Anterior corneal and internal contributions to peripheral aberrations of human eyes

Anterior corneal and internal component contributions to overall peripheral aberrations of five human eyes were determined, based on corneal topography and overall aberration measurements. Anterior corneal position and orientation (tilt) were referenced to the line of sight. Ray tracing was performed through the anterior cornea for 6-mm-diameter pupils at angles out to 40 degrees in both the temporal and the nasal visual fields. In general, both component and overall Zernike aberrations were greater for the nasal than for the temporal visual field. In general, the anterior corneal aberration components were considerably higher than the overall aberrations across the visual field and were balanced to a considerable degree by the internal ocular aberration components. The component and overall levels of Zernike third-order aberrations showed linear trends away from the fixation axis, and the component levels of Zernike fourth-order aberrations showed quadratic trends away from the fixation axis. The second-order, but not higher-order, aberration components were susceptible to the choice of image radius of curvature, while disregarding corneal position and orientation affected second- and higher-order aberration components.     

Hartmann-Shack technique and refraction across the horizontal visual field

We compared refractions across the horizontal visual field, based on different analyses of wave aberration obtained with a Hartmann-Shack instrument. The wave aberrations had been determined for 6-mm-diameter pupils up to at least the sixth Zernike order in five normal subjects [J. Opt. Soc. Am. A 19, 2180 (2002)]. The polynomials were converted into refractions based on 6-mm pupils and second-order Zernike aberrations (6 mm/2nd order), 3-mm pupils and second-order aberrations (3 mm/2nd order), 1-mm pupils and second-order aberrations (1 mm/2nd order), and 6-mm pupils with both second- and fourth-order aberrations (6 mm/4th order). The 3-mm/2nd-order and 6-mm/2nd-order refractions differed by as much as 0.9 D in mean sphere on axis, but the differences reduced markedly toward the edges of the visual field. The cylindrical differences between these two analyses were small at the center of the visual field (<0.3 D) but increased into the periphery to be greater than 1.0 D for some subjects. Much smaller differences in mean sphere and cylinder were found when 3-mm/2nd-order refractions and either the 1-mm/2nd-order refractions or the 6-mm/4th-order refractions were compared. The results suggest that, for determining refractions based on wave aberration data with large pupils, similar results occur by either restricting the analysis to second-order Zernike aberrations with a smaller pupil such as 3 mm or using both second- and fourth-order Zernike aberrations. Since subjective refraction is largely independent of the pupil size under photopic conditions, objective refractions based on either of these analyses may be the most useful.     

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.     

Measuring ocular aberrations in the peripheral visual field using Hartmann-Shack aberrometry

We have used the Hartmann-Shack technique previously to measure ocular aberrations along the horizontal meridian of the visual field. This requires considerable modifications from the technique for measuring the aberrations corresponding to the fovea. We now further develop the technique so that it can be used for any meridian of the visual field. Allowance is made for any auxiliary optics placed in front of the eye to compensate for the limited range of the Hartmann-Shack technique and for the case where aberrations are estimated at a wavelength other than the measuring wavelength. Zernike wave aberrations are converted to peripheral refractions. Examples are presented showing the developments, and we discuss change in wave aberrations when converting from a circular to an elliptical pupil.     

Application of the see-saw method to all refracting optical systems

The optical see-saw diagram is a method that describes image correction to third-order approximation over a finite field of view in rotationally symmetric systems that employ aspheric surfaces. The aim of this paper is to describe the correction of aberrations caused by plane surfaces in all refracting optical systems in terms of the see-saw diagram. A lens correction algorithm based on the see-saw method is described to correct analytically the Seidel aberrations, primary spherical aberration, coma, astigmatism, and distortion, in such systems. We then apply this lens correction algorithm to the design of equivalent configurations by aspherizing different surfaces of the system, and the high-order aberrations of the equivalent configurations are evaluated by means of transverse-ray-aberration plots. Results indicate that this method gives information on what the contribution must be to the third-order aberrations that each component should provide to the system to give a better balance of high-order aberrations. Examples of the lens correction algorithm applied to lenses with six refracting surfaces and working for both finite and infinite object conjugates are given.     

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.     

Weak correlation between the aberration dynamics of the human eye and the cardiopulmonary system

It is fairly well established that the higher-order aberrations of the eye fluctuate over relatively short time periods, but as yet there is no conclusive evidence regarding the origin of these fluctuations. We measured the aberrations and the pulse pressure wave simultaneously for five subjects. The aberrations were measured by using a Shack-Hartmann sensor sampling at 21.2 Hz. We decomposed the aberration data into Zernike coefficients up to and including fifth order and also calculated the rms wave-front error. From the pulse data the heart rate variability signal was also derived. Coherence function analysis showed that for all subjects there was a weak correlation between many of the aberrations and the pulse and the derived heart rate variability. The pulse and the heart rate variability can account for only 11% +/- 2% and 20% +/- 2%, respectively, of the aberration dynamics.     

Weighted Zernike expansion with applications to the optical aberration of the human eye

We propose to use weighted Zernike functions to represent the aberrations of an eye. Methods for computing the phase of an aberrated ophthalmic wavefront in terms of weighted Zernike functions are discussed. In particular, we consider several options for integrating the phase out of its measured slopes. The weighted functions involve a free parameter. Clinical data on subjective refraction and aberration maps of individual subjects are used to determine an estimate for this parameter.     

Multifractality in steady-state accommodation is robust to dynamic correction of aberrations using adaptive optics

When fixating on a stationary object, the human eye exhibits microfluctuations in accommodation. Changes in the magnitude of these fluctuations reflect changes in the accommodation control system. We used adaptive optics to determine the effect of monochromatic aberration dynamics on the control of steady-state accommodation of four subjects. The subjects viewed a stationary stimulus at 2 D while selective Zernike aberrations were corrected. The fluctuations in accommodation were characterised using a wavelet-based multifractal formalism approach. We found that for all subjects, and all experimental conditions, the accommodative fluctuations were multifractal. For one subject, we found that the width of the multifractal spectrum was statistically significantly larger when even-order aberrations were corrected as compared to no aberrations corrected. Hence, in general, for the subjects tested, the multifractal nature of steady-state accommodation control is unaffected by the manipulation of monochromatic aberration dynamics. Averaging across all subjects and experimental conditions, the mean spectrum was right-skewed with a most frequently occurring Hölder exponent of 0.31?±?0.08. Future applications of multifractal analysis to accommodation control are discussed.     

The design of a triplet with a shape-dependent third-order aberration optimization technique

Abstract

We have derived a series of shape-dependent third-order aberration equations involving lens thickness. These equations are used in a simple and direct method to correct real aberrations and to find the minimum aberration for triple design that has been proposed. These calculated third-order aberrations can exactly meet the target values for each stage by means of the damped least-squares method. The shape of the three elements permits the control of three third-order aberrations: spherical aberration, coma and astigmatism. The spherical aberration is targeted first. The minimal value of the real on-axis aberration is obtained. Similarly, the coma and astigmatism target values are adjusted so as to force the full field angle real off-axis aberrations down to a minimum. Finally, the manual adjustment of the lens thickness and air spacing is used to attain the minimum aberration at the 0.7 field angle. To verify the method, two triplet design examples are presented.     

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.     

Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials

A general wavefront fitting procedure with Zernike annular polynomials for circular and annular pupils is proposed. For interferometric data of typical annular wavefronts with smaller and larger obscuration ratios, the results fitted with Zernike annular polynomials are compared with those of Zernike circle polynomials. Data are provided demonstrating that the annular wavefront expressed with Zernike annular polynomials is more accurate and meaningful for the decomposition of aberrations, the calculation of Seidel aberrations, and the removal of misalignments in interferometry. The primary limitations of current interferogram reduction software with Zernike circle polynomials in analyzing wavefronts of annular pupils are further illustrated, and some reasonable explanations are provided. It is suggested that the use of orthogonal basis functions on the pupils of the wavefronts analyzed is more appropriate.     

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.     

Variable power phase lenses

A method is given to construct a phase lens capable of creating an optical aberration of variable power that is described by a single Zernike polynomial function whose meridional index is 1 or greater. The phase lens is created from two identical phase elements, each creating a single Zernike aberration, that can be rotated with respect to each other, thereby increasing the aberration effect from zero to twice the value of either. This is possible because these aberrations are vectorlike. Results are given from the testing of an example that was manufactured and designed to produce coma (Zernike term Z(3,1)).     

Adaptive optics for peripheral vision

Understanding peripheral optical errors and their impact on vision is important for various applications, e.g. research on myopia development and optical correction of patients with central visual field loss. In this study, we investigated whether correction of higher order aberrations with adaptive optics (AO) improve resolution beyond what is achieved with best peripheral refractive correction. A laboratory AO system was constructed for correcting peripheral aberrations. The peripheral low contrast grating resolution acuity in the 20° nasal visual field of the right eye was evaluated for 12 subjects using three types of correction: refractive correction of sphere and cylinder, static closed loop AO correction and continuous closed loop AO correction. Running AO in continuous closed loop improved acuity compared to refractive correction for most subjects (maximum benefit 0.15?logMAR). The visual improvement from aberration correction was highly correlated with the subject's initial amount of higher order aberrations (p?=?0.001, R ^2?=?0.72). There was, however, no acuity improvement from static AO correction. In conclusion, correction of peripheral higher order aberrations can improve low contrast resolution, provided refractive errors are corrected and the system runs in continuous closed loop.     

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.     

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.     

Space-charge and third-order aberrations in quadrupole focusing systems

Our purpose is to consider third-order aberrations due to space charge. We derive the single-particle equations of transverse motion for a beam of elliptical cross section that propagates down a focusing channel. These equations include electric and magnetic fields caused by space charge and magnetic fields caused by quadrupoles and octupoles. We then calculate the third-order aberrations of a point-to-parallel system. We suggest that beam sizes should be minimized in order to control aberrations.     

Techniques for measuring aberrations in lenses used in photolithography with printed patterns

In optical lithography, it is a serious problem that aberrations in projection lenses reduce the imaging quality. Therefore techniques to measure the aberrations are required that will predict the adverse effects of aberrations on lithographic imagery and reduce them. We present a measurement method that uses a fine grating and its imaging condition to quantify coma, astigmatism, and spherical aberration. With this method, these aberrations can be described with simple expressions from the measured results. Application of this method revealed the coma of Zernike polynomials for our krypton fluoride (KrF) excimer-laser scanner.