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.     

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.     

Gaussian weighting of ocular wave-front measurements

The measurement of ocular wave-front error gives insight into the optical performance of the eye and possibly a means for assessing visual performance. The visual system responds not only to the quality of the optical image formed on the retina but also to the processing that occurs in the retina and the brain. To develop a metric of visual performance based on wave-front error measurements, these latter processes must somehow be incorporated. In representing the wave-front error in terms of Zernike polynomials, it appears that terms with lower angular frequency have a greater deleterious effect on visual performance than higher-angular-frequency terms. A technique for weighting the pupil function of the eye with a Gaussian filter is demonstrated. It is further demonstrated that the variance of the Gaussian-weighted wave-front error is well correlated with visual performance.     

高级像差对人眼成像质量和视觉的影响

定量地研究高级像差对人眼成像质量和视觉的影响对人眼像差矫正具有重要的实验和临床意义.利用Hartmann-Shack波前传感器人眼像差仪测量了正常人眼6mm瞳孔的波前像差,由波前像差计算出人眼光学系统的光学调制传递函数MTF和Strehl比率,并由MTF和视网膜空间像调制度AIM曲线计算出人眼视锐度和对比敏感度函数CSF.根据MTF和Strehl比率分析了高级像差对人眼成像质量的影响,根据视锐度和对比敏感度函数CSF分析了高级波像差对视觉的影响.研究表明Zernik前6级像差对人眼成像质量和视觉的影响是不可忽略的,更高级的像差对人眼成像质量和视觉的影响较小,甚至可以忽略.对Zemik前6级像差进行矫正,可以得到相当好的视觉.     

General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes

Zernike polynomials have been widely used to describe the aberrations in wavefront sensing of the eye. The Zernike coefficients are often computed under different aperture sizes. For the sake of comparison, the same aperture diameter is required. Since no standard aperture size is available for reporting the results, it is important to develop a technique for converting the Zernike coefficients obtained from one aperture size to another size. By investigating the properties of Zernike polynomials, we propose a general method for establishing the relationship between two sets of Zernike coefficients computed with different aperture sizes.     

Wave-front aberrations in the anterior corneal surface and the whole eye

In order to investigate the sources of wave-front aberrations in the human eye, we have measured the aberrations of the anterior cornea and the whole eye using a topographic system and a psychophysical wave-front sensor. We have also calculated the aberrations for the internal optics of both eyes of 45 young subjects (aged 9 to 29 years). The mean rms for the anterior cornea was similar to that for the internal optics and thewhole eye when astigmatism was included, but less than that for both the internal optics and the whole eye with astigmatism removed. For eyes with low whole-eye rms values, mean rms for the anterior cornea was greater than that for the whole eye, suggesting that the anterior corneal aberration is partially compensated by the internal optics of the eye to produce the low whole-eye rms. For eyes with larger whole-eye rms values, the rms values for both the anterior cornea and the internal optics were less than that for the whole eye. Thus the aberrations for the two elements tend to be primarily additive. This pattern exists whether or not astigmatism was included in the wave-front aberration rms. For individual Zernike terms, astigmatism and spherical aberration in the anterior cornea were partially compensated by internal optics, while some other Zernike terms showed addition between the anterior cornea and internal optics. Individual eyes show different combinations of compensation and addition across different Zernike terms. Our data suggest that the reported loss of internal compensation for anterior corneal aberrations in elderly eyes with large whole-eye aberrations [J. Opt. Soc. Am. A 19, 137 (2002)] may also occur in young eyes.     

Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils

In eye aberrometry it is often necessary to transform the aberration coefficients in order to express them in a scaled, rotated, and/or displaced pupil. This is usually done by applying to the original coefficients vector a set of matrices accounting for each elementary transformation. We describe an equivalent algebraic approach that allows us to perform this conversion in a single step and in a straightforward way. This approach can be applied to any particular definition, normalization, and ordering of the Zernike polynomials, and can handle a wide range of pupil transformations, including, but not restricted to, anisotropic scalings. It may also be used to transform the aberration coefficients between different polynomial basis sets.     

Measuring eye aberrations with Hartmann-Shack wave-front sensors: should the irradiance distribution across the eye pupil be taken into account?

A usual approximation in Hartmann-Shack aberrometry is that the centroid displacements are proportional to the spatial averages of the wave-front slopes at the sampling subapertures. However, these spatial averages are actually weighted by the local irradiance distribution across each microlens. The irradiance across the eye pupil is not uniform in usual reflectometric aberrometers, which is due to several factors including retinal scattering and cone waveguiding directionality. It is shown that neglecting this fact in usual least-squares reconstruction procedures gives rise to a biased estimation of the aberration coefficients. The magnitude of this bias depends on the actual irradiance distribution across the eye pupil, the mode being estimated, the detailed modal composition of the aberrated wave front, and the geometry of the wave-front sampling array. Order-of-magnitude calculations suggest that this bias may well be in the range 5%-10% for relatively smooth irradiance distributions. The systematic nature of this error makes it advisable to check for its presence and, if required, to compensate for it by an adequate choice of the least-squares reconstruction matrix.     

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.     

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.     

Measurement of telescope aberrations through atmospheric turbulence by use of phase diversity

We conduct computer simulations of the reconstruction of a wave front at a telescope pupil with the phase-diversity method. An instantaneous wave front is reconstructed from focused and defocused specklegrams of a point star. In the wave-front reconstruction we do not fit the wave front to Zernike polynomials but retrieve the phase with a phase-unwrapping procedure. Averaging over many atmospherically perturbed wave fronts leads to the residual phase error, namely, the aberration of the telescope. The scintillation effect, nonuniformity of amplitude on a telescope pupil, is also discussed.     

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.     

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.     

Dynamics of the eye's wave aberration

It is well known that the eye's optics exhibit temporal instability in the form of microfluctuations in focus; however, almost nothing is known of the temporal properties of the eye's other aberrations. We constructed a real-time Hartmann-Shack (HS) wave-front sensor to measure these dynamics at frequencies as high as 60 Hz. To reduce spatial inhomogeneities in the short-exposure HS images, we used a low-coherence source and a scanning system. HS images were collected on three normal subjects with natural and paralyzed accommodation. Average temporal power spectra were computed for the wave-front rms, the Seidel aberrations, and each of 32 Zernike coefficients. The results indicate the presence of fluctuations in all of the eye's aberration, not just defocus. Fluctuations in higher-order aberrations share similar spectra and bandwidths both within and between subjects, dropping at a rate of approximately 4 dB per octave in temporal frequency. The spectrum shape for higher-order aberrations is generally different from that for microfluctuations of accommodation. The origin of these measured fluctuations is not known, and both corneal/lenticular and retinal causes are considered. Under the assumption that they are purely corneal or lenticular, calculations suggest that a perfect adaptive optics system with a closed-loop bandwidth of 1-2 Hz could correct these aberrations well enough to achieve diffraction-limited imaging over a dilated pupil.     

Effect of aging on the monochromatic aberrations of the human eye.

We measured the contrast sensitivity (CS) of a group of older subjects through natural pupils and compared the results with those from a group of younger subjects. We also measured each subject's monochromatic ocular wave-front aberrations using a crossed-cylinder aberroscope and calculated their modulation transfer functions (MTF's) and root-mean-squared (RMS) wave-front aberrations for fixed pupil diameters of 4 mm and 6 mm and for a natural pupil diameter. The CS at a natural pupil diameter and the MTF computed for a fixed pupil diameter were found to be significantly poorer for the older group than for the younger group. However, the older group showed very similar MTF's and significantly smaller RMS wave-front aberrations compared with the younger group at their natural pupil diameters, owing to the effects of age-related miosis. These results suggest that although monochromatic ocular wave-front aberrations for a given pupil size increase with age, the reduction in CS with age is not due to this increase.     

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.     

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.     

Wind-induced deformations in a segmented mirror

A Zernike expansion of wind-induced deformations in a segmented mirror is described. The wind model is a frozen turbulent field with a Kolmogorov spectrum for scales smaller than the outer scale and a flat spectrum for scales larger than the outer scale. The approach allows a mode-by-mode comparison of the wave-front error contributions from atmospheric phase distortions, wind-induced deformations, and the mirror control system noise. This is used to design a controller that minimizes the mirror surface errors by application of corrections based on edge sensor measurements and wave-front measurements on a guide star.     

System for the design, manufacture, and testing of custom lenses with known amounts of high-order aberrations

Now that excimer laser systems can be programmed to correct complex aberrations of the eye on the basis of wave-front measurements, a method is needed to test the accuracy of the system from measurement through treatment. A closed-loop test method was developed to ensure that treatment plans generated by a wavefront measuring system were accurately transferred to and executed by the excimer laser. A surface was analytically defined, and a Shack-Hartmann-based wave-front system was used to formulate a treatment plan, which was downloaded to an excimer laser system. A plastic lens was ablated by the laser and then returned to the wave-front device, where it was measured and compared with the analytically defined wave-front surface. The two surfaces agreed up to 6th-order Zernike terms, validating the accuracy of the system.