Showing papers in "Journal of The Optical Society of America A-optics Image Science and Vision in 1994"

Discrete-Dipole Approximation For Scattering Calculations

Abstract: The discrete-dipole approximation (DDA) for scattering calculations, including the relationship between the DDA and other methods, is reviewed. Computational considerations, i.e., the use of complex-conjugate gradient algorithms and fast-Fourier-transform methods, are discussed. We test the accuracy of the DDA by using the DDA to compute scattering and absorption by isolated, homogeneous spheres as well as by targets consisting of two contiguous spheres. It is shown that, for dielectric materials (|m| ≲ 2), the DDA permits calculations of scattering and absorption that are accurate to within a few percent.

Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor

Abstract: A Hartmann-Shack wave-front sensor is used to measure the wave aberrations of the human eye by sensing the wave front emerging from the eye produced by the retinal reflection of a focused light spot on the fovea. Since the test involves the measurements of the local slopes of the wave front, the actual wave front is reconstructed by the use of wave-front estimation with Zernike polynomials. From the estimated Zernike coefficients of the tested wave front the aberrations of the eye are evaluated. It is shown that with this method, using a Hartmann-Shack wave-front sensor, one can obtain a fast, precise, and objective measurement of the aberrations of the eye.

Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods

Abstract: Two-dimensional (2D) phase unwrapping continues to find applications in a wide variety of scientific and engineering areas including optical and microwave interferometry, adaptive optics, compensated imaging, and synthetic-aperture-radar phase correction, and image processing. We have developed a robust method (not based on any path-following scheme) for unwrapping 2D phase principal values (in a least-squares sense) by using fast cosine transforms. If the 2D phase values are associated with a 2D weighting, the fast transforms can still be used in iterative methods for solving the weighted unwrapping problem. Weighted unwrapping can be used to isolate inconsistent regions (i.e., phase shear) in an elegant fashion.

Boundary conditions for the diffusion equation in radiative transfer

Abstract: Using the method of images, we examine the three boundary conditions commonly applied to the surface of a semi-infinite turbid medium. We find that the image-charge configurations of the partial-current and extrapolated-boundary conditions have the same dipole and quadrupole moments and that the two corresponding solutions to the diffusion equation are approximately equal. In the application of diffusion theory to frequency-domain photon-migration (FDPM) data, these two approaches yield values for the scattering and absorption coefficients that are equal to within 3%. Moreover, the two boundary conditions can be combined to yield a remarkably simple, accurate, and computationally fast method for extracting values for optical parameters from FDPM data. FDPM data were taken both at the surface and deep inside tissue phantoms, and the difference in data between the two geometries is striking. If one analyzes the surface data without accounting for the boundary, values deduced for the optical coefficients are in error by 50% or more. As expected, when aluminum foil was placed on the surface of a tissue phantom, phase and modulation data were closer to the results for an infinite-medium geometry. Raising the reflectivity of a tissue surface can, in principle, eliminate the effect of the boundary. However, we find that phase and modulation data are highly sensitive to the reflectivity in the range of 80-100%, and a minimum value of 98% is needed to mimic an infinite-medium geometry reliably. We conclude that noninvasive measurements of optically thick tissue require a rigorous treatment of the tissue boundary, and we suggest a unified partial-current--extrapolated boundary approach.

Coupled-mode theory for optical waveguides: an overview

Abstract: The coupled-mode theory (CMT) for optical waveguides is reviewed, with emphasis on the analysis of coupled optical waveguides. A brief account of the recent development of the CMT for coupled optical waveguides is given. Issues raised in the debates of the 1980’s on the merits and shortcomings of the conventional as well as the improved coupled-mode formulations are discussed. The conventional coupled-mode formulations are set up in a simple, intuitive way. The rigorous CMT is established on the basis of a linear superposition of the modes for individual waveguides. The cross-power terms appear logically as a result of modal nonorthogonality. The cross power is necessary for the self-consistency of the CMT for dissimilar waveguides. The nonorthogonal CMT, though more complicated, yields more-accurate results than the conventional orthogonal CMT for most practical applications. It also leads to the prediction of cross talk in directional couplers. The conventional orthogonal CMT is, however, reliably accurate for describing the power coupling between two weakly coupled, nearly identical waveguides. For dissimilar waveguides, a self-consistent orthogonal CMT can be derived by a redefinition of the coupling coefficients, and it predicts the coupling length and therefore the power exchange between the waveguides accurately if the two waveguides are far apart. Three typical coupler configurations—the uniform, the grating-assisted, and the tapered—are examined in detail. The accuracy, scope of validity, limitations, and extensions of the coupled-mode formulations are discussed in conjunction with each configuration. To verify the arguments in the discussions, comparisons with the exact analytical solutions and the rigorous numerical simulations are made.

Human luminance pattern-vision mechanisms: masking experiments require a new model.

Abstract: A widely used model of simultaneous luminance pattern masking is based on mechanisms that sum inputs linearly and produce a response that is an S-shaped function of that sum This model makes two predictions about masking: (1) Changing the masker spatial waveform will shift the threshold-versus-masker contrast function horizontally by a multiplicative constant (2) Adding a second fixed-contrast masker will shift this function horizontally by an additive constant Experimental tests do not support these predictions The results can be explained by a new model that incorporates broadband divisive inhibition, consistent with physiology

Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms

Abstract: A concise introduction to the concept of fractional Fourier transforms is followed by a discussion of their relation to chirp and wavelet transforms. The notion of fractional Fourier domains is developed in conjunction with the Wigner distribution of a signal. Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.

Direct phase determination in hologram interferometry with use of digitally recorded holograms

Abstract: A new method of phase determination in hologram interferometry is described. The Fresnel holograms, which represent the undeformed and the deformed states of the object, are generated on a CCD target and stored electronically. No lens or other imaging device is used. The reconstruction is done from the digitally stored holograms with mathematical methods. It is shown that the intensity as well as the phase can be calculated from the digitally sampled holograms. A comparison of the phases of the undeformed and the deformed states permits direct determination of the interference phase.

Calculation of total cross sections of multiple-sphere clusters

Abstract: A method for calculating the extinction, absorption, and scattering cross sections of clusters of neighboring spheres for both fixed and random orientations is developed. The analysis employs the superposition formulation for radiative interactions among spheres, in which the total field from the cluster is expressed as a superposition of vector spherical harmonic expansions about each of the spheres in the cluster. Through the use of addition theorems a matrix equation for the expansion coefficients is obtained. Further application of addition theorems on the inverse of the coefficient matrix is shown to yield analytical expressions for the orientation-averaged total cross sections of the sphere cluster. Calculations of the cross sections of pairs of spheres and fractal aggregates of several spheres are presented. It is found that a dipole representation of the field in each sphere does not adequately predict the absorption cross section of clusters of small-size-parameter spheres when the spheres are highly conducting. For this situation several multipole orders are required for an accurate calculation of the absorption cross section. In addition, the predicted absorption of sphere clusters can be significantly greater than that estimated from the sum of the isolated-sphere cross sections.

Spectral sharpening: sensor transformations for improved color constancy

Abstract: We develop sensor transformations, collectively called spectral sharpening, that convert a given set of sensor sensitivity functions into a new set that will improve the performance of any color-constancy algorithm that is based on an independent adjustment of the sensor response channels. Independent adjustment of multiplicative coefficients corresponds to the application of a diagonal-matrix transform (DMT) to the sensor response vector and is a common feature of many theories of color constancy, Land’s retinex and von Kries adaptation in particular. We set forth three techniques for spectral sharpening. Sensor-based sharpening focuses on the production of new sensors as linear combinations of the given ones such that each new sensor has its spectral sensitivity concentrated as much as possible within a narrow band of wavelengths. Data-based sharpening, on the other hand, extracts new sensors by optimizing the ability of a DMT to account for a given illumination change by examining the sensor response vectors obtained from a set of surfaces under two different illuminants. Finally in perfect sharpening we demonstrate that, if illumination and surface reflectance are described by two- and three-parameter finite-dimensional models, there exists a unique optimal sharpening transform. All three sharpening methods yield similar results. When sharpened cone sensitivities are used as sensors, a DMT models illumination change extremely well. We present simulation results suggesting that in general nondiagonal transforms can do only marginally better. Our sharpening results correlate well with the psychophysical evidence of spectral sharpening in the human visual system.

Scattering by a random set of parallel cylinders

Abstract: A theory of scattering by a finite number of cylinders of arbitrary cross section is presented. This theory is based on a self-consistent approach that identifies incident and scattered fields around each cylinder and then uses the notion of a scattering matrix in order to get a linear system of equations. Special attention is paid to the simplified case of a sparse distribution of small cylinders for low frequencies. Surprisingly, it is found that the classical rules of homogenization must be modified in that case. The phenomenon of enhanced backscattering of light is investigated from numerical data for a dense distribution of cylinders.

Orthogonal Fourier–Mellin moments for invariant pattern recognition

Abstract: We propose orthogonal Fourier–Mellin moments, which are more suitable than Zernike moments, for scaleand rotation-invariant pattern recognition. The new orthogonal radial polynomials have more zeros than do the Zernike radial polynomials in the region of small radial distance. The orthogonal Fourier–Mellin moments may be thought of as generalized Zernike moments and orthogonalized complex moments. For small images, the description by the orthogonal Fourier–Mellin moments is better than that by the Zernike moments in terms of image-reconstruction errors and signal-to-noise ratio. Experimental results are shown.

Change of polarization of light beams on propagation in free space

Abstract: It is shown by use of a simple model that in general the state of polarization of a light beam generated by a partially coherent source changes as the beam propagates in free space.

Photometric stereo under a light source with arbitrary motion

Abstract: A new photometric-stereo method for estimating the surface normal and the surface reflectance of objects without a priori knowledge of the light-source direction or the light-source intensity is proposed. First, I construct a p × f image data matrix I from p pixel image intensity data through f frames by moving a light source arbitrarily. Under the Lambertian assumption the image data matrix I can be written as the product of two matrices S and L, with S representing the surface normal and the surface reflectance and L representing the light-source direction and the light-source intensity. Using this formulation, I show that the image data matrix I is of rank 3. On the basis of this observation, I use a singular-value decomposition technique and useful constraints to factorize the image data matrix. This method can also be used to treat cast shadows and self-shadows without assumptions. The effectiveness of this method is demonstrated through performance analysis, laboratory experiment, and out-of-laboratory experiment.

Algorithm for the rigorous coupled-wave analysis of grating diffraction

Abstract: Diffraction of light by periodic gratings is analyzed with a characteristic-matrix formalism based on a rigorous coupled-wave approach. This formalism is particularly convenient for modeling the diffraction by nonuniform periodic structures. In order to overcome numerical difficulties that are due to inhomogeneous eigenmodes, we propose a new algorithm that remains stable for gratings of any thickness. We obtain the stability by distinguishing in the computation the growing and the decaying inhomogeneous modes. Numerical examples and comparisons with previous results are given.

Homogeneous and inhomogeneous Jones matrices

Abstract: The classification of polarization properties of polarization elements is studied to derive data-reduction equations for extracting the diattenuation, retardance, and other polarization properties from their Jones matrices. Polarization elements, and Jones matrices as well, are divided into two classes: homogeneous, with orthogonal eigenpolarizations, and inhomogeneous, with nonorthogonal eigenpolarizations. The basic polarization properties, diattenuation and retardance, of homogeneous polarization elements are straightforward and well known; these elements are characterized by their eigenvalues and eigenpolarizations. Polarization properties of inhomogeneous polarization elements are not so evident. By applying polar decomposition, the definitions of diattenuation and retardance are generalized to inhomogeneous polarization elements, providing an understanding of their polarization characteristics. Furthermore, an inhomogeneity parameter is introduced to describe the degree of inhomogeneity in a polarization element. These results are then adapted to degenerate polarization elements, which have only one linearly independent eigenpolarization.

Phase-shifting masks for microlithography: automated design and mask requirements

Abstract: The problem of automated design of phase-shifting masks for enhanced-resolution optical lithography is examined. We propose a computationally viable algorithm for the rapid design of phase-shifting masks for arbitrary two-dimensional patterns. Our approach is based on the use of a class of optimal coherent approximations to partially coherent imaging systems described by the Hopkins model. These approximations lead to substantial computational and analytical benefits, and, in addition, the resultant approximation error can be quite small for imaging systems with coherence factor σ ≤ 0.5. These approximate models allow us to reduce the mask-design problem to the classical phase-retrieval problem in optics. A fast iterative algorithm, closely related to the Gerchberg–Saxton algorithm, is then applied to generate (suboptimal) phase-shifting masks. Analytical results related to practical requirements for phase-shifting masks are also presented. These results address questions related to the number of discrete phase levels required for arbitrary patterns and provide some insight into alternative strategies for the use of phase-shifting masks. A number of simulated phase-shifting mask-design examples are provided to illustrate the methods and ideas presented.

Light propagation through nanometer-sized structures: the two-dimensional-aperture scanning near-field optical microscope

Abstract: The propagation of light through nanometer-sized structures is studied computationally by use of multiple-multipole method. A two-dimensional scanning near-field optical microscope structure is chosen as an example. The relevant near and far fields as well as some imaging properties are determined for the two principal polarizations. Strikingly different results are obtained for the two principal polarizations: for s polarization, strong field confinement in the gap region, high sensitivity of the radiation pattern to the presence of an object, and high contrast; for p polarization, higher signal level with low contrast. At small gap widths a substantial amount of radiation is coupled into the substrate at angles larger than the critical angle. Line scan simulations for λ = 488 nm indicate a resolution of approximately two times the optical slit width. Resolution and contrast can be optimized by the appropriate choice of detector orientation and angle of acceptance. Coherent superposition of the radiation emitted into different directions permits further improvements.

Rigorous Justification of the Localized Approximation to the Beam-Shape Coefficients in Generalized Lorenz-Mie Theory

Abstract: Generalized Lorenz-Mie theory describes electromagnetic scattering of an arbitrary light beam by a spherical particle. The computationally most expensive feature of the theory is the evaluation of the beam-shape coefficients, which give the decomposition of the incident light beam into partial waves. The so-called localized approximation to these coefficients for a focused Gaussian beam is an analytical function whose use greatly simplifies Gaussian-beam scattering calculations. A mathematical justification and physical interpretation of the localized approximation is presented for on-axis beams.

First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes.

Abstract: An approach is presented for evaluating the performance achieved by a closed-loop adaptive-optics system that is employed with an astronomical telescope. This method applies to systems incorporating one or several guide stars, a wave-front reconstruction algorithm that is equivalent to a matrix multiply, and one or several deformable mirrors that are optically conjugate to different ranges. System performance is evaluated in terms of residual mean-square phase distortion and the associated optical transfer function. This evaluation accounts for the effects of the atmospheric turbulence Cn2(h) and wind profiles, the wave-front sensor and deformable-mirror fitting error, the sensor noise, the control-system bandwidth, and the net anisoplanatism for a given constellation of natural and/or laser guide stars. Optimal wave-front reconstruction algorithms are derived that minimize the telescope’s field-of-view-averaged residual mean-square phase distortion. Numerical results are presented for adaptive-optics configurations incorporating a single guide star and a single deformable mirror, multiple guide stars and a single deformable mirror, or multiple guide stars and two deformable mirrors.

Color constancy - Generalized diagonal transforms suffice

Abstract: This study’s main result is to show that under the conditions imposed by the Maloney–Wandell color constancy algorithm, whereby illuminants are three dimensional and reflectances two dimensional (the 3–2 world), color constancy can be expressed in terms of a simple independent adjustment of the sensor responses (in other words, as a von Kries adaptation type of coefficient rule algorithm) as long as the sensor space is first transformed to a new basis. A consequence of this result is that any color constancy algorithm that makes 3–2 assumptions, such as the Maloney–Wandell subspace algorithm, Forsyth’s MWEXT, and the Funt–Drew lightness algorithm, must effectively calculate a simple von Kries-type scaling of sensor responses, i.e., a diagonal matrix. Our results are strong in the sense that no constraint is placed on the initial spectral sensitivities of the sensors. In addition to purely theoretical arguments, we present results from simulations of von Kries-type color constancy in which the spectra of real illuminants and reflectances along with the human-cone-sensitivity functions are used. The simulations demonstrate that when the cone sensor space is transformed to its new basis in the appropriate manner a diagonal matrix supports nearly optimal color constancy.

Relationships between the radon-wigner and fractional fourier transforms

Abstract: Two recently described transforms are shown to be related. The Radon–Wigner transform is the squared modulus of the fractional Fourier transform. This new theorem may serve to translate signal and image processing results between different signal representations. Some consequences regarding moments are presented, including a new fractional-Fourier-transform uncertainty relation. Implications for processing are suggested.

Global color constancy: recognition of objects by use of illumination-invariant properties of color distributions

Abstract: Color pixel distributions provide a useful cue for object recognition but are dependent on scene illumination. We develop an algorithm that assigns color descriptors to an object that depend on the surface properties of the object and not on the illumination. An object is defined by a set of possibly textured surfaces and gives rise to a color pixel distribution. For a trichromatic system, the algorithm assumes a three-dimensional linear model for surface spectral reflectance. There are no assumptions about the contents of the scene and only weak constraints on the illumination. The global color invariants can be computed in an amount of time that is proportional to the number of pixels that define an object. A set of experiments on complex scenes under various illuminants demonstrates that the global color constancy algorithm performs significantly better than previous recognition algorithms based on color distribution.

Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams

Abstract: Generalized Lorenz–Mie theory describes electromagnetic scattering of an arbitrary light beam by a spherical particle. The computationally most expensive feature of the theory is the evaluation of the beam-shape coefficients, which give the decomposition of the incident light beam into partial waves. The so-called localized approximation to these coefficients for a focused Gaussian beam is an analytical function whose use greatly simplifies Gaussian-beam scattering calculations. A mathematical justification and physical interpretation of the localized approximation is presented for on-axis beams.

Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles

Abstract: We extend the rigorous eigenmode theory of binary surface-relief gratings to accommodate three-dimensional-modulation profiles, to formulate a synthesis problem for doubly periodic resonance-domain diffractive elements, and to demonstrate some of the problem’s symmetry properties. Several solutions for multiple beam splitters with ~90% transmission-mode diffraction efficiency are obtained by nonlinear parametric optimization. The polarization sensitivity and the required fabrication accuracy are analyzed for some solutions.

Generalized method of moments for three-dimensional penetrable scatterers

Abstract: We outline a generalized form of the method-of-moments technique. Integral equation formulations are developed for a diverse class of arbitrarily shaped three-dimensional scatterers. The scatterers may be totally or partially penetrable. Specific cases examined are scatterers with surfaces that are perfectly conducting, dielectric, resistive, or magnetically conducting or that satisfy the Leontovich (impedance) boundary condition. All the integral equation formulations are transformed into matrix equations expressed in terms of five general Galerkin (matrix) operators. This allows a unified numerical solution procedure to be implemented for the foregoing hierarchy of scatterers. The operators are general and apply to any arbitrarily shaped three-dimensional body. The operator calculus of the generalized approach is independent of geometry and basis or testing functions used in the method-of-moments approach. Representative numerical results for a number of scattering geometries modeled by triangularly faceted surfaces are given to illustrate the efficacy and the versatility of the present approach.

Confocal microscopy in turbid media.

Abstract: We examine the performance of confocal microscopes designed for probing structures embedded in turbid media. A heuristic scheme is described that combines a numerical Monte Carlo simulation of photon transport in a turbid medium with a geometrical ray trace through the confocal optics. To show the effects of multiple scattering on depth discrimination, we compare results from the Monte Carlo simulations and scalar diffraction theory. Experimental results showing the effects of the pinhole diameter and other variables on imaging performance at various optical depths in suspensions of polystyrene microspheres were found to correspond well with the Monte Carlo simulations. The major conclusion of the paper is that the trade-off between signal level and background scattered-light rejection places a fundamental limit on the sectioning capability of the microscope.

Temporal-color space analysis of reflection

Abstract: We propose a novel method to analyze a sequence of color images. A series of color images is examined in a four-dimensional space, which we call the temporal-color space, whose axes are the three color axes red, green, and blue and one temporal axis. The significance of the temporal-color space lies in its ability to represent the change of image color with time. A conventional color space analysis yields a histogram of the colors in an image, only for an instant of time. Conceptually, the two reflection components from the dichromatic-reflection model, the specular-reflection component and the body-reflection component, form two subspaces in temporal-color space. These two components can be extracted at each pixel in the image locally. Using this fact, we analyzed real color images and separated the two reflection components successfully. We did not make any assumptions about surface properties or the global distribution of surface normals. Finally, object shape was recovered.

Propagation through nonuniform grating structures

Abstract: We consider linear propagation through shallow, nonuniform gratings, such as those written in the core of photosensitive optical fibers. Though, of course, the coupled-mode equations for such gratings are well known, they are often derived heuristically. Here we present a rigorous derivation and include effects that are second order in the grating parameters. While the resulting coupled-mode equations can easily be solved numerically, such a calculation often does not give direct insight into the qualitative nature of the response. Here we present a new way of looking at nonuniform gratings that immediately does yield such insight and, as well, provides a convenient starting point for approximate treatments such as WKB analysis. Our approach, which is completely within the context of coupled-mode theory, makes use of an effective-medium description, in which one replaces the (in general, nonuniform) grating by a medium with a frequency-dependent refractive index distribution but without a grating.

Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings

Abstract: Two-dimensional symmetric and asymmetric subwavelength binary gratings are investigated. A method for determining the three effective indices of a two-dimensional (2-D) subwavelength grating is presented, as well as a theoretical formalization for the effective index parallel with the normal to the surface. It is shown that a 2-D asymmetric binary grating on the surface of a dielectric substrate is analogous to a biaxial thin film. If the grating is symmetric, then the two effective indices perpendicular to the normal are equal, and the grating is analogous to a uniaxial thin film. Using these effective indices and the quarter-wave Tschebyscheff synthesis technique, we designed two- and three-level binary gratings to suppress reflections over a broad band. It is shown that for a substrate index of ns = 3.0 a three-level 2-D binary grating reduced reflections below 0.1% from 8 μm to 12 μm.