Extended Nijboer-Zernike approach to aberration and birefringence retrieval in a high-numerical-aperture optical system

The judgment of the imaging quality of an optical system can be carried out by examining its through-focus intensity distribution. It has been shown in a previous paper that a scalar-wave analysis of the imaging process according to the extended Nijboer-Zernike theory allows the retrieval of the complex pupil function of the imaging system, including aberrations as well as transmission variations. However, the applicability of the scalar analysis is limited to systems with a numerical aperture (NA) value of the order of 0.60 or less; beyond these values polarization effects become significant. In this scalar retrieval method, the complex pupil function is represented by means of the coefficients of its expansion in a series involving the Zernike polynomials. This representation is highly efficient, in terms of number and magnitude of the required coefficients, and lends itself quite well to matching procedures in the focal region. This distinguishes the method from the retrieval schemes in the literature, which are normally not based on Zernike-type expansions, and rather rely on point-by-point matching procedures. In a previous paper [J. Opt. Soc. Am. A 20, 2281 (2003)] we have incorporated the extended Nijboer-Zernike approach into the Ignatowsky-Richards/Wolf formalism for the vectorial treatment of optical systems with high NA. In the present paper we further develop this approach by defining an appropriate set of functions that describe the energy density distribution in the focal region. Using this more refined analysis, we establish the set of equations that allow the retrieval of aberrations and birefringence from the intensity point-spread function in the focal volume for high-NA systems. It is shown that one needs four analyses of the intensity distribution in the image volume with different states of polarization in the entrance pupil. Only in this way will it be possible to retrieve the "vectorial" pupil function that includes the effects of birefringence induced by the imaging system. A first numerical test example is presented that illustrates the importance of using the vectorial approach and the correct NA value in the aberration retrieval scheme.     

Statistical variation of aberration structure and image quality in a normal population of healthy eyes

A Shack-Hartmann aberrometer was used to measure the monochromatic aberration structure along the primary line of sight of 200 cyclopleged, normal, healthy eyes from 100 individuals. Sphero-cylindrical refractive errors were corrected with ophthalmic spectacle lenses based on the results of a subjective refraction performed immediately prior to experimentation. Zernike expansions of the experimental wave-front aberration functions were used to determine aberration coefficients for a series of pupil diameters. The residual Zernike coefficients for defocus were not zero but varied systematically with pupil diameter and with the Zernike coefficient for spherical aberration in a way that maximizes visual acuity. We infer from these results that subjective best focus occurs when the area of the central, aberration-free region of the pupil is maximized. We found that the population averages of Zernike coefficients were nearly zero for all of the higher-order modes except spherical aberration. This result indicates that a hypothetical average eye representing the central tendency of the population is nearly free of aberrations, suggesting the possible influence of an emmetropization process or evolutionary pressure. However, for any individual eye the aberration coefficients were rarely zero for any Zernike mode. To first approximation, wave-front error fell exponentially with Zernike order and increased linearly with pupil area. On average, the total wave-front variance produced by higher-order aberrations was less than the wave-front variance of residual defocus and astigmatism. For example, the average amount of higher-order aberrations present for a 7.5-mm pupil was equivalent to the wave-front error produced by less than 1/4 diopter (D) of defocus. The largest pupil for which an eye may be considered diffraction-limited was 1.22 mm on average. Correlation of aberrations from the left and right eyes indicated the presence of significant bilateral symmetry. No evidence was found of a universal anatomical feature responsible for third-order optical aberrations. Using the Marechal criterion, we conclude that correction of the 12 largest principal components, or 14 largest Zernike modes, would be required to achieve diffraction-limited performance on average for a 6-mm pupil. Different methods of computing population averages provided upper and lower limits to the mean optical transfer function and mean point-spread function for our population of eyes.     

On the computation of the nijboer-zernike aberration integrals at arbitrary defocus

Abstract

We present a new computation scheme for the integral expressions describing the contributions of single aberrations to the diffraction integral in the context of an extended Nijboer-Zernike approach. Such a scheme, in the form of a power series involving the defocus parameter with coefficients given explicitly in terms of Bessel functions and binomial coefficients, was presented recently by the authors with satisfactory results for small-to-medium-large defocus values. The new scheme amounts to systemizing the procedure proposed by Nijboer in which the appropriate linearization of products of Zernike polynomials is achieved by using certain results of the modern theory of orthogonal polynomials. It can be used to compute point-spread functions of general optical systems in the presence of arbitrary lens transmission and lens aberration functions and the scheme provides accurate data for any, small or large, defocus value and at any spatial point in one and the same format. The cases with high numerical aperture, requiring a vectorial approach, are equally well handled. The resulting infinite series expressions for these point-spread functions, involving products of Bessel functions, can be shown to be practically immune to loss of digits. In this respect, because of its virtually unlimited defocus range, the scheme is particularly valuable in replacing numerical Fourier transform methods when the defocused pupil functions require intolerably high sampling densities.     

Zernike coefficients of a scaled pupil

By expressing a scaled Zernike radial polynomial as a linear combination of the unscaled radial polynomials, we give a simple derivation for determining the Zernike coefficients of an aberration function of a scaled pupil in terms of their values for a corresponding unscaled pupil.     

Reduced-complexity representation of the coherent point-spread function in the presence of aberrations and arbitrarily large defocus

We introduce a method to analyze the diffraction integral for evaluating the point-spread function. Our method is based on the use of higher-order Airy functions along with Zernike and Taylor expansions. Our approach is applicable when we are considering a finite, arbitrary number of aberrations and arbitrarily large defocus simultaneously. We present an upper bound for the complexity and the convergence rate of this method. We also compare the cost and accuracy of this method with those of traditional ones and show the efficiency of our method through these comparisons. In particular, we rigorously show that this method is constructed in a way that the complexity of the analysis (i.e., the number of terms needed for expressing the light disturbance) does not increase as either defocus or resolution of interest increases. This has applications in several fields such as biological microscopy, lithography, and multidomain optimization in optical systems.     

Orthonormal polynomials in wavefront analysis: error analysis

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.     

Orthonormal polynomials in wavefront analysis: analytical solution

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.     

Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils

Zernike polynomials and their associated coefficients are commonly used to quantify the wavefront aberrations of the eye. When the aberrations of different eyes, pupil sizes, or corrections are compared or averaged, it is important that the Zernike coefficients have been calculated for the correct size, position, orientation, and shape of the pupil. We present the first complete theory to transform Zernike coefficients analytically with regard to concentric scaling, translation of pupil center, and rotation. The transformations are described both for circular and elliptical pupils. The algorithm has been implemented in MATLAB, for which the code is given in an appendix.     

Scalar and electromagnetic diffraction point-spread functions

We describe an accurate technique for computing the diffraction point-spread function for optical systems. The approach is based on the combined method of ray tracing and diffraction, which implies that the computation is accomplished in a two-step procedure. First, ray tracing is employed to compute the wave-front error in a reference plane on the image side of the system and to determine the shape of the vignetted pupil. Next the Rayleigh-Sommerfeld diffraction theory, combined with the Kirchhoff approximation and the Stamnes-Spjelkavik-Pedersen method for numerical integration, is applied to compute the field in the region of the image. The method does not rely on small-angle approximations and works well for a pupil of general shape. Both scalar and electromagnetic computations are discussed and numerical results are presented.     

Scaling Zernike expansion coefficients to different pupil sizes

Recent developments in technologies to correct aberrations in the eye have fostered extensive research in wave-front sensing of the eye, resulting in many reports of Zernike expansions of wave-front errors of the eye. For different reports of Zernike expansions, to be compared, the same pupil diameter is required. Since no standard pupil size has been established for reporting these results, a technique for converting Zernike expansion coefficients from one pupil size to another is needed. This investigation derives relationships between the Zernike expansion coefficients for two different pupil sizes.     

Annular pupils, radial polarization, and superresolution

An annular pupil, which can be used to produce a Bessel beam, when combined with radially polarized illumination promises improvements in microscope resolution, increased packing density for optical storage, and finer optical lithography. When combined with a circular detection pupil in confocal microscopy a point-spread function 112 nm wide results (lambda = 488 nm). Radially polarized annular illumination of a solid-immersion lens can yield a focal spot smaller than 100 nm for lambda = 488 nm. Use of radially polarized illumination with pupil masks is discussed.     

Defocus transfer function for circularly symmetric pupils

We present a two-dimensional function that graphically illustrates the effects of defocus on the optical transfer function (OTF) associated with a circularly symmetric pupil function. We call it the defocus transfer function (DTF). The function is similar in application to the ambiguity function, which can be used to display the OTF associated with a defocused rectangularly separable pupil function. The properties of the DTF make it useful for analyzing optical systems with circularly symmetric pupils when one is interested in the OTF as a function of defocus. In addition to presenting these properties, we give examples of the DTF for systems with clear, bifocal, and annular pupil functions.     

Zernike annular polynomials and atmospheric turbulence

Imaging through atmospheric turbulence by systems with annular pupils is discussed using the Zernike annular polynomials. Fourier transforms of these polynomials are derived analytically to facilitate the calculation of variance and covariance of the aberration coefficients. Zernike annular shape functions are derived and used to calculate the Strehl ratio and the residual phase structure and mutual coherence functions when a certain number of modes are corrected using, say, a deformable mirror. Special cases of long- and short-exposure images are also considered. The results for systems with a circular pupil are obtained as a special case of the annular pupil.     

Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions

We assess the validity of an extended Nijboer-Zernike approach [J. Opt. Soc. Am. A 19, 849 (2002)], based on ecently found Bessel-series representations of diffraction integrals comprising an arbitrary aberration and a defocus part, for the computation of optical point-spread functions of circular, aberrated optical systems. These new series representations yield a flexible means to compute optical point-spread functions, both accurately and efficiently, under defocus and aberration conditions that seem to cover almost all cases of practical interest. Because of the analytical nature of the formulas, there are no discretization effects limiting the accuracy, as opposed to the more commonly used numerical packages based on strictly numerical integration methods. Instead, we have an easily managed criterion, expressed in the number of terms to be included in the Bessel-series representations, guaranteeing the desired accuracy. For this reason, the analytical method can also serve as a calibration tool for the numerically based methods. The analysis is not limited to pointlike objects but can also be used for extended objects under various illumination conditions. The calculation schemes are simple and permit one to trace the relative strength of the various interfering complex-amplitude terms that contribute to the final image intensity function.     

Self-apodization of low-resolution pixelated lenses

We show that a pixelated lens with appropriate parameters exhibits an apodized point-spread function that originates in the finite size of the pixel's pupil. We evaluate numerically the degree of apodization and the enlargement associated with the point-spread function in terms of the parameters that characterize the pixelated lens.     

The Rapid Calculation of the Optical Transfer Function for On-axis Systems Using the Orthogonal Properties of Tchebycheff Polynomials

A simple and rapid numerical quadrature is developed for the evaluation of the diffraction-based optical transfer function for on-axis systems, using a Tchebycheff polynomial expansion of the pupil function. The integration of the autocorrelation integral of the pupil function is replaced by once and for all evaluations of the cross-correlation of respective polynomials. However, the expansion coefficients themselves of the Tchebycheff series are linear sums of the sampled pupil function and thus a series of coefficients can be generated that weight the pupil function at various points to give the resultant OTF. The coefficients for a tenth-order Tchebycheff expansion are included in the paper, and a set of tables of OTF values calculated with these coefficients and 64 2 64 Gaussian quadrature for a diffraction-limited system, and one with one wavelength of primary spherical aberration.     

Experimental study of an off-axis three mirror anastigmatic system with wavefront coding technology

Wavefront coding (WFC) is a kind of computational imaging technique that controls defocus and defocus related aberrations of optical systems by introducing a specially designed phase distribution to the pupil function. This technology has been applied in many imaging systems to improve performance and/or reduce cost. The application of WFC technology in an off-axis three mirror anastigmatic (TMA) system has been proposed, and the design and optimization of optics, the restoration of degraded images, and the manufacturing of wavefront coded elements have been researched in our previous work. In this paper, we describe the alignment, the imaging experiment, and the image restoration of the off-axis TMA system with WFC technology. The ideal wavefront map is set to be the system error of the interferometer to simplify the assembly, and the coefficients of certain Zernike polynomials are monitored to verify the result in the alignment process. A pinhole of 20 μm diameter and the third plate of WT1005-62 resolution patterns are selected as the targets in the imaging experiment. The comparison of the tail lengths of point spread functions is represented to show the invariance of the image quality in the extended depth of focus. The structure similarity is applied to estimate the relationship among the captured images with varying defocus. We conclude that the experiment results agree with the earlier theoretical analysis.     

Generalized Fried parameter after adaptive optics partial wave-front compensation

Atmospheric turbulence imposes the resolution limit attainable by large ground-based telescopes. This limit is lambda/r(0), where r(0) is the Fried parameter or seeing cell size. Working in the visible, adaptive optics systems can partially compensate for turbulence-induced distortions. By analogy with the Fried parameter, r(0), we have introduced a generalized Fried parameter, rho(0), that plays the same role as r(0) but in partial compensation. Using this parameter and the residual phase variance, we have described the phase structure function, estimated the point-spread function halo size, and derived an expression for the Strehl ratio as a function of the degree of compensation. Finally, it is shown that rho(0) represents the diameter of the coherent cells in the pupil domain.     

Analysis of Seidel aberration by use of the discrete wavelet transform

Seidel aberration coefficients can be expressed by Zernike coefficients. The least-squares matrix-inversion method of determining Zernike coefficients from a sampled wave front with measurement noise has been found to be numerically unstable. We present a method of estimating the Seidel aberration coefficients by using a two-dimensional discrete wavelet transform. This method is applied to analyze the wave front of an optical system, and we obtain not only more-accurate Seidel aberration coefficients, but we also speed the computation. Three simulated wave fronts are fitted, and simulation results are shown for spherical aberration, coma, astigmatism, and defocus.