EDITORIAL ADVISORY BOARD G. S. AGARWAL,
Ahmedabad, India
T. ASAKURA,
Sapporo, Japan
A. ASPECT,
Orsay, France
M.V. ...
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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 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 >~ 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 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
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PERTURBATION EFFECTS ON MICROSPHERE RESONANCES
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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 ~
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PERTURBATION EFFECTS ON MICROSPHERE RESONANCES
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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.
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NONLINEAR OPTICS IN MICROSPHERES
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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
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CAVITY-MODIFIED OPTICALPROCESSESIN MICROSPHERES
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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
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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
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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.
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NONLINEAR OPTICS IN MICROSPHERES
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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).
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CAVITY-MODIFIED OPTICAL PROCESSES IN MICROSPHERES
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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.
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NONLINEAR OPTICS IN MICROSPHERES
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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
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CAVITY-MODIFIED OPTICAL PROCESSES IN MICROSPHERES
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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
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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 ( Ms2Hc2,
(5)
where aws is the interface wall energy between the writing layer and the
142
PRINCIPLES OF OPTICAL DISK DATASTORAGE
[2, w 6
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
2, w 6]
MAGNETO-OPTICAL(MO) RECORDING
143
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
144
PRINCIPLES OF OPTICAL DISK DATA STORAGE
[2, w 6
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,
2, w 6]
MAGNETO-OPTICAL(MO) RECORDING
145
(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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PRINCIPLESOF OPTICALDISK DATASTORAGE
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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
. . . . . . . . . . . . . .
211 235
. . . . . . . . . . . . . . . . . . . .
276
. . . . . . . . . . . . . . . . . . .
278
. . . . . . . . . . . . . . . . . . . . . . .
278
ACKNOWLEDGMENTS REFERENCES
183
. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
ELLIPSOMETRIC SYSTEMS
. . . . . . . . . . . . .
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.
3, w 3]
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.
188
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.
192
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)
3, w 4]
THEORY OF MEASUREMENTS IN ELLIPSOMETRY
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.
3, w 4]
THEORY OF MEASUREMENTS IN ELLIPSOMETRY
195
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]
ELLIPSOMETRIC QUANTITIES OF IDEAL THIN FILM SYSTEMS
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]
ELLIPSOMETRIC QUANTITIESOF IDEALTHIN FILM SYSTEMS
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]
ELLIPSOMETRIC QUANTITIES OF IDEAL THIN FILM SYSTEMS
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
l__.l
Conu'oIler Optoelectronic T/R Modules
' Isiaole ~ d * - I ,| F-Bit
I
!
Hop=~V"----"--!
...... t, ,
Signal _J_. ~
Antenna
~" ,: J, i~ t' ' ',
,'
,1~o.~1
I~=~~
+ ~ { 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 ( ]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]).
~- 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 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.
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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-
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QUANTUM STATISTICS OF NONLINEAR OPTICAL COUPLERS
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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]).
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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 ( 0 is a constant. The conditions described in eqs. (7.12) are optimum for the generation of squeezed light.
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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
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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
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413
1
~o,
~s5
~ 0.6
~25
-..
. .
>
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
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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 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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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
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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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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
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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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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
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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,
6, w3]
QUANTUMRELATIVEPHASEFORMALISMS
437
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), 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]).
6, w3]
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 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~
,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)
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OPTICALSOLITONSIN MEDIAWITHA QUADRATICNONLINEARITY
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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
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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"
s2tcs.
(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 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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OPTICALSOLITONSIN MEDIAWITHA QUADRATICNONLINEARITY
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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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SOLITONS IN PLANAR WAVEGUIDES
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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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SOLITONS IN PLANAR WAVEGUIDES
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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].
522
OPTICAL SOLITONS IN MEDIA WITH A QUADRATIC NONLINEARITY
[7, w 4
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
[7, w 4
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
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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
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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
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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
7, w 4]
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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
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OPTICAL SOLITONSIN MEDIAWITHA QUADRATICNONLINEARITY
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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)
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SOLITONSIN PLANARWAVEGUIDES
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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 ]).
7, w 5]
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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 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] 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]
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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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OPTICALSOLITONSIN MEDIAWITHA QUADRATICNONLINEARITY
[7
Torruellas, W .E., G. Assanto, B.L. Lawrence, R.A. Fuerst and G.I. Stegeman, 1996, Appl. Phys. Lett. 78, 1449. Torruellas, W.E., Z. Wang, D.J. Hagan, E.W. Van Stryland, G.I. Stegeman, L. Torner and C.R. Menyuk, 1995, Phys. Rev. Lett. 74, 5036. Torruellas, W.E., Z. Wang, L. Torner and G.I. Stegeman, 1995, Opt. Lett. 20, 1949. Tran, H.T., 1995, Opt. Commun. 118, 581. Trillo, A., and P. Ferro, 1995, Opt. Lett. 20, 438. Trillo, S., A.V. Buryak and Yu.S. Kivshar, 1996, Opt. Commun. 122, 200. Trillo, S., and M. Haelterman, 1998, Opt. Lett. 23, 1514. Turitsyn, S.K., 1995, Pis'ma Zh. Eksp. Teor. Fiz. 61,458. Vakhitov, M.G., and A.A. Kolokolov, 1973, Sov. J. Radiophys. Quantum Electron. 16, 783. Werner, M.J., and P.D. Drummond, 1993, J. Opt. Soc. Am. B 10, 2390.
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