ -ff~
OZ
UI + QUI
+ U2 -
k U~ U2 + A[UI 12 UI = 0,
(5.4)
g2 + n g2 + g~ - k * g 2 u~ + z~ l g212 g2 = o
For the local approximation to be valid the SH amplitudes should also be small compared to the FH ones. Since the SH intensity scales inversely with I~c[2 - ~2 this condition may be formulated as
ll'l From eq. (5.5) we may derive two different limiting cases where a local approximation is valid: (1) For soliton formation in an uncorrugated waveguide we may assume a large detuning (~2 >> ]x.i2 or IA[ >> Ikl) as it was discussed first by Kivshar [1995] for Bragg waveguides. Now self-phase modulation is the dominant term of the effective third-order nonlinearity. As an interesting aspect, it was shown numerically that the coupling for the SH and the corresponding stop-band may even vanish (Conti, Assanto and Trillo [ 1997], Conti, De Rossi and Trillo [1998]). This finding apparently contradicts the usual assumption that stable solitons may not couple to freely propagating linear waves. However, it is known that some isolated solitary waves exist inside the continuous spectrum of the linear modes (Champneys, Malomed and Friedman [1998]). Very recently, a
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SOLITONS IN PERIODIC WAVEGUIDE STRUCTURES - BRAGG SOLITONS
537
large number of such solitons embedded into the continuous spectrum of the SH have been found by Champneys and Malomed [2000] in the above-mentioned effectively three-wave model (Mak, Malomed and Chu [1998c]) in which the FH components are coupled via the Bragg grating, whereas the SH wave is not affected by the grating. A partial explanation for the stability of the solitons observed by Conti, Trillo and Assanto [1998a,b] and Conti, De Rossi and Trillo [1998] may be the fact that on one hand the SH amplitude is always small if we assume a local third-order nonlinearity. On the other hand, the generation of SH waves increases quadratically with the FH amplitude, while the down-conversion depends on it linearly. Thus, SH is generated mainly in the center of the solitary wave, while it is down-converted in its tails. This effect shields the SH field generated by the soliton against the linear waves and reduces the probability for an SH wave to escape. In particular, this shielding effect is demonstrated by the exact soliton solution (Conti, Trillo and Assanto [ 1998a]), especially if the frequency approaches zero (see fig. 5.1 d). Far from the soliton center the SH amplitudes decay exponentially like the square of the FH amplitudes. (2) In the second limiting case the coupling between both SH waves is the dominant e f f e c t (~2 << IK.]2 or IA[ << Ikl). Both FH and SH frequencies are situated within the corresponding stop-bands. In this case the energy exchange is the dominant effect in the effective nonlinearity. This case was discussed in detail by T. Peschel, Peschel, Lederer and Malomed [ 1997]. A third, different limiting case emerges from the model discussed by Mak, Malomed and Chu [ 1998c]. Neglecting the propagation for the SH gives rise to a cross-phase modulation in the FH corresponding to the (integrable) massive Thirring model. We will not discuss this case further here, since the influence of cross- and self-phase modulation on solitary waves in Bragg systems may be treated similarly (Feng and Kneubfihl [1993]).
5.2. B R A G G SOLITONS IN THE L O C A L A P P R O X I M A T I O N
For the sake of physical insight we mainly restrict ourselves to the local approximation (5.4) because there we have access to analytical rather than numerical solutions. The simplified system (5.4) yields the conserved quantities energy Q, Hamiltonian H and momentum P:
538
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
1f
H = -~
dZ
[i ( U ~ --ff OUl 2- -
og2)
+ ~ (,Ul]2
U2 -~
[7, w 5
+ Iu212) + 2U1W2* (5.7)
- k *u ?g; 2 -1-~..(i..i 4 4-IN214)] + c . c . , i/ (OU10U2) P= -~ dZ U?--~- + U;--~-
(5.8)
+c.c.
In looking for soliton solutions moving with the velocity v we may write U~ (z, t) = a, (x) exp [iqg~(x)] ,
n = 1,2,
(5.9)
where x = Z - vT. From the conservation of the pulse energy (5.6) we derive a condition for the soliton amplitudes
I(x)
= (1
- v) a2(x)
= (1
+ v) a~(x) ,
(5.10)
which may be expressed by the local intensity I(x). Moreover, eq. (5.7) can be used to express the intensity I(x) in terms of the phase difference ~p(x) between forward and backward propagating fields: i(x) = 2 / 1 - v 2
o + cos [lp(x)] - arg(k)] - 6 '
(5.11)
Ikl cos [~(x)
where we have introduced 0 = ~2/v/1 - v 2 for the frequency and 6 = A(1 + v2)/ ( 1 - o 2) for the detuning. Obviously the constraint that the FH lies in the respective stop-band implies -1 < 0 < 1. The equations for the FH phases may be simplified to a single equation for the phase difference ~p(x): d~p(x)
dx
2
x,/1 _ v 2 {o
+ cos [~p(x)]} .
(5.12)
Depending on the initial conditions (~p(0) = 0 or ~p(0) - Jr), we get the solutions
/Pl (X) = - 2 arctan
1 + o tanh i - o
x
1 - 0 2
1
U2
(5.13) ~ ( x ) = sr + 2 arctan
[ ~ / 11 -+~ o1 7 6
1
v2
"
For obvious reasons we term ~Pl in-phase and ~ anti-phase solution. As expected, the asymptotic behavior of the phase difference Op(+oc) = q: a r c c o s ( - o ) ) entails vanishing intensities at x = + o c because of eq. (5.11).
7, w 5]
SOLITONS IN PERIODIC WAVEGUIDE STRUCTURES- BRAGG SOLITONS
539
Finally, the individual phases of the FH waves are determined through the phase difference by an additional integration and we get ~p(x) qg+(x) = -+---~ + I(y)
o)o
1_o2
= (1 + v 2 ) v / I k l 2 -
x
- f [~p(x) arg(k)/2] + f [~p(0) arg(k)/2] -
arctan
_
k - 6
1
,
(5.14) For decoupled SH waves (Conti, Assanto and Trillo [1997], Conti, Trillo and Assanto [1998a,b]) or if the detuning dominates over SH coupling (local limit 1), one type of solitons may exist inside the whole stop-band. The sign of the detuning determines whether the solitons are in-phase or anti-phase. A completely different behavior emerges if SH coupling prevails against the detuning (local limit 2) (for details see T. Peschel, Peschel, Lederer and Malomed [ 1997]). Solitons exist only in well-defined regions of the gap. Furthermore, inphase and anti-phase solitons may coexist inside the gap with the corresponding regions touching each other only at particular points. The condition I ( x ) > 0 [see eq. (5.11)] determines the domains where solitons may exist. First we consider the case of a real and positive SH coupling constant: (1) 6 < Ikl (in-phase solutions). The denominator in eq. (5.11) is always positive. Hence, cos(~) > -o" is required which implies an in-phase solution ~l(x) (eq. 5.13). Two cases have to be distinguished: (a) 6 < - I k l : here, there are no singularities and bright solitons may exist for all frequencies and velocities provided that 6 < - I k l holds. (b) -Ikl < 6 < Ikl: in this case singularities may occur and the existence of solutions requires a < -V/(lk] + 6)/(2lkl) < O. (2) 6 > ]k[ (anti-phase solutions). For the anti-phase solution ~Pz(x) (eq. 5.13) the denominator in eq. (5.11) is always negative and consequently there are no singularities. The domains corresponding to both cases are uniquely defined (see fig. 5.2). In particular, regions in the frequency-velocity plane corresponding to in-phase or anti-phase solutions do not overlap provided that the local approximation holds. In contrast to this numerical solutions of the complete system of eqs. (2.25) may share parts of their domains, hence giving rise to a certain kind of bistability (T. Peschel, Peschel, Lederer and Malomed [1997]). An interesting aspect only marginally dealt with in the literature concerns the effect of a nonvanishing phase of the SH coupling constant k. From eq. (5.11) it follows that the intensity distribution of the soliton is no longer symmetric. Furthermore the maximum of the FH intensity is shifted with respect to that of
540
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 5
Fig. 5.2. Domains of existence of solitons in the (v, f2)-plane in the local limit for different values of the dephasing between FH and SH coupling constants, for q = -3, c = 1 and I~:1 - 10 (gray: in-phase solution, black: anti-phase solution). the SH. At first glance this behavior seems to be quite unusual. However, the non-zero phase of the coupling constant corresponds to a nonsymmetric shape of the grating, which breaks the inversion symmetry of the system. At least for small dephasing angles, numerical investigations indicate that the corresponding solitons represent stable solutions of eqs. (5.1) though the FH and SH components are shifted with respect to each other. An example of such a soliton solution is displayed in fig. 5.3. Stable propagation has been observed over a distance of Z = 1000. The domains of existence which correspond to ]6] > ]k] are not affected by the phase of the SH coupling constant. Domains with 16[ < Ikl are maximum when arg(k) = 0 or arg(k) = Jr. The corresponding results are depicted in fig. 5.2. An interesting additional property of Bragg solitons shows up when the sign of the complex coupling constant for the SH waves changes (first noted by Conti, Trillo and Assanto [ 1998a]). The solitary wave solutions found in both situations and their domains of existence are identical if frequency and phase mismatch change their signs too. Additionally in-phase solitons are exchanged with antiphase ones and vice versa (see fig. 5.4). Numerical calculations indicate that
7, w 5]
SOLITONS IN PERIODIC WAVEGUIDE STRUCTURES - BRAGG SOLITONS
541
Fig. 5.3. Solitons in the local limit for f2 = -0.75, u = 0, q = -3, c - 1, and tr = 10exp(0.4i). (a) stationary solutions (solid line: amplitude of the forward component, short dashed line: phase of the forward component, long dashed line: phase difference between forward and backward component). (b) propagation (despite of the large dephasing between FH and SH coupling constants the solitary wave is numerically robust).
the stability behavior remains unchanged. This symmetry permits some kind of "soliton-type engineering" by manipulating the grating shape and therefore the coupling constant of the SH. A comparison of the approximate solutions with the exact ones derived by Conti, Trillo and Assanto [ 1998a] yields an interesting result. The approximate solution given by eqs. (5.11), (5.13) and (5.14) coincides with the exact one if we formally set v = 0, k = -1/(4g2) and A = 1/(4s This choice of the parameters obviously violates the condition of eq. (5.5) for the validity of the local approximation. However, the fact that both solutions share the same functional structure implies that the approximate solutions are relatively close to the (unknown) exact ones. This also is demonstrated by the numerical simulations which show only slight changes when an approximate solution is
542
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 6
in-phase 3ip
b
~
]
FHI I
2 ~
~
-1 ~
]
I 1 . -"i 3 ........................ 2
I
I SH
/'//i
-1 !
-4
-2
0
2
4
'
-4
Z
-2
0
,
2
4
Z anti-phase
3 ! ' ~ ...// I........... i
3-FH
2~
SH
2 1 0 i
E
-1!
,-,,, ",..
-1
-2~
,/f~-
-3! -4
-2
...... 1
V"
t
~ t
0
2
4
Z
-3
_ .........i............J -4 -2 0
i 2
J ~ 4
Z
Fig. 5.4. Comparison between equivalent in-phase and anti-phase solitons for s q=• c=l andtr
= +0.7, u = 0,
fed into the exact system of equations (see, e.g., T. Peschel, Peschel, Lederer and Malomed [ 1997]).
w 6. Solitons and Their Bifurcations in Nonlinear Couplers
Apart from potential future applications, a quadratically nonlinear coupler or dual-core waveguide is a system of fundamental interest by itself. Here we consider the case of a basically symmetric structure (for asymmetric dualcore waveguides see Mak, Malomed and Chu [1998c]). Previous experimental (Schiek, Baek, Krijnen, Stegeman, Baumann and Sohler [1996]) and theoretical (Assanto, Stegeman, Sheik-Bahae and Van Stryland [ 1993] and Assanto, LauretiPalma, Sibilia and Bertolotti [1994]) investigations of quadratically nonlinear couplers focused on the switching behavior in the cw-regime. It turns out
7, w 6]
SOLITONS AND THEIR BIFURCATIONSIN NONLINEAR COUPLERS
543
that most of the cw states are modulationally unstable (see w3). Here we are interested in stationary field distributions and focus on the formation of solitons (Mak, Malomed and Chu [1998a,b]). Starting from eqs. (2.31) and using a transformation similar to eq. (4.10), _rvv--_ 2itcK~Z gl,2 = V'KlOl,2e ,
U1,2 = V~lUl,2 eizcK1Z ,
Z-
Z
K1 '
T-
t (6.1)
leads to the evolution equations 90ul
1
02/,/1
1---~- + ~ Ot--y- - ~CUl + U~Vl + u2 - O,
001
t70Zu1
i-&z -~ 2 0 t 2
(fi 4- 2K')U1 4- bt2 4- ko 2 = 0 ,
1 02b/2 1---~- 4- -~ at-----f- -- 1CU2 4- bl~O2 4- Ul = O,
(6.2)
9O/d2
002
O" 0202
i-&z ~ 2 0 t 2
(fi + 2to)v2 + u22+ kvl = 0,
where, as before, tc denotes the soliton parameter, fi = fi/K1 measures the phase mismatch between the two harmonics and k = Kz/K1 is the relation between the coupling constants at the FH and SH frequencies which is always positive. Above notation pertains to the evolution in the temporal domain. Thus we assume stationary pulses propagating in two parallel waveguides which are coupled by the evanescent fields. The temporal walk-off is neglected in eqs. (6.2). The influence of spatial walk-off was studied in detail by Mak, Malomed and Chu [ 1998b]. It turned out that no qualitative changes are introduced by the presence of this term. In most realistic configurations the coupling between the FHs is much stronger than that between the SH fields, i.e., k < 1. To characterize the solitary wave solutions we use the total energy, O=
fdt (lu,
]2 4- lU2[2 4- Iv1 ]2 4- iO212) ,
(6.3)
and the energy imbalance between the FH waves,
f d t (lUl [2 -lu212) CH = f d t ([ul [2 + lu212)
9
(6.4)
6.1. STATIONARY SOLUTIONS
Stationary solutions are obtained by equating the z-derivatives in eq. (6.2) to zero and by solving the resulting system of ordinary differential equations
544
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 6
numerically. Alternatively a variational approach can be applied as it was done by Mak, Malomed and Chu [ 1998a,b]. In this case a system of coupled algebraic equations is obtained which also has to be solved numerically. Here we follow the first option. Two obvious solutions of the system can be found immediately. Note that the two coupled cores are assumed to be identical. Therefore symmetric (U 1 = U2, U 1 = U2) and antisymmetric (ul = -u2, U1 = U2) states are supported. The energy imbalance of these obvious solutions vanishes (CH = 0) and the problem is reduced to the case already investigated in w4. To apply these results (scalar interaction without walk-off) to the coupler geometry an effective wave vector K'eff and an effective mismatch/3elf have to be introduced as tCeff = tr T 1,
fleff = f l - k + 2 ,
(6.5)
where the upper and lower signs account for the symmetric and antisymmetric solutions, respectively. Consequently the range of existence of these obvious solutions is modified. Symmetric bright soliton solutions exist for tr > m a x [ 1 , ( k - fi)/2] and antisymmetric ones for tr > max[-1, ( k - fl)/2]. The effective mismatch fieff is positive for fl > k - 2 in the case of symmetric solutions and for fl > k + 2 in the case of antisymmetric ones. Both solutions show the behavior already observed for the common case (cf. w4). In particular, for negative effective mismatches (/3elf < 0) they have a minimum in the energy vs. wavenumber diagram and therefore a finite energy threshold (cf. fig. 6.1c). Thus the branch with a large SH content of the solution is unstable in this case. The most interesting question is whether nontrivial solutions exist, which do not obey the above symmetry relation and which are thus specific for the twocore geometry. In particular we are looking for spontaneous symmetry breaking, or a bifurcation which generates asymmetric solitons. Those asymmetric solutions are known from dual-core waveguides with a cubic nonlinearity (Malomed, Skinner, Chu and Peng [1996], Ankiewicz and Akhmediev [1996] and references therein). For the Kerr-nonlinearity nontrivial solutions are found to bifurcate from the trivial ones above a certain threshold. A similar behavior is found for the quadratically nonlinear coupler. For every set of system parameters (/3 and k) asymmetric solitons with a nonsymmetric intensity distribution are found. New branches bifurcate supercritically from both symmetric and antisymmetric solutions (see fig. 6.1). This is in contrast to the cubic case where the bifurcation from the symmetric state is subcritical (for definitions see, e.g., Seydel [ 1988]). Above a certain wavenumber asymmetric solutions coexist with the trivial ones (see an example of respective field shapes in fig. 6.2). In case
7, w 6]
SOLITONS AND THEIR BIFURCATIONS IN NONLINEAR COUPLERS
60t ,a,
545
,]
40
20
-
(b)
_
CFH
0
_
-1 150
-
(c)
100 i i t
s /
..
**...2."
'~t 0
~ T
0
r
1
T
T
2
I
3
4
K"
Fig. 6.1. Bifurcation diagrams displaying symmetry breaking in terms of the wavenumber tc for the total energy Q and the imbalance CFI-I,starting from (a), (b) symmetric and (c) antisymmetric branches (k = 0.5,/3 =-1, o = 0.5, solid lines: stable, dashed lines: unstable).
of negative effective mismatch the bifurcation point is always above (i.e., for higher wavenumbers to) the minimum in the energy vs. wavenumber diagram which usually marks the onset of the instability for conventional solitons (see fig. 6. lc). The fact that symmetry breaking is found above certain wavenumbers and thus above an energy threshold can be understood by means of a simple physical picture. Formally the propagation constant can be kept constant by dividing
546
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 6
(a) Un, Vn
l (b)
U n, Vn
'
-8
I
0
Fig. 6.2. Typical (a) symmetric and (b) asymmetric solitons in the coupler for k = 0.5,/3 = - 1 , K" = 2.5 and o = 0.5. (solid lines: FH, dashed lines: SH). Bold and thin lines in (b) correspond to different cores.
eqs. (6.2) by K'. Consequently an increase of the propagation constant and therefore of the soliton energy corresponds to a reduced effective coupling between the cores. Symmetry breaking occurs if the coupling is diminished and if the respective fields can develop independently (see Mak, Malomed and Chu [1998a,b]). In fact the amplitudes and the spatial dimensions are also influenced by this rescaling procedure. Therefore care has to be taken when drawing quantitative conclusions from this argument. 6.2. STABILITY BEHAVIOR
In the case of a dual-core coupler with a cubic nonlinearity all antisymmetric solutions are already unstable. This also seems to be true for the quadratically nonlinear dual-core coupler. Again the entire branch of antisymmetric solutions is unstable. Thus the asymmetric branches which bifurcate from the antisymmetric one also are unstable (fig. 6.2c). In what follows we concentrate on the symmetric branch. It destabilizes at the bifurcation point. In contrast to the case of antisymmetric solutions the asymmetric solutions which emerge
7, w 7]
DISCRETE SOLITONS IN WAVEGUIDE ARRAYS
547
from the symmetric branch at the bifurcation point remain stable for all parameter values (fig. 6.2a,b). Just above the bifurcation point the growth of the eigenmode of the corresponding linear problem which causes the instability of the symmetric trivial solutions results in a transition to the asymmetric branch. Thus spontaneous symmetry breaking occurs. The difference between symmetric and asymmetric solutions is proportional to the unstable eigenmode. In the limit of large wavenumbers or small effective coupling this eigenmode converges to an antisymmetric linear combination of the bound states of the individual solitons.
w 7. Discrete Solitons in Waveguide Arrays Since the pioneering work of Fermi, Pasta and Ulam [1955] the study of nonlinear dynamics in discrete systems is one of the major issues in basic nonlinear physics. The most relevant subject of these studies is how the very discreteness of the system affects the dynamical behavior of excitations beyond the continuum approximation. In this context questions that concern the existence and properties of intrinsically localized nonlinear solutions, frequently referred to as discrete solitons or localized modes, attract a steadily growing interest. Primarily discrete solitons have been studied in discrete lattices with an onsite cubic nonlinearity and other nonoptical systems. But very early it turned out that this concept can be successfully extended towards arrays of coupled optical waveguides. The latter system represents a convenient laboratory for the experimental verification of numerous theoretical predictions and constitutes the subject of this section. Moreover, nonlinear waveguide arrays may have a fair potential in future all-optical switching and routing schemes (see e.g., Lederer and Aitchison [ 1999]). The aim here is to provide a basic understanding of effects like "discrete diffraction", the mutual interplay of this controllable kind of diffraction with nonlinearly induced localization, which leads to soliton formation and the competition between soliton motion across the array and self-trapping. Until recently, discrete soliton formation was only shown to appear in arrays with cubic nonlinearities (Christodoulides and Joseph [ 1988], for a survey on potential effects, see, e.g., Aceves, De Angelis, Peschel, Muschall, Lederer, Trillo and Wabnitz [1996], and for the first experiments, see Millar, Aitchison, Kang, Stegeman, Villeneuve, Kennedy and Sibbett [1997] and Eisenberg, Silberberg, Morandotti, Boyd and Aitchison [1998]). Because the field dynamics in an array of N nonlinear waveguides, exhibiting nearest-neighbor interaction, can
548
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 7
be considered as a general case covering the two limits of nonlinear trapping in a two-core coupler (see w6) and spatial soliton formation in a film waveguide (see w it is intuitively clear that discrete solitons can be formed in a quadratic nonlinear environment. Moreover it can be anticipated that there is a much richer diversity of solutions than in the cubic case because the number of dynamical variables (two- or three-component system) and parameters (mismatch, two different coupling coefficients) is larger. We start with the discussion of the dispersion relations for plane-wave solutions. In doing this we can identify potential domains in parameter space where bright solitons may form. Moreover, we emphasize the peculiarities of "discrete diffraction", being a synonym for coupling in an array, and the consequences for the formation of different types of discrete solitons. Then we discuss various types of solutions, viz., moderately and strongly localized ones, and their stability. 7.1. DISCRETEDIFFRACTIONAND DISPERSION RELATIONS In order to drop boundary effects we assume an infinite array (N -+ zx~). As shown in w2 the evolution of the FH (Un) and SH (Vn) envelopes in the nth guide can be described by eqs. (2.34)"
dU.
i--d-~-+ Cu (u,,+ 1 + u , , _ l ) + u ~ , v , , = o , dV,
(7.1)
i---d~ + Cv (v,,+l + v,,_l) - flv,, + ui~ = o ,
where we replaced/3 by/3. Eqs. (7.1) exhibit two conserved quantities, the total energy Q and the Hamiltonian H" Oc
t/=-Oc
H = -
2cuU~,U,,+l + c v V ~ , V , , + l -
IV,,
+ U;, V;, +c.c.
11=--0C
The lack of momentum conservation across the array is a strong indication for self-trapping of moving localized solutions. This is the primary difference in the corresponding continuum limit of eqs. (7.1) which can be obtained by expanding (Un+l -k- U n _ l )
'~
2U(Y)+ h202 U/OY z + . . .
,
(7.4)
with Y = hn. After a trivial phase transformation this leads to the scalar version of eq. (2.21) with both zero walk-off and group-velocity dispersion. We are not interested in that issue here because this limit reproduces the results obtained
7, w 7]
DISCRETE SOLITONS IN WAVEGUIDEARRAYS
549
Fig. 7.1. Dispersion relation 2,(q) of continuous waves for two mismatches/3 and Cu = Cv = 1 (dark gray: staggered nonlinear waves, light gray: unstaggered nonlinear waves, bold lines: linear waves).
for one-dimensional scalar solitons. On the contrary we search for the very discreteness effects. The simplest solutions of eqs. (7.1) are nonlinear plane waves U,(Z) = U0 exp [i (~wZ + qn)] ,
V,,(Z) = Vo exp [2i (~wZ + qn)] ,
(7.5) where U0 and V0 can be assumed real and ~w has to obey the dispersion relation (T. Peschel, Peschel and Lederer [1998]) [~w - Cv cos(2q) + 89
[~w - 2Cu cos(q)] m l~ U2 - 0,
(7.6)
and q is the transverse wave vector or likewise the phase difference between adjacent guides. The solution of eq. (7.6) is (Darmanyan, Kobyakov and Lederer [1998]) ~w = 2Cu cos(q) + V0, (7.7) U2 - 2 V02+ V0 [4Cu c o s ( q ) - 2Cv cos(2q) +/3] > 0. Figure 7.1 shows the domains of existence of linear and nonlinear waves. The inspection of the linear limit of the dispersion relation (U 2 = 0) discloses the peculiar character of "discrete diffraction". The usual diffraction in a continuous system (r w = 2Cu(1 -q2/2), r w + / 3 / 2 = Cv(1- 2q2)) occurs only for q] << 1 (in-phase or unstaggered solution). If q = Jr (out-of-phase or staggered solution) the diffraction changes sign. This is a genuine effect of discreteness. In the cubic case this has the consequence that bright solitons can form in a medium with defocusing nonlinearity. Obviously for discrete solitons to be formed requires
550
OPTICAL SOLITONS IN MEDIA WITH A QUADRATICNONLINEARITY
[7, w 7
two prerequisites, viz., (a) their wavenumbers Z have to be situated in regions of the (~., q)-plane where linear waves must not exist and (b) nonlinear plane waves are allowed to propagate but should be modulationally unstable (for a detailed discussion of modulational instability in discrete quadratic media see Miller and Bang [1998], Darmanyan, Kobyakov and Lederer [1998]), i.e., ~. > max (2Cu, Cv -
fl/2)
(7.8)
,
or
< min (-2Cu,-Cv -
ill2)
(7.9)
.
Obviously the mismatch crucially affects the existence conditions for discrete solitons. 7.2. VARIOUS TYPES OF DISCRETE SOLITON SOLUTIONS
We are searching for discrete solitons with real-valued amplitudes and wavenumbers, Un(Z)
= u,,
exp(iXZ),
V , , ( Z ) = v,,
exp(2iXZ),
(7.10)
and get from eq. (7.1) * =0 --~U n + Cu (Un+l + Un-1 ) + UnVn
,
(7.11) --2/~Un + Cv (Un+l + On-l) -- [~Un + U21 = O.
Like in the cubic case eqs. (7.11) can be solved only numerically. But in addition to this numerical approach we discuss some limiting cases where analytical solutions are accessible or we can take advantage of known models. We distinguish two basic types of discrete solitons, namely moderately and strongly localized ones. The former type will appear in situations where the normalized amplitudes are comparable or smaller than the coupling constants, i.e., linear coupling and nonlinear effects are of the same order of magnitude and the excitation spreads over many waveguides (T. Peschel, Peschel and Lederer [1998]). The latter type requires higher powers, where nonlinear effects dominate the linear ones. Thus the excitation is restricted to 3 or 4 guides (Darmanyan, Kobyakov and Lederer [1998]).
7, w 7]
DISCRETE SOLITONS IN WAVEGUIDEARRAYS
551
7.2.1. Moderately localized discrete solitons This case is the more general one. Formally it also includes strongly localized solutions, which are discussed below. As is well-known, soliton formation relies always on a balance of a linear effect that tends to spread the excitation (diffraction, coupling, diffusion, dispersion) and a nonlinear effect that evokes localization. In discrete systems there are staggered and unstaggered linear waves that are situated either at the top or the bottom of the linear band and thus exhibit a different sign of "discrete" diffraction. The nonlinearity has to shift the solution beyond the band. For a moderate nonlinearity, and thus moderate localization, this can only be achieved by reducing (increasing) the wavenumber for staggered (unstaggered) solutions (see fig. 7.1). Thus these solitons have a smaller (larger) wavenumber than linear waves. Two limiting cases can be considered, viz., vanishing coupling of the SH or of the FH waves (the residual coupling constant is scaled to 1 for convenience): (1) No coupling of the second harmonics (Cu = 1,Cv = 0). From eqs. (7.11) we get i/An 2
--~,Un + Cu (Un+ 1 + Un_l)-t- 2X + fi u, = 0,
Vn
--
2
un
(7 12)
2,,l + fi"
The remaining equation for the FH wave to be solved is obviously identical to the cubic nonlinear case. Because this situation is well investigated (see, e.g., Christodoulides and Joseph [1988] and Aceves, De Angelis, Peschel, Muschall, Lederer, Trillo and Wabnitz [1996]), we will not go into details. We mention only that it is possible to get bright staggered (2Jl + fi < 0) and unstaggered (2Jl + fi > 0) solitons in the same waveguide array depending on the sign of mismatch and wavenumber. Moreover, in this case strongly localized solutions also can be found (see Page [1990]). (2) No coupling of the fundamental waves (Cu = 0, Cv = 1). Because the energy exchange relies completely on SH waves it can be anticipated that generic features of the quadratic nonlinearity are more pronounced in this approximation. The system (7.11) can be solved resulting in a one-parameter family of even (virtual soliton peak is centered between waveguides) and odd solitons (soliton peak is centered at a waveguide) with different topologies (staggered, unstaggered). The solutions are (T. Peschel, Peschel and Lederer [19981) U(n~ U(even)
= bn,0V/2~,(~,=
Ot +
fi/2),
(On,0 + 6n,1)V/2X [~, _ ( a
V(n~ _
JlotI~l ,
/3 + 1)/2],
(7.13) _ (even) = xa[n-l[- 89
o n
,
(7.14)
552
10U n, V n
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
(a)
t 1,,
sHFH(b)
'1
,
!
i
t'
[7, w 7
Itt'! i'
't
0 I0 -
(c)
(d)
Un, V n
0 i
t i i i t
-10
~[----~r
1
-5
0
r---'--7 ~
'
n
5
-5
r
I
0
n
5
Fig. 7.2. Numerically calculated discrete soliton solutions for Cu = Cv = 1,/3 = - 4 and ~. = 8. (a) odd, unstaggered, (b) even, unstaggered, (c) odd, staggered and (d) even, staggered.
with
1-
1-(2~.+/3) 2
"
Lifting the requirement of vanishing coupling between either FH or SH components and solving eqs. (7.11) numerically, the solutions are somewhere in between cases 1 and 2. Typical effects of the quadratic nonlinearity (as, e.g., finite energy thresholds for certain values of the parameters) are more pronounced in the second limiting case. Frequently the stability of discrete solitons is evaluated by using the PeierlsNabarro potential (PNP) which is the Hamiltonian (7.3) of a resting discrete soliton. This Hamiltonian is periodic with the period "1" and shows maxima/minima for inter/on-site location of the soliton peak. The difference is called Peierls-Nabarro barrier. Stronger localization increases the barrier and traps the
7, w 7]
DISCRETE SOLITONS IN WAVEGUIDE ARRAYS
553
soliton, i.e., it cannot move across the array. Following the arguments of Cai, Bishop and Gronbech-Jensen [ 1994] odd and even solitons with the same energy (eq. 7.2) and the same topology (staggered, unstaggered) can be considered as two realizations of one soliton centered either on- or inter-site. Examples for Cu = Cv = 1 are displayed in fig. 7.2. Stable unstaggered solitons settle always in PNP minima whereas staggered solutions prefer PNP maxima. Thus all even solutions are unstable and transform to their odd counterparts, whereas odd solitons are stable at least above a power threshold. Both analytical and numerical solutions yield similar results with respect to stability. A rich collision scenario and soliton self-trapping can be observed for these moderately localized solitons (T. Peschel, Peschel and Lederer [1998]).
7.2.2. Strongly localized discrete solitons Similarly to the cubic case (see Page [1990]) strong localization also occurs in quadratic media provided that for the peak amplitudes (e.g., at n = 0) lu0] 2 >> cu or Iv0]2 >> c~ holds (as a matter of fact, e.g. ]u0 2/c, > 15 suffices). Because of strong localization the PNB is very large and all solutions are at rest. The approximate analytical solutions can be derived in assuming that for odd solitons essentially one guide is strongly excited. If the amplitudes of the nearest neighbors can be neglected we obtain ~ = v0 and u~ - 2v 2 +/3v0. The two adjacent guides then exhibit secondary amplitudes proportional to Cu/VO and Cv/(2vo + [3). For even solitons the two central guides carry the maximum amplitude. Beyond this categorization into even and odd solitons we have symmetric and antisymmetric as well as staggered and unstaggered solutions. Calculating the PNP one can infer that again the odd solutions are stable and the even ones unstable. A novel type of solitons, viz., twisted solutions, appears (for details see Darmanyan, Kobyakov and Lederer [1998]). In contrast to staggered and unstaggered solutions even twisted solitons are stable below a certain instability threshold given by the secondary amplitudes. This controllable instability can be exploited for steering and switching operations. In fig. 7.3 the evolution of a typical stable odd, an unstable even and a stable twisted soliton is shown. It remains to mention that solitons of different topologies can form bound states. Recently it was shown by Kobyakov, Darmanyan, Pertsch and Lederer [1999] that quasi-rectangular strongly localized solitons may exist which are basically a superposition of two step-like fronts. The stability of these bound states can be controlled by slightly changing the amplitudes of excitation.
554
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 8
Fig. 7.3. Evolution of strongly localized bright solitons for Cu = Cv = 0.15, /3 = 0 and v0 = 2 (fundamental waves). (a) stable odd soliton, (b) unstable even, unstaggered soliton and (c) stable even twisted soliton.
w 8. Multidimensional Solitons The concept o f spatio-temporal optical solitons, or light bullets (Silberberg [ 1990]), which are supported by the interplay o f nonlinearity, spatial diffraction and temporal dispersion, has attracted a lot o f attention as a unique opportunity to create a self-supporting fully localized object freely propagating in a nonlinear
7, w 8]
MULTIDIMENSIONAL SOLITONS
555
medium. However, it is well known that bullets in media with a Kerr (cubic) nonlinearity are unstable, being subject to a wave collapse (see a review by Kuznetsov, Rubenchik and Zakharov [ 1986]). Nevertheless, the collapse does not take place in two- and three-dimensional (2D and 3D) quadratically nonlinear media, which was demonstrated long ago by Kanashov and Rubenchik [1981] and later confirmed numerically by Hayata and Koshiba [1993] (see also recent works by Bergr, Mezentsev, Rasmussen and Wyller [ 1995] and Turitsyn [1995], where the collapse problem was studied in detail in the context of multidimensional models with quadratic nonlinearities). Moreover, Kanashov and Rubenchik [ 1981 ] had rigorously proved, by means of variational estimates, that the Hamiltonian of the 3D-model attains, at a fixed value of the energy (also called the norm, or "number of quanta"), a nontrivial minimum corresponding to a fully localized state. This fact is tantamount to a rigorous proof of the existence of stable multidimensional LBs. The same is true in the 2D case. In an explicit form, approximate LB solutions were constructed analytically (within the framework of the variational approximation, VA) and numerically (for the 2D case only) by Malomed, Drummond, He, Berntson, Anderson and Lisak [1997]. Additional numerical results for the 2D and 3D cases (including the shape of the stationary LBs) were obtained, respectively, by Mihalache, Mazilu, Malomed and Torner [1998, 1999] and Mihalache, Mazilu, D/Srring and Torner [1999]. The first experimental observation of a spatio-temporal optical soliton in a quadratically nonlinear medium was reported by Liu, Qian and Wise [ 1999] very recently. This spatio-temporal soliton was not a 3D one, but rather a quasi-2D "bullet", i.e., it was localized in the propagation direction and one transverse direction, but delocalized in the other transverse direction. An observation of 3D solitons has not yet been reported. Equations describing spatio-temporal evolution of the FH and SH waves in the SHG medium were put forward by Kanashov and Rubenchik [1981]. Actually, this is a particular case of the more general eqs. (2.12) the derivation of which was given above. Here we use the scaling of eqs. (4.34) neglecting walk-off and taking into account dispersion and diffraction: 9o%1
1
1 Oq2Ul
. Obt 2
1
o" oq2u2
1---~ + -~V2Ul -Jr--~ aT----~ - Ul h- u~u2 -- O, 1--~ q- ~ V 2 lg2 q 2 0 T 2
(8.1) a u g -+- U2 = O,
where the gradient V• is d-dimensional with d = D - 1 being the transverse spatial dimension. Note that the exact-matching point corresponds, in the present
556
OPTICAL SOLITONSIN MEDIA WITH A QUADRATICNONLINEARITY
[7, w 8
notation, to an effective mismatch parameter a = 2. An example where the walk-off between the harmonics is taken into account was treated in Mihalache, Mazilu, Malomed and Tomer [ 1999]. Stationary (Z-independent) solutions of eqs. (8.1) are determined by the following equations for the real functions ul and u2: 1 [O2Ul OUl 02Ul] -~-RT + ( d - 1)R-l-0-~- + OT 2 ] - ul + ul u2 -- 0 , 1
02
-~ ~
U2
0122 ]
+ (d - 1 ) R - l - ~
(8.2)
(7 0 2u2
-t 2 0 T 2
otu2 + u 2 = O,
where R is the radial coordinate in the case d 1> 2 (axial symmetry is assumed), or the single transverse coordinate X in the case d - 1. In the case d = 1 numerical solutions of eqs. (8.2) describing fully localized spatio-temporal solitons were obtained by Mihalache, Mazilu, Malomed and Torner [ 1999], and in the case d = 2 by Mihalache, Mazilu, D6rring and Torner [1999]. 8.1. ANALYTICAL RESULTS (VARIATIONAL APPROXIMATION)
Before proceeding to the presentation of the numerical solutions, it is relevant to consider an analytical approach based on VA, developed in detail by Malomed, Drummond, He, Berntson, Anderson and Lisak [1997]. Still earlier, it was demonstrated by Steblina, Kivshar, Lisak and Malomed [ 1995] that VA provides high accuracy in considering a related but simpler object, viz., spatial solitons in 2D and 3D media with a quadratic nonlinearity. More examples attesting for the usefulness of VA in the theoretical study of solitons in quadratically nonlinear media can be found above in w4. VA is based on a certain ansatz or trial function for the solution. A general property of solitons in SHG models is a difference in spatial and temporal widths of their FH and SH components. The only tractable ansatz which can accommodate this property is based on the Gaussians u l = A exp (-aiR2
_
b l T 2) ,
U2 = B exp (-a2 R2 - b2 T 2) .
(8.3)
In ansatz (8.3), the arbitrary parameters ai,bl, a2, b2 and A , B represent, respectively, the inverse squared spatial and temporal widths and amplitudes of the FH and SH components of the LB. The next ingredient of the variational technique is the Lagrangian corresponding to eqs. (8.2), L = f + ~ d R f + ~ d T E
7, w 8] in the case d = 1, or L = density
1
MULTIDIMENSIONALSOLITONS
f~dR R f + ~ d T
s in the case d - 2, with the Lagrangian
1/ 1)2 37
557
a -b-F
(8.4) Given the ansatz of eq. (8.3) for the solutions, it is inserted into eq. (8.4). Integrating the resulting expression over R and T it is straightforward to find the corresponding effective Lagrangian. Finally, the equations for the parameters of the ansatz (which can be found in a full form Malomed, Drummond, He, Berntson, Anderson and Lisak [1997]) are obtained by equating the variations of the effective Lagrangian with respect to the parameters al, a2, bl, b2, and A,B to zero. Physical solutions are those for which the inverse squared spatial and temporal widths of the soliton, al,bl and a2, b2, are real and positive. The following results were obtained from the VA-generated equations for the parameters of the ansatz (8.3): (i) for all positive a and o there is exactly one physical solution (for both 2D and 3D cases); (ii) for d = 1 there is exactly one solution for negative o too, i.e., for the case when the anomalous dispersion at FH coexists with the normal dispersion at SH; (iii) for d - 2 solutions for negative ty were found only in a very narrow stripe, e.g., for ty > - 0 . 0 3 4 if a - 1/2, where VA produces two different solutions (for d - 2 only); (iv) in both cases d - 2 and d - 3, there is, generally, a finite energy threshold for the existence of LBs (i.e., their energy, considered at fixed a (and d) as a function of ty, attains a minimum nonzero value). Feature (iii) is illustrated in fig. 8.1, where the widths of the 3D LBs, as predicted by VA, are shown vs. the relative dispersion o. The feature (iv) implies a drastic difference from the 1D case, where there is no finite energy threshold (Buryak and Kivshar [1995a], Steblina, Kivshar, Lisak and Malomed [1995]). Malomed, Drummond, He, Berntson, Anderson and Lisak [1997] ran direct simulations of the full dynamical eqs. (8.1) (but only for the 2D case), using the VA-predicted LB shapes as initial condition. On the basis of the numerical results, it was found that the above property (i) is correct, the shape of the numerically found solitons being quite close to that predicted by ansatz (8.3) (including the case ty - 0). The property (iii), pertaining to the 3D case, has not yet been checked vs. direct simulations. As for the property (ii), the simulations
558
OPTICAL SOLITONS IN MEDIA WITH A QUADRATICNONLINEARITY
[7, w 8
4
i
2 ',,,.b 2
0 -0.1
] 0.1
r 0.3
0.5
Fig. 8.1. The inverse-squared temporal and spatial widths of the 3D light bullet (spatio-temporal soliton, see eq. 8.3) vs. the relative dispersion parameter o for a = 1/2, as predicted by the variational approximation.
demonstrate that for a < 0 LBs do not exist (they decay into radiation). This discrepancy is due to the fact that the Gaussian ansatz, essentially, chops off the exponentially decaying tails of LBs, while the decay of LBs at a < 0 is accounted for just by their tails. Nevertheless, if a is negative but small, the decay rate of the LB is numerically found to be so small that it should be regarded as a quasistable (and therefore physically meaningful) state. This quasistability is enhanced if the mismatch parameter a is large enough. Finally, numerically exact solutions of the stationary equations (8.2) clearly corroborate the existence of the nonzero energy threshold for the multidimensional LBs, in agreement with the property (iv) in the above list (Mihalache, Mazilu, Malomed and Torner [1998, 1999]). An essential peculiarity missed by VA is that the threshold vanishes at the exact-matching point (a = 2). The numerically found threshold monotonically increases with a (in both the 2D and 3D cases, i.e., d = 1 and d - 2). In agreement with the prediction of VA, the threshold is essentially higher in the 3D case than in the 2D one. Note that eqs. (8.1) have four integrals of motion (dynamical invariants). One of them is the momentum, which is zero for the ansatz (8.3). Other invariants are the above-mentioned energy (norm) Q, which is the same as in the 1D cases, i.e., it has the density Q = [u I 12+ [U2[2, and the Hamiltonian (for real Z-independent solutions, it coincides with above Lagrangian). The last dynamical invariant is (in the 3D case) the angular momentum in the transverse plane. Some conclusions about the stability of solitons can also be obtained within the framework of VA, using the known necessary stability criterion put forward by Vakhitov and Kolokolov [ 1973] for solitons of the NLS type. According to this
7, w 8]
MULTIDIMENSIONAL SOLITONS 0.6
559
(a) stable <>
0.4
0.2 unstable
0.0
'
I
0
0.6
0
(b)
0.5 stable
0.4 no solution
0C
0.3 unstable
0.2 i
i
0
I
1
i
O
2
Fig. 8.2. The stability boundary for the (a) 2D and (b) 3D light bullets in the parametric plane (o, or), as predicted by the variational approximation. In (a) the circles and rhombuses correspond to the cases for which direct simulations have been run to check the predictions (filled rhombuses: decay, open rhombuses: stable propagation, open circles: ambiguous behavior).
criterion (which was applied to the above-mentioned stationary solitary beams by Buryak, Kivshar and Steblina [ 1995]), a necessary condition for the stability of solitons is a positive slope of the soliton energy as a function of its propagation constant. In eqs. (8.3) this is scaled out (cf. w 4) and the solitons are described by the parameter a. For the LB solutions, for each value of d and a, there is a single critical value act of a which separates the stable LBs (a > acr) from unstable ones (a < acr). The critical values for d = 2 are acr -~ 0.33 for both o = 1/2 and a = 0. At d = 1, they are act --- 0.19 for a = 1/2, and acr ~" 0.26 for o = 0. The predictions for the existence of stable solitons in the cases d = 1 and d = 2, generated by a combination of the VA and techniques used in Vakhitov and Kolokolov [ 1973], are summarized in fig. 8.2, where a boundary curve separating stable and unstable solitons in the (o, a)-plane is shown (for d = 2 and a - - 1/2 cf. also Skryabin and Firth [1998]).
560
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 8
Fig. 8.3. Example of a numerically simulated evolution of a 2D light bullet for o = 0.072 and a = 4.144. The initial configuration is taken as per the variational ansatz (8.3). Shown is the cross section T = const, of the evolution of the fundamental harmonic's intensity u2(R, T). 8.2. N U M E R I C A L RESULTS
Several typical examples of the evolution of 2D LBs, generated by VA-predicted initial conditions, are displayed below. In fig. 8.3, the case of small o and relatively large a is presented where a stable LB with some internal vibrations emerges. A noticeable feature of the vibrations is that, while they are not growing and hence do not give rise to an instability, they do not show a pronounced radiative damping either. This property is quite close to that of 1D solitons, where the existence of a genuine internal mode and of a quasimode (which is embedded into the continuous spectrum, but, nevertheless, tums out to be fairly robust in the simulations) explains extremely stable internal vibrations of the solitons triggered by a strong perturbation (Etrich, Peschel, Lederer, Malomed and Kivshar [1996], see also w4 of this review). As mentioned above, VA produces a solution for 2D LB at all negative values of ty, while the direct simulations do not support this prediction. However, if lty [ is small the pulse generated from a VA-ansatz seems practically stable: in this case, the numerically observed LB does not have any visible difference from the stable solitons found for o > 0. An example for o = -0.005 and ct -- 0.43 is shown in fig. 8.4. It is relevant to stress that for o = 0 (SH exactly at the zerodispersion point) LB still exists and is very close to that shown in fig. 8.4. As a general trend, a larger a helps to stabilize the LB for negative ty. For a larger normal dispersion of the SH the decay of the pulse becomes fast (fig. 8.5). More simulations have been performed to check the stability boundary predicted by VA and shown in fig. 8.2. To this end a string of points along the line
7, w 8]
MULTIDIMENSIONALSOLITONS
561
Fig. 8.4. The same as in fig. 8.3 for a =-0.005 and a = 0.43.
Fig. 8.5. The same as in fig. 8.4 for u =-0.5 and a = 5. a = 0.35 (which is expected to intersect this stability boundary curve twice) was selected. The results are indicated by the corresponding symbols in fig. 8.2. Some points which are located close to the boundary show an ambiguous behavior, viz., vibrations with very large amplitudes, but without decay. The stability boundary predicted by VA proves to be "fuzzy" compared to the direct simulations, but nevertheless this boundary is meaningful. It should be stressed that for ~ < 0 the stability boundary predicted by VA has no meaning. The direct numerical investigation of the stability in both cases, d = 1 (Malomed, Drummond, He, Berntson, Anderson and Lisak [1997]) and d = 2 (Mihalache, Mazilu, D6rring and Torner [1999]), show that LBs in quadratically nonlinear media are stable in a large domain of parameter space, provided that the
562
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 8
4-
lull 2
2
0 0
x,t
8
Fig. 8.6. A typical example of cross sections along the temporal (solid) and transverse spatial (dashed) coordinates of the 2D stationary spatio-temporal soliton, obtained from direct numerical solution of eqs. (8.2) with a = 0 and a -- 0.938. (a) and (b) show the FH and SH components, respectively. dispersion is anomalous for both harmonics, FH and SH. Note that Mihalache, Mazilu, Drrring and Torner [ 1999] have also developed a numerical analysis of the stability in the framework o f the underlying eqs. (8.1) linearized around the stationary LB solutions, eventually arriving at the same conclusions. A noticeable feature of the LBs in the general case ( a ~ 1/2) is their spatio-temporal asymmetry. This feature is illustrated by fig. 8.6, which shows the transverse temporal and spatial cross sections of an LB obtained by direct numerical solution of the stationary equations (8.2) in the case d = 1 (Mihalache, Mazilu, Malomed and Torner [ 1998], for d = 2 see Mihalache, Mazilu, Drrring and Torner [ 1999]). 8.3. SPINNING SPATIO-TEMPORAL SOLITONS AND OTHER GENERALIZATIONS The theory of LBs in quadratically nonlinear media has been further developed in various directions. Multidimensional media with a Bragg grating were considered in He and D r u m m o n d [1998]. Here the theoretical consideration was based on the so-called effective-mass approximation which essentially reduces
7, w 9]
CONCLUSIONS
563
the model to eqs. (8.1). Taking into account the temporal walk-off between the harmonics more general families of "walking" LBs are obtained (Mihalache, Mazilu, Malomed and Torner [1999]). In particular, the walk-off makes the LB solutions chirped, but does not destroy them or make them unstable. An interesting generalization is to consider "spinning" spatial and spatiotemporal solitons. Spatial cylindrical solitons can be sought for in the form Ul ( Y , Y, Z) = U(R) e ikZ+iSO ,
u 2 ( X , Y, Z) = V(R) e 2ikZ+2iSO,
(8.5)
where R and 0 are the polar coordinates in the transverse (X, Y)-plane, the integer S is the "spin" of the soliton, and the functions U(R) and V(R) are real. This is a soliton with a hole in the middle, as the functions U and V have an obvious asymptotic form U ~ R s and V ~ R 2S for R ~ 0. The spatial dynamics of such solitons was simulated in detail by Skryabin and Firth [1997] and by Petrov and Torner [1997]. It has been found that all the solitons with S ~ 0 are subject to a strong instability against azimuthal (i.e., 0-dependent) perturbations. As a result of the development of the instability, the soliton explodes. It splits into several moving solitons with S = 0, such that its original intemal angular momentum is transformed into the orbital momentum of the remains. The instability of the spinning soliton and its splitting into usual moving solitons has also been observed experimentally (Petrov, Torner, Martorell, Vilaseca, Tortes and Cojocaru [1998]). A similar spinning soliton (S = 1) was shown to be very stable in another nonlinear model of optical origin which is also collapse-flee, viz., the one-component 2D nonlinear Schr6dinger equation with a cubic focusing and a quintic defocusing term (Quiroga-Teixeiro and Michinel [ 1997]). Spinning 3D spatio-temporal solitons in a quadratically nonlinear model were very recently considered by Bakman and Malomed [2000]. In this case the solution is sought for in the form of eq. (8.5) with U and V being functions of R and T [cf. eq. (8.3)]. The solution with S = 1 was found both numerically and by means of VA, the analytical results being close to the numerical ones. Stability simulations of the spinning bullets have not yet been completed.
w 9. Conclusions As is evident from the reference list, which is by no means complete, the study of quadratic solitons has been a very active and dynamic field of nonlinear optics in the last five years. Thus, the aim of the present review was threefold, viz.,
564
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7
to make the reader familiar with the basic theoretical concepts of this area of research, to provide the reader with a guide through the numerous literature, and to invite the reader to join the community in the adventure to tackle the many challenging problems that are left unsolved to date. As usual, a bundle of theoretical concepts is waiting for experimental verification. Although considerable progress has been achieved regarding experiments in bulk crystals, in particular in potassium-titanyl-phosphate (KTP), there are only a few attempts towards the demonstration of soliton effects in waveguides with a quadratic nonlinearity. These experiments would be of primary interest for all-optical applications, which usually require chip dimensions. In this respect, the recent advances in quasi-phase matching technology could lead to a major breakthrough because the effectively acting nonlinearities can be not only optimized but also appropriately tailored. If this or related techniques can be efficiently applied to materials with originally large nonlinear coefficients, such as polymers or semiconductors, then even photonic data processing may become real. As far as the fundamental aspects of the field are concerned, the experimental verification of quadratic Bragg and discrete solitons is still a challenge. In this connection, we believe that an experimental observation of double-humped solitons, predicted to be stable in the Bragg waveguide, would be especially interesting. Another problem that has to be solved is to achieve complete understanding of the excitation of various types of quadratic solitons, in particular, if only one component of the soliton is launched. Little investigated up to now is the soliton formation in quadratic media when a dc field is involved. These few examples emphasize that the study of quadratically nonlinear systems will remain an interesting subject of research in the future. Most of the techniques introduced here, as, e.g., the approaches to the investigation of internal modes of the solitons, are rather general and can be used to study the dynamics of other nonlinear systems of completely different origin.
Acknowledgments The authors gratefully acknowledge a long-term grant of the Deutsche Forschungsgemeinschaft, generously supporting the research activities in the field of quadratic nonlinearities, in the framework of the Sonderforschungsbereich 196. B.A.M. is particularly grateful to the same organization for providing him with a scholarship during his stay at Friedrich-Schiller-Universit~it Jena, where most of the work was performed.
7]
REFERENCES
565
We appreciate valuable collaborations with coauthors of our original papers, namely, P.L. Chu, L.C. Crasovan, S. Darmanyan, P.D. Drummond, H. He, Y.S. Kivshar, A. Kobyakov, W.C.K. Mak, D. Mazilu, D. Mihalache and L. Torner.
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A U T H O R I N D E X F O R V O L U M E 41
A Aakjer, T. 363 Abdalla, M.S. 365, 366, 392 Abel+s, E 228 Abraham, M. 236 Aceves, A. 547, 551 Aceves, A.B. 534 Acher, O. 195 Acker, W.P. 12, 60, 61, 72, 76, 79-81 Ackerman, E. 309 Acrivos, A. 20 Adam, P. 363, 365, 390, 392, 395 Agarwal, G.S. 365, 454, 460, 463 Agranovich, V.M. 521 Agrawal, G.P. 364, 506 Aharonov, Y. 458, 469 Ahmend, M.M.A. 365 Aitchison, J.S. 547 Akahira, N. 101, 148, 149, 172 Akasaka, H. 142 Akasaka, M. 139 Akhmediev, N.N. 364, 529, 544 Akiyama, T. 150, 151 A1-Homidan, S. 365 Alameh, K.E. 350 Albert, D.Z. 469 Alexander, D.R. 16, 72 Alexandrou, I. 245, 247 Alink, R.B.J.M. 115-117 Alodjants, A.P. 428, 440, 460, 463 Amato, J.P. 236 An, D. 335 An, I. 235 Anandan, J. 458 Anderson, D. 555-557, 561 Andreoni, A. 486, 489 Andres, M.V. 314 Ankiewicz, A. 364, 544
Anma, H. 253 Antoine, J. 297 Aragone, C. 460 Arai, S. 108 Arakelian, S.M. 428, 440, 460, 463 Aratani, K. 139 Archer, R.J. 236 Arecchi, ET. 429, 431 Ariunbold, G. 365, 395, 396 Armstrong, J.A. 485, 487 Armstrong, R.L. 53, 64, 68, 70-72, 74, 76, 86, 87, 89 Arnold, S. 16, 50, 85 Arraf, A. 509 Artigas, D. 364, 529 Arwin, H. 251 Ashkin, A. 3, 70 Ashrit, P.V. 265 Aspnes, D.E. 193, 195, 214, 246, 249, 250, 261,267, 276 Assanto, G. 363, 364, 489, 490, 533-537, 539-542 Atatiire, M. 447 Atkins, P.W. 429 Au, C.K. 469 Awano, H. 139, 146 Aytfir, O. 440 Azzam, R.M. 425 Azzam, R.M.A. 186, 192, 193, 195, 196, 262, 276
B Baboiu, D.M. 488, 489 Bachor, H.-A. 364, 457, 459 Bader, G. 265 Baek, Y. 485, 487, 489, 516, 536, 542 Baer, T. 57, 63 569
570
AUTHOR INDEX FOR VOLUME 41
Bajer, J. 365, 377, 378, 384-386, 439, 454456 Baker, G.L. 363 Bakman, u 563 Baldwin, K.C. 303 Ball, G.A. 313 Baltog, I. 364 Ban, M. 448, 449, 468, 470 Banaszek, K. 439 Bandilla, A. 375, 454 Bandyopadhyay, A. 363, 461 Bang, O. 490, 550 Bann, S. 297 Banyai, W.C. 363 Bao, Z. 68 Barber, P.W. 10, 16, 19, 21-26, 28, 32, 33, 40, 48, 49, 53, 56, 63, 68, 70, 72, 82, 85 Baril, M. 297 Barnes, M.D. 46, 49, 50 Barnett, S.M. 365, 366, 395, 397, 398, 425, 433, 435,438, 445-447, 450, 452, 460, 463, 465, 470, 471 Barthelemy, A. 488 Bartlett, C.L. 111 Barton, J.P. 16, 63, 72 Baseia, B. 452 Bashara, N.M. 186, 192, 193, 195, 239, 243, 244, 262, 276, 425 Bashir, M.A. 365 Baumann, I. 542 Bechtel, J.H. 334 Beck, M. 439, 440, 442, 443, 467 Beckmann, P. 218 Belanger, M. 330 Belcher, M. 325 Belisle, C. 330 Belkind, A. 252 Belsley, M. 440 Ben-Aryeh, Y. 370, 460, 461 Bennemann, K.H. 81 Benner, R.E. 4, 7, 11, 12, 40, 48, 53, 82, 84 Bennett, H.E. 263 Bennett, J.M. 263 Bennion, I. 313 Berg6, L. 555 Berggren, M. 68 Bergou, J. 425 Bernacki, B.E. 106, 108, 109, 125 Berning, P.H. 229 Bernstein, N. 305
Berntson, A. 555-557, 561 Berquist, J.C. 459 Bertet, P. 464 Bertolotti, M. 363-365, 390, 392, 395, 542 Berzanskis, A. 365 Best, M.E. 173 Betzig, E. 173 Bhaumik, D. 446 Bi, W.G. 52, 68, 69 Bialynicka-Birula, Z. 432 Bialynicki-Birula, I. 432, 456 Bigan, E. 195 Bigelow, N.P. 463 Birks, T.A. 16 Bishop, A.R. 553 Biswas, A. 71, 72, 74, 76 Bj6rk, G. 446, 447, 458, 464, 465 Blau, G. 364 Blaudez, D. 270, 272 Block, D.G. 115 Bloembergen, N. 485, 487 Boardman, A.D. 364, 488, 510, 521 Boatner, L.A. 270 Bogoliubov, N.M. 453 B6hmer, B. 433,450, 451,454 Bollmger, J.J. 459, 460, 463, 464 Bondurant, R.S. 461 Bontemps, P. 510 Bootsma, G.A. 237 Born, M. 205, 207, 217, 425, 426 B6rner, M. 497, 498 B6rnstein, R. 256 Bosenberg, W.R. 364 Boughton, R.S. 302 Bourliaguet, B. 488 Boutou, V. 81 Bouwhuis, G. 99, 101, 106, 109, 111 Bouyer, P. 463 Boyd, A.R. 547 Boyd, R.W. 364 Braat, J. 119 Braat, J.J.M. 101, 106, 109, 124, 164 Braginsky, V.B. 52, 463 Braunstein, D. 71 Braunstein, M. 244 Braunstein, S.L. 439, 455, 458, 459, 469 Bricot, C. 108, 110 Briegel, H.-J. 464 Brif, C. 460--462 Broers, R. 115
AUTHOR INDEX FOR VOLUME41 Brookner, E. 285, 286, 288 Brooks, D. 364 Brorson, S.D. 40, 47, 52 Bruggeman, D.A.G. 214 Brtme, M. 52, 58, 464 Btichler, U. 431,452, 471 Bunkin, N.E 85 Burgstede, P. 111 Burnett, K. 435, 462, 463, 472 Buryak, A.V. 488-490, 506, 512, 517, 530, 557, 559 Busch, P. 433, 450, 472 Butcher, P.N. 495 Bu2ek, V. 363, 365, 446, 462 Byer, R.L. 10, 364 C Cai, D. 553 Caironi, D. 489 Campillo, A.J. 10, 16, 22, 47-51, 53-57, 59, 60, 63, 70-72, 75, 85, 86 Campos, R.A. 429, 460, 462, 465 Canal, E 364 Candela, G.A. 242 Cantrell, C.D. 84 Canva, M.T.G. 489 Capasso, E 34, 69 Capmany, J. 314 Carcia, P.E 137 Cardone, L. 304 Carline, R.T. 246 Carroll, J.E. 436 Carruthers, P. 425, 432, 451,453, 471 Castin, Y. 454, 462 Caucuitto, M.J. 303 Caulfield, H.J. 301 Caves, C.M. 455, 458, 459, 461 Cerullo, G. 364 Chalbaud, E. 460 Champneys, A.R. 536, 537 Chandezon, J. 160 Chandler-Horowitz, D. 242, 244 Chang, and R.K. 56, 57 Chang, C.-H. 173 Chang, H.-R. 174 Chang, J.-S. 34 Chang, M.-W. 174 Chang, R.K. 11, 12, 23, 26-28, 31-35, 38, 40, 48, 49, 53, 56, 57, 59-68, 70-89 Chang, S.S. 38
571
Chang, T.-K. 173 Chang, Y. 302, 317 Chapman, J.N. 137 Chase, S. 134 Chefles, A. 365, 395, 397, 398 Chemla, Y.R. 26, 61-63 Chen, C.-L. 174 Chen, G. 26, 34, 35, 38, 60-63, 66, 67, 72-76, 84, 86-89 Chen, L.Y. 195 Chen, M. 149 Chen, R.T. 335, 337-342 Chenard, E 330 Cheng, L. 157 Cheong, B.-K. 171 Chetkin, M.V. 207 Cheung, G. 16 Cheung, J.L. 70, 74, 85, 89 Chew, H. 40--43, 53, 67, 70 Chickarmane, V.S. 365 Chin, M.-K. 52, 53, 68, 69 Chinaglia, W. 486, 489 Chindaudom, P. 248 Ching, E.S.C. 4, 19, 20, 24, 40, 44, 46, 53, 54 Chirkin, A.S. 463 Chitanvis, S.M. 84 Chizhov, A.V. 364, 444, 472 Cho, A.Y. 34, 69 Cho, H.M. 252 Cho, Y.J. 252 Chowdhury, D.Q. 16, 20-22, 27-29, 31, 33, 70, 72, 86-89 Christian, J.W. 156, 157 Christodoulides, D.N. 533, 547, 551 Chu, B.T. 76 Chu, D.Y. 52, 53, 68, 69 Chu, EL. 363, 364, 366, 490, 491,533, 537, 542-544, 546 Chuang, I. 465 Chuang, R.-N. 174 Chuang, S. 503 Chumakov, S.M. 470 Chung, H.K. 252 Chvostovfi, D. 254 Ch~,lek, E 11, 53-55, 60, 74 Cibils, M.B. 432 Cirac, I. 464 Cirac, J.I. 464 Clausen, C.B. 490
572
AUTHOR INDEX FOR VOLUME 41
Clauser, M.J. 239 Coey, J.M.D. 132 Cohen, D. 317 Cohen, D.A. 317 Cohen, D.K. 105, 108 Cojocaru, C. 563 Collet, M.J. 462 Collins, R.W. 235 Collot, L. 52, 58 Constantini, B. 488 Conti, C. 490, 533-537, 539-541 Cooke, D.D. 41, 42, 70 Cooper, D.G. 311 Cooper, J. 439, 440, 442, 443, 467 Corey, L.E. 325 Comet, G. 160 Comey, J.E 464 Como, J. 215 Cotter, D. 82, 495 Courtens, E. 429, 431 Cousins, R.D. 455 Cram6r, H. 211 Crasovan, L.-C. 488, 491,523, 528 Creegan, E. 71, 74 Cromer, A.H. 432 Crosignani, B. 363 Cruz, J.L. 314 Cuche, Y. 432 Cullen, T.J. 363 D
da Silva, M.P. 266 Dahmani, R. 244 Dakss, M.K. 166 Dalibard, J. 454, 462 Danielus, R. 489 D'Ariano, G.M. 437, 446, 452, 456, 457, 464, 467, 468, 470, 472 Darmanyan, S. 490, 549, 550, 553 Darmanyan, S.A. 490, 521,553 Daryoush, A.S. 323 Das, K.K. 448 Davidovi6, D.M. 470 Davies, D.K. 307, 347 Davis, E.J. 70 De Angelis, C. 547, 551 De Angelis, C.A. 488 De La Torre, A.C. 445 de Nijs, J.M.M. 193, 276 De Renzi, V. 472
De Rossi, A. 536, 537 De Silvestri, S. 364 De Sterke, C.M. 533, 535 de Sterke, C.M. 509 den Boef, A.J. 137 den Engelsen, D. 237 deNeufville, J. 147 DeNicola, R.O. 244 Denk, W. 173 Depine, R.A. 160 Derka, R. 446 DeSalvo, R. 486 Detry, R.J. 239 DeVoe, R.G. 461 Devore, S.L. 108, 109, 111 Dewitz, J.-P. 81 Dexter, J.L. 311 Dexter, L. 311 Di Porto, P. 363 Di Trapani, P. 486, 489 Dicke, R.H. 431 Dill, EH. 195 Dinges, H.W. 237 Dios, E 364 Dobson, J.C. 429 Dodabalapur, A. 68 Dolfi, D. 297 Dong, L. 314 Doremus, R.H. 256 Dora, R. 237 D6rring, J. 555, 556, 561,562 Dowling, J.P. 459, 460, 462 Dr6villon, B. 195 Driver, H.S.T. 63, 68 Drude, P. 231 Druger, N.L. 50 DriJhl, K. 364 Drummond, P. 509, 555-557, 561 Drummond, P.D. 488-490, 506, 508, 509, 516, 526, 528, 534, 562 Dubietis, A. 489 Dubreuil, N. 16 Ducuing, J. 485, 487 Dunnigham, J.A. 435 Dtmningham, J.A. 462 Dupertuis, M.A. 460 Dupertuis, M.-A. 460, 463 Dupuis, M.T. 160 Du~ek, M. 442 Dutta, P. 242
AUTHOR INDEX FOR VOLUME 41 Dziedzic, J.M.
3
E
Earman, A. 104 Eberly, J.H. 460 Eckardt, R.C. 364 Edge, C. 313 Eickmans, J.H. 76 Eisenberg, H.S. 547 Ejnisman, R. 463 Ekert, A.K. 446, 464 E1-Orany, EA.A. 366, 392 Elizalde, E. 265 Ellinas, D. 445-447, 470 Elliott, J.E. 109 Elman, J.E 273 Elshazly-Zaghloul, M. 262 Ema, K. 58, 60, 63 Emslie, A.G. 224, 226 Englert, B.-G. 425, 437 Erdmann, R. 310 Erickson, L.E. 63 Erman, M. 246 Erwin, J.K. 109 Esman, R.D. 311, 312 Espiau, EM. 302 Etemad, S. 363 Etrich, C. 486, 488-491,512, 516, 517, 519, 521,522, 525, 528, 530, 532, 560 Eversole, J.D. 10, 16, 22, 47-51, 53-56, 59, 60, 63, 70-72, 85, 86 F
Facchi, P. 380 Faist, J. 34, 69 Fan, H. 468 Fano, U. 532 Fazio, E. 364 Feinleib, R.E. 302 Feinlib, J. 147 Fejer, M.M. 364, 485 Feller, K.-H. 365 Feng, D.H. 366 Feng, J. 534, 537 Fermi, E. 547 Fernandez, G. 71, 74 Fernandez, G.L. 68, 86 Ferr6, J. 273 Ferro, P. 489, 506, 528 Fert, A.R. 274
573
Feshbach, H. 373 Fetterman, H.R. 302, 317 Fields, M.H. 64-67, 74, 76-78 Finlayson, N. 363 Finn, P.L. 173 Firth, W.J. 559, 563 Fiur~i~ek, J. 366, 375, 390, 395-398, 400, 401,403, 404, 408-412 Flamme, B. 237 Fleming, J.W. 78 Flevaris, N.K. 274 Folan, L.M. 85 Fonda, L. 429 Fonseca, E.J.S. 465 Fontenelle, M.T. 439, 469 Forrest, S.R. 302, 343, 345 Forward, R.L. 457 Foug+res, A. 430, 435, 436, 438, 439 Fourikis, N. 350 Frank, J. 289 Frankel, M.Y. 312 Frankel, M.Y.J. 311 Franken, P.A. 485 Franta, D. 215-217, 266, 267, 269 Frazee, R.E. 244 Freeman, M.O. 174 Freitag, P.M. 343 Freyberger, M. 453,456, 467-469 Friedman, M.J. 536 Frigerio, J.M. 265 Froehlich, EE 173 Fu, H. 120 Fu, J. 301 Fu, Z. 335, 338, 339, 341,342 Fuerst, R.A. 489 Fujii, E. 171 Fujii, N. 245 Fujiwara, H. 68 Fukami, T. 141, 142 Fukumoto, A. 139 Fuller, K.A. 67 Fung, K.H. 71 Furukawa, S. 150, 151 Furuki, M. 117 Furuta, M. 117 Fuss, I.G. 452, 468 G Gaillyov~i, Y. 240, 241 Gale, G.M. 364
574
AUTHOR INDEX FOR VOLUME 41
Gallot, G. 364 Gamino, R.J. 171 Gantsog, Ts. 375, 425, 435, 450, 452 Gardiner, C.W. 464 Garland, J.W. 195 Garret, C.G.B. 4, 56 Gea-Banacloche, J. 461 Gee, W.H. 108 Gennaro, G. 428 Gerber, R.E. 109, 111, 124, 159, 164, 167 Gerhardt, H. 431,452, 471 Gerry, C.C. 452, 460, 464 Gesell, L.H. 302, 320 Giles, R. 130, 132 Gillespie, J.B. 31, 32, 59, 66, 67, 76, 77 Gilligan, J.M. 459 Gilmore, R. 366, 429, 431 Gimeno, B. 314 Girouard, EE. 265 Glauber, R.J. 429 Glenn, W.H. 313 Glogower, J. 451 Gmachl, C. 34, 69 Goddard, N.L. 50 Goiran, M. 274 Goldberg, L. 311 Goldberg, W. 326 Goldhirsch, I. 445 Goldstein, H. 207 Goobar, E. 447 Goodell, J.B. 109 Goodman, J.W. 440, 442 Goodman, T.D. 110, 119, 121 Gopinath, A. 326 Gorodetsky, M.L. 52, 463 Gouesbet, G. 16 Goutzoulis, A.P. 307, 347 Grabowski, M. 433, 434, 450, 451,453, 472 Graham, R. 366 Granger, P. 297 Grangier, P. 461 Gray, M.B. 457 Grayson, T.P. 436 Green, L. 352, 354 Greener, J. 273 Gr~han, G. 16 Griffel, G. 16 Gronbech-Jensen, N. 553 Grossman, H.L. 34 Grygiel, K. 401
Gu, J. 64, 68, 86, 87, 89 Guerri, G. 460 Guo, G.-C. 464 Guo, S. 251 Gustafsson, G. 251 Gyorgy, E.M. 173 I-I Ha, Y. 462 Hache, E 364 Haderka, O. 442 Haelterman, M. 491,528 Hagan, D.J. 485-487, 489 Hagel, O.J. 251 Haggerty, J.S. 224, 226 Haitjema, H. 240, 241 Hajjar, R.A. 134 Haken, H. 366 Hakio(glu, T. 439, 440, 446 Hald, J. 459, 461,463 Hall, J.L. 461-463 Hall, M.J.W. 434, 451,452, 456, 468 Hamada, K. 108 Han, Z. 335 Hanna, D.C. 82 Hansen, P. 127, 171 Hansen, P.B. 363 Hare, J. 6, 16, 27, 41, 52, 53, 58, 59 Haroche, S. 6, 27, 41, 52, 53, 58, 59, 63,464 Harrington, J.D. 30 Harris, J.B. 160 Harris, R.D. 364 Hartings, J.M. 70, 74, 77, 85, 89 Hartman, N.E 325 Hartmann, M. 119 Hasegawa, A. 486 Hasegawa, M. 130 Hasegawa, Y. 455 Hashimoto, M. 139 Hatami-Hanza, H. 363, 364 Hauge, P.S. 195 Haus, H.A. 7, 302, 453, 466, 467, 469 Haus, J.W. 364 Hayata, K. 488, 555 Hayes, R. 328, 332 Hazel, R. 112 He, H. 489, 490, 506, 508, 509, 526, 528, 534, 555-557, 561,562 Heavens, O.S. 229 Heckens, S. 248
AUTHOR INDEX FOR VOLUME 41 Heidrich, P.E 166 Heinzen, D.J. 459, 460, 463,464 Heller, L.M. 247 Helstrom, C.W. 433, 434, 451,452, 455, 468 Hendriks, B.H.W. 124 Hendrych, M. 442 Heni, M. 467, 468 Herczfeld, P.R. 302, 323 Herec, J. 392, 393, 395 Herzinger, C.M. 193, 253, 273 Hibbs-Brenner, M.K. 326 Hidaka, Y. 130 Hightower, R.L. 21, 22 Hilgevoord, J. 458 Hill, A.E. 485 Hill, S.C. 4, 7, 10-12, 16, 19-29, 31-33, 42, 46, 61-63, 70, 72, 75, 76, 79, 82, 84 Hillery, M. 375, 456, 458, 460-462 Hirao, K. 63 Ho, S.T. 52, 68 Ho, S.-T. 52, 53, 68, 69 Hodgkinson, I.J. 233 Hoekstra, A. 193 Hoffmann, H. 215 Holevo, A.S. 451,452, 456 Holland, M.J. 462, 463, 472 Hollberg, L.W. 461 Holler, S. 50 Holtslag, A.H.M. 193 Hong, C.K. 462 Hong, Ch.K. 462 Hong-Yi, E 468 Hope, D.A.O. 246 Hopkins, H.H. 161 Hora, J. 266 Horai, K. 151, 171 Horfik, R. 395 Horesh, N. 458, 464 Horowitz, E 233 Hosokawa, T. 139, 142 Hottier, E 214 Hou, H.Q. 52, 69 Hozumi, Y. 171 Hradil, Z. 366, 395, 425, 437, 439, 442, 454-456, 458, 462, 467, 468, 472 Hrycak, P. 347 Hsieh, W-E 59 Hsieh, W.-E 51, 52, 72, 74-76, 86-89 Hsieh, Y.C. 119 Hsieh, Y.-C. 121
Hiibner, W. 81 Huelga, S.E 464 Huignard, J.P. 297 Huignard, J.-P. 297 Hulse, J.E. 247 Humli~ek, J. 240, 275 Huston, A.L. 10, 49, 53-55, 57, 60 Huston, B.L. 85, 86 Hutcheon, R.J. 68 Huting, W.A. 289 Hutley, M.C. 160 Huttner, B. 465 I
Ianno, N.J. 254 Ibach, H. 237 Ibrahim, A.M.A. 365, 395 Ibrahim, M.M. 239, 243 Ide, T. 149, 154 Iguain, J.L. 445 Iida, H. 142 Iketani, N. 144 Ilchenko, V.S. 27, 41, 52, 53 Imaino, W.I. 173 Imanaka, R. 151 Imoto, N. 465 Imre, D.G. 71 Inaba, H. 64, 66 Inchaussandague, M.E. 160 Inoue, K. 150, 151 Inoue, S. 101 Irene, E.A. 246, 258, 259 Ironside, C.N. 363 Isaacson, M. 173 Ishikawa, T. 144 Itano, W.M. 459, 460, 463, 464 Ito, K. 109 Itoh, H. 144 Ivanova, I.C. 256 Iwasa, N. 124 Iwasaki, M. 172 Izergin, A.G. 453 Izutsu, M. 293 J
Jackel, J.L. 363 Jackson, J.D. 7 Jacobs, B. 119 Jacobs, B.A.J. 137 Jacobson, J. 465
575
576
AUTHOR INDEX FOR VOLUME 41
Jacobson, R. 224, 228, 229 Jacques, E 16 Jaffres, H. 274 Jain, E 352, 354 Jakeman, E. 462, 465 Jaksch, D. 464 Jakubczyk, Z. 330 Jalali, B. 330 Jan6a, J. 276 Janda, P. 266 Janis, J. 463 Janszky, J. 363, 365, 390, 392, 395 Jarzembski, M. 74 Jarzembski, M.A. 86 Jastrabik, L. 254 Jauch, J.M. 425 Javanainen, J. 445 Jaynes, E.T. 431 Jegorova, G.A. 256 Jelinek, V. 392 Jellison Jr, G.E. 196, 270 Jemison, W.D. 302 Jeong, S.Y. 114 Jericha, E. 453 Jian, P.-S. 491 Joffre, P. 297 Johns, S. 315 Johns, S.T. 310 Johnson, A. 347 Johnson, B.R. 10, 17, 19 Johnson, J.A. 244 Johs, B. 193, 252, 253, 273 Jones, K.R.W. 456 Jones, V. 329 Jones, V.I. 306 Jones, V.L. 306 Jonker, B.T. 244 Jordan, R.H. 68 Joseph, R.I. 533, 547, 551 Ju, J.-J. 173 Jundt, D.H. 364 Jur6o, B. 366, 425 Justus, B.L. 57, 60 K
Kaiser, W. 4, 56 Kalili, EY. 463 Kalmykov, S.Yu. 364 Kalweit, E. 326 Kamada, K. 67
Kamchatnov, A. 490 Kamchatnov, A.M. 521 Kanashov, A.A. 487, 555 Kaneda, Y. 117 Kaneko, M. 142, 144 Kaneno, M. 139 Kang, J.U. 547 Kao, S.-C. 256 Kar, T.K. 446 Karamzin, Yu.N. 487, 515 Karpati, A. 390 Karpierz, M. 364 Karpierz, M.A. 490, 515 Karpov, V.B. 85 Kfirskfi, M. 373, 374 Kasemset, D. 309 Kasevich, M.A. 463 Kashihara, T. 151, 171 Kashiwagi, T. 117 Kasono, O. 117 Kasparian, J. 81 Kato, K. 109 Katsumura, M. 117 Katz, H.E. 68 Kaup, D.J. 366, 488 Kawahara, K. 172 Kawano, T. 144 Kawano, Y. 142 Kay, D. 111 Keefer, C.W. 310 Kelly, J.R. 302 Kenan, R.P. 325 Kennedy, G.T. 547 Kerker, M. 21, 22, 41, 42, 67, 70 Kermene, V. 488 Khaled, E.E.M. 16, 72 Khazanov, A.M. 71 Kheruntsyan, K.V. 365 Kiefer, W. 70, 71 Kiehl, J.J. 60 Kielar, P. 274 Kieli, M. 323 Kielich, S. 364, 428, 435 Kielpinski, D. 464 Killip, R.B. 471 Kim, C.C. 195 Kim, G.-H. 34 Kim, H. 462 Kim, J.B. 114 Kim, J.Y. 114
AUTHOR INDEX FOR VOLUME41 Kim, K. 462 Kim, S.G. 140, 171 Kim, S.J. 252 Kim, S.Y. 248, 252, 276 Kim, T. 462, 463 Kim, W.M. 171 Kim, Y.S. 366 Kimble, H.J. 41, 52, 53, 461 Kindlmann, RJ. 67 King, B.E. 464 Kino, G.S. 165 Kinosita, K. 238 Kireev, A.N. 460 Kitagawa, M. 431,459, 460, 463 Kitahara, H. 117 Kitanine, N.A. 453 Kivshar, Yu.S. 486, 488-490, 506, 512, 517, 530, 532, 536, 556, 557, 559, 560 Kiyoku, H. 124 Klauder, J.R. 366, 429, 456, 457, 461,462, 468 Klimov, A.B. 470 Kloch, A. 363 Knausenberger, W.H. 256 Kneubiihl, EK. 534, 537 Knight, J.C. 6, 16, 41, 52, 53, 63, 68 Knight, EL. 363, 365 Knittl, Z. 199, 226, 227, 231 Knorr, K. 237 Ko, K-H. 34 Ko, M.K.W. 60 Kobayashi, T. 138, 139, 142 Kobayashi, Y. 144 Kobyakov, A. 364, 486, 487, 490, 549, 550, 553 Kodama, Y. 486 Koepf, G.A. 291,292 Koganov, G.A. 71 Kojima, R. 151, 171 Kojima, Y. 117 Kokai, S. 142 Kolokolov, A.A. 519, 558, 559 Kondo, K. 117 Konev, VA. 244 Korolkova, N. 365, 388, 390, 395, 396, 399401,412-414 Korpel, A. 99 Koshiba, M. 488, 555 Kotov, VA. 210, 273 Kowarz, M.W. 160
577
Koynov, K. 364 Kr~imer, B. 81 K~epelka, J. 363, 371, 374, 395-398, 400, 460, 467, 468, 471 Krijnen, G. 542 Krinchik, G.S. 207 Krishen, K. 216 Krishnan, R. 273,274 Kryder, M.H. 173 Kryuchkyan, GNu. 365 Kuang, L.-M. 401 Kubota, S. 117 Kuhn, L. 166 Kuleshov, E.M. 244 Kumar, A. 302 Kung, C-Y. 50 Kurokawa, K. 117 Kuwata-Gonokami, M. 58, 60, 63, 68, 463 Kuzmich, A. 462, 463 Kuznetsov, E.A. 555 Kwok, A.S. 59, 77, 78 L LaBudde, E. 112 Lafuse, J.L. 302 Lahti, EJ. 433,450, 472 Lai, H.M. 19-21, 24-26, 28, 32, 40, 41, 44, 46, 48, 53, 54, 82, 84, 85, 89 Lai, W.K. 363 Lai, Y. 453, 466, 467, 469 Lakoba, T.I. 366 Lakshmi, P.A. 452 Lalovid, D.I. 470 Lam, C.C. 10, 11, 19, 21, 24-26, 28 Lam, P.K. 364 Landau, L.D. 207 LandoR, H. 256 Lane, A.S. 455, 458, 459 Langbein, U. 363 Langer, C. 464 Lanz, M. 173 LaPorta, A. 461 Larchuk, T. 462, 465 Laskowski, E. 68 Latifi, H. 71, 72, 74 Latta, M.R. 106 Laureti-Palma, A. 363, 542 Lawrence, B.L. 489 Le Carvennec, E 108, 110 Leach, D.H. 12, 26, 27, 72, 74, 76, 79-81, 84
578
AUTHOR INDEX FOR VOLUME41
Lecourt, B. 270, 272 Lederer, E 363,364, 486-491,512, 516, 517, 519, 521,522, 525, 528, 530, 532, 533, 537, 539, 542, 547, 549-551,553, 560 Lederich, R.J. 237 Lee, H.Y. 252 Lee, I.W. 252 Lee, J.-H. 34 Lee, J.J. 305, 306, 329 Lee, R. 340 Lee, S.K. 140, 171,252 Lee, Y.W. 252 Lefebvre, P.R. 259 Lef6vre-Seguin, V. 6, 16, 27, 41, 52, 53, 58, 59, 63 Leggett, A.J. 454 Lehureau, J.C. 108, 110 Leisner, T. 81 Leo, G. 364 Leonardi, C. 428 Leonhardt, U. 429, 433, 450, 451,453, 454, 467, 469 Leskova, T.A. 521 Leslie, J.D. 237 Leuchs, G. 395, 461 Leung, P.T. 4, 10, 11, 19-21, 24-26, 28, 32, 41, 44, 46, 48-50, 82, 84, 85, 89 Leutheuser, V. 363 Levenson, J.A. 365 Levenson, M.D. 461 Leventhal, D.K. 16 Levi, A.EJ. 317 Levin, B.R. 213 Levine, A.M. 304 L6vy-Leblond, J.M. 445, 451,456, 458, 469 Lewis, A. 173 Lewis, J. 306, 329 Lewis, J.B. 306 Lewkowicz, J. 108 Li, L. 159-161, 166-168, 170 Li, R. 337-339 Li, Y. 235 Li, Y.Q. 49, 58 Libera, M. 149 Libezn~,, M. 256, 258-261,264, 265 Lifshitz, E.M. 207 Lillo, E 428 Lin, C. 238 Lin, H.-B. 10, 16, 22, 47-51, 53-57, 59, 60, 63, 70-72, 75, 85, 86
Lin, K.-H. 51, 52 Lin, S.-C. 302 Lin, W. 334 Lindblad, G. 437 Lisak, M. 555-557, 561 Lison~k, P. 385 Lissberger, P.H. 209 Litfin, G. 431,452, 471 Liu, J. 340 Liu, J.-S. 173 Liu, P.-Y. 173 Liu, Q. 258, 259 Liu, S.Y. 10, 19, 20, 24, 44, 49 Liu, T. 332 Liu, W.-C. 160 Liu, X. 486, 555 Liu, Y.S. 49, 58 Livingston, S. 306, 329 Lock, J.A. 16 Loescher, D.H. 239 Loewen, E.G. 160 Logothetidis, S. 245, 247 Long, M.B. 23, 56, 57, 63, 85 Long, W.L. 4, 56 Loo, R. 329 Loo, R.Y. 306 Lopu~nik, R. 273 L6schke, K. 239 Lu, B.L. 49, 58 Lu, Y. 235 Lubinskaya, RT 257 Ludeke, M. 108 Luff, B.J. 364 Luis, A. 366, 370, 375, 381,435, 438, 440, 444-446, 448, 450, 452, 456, 460, 470 Luke,, E 218, 219, 222, 237, 242, 243, 255, 256, 259, 262, 263 Luke, A. 363, 374, 375, 425, 432, 435, 450, 451,456, 458, 460, 467, 468, 471 Liith, H. 237 Lynch, D.W. 195 Lynch, R. 425, 435, 452, 471 M
Mabuchi, H. 41, 52, 53 Macchiavello, C. 446, 452, 456, 464, 470 MacCrackin, EL. 120 Macleod, H.A. 233 Madamopoulos, N. 352, 354 Maeda, H. 363
AUTHOR INDEX FOR VOLUME41 Maekawa, N. 363 Magel, G.A. 364 Magni, V. 364 Maheu, B. 16 Mailloux, R.J. 296 Maischberger, K. 457 Mak, W.C.K. 490, 533, 537, 542-544, 546 Maleki, L. 302 Malomed, B. 490, 491,512, 517, 519, 521 Malomed, B.A. 364, 366, 486, 488-491,506, 508, 509, 516, 517, 522, 525, 530, 532, 533, 536, 537, 539, 542-544, 546, 555-558, 560-563 Mamin, H.J. 165, 173 Mandel, L. 366, 425, 429, 430, 435, 436, 438, 439, 462, 463, 466, 469 Manko~-Bor~mik, N. 429 Mann, A. 458, 460-462, 464 Mansfield, S.M. 165, 166 Manson, P.J. 459 Mansuripur, M. 101, 106, 108-111, 114, 119-121,124, 125, 127, 129, 130, 132, 134, 157, 159, 161, 166-168, 170 Marburger III, J.H. 448 Marchant, A.B. 104, 106, 109, 110, 163 Marchiando, J.E 242 Mardezhov, A.S. 257 Marinilli, A. 352, 354 Marta, T. 326 Marte, M.A.M. 364 Martorell, J. 563 Marvulle, V. 432 Marx, D.S. 160 Masetti, E. 266 Massar, S. 446 Masuhara, H. 68 Mather, A. 302 Matsumoto, H. 142 Matsumoto, K. 144, 293 Matsumoto, M. 139 Matsunaga, T. 151, 171, 172 Matsushita, T. 124 Matthys, D.R. 431 Mauhara, H. 67 Maxwell Garnett, J.C. 214 Maystre, D. 160 Mazilu, D. 488, 489, 522, 523,528, 555, 556, 558, 561-563 Mazumder, M.M. 20, 21, 25-29, 31-33, 60-63, 66, 67, 70, 76, 77
579
McCall, S.L. 429, 456, 457, 461,462 McClelland, D.E. 364, 457 McCrackin, EL. 237, 255 McGahan, W.A. 252, 253 McGuire, T.R. 171 McLeod, J.H. 109 McMarr, P.J. 249 McNulty, EJ. 41, 42, 70 Mekis, A. 34, 35, 38 Memarzadeh, K. 252 Menyuk, C.R. 485-489 Mergel, D. 171 Merkt, U. 192 Merritt, C.D. 50 Mertz, J.C. 461 Mertz, L. 440 Meyer, E 237 Meyer, V. 464 Mezentsev, V.K. 555 Michel-Gabriel, E 297 Michinel, H. 563 Mickelson, L. 109 Mihalache, D. 488, 489, 491,522, 523, 528, 555, 556, 558, 561-563 Milburn, G.J. 366, 447, 451,452, 456, 462, 464, 472 Millar, P. 547 Miller, L.R. 457 Miller, P.D. 550 Milster, T.D. 108, 173 Minasian, R.A. 350 Minford, M. 309 Miranowicz, A. 364, 375, 425, 452 Mi~ta Jr, L. 365, 366, 380, 382, 384, 387, 392-395 Mitsunaga, N. 363 Miuram, K. 63 Miyagawa, N. 148, 172 Miyaoka, Y. 171 Miyata, K. 139 Mlodinow, L. 458, 460, 461 Mlynek, J. 364 Modine, EA. 196, 270 Mogilevtsev, D. 365, 412-414, 439 Mohideen, U. 52, 53, 68 Molmer, K. 463, 464 Molony, A. 313 Monguzzi, A. 364 Monken, C.H. 436, 465 Monroe, C. 464
580
AUTHOR INDEX FOR VOLUME 41
Monsay, E.H. 303 Monsma, M.J. 311 Montecchi, M. 266 Moon, H.-J. 34 Moon, S.Y. 252 Moore, EL. 459, 460, 463 Morandotti, R. 547 Morey, W.W. 313 Moil, T. 245 Morita, S. 117, 172 Moroga, K. 130 Morse, P.M. 373 Moss, G.E. 457 Moss, S.C. 147 Mostofi, A. 364, 366 Mouradyan, N.T. 365 Moussa, M.H.Y. 452 Moy, Y.-P. 257 Mukherjee, S.D. 326 Miiller, R. 497, 498 Munkelwitz, H.R. 71 Murakami, T. 109 Murakami, Y. 145 Murao, N. 172 Miirau, P.C. 246 Muschall, R. 547, 551 Musha, T. 109 Musslimani, Z. 509 Miistecaplio~lu, (~.E. 443, 444 Muto, Y. 142 Myatt, C.J. 464 Myers, L.E. 364 My~ka, R. 454-456, 462, 472 N Nagahama, S. 124 Nagata, K. 150, 151 Nakajima, J. 144 Nakaki, Y. 141, 142 Nakamura, S. 124, 150, 151 Nakaoki, A. 142, 144 Nalamasu, O. 68 Narayan, J. 249 Narayanan, A. 328, 332 Narimanov, E.E. 34, 69 Nassar, T. 453 Navr~til, K. 211, 218, 242, 266 Nevot, L. 215 Newberg, I.L. 305, 317 Newton, R.G. 470
Ng, C.K. 85 Ng, W. 328, 329, 332 Ng, W.W. 305 Ngo, D. 74 Nguyen, H.V. 235 Nguyen, N.V. 244 Nguyen Van Dau, E 274 Nienhuis, G. 445 Nieto, M.M. 425, 432, 451,453, 469, 471 Niihara, T. 136 Nishibori, M. 238 Nishimura, N. 146 Nishiuchi, K. 148, 149 Nishiyama, M. 117, 172 N6ckel, J.U. 21, 34-38, 69 Nogues, G. 464 Noh, J. 462, 463 Noh, J.W. 430, 435, 436, 438, 439 Noh, T.G. 462 Noh, T.-G. 462 Noh, Y.-C. 34 Noirie, L. 365 Norton, D.A. 310 Noz, M.E. 366 N~vlt, M. 273, 274 O O'Connell, R.E 469 Ogawa, K. 108 Ogilvy, J.A. 218 Ohara, S. 101 Ohkubo, S. 149, 154 Ohlidal, I. 211, 215-219, 221,222, 240, 242, 243, 256, 258-267, 269, 276 Ohlidal, M. 211,221,267, 269 Ohno, E. 148, 149, 151, 171 Ohnuki, S. 139, 146 Ohta, H. 172 Ohta, K. 144 Ohta, M. 139 Ohta, N. 136, 139, 146 Ohta, T. 150, 151, 171 Ohtaki, S. 172 Oka, M. 117 Okabayashi, S. 151, 171 Okada, M. 130, 149, 154 Okada, O. 130 Okamuro, A. 139, 142 Okumura, H. 253 Olivik, M. 366, 412
AUTHOR INDEX FOR VOLUME41 Opatrn~,, T. 435, 454-456, 458 Orszag, M. 452 Ortega, B. 314 Osato, K. 165 Osgood Jr, R.M. 302 Osnaghi, S. 464 Otoba, M. 172 Ou, Z.Y. 446, 458, 462, 472 Ovshinsky, S.R. 147, 149 Owen, J.E 40, 48, 49, 53, 56, 68, 70 Owrutsky, J.C. 78 Ozawa, S. 68 P
Pfidua, S. 465 Page, J.B. 551,553 Paneva, A. 240 Pang, H.Y. 252 Papadopoulos, A. 245 Paquet, S. 330 Pardo, B. 215 Parent, M.G. 311, 312 Paris, M.G.A. 444, 456, 457, 461,467, 472 Pa~izek, V. 274 Park, G. 462 Parker, M.R. 209 Pascazio, S. 380, 437, 442, 456 Pasman, J. 115, 161, 163 Passaglia, E. 120, 237, 255 Pasta, J. 547 Pastemack, L. 78 Pastor, D. 314 Paul, H. 375, 433, 450, 451,453, 454, 467, 469 Paulson, W. 253 Payson, P. 315 Peckerar, M.C. 242 Pedinoff, M.E. 239, 244 Pegg, D.T. 425, 433,435, 438, 445-447, 450, 452, 469-471 Pelinovsky, D.E. 488, 530, 531 Pellizzari, T. 464 Pendleton, J.D. 42, 86, 89 Peng, C. 114, 140, 157, 171 Peng, G.D. 364, 366, 544 Peng, G.-D. 491 P6nissard, G. 273 Pennings, E.C.M. 364 Perelomov, A. 429 Perelomov, A.M. 366
581
Peres, A. 433, 434, 445 Pefina, J. 364-366, 370-392, 395-414, 425, 432, 435, 437-440, 442, 446, 448, 455, 456, 458, 460, 461,467, 468, 472 Pefina Jr, J. 365, 366, 370, 376-379, 381383, 387, 389-391,401,404-407 Pefinovfi, V 363, 374, 375, 425, 432, 435, 450, 451,456, 458, 460, 467, 468, 471 Perlmutter, S.H. 461 Pershan, P.S. 133, 485, 487 Pertsch, T. 490, 553 Peschel, T. 490, 533, 537, 539, 542, 547, 549-551,553 Peschel, U. 486-491, 512, 516, 517, 519, 521,522, 525, 528, 530, 532, 533, 537, 539, 542, 549-551,553, 560 Petak, A. 365, 392, 395 Peters, C.W. 485 Petit, R. 160 Petrosyan, K.G. 365 Petrov, D.V 563 Pfister, O. 462, 463 Philippet, D. 297 Phillips, L.S. 452 Physica Scripta T 48 425 Picciau, M. 364 Pickering, C. 246 Pieczonkovfi, A. 405 Pierce, J.W. 364 Pin6ik, E. 269 Pinnick, R.G. 64, 68, 71, 72, 74, 76, 86, 87, 89 Pisarkiewicz, P. 234 Piskarskas, A. 489 Pitaevski, L.P. 207 Pittal, S. 254 Plant, D.V. 302 Plenio, M.B. 464 Pohl, D.W. 173 Pokrowsky, P. 236 Polzik, E.S. 459, 461,463 Pons, C. 109 Poon, K.L. 85, 89 Popov, E. 160 Popp, J. 64-67, 71, 78 Postava, K. 274 Potapov, E.V. 256 Preiss, J. 352, 354 Preist, T.W. 160 Primeau, N. 364
582
AUTHOR INDEX FOR VOLUME 41
Probert-Jones, J.R. 11 Prosser, V. 273 Psaltis, D. 160 Pu, X. 74, 77, 89 Puech, C. 108, 110 Ptmko, N.N. 244 Purcell, E.M. 40, 45 Puri, R.R. 460, 463
Q Qian, L.J. 486, 555 Qian, S.-X. 11, 57, 71, 72, 81, 82 Quiroga-Teixeiro, M. 563 Quyang, Z.-W. 401 R
Raasch, D. 171 Raccah, P.M. 195 Radcliffe, J.M. 429 Radmore, P.M. 366 Rai, J. 363,461 Rai, R. 255, 256 Raimond, J.M. 27, 52, 58 Raimond, J.-M. 6, 41, 52, 53, 58, 59, 464 Rairoux, P. 81 Raizen, M.G. 459 Rajagopal, A.K. 469 Rakov, A.V. 256 Ralston, A. 226 Ramsey, J.M. 46, 49, 50 Ramsey, N.E 459 Rao, C.N.R. 156, 157 Rao, K.J. 156, 157 Raoult, G. 160 Rarity, J.G. 462, 465 Rashid, M.A. 460 Rasmussen, J.J. 555 Rasmussen, T. 363 Rauch, H. 437, 442, 453,455, 456 Rauschenbeutel, A. 464 Ravindran, P. 70 Ray, A.K. 70 Raymer, M.G. 439, 440, 442, 443,467 l~ehfi6ek, J. 366, 380, 382, 384, 392, 437, 442, 456 Reid, M.E 452, 454 Reinisch, R. 364 Renard, D. 273 Rezek, B. 267 Rhee, J.-K. 462 Rheinl~inder, B. 193
Rice, S.O. 215 Richardson, C.B. 21, 22 Richter, T. 439, 440 Rilum, J.H. 115 Ritze, H.-H. 471 Rivory, J. 265 Riza, N.A. 299, 301,302, 319, 352, 354 Robbins, D.J. 246 Robertson, G.N. 63, 68 Rocca, E 448 Roch, J.E 27, 52 Rohrlich, F. 425 Rolfe, S.J. 247 Rosen, H.J. 173 Rosenvold, R. 134 Rosma, M. 429 Rowe, M. 464 Royer, A. 469 Ruane, M. 134 Rubenchik, A.M. 487, 555 Rubin, K.A. 173 Rubin, M.W. 320 Riidiger, A. 457 Ruekgauer, T.E. 64, 68, 74, 86, 87, 89 Rugar, D. 165, 173 Ruschin, S. 364, 460 Russell, M. 352, 354 Ryan, D.H. 132 S Saavedra, C. 452 Sacchi, M.E 446, 452, 456, 468, 470, 472 Sackett, C.A. 464 Saifi, M.A. 244 Saito, H. 463 Saito, J. 139, 142 Saitoh, T. 245 Sakamoto, M. 117 Sakaue, Y. 148 Salamanca-Riba, L. 244 Salam6, S. 460 Saleh, B.E.A. 363,429, 462, 465 Saleheen, H.I. 46 Saletan, E.J. 432 Saltiel, S. 364 Sambles, J.R. 160 Sanchez-Morcillo, V.J. 491 Sfinchez-Soto, L.L. 375, 435, 444-446, 450, 452, 470 Sanders, B.C. 447, 451,452, 456, 462, 472
AUTHOR INDEX FOR VOLUME 41 Sandoghdar, V. 6, 16, 41, 52, 53, 58, 59 Sangarpaul, A. 488 Santamaura, M. 453 Santhanam, T.S. 445-447 Sasaki, K. 67, 68 Sato, M. 118, 139 Satoh, I. 101, 151 Scarlat, D. 364 Schamschula, M. 301 Schaschek, K. 31, 32, 71, 76, 77 Schaub, S.A. 16, 72 Schiek, R. 485,487, 489, 497, 498, 516, 536, 542 Schiffer, R. 217 Schiffrin, D.J. 364 Schiller, S. 10, 364 Schilling, M.L. 68 Schilling, R. 457 Schiortino, P. 68 Schleich, W. 378, 456, 471,472 Schleich, W.P. 439, 453, 460, 467-469 Schmalzbauer, K. 215 Schmidt, E. 240, 264, 265 Schnupp, L. 457 Schoemaker, D. 457 Schori, C. 463 Schubert, M. 193, 273, 363 Schweiger, G. 70 Schwinger, J. 427 Scott, B.A. 166 Scott, D.C. 302 Sculley, M. 41, 42, 70 Scully, M.O. 366, 459, 463 Sczaniecki, L. 366 Seaton, C.T. 363 Segala, D. 364 Semenenko, A.I. 237 Semenenko, L.V. 237 Senesi, E 364 Seno, R. 472 Senoh, M. 124 Sergienko, A.V. 447 Serpengiizel, A. 16, 26, 59, 61-63, 72, 74, 75 Seydel, R. 544 Shapiro, J.H. 425, 434, 451,453, 455, 456, 461,467, 468, 470 Sheik-Bahae, M. 486, 542 Shelburne III, J.A. 363 Shelby, R.M. 461
583
Sheldon, B. 224, 226 Shen, Y.R. 363 Shepard, S.R. 425, 434, 451,455, 456, 470 Shi, Y. 334 Shieh, H.-P.D. 173 Shigematsu, K. 101 Shih, H.-E 174 Shimouma, T. 142 Shin, J. 462 Shirai, H. 139, 146 Shiratori, T. 171 Shono, K. 144 Shuker, R. 71 Shumovsky, A.S. 435, 440, 443, 444 Shvets, V.A. 257 Sibbett, W. 547 Sibilia, C. 363-365, 390, 392, 395, 542 Siegel, S. 323 Sikkens, M. 233 Silberberg, Y. 547, 554 Silver, S. 221 Simondet, E 246 Simonis, G.J. 302 Sinatra, A. 454 Singh, R.P. 454 Sipe, J.E. 531,533, 535 Sirtori, C. 69 Sirugue, M. 448 Sivco, D.L. 34, 69 Sizmann, A. 395 Skagerstam, B.-S. 366 Skinner, I. 544 Skinner, I.M. 366 Skryabin, D. 559, 563 Slusher, R.E. 52, 53, 68, 461 Smithey, D.T. 440 Snow, J.B. 57, 71, 81, 82 Snyder, P.G. 254 Sobota, J. 254 S6derholm, J. 446, 447, 458, 464, 465 Sohler, W. 489, 542 Sokolov, A.V. 207 Sokolov, V.K. 237 Soldano, L.B. 364 Solimeno, S. 363 Sols, E 454 Soref, R. 323 Soref, R.A. 306, 310, 315 Sorensen, A. 463, 464 Sorensen, J.L. 459, 461,463
584
AUTHOR INDEX FOR VOLUME 41
Spizzichino, A. 218 Spock, D.E. 85 Sprokel, G.L. 133 Spruit, J.H.M. 137 Srinivasan, R. 255, 256 Srivastava, V. 71, 74, 86 Stabinis, A. 365 Stafsudd, O.M. 239, 244 Staliunas, K. 491 Steblina, V. 556, 557 Steblina, V.V. 559 Stegeman, G. 542 Stegeman, G.I. 363,485-489, 516, 536, 542, 547 Steier, W.H. 302 Steinbach, J. 464 Steinberg, H.L. 120, 237, 255 Stenholm, S. 467, 469 Stenkamp, B. 236 Sterpi, N. 464 Steuemagel, O. 444 Stevenson, A.J. 457 Stilwell, D. 311 Stinson, D. 99, 100 Stone, A.D. 21, 34-38, 69 Strand, T.C. 106 Strand, T.S. 173 Stratton, J.A. 7 Streed, E.W. 41, 52, 53 Striccoli, M. 69 Stromberg, R.R. 120, 237, 255 Studenmund, W.R. 165 Studna, A.A. 267 Sueta, T. 293 Sugaya, S. 119 Sugimoto, Y. 124 Sukhorukov, A.P. 487, 515 Sullivan, C.T. 326 Sumi, S. 139 Summhammer, J. 453 Sun, J.W. 252 Susskind, L. 451 Suzuki, K. 142 Svitashov, K.K. 237, 257 Swain, S. 452 Swindal, J.C. 26, 27, 60, 76 Sypek, M. 515 Szabo, A. 63 Szczyrbowski, J. 215 Szlachetka, P. 401
T Taguchi, M. 141 Takahashi, A. 144, 145 Takahashi, M. 136 Takeda, K. 58, 60, 63 Takeda, M. 117 Takenaga, M. 101 Tamir, T. 168 Tanaka, K. 63 Tana~, R. 375, 425, 428, 435, 445, 450, 452, 470 Tanev, S. 364 Tang, I.N. 71 Tang, S. 335 Tangonan, G.L. 305, 306 Tani, J.L. 460 Taniguchi, H. 64, 66 Tanosaki, S. 64, 66 Tapster, P.R. 462, 465 Tarasenko, A.A. 254 Tasgal, R.S. 366 Tatsuki, K. 117 Taubman, M.S. 364 Tauc, J. 147, 156, 157 Taylor, R. 345 Taylor, T.D. 20 Teich, M.C. 363, 429, 462, 465 Tekumalla, A.R. 446 Terashima, S. 145 Terris, B.D. 165, 173 Tetu, M. 330 Theeten, J.B. 214, 246, 250, 261 Thomas, H. 429, 431 Thorsten, N. 309 Thurn, R. 70 Tiberio, R.C. 52, 68 Tirnko, A. 68 Toba, H. 118 Tokuhara, S. 118 Tokunaga, T. 141, 142 Tomisawa, H. 66 Tong, D.K.T. 346, 353 Tong, S.S. 19 Torazawa, K. 139 Torelli, I. 364 Toren, M. 370 Torgerson, J.R. 436, 439, 469 Tomer, L. 485-489, 491,511,522, 523, 529, 555, 556, 558, 561-563 Torres, J.P. 563
AUTHOR INDEX FOR VOLUME 41 Torruellas, W .E. 489 Torruellas, W.E. 485, 487, 489, 491 Toughlian, E.N. 303, 305, 315 Townsend, P.D. 363 Tran, H.T. 490 Trautman, J.K. 173 Treussart, E 27, 52, 58, 59 Trifonov, A. 447, 458, 464, 465 Trifonov, D.A. 447, 460 Trillo, A. 489, 506 Trillo, S. 489-491,512, 517, 528, 533-537, 539-541,547, 551 Trinh, P.D. 330 Trommer, G. 497, 498 Truong, V.-V. 265 Tsai, S.-T. 173 Tsap, B. 317 Tsegaye, T. 447, 458, 464, 465 Tsui, Y.K. 452, 454 Tsui, Y.-K. 453 Tsuji, H. 138, 139, 142 Tsujita, K. 64, 66 Tsukane, N. 118 Tsunashima, S. 138, 139, 142, 146 Tsutsumi, K. 141, 142 Tu, C.W. 52, 68, 69 Turchette, Q.A. 464 Turitsyn, S.K. 555 Turlet, J.M. 270, 272 Turpin, T.M. 302, 320 Tzeng, H.M. 23, 56, 57, 63, 85 Tzeng, H.-M. 57 U Uchida, M. 150, 151 Uchiyama, S. 138, 139, 142 Ueda, M. 431,459, 460, 463 Uffink, J.M.B. 458 Ukita, H. 109 Ulam, S. 547 Umarov, B.A. 365, 395 Urban, R. 273 Urbansky, K.E. 452 Umer-Wille, M. 127 Usami, S. 101 V Vaccaro, J.A. 433, 447, 450, 451,454, 469, 470 Vaglica, A. 428, 453
585
Vaidman, L. 458 Vajda, S. 81 Vakhitov, M.G. 519, 558, 559 Valiulis, G. 486, 489 Valley, J.E 461 van der Pauw, L.J. 135 Van Enk, S.J. 445 van Kampen, N.G. 211 van Kesteren, H.W. 137 van Rosmalen, G. 104 van Silfhout, A. 193, 276 Van Stryland, E. 542 Van Stryland, E.W. 485-487, 489 Vanherzeele, H. 486 Vaw A. 199, 229, 231 Vedam, K. 248, 249, 255, 256, 276 Veisman, M.E. 364 Vernooy, D.W. 41, 52, 53 Vetri, G. 428, 453 Vezin, B. 81 Vidakovi6, P. 365 Vilaseca, R. 563 Villeneuve, A. 547 Vi~fiovsk~, S. 208, 273,274 Vitrant, G. 364 Vi~_da, E 221 Vlieger, J. 217 Vogel, K. 453,469 Vogel, W. 366, 425, 430, 439, 466, 471,472 Voigt, W. 209 Vonsovskii, S.V. 132 Vourdas, A. 445 Vouroutzis, N. 247 W
Wabnitz, S. 489, 534, 547, 551 Wada, Y. 117 Wagner, S.S. 453, 467, 468 Wahiddin, M.R.B. 365, 395 Waiterson, R. 326 Wakabayashi, T. 463 Walker, N.G. 436, 440, 467, 469 Walkup, J.E 440, 442 Wall, J.E 259 Wall, K.E 56, 57 Wallentowitz, S. 439 Walls, D.E 366, 460-463 Walston, A. 328 Walston, A.A. 305 Wang, D.-S. 22, 67
586
AUTHOR INDEX FOR VOLUME 41
Wang, H. 233 Wang, J.-K. 174 Wang, L.J. 462 Wang, M.S. 108 Wang, W. 334 Wang, Y.Z. 49, 58 Wang, Z. 485, 487, 489 Wanuga, S. 309 Watabe, A. 109 Watanabe, K. 139 Watson, J. 309 Weber, J. 266 Wechsberg, M. 306 Wedding, K. 440 Wei, Y. 238 Weinert-Raczka, E. 363 Weinreich, G. 485 Weiss, D.S. 6, 41, 52, 53 Welsch, D.-G. 366, 425, 430, 466 Wemer, M.J. 488, 516, 526, 528 Wettling, W. 207 Wharton, J.J. 233 Wheeler, A. 378 Whipple, B. 102, 171 White, A.G. 364 Whitten, W.B. 46, 49, 50 Wijn, J.M. 115-117 Wilf, H.S. 226 Wilhelmi, B. 363 Wilkens, M. 445 Wilkinson, J.S. 364 Wilkinson, R. 115 Wilson, R. 364 Wind, M.M. 217 Windenberger, C. 364 Wineland, D.J. 459, 460, 463, 464 Winkler, W. 457 Winterbottom, A.B. 255 Wise, EW. 486, 555 Wiseman, H.M. 471 Witter, K. 127 W6dkiewicz, K. 437, 439, 460, 469 Woerlee, G.E 240, 241 Wolf, E. 205, 207, 217, 366, 425, 426, 429, 436, 466 Wolf, J.P. 81 Wolfe, R. 173 Wolifiski, T.R. 364 Wolniansky, P. 134 Wong, N.C. 451,455
Wong, T. 462 Wood, C.E 76 Wood, E.L. 160 Wood, R.W. 166 Woollam, J.A. 193, 248, 252, 253 Wootters, W.K. 437 W6ste, L. 81 Wreszinski, W.E 432 Wright, E.M. 363 Wu, H. 461 Wu, L. 335 Wu, L.-A. 461 Wu, M. 302 Wu, M.C. 346, 353 Wu, Q.H. 233 Wu, S. 52, 53, 68, 444 Wu, S.L. 52, 68 Wu, Y. 143 Wyller, J. 555 X Xiao, M. 461,468 Xie, J.-G. 64, 68, 74, 86, 87, 89 Xie, K. 488, 510 Xiong, Y.-M. 245 Xu, L. 345 Y Yablinovich, E. 51 Yakovlev, V.A. 258, 259 Yamada, N. 149, 151, 171, 172 Yamada, T. 124 Yamaguchi, A. 139 Yamaguchi, T. 253 Yamamoto, M. 109, 117 Yamamoto, Y. 465 Yamatsu, H. 117 Yan, Z. 120 Yang, J. 531 Yang, T.-M. 173 Yao, X.S. 302 Yap, D. 328, 332 Yariv, A. 28, 363 Yasuda, H. 58, 60 Yasumoto, K. 363 Yegnanarayanan, S. 330 Yeh, C. 21,22 Yeh, P. 205, 363 Yeh, W.-H. 114, 159, 166, 170 Yen, H.W. 306
AUTHOR INDEX FOR VOLUME41 Yeong, K.C. 364 Yokoyama, H. 40, 47, 52 Yoshida, S. 253 Yoshida, Y. 144 Yoshioka, K. 150, 151 Young, K. 4, 10, 11, 19-21, 24-28, 32, 40, 41, 44, 46, 48-50, 53, 54, 82, 84, 85, 89 Young, Y.E. 463 Yu, S. 444, 446, 466, 470 Yuen, H.P. 433, 434, 437, 464, 467 Yurakami, Y. 144 Yuratich, M.A. 82 Yurke, B. 429, 456, 457, 461,462 Yussof, B.M.N. 143 Z Zaji6kovfi, L. 276 Zakharov, V.E. 555 Zambuto, J.J. 109 Zavislan, J.M. 106 Zawisky, M. 437, 442, 453, 455, 456 Z~boulon, A. 364 Zeper, W.B. 137
Zetterer, T. 236 Zhang, J.P. 52, 68 Zhang, J.-P. 52, 53, 68 Zhang, J.-Z. 73, 74, 82-85, 89 Zhang, W. 464 Zhang, W.M. 366 Zhang, Y. 444 Zhang, Z. 447, 452, 456 Zhao, C. 259, 340 Zheng, J.-B. 74-76 Zheng, S.-B. 464 Zhou, C. 341,342 Zhou, EL. 134 Zhu, R. 238 Zmuda, H. 303, 305, 315 Zoller, P. 460, 463, 464 Zomp, J. 347 Zomp, J.M. 307 Zou, M. 462 Zou, X.Y. 462 Zubairy, M.S. 366 Zurek, W.H. 437 Zvezdin, A.K. 210, 273
587
SUBJECT
INDEX
FOR
A acousto-optic Bragg cell 298 - - true-time delay line 302 Airy function 10 amorphization kinetics 156 angular momentum operator 427 anti-Stokes field 410 antibunching of photons 370 astigmatic lens method 108 atomic coherent state 429 autocorrelation function 211 - length 212
VOLUME
41
-, CD-ReWritable drive 99-101, 110 -, CD-ROM 99-102, 110, 111 chirp grating 315 chromatic dispersion 311 coherent state 388, 390, 391,430, 436, 453, 454, 463 colloid chemistry 3 complementarity 446 correlated-emission laser 452, 463 coupler, codirectional 384, 386, 387, 392 -, contradirectional 380, 390 -, directional 487 -, nonlinear 542 -, waveguide 503 cross-correlation coefficient 213 - spectral density 212 crystallization 156 Curie temperature 141
B
bandgap coupler 412 beam squint 289 bi-prism method 106 biaxial crystal 496 birefringence 119, 120 Bose-Einstein condensate 435, 445, 463,464 Bragg cell 302, 303 - condition 502 - grating 490, 509 - - mirror 321 - reflection grating 313 - waveguide 491,500, 536, 564 Brillouin coupler 411 -scattering 366, 401,402, 405, 415 - - , stimulated 82 Bruggeman formula 214, 232, 246, 261
D
difference frequency generation 485 differential phase-detection method 113 Digital Versatile Disk (DVD) 100-102, 110, 113, 173 distributed feedback lasers 302 Dolph-Chebyshev weighting 286 down-conversion 291,486, 487, 489, 490 - - , parametric 472 Drude approximation 231 - model 246 E
C caustic region 5 cavity, leaky 19, 44 -quantum electrodynamics 4, 40, 47, 48 CD, audio 99 -, CD-Recordable drive 99-101, 110
Einstein A coefficient 46, 53 - B coefficient 46, 56 electro-optic modulator 322 - - polymer 322, 326 - - switch 335 ellipsometer 120-122, 126 589
590
SUBJECT INDEX FOR VOLUME 41
ellipsometry 183, 188, 235, 237-250, 252 -, multiple angle of incidence 239-244, 250255, 257-259, 261,263, 267, 268, 270, 275, 277 - , n u l l 189, 192 -, oscillating analyzer 195 -, phase modulated 193 -, principles of 187-189 -, return-path 196 -, rotating analyzer 192, 193, 195 , compensator 195 -, spectroscopic 244, 265 -, theory of measurements in 189-196 -, two-modulator generalized 196 Euler's angles 207 evanescent wave 363 F
Fabry-Perot cavity 5, 34, 456, 457 interferometer 429, 456 Faraday effect 210, 273 Fermi's golden rule 44, 45 fiber, polarization maintaining 346 -, step index multimode 324 amplifier, erbium doped 341,344 - delay line 307-310 --prism 312 grating prism 315 optic communication 336 - transmission line 311 fluctuation-dissipation theorem 368 Fourier optics 291 Fresnel coefficient 188, 196, 198, 200, 216, 219, 220, 222, 226, 231 fuel dynamics 3 - -
G geometric optics 5, 106 Glauber-Sudarshan quasidistribution 372, 374, 411 Global Broadcasting System (GBS) 287 grating, theory of 160 H
Hall resistivity 135 - voltage 135 Hankel function 20 Heisenberg equation 376, 385, 396 Heisenberg-Langevin equation 368 Helmholtz wave equation 17
heterodyne detection holographic grating homodyne detection - - , eight-port 435, - - , many-port 440 - - , six-port 444 detector 430
467 337 436, 439, 471 448, 452, 466
J
Jaynes-Cummings model 443 Jones formalism 184 - m a t r i x 185, 186, 191, 192 - vector 184, 185, 190, 191 Josephson junction 449, 453 K
Kerr effect 136, 210, 273, 365 , couplers based on 395 medium 414 - nonlinearity 52, 486, 555 rotation 142 - signal 132, 145 Kirchhoff approximation 218 Kolmogorov-Arnold-Moser theory Kramers-Kronig relations 254 L Lagrangian 520 Laguerre polynomials 373 Langmuir-Blodgett film 271 laser, blue diode 170 -, erbium-doped fiber ring laser -, microcavity 68 -, microring 68 -, photonic wire 68 -, Q-switched 73 - gyroscope 423 Lorentz-Lorenz formula 214 Lorenz-Mie formalism 3
4, 35
311
M
Mach-Zehnder interferometer 293, 429, 456, 487 - - modulator 311,313, 334, 345 magneto-optical constant 209 - disk 100 - - effect 209, 273 - - loop tracer 134 - - recording 127-147 - - tensor 207, 208
SUBJECT INDEX FOR VOLUME 41 magnetometer, vibrating sample 133 Mandel photodetection formula 372 Maxwell equations 7, 168, 196, 201 - -Garnett formula 214 Michelson interferometer 429, 456 microscopy, polarized light 135-137 microsphere 5, 13 -, cavity modes of 4 , modified optical processes in 40-53 , QED effects in 48 -, dispersive optical bistability 27 -, dye-doped 67 -, fluorescence and lasing in 53 -, laser 56, 59, 63 -, stimulated Raman scattering in 50 - resonance 20 modulational instability 489, 506 N
Neumann function 29 neutron interferometry 455 nonlinear waveguide 363 numerical aperture 102, 117, 168 O Onsager relation 208 optical amplifier, semiconductor - bistability, absorptive 31 - computing 416 - coupler 363, 364 - data storage 148, 149, 170 - disk reader 99 - fiber 52 - - amplifier 343 - - true-time delay line 304 - parametric process 365 , nondegenerate 364, 392 - phase-shifter 293, 354 - waveguide 293, 324 , silica based 329 optoelectronics 363
591
phased array antenna 285-289, 302-304, 315, 334, 354 , multiwavelength optical-controlled 346 , photonic technology in 290-297 plasmon, surface 166 Poincar6 sphere 320, 426, 427, 432, 453,460 - surface 36 Poisson bracket 432 polarization transfer function 185 pupil obscuration method 106 push-pull method 110, 163
Q Q function 450, 453, 469 Q-switching 63 quadrature component 374 - operator 423 quantum beat laser 452 - clock 445 - nondemolition measurement 463 - p h a s e 423, 431,467 - - difference 431,436 quarter-wave compensator 190 --plate 103 quasi-normal modes 19
344 R
Raman coupler 410, 411 gain 71 - scattering 40, 42, 365, 401,402, 407, 415 , coherent anti-Stokes 42, 81 , spontaneous 70, 77 , stimulated 64, 71, 81 Ramsey method 459 ray optics 38 Rayleigh-Rice theory 215 Ricatti-Bessel functions 8 Ricatti-Hankel functions 8 ring-toric lens method 109 -
S P
parametric approximation 367, 377, 381,393 Peierls-Nabarro barrier 552 - - potential 552 phase difference operator 444 --state 451 - m a t c h i n g 3, 388, 390 - - , quasi 364 - operator 448
sample-servo method 112 saturable absorber 67 Schr6dinger-cat state 365, 414, 415 - equation, nonlinear 487, 516 second-harmonic generator 375, 485 semiconductor laser 52 soliton 486, 512, 518, 519, 524, 525, 531, 542, 547, 548, 551,553 -, Bragg 490, 533, 534, 537, 540
592
SUBJECT INDEX FOR VOLUME 4t
soliton (cont'd) -, in planar waveguide 510 -, multidimensional 554 -, quantum gap 490 -, spatio-temporal 485, 562 -, stability of 516 squeezed light 377, 378, 386, 391,394, 400, 404, 407, 408, 413,414 state, phase 383, 392 vacuum state 472 stationary point 367 Stokes field 410 - operator 425,427, 429, 431,434, 441,443, 451,463, 473 , polar decomposition of 444, 469 - parameter 425, 426 , phase difference from 435 Stratton-Chu-Silver integral 221 sub-Poissonian light 377, 386, 394, 407 statistics 370, 379, 382, 385, 394, 404, 405, 407, 412, 413,415 sum-frequency generation 79, 485 super-Poissonian statistics 382, 407
Susskind-Glogower phase operator ---state 451,469, 470, 472
451,453
T tapered-Taylor weighting 286 thermomagnetic recording 129 thin film 183, 196, 211, 216-218, 233, 234 T-matrix 22, 32, 33
-
-
U up-conversion
486, 487
V vacuum fluctuations 370 variational approach 520 W
waveguide array 505 wavelength division multiplexing 347, 350 wax-wane method 108 whispering-gallery modes 3 Wigner-Weisskopf approximation 368 WKBJ method 227
CONTENTS O F P R E V I O U S
VOLUMES*
VOLUME 1 (1961) I II III
The Modern Development of Hamiltonian Optics, R.J. PEGIS Wave Optics and Geometrical Optics in Optical Design, K. MIYAMOTO The Intensity Distribution and Total Illumination of Aberration-Free Diffraction Images, R. BARAKAT IV Light and Information, D. GABOR V On Basic Analogies and Principal Differences between Optical and Electronic Information, H. WOLTER VI Interference Color, H. KUBOTA VII Dynamic Characteristics of Visual Processes, A. FIORENTINI VIII Modern Alignment Devices, A.C.S. VAN HEEL
1- 29 31- 66 67-108 109-153 155-210 211-251 253-288 289-329
VOLUME 2 (1963) I
Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy, G.W. STROKE II The Metrological Applications of Diffraction Gratings, J.M. Bt;RCH III Diffusion Through Non-Uniform Media, R.G. GIOVANELLI IV Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering, J. TSUJIUCHI V Fluctuations of Light Beams, L. MANDEL VI Methods for Determining Optical Parameters of Thin Films, E ABEL'S
1- 72 73-108 109-129 131-180 181-248 249-288
VOLUME 3 (1964) I II III
The Elements of Radiative Transfer, E KOTTLER Apodisation, P. JACQUrNOT,B. ROIZEN-DOSSlER Matrix Treatment of Partial Coherence, H. GAMO
1- 28 29-186 187-332
VOLUME 4 (1965) I II III IV
Higher Order Aberration Theory, J. FOCKE Applications of Shearing Interferometry, O. BRYNGDAHL Surface Deterioration of Optical Glasses, K. KINOSITA Optical Constants of Thin Films, P. ROUARD,P. BOUSQUET
* Volumes I-XL were previously distinguished by roman rather than by arabic numerals. 593
1- 36 37- 83 85-143 145-197
594
CONTENTS OF PREVIOUSVOLUMES
V The Miyamoto-Wolf Diffraction Wave, A. RUB1NOWICZ VI Aberration Theory of Gratings and Grating Mountings, W.T. WELFORD VII Diffraction at a Black Screen, Part I: Kirchhoff's Theory, E KOTTLER
199-240 241-280 281-314
VOLUME 5 (1966) I II III IV V VI
Optical Pumping, C. CormN-TnyyotmJi, A. KASTLER Non-Linear Optics, P.S. PERSnAY Two-Beam Interferometry, W.H. STEEL Instruments for the Measuring of Optical Transfer Functions, K. Mtn~TA Light Reflection from Films of Continuously Varying Refractive Index, R. JACOBSSON X-Ray Crystal-Structure Determination as a Branch of Physical Optics, H. LIPSON, C.A. TAYLOR VII The Wave of a Moving Classical Electron, J. PICHT
1-- 81 83--144 145--197 199--245 247-286 287--350 351-370
VOLUME 6 (1967) I II III IV V VI
Recent Advances in Holography, E.N. LEITH, J. UPATNIEKS Scattering of Light by Rough Surfaces, P. BECKMANN Measurement of the Second Order Degree of Coherence, M. FRANqON, S. MALLICK Design of Zoom Lenses, K. YAMAJI Some Applications of Lasers to Interferometry, D.R. HERRIOT Experimental Studies of Intensity Fluctuations in Lasers, J.A. ARMSTRONg, A.W. SMITH VII Fourier Spectroscopy, G.A. VANASSE,H. SAr~AI VIII Diffraction at a Black Screen, Part II: Electromagnetic Theory, E KOaq'LER
1- 52 53- 69 71-104 105-170 171-209 211-257 259-330 331-377
VOLUME 7 (1969) I
II III IV V VI VII
Multiple-Beam Interference and Natural Modes in Open Resonators, G. KOPPELMAN Methods of Synthesis for Dielectric Multilayer Filters, E. DELANO,R.J. I~GIS Echoes at Optical Frequencies, I.D. ABELLA Image Formation with Partially Coherent Light, B.J. THOMPSON Quasi-Classical Theory of Laser Radiation, A.L. MIKAELIAY,M.L. TER-MII~LIAN The Photographic Image, S. Ootm Interaction of Very Intense Light with Free Electrons, J.H. EBEm~Y
1- 66 67-137 139-168 169-230 231-297 299-358 359-415
VOLUME 8 (1970) I II III IV V VI
Synthetic-Aperture Optics, J.W. GOOOMAN The Optical Performance of the Human Eye, G.A. FRY Light Beating Spectroscopy, H.Z. CUMMINS,H.L. SWINNEY Multilayer Antireflection Coatings, A. MUSSET, A. THELEN Statistical Properties of Laser Light, H. RaSKEN Coherence Theory of Source-Size Compensation in Interference Microscopy, T. YAMAMOTO VII Vision in Communication, L. LEVI VIII Theory of Photoelectron Counting, C.L. MEHTA
1- 50 51-131 133-200 201-237 239-294 295-341 343-372 373-440
CONTENTS OF PREVIOUS VOLUMES
595
VOLUME 9 (1971) I 11 III IV V VI VII
Gas Lasers and their Application to Precise Length Measurements, A.L. BLOOM Picosecond Laser Pulses, A.J. DEMARIA Optical Propagation Through the Turbulent Atmosphere, J.W. STROHBEHN Synthesis of Optical Birefringent Networks, E.O. AMMANN Mode Locking in Gas Lasers, L. ALLEN, D.G.C. JONES Crystal Optics with Spatial Dispersion, V.M. AGRANOVICH,V.L. GrNZBURG Applications of Optical Methods in the Diffraction Theory of Elastic Waves, K. GNIADEK,J. PETYKIEWICZ VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions, B.R. FRIEDEN
1- 30 31- 71 73-122 123-177 179-234 235-280 281-310 311-407
VOLUME 10 (1972) I II III IV V VI VII
Bandwidth Compression of Optical Images, T.S. HUAN6 The Use of Image Tubes as Shutters, R.W. SMITH Tools of Theoretical Quantum Optics, M.O. SCULLY, K.G. WHITNEY Field Correctors for Astronomical Telescopes, C.G. WYNNE Optical Absorption Strength of Defects in Insulators, D.Y. SMITH, D.L. DEXTER Elastooptic Light Modulation and Deflection, E.K. SITTIG Quantum Detection Theory, C.W. HELSTROM
1- 44 45- 87 89-135 137-164 165-228 229-288 289-369
VOLUME 11 (1973) I II III IV V VI VII
Master Equation Methods in Quantum Optics, G.S. AG~a~WAL Recent Developments in Far Infrared Spectroscopic Techniques, H. YOSHINAGA Interaction of Light and Acoustic Surface Waves, E.G. LEAN Evanescent Waves in Optical Imaging, O. BRYNCI)Am~ Production of Electron Probes Using a Field Emission Source, A.V. CREWE Hamiltonian Theory of Beam Mode Propagation, J.A. ARNAUD Gradient Index Lenses, E.W. MARCHAND
1- 76 77-122 123-166 167-221 223-246 247-304 305-337
VOLUME 12 (1974) I
Self-Focusing, Self-Trapping, and Self-Phase Modulation of Laser Beams, O. SVELTO
II III IV V VI
Self-Induced Transparency, R.E. SLUSHER Modulation Techniques in Spectrometry, M. HARWIT, J.A. DECKERJR Interaction of Light with Monomolecular Dye Layers, K.H. DREXHAGE The Phase Transition Concept and Coherence in Atomic Emission, R. GRAHAM Beam-Foil Spectroscopy, S. BASHION
I
On the Validity of Kirchhoff's Law of Heat Radiation for a Body in a Nonequilibrium Environment, H.P. BALTES The Case For and Against Semiclassical Radiation Theory, L. MANDEL Objective and Subjective Spherical Aberration Measurements of the Human Eye, W.M. ROSENBLUM,J.L. CHRISTENSEN Interferometric Testing of Smooth Surfaces, G. SCHULZ,J. SCHWIDER
1- 51 53-100 101-162 163-232 233-286 287-344
VOLUME 13 (1976)
II III IV
1- 25 27- 68 69- 91 93-167
596 V VI
CONTENTS OF PREVIOUSVOLUMES Self-Focusing of Laser Beams in Plasmas and Semiconductors, M.S. SODHA, A.K. GHATAK,V.K. TRIPATHI Aplanatism and Isoplanatism, W.T. WELFORD
169--265 267--292
VOLUME 14 (1976) I II III IV V VI VII
The Statistics of Speckle Patterns, J.C. DAINTY High-Resolution Techniques in Optical Astronomy, A. LA~EVRIE Relaxation Phenomena in Rare-Earth Luminescence, L.A. RaSEBERG,M.J. WEBER The Ultrafast Optical Kerr Shutter, M.A. DUGUAV Holographic Diffraction Gratings, G. SCHMAHL,D. RUDOLPH Photoemission, P.J. VERNmR Optical Fibre Waveguides- A Review, P.J.B. CLARRICOATS
1- 46 47- 87 89-159 161-193 195-244 245-325 327--402
VOLUME 15 (1977) I II III IV V
1-75 Theory of Optical Parametric Amplification and Oscillation, W. BRUNNER,H. PAUL 77-137 Optical Properties of Thin Metal Films, P. ROUARD,A. MEESSEN 139-185 Projection-Type Holography, T. OKOSHI 187-244 Quasi-Optical Techniques of Radio Astronomy, T.W COLE Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media, 245-350 J. VAN KRANENDONK,J.E. SIPE
VOLUME 16 (1978) I Laser Selective Photophysics and Photochemistry, V.S. LETOKHOV II Recent Advances in Phase Profiles Generation, J.J. CLAIR,C.I. ABITBOL III Computer-Generated Holograms: Techniques and Applications, W.-H. LEE IV Speckle Interferometry, A.E. ENNOS V Deformation Invariant, Space-Variant Optical Pattern Recognition, D. CASASENT, D. PSALTIS VI Light Emission From High-Current Surface-Spark Discharges, R.E. BEVERLYIII VII Semiclassical Radiation Theory Within a Quantum-Mechanical Framework, I.R. SENITZKY
1- 69 71-117 119-232 233-288 289-356 357--411 413--448
VOLUME 17 (1980) I II III
Heterodyne Holographic Interferometry, R. D.~dqDLIKER Doppler-Free Multiphoton Spectroscopy, E. GtACOBrNO,B. CAGNAC The Mutual Dependence Between Coherence Properties of Light and Nonlinear Optical Processes, M. SCHUBERT,B. WILHELMI IV Michelson Stellar Interferometry, W.J. TANGO,R.Q. TWlSS V Self-Focusing Media with Variable Index of Refraction, A.L. MIKAELIAN
1-- 84 85--161
163--238 239--277 279--345
VOLUME 18 (1980) Graded Index Optical Waveguides: A Review, A. GHATAK,K. THYAGARAJAN Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media, J. PE~aNA
1-126 127-203
CONTENTS OF PREVIOUS VOLUMES
Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium, VI. TATARSKII,V..U. ZAVOROTNYI IV Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns, M.V BERRY, C. UPSTILL
597
III
204-256 257-346
VOLUME 19 (1981) I
III
Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence, B.R. MOLLOW Surface and Size Effects on the Light Scattering Spectra of Solids, D.L. MILLS, K.R. SUBBASWAMY Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids,
IV V
Principles of Optical Data-Processing, H.J. BUTTERWECK The Effects of Atmospheric Turbulence in Optical Astronomy, E RODDIER
II
S. USHIODA
1-43 45-137 139-210 211-280 281-376
VOLUME 20 (1983) I II
Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects, G. COURT~S,P. CRUVELLIER,M. DETAILLE, M. SA~SSE Shaping and Analysis of Picosecond Light Pulses, C. FROEHLY,B. COLOMBEAU, M. VAMPOUILLE
III IV V
Multi-Photon Scattering Molecular Spectroscopy, S. KIELICH Colour Holography, P. HARIHARAN Generation of Tunable Coherent Vacuum-Ultraviolet Radiation, W. JAMROZ, B.P. STOICHEFF
1-61 63-153 155-261 263-324 325-380
VOLUME 21 (1984) I II II! IV V
Rigorous Vector Theories of Diffraction Gratings, D. MAYSTRE 1-- 67 Theory of Optical Bistability, L.A. LUGIATO 69--216 The Radon Transform and its Applications, H.H. BARRETT 217--286 Zone Plate Coded Imaging: Theory and Applications, N.M. CmLIO, D.W. SWEENEY 287--354 Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity, J.C. ENGLUND,R.R. SNAPP, W.C. SCHIEVE 355-428 VOLUME 22 (1985)
I Optical and Electronic Processing of Medical Images, D. MALACARA 1-- 76 II Quantum Fluctuations in Vision, M.A. BOUMAN,W.A. VAN DE GRIND,P. ZUIDEMA 77--144 Ill Spectral and Temporal Fluctuations of Broad-Band Laser Radiation, A.V. MASALOV 145--196 IV Holographic Methods of Plasma Diagnostics, G.V. OSTROVSKAYA,Yu.I. OSTROVSKY 197--270 V Fringe Formations in Deformation and Vibration Measurements using Laser Light, I. YAMAGUCHI 271-340 VI Wave Propagation in Random Media: A Systems Approach, R.L. FANTE 341--398 VOLUME 23 (1986) I II
Analytical Techniques for Multiple Scattering from Rough Surfaces, J.A. DESANTO, G.S. BROWN Paraxial Theory in Optical Design in Terms of Gaussian Brackets, K. TANAKA
1- 62 63-111
598 III IV V
CONTENTS OF PREVIOUSVOLUMES Optical Films Produced by Ion-Based Techniques, P.J. 1VIARTIN,R.P. NETTERfiELD Electron Holography, A. TONOMURA Principles of Optical Processing with Partially Coherent Light, ET.S. Yu
113-182 183-220 221-275
VOLUME 24 (1987) I II III IV V
Micro Fresnel Lenses, H. NISHIHARA,T. SUHARA Dephasing-Induced Coherent Phenomena, L. ROTHBERG Interferometry with Lasers, P. HARIHARAN Unstable Resonator Modes, K.E. OUGHSTUN Information Processing with Spatially Incoherent Light, I. GLASER
1- 37 39-101 103-164 165-387 389-509
VOLUME 25 (1988) I II Ill IV
Dynamical Instabilities and Pulsations in Lasers, N.B. ABRAHAM, P. MANDEL, L.M. NARDUCCI Coherence in Semiconductor Lasers, M. OHTSU, T. TAKO Principles and Design of Optical Arrays, WANG SHAOMIN,L. RONCHI Aspheric Surfaces, G. SCnVLZ
1-190 191-278 279-348 349-415
VOLUME 26 (1988) I Photon Bunching and Antibunching, M.C. TEICH,B.E.A. SALEH II Nonlinear Optics of Liquid Crystals, I.C. KHOO III Single-Longitudinal-Mode Semiconductor Lasers, G.E AGRAWAL IV Rays and Caustics as Physical Objects, Yu.A. KRAVTSOV V Phase-Measurement Interferometry Techniques, K. CREATH
1-104 105-161 163-225 227-348 349-393
VOLUME 27 (1989) I II III
The Self-Imaging Phenomenon and Its Applications, K. PATORSKI Axicons and Meso-Optical Imaging Devices, L.M. SOROKO Nonimaging Optics for Flux Concentration, I.M. BASSETT, W.T. WELFORD, R. WINSTON IV Nonlinear Wave Propagation in Planar Structures, D. MIHALACHE,M. BERTOLOTTI, C. SIBILIA V Generalized Holography with Application to Inverse Scattering and Inverse Source Problems, R.P. PORTER
1-108 109-160 161-226 227-313 315-397
VOLUME 28 (1990) I II
III IV V
1- 86 Digital Holography- Computer-Generated Holograms, O. BRYNGDAHL,E WYROWSKI Quantum Mechanical Limit in Optical Precision Measurement and Communication, Y. YAMAMOTO,S. ]k,IACHIDA, S. SAITO, N. IMOTO, T. YANAGAWA,M. KITAGAWA, 87-179 G. BJORK The Quantum Coherence Properties of Stimulated Raman Scattering, M.G. RAYMER, 181-270 I.A. WALMSLEY 271-359 Advanced Evaluation Techniques in Interferometry, J. SCHWIDER 361--416 Quantum Jumps, R.J. COOK
CONTENTS OF PREVIOUSVOLUMES
599
VOLUME 29 (1991) I
Optical Waveguide Diffraction Gratings: Coupling between Guided Modes, D.G. HALL II Enhanced Backscattering in Optics, Yu.N. BARABANENKOV,Yu.A. KRAVTSOV, V.D. OZRIN, A.I. SAICHEV IlI Generation and Propagation of Ultrashort Optical Pulses, I.P. CHRISTOV IV Triple-Correlation Imaging in Optical Astronomy, G. WEIGELT V Nonlinear Optics in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics, C. FLYTZANIS,E HACHE, M.C. KLEIN, D. RICARD,PH. ROUSSIGNOL
1-- 63 65--197 199--291
293--319 321-411
VOLUME 30 (1992) I
Quantum Fluctuations in Optical Systems, S. RE~q,JAUD,A. HEIDMANN,E. GIACOBINO, C. FABRE 1- 85 II Correlation Holographic and Speckle Interferometry, Yu.I. OSTROVSKY, V.P. SHCHEP1NOV 87-135 III Localization of Waves in Media with One-Dimensional Disorder, V.D. FREILIKHER, S.A. GREDESKUL 137-203 IV Theoretical Foundation of Optical-Soliton Concept in Fibers, Y. KODAMA, A. HASEGAWA 205-259 V Cavity Quantum Optics and the Quantum Measurement Process, P. MEYSTRE 261-355
VOLUME 31 (1993) I II
Atoms in Strong Fields: Photoionization and Chaos, P.W. MILONNI,B. StrNDARAM Light Diffraction by Relief Gratings: A Macroscopic and Microscopic View, E. PoPov III Optical Amplifiers, N.K. DUTTA, J.R. SIMPSON IV Adaptive Multilayer Optical Networks, D. PSALTIS,u QIAO V Optical Atoms, R.J.C. SPREEUW,J.P. WOERDMAN VI Theory of Compton Free Electron Lasers, G. DATTOLI,L. G~AN~VESS~,A. RENIERI, A. TORRE
1-137 139-187 189-226 227-261 263-319 321-412
VOLUME 32 (1993) Guided-Wave Optics on Silicon: Physics, Technology and Status, B.P. PAL II Optical Neural Networks: Architecture, Design and Models, ET.S. Yu III The Theory of Optimal Methods for Localization of Objects in Pictures, L.P. YAROSLAVSKY IV Wave Propagation Theories in Random Media Based on the Path-Integral Approach, M.I. CHARNOTSKII,J. GOZANI,V.I. TATARSKII,V.U. ZAVOROTNY V Radiation by Uniformly Moving Sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. GINZBURG VI Nonlinear Processes in Atoms and in Weakly Relativistic Plasmas, G. MAINFRAY, C. MANUS I
1- 59 61-144 145-201 203-266 267-312 313-361
600
CONTENTS OF PREVIOUSVOLUMES VOLUME 33 (1994)
I
The Imbedding Method in Statistical Boundary-Value Wave Problems, VI. KLYATSKIN II Quantum Statistics of Dissipative Nonlinear Oscillators, V PENNOV.~,A. LUKS III Gap Solitons, C.M. DE STERKE, J.E. SIPE IV Direct Spatial Reconstruction of Optical Phase from Phase-Modulated Images, VI. gLAD, D. MALACARA V Imaging through Turbulence in the Atmosphere, M.J. BERAN, J. Oz-VOGT VI Digital Halftoning: Synthesis of Binary Images, O. BRVNCDAHL,T. SCHEERMESSER, E WYROWSKI
1-127 129-202 203-260 261-317 319-388 389-463
VOLUME 34 (1995) I
Quantum Interference, Superposition States of Light, and Nonclassical Effects, V. BUT.EK,P.L. KNIGHT II Wave Propagation in Inhomogeneous Media: Phase-Shift Approach, L.P. PRESNYAKOV III The Statistics of Dynamic Speckles, T. OKAMOTO,T. ASAKURA IV Scattering of Light from Multilayer Systems with Rough Boundaries, I. OHLiDAL, K. NAVRATIL,M. OHLiDAL V Random Walk and Diffusion-Like Models of Photon Migration in Turbid Media, A.H. GANDJBAKHCHE,G.H. IvVEISS
1-158 159-181 183-248 249-331 333-402
VOLUME 35 (1996) I II
Transverse Patterns in Wide-Aperture Nonlinear Optical Systems, N.N. ROSANOV Optical Spectroscopy of Single Molecules in Solids, M. ORRIT, J. BERNARD, R. BROWN,B. LOUNIS III Interferometric Multispectral Imaging, K. ITOH IV Interferometric Methods for Artwork Diagnostics, D. PAOLETTI, G. SCHIRRIPA SPAGNOLO V Coherent Population Trapping in Laser Spectroscopy, E. ARIMONDO VI Quantum Phase Properties of Nonlinear Optical Phenomena, R. TANA~, A. MIRANOWICZ,Ts. GANTSOG
1- 60 61-144 145-196 197-255 257-354 355-446
VOLUME 36 (1996) I
Nonlinear Propagation of Strong Laser Pulses in Chalcogenide Glass Films, V. CHUMASH,I. COJOCARU,E. FAZIO,E MICHELOTTI,M. BERTOLO'lq'I II Quantum Phenomena in Optical Interferometry, P. HARIHARAN,B.C. SANDERS III Super-Resolution by Data Inversion, M. BERTERO,C. DE MOL IV Radiative Transfer: New Aspects of the Old Theory, Yu.A. K~VTSOV, L.A. APRESYAN V Photon Wave Function, I. BIALVNICKbBmt~A
1-- 47 49--128 129-178 179--244 245--294
VOLUME 37 (1997) I II III
The Wigner Distribution Function in Optics and Optoelectronics, D. D~COMAN Dispersion Relations and Phase Retrieval in Optical Spectroscopy, K.-E. PHPONEN, E.M. VARTIArNEN,T. ASAKtmA Spectra of Molecular Scattering of Light, I.L. FABELINS~I
1- 56 57- 94 95-184
CONTENTS OF PREVIOUSVOLUMES IV Soliton Communication Systems, R.-J. ESSIAMBRE,G.R AGRAWAL V Local Fields in Linear and Nonlinear Optics of Mesoscopic Systems, O. KELLER VI Tunneling Times and Superluminality, R.Y. CHIAO, A.M. STEINBERG
601 185--256 257--343 345-405
VOLUME 38 (1998) I Nonlinear Optics of Stratified Media, S. DUTTA GUPTA II Optical Aspects of Interferometric Gravitational-Wave Detectors, P. HELLO III Thermal Properties of Vertical-Cavity Surface-Emitting Semiconductor Lasers, W. NAKWASgJ,M. Osr~srd IV Fractional Transformations in Optics, A.W. LOHMANN,D. MZNDLOVIC,Z. ZALEVSKY V Pattern Recognition with Nonlinear Techniques in the Fourier Domain, B. JAvmI, J.L. HORNER VI Free-space Optical Digital Computing and Interconnection, J. JAHNS
1-- 84 85--164 165-262 263--342 343-4 18 419--513
VOLUME 39 (1999) I
Theory and Applications of Complex Rays, Yu.A. KRAVTSOV, G.W. FORBES, 1-- 62 A.A. ASATRYAN II Homodyne Detection and Quantum-State Reconstruction, D.-G. WELSCH,W. VOGEL, 63--211 T. OPATRNY III Scattering of Light in the Eikonal Approximation, S.K. SHARMA,D.J. SOMERFORD 213--290 291--372 IV The Orbital Angular Momentum of Light, L. ALLEN,M.J. PADGETT,M. BABIKER 373-469 V The Optical Kerr Effect and Quantum Optics in Fibers, A. SIZMANN,G. LEUCHS VOLUME 40 (2000) I Polarimetric Optical Fibers and Sensors, T.R. WOLriqsKI II Digital Optical Computing, J. TANIDA,Y. ICHIOKA III Continuous Measurements in Quantum Optics, V. PE~aNOV~,A. LtsK~ IV Optical Systems with Improved Resolving Power, Z. ZALEVSI(V,D. MENDLOVIr A.W. LOHMANN V Diffractive Optics: Electromagnetic Approach, J. TURUNEN, M. KUITTINEN, E WYROWSKI VI Spectroscopy in Polychromatic Fields, Z. FICEKAND H.S. FREEDHOFF
1- 75 77-114 115-269 271-341 343-388 389-441
CUMULATIVE
INDEX - VOLUMES
1-41"
ABELI~S,E, Methods for Determining Optical Parameters of Thin Films ABELLA,I.D., Echoes at Optical Frequencies ABITBOL, C.I., s e e Clair, J.J. ABRAHAM,N.B., P. MANDEL,L.M. NARDUCCX,Dynamical Instabilities and Pulsations in Lasers AGARWAL,G.S., Master Equation Methods in Quantum Optics AGRANOVICH,V.M., V.L. GINZBURG,Crystal Optics with Spatial Dispersion AGRAWAL,G.P., Single-Longitudinal-Mode Semiconductor Lasers AGRAWAL,G.P., s e e Essiambre, R.-J. ALLEN, L., D.G.C. JONES,Mode Locking in Gas Lasers ALLEN, L., M.J. PADGETT,M. BABIKER,The Orbital Angular Momentum of Light AMMANN,E.O., Synthesis of Optical Birefringent Networks ANDERSON,R., s e e Carriere, J. APRESVAN,L.A., s e e Kravtsov, Yu.A. ARIMONDO,E., Coherent Population Trapping in Laser Spectroscopy ARMSTRONC,J.A., A.W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers ARNAtJD, J.A., Hamiltonian Theory of Beam Mode Propagation ASAKURA,T., s e e Okamoto, T. ASAKURA,T., s e e Peiponen, K.-E. ASATRYAN,A.A., s e e Kravtsov, Yu.A. BABIKER,M., s e e Allen, L. BALTES, H.R, On the Validity of Kirchhoff's Law of Heat Radiation for a Body in a Nonequilibrium Environment BARABANENKOV, Yu.N., Yu.A. KRAVTSOV, VD. OZRrN, A.I. SAtCHEV, Enhanced Backscattering in Optics BARAKAT, R., The Intensity Distribution and Total Illumination of Aberration-Free Diffraction Images BARRETT, H.H., The Radon Transform and its Applications BASHKIN, S., Beam-Foil Spectroscopy BASSETT,I.M., W.T. WELFORD,R. WINSTON,Nonimaging Optics for Flux Concentration BECKMANN,R, Scattering of Light by Rough Surfaces BERAN, M.J., J. Oz-VOGT, Imaging through Turbulence in the Atmosphere BERNARD,J., s e e Orrit, M.
* Volumes I-XL were previously distinguished by roman rather than by arabic numerals. 603
2, 249 7, 139 16, 71 25, 11, 9, 26, 37, 9, 39, 9, 41, 36, 35, 6, 11, 34, 37, 39,
1 1
235 163 185 179 291 123 97 179 257 211 247 183 57 1
39, 291 13,
1
29, 65 1, 21, 12, 27, 6, 33, 35,
67 217 287 161 53 319 61
604
CUMULATIVEINDEX- VOLUMES 1-41
BERRY, M.V., C. UPSTILL, Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns BERTERO, M., C. DE MOL, Super-Resolution by Data Inversion BERTOLOTI"I,M., s e e Mihalache, D. BERTOLOTTI,M., s e e Chumash, V. BEVERLYIII, R.E., Light Emission From High-Current Surface-Spark Discharges BIALVNICm-BIRtJLA,I., Photon Wave Function BJORK, G., s e e Yamamoto, Y. BLOOM, A.L., Gas Lasers and their Application to Precise Length Measurements BOtrMaN, M.A., W.A. VAN DE GRIND,P. ZUDEMA, Quantum Fluctuations in Vision BOUSQUET, P., s e e Rouard, P. BROWN, G.S., s e e DeSanto, J.A. BROWN, R., s e e Orrit, M. BRtYNNER,W., H. PAUL, Theory of Optical Parametric Amplification and Oscillation BRVNODAHL,O., Applications of Shearing Interferometry BRVNfiDAHL, O., Evanescent Waves in Optical Imaging BRVNGDAHL,O., E WYROWSra,Digital Holography- Computer-Generated Holograms BRYNGDAHL,O., T. SCHEERMESSER,E WYROWSKI,Digital Halftoning: Synthesis of Binary Images BURCH, J.M., The Metrological Applications of Diffraction Gratings BUTTERWECK,H.J., Principles of Optical Data-Processing BU2EK, V., P.L. KNIGHT, Quantum Interference, Superposition States of Light, and Nonclassical Effects
18, 36, 27, 36, 16, 36, 28, 9, 22, 4, 23, 35, 15, 4, 11, 28,
CAGNAC,B.,
17, 85
s e e Giacobino, E. CARRIERE,J., R. NARAYAN,W-H. YEa, C. PENG,P. KHULBE,L. LI, R. ANDERSON,J. CHOI, M. MANSURIPUR,Principles of Optical Disk Data Storage CASASENT, D., O. PSALTIS, Deformation Invariant, Space-Variant Optical Pattern Recognition CEGLIO,N.M., D.W. SWEENEY,Zone Plate Coded Imaging: Theory and Applications CHANG, R.K., s e e Fields, M.H. CHARNOTSKII, M.I., J. GOZANI, V.I. TATARSKII,V.U. ZAVOROTNY,Wave Propagation Theories in Random Media Based on the Path-Integral Approach CHEN, R.T., Z. Fo, Optical True-Time Delay Control Systems for Wideband Phased Array Antennas CHIAO,R.Y., A.M. STEINBERG,Tunneling Times and Superluminality CHOI, J., s e e Carriere, J. CHRISTENSEN,J.L., s e e Rosenblum, WM. CHRISa'OV,I.P., Generation and Propagation of Ultrashort Optical Pulses CHUMASH, V., I. COJOCARU, E. FAZIO, F. MICHELOTTI, M. BERTOLOTTI, Nonlinear Propagation of Strong Laser Pulses in Chalcogenide Glass Films CLAIR, J.J., C.I. ABIXBOL,Recent Advances in Phase Profiles Generation CLARRICOATS,P.J.B., Optical Fibre Waveguides- A Review COHEN-TANNOUDJI,C., A. KASTLER,Optical Pumping COJOCARU, I., s e e Chumash, V. COLE, T.W., Quasi-Optical Techniques of Radio Astronomy COLOMBEAU,B., s e e Froehly, C. COOK, R.J., Quantum Jumps COURTI~S,G., P. CRUVELLIER,M. DETAILLE,M. SAYSSE,Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects CREATH, K., Phase-Measurement Interferometry Techniques
257 129 227 1 357 245 87 1 77 145 1 61 1 37 167 1
33, 389 2, 73 19, 211 34,
41,
1
97
16, 289 21, 287 41, 1 32, 203 41, 37, 41, 13, 29,
283 345 97 69 199
36, 16, 14, 5, 36, 15, 20, 28,
1 71 327 1 1 187 63 361
20, 1 26, 349
CUMULATIVEINDEX- VOLUMES1-4t
605
CREWE, A.V., Production of Electron Probes Using a Field Emission Source CRUVELLIER,P., s e e Courtbs, G. CUMMINS, H.Z., H.L. SWlNNEY,Light Beating Spectroscopy
11, 223 20, 1 8, 133
DAINTY, J.C., The Statistics of Speckle Patterns D)~NDLIKER,R., Heterodyne Holographic Interferometry DATTOLI, G., L. GIANNESSI,A. RENIERI, A. TORRE, Theory of Compton Free Electron Lasers DE MOL, C., s e e Bertero, M. DE STERKE,C.M., J.E. SIPE, Gap Solitons DECKER JR, J.A., s e e Harwit, M. DELANO, E., R.J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters DEMARIA, A.J., Picosecond Laser Pulses DESANTO,J.A., G.S. BROWN,Analytical Techniques for Multiple Scattering from Rough Surfaces DETAILLE,M., s e e Courtbs, G. DEXTER, D.L., s e e Smith, D.Y. DRAGOMAN,D., The Wigner Distribution Function in Optics and Optoelectronics DREXHAGE,K.H., Interaction of Light with Monomolecular Dye Layers DUGUAY,M.A., The Ultrafast Optical Kerr Shutter DUTTA, N.K., J.R. SIMPSON,Optical Amplifiers DUTTA GUPTA, S., Nonlinear Optics of Stratified Media
14, 17,
1 1
31, 36, 33, 12, 7, 9,
321 129 203 101 67 31
23, 20, 10, 37, 12, 14, 31, 38,
1 1 165 1 163 161 189 1
EBERLY, J.H., Interaction of Very Intense Light with Free Electrons ENGLUND,J.C., R.R. SNAPP, W.C. SCHIEVE,Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity ENNOS, A.E., Speckle Interferometry ESSIAMBRE,R.-J., G.R AGRAWAL,Soliton Communication Systems ETRICH, C., E LEDERER,B.A. MALOMED,T. PESCHEL,U. PESCHEL,Optical Solitons in Media with a Quadratic Nonlinearity FABELINSKII,I.L., Spectra of Molecular Scattering of Light FABRE, C., s e e Reynaud, S. FANTE, R.L., Wave Propagation in Random Media: A Systems Approach FAzIO, E., s e e Chumash, V FICEK, Z. AND H.S. FREEDHOFF,Spectroscopy in Polychromatic Fields FIELDS, M.H., J. PoPP, R.K. CHANG,Nonlinear Optics in Microspheres FIORENTINI,A., Dynamic Characteristics of Visual Processes FLYTZANIS,C., E HACHE,M.C. KLEIN,D. RICARD,PH. ROUSSIGNOL,Nonlinear Optics in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics FocKE, J., Higher Order Aberration Theory FOPd3ES, G.W, s e e Kravtsov, Yu.A. FRANqON, M., S. MALLICK,Measurement of the Second Order Degree of Coherence FRANTA, D., s e e Ohlidal, I. FREEDHOFF,H.S., s e e Ficek, Z. FREILIKHER, V.D., S.A. GREDESKUL, Localization of Waves in Media with OneDimensional Disorder FRIEDEN, B.R., Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions FROEHLY, C., B. COLOMBEAU,M. VAMPOUILLE,Shaping and Analysis of Picosecond Light Pulses
7, 359 21, 355 16, 233 37, 185 41, 483 37, 30, 22, 36, 40, 41, 1,
95 1 341 1 389 1 253
29, 4, 39, 6, 41, 40,
321 1 1 71 181 389
30, 137 9, 311 20,
63
606
CUMULATIVEINDEX- VOLUMES 1-41
FRY, G.A., The Optical Performance of the Human Eye Fu, Z., s e e Chen, R.T. GABOR, D., Light and Information GAMO, H., Matrix Treatment of Partial Coherence GANDJI3AKHCHE,A.H., G.H. WEISS,Random Walk and Diffusion-Like Models of Photon Migration in Turbid Media GANTSOG,TS., s e e Tanag, R. GHATAK, A., K. THYAGARAJAN,Graded Index Optical Waveguides: A Review GHATAK, A.K., s e e Sodha, M.S. GIACOBINO,E., B. CAGNAC,Doppler-Free Multiphoton Spectroscopy GIACOBINO,E., s e e Reynaud, S. GIANNESSI,L., s e e Dattoli, G. GINZBURG,V.L., s e e Agranovich, V.M. GINZBURG, V.L., Radiation by Uniformly Moving Sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena GIOVANELLI,R.G., Diffusion Through Non-Uniform Media GLASER, I., Information Processing with Spatially Incoherent Light GNIADEK,K., J. PETYKIEWICZ,Applications of Optical Methods in the Diffraction Theory of Elastic Waves GOODMAN,J.W., Synthetic-Aperture Optics GOZANI, J., s e e Charnotskii, M.I. GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission GREDESKUL,S.A., s e e Freilikher, V.D.
8, 51 41, 283 1, 109 3, 187 34, 35, 18, 13, 17, 30, 31, 9,
333 355 1 169 85 1 321 235
32, 267 2, 109 24, 389 9, 8, 32, 12, 30,
281 1 203 233 137
HACHE, E, s e e Flytzanis, C. HALL, D.G., Optical Waveguide Diffraction Gratings: Coupling between Guided Modes HARIHARAN,P., Colour Holography HARIHARAN,P., Interferometry with Lasers HARIHARAN,P., B.C. SANDERS,Quantum Phenomena in Optical Interferometry HARWIT, M., J.A. DECKERJR, Modulation Techniques in Spectrometry HASEGAWA,A., s e e Kodama, Y. HEIDMANN,A., s e e Reynaud, S. HELLO, P., Optical Aspects of Interferometric Gravitational-Wave Detectors HELSTROM, C.W., Quantum Detection Theory HERRIOT, D.R., Some Applications of Lasers to Interferometry HORNER, J.L., s e e Javidi, B. HUANG, T.S., Bandwidth Compression of Optical Images
29, 29, 20, 24, 36, 12, 30, 30, 38, 10, 6, 38, 10,
321 1 263 103 49 101 205 1 85 289 171 343 1
ICHIOKA,Y., s e e Tanida, J. IMOTO,N., s e e Yamamoto, Y. ITOH, K., Interferometric Multispectral Imaging
40, 77 28, 87 35, 145
JACOBSSON,R., Light Reflection from Films of Continuously Varying Refractive Index JACQUINOT,P., B. ROIZEN-DOSSIER,Apodisation JAHNS, J., Free-space Optical Digital Computing and Interconnection JAMROZ, W., B.P. STOICHEFF, Generation of Tunable Coherent Vacuum-Ultraviolet Radiation JAVIDI, B., J.L. HORNER, Pattern Recognition with Nonlinear Techniques in the Fourier Domain JONES, D.G.C., s e e Allen, L.
5, 247 3, 29 38, 419 20, 325 38, 343 9, 179
CUMULATIVEINDEX- VOLUMES141
KASTLER,A., s e e Cohen-Tannoudji, C. KELLER,O., Local Fields in Linear and Nonlinear Optics of Mesoscopic Systems KHOO, I.C., Nonlinear Optics of Liquid Crystals KHULBE, P., s e e Carriere, J. KIELICH, S., Multi-Photon Scattering Molecular Spectroscopy K1NOSITA,K., Surface Deterioration of Optical Glasses KITAGAWA,M., s e e Yamamoto, Y. KLEIN, M.C., s e e Flytzanis, C. KLYATSKIN,V.I., The Imbedding Method in Statistical Boundary-Value Wave Problems KNIGHT,EL., s e e Bu~ek, V. KODAMA,Y., A. HASECAWA,Theoretical Foundation of Optical-Soliton Concept in Fibers KOPPELMAN,G., Multiple-Beam Interference and Natural Modes in Open Resonators KOTTLER,E, The Elements of Radiative Transfer KOTTLER,E, Diffraction at a Black Screen, Part I: Kirchhoff's Theory KOTTLER,E, Diffraction at a Black Screen, Part II: Electromagnetic Theory KRAVTSOV,Yu.A., Rays and Caustics as Physical Objects KRAVTSOV,YtJ.A., s e e Barabanenkov, Yu.N. KRAVTSOV,Yu.A., L.A. APRESYAN,Radiative Transfer: New Aspects of the Old Theory KRAVTSOV,Yu.A., G.W. FORBES,A.A. ASATRYAN,Theory and Applications of Complex Rays KtmOTA, H., Interference Color KUIaq'INEN, M., s e e Turunen, J.
607 5, 37, 26, 41, 20, 4, 28, 29, 33, 34, 30, 7, 3, 4, 6, 26, 29, 36,
1 257 105 97 155 85 87 321 1 1 205 1 1 281 331 227 65 179
39, 1 1, 211 40, 343
LABEYRIE,A., High-Resolution Techniques in Optical Astronomy LEAN, E.G., Interaction of Light and Acoustic Surface Waves LEDERER, E, s e e Etrich, C. LEE, W-H., Computer-Generated Holograms: Techniques and Applications LEITH, E.N., J. UPATNIEKS,Recent Advances in Holography LETOKHOV,V.S., Laser Selective Photophysics and Photochemistry LEUCHS, G., s e e Sizmann, A. LEVI, L., Vision in Communication L I , L . , s e e Carriere, J. LIPSON, H., C.A. TAYLOR, X-Ray Crystal-Structure Determination as a Branch of Physical Optics LOHMANN,A.W, D. MENDLOVIC,Z. ZALEVSKY,Fractional Transformations in Optics LOHMANN,A.W, s e e Zalevsky, Z. LOUNIS, B., s e e Orrit, M. LtJGL~TO,L.A., Theory of Optical Bistability LLrIS, A., L.L. S~'qCHEZ-SOTO, Quantum Phase Difference, Phase Measurements and Stokes Operators L U K e , A . , s e e Pe~inov~, V. L u K e , A . , s e e Pe~inov~i, V.
14, 11, 41, 16, 6, 16, 39, 8, 41,
47 123 483 119 1 1 373 343 97
5, 38, 40, 35, 21,
287 263 271 61 69
MACHIDA, S., s e e Yamamoto, Y. M A I ~ Y , G., C. ~ s , Nonlinear Processes in Atoms and in Weakly Relativistic Plasmas M~ACM~A, D., Optical and Electronic Processing of Medical Images MALACARA,D., s e e Vlad, V.I. MALLICK, S., s e e Fran9on, M. MALOMED,B.A., s e e Etrich, C. MANDEL, L., Fluctuations of Light Beams
28,
87
32, 22, 33, 6, 41, 2,
313 1 261 71 483 181
41, 419 33, 129 40, 115
608
CUMULATIVEINDEX- VOLUMES 1-41
M_ANDEL,L., The Case For and Against Semiclassical Radiation Theory MANDEL, P., s e e Abraham, N.B. MANSURIPUR~M., s e e Carriere, J. M A N U S , C . , s e e Mainfray, G. MARCHAND,E.W., Gradient Index Lenses MARTIN, P.J., R.E NETrERfiELD,Optical Films Produced by Ion-Based Techniques MASALOV,A.V., Spectral and Temporal Fluctuations of Broad-Band Laser Radiation MAVSTRE, D., Rigorous Vector Theories of Diffraction Gratings /VI__EESSEN,A., s e e Rouard, P. MEHTA, C.L., Theory of Photoelectron Counting 1VIENDLOVIC,D., s e e Lohmann, A.W. 1VIENDLOVIC,O., s e e Zalevsky, Z. MEYSTRE, E, Cavity Quantum Optics and the Quantum Measurement Process MICHELOTI"I,E, s e e Chumash, V. MJHALACHE, D., M. BERTOLOTTI,C. SIBILIA,Nonlinear Wave Propagation in Planar Structures MIKAELIAN,A.L., M.L. TER-MIKAELIAN,Quasi-Classical Theory of Laser Radiation MIKAELIAN,A.L., Self-Focusing Media with Variable Index of Refraction MILLS, D.L., K.R. SUBBASWAMY,Surface and Size Effects on the Light Scattering Spectra of Solids MILONNI, PW., B. SUNDARAM,Atoms in Strong Fields: Photoionization and Chaos MIRANOWlCZ,A., s e e Tana~, R. MIYAMOTO,K., Wave Optics and Geometrical Optics in Optical Design MOLLOW, B.R., Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence MURATA, K., Instruments for the Measuring of Optical Transfer Functions MUSSET, A., A. THELEN,Multilayer Antireflection Coatings NAKWASKI, W., M. OSI~SKI, Thermal Properties of Vertical-Cavity Surface-Emitting Semiconductor Lasers NARAVAN,R., s e e Carriere, J. NARDUCCI, L.M., s e e Abraham, N.B. NAVRATIL,K., s e e Ohlidal, I. NETTERfiELD, R.P., s e e Martin, P.J. NISHIHARA,H., T. SUHARA,Micro Fresnel Lenses OHLiDAL, I., K. NAVRATIL,M. OHLiDAL,Scattering of Light from Multilayer Systems with Rough Boundaries OHLiDAL, I., O. FRANTA,Ellipsometry of Thin Film Systems OHLiDAL, M., s e e Ohlidal, I. OHTSU, M., T. TAKO,Coherence in Semiconductor Lasers OKAMOTO,T., T. ASAKURA,The Statistics of Dynamic Speckles OKOSHI, T., Projection-Type Holography OOUE, S., The Photographic Image OPATRN~', T., s e e Welsch, D.-G. ORRIT, M., J. BERNARD,R. BROWN,B. LOUNIS,Optical Spectroscopy of Single Molecules in Solids OSI~SKI, M., s e e Nakwaski, W. OSTROVSKAYA,G.V., Yu.I. OSTROVSKY,Holographic Methods of Plasma Diagnostics OSTROVSKY,Yu.I., s e e Ostrovskaya, G.V. OSTROVSI(u Yu.I., V.E SHCHEPINOV,Correlation Holographic and Speckle Interferometry
13, 25, 41, 32, 11, 23, 22, 21, 15, 8, 38, 40, 30, 36,
27 1 97 313 305 113 145 1 77 373 263 271 261 1
27, 227 7, 231 17, 279 19, 45 31, 1 35, 355 1, 31 19, 1 5, 199 8, 201
38, 165 41, 97 25, 1 34, 249 23, 113 24, 1
34, 41, 34, 25, 34, 15, 7, 39,
249 181 249 191 183 139 299 63
35, 61 38, 165 22, 197 22, 197 30, 87
CUMULATIVEINDEX- VOLUMES 1-41
OUGHSTUN,K.E., Unstable Resonator Modes Oz-VOGT, J., s e e Beran, M.J. OzR_rn, V.D., s e e Barabanenkov, Yu.N.
609 24, 165 33, 319 29, 65
PADGETT, M.J., s e e Allen, L. PAL, B.R, Guided-Wave Optics on Silicon: Physics, Technology and Status PAOLETTI,D., G. SCHIRRIPASPAGNOLO,Interferometric Methods for Artwork Diagnostics PATORSrd, K., The Self-Imaging Phenomenon and Its Applications PAUL, H., s e e Brunner, W. PEGIS, R.J., The Modern Development of Hamiltonian Optics PEGIS, R.J., s e e Delano, E. PEIPONEN, K.-E., E.M. VARTIArNEN, T. ASAKURA, Dispersion Relations and Phase Retrieval in Optical Spectroscopy PENG, C., s e e Carriere, J. I~NNA, J., Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media I ~ d N A , J., s e e Pefina Jr, J. PENNA JR, J., J. PENNA, Quantum Statistics of Nonlinear Optical Couplers PENNOV~, V, A. LUKg, Quantum Statistics of Dissipative Nonlinear Oscillators I~,INOV~, V, A. Ltm~, Continuous Measurements in Quantum Optics PERSHAN, RS., Non-Linear Optics PESCHEL,T., s e e Etrich, C. I~SCI-mL, U., s e e Etrich, C. I~TYKIEWICZ,J., s e e Gniadek, K. PICHT, J., The Wave of a Moving Classical Electron PoFov, E., Light Diffraction by Relief Gratings: A Macroscopic and Microscopic View PoPP, J., s e e Fields, M.H. PORTER, R.P., Generalized Holography with Application to Inverse Scattering and Inverse Source Problems PRESNYAKOV,L.P., Wave Propagation in Inhomogeneous Media: Phase-Shift Approach PSALTIS, D., s e e Casasent, D. PSALTIS, D., Y. QIAO, Adaptive Multilayer Optical Networks
39, 291 32, 1 35, 197 27, 1 15, 1 1, 1 7, 67
QIAO, Y., s e e Psaltis, D.
31, 227
RAYMER, M.G., I.A. WALMSLEY,The Quantum Coherence Properties of Stimulated Raman Scattering RENIERI, A., s e e Dattoli, G. REYNAUD,S., A. HEIDMANN,E. GIACOBINO,C. FABle, Quantum Fluctuations in Optical Systems RICARD, D . , s e e Flytzanis, C. PdSEBER6, L.A., M.J. WEBER,Relaxation Phenomena in Rare-Earth Luminescence RJSI~N, H., Statistical Properties of Laser Light RODDIER, E, The Effects of Atmospheric Turbulence in Optical Astronomy ROIZEN-DOSSlER,B., s e e Jacquinot, P. RONCHI, L., s e e Wang Shaomin ROSANOV,N.N., Transverse Patterns in Wide-Aperture Nonlinear Optical Systems ROSENBLUM, WM., J.L. CHRISTENSEN,Objective and Subjective Spherical Aberration Measurements of the Human Eye ROTrmERG, L., Dephasing-Induced Coherent Phenomena ROUARD,P., P. BOUSQUET,Optical Constants of Thin Films
37, 57 41, 97 18, 41, 41, 33, 40, 5, 41, 41, 9, 5, 31, 41,
127 359 359 129 115 83 483 483 281 351 139 1
27, 34, 16, 31,
315 159 289 227
28, 181 31, 321 30, 29, 14, 8, 19, 3, 25, 35,
1 321 89 239 281 29 279 1
13, 69 24, 39 4, 145
610
CUMULATIVEINDEX- VOLUMES1-41
ROUARD, P., A. MEESSEN, Optical Properties of Thin Metal Films ROUSSIGNOL,PH., s e e Flytzanis, C. RtJBINOWlCZ,A., The Miyamoto-Wolf Diffraction Wave RUDOLPH, O., s e e Schmahl, G.
15, 29, 4, 14,
77 321 199 195
SAICHEV,A.I., s e e Barabanenkov, Yu.N. SAI'SSE,M., s e e Court~s, G.
29, 20, 28, 6, 26, 41, 36, 33, 21, 35, 14,
65 1 87 259 1 419 49 389 355 197 195
17, 13, 25, 13, 28, 10,
163 93 349 93 271 89
16, 39, 30, 27, 31, 15, 33, 10, 39, 12, 6, 10, 10, 21,
413 213 87 227 189 245 203 229 373 53 211 165 45 355
13, 39, 27, 31, 5, 37, 20, 9,
169 213 109 263 145 345 325 73
SAITO, S., s e e Yamamoto, Y. SAKAI, H., s e e Vanasse, G.A. SALEH, B.E.A., s e e Teich, M.C. SANCHEZ-SOTO,L.L., s e e Luis, A. SANDERS, B.C., s e e Hariharan, P. SCHEERMESSER,T., s e e Bryngdahl, O. SCHIEVE, W.C., s e e Englund, J.C. SCHIRRIPASPAGNOLO,G., s e e Paoletti, D. SCHMAn~, G., D. RUDOLPH,Holographic Diffraction Gratings SCHUBERT,M., B. WILHELMI,The Mutual Dependence Between Coherence Properties of Light and Nonlinear Optical Processes ScmrLz, G., J. SCmVIOER,Interferometric Testing of Smooth Surfaces Scm~z, G., Aspheric Surfaces SCnWmER, J., s e e Schulz, G. SCmVIDER, J., Advanced Evaluation Techniques in Interferometry Sct~LY, M.O., K.G. WmTYEV, Tools of Theoretical Quantum Optics SENITZKY, I.R., Semiclassical Radiation Theory Within a Quantum-Mechanical Framework SHARMA, S.K., D.J. SOMERVORD,Scattering of Light in the Eikonal Approximation SHCHEPINOV,V.P., s e e Ostrovsky, Yu.I. SIBILIA, C., s e e Mihalache, D. SIMPSON,J.R., s e e Dutta, N.K. SIPE, J.E., s e e Van Kranendonk, J. SWE, J.E., s e e De Sterke, C.M. SITI'IG, E.K., Elastooptic Light Modulation and Deflection SIZMANN,A., G. LEUCHS,The Optical Kerr Effect and Quantum Optics in Fibers SLUSHER, R.E., Self-Induced Transparency SMITH,A.W., s e e Armstrong, J.A. SMITH, D.Y., D.L. DEXTER, Optical Absorption Strength of Defects in Insulators SMITH, R.W., The Use of Image Tubes as Shutters SNAPP, R.R., s e e Englund, J.C. SODHA, M.S., A.K. GHATAK,V.K. TRIPATHI,Self-Focusing of Laser Beams in Plasmas and Semiconductors SOMERFORD,D.J., s e e Sharma, S.K. SOROKO, L.M., Axicons and Meso-Optical Imaging Devices SPREEtrW, R.J.C., J.P. WOERDMAN,Optical Atoms STEEL,W.H., Two-Beam Interferometry STEINBERG,A.M., s e e Chiao, R.Y. STOICHEFF,B.P., s e e Jamroz, W. STROHBEHN,J.W., Optical Propagation Through the Turbulent Atmosphere STROKE, G.W., Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy SUBBASWAMY,K.R., s e e Mills, D.L. S U H A R A , T . , s e e Nishihara, H.
2, 1 19, 45 24, 1
611
CUMULATIVEINDEX- VOLUMES 1-41 SUNDARAM,B., s e e Milonni, E W SVELTO, O., Self-Focusing, Self-Trapping, and Self-Phase Modulation of Laser Beams SWEENEY,D.W., s e e Ceglio, N.M. SW~NEY, H.L., s e e Cummins, H.Z.
31, 1 12, 1 21, 287 8, 133
TAKO, T., s e e Ohtsu, M. TANAKA,K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets TANAS, R., A. MIRANOWICZ,TS. GANTSOG,Quantum Phase Properties of Nonlinear Optical Phenomena TANGO,WJ., R.Q. TwIss, Michelson Stellar Interferometry TANIDA, J., Y. ICHIOKA,Digital Optical Computing TATARSKII, V..I., V.U. ZAVOROTNYI, Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium TATARSKII,V.I., s e e Chamotskii, M.I. TAYLOR,C.A., s e e Lipson, H. TEICH, M.C., B.E.A. SALEH,Photon Bunching and Antibunching TER-MIKAELIAN,M.L., s e e Mikaelian, A.L. THELEN, A., s e e Musset, A. THOMPSON,B.J., Image Formation with Partially Coherent Light THYAGARAJAN,K., s e e Ghatak, A. TONOMtn~, A., Electron Holography TORRE, A., s e e Dattoli, G. TRIPATHI,V.K., s e e Sodha, M.S. TsuJIUCHI, J., Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering TURUNEN,J., M. KUITT1NEN,E WYROWSrO,Diffractive Optics: Electromagnetic Approach TwIss, R.Q., s e e Tango, W.J.
25, 191 23, 63
UPATNIEKS,J., s e e Leith, E.N. UPSTILL, C., s e e Berry, M.V USHIODA,S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids
6, 1 18, 257 19, 139
VAMPOUILLE,M., s e e Froehly, C. VAN DE GRIND,W.A., s e e Bouman, M.A. VAN HEEL, A.C.S., Modem Alignment Devices VANKRANENDONK,J., J.E. SIPE,Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media VANASSE, G.A., H. SAKAI,Fourier Spectroscopy VARTIAINEN,E.M., s e e Peiponen, K.-E. VERNIER,EJ., Photoemission VLAO, V.I., D. MALACAI~,Direct Spatial Reconstruction of Optical Phase from PhaseModulated Images V O G E L , W . , s e e Welsch, D.-G.
20, 63 22, 77 1, 289
WALMSLEY,I.A., s e e Raymer, M.G. WANG SHAOMIN,L. RONCHI,Principles and Design of Optical Arrays WEBER, M.J., s e e Riseberg, L.A. WEIGELT,G., Triple-Correlation Imaging in Optical Astronomy WHss, G.H., s e e Gandjbakhche, A.H. WELFORD,W.T., Aberration Theory of Gratings and Grating Mountings WELFORD,WT., Aplanatism and Isoplanatism
28, 25, 14, 29, 34, 4, 13,
35, 355 17, 239 40, 77 18, 32, 5, 26, 7, 8, 7, 18, 23, 31, 13,
204 203 287 1 231 201 169 1 183 321 169
2, 131 40, 343 17, 239
15, 6, 37, 14,
245 259 57 245
33, 261 39, 63 181 279 89 293 333 241 267
612
CUMULATIVEINDEX- VOLUMES 1-41
~tVELFORD,W.T., s e e Bassett, I.M. V~LSCH, D.-G., W. VOGEL, Z. OPATRNY, Homodyne Detection and Quantum-State Reconstruction
WHITNEY,K.G., s e e Scully, M.O. WILHELMI,B., s e e Schubert, M. WINSTON,R.,
see
Bassett, I.M.
WOERDMAN,J.P., s e e Spreeuw, R.J.C. WOL~SKI, T.R., Polarimetric Optical Fibers and Sensors WOLXER, H., On Basic Analogies and Principal Differences between Optical and Electronic Information WYNNE, C.G., Field Correctors for Astronomical Telescopes WYROWSKI,E, s e e Bryngdahl, O. WYROWSrd, E, s e e Bryngdahl, O. WYROWSKI,E, s e e Turunen, J. YAMAGUCHI,I., Fringe Formations in Deformation and Vibration Measurements using Laser Light YAMAJI, K., Design of Zoom Lenses YAMAMOTO, T., Coherence Theory of Source-Size Compensation in Interference Microscopy YAMAMOTO,Y., S. MACHIDA,S. SAITO,N. IMOTO,T. YANAGAWA,M. KITAGAWA,G. BJtRK, Quantum Mechanical Limit in Optical Precision Measurement and Communication YANAGAWA,T., s e e Yamamoto, Y. YAROSLAVSKY,L.P., The Theory of Optimal Methods for Localization of Objects in Pictures YEH, W.-H., s e e Carriere, J. YOSHINAGA,H., Recent Developments in Far Infrared Spectroscopic Techniques Yu, ETS., Principles of Optical Processing with Partially Coherent Light Yu, ET.S., Optical Neural Networks: Architecture, Design and Models ZALEVSKY,Z., s e e Lohmann, A.W. ZALEVSKY, Z., D. MENDLOVIC, A.W. LOHMANN, Optical Systems with Improved Resolving Power ZAVOROTNY,V.U., s e e Charnotskii, M.I. ZAVOROTNYI,V.U., s e e Tatarskii, V.I. ZUIDEMA,P., s e e Bouman, M.A.
27, 161 39, 10, 17, 27, 31, 40,
63 89 163 161 263 1
1, 10, 28, 33, 40,
155 137 1 389 343
22, 271 6, 105 8, 295 28, 28,
87 87
32, 41, 11, 23, 32,
145 97 77 221 61
38, 263 40, 32, 18, 22,
271 203 204 77
