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.     

Calculated impact of higher-order monochromatic aberrations on retinal image quality in a population of human eyes.

We calculated the impact of higher-order aberrations on retinal image quality and the magnitude of the visual benefit expected from their correction in a large population of human eyes. Wave aberrations for both eyes of 109 normal subjects and 4 keratoconic patients were measured for 3-, 4-, and 5.7-mm pupils with a Shack-Hartmann sensor. Retinal image quality was estimated by means of the modulation transfer function (MTF) in white light. The visual benefit was calculated as the ratio of the MTF when the monochromatic higher-order aberrations are corrected to the MTF corresponding to the best correction of defocus and astigmatism. On average, the impact of the higher-order aberrations for a 5.7-mm pupil in normal eyes is similar to an equivalent defocus of approximately 0.3 D. The average visual benefit for normal eyes at 16 c/deg is approximately 2.5 for a 5.7-mm pupil and is negligible for small pupils (1.25 for a 3-mm pupil). The benefit varies greatly among eyes, with some normal eyes showing almost no benefit and others a benefit higher than 4 at 16 c/deg across a 5.7-mm pupil. The benefit for keratoconic eyes is much larger. The benefit at 16 c/deg is 12 and 3 for 5.7- and 3-mm pupils, respectively, averaged across four keratoconics. These theoretical benefits could be realized in normal viewing conditions but only under specific conditions.     

Calculated impact of higher-order monochromatic aberrations on retinal image quality in a population of human eyes

We calculated the impact of higher-order aberrations on retinal image quality and the magnitude of the visual benefit expected from their correction in a large population of human eyes. Wave aberrations for both eyes of 109 normal subjects and 4 keratoconic patients were measured for 3-, 4-, and 5.7-mm pupils with a Shack-Hartmann sensor. Retinal image quality was estimated by means of the modulation transfer function (MTF) in white light. The visual benefit was calculated as the ratio of the MTF when the monochromatic higher-order aberrations are corrected to the MTF corresponding to the best correction of defocus and astigmatism. On average, the impact of the higher-order aberrations for a 5.7-mm pupil in normal eyes is similar to an equivalent defocus of approximately 0.3 D. The average visual benefit for normal eyes at 16 c/deg is approximately 2.5 for a 5.7-mm pupil and is negligible for small pupils (1.25 for a 3-mm pupil). The benefit varies greatly among eyes, with some normal eyes showing almost no benefit and others a benefit higher than 4 at 16 c/deg across a 5.7-mm pupil. The benefit for keratoconic eyes is much larger. The benefit at 16 c/deg is 12 and 3 for 5.7- and 3-mm pupils, respectively, averaged across four keratoconics. These theoretical benefits could be realized in normal viewing conditions but only under specific conditions.     

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.     

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.     

Monochromatic aberrations provide an odd-error cue to focus direction

Monochromatic aberrations that exist in the human eye will cause differences in the appearance of the point-spread function (PSF) depending on whether there is positive or negative defocus. We establish whether it is possible to use these differences in the PSF to distinguish the direction of defocus. The monochromatic aberrations of eight subjects were measured with a Hartmann-Shack wave-front sensor. Subjects also performed a forced-choice psychophysical task in which they decided whether a blurred target was defocused in front of or behind the retina. The optical system for the psychophysical task was designed to isolate the blur due to monochromatic aberrations as the only odd-error cue to the direction of defocus. Shack-Hartmann measurements showed that monochromatic aberrations increase as the pupil size increases. On average, the correct/incorrect responses for discriminating differences in the PSF for different directions of defocus were 54/46 for a 1-mm pupil and 83/17 for a 5-mm pupil, representing more than an eight-fold increase in discriminability. This discriminability extended for large amounts of defocus and also for more complex targets, such as letters. Sensitivity to the differences in the PSF for different directions of defocus increased as monochromatic aberrations increased, particularly for the even-order aberrations, which give rise to an odd-error focus cue. It was found that the ability to discriminate PSFs for different directions of defocus varied among individuals but, in general, depended on the magnitude of monochromatic aberrations.     

Wavefront-guided correction of ocular aberrations: are phase plate and refractive surgery solutions equal?

Wavefront-guided laser eye surgery has been recently introduced and holds the promise of correcting not only defocus and astigmatism in patients but also higher-order aberrations. Research is just beginning on the implementation of wavefront-guided methods in optical solutions, such as phase-plate-based spectacles, as alternatives to surgery. We investigate the theoretical differences between the implementation of wavefront-guided surgical and phase plate corrections. The residual aberrations of 43 model eyes are calculated after simulated refractive surgery and also after a phase plate is placed in front of the untreated eye. In each case, the current wavefront-guided paradigm that applies a direct map of the ocular aberrations to the correction zone is used. The simulation results demonstrate that an ablation map that is a Zernike fit of a direct transform of the ocular wavefront phase error is not as efficient in correcting refractive errors of sphere, cylinder, spherical aberration, and coma as when the same Zernike coefficients are applied to a phase plate, with statistically significant improvements from 2% to 6%.     

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.     

Effect of rotation and translation on the expected benefit of an ideal method to correct the eye's higher-order aberrations

An ideal correcting method, such as a customized contact lens, laser refractive surgery, or adaptive optics, that corrects higher-order aberrations as well as defocus and astigmatism could improve vision. The benefit achieved with this ideal method will be limited by decentration. To estimate the significance of this potential limitation we studied the effect on image quality expected when an ideal correcting method translates or rotates with respect to the eye's pupil. Actual wave aberrations were obtained from ten human eyes for a 7.3-mm pupil with a Shack-Hartmann sensor. We computed the residual aberrations that appear as a result of translation or rotation of an otherwise ideal correction. The model is valid for adaptive optics, contact lenses, and phase plates, but it constitutes only a first approximation to the laser refractive surgery case where tissue removal occurs. Calculations suggest that the typical decentrations will reduce only slightly the optical benefits expected from an ideal correcting method. For typical decentrations the ideal correcting method offers a benefit in modulation 2-4 times higher (1.5-2 times in white light) than with a standard correction of defocus and astigmatism. We obtained analytical expressions that show the impact of translation and rotation on individual Zernike terms. These calculations also reveal which aberrations are most beneficial to correct. We provided practical rules to implement a selective correction depending on the amount of decentration. An experimental study was performed with an aberrated artificial eye corrected with an adaptive optics system, validating the theoretical predictions. The results in a keratoconic subject, also corrected with adaptive optics, showed that important benefits are obtained despite decentrations in highly aberrated eyes.     

Influences of reference plane and direction of measurement on eye aberration measurement

We explored effects of measurement conditions on wave aberration estimates for uncorrected, axially myopic model eyes. Wave aberrations were initially referenced to either the anterior corneal pole or the natural entrance pupil of symmetrical eye models, with rays traced into the eye from infinity (into the eye) to simulate normal vision, into the eye from infinity and then back out of the eye from the retinal intercepts (into/out of the eye), or out of the eye from the retinal fovea (out of the eye). The into-the-eye and out-of-the-eye ray traces gave increases in spherical aberration as myopia increased, but the into/out-of-the-eye ray trace showed little variation in spherical aberration. Reference plane choice also affected spherical aberration. Corresponding residual aberrations were calculated after the models had been optically corrected, either by placing the object or image plane at the paraxial far point or by modifying corneas to simulate laser ablation corrections. Correcting aberrations by ablation was more complete if the original aberrations were referenced to the cornea rather than to the entrance pupil. For eyes corrected by spectacle lenses, failure to allow for effects of pupil magnification on apparent entrance pupil diameter produced larger changes in measured aberrations. The general findings regarding choice of reference plane and direction of measurement were found to be equally applicable to eyes that lacked rotational symmetry.     

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.     

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.     

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.     

Contribution of the cornea and internal surfaces to the change of ocular aberrations with age.

We studied the age dependence of the relative contributions of the aberrations of the cornea and the internal ocular surfaces to the total aberrations of the eye. We measured the wave-front aberration of the eye with a Hartmann-Shack sensor and the aberrations of the anterior corneal surface from the elevation data provided by a corneal topography system. The aberrations of the internal surfaces were obtained by direct subtraction of the ocular and corneal wave-front data. Measurements were obtained for normal healthy subjects with ages ranging from 20 to 70 years. The magnitude of the RMS wave-front aberration (excluding defocus and astigmatism) of the eye increases more than threefold within the age range considered. However, the aberrations of the anterior corneal surface increase only slightly with age. In most of the younger subjects, total ocular aberrations are lower than corneal aberrations, while in the older subjects the reverse condition occurs. Astigmatism, coma, and spherical aberration of the cornea are larger than in the complete eye in younger subjects, whereas the contrary is true for the older subjects. The internal ocular surfaces compensate, at least in part, for the aberrations associated with the cornea in most younger subjects, but this compensation is not present in the older subjects. These results suggest that the degradation of the ocular optics with age can be explained largely by the loss of the balance between the aberrations of the corneal and the internal surfaces.     

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.     

Analysis of refractive errors in the eyes of young Indians

The refractive errors of normal, healthy eyes were measured using an indigenously developed Shack–Hartmann aberrometer. Measurement was made in both right and left eyes after dilation for a 6?mm pupil size. The power vector method was used to represent the sphero-cylindrical errors. Analysis was done for astigmatism with the rule, oblique astigmatism and defocus between the right and left eyes of the subjects, which showed a negative, positive and zero correlation respectively. No correlation could be detected for RMS values between right and left eyes, though Zernike between right and left showed bilateral symmetry in our subjects. It was found that with an increase of spherical aberration, defocus decreased slightly. The validity and repeatability of our Shack–Hartmann aberrometry in measuring the refractive error was analysed and repeatability coefficient calculated. Optimal correction for greater retinal image quality has been discussed and far-point vergence for detecting the point of maximum retinal image quality is suggested.     

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.     

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.     

高精度球面检测中调整误差的精确校正

针对高精度球面检测中球面的调整误差影响,分别研究了波前离焦和倾斜所引入的Zernike像差项,并以此为基础,提出了基于Zernike系数的高精度球面调整误差校正方法.利用Zygo干涉仪对大数值孔径的待测球面进行了检测,实验中对待测球面同时引入波前离焦和倾斜误差,再分别运用所提出的校正方法和传统的消倾斜离焦校正方法得到波面数据并进行对比,进而验证了所提出的校正方法的可行性.实验结果表明,利用该方法可有效地校正大数值孔径球面存在调整误差时所引入的球面像差,并且实验中的校正精度达到了0.002λ（λ为检测光波长）.提出的基于Zernike系数的球面调整误差校正方法可降低对于调节机构的精度以及检测人员的经验要求,在高精度球面检测中具有非常高的实际应用价值.