Rapid laser solver for the phase retrieval problem

Tailored physical systems were recently exploited to rapidly solve hard computational challenges, such as spin simulators, combinatorial optimization, and focusing through scattering media. Here, we address the phase retrieval problem where an object is reconstructed from its scattered intensity distribution. This is a key problem in many applications, ranging from x-ray imaging to astrophysics, and currently, it lacks efficient direct reconstruction methods: The widely used indirect iterative algorithms are inherently slow. We present an optical approach based on a digital degenerate cavity laser, whose most probable lasing mode rapidly and efficiently reconstructs the object. Our experimental results suggest that the gain competition between the many lasing modes acts as a highly parallel computer that could rapidly solve the phase retrieval problem. Our approach applies to most two-dimensional objects with known compact support, including complex-valued objects, and can be generalized to imaging through scattering media and other hard computational tasks.

Coupled lasers: phase versus chaos synchronization

The synchronization of chaotic lasers and the optical phase synchronization of light originating in multiple coupled lasers have both been extensively studied. However, the interplay between these two phenomena, especially at the network level, is unexplored. Here, we experimentally compare these phenomena by controlling the heterogeneity of the coupling delay times of two lasers. While chaotic lasers exhibit deterioration in synchronization as the time delay heterogeneity increases, phase synchronization is found to be independent of heterogeneity. The experimental results are found to be in agreement with numerical simulations for semiconductor lasers.

Efficient method for controlling the spatial coherence of a laser

An efficient method to tune the spatial coherence of a degenerate laser over a broad range with minimum variation in the total output power is presented. It is based on varying the diameter of a spatial filter inside the laser cavity. The number of lasing modes supported by the degenerate laser can be controlled from 1 to 320,000, with less than a 50% change in the total output power. We show that a degenerate laser designed for low spatial coherence can be used as an illumination source for speckle-free microscopy that is nine orders of magnitude brighter than conventional thermal light.

Phase locking of lasers with self-stabilized minimal coupling

A novel configuration for phase locking two ring lasers with self-stabilized minimal exchange of power between them is presented. We show experimentally that losses introduced between the lasers are self compensated in order to maintain minimal power exchange between them. The experimental results are in good agreement with numerical results.

Enhanced coherence of weakly coupled lasers

We present the phase-locking and coherence properties between two weakly coupled lasers. We show how the degree of coherence between the two lasers can be enhanced by nearly 1 order of magnitude after taking into account the effects of coupling on both their phases as well as their amplitudes. Specifically, correlations between synchronized spikes in the amplitude dynamics and the phase dynamics of the lasers allow for an interference pattern with a fringe visibility of 90%, even when the coupling strength is far below the critical value and they are not phase locked.

Phase locking of lasers with intracavity polarization elements

New configurations for phase locking several laser beams with intracavity polarization elements are presented. With this configuration we demonstrated efficient phase lock of up to 24 ND:YAG laser beams with only two polarization beam displacers.

Anisotropic Poisson's ratio and compression modulus of cortical bone determined by speckle interferometry

Young's modulus and Poisson's ratios of 6mm-sized cubes of equine cortical bone were measured in compression using a micro-mechanical loading device. Surface displacements were determined by electronic speckle pattern-correlation interferometry. This method allows for non-destructive testing of very small samples in water. Analyses of standard materials showed that the method is accurate and precise for determining both Young's modulus and Poisson's ratio. Material properties were determined concurrently in three orthogonal anatomic directions (axial, radial and transverse). Young's modulus values were found to be anisotropic and consistent with values of equine cortical bone reported in the literature. Poisson's ratios were also found to be anisotropic, but lower than those previously reported. Poisson's ratios for the radial-transverse and transverse-radial directions were 0.15+/-0.02, for the axial-transverse and axial-radial directions 0.19+/-0.04, and for the transverse-axial and radial-axial direction 0.09+/-0.02 (mean+/-SD). Cubes located only millimetres apart had significantly different elastic properties, showing that significant spatial variation occurs in equine cortical bone.

Color correction in planar optics configurations

Color correction in planar optics configurations can be achieved by resorting to gradient-index rather than uniform-refractive-index substrates. The basic configuration, principle of correction, and calculated and experimental results are presented. The results reveal that, with an appropriate refractive index distribution along the thickness of the substrates, the color can be corrected over a wavelength range up to 155 nm depending on incidence angles.

Intracavity coherent addition of 16 laser distributions

The efficient intracavity coherent addition of 16 separate laser Gaussian mode distributions is presented. The coherent addition is achieved in a multichannel pulsed Nd:YAG laser resonator by use of four intracavity interferometric beam combiners. The results reveal 88% combining efficiency with a combined output beam of nearly pure Gaussian distribution.

Intracavity coherent addition of single high-order modes

We report on efficient intracavity coherent addition of several single high-order mode distributions in a multichannel laser resonator. The phase locking and coherent addition is achieved by using an intracavity interferometric beam combiner. The principle, configuration, and experimental results with pulsed Nd:YAG Laguerre-Gaussian TEM01 and TEM02 laser beam distributions are presented. The results reveal more than 95% combining efficiency with a nearly pure high-order mode output beam distribution in both free-running and Q-switched operation.

Enhanced two-photon fluorescence excitation by resonant grating waveguide structures

Enhanced two-photon fluorescence (TPF) spectroscopy with novel high-finesse resonant polymeric grating waveguide structures (GWSs) is presented. Under resonant conditions the field enhancement at the surface of a GWS can be exploited for TPF spectroscopy without the need for highly focused laser excitation light. We compare the TPF obtained by placing a drop of tetramethylrhodamine (TMR) on top of a GWS with that obtained with TMR on top of a glass substrate. Our procedure and results indicate that the detection of TPF can be improved by a factor of 10 with resonant GWSs.

Conversion of a high-order mode beam into a nearly Gaussian beam by use of a single interferometric element

We present a new, compact, and practical optical mode converter that efficiently transforms a high-order Hermite-Gaussian (HG) laser beam into a nearly Gaussian beam. The mode converter is based on coherently adding different transverse parts of the high-order mode beam by use of a single planar interferometric element. The method, configuration, and experimental results obtained with a pulsed Nd:YAG HG TEM10 laser beam are presented. The results reveal that the efficiency of conversion of a HG beam to a nearly Gaussian beam can be as high as 90%.

Efficient formation of a high-quality beam from a pure high-order Hermite-Gaussian mode

We present a relatively simple method for efficiently transforming a single high-order mode into a nearly Gaussian beam of much higher quality. The method is based on dividing the mode into equal parts that are then combined coherently. We illustrate the method by transforming a Hermite-Gaussian (1, 0) mode with M(x)(2)=3 into a nearly Gaussian beam with M(x)(2)=1. 045 . Experimental results are presented and compared with theoretical results.

Flatland optics. III. Achromatic diffraction

In the previous two sections of "Flatland optics" [J. Opt. Soc. Am. A 17, 1755 (2000); 18, 1056 (2001)] we described the basic principles of two-dimensional (2D) optics and showed that a wavelength lambda in three-dimensional (3D) space (x, y, z) may appear in Flatland (x, z) as a wave with another wavelength Lambda=lambda/cos alpha. The tilt angle alpha can be modified by a 3D-Spaceland individual, who then is able to influence the 2D optics in a way that must appear to be magical to 2D-Flatland individuals-in the spirit of E. A. Abbott's science fiction story of 1884 [Flatland, a Romance of Many Dimensions, 6th ed. (Dover, New York, 1952)]. Here we show how the light from a white source can be perceived in Flatland as perfectly monochromatic, so diffraction with white light will be free of color blurring and the contrast of interference fringes can be 100%. The basic considerations for perfectly achromatic diffraction are presented, along with experimental illustration of Talbot self-imaging performed with broadband illumination.

Flatland optics. II. Basic experiments

In "Flatland optics: fundamentals" [J. Opt. Soc. Am. A 17, 1755 (2000)] we described the basic principles of two-dimensional (2D) optics and showed that a wavelength lambda in three-dimensional (3D) space (x,y,z) may appear in Flatland (x,z) as a wave with another wavelength, lambda = lambda/cosalpha. The tilt angle alpha can be modified by a 3D (Spaceland) individual who then is able to influence the 2D optics in a way that must appear to be magical to 2D Flatland individuals-in the spirit of E. A. Abbott's science fiction story [Flatland, a Romance of Many Dimensions, 6th ed. (Dover, New York, 1952)] of 1884. We now want to establish the reality or objectivity of the 2D wavelength lambda by some basic experiments similar to those that demonstrated roughly 200 years ago the wave nature of light. Specifically, we describe how to measure the 2D wavelength lambda by mean of five different arrangements that involve Young's biprism configuration, Talbot's self-imaging effect, measuring the focal length of a Fresnel zone plate, and letting light be diffracted by a double slit and by a grating. We also performed experiments with most of these arrangements. The results reveal that the theoretical wavelength, as predicted by our Flatland optics theory, does indeed coincide with the wavelength lambda as measured by Flatland experiments. Finally, we present an alternative way to understand Flatland optics in the spatial frequency domains of Flatland and Spaceland.

Achromatic phase retarder by slanted illumination of a dielectric grating with period comparable with the wavelength

We discuss some properties of dielectric gratings with period comparable with the illuminating wavelength for slanted illumination (this illumination geometry is often referred to as concical mounting). We demonstrate the usefulness of such an illuminating geometry. We show that the threshold period (under which only the zeroth transmission and reflection orders are nonevanescent) can be significantly higher, thereby easing fabrication constraints, and that this illumination setup makes it possible to design achromatic phase retarders. Such a design, for an achromatic quarter-wave plate with lambda/60 uniformity of the retardation phase in the 0.47-0.63-mum wavelength interval, is demonstrated.

Wigner formulation of optical processing with light of arbitrary coherence

A unified mathematical formulation for designing and analyzing even the most general optical processor is presented. It exploits the Wigner distribution function to characterize the illumination, the input, the inherent filter, and the output results. To characterize the propagation of the light through the optical processor setup, we exploit the Wigner matrix formalism, which is appealing because it allows simple geometric analysis. The Wigner distribution function was extended to include illumination of arbitrary coherence so that processors using either coherent light or partially coherent light can be designed and analyzed with the same Wigner formalism. The basic principles, design, and analysis of the imaging and Fourier-transform operations and use of the Wigner formalism to evaluate the performance and tolerances of optical processors are presented.

General linear optical coordinate tranformations

New optical configurations for performing general coordinate transformation operations of shear, rotation, and their combination are presented. These configurations consist of refractive spherical and cylindrical lenses that are readily available. Typically, high-resolution imagery can be obtained, depending on the size of the input object, the illumination wavelength, and the f-number of the lenses. Basic and more general configurations are presented, along with experimental results clearly showing image shearing, rotation, and a combination of these with high-quality output imagery.

Flatland optics: fundamentals

"Flatland" is the title of a 120-year-old science fiction story. It describes the life of creatures living in a two-dimensional (2D) Flatland. A superior creature living in the three-dimensional (3D) spaceland, as we do, can easily inspect, for example, the inside of a Flatland house, as well as the content of a flat man's stomach without leaving any trace. Furthermore, the 3D person has supernatural powers that enable him to change the laws of physics in Flatland. We present here the concept of a 2D Flatland optics with one transversal coordinate x and one longitudinal coordinate z. The other transversal coordinate y allows total inspection of Flatland optics, and the freedom to change the wavelength, without using something like nonlinear optics or a Doppler shift. Monochromatic 3D light can be converted reversibly into polychromatic 2D light. A large variety of 2D systems and 2D effects will be presented here and in follow-up contributions. An epilogue faces the question, how "real" is Flatland optics?

Continuous-phase elements can improve laser beam quality

Siegman [Opt. Lett. 18, 675 (1993)] showed that binary-phase plates cannot improve laser beam quality. We demonstrate that continuous spiral phase elements can improve the quality of beams that originate from a laser operating with a pure high-order transverse mode. A theoretical analysis is presented, along with experimental results obtained with a CO(2) laser. The results reveal that a nearly optimal Gaussian output beam can be obtained with only a small decrease in the output power.

Apochromatic optical correlation

Optical correlation, or matched filtering, can now be applied more widely than before, because the light is now allowed to be totally incoherent, spatially and spectrally. Two such correlators were demonstrated recently. Their state of chromatic correction can be called achromatic, since the scaling error has two zero crossings within the visible range of wavelengths. We present a new apochromatic correlator, in which the scaling error has three zero crossings. The maximum error and the rms error are reduced by a factor of 5. Our apochromatic correlator is composed of two highly dispersive heavy flint lenses that are in contact with two diffractive lenses and two chromatic corrected refractive lenses. The uncommon combination of flint dispersion and diffractive dispersion enabled us to achieve apochromatic correction of the scaling factor of the correlator.

Anomaly in a high-numerical-aperture diffractive focusing lens

We show an anomalous behavior in a diffractive lens in which the spot size at the focus reaches a minimum at a numerical aperture of ~0.5 and then increases significantly at higher values. Theoretical and experimental results are presented, along with a comparison with refractive aplanatic lenses, in which the anomaly does not appear to exist.

Optical correlation with totally incoherent light

An optical correlator that can operate with totally incoherent light is presented. Such a correlator can be designed to compensate completely for the inherent chromatic aberrations by resorting to elements with specialized, possibly impractical, dispersion characteristics. Nevertheless, a practical configuration that exploits available achromatic lenses and Fresnel zone plates was designed, built, and tested experimentally. The results reveal that detectable correlation peaks can be obtained with totally incoherent white light. The designs, experimental procedures, and results are presented.

Three-dimensional optical metrology with color-coded extended depth of focus

A novel method of rapid three-dimensional optical metrology that is based on triangulation of a configuration of color-coded light stripes is presented. The method exploits polychromatic illumination and a combined diffractive-refractive element, so the incident light is focused upon a stripe that is axially dispersed, greatly increasing the depth-measuring range without any decrease in the axial or the lateral resolution. The discrimination of each color stripe is further improved by spectral coding and decoding techniques. An 18-fold increase in the depth of focus was experimentally obtained while diffraction-limited light stripes were completely maintained.

Computer-originated planar holographic optical elements

We present novel, to our knowledge, methods for the analytical design and recording of planar holographic optical elements in thick materials. The recording of each planar holographic element is done by interference of two aspherical waves that are derived from appropriately designed computer-generated holograms such that the element has the desired grating function for minimizing aberrations and closely fulfills the Bragg condition over its entire area. The design and recording methods are described, along with calculated results of representative elements.

Compact planar optical correlator

A compact planar configuration for performing optical correlation is described. It comprises two pairs of identical holographic lenses and a holographic filter. Each pair of lenses is recorded on a single substrate; together the lenses perform exact Fourier transformation. The light between the lenses propagates inside the substrate by total internal reflection. The design and recording considerations for each of the elements along with experimental results for the overall planar correlator are presented.

Modeling supra-molecular helices: extension of the molecular surface recognition algorithm and application to the protein coat of the tobacco mosaic virus

Geometric matching of molecular surfaces appears to be essential for the formation of binary molecular complexes and of supra-molecular aggregates. The structure of a binary complex is characterized by the best geometric match, whereas the structure of an aggregate is characterized by the best combined match, i.e. the sum of all the internal matches in the system. We describe a method to identify and quantify the binary matches between molecules and then use them to form the supra-molecular helices and evaluate them. This method is applied to the single protein subunit of tobacco mosaic virus. It successfully predicts the structure of the helical protein coat of the virus and the structure of the disk that is formed as the initial step in the virus assembly process. It also predicts structural intermediates, between disk and helix, which explain how the disk can transform into a helix without dissociating into subunits.

Compact optical crossbar switch

A novel compact holographic crossbar architecture based on planar optics is presented. It consists of a pair of identical planar holographic elements, a two-dimensional array (N x N) transmission mask, a one-dimensional array (N) of input light sources, and a one-dimensional array (N) of detectors. Each planar element contains two cylindrical holographic lenses, both of which are recorded on a single glass substrate. The design of the overall compact configuration is presented along with experimental results.

Light modulation with resonant grating-waveguide structures

Resonant grating-waveguide structures formed with InP/InGaAsP semiconductor materials were tested to show light modulation at a wavelength of 1.55 microm. Narrow, subnanometer resonant spectral bandwidths and a ratio of ref lected intensities between resonance and away from resonance of greater than 50 were measured. For a resonant structure with an area of 3 mm x 3 mm, the modulation frequency reached 5 MHz.

Visor-display design based on planar holographic optics

A method for designing and recording visor displays based on planar holographic optics is presented. This method can deal with the problem of recording-readout wavelength shift. The display system is composed of two holographic optical elements that are recorded on the same substrate. One element collimates the waves from each data point in the display into a plane wave that is trapped inside the substrate by total internal reflection. The other diffracts the plane waves into the eye of an observer. Because the chromatic dispersion of the first element can be corrected by the dispersion of the second, this configuration is relatively insensitive to source wavelength shifts. The method is illustrated by the design, recording, and testing of a compact holographic doublet visor display. The recording was at a wavelength of 458 nm, and readout was at 633 nm. The results indicate that diffraction-limited performance and relatively low chromatic dispersion over a wide field of view can be obtained.

Fourier transformation with a planar holographic doublet

The use of a single off-axis holographic lens for Fourier transformation results in phase errors that degrade its performance. A configuration of two identical off-axis holographic elements is proposed for performing Fourier transformation without phase errors. Such a configuration can be readily folded to form a compact and cascadable element that can be conveniently incorporated into optical correlators. The grating functions of the holographic elements needed for performing the desired transformation were used to record the planar configuration, which was then experimentally evaluated. The results reveal that phase errors were indeed eliminated, in close agreement with the calculations.

Spatial-frequency response of holographic gratings photodeposited from inorganic colloids

Surface photodeposition is a photon-assisted process by which thin films are formed on substrates immersed in colloid solutions. We experimentally evaluate the resolution capabilities of the photodeposition process with amorphous selenium colloids by recording holographic gratings at different spatial frequencies, up to 2200 lines/mm. The experimental diffraction efficiencies are analyzed in terms of a theoretical model, which relates the spatial-frequency response to optical recording parameters and colloid particle sizes. The maximal experimental diffraction efficiency reaches 13% with a spatial frequency of f = 1100 lines/mm. The diffraction efficiencies decrease monotonically with spatial frequency, and drop to half of the maximal diffraction efficiency at f ≈ 1500 lines/mm. These resolution capabilities are achieved with colloid particle sizes extending up to 80 nm. The theoretical derivation indicates that to obtain spatial frequencies above 3000 lines/mm, one should restrict the colloid particle size to a(max) ≤ 30 nm.

Reflective and refractive systems for general two-dimensional beam transformations

A method for designing reflective and refractive surfaces that perform general transformations on two-dimensional beams is presented. In some cases the shape of the surfaces is represented by a simple integral of an analytic expression, whereas in other specific cases it is represented as a solution of a Poisson-like equation. Finally, the possible use of noncontinuous surfaces (facets) is discussed and evaluated quantitatively. Some of the novel techniques developed are also applicable for beam transformations that are realized with diffractive systems.

Analytic design of hybrid diffractive-refractive achromats

An analytic design of hybrid achromats that combine refractive and diffractive elements is presented. The design procedure does not rely on paraxial approximations and involves two separate stages. In the first stage the chromatic aberrations are corrected for the paraxial rays, and in the second stage the spherical aberrations are corrected by addition of an aspherical phase function to the diffractive element. The residual spherochromatic aberrations of the achromat are evaluated both analytically and numerically, with good agreement between the results. Finally, we illustrate the design procedure by designing a plano-convex achromat for IR radiation with little chromatic dispersion.

Compact holographic beam expander

A compact planar beam expander, composed of two holographic lenses that are recorded on a signal holographic plate, is presented. The smaller lens converts a narrow input beam into a diverging spherical wave at a high offaxis angle. This wave propagates toward the second lens, undergoing total internal reflections within the plate, and emerges from the larger lens as a magnified plane wave. One such expander, having a magnification of 3.5, was designed and recorded. The design and the recording procedure, along with the experimental results, are given.

White-light holographic display based on planar optics

A novel method for designing, recording, and reconstructing a planar-optics display holographic doublet, composed of two holograms on the same plate, is presented. The first hologram traps a light from a white-light source into the substrate by total internal reflection, whereas the second hologram couples out the trapped light from the plate to form the reconstructed three-dimensional image. The method is illustrated with a holographic doublet recorded at 488 nm and read out with a white-light source. A sharp image with high efficiency was reconstructed.

On the limits of optical interconnects

The density capabilities of free-space optical interconnects are analyzed by applying Gabor's theory of information. It is shown that it is possible to increase the space-bandwidth product capabilities of space-variant interconnect schemes if they have symmetry properties. Several examples of such symmetries (locality, separability and smoothness) are discussed in detail, together with some experimental results.

Realization of perfect shuffle and inverse perfect shuffle transforms with holographic elements

Techniques for implementing perfect shuffle and inverse perfect shuffle operations with the aid of a single holographic optical element are presented. The element is composed of subholographic lenses which operate on a different input area. For the inverse perfect shuffle operation, polarization coding is added in order to separate the input into distinct groups. Experimental results illustrating the effectiveness of the proposed techniques are given.

Molecular surface recognition: determination of geometric fit between proteins and their ligands by correlation techniques

A geometric recognition algorithm was developed to identify molecular surface complementarity. It is based on a purely geometric approach and takes advantage of techniques applied in the field of pattern recognition. The algorithm involves an automated procedure including (i) a digital representation of the molecules (derived from atomic coordinates) by three-dimensional discrete functions that distinguishes between the surface and the interior; (ii) the calculation, using Fourier transformation, of a correlation function that assesses the degree of molecular surface overlap and penetration upon relative shifts of the molecules in three dimensions; and (iii) a scan of the relative orientations of the molecules in three dimensions. The algorithm provides a list of correlation values indicating the extent of geometric match between the surfaces of the molecules; each of these values is associated with six numbers describing the relative position (translation and rotation) of the molecules. The procedure is thus equivalent to a six-dimensional search but much faster by design, and the computation time is only moderately dependent on molecular size. The procedure was tested and validated by using five known complexes for which the correct relative position of the molecules in the respective adducts was successfully predicted. The molecular pairs were deoxyhemoglobin and methemoglobin, tRNA synthetase-tyrosinyl adenylate, aspartic proteinase-peptide inhibitor, and trypsin-trypsin inhibitor. A more realistic test was performed with the last two pairs by using the structures of uncomplexed aspartic proteinase and trypsin inhibitor, respectively. The results are indicative of the extent of conformational changes in the molecules tolerated by the algorithm.

Optical coordinate transformations

A novel technique for designing holographic optical elements that can perform general types of coordinate transformation is presented. The design is based on analytic ray-tracing techniques for finding the grating vector of the element, from which the holographic grating function is obtained as a solution of a Poissonlike equation. The grating function can be formed either as a computer-generated or as a computer-originated hologram. The design and realization procedure are illustrated for a specific holographic element that performs a logarithmic coordinate transformation on two-dimensional patterns.

All-optical bipolar neural network with polarization-modulating neurons

A general all-optical implementation scheme for bipolar neural networks is presented. Bipolar operation is realized by exploiting two orthogonal light polarizations. A three-layer feed-forward bipolar network is demonstrated experimentally by using a liquid-crystal light valve for implementing an array of polarization modulating neurons.

Rotation-invariant correlation with incoherent light

We have incorporated an algorithm for rotation invariance, based on circular-harmonic expansion, into a pattern recognition system that uses incoherent light. Simulations and experimental results reveal that the algorithm can provide adequate peak-to-background ratios regardless of the input rotation.

Heterostructure multilevel binary optics

A method for forming multilevel diffractive elements (kinoforms) that have highly accurate level heights so as to obtain high diffraction efficiencies is presented. The method, which leads to heterostructure multilevel binary optical elements, relies on conventional deposition technology, selective etching, and multimask lithography. As an illustration, a reflective multilevel element for 10.6-microm radiation is designed, recorded, and tested.

Holographic axilens: high resolution and long focal depth

We report a novel aspheric holographic optical element, the holographic axilens, for achieving extended focal depth while keeping high lateral resolution. The element is designed according to special optimization techniques and recorded as a computer-generated hologram. The results for a specific element, which has a depth of focus of 30 mm, a lateral resolution of 80 microm, a focal length of 1250 mm, and a diameter of 12.5 mm at a wavelength of 633 nm, are presented.

Efficient multilevel phase holograms for CO(2) lasers

Multilevel phase holograms for monochromatic radiation at a wavelength of 10.6 microm are recorded as surface relief gratings with multilevel discrete binary steps. Our experiments show that diffraction efficiencies close to 90% can be achieved both for transmissive and reflective elements. The reduction of efficiency due to errors in the depth and the width of the step levels is considered.

Cross-induced polarization changes in single-mode fibers

The polarization changes induced by a strong wave on a copropagating weaker wave in a single-mode birefringent fiber are investigated. Quantitative analysis and measurements are presented for the dependence of these polarization changes on the relative input state of polarization of the two waves as well as on their frequency separation.