Showing papers in "Journal of The Optical Society of America A-optics Image Science and Vision in 1995"

Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings

Abstract: The rigorous coupled-wave analysis technique for describing the diffraction of electromagnetic waves by periodic grating structures is reviewed. Formulations for a stable and efficient numerical implementation of the analysis technique are presented for one-dimensional binary gratings for both TE and TM polarization and for the general case of conical diffraction. It is shown that by exploitation of the symmetry of the diffraction problem a very efficient formulation, with up to an order-of-magnitude improvement in the numerical efficiency, is produced. The rigorous coupled-wave analysis is shown to be inherently stable. The sources of potential numerical problems associated with underflow and overflow, inherent in digital calculations, are presented. A formulation that anticipates and preempts these instability problems is presented. The calculated diffraction efficiencies for dielectric gratings are shown to converge to the correct value with an increasing number of space harmonics over a wide range of parameters, including very deep gratings. The effect of the number of harmonics on the convergence of the diffraction efficiencies is investigated. More field harmonics are shown to be required for the convergence of gratings with larger grating periods, deeper gratings, TM polarization, and conical diffraction.

Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach

Abstract: An enhanced, numerically stable transmittance matrix approach is developed and is applied to the implementation of the rigorous coupled-wave analysis for surface-relief and multilevel gratings. The enhanced approach is shown to produce numerically stable results for excessively deep multilevel surface-relief dielectric gratings. The nature of the numerical instability for the classic transmission matrix approach in the presence of evanescent fields is determined. The finite precision of the numerical representation on digital computers results in insufficient accuracy in numerically representing the elements produced by inverting an ill-conditioned transmission matrix. These inaccuracies will result in numerical instability in the calculations for successive field matching between the layers. The new technique that we present anticipates and preempts these potential numerical problems. In addition to the full-solution approach whereby all the reflected and the transmitted amplitudes are calculated, a simpler, more efficient formulation is proposed for cases in which only the reflected amplitudes (or the transmitted amplitudes) are required. Incorporating this enhanced approach into the implementation of the rigorous coupled-wave analysis, we obtain numerically stable and convergent results for excessively deep (50 wavelengths), 16-level, asymmetric binary gratings. Calculated results are presented for both TE and TM polarization and for conical diffraction.

Karhunen–Loève procedure for gappy data

Abstract: The problem of using the Karhunen–Loeve transform with partial data is addressed. Given a set of empirical eigenfunctions, we show how to recover the modal coefficients for each gappy snapshot by a least-squares procedure. This method gives an unbiased estimate of the data that lie in the gaps and permits gaps to be filled in a reasonable manner. In addition, a scheme is advanced for finding empirical eigenfunctions from gappy data. It is shown numerically that this procedure obtains spectra and eigenfunctions that are close to those obtained from unmarred data.

Blind deconvolution by means of the Richardson-Lucy algorithm.

Abstract: A blind deconvolution algorithm based on the Richardson–Lucy deconvolution algorithm is presented. Its performance in the presence of noise is found to be superior to that of other blind deconvolution algorithms. Results are presented and compared with results obtained from implementation of a Weiner filter blind deconvolution algorithm. The algorithm is developed further to incorporate functional forms of the point-spread function with unknown parameters. In the presence of noise the point-spread function can be evaluated with 1.0% error, and the object can be reconstructed with a quality near that of the deconvolution process with a known point-spread function.

Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation

Abstract: The diffraction of electromagnetic waves for light focused by a high numerical aperture lens from a first material into a second material is treated. The second material has a different refractive index from that of the first material and introduces spherical aberration. We solve the diffraction problem for the case of a planar interface between two isotropic and homogeneous materials with this interface perpendicular to the optical axis. The solution is obtained in a rigorous mathematical manner, and it satisfies the homogeneous wave equation. The electric and magnetic strength vectors are determined in the second material. The solution is in a simple form that can be readily used for numerical computation. A physical interpretation of the results is given, and the paraxial approximation of the solution is derived.

Stratification of three-dimensional vision: projective, affine, and metric representations

Abstract: A conceptual framework is provided in which to think of the relationships between the three-dimensional structure of physical space and the geometric properties of a set of cameras that provide pictures from which measurements can be made. We usually think of physical space as being embedded in a three-dimensional Euclidean space, in which measurements of lengths and angles do make sense. It turns out that for artificial systems, such as robots, this is not a mandatory viewpoint and that it is sometimes sufficient to think of physical space as being embedded in an affine or even a projective space. The question then arises of how to relate these models to image measurements and to geometric properties of sets of cameras. It is shown that, in the case of two cameras, a stereo rig, the projective structure of the world can be recovered as soon as the epipolar geometry of the stereo rig is known and that this geometry is summarized by a single 3 × 3 matrix, which is called the fundamental matrix. The affine structure can then be recovered if to this information is added a projective transformation between the two images that is induced by the plane at infinity. Finally, the Euclidean structure (up to a similitude) can be recovered if to these two elements is added the knowledge of two conics (one for each camera) that are the images of the absolute conic, a circle of radius -1 in the plane at infinity. In all three cases it is shown how the three-dimensional information can be recovered directly from the images without explicit reconstruction of the scene structure. This defines a natural hierarchy of geometric structures, a set of three strata that is overlaid upon the physical world and that is shown to be recoverable by simple procedures that rely on two items, the physical space itself together with possibly, but not necessarily, some a priori information about it, and some voluntary motions of the set of cameras.

Fractional Fourier optics

Abstract: There exists a fractional Fourier-transform relation between the amplitude distributions of light on two spherical surfaces of given radii and separation. The propagation of light can be viewed as a process of continual fractional Fourier transformation. As light propagates, its amplitude distribution evolves through fractional transforms of increasing order. This result allows us to pose the fractional Fourier transform as a tool for analyzing and describing optical systems composed of an arbitrary sequence of thin lenses and sections of free space and to arrive at a general class of fractional Fourier-transforming systems with variable input and output scale factors.

Image merging and data fusion by means of the discrete two-dimensional wavelet transform

Abstract: A new technique is developed for the merging and data fusion of two images. Two spatially registered images with differing spatial resolutions and color content are merged by combining multiresolution wavelet-decomposition components from each and then reconstructing the merged image by means of the inverse wavelet transform. The wavelet merger can employ a variety of wavelet bases, but in presentation of the concept, simple orthonormal sets—Haar and Daubechies wavelets—are explored. The wavelet technique is compared with the intensity–hue–saturation merging technique by means of multispectral and panchromatic test images. The results of the comparison show the wavelet merger performing better in combining and preserving spectral–spatial information for the test images.

Boundary conditions for diffusion of light.

Abstract: In connection with recent work on remote imaging of random media by light, a straightforward generalization of the proper diffusion boundary conditions is presented that takes into account Fresnel reflection. The Milne problem at exterior boundaries is solved for various values of index of refraction, absorption, and scattering anisotropy parameters to yield extrapolated end points and extrapolation distances. A generalized interface condition is derived to replace the usual condition of continuity of intensity. Benchmark-quality numerical results are given for the extrapolation distance and for the new index-dependent parameter in the interface conditions. Difficulties in using the extrapolated end point when the index is sufficiently large are discussed, and a new image procedure suitable for this case is presented.

Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance

Abstract: Figures of merit for image quality are derived on the basis of the performance of mathematical observers on specific detection and estimation tasks. The tasks include detection of a known signal superimposed on a known background, detection of a known signal on a random background, estimation of Fourier coefficients of the object, and estimation of the integral of the object over a specified region of interest. The chosen observer for the detection tasks is the ideal linear discriminant, which we call the Hotelling observer. The figures of merit are based on the Fisher information matrix relevant to estimation of the Fourier coefficients and the closely related Fourier crosstalk matrix introduced earlier by Barrett and Gifford [Phys. Med. Biol. 39, 451 (1994)]. A finite submatrix of the infinite Fisher information matrix is used to set Cramer-Rao lower bounds on the variances of the estimates of the first N Fourier coefficients. The figures of merit for detection tasks are shown to be closely related to the concepts of noise-equivalent quanta (NEQ) and generalized NEQ, originally derived for linear, shift-invariant imaging systems and stationary noise. Application of these results to the design of imaging systems is discussed.

Optimal design for antireflective tapered two-dimensional subwavelength grating structures

Abstract: Techniques for the design of continuously tapered two-dimensional (2D) subwavelength surface-relief grating structures for broadband antireflection surfaces are investigated. It has been determined that the Klopfenstein taper [ Proc. IRE44, 31 ( 1956)] produces the optimum graded-index profile with the smallest depth for any specified minimum reflectance. A technique is developed to design the equivalent tapered subwavelength surface-relief grating structure by use of 2D effective-medium theory. An optimal Klopfenstein tapered 2D subwavelength grating is designed to reduce the Fresnel reflections by 20 dB over a broad band from an air–substrate (ns = 3.0) interface. The performance is verified by use of both a 2D effective-medium-theory simulation algorithm and rigorous coupled-wave analysis. These structures are also shown to achieve this low reflectance over a wide field of view (θFOV > 110°). The pyramidal spatial profile, which has generally been assumed to produce the optimal broadband antireflection grating structure, is shown to require a significantly larger depth to achieve the same performance as a Klopfenstein-designed tapered antireflection grating structure.

Phase shifting for nonsinusoidal waveforms with phase-shift errors

Abstract: In phase measurement systems that use phase shifting techniques, phase errors that are due to nonsinusoidal waveforms can be minimized by applying synchronous phase shifting algorithms with more than four samples. However, when the phase shift calibration is inaccurate, these algorithms cannot eliminate the effects of nonsinusoidal characteristics. It is shown that, when a number of samples beyond one period of a waveform such as a fringe pattern are taken, phase errors that are due to the harmonic components of the waveform can be eliminated, even when there exists a constant error in the phase shift interval. A general procedure for constructing phase shifting algorithms that eliminate these errors is derived. It is shown that 2j + 3 samples are necessary for the elimination of the effects of higher harmonic components up to the jth order. As examples, three algorithms are derived, in which the effects of harmonic components of low orders can be eliminated in the presence of a constant error in the phase shift interval.

Partially coherent fields, the transport-of-intensity equation, and phase uniqueness

Abstract: Recent papers have shown that there are different coherent and partially coherent fields that may have identical intensity distributions throughout space. On the other hand, the well-known transport-of-intensity equation allows the phase of a coherent field to be recovered from intensity measurements, and the solution is widely held to be unique. A discussion is given on the recovery of the structure of both coherent and partially coherent fields from intensity measurements, and we reconcile the uniqueness question by showing that the transport-of-intensity equation has a unique solution for the phase only if the intensity distribution has no zeros.

Wave-front temporal spectra in high-resolution imaging through turbulence

Abstract: A general formulation is given for the derivation of theoretical temporal power spectra of quantities related to turbulent wave-front phase. These temporal power spectra and their asymptotic power laws and cutoff frequencies are presented for various quantities of interest in the field of interferometry (differential piston), wave-front sensing (Shack–Hartmann and curvature sensor), adaptive optics (Zernike polynomials), and seeing monitoring (differential angle of arrival). We show that the differential piston spectrum has two cutoff frequencies and exhibits a very steep decrease at high frequencies. The curvature sensor is shown to be much less sensitive than the Shack–Hartmann sensor to the low temporal frequencies. A study of the Zernike temporal power spectra shows that their cutoff frequencies increase with the polynomial radial degree. Both single-layer and multilayer plane and spherical waves are considered. The effect of wind direction is also taken into account. We point out the influence of the cone effect on the temporal power spectra when Rayleigh or sodium laser guide stars are used for wave-front sensing. The cone effect results in a temporal decorrelation between natural and laser guide star wave fronts. Finally, we demonstrate that in adaptive optics systems low-order modes require higher servoloop bandwidths than do high-order modes in order for the residual variance to be balanced between the corrected modes. The same conclusion applies to fringe tracking in large telescope interferometers equipped with adaptive optics systems.

Representation of videokeratoscopic height data with Zernike polynomials.

Abstract: Videokeratoscopic data are generally displayed as a color-coded map of corneal refractive power, corneal curvature, or surface height. Although the merits of the refractive power and curvature methods have been extensively debated, the display of corneal surface height demands further investigation. A significant drawback to viewing corneal surface height is that the spherical and cylindrical components of the cornea obscure small variations in the surface. To overcome this drawback, a methodology for decomposing corneal height data into a unique set of Zernike polynomials is presented. Repeatedly removing the low-order Zernike terms reveals the hidden height variations. Examples of the decomposition-and-display technique are shown for cases of astigmatism, keratoconus, and radial keratotomy.

Time course of chromatic adaptation for color-appearance judgments.

Abstract: Observer production of achromatic appearance has previously been used to measure the time course of chromatic adaptation for changes from daylight to incandescent illuminants at constant luminance, indicating an exponential decay of chromatic adaptation with a time constant of the order of 10 s. The work extends previous results in several ways. The psychophysical technique was significantly improved to provide more reliable estimates of color appearance as a function of adaptation duration, and the time course of chromatic adaptation was measured for six chromaticity changes. Three observers tracked achromatic appearance on a computer-controlled CRT display during transitions of 2-min duration between the various chromaticities. The results indicate that observer differences are statistically significant. However, differences in time course for different chromaticity changes are not statistically significant (within observer). Single or piecewise exponential decay functions cannot be fitted to the data. However, sum-of-two-exponentials functions provided accurate descriptions of the data. The results suggest two stages of adaptation: one extremely rapid (a few seconds) and the other somewhat slower (approximately 1 min). Chromatic adaptation at constant luminance was 90% complete after approximately 60 s.

Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings

Abstract: A computationally efficient implementation of rigorous coupled-wave analysis is presented The eigenvalue problem for a one-dimensional grating in a conical mounting is reduced to two eigenvalue problems in the corresponding nonconical mounting This reduction yields two n × n matrices to solve for eigenvalues and eigenvectors, where n is the number of orders retained in the computation For a two-dimensional grating, the size of the matrix in the eigenvalue problem is reduced to 2n × 2n These simplifications reduce the computation time for the eigenvalue problem by 8–32 times compared with the original computation time In addition, we show that with rigorous coupled-wave analysis one analytically satisfies reciprocity by retaining the appropriate choice of spatial harmonics in the analysis Numerical examples are given for metallic lamellar gratings, pulse-width-modulated gratings, deep continuous surface-relief gratings, and two-dimensional gratings

Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials

Abstract: A new technique is proposed for the recovery of optical phase from intensity information. The method is based on the decomposition of the transport-of-intensity equation into a series of Zernike polynomials. An explicit matrix formula is derived, expressing the Zernike coefficients of the phase as functions of the Zernike coefficients of the wave-front curvature inside the aperture and the Fourier coefficients of the wave-front boundary slopes. Analytical expressions are given, as well as a numerical example of the corresponding phase retrieval matrix. This work lays the basis for an effective algorithm for fast and accurate phase retrieval.

Odd aberrations and double-pass measurements of retinal image quality

Abstract: We investigated the formation of the aerial image in the double-pass method to measure the optical quality of the human eye. We show theoretically and empirically that the double pass through the eye’s optics forces the light distribution in the aerial image to be an even-symmetric function even if the single-pass point-spread function is asymmetric as a result of odd aberrations in the eye. The reason for this is that the doublepass imaging process is described by the autocorrelation rather than the autoconvolution of the single-pass point-spread functions, as has been previously assumed. This implies that although the modulation transfer function can be computed from the double-pass aerial image, the phase transfer function cannot. We also show that the lateral chromatic aberration of the eye cannot be measured with the double-pass procedure because it is canceled by the second pass through the eye’s optics.

Depth profiling of laser-heated chromophores in biological tissues by pulsed photothermal radiometry

Abstract: A solution method is proposed to the inverse problem of determining the unknown initial temperature distribution in a laser-exposed test material from measurements provided by infrared radiometry. A Fredholm integral equation of the first kind is derived that relates the temporal evolution of the infrared signal amplitude to the unknown initial temperature distribution in the exposed test material. The singular-value decomposition is used to demonstrate the severely ill-posed nature of the derived inverse problem. Three inversion methods are used to estimate solutions for the initial temperature distribution. A nonnegatively constrained conjugate-gradient algorithm using early termination is found superior to unconstrained inversion methods and is applied to image the depth of laser-heated chromophores in human skin.

Compensation for readout noise in CCD images

Abstract: Data acquired with a CCD camera are modeled as an additive Poisson–Gaussian mixture, with the Poisson component representing cumulative counts of object-dependent photoelectrons, object-independent photoelectrons, bias electrons, and thermoelectrons and the Gaussian component representing readout noise. Two methods are examined for compensating for readout noise. One method relies on approximating the Gaussian readout noise by a Poisson noise and then using a modified Richardson–Lucy algorithm to effect the compensation. This method has been used for restoring images acquired with CCD’s in the original Wide-Field/Planetary Camera aboard the Hubble Space Telescope. The second method directly uses the expectation-maximization algorithm derived for the Poisson–Gaussian mixture data. This requires the determination of the conditional-mean estimate of the Poisson component of the mixture, which is accomplished by the evaluation of a nonlinear function of the data. The second method requires more computation than the first but is more accurate mathematically and yields modest improvements in the quality of the restorations, particularly for fainter objects. As a specific example, we compare the two methods in restorations of images representative of those acquired with that camera; they contain excess blurring that is due to spherical aberration and a rms readout noise level of 13 electrons.

Vibration in phase-shifting interferometry

Abstract: Unexpected mechanical vibrations can significantly degrade the otherwise high accuracy of phase-shifting interferometry. Fourier analysis of phase-shift algorithms is shown to provide the analytical means of predicting measurement errors as a function of the frequency, the phase, and the amplitude of vibrations. The results of this analysis are concisely represented by a phase-error transfer function, which may be multiplied by the noise spectrum to predict the response of an interferometer to various forms of vibration. Analytical forms for the phase error are derived for several well-known algorithms, and the results are supported by numerical simulations and experiments with an interference microscope.

Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models

Abstract: We have developed a finite-difference time domain (FDTD) method and a novel geometric ray-tracing model for the calculation of light scattering by hexagonal ice crystals. In the FDTD method we use a staggered Cartesian grid with the implementation of an efficient absorbing boundary condition for the truncation of the computation domain. We introduce the Maxwell–Garnett rule to compute the mean values of the dielectric constant at grid points to reduce the inaccuracy produced by the staircasing approximation. The phase matrix elements and the scattering efficiencies for the scattering of visible light by two-dimensional long circular ice cylinders match closely those computed from the exact solution for size parameters as large as 60, with maximum differences less than 5%. In the new ray-tracing model we invoke the principle of geometric optics to evaluate the reflection and the refraction of localized waves, from which the electric and magnetic fields at the particle surface (near field) can be computed. Based on the equivalence theorem, the near field can subsequently be transformed to the far field, in which the phase interferences are fully accounted for. The phase functions and the scattering efficiencies for hexagonal ice crystals computed from the new geometric ray-tracing method compare reasonably well with the FDTD results for size parameters larger than approximately 20. When absorption is involved in geometric ray tracing, the adjusted real and imaginary refractive indices and Fresnel formulas are derived for practical applications based on the fundamental wave theory.

Mathematical studies in rigorous grating theory

Abstract: We consider the diffraction of a time-harmonic wave incident upon a grating (or periodic) structure. We study mathematical issues that arise in the direct modeling, inverse, and optimal design problems. Particular attention is paid to the variational approach and to finite-element methods. For the direct problem various results on existence, uniqueness, and numerical approximations of solutions are presented. Convergence properties of the variational method and sensitivity to TM polarization are examined. Our recent research on inverse diffraction problems and optimal design problems is also discussed.

Achromatic three-wave (or more) lateral shearing interferometer

Abstract: A new kind of lateral shearing interferometer, called the three-wave lateral shearing interferometer, was previously described [ Appl. Opt.32, 6242 ( 1993)]. As this instrument was monochromatic and its usable light efficiency was poor, the proposed setup was well suited only for a class of wave-front sensing problems, such as optical testing, in which the source can be easily adapted. A new achromatic setup adapted to low light level applications is presented. Three replicas of the analyzed wave front are obtained by Fourier filtering of the orders diffracted by a microlens array. An important feature of these new devices is their great similarity to another class of wave-front sensors based on the Hartmann test.

Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I

Abstract: We consider the electromagnetic diffraction occurring when light is focused by a lens without spherical aberration through a planar interface between materials of mismatched refractive indices, which focusing produces spherical aberration. By means of a rigorous vectorial electromagnetic treatment developed previously for this problem by Torok et al. [ J. Opt. Soc. Am. A12, 325 ( 1995)], the time-averaged electric energy density distributions in the region of the focused probe are numerically evaluated for air–glass and air–silicon interfaces as functions of lens numerical aperture and probe depth. Strehl intensity, lateral and axial sizes, and axial location of the probe are shown to be regular functions for low numerical apertures and probe depths but irregular functions for high numerical apertures and probe depths. An explanation to account for these occurrences is presented that also explains some previous experimental results of confocal microscopy.

Quadratic regularization functionals for phase unwrapping

Abstract: The problem of unwrapping a noisy principal-value phase field or, equivalently, reconstructing an unwrapped phase field from noisy and possibly incomplete phase differences may be considered ill-posed in the sense of Hadamard. We apply the Thikonov regularization theory to find solutions that correspond to minimizers of positive-definite quadratic cost functionals. These methods may be considered generalizations of the classical least-squares solution to the unwrapping problem; the introduction of the regularization term permits the reduction of noise (even if this noise does not generate integration-path inconsistencies) and the interpolation of the solution over regions with missing data in a stable and controlled way, with a minimum increase of computational complexity. Algorithms for finding direct solutions with transform methods and implementations of iterative procedures are discussed as well. Experimental results on synthetic test images are presented to illustrate the performance of these methods.

Scattering-matrix approach to multilayer diffraction

Abstract: A new modeling system to determine the optical response function of a multilayer structure with imposed periodicity in the plane of the layers, a multilayer diffraction grating, is described This new model has two essential ingredients This model is based on the well-established coordinate transformation procedure developed by Chandezon et al [ J Opt Soc Am72, 839– 846 ( 1982)] in which a periodically modulated surface is transformed into a frame in which it is flat, permitting simpler use of Maxwell’s boundary conditions Then, instead of using the conventional transfer-matrix method, we developed a scattering-matrix technique that permits the modeling of very thick (of the order of 1 μm or greater) multilayer systems with many field components without numerical instability Model programs have been developed based on this new scattering-matrix approach and tested by comparison with other models and experimental data

Numerical modeling of the cornea’s lamellar structure and birefringence properties

Abstract: A model of the cornea's lamellar structure is proposed that is capable of explaining experimental results obtained for the transmission of normal-incidence polarized light through rabbit and bovine cornea. The model consists of a large number of planar lamellae, each approximated as a uniaxial birefringent layer, stacked one upon another with various angular orientations. Polarized light transmission through the composite system is modeled theoretically by use of the Jones matrix formalism. The light transmission is calculated numerically for a large number of model lamellae arrangements, each generated from a statistical description, and histograms are constructed of various properties of the light transmission, including the minimum and maximum cross-polarized output intensities. It is demonstrated that various structural and optical parameters of the lamellae arrangements of actual corneas may be estimated by comparison of the calculations with detailed experimental data. Certain characteristics of the histograms are identified that permit a clear distinction between random and partially ordered systems. Comparisons with previously published experimental data provide strong evidence that the lamellae orientations are not entirely random, but rather a significant fraction are oriented in a fixed, preferred direction.

Double-pass measurements of the retinal image quality with unequal entrance and exit pupil sizes and the reversibility of the eye's optical system

Abstract: We have used a modified double-pass apparatus with unequal entrance and exit pupil sizes to measure the optical transfer function in the human eye and have applied the technique to three different problems. First, we confirm that in the eye the double-pass spread function is the cross correlation of the input spread function with the output spread function [J. Opt. Soc. Am. A 12, 195 (1995)]. Consequently, when entrance and exit pupil sizes are equal, phase information is lost from the double-pass images. Second, we show that in double-pass measurements the eye behaves like a reversible optical system. That is, when entrance and exit pupils are equal, the double-pass image results from two passes through an optical system having a transfer function that is the same in both directions. To test for reversibility in the living eye we have used a double-pass apparatus with different exit and entrance pupil sizes (one of them small enough to consider the eye diffraction limited), so that the ingoing and the outgoing transfer functions are different. The measured image quality was unchanged when the pupils were interchanged, i.e., when the first-pass entrance pupil size becomes the second-pass exit pupil size, and vice versa. Third, the technique provides a means for inferring the complete optical transfer function of the eye, including the phase transfer function, and the shape of the point-spread function.