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.     

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.     

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.     

High-order optical aberration coefficients: extension to finite objects and to telecentricity in object space

We extend the method for the automatic computation of high-order optical aberration coefficients to include (1) a finite object distance and (2) an infinite entrance pupil position (telecentricity in object space). We present coefficients of the power series expansion of the transverse aberration vector with respect to the normalized aperture and field coordinates. Aberration coefficients of very high order (e.g., 21) can be computed easily and--as shown by comparisons with trigonometric ray tracing--reliably.     

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.     

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.     

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.     

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.     

Wave-front measurement errors from restricted concentric subdomains

In interferometry and optical testing, system wave-front measurements that are analyzed on a restricted subdomain of the full pupil can include predictable systematic errors. In nearly all cases, the measured rms wave-front error and the magnitudes of the individual aberration polynomial coefficients underestimate the wave-front error magnitudes present in the full-pupil domain. We present an analytic method to determine the relationships between the coefficients of aberration polynomials defined on the full-pupil domain and those defined on a restricted concentric subdomain. In this way, systematic wave-front measurement errors introduced by subregion selection are investigated. Using vector and matrix representations for the wave-front aberration coefficients, we generalize the method to the study of arbitrary input wave fronts and subdomain sizes. While wave-front measurements on a restricted subdomain are insufficient for predicting the wave front of the full-pupil domain, studying the relationship between known full-pupil wave fronts and subdomain wave fronts allows us to set subdomain size limits for arbitrary measurement fidelity.     

Asymptotic behavior of the spatial frequency response of an optical system with defocus and spherical aberration

Asymptotic expressions are derived for the two-dimensional incoherent optical transfer function (OTF) of an optical system with defocus and spherical aberration. The two-dimensional stationary phase method is used to evaluate the aberrated OTF at large and moderately large defocus and spherical aberration. For small aberrations, the OTF is approximated by a power series in the aberration coefficients. An accurate approximation (in elementary functions) to the OTF is obtained for a defocused optical system with a circular pupil. We experimentally demonstrate the validity of the OTF approximations in sharp-focus image restoration from two defocused images. A digital focusing method is presented.     

Systematic comparison of the use of annular and Zernike circle polynomials for annular wavefronts

The theory of wavefront analysis of a noncircular wavefront is given and applied for a systematic comparison of the use of annular and Zernike circle polynomials for the analysis of an annular wavefront. It is shown that, unlike the annular coefficients, the circle coefficients generally change as the number of polynomials used in the expansion changes. Although the wavefront fit with a certain number of circle polynomials is identically the same as that with the corresponding annular polynomials, the piston circle coefficient does not represent the mean value of the aberration function, and the sum of the squares of the other coefficients does not yield its variance. The interferometer setting errors of tip, tilt, and defocus from a four-circle-polynomial expansion are the same as those from the annular-polynomial expansion. However, if these errors are obtained from, say, an 11-circle-polynomial expansion, and are removed from the aberration function, wrong polishing will result by zeroing out the residual aberration function. If the common practice of defining the center of an interferogram and drawing a circle around it is followed, then the circle coefficients of a noncircular interferogram do not yield a correct representation of the aberration function. Moreover, in this case, some of the higher-order coefficients of aberrations that are nonexistent in the aberration function are also nonzero. Finally, the circle coefficients, however obtained, do not represent coefficients of the balanced aberrations for an annular pupil. The various results are illustrated analytically and numerically by considering an annular Seidel aberration function.     

The Computation of Diffraction Patterns in the Presence of Aberrations

The diffraction integral for the disturbance produced in the image plane normal to the optical axis by an extra-axial pencil has been shown to lead to a Fourier transform provided the exit pupil surface is taken to be that of the reference sphere. It has been shown also that, except for small aperture and field sizes, the effect on the wave-front aberration of a shift of the image plane is not represented with sufficient accuracy merely by a term proportional to the aperture squared. Both of these results have been respected in formulating a numerical technique for the calculation of point spread functions. The diffraction integral is evaluated in polar coordinates, and is such that no error is made in approximating the domain of the exit pupil in cases where this may be represented by an ellipse. A study of the accuracy obtained has shown that, if each quadrant of the pupil is divided into a 20 × 20 mesh of elementary areas, the error in the intensity is not expected to exceed 0·8 per cent of the intensity at the focus of a diffraction limited system. The method takes account of the first derivatives of the wave aberration at each mesh point, and the results are therefore expected to be more accurate than those obtained by merely replacing the integral by a simple sum. Results are given for a case of primary and secondary coma, and of a study of the influence of secondary spectrum and spherical aberration on the images formed by 2 mm achromatic microscope objectives of numerical aperture equal to 1·40.     

Zernike representation and Strehl ratio of optical systems with variable numerical aperture

We consider optical systems with variable numerical aperture (NA) on the level of the Zernike coefficients of the correspondingly scalable pupil function. We thus present formulas for the Zernike coefficients and their first two derivatives as a function of the scaling factor ε ≤ 1, and we apply this to the Strehl ratio and its derivatives of NA-reduced optical systems. The formulas for the Zernike coefficients of NA-reduced optical systems are also useful for the forward calculation of point-spread functions and aberration retrieval within the Extended Nijboer–Zernike (ENZ) formalism for optical systems with reduced NA or systems that have a central obstruction. Thus, we retrieve a Gaussian, comatic pupil function on an annular set from the intensity point-spread function in the focal region under high-NA conditions.     

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.     

Aberrations of anamorphic optical systems III: the primary aberration theory for toroidal anamorphic systems

We apply a new method for optical aberration derivation to anamorphic systems made from toroidal surfaces and obtain a complete set of primary aberration coefficients. This set is written in a form similar to the well-known Seidel aberrations for rotationally symmetrical optical systems and includes first-order quantities only, thus it can be easily applied to anamorphic lens design practice. By tracing four nonskew paraxial marginal and chief rays, the 16 anamorphic primary aberration coefficients can be easily calculated.     

Pupil exploration and wave-front-polynomial fitting of optical systems

Pupil exploration and wave-front-polynomial fitting algorithms are tools that are often employed in image-quality evaluation techniques, such as optical-transfer-function and point-spread-function calculations. These techniques require that aberration data be determined for a large number of points across the pupil. With optical systems increasing in complexity, it is necessary that these algorithms become more sophisticated to ensure that the proper pupil shapes and aberration maps are used to represent the wave fronts. Such algorithms are described. These algorithms can handle systems that not only lack the symmetry found with the more conventional lens systems but those that also have apertures with unusual shapes. As practical demonstrations the treatments employed in the pupil exploration and the wave-front-polynomial fitting have been applied to various lens arrangements and the results discussed.     

Spherical aberration gauge for human vision

Spherical aberration affects vision in varying degrees depending on pupil size, accommodation, individual eye characteristics, and interpretations by the brain. We developed a spherical aberration gauge to help evaluate the correction potential of spherical aberration in human vision. Variable aberration levels are achieved with laterally shifted polynomial plates from which a user selects a setting that provides the best vision. The aberration is mapped into the pupil of the eye using a simple telescope. Calibration data are given.     

Characteristic functions of Hartmann-Shack wavefront sensors and laser-ray-tracing aberrometers

It is shown that the aberration estimated at any point of the pupil using wavefront slope aberrometers such as Hartmann-Shack wavefront sensors or laser ray tracers is a spatial average of the actual aberration weighted by a characteristic function that depends on the aberrometer design and on the estimation procedure. This characteristic function, whose explicit form is given here for wavefront slope aberrometers using either modal or zonal estimators, may be useful in analyzing some basic aspects of the aberrometer performance. It is also instrumental in establishing the links between the statistical properties of the actual and the estimated aberrations. Explicit formulas are given to show in terms of this function how the bias arises in the first- and second-order statistics of the retrieved aberrations. This approach is mathematically equivalent to the analysis of the effects of modal coupling (cross-coupling and aliasing). It may provide, however, some complementary insight.     

Evaluation of least-squares phase-diversity technique for space telescope wave-front sensing

Because of mechanical aspects of fabrication, launch, and operational environment, space telescope optics can suffer from unforeseen aberrations, detracting from their intended diffraction-limited performance goals. We give the results of simulation studies designed to explore how wave-front aberration information for such near-diffraction-limited telescopes can be estimated through a regularized, low-pass filtered version of the Gonsalves (least-squares) phase-diversity technique. We numerically simulate models of both monolithic and segmented space telescope mirrors; the segmented case is a simplified model of the proposed next generation space telescope. The simulation results quantify the accuracy of phase diversity as a wave-front sensing (WFS) technique in estimating the pupil phase map. The pupil phase is estimated from pairs of conventional and out-of-focus photon-limited point-source images. Image photon statistics are simulated for three different average light levels. Simulation results give an indication of the minimum light level required for reliable estimation of a large number of aberration parameters under the least-squares paradigm. For weak aberrations that average a 0.10lambda pupil rms, the average WFS estimation errors obtained here range from a worst case of 0.057lambda pupil rms to a best case of only 0.002lambda pupil rms, depending on the light level as well as on the types and degrees of freedom of the aberrations present.     

Combined amplitude and phase filters for increased tolerance to spherical aberration

Abstract

Analysis of the expression for Strehl ratio for a circularly symmetric pupil allows one to design complex filters that offer reduced sensitivity to spherical aberration. It is shown that filters that combine hyper-Gaussian amplitude transmittance with hyper-Gaussian phase modulation provide five-fold reduction in sensitivity to spherical aberration. Furthermore, this is achieved without the introduction of zeros into the modulation transfer function and deconvolution can restore the transfer function to that of a diffraction-limited imager. The performance of the derived combined amplitude and phase filter is illustrated through the variation of its axial intensity versus spherical aberration. This technique is applicable to imaging in the presence of significant amounts of spherical aberration as is encountered in, for example, microscopy.