Aberrations and retinal image quality of the normal human eye

2874 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 J. Liang and D. R. WilliamsFig. 1. Hartmann–Shack wave-front sensor for the eye. Lightfrom a He–Ne laser produces a compact point source on theretina. If the eye has aberrations, the wave front of the lightreturning from the retina forms a distorted wave front at the pupilplane. This wave front is recreated by lenses L 3 ,L 2 ,L 4 andL 5 at the plane of lenslet array. The two-dimensional lenslet arraysamples this warped wave front and forms an array of focusedspots on a CCD array. Each of the spots from the lensletsis displaced on the CCD array in proportion to the slope of thewave front; the wave aberration itself can be calculated from thisdisplacement.lenslet had an aperture of 0.5 mm and a focal length of 97mm. The wave aberration was measured across either a3.4-mm pupil or a 7.3-mm pupil. For the 3.4-mm pupilmeasurements, the pupil was magnified 2.5 times, so thatthe pupil was sampled with a center-to-center spacing of0.2 mm. For the 7.3-mm pupil measurements, lens L 5was changed so that the pupil was magnified 1.17 times,and the center-to-center spacing in the pupil was 0.43mm. Each lenslet forms an aerial image of the retinalpoint source on a cooled, scientific-grade CCD camera.The wave-front sensor measures an aberrated wave frontemerging from the eye in reference to a perfect planewave at the eye’s entrance pupil. This is equivalent tomeasuring the wave-front error of the eye at the exit pupilin reference to a perfect reference sphere. 9C. Measurements with Small PupilsTwo sets of measurements of the wave aberration weremade with 3.4-mm pupils. The first set was made on theeyes of three subjects whose modulation transfer functions(MTF’s) were measured before with the double-passand interferometric techniques. 12 The second set wasmade on nine other eyes with a slightly different procedure.For comparison with double-pass and interferometricMTF’s, measurements were made on the right eyes ofRNB, DRW, and DHB, whose ages were 36, 40, and 34,respectively. These observers were mildly myopic [0.2,1.6, and 0.4 diopters (D), respectively]. In addition, DRWhad 0.8 D of astigmatism. In measuring these three eyeswe used the same alignment procedure as in the earlierstudy. 12 Accommodation was paralyzed with two dropsof cyclopentolate hydrochloride (1%). During the alignmentprocedure before the measurements, the observeradjusted the horizontal and vertical positions of his eye tooptimize the image quality of an 18-c/deg horizontalsquare-wave grating. We chose this criterion for alignmentbecause the goal of the earlier study was to measurethe best image quality possible in the human eye. A mirrortemporarily placed between the spatial filter pinholeand lens L 1 allowed the observer to view the grating,which was sandwiched against a diffuser and was backlitwith 630-nm light. The grating lay at the same opticaldistance as the pinhole from the eye. Lens L 3 was attachedto the bite-bar mount so that the observer could focusthe grating by translating his eye together with thelens along the optical axis. Corrective lenses were notworn during the experiment. The entrance pupil for thebeam was 3 mm and was determined by an artificial pupilin the back focal plane of L 2 . The observer’s head wasstabilized with a bite bar.For each observer, 20 images were obtained, each correspondingto an exposure of 2sontheretina. The use ofthis long exposure reduced the speckle in the images, becauseeye movements cause slightly different retinal regionsto be illuminated by the point source over time,which alters the speckle pattern. This has the effect ofimproving signal-to-noise ratio. Nonetheless, we foundsimilar results with the shortest exposures that we tried,which were 100 ms. All images were obtained in a singleexperimental session.The measurements made for 3.4-mm pupils on nineother observers used a similar procedure, except for thefollowing. Each eye was aligned, not for optimum imagequality as before, but with respect to the center of thenatural pupil. For these measurements no drug wasused to dilate the pupil. At the beginning of the measurementon each eye, the subject adjusted his horizontalposition until the left side of the pupil occluded his view ofthe point source he was fixating. He repeated this taskusing the right side, the top, and the lower margin of thepupil. The average of two settings in each of these fourlocations was taken as the center of the entrance pupil.This served as the origin of the coordinate system inwhich the wave aberration was defined. The standarddeviation for the center of the entrance pupil with thisalignment technique was less than 0.1 mm. We reducedthe eye’s defocus by asking the observer to translate hiseye together with the lens (L 3 ) along the optical axis tooptimize image quality of the point source. The diameterof the laser beam at the entrance pupil was 1.5 mm in-

J. Liang and D. R. Williams Vol. 14, No. 11/November 1997/J. Opt. Soc. Am. A 2875stead of 3 mm. Subject age ranged from 21 to 38. Allhad normal visual acuity and required a correction for defocusand astigmatism of less than 3 D.D. Measurements with Large PupilsUsing the same procedures used on the nine observers describedabove, we made measurements on 14 eyes (nineobservers) for a 7.3-mm pupil. The pupil was dilatedwith tropicamide (1%). The exposure duration was 1 s.Three images, taken within 60 s, were averaged. Subjectage ranged from 21 to 38. All had normal visual acuityand had less than 3Dofdefocus and astigmatism.3. RESULTSA. Wave Aberration of the EyeFigure 2 shows results for the 3.4-mm pupil. Figure 2a.shows the expected result if an eye with perfect optics hadbeen used. It was obtained by introducing a plane waveinto the imaging path at the point where the eye’s pupilwould normally have been. The image consists of ahighly regular array of spots, one spot for each lenslet ofthe wave-front sensor. Figures 2b. and 2c. show the imagesobtained from real eyes. Aberrations in real eyesdisplaced each spot relative to the corresponding spot inthe reference image obtained with a planar wave front.The displacement is proportional to the local slope of thewave front at that lenslet. The local wave-front slope inthe x and y directions was measured at 217 locations simultaneouslyacross the pupil. The wave aberration wascomputed from the array of local slopes with a leastsquarestechnique. 9,13 We represented the wave aberrationwith the sum of 65 Zernike polynomials, correspondingto aberrations up to and including tenth order. 14Figures 2e. and 2f. show contour plots of the reconstructedwave aberration for each observer, averagedacross 20 exposures. The wave aberration shown hasbeen truncated to a pupil diameter of 3 mm. The spacingbetween contour lines is 0.15 m, which is roughly /4 at632.8 nm.Figures 3a.–c. show the images obtained with the large(7.3-mm) pupil from three real eyes. Figures 3d.–f. showthe contour plots of the wave aberration for three of theobservers with the 7.3-mm pupil. The raw images aswell as the reconstructed wave aberration from real eyesreveal substantial local, irregular aberrations that are notevident with the smaller pupil. For example, subject MLhas an arc-shaped ridge on the upper third of the plot thatcorresponds to the location where his eyelid normallyrests against the cornea.Observer JL was atypical in that he showed a substantialamount of spherical aberration. The spherical aberrationin his eye, like that in almost all eyes that had anyappreciable spherical aberration, was in the directionthat the rays entering the margin of the pupil experiencedhigher power than those entering near the center. Allthe eyes revealed less spherical aberration than would beexpected from a simplified eye with a single spherical refractingsurface. Approximately 1/3 of the eyes showedalmost no spherical aberration.B. Repeatability of the Wave-Aberration MeasurementsFigure 4a. shows with square symbols a vertical cross sectionthrough the wave aberration for one observer, RNB,measured with the small (3.4-mm) pupil. The error barsrepresent 1 standard deviation based on the 20 exposurestaken within a single session. The standard deviationaverages 0.046 m or/14 ( 0.633 m), indicatingthat the wave-aberration measurements are highlyrepeatable. The wave aberration obtained when a planewave passing through a 0.5-D trial lens replacedFig. 2. Wave-front-sensor images and wave aberration of eyes for a small 3-mm pupil. a. The image from the wave-front sensor foran ideal eye on the left, which corresponds to no phase error across the pupil, as shown in the wave aberration on the right. b. and c.show the wave-front-sensor images for two real eyes along with the calculated wave aberration. The contour interval in the waveaberrationplots (d.–f.) is 0.15 m. The pupil was sampled with a center-to-center spacing of 0.2 mm.

2876 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 J. Liang and D. R. WilliamsFig. 3. Wave-front-sensor images and wave aberration of eyes for a 7.3-mm pupil. a.–c. are the wave-front-sensor images for threeobservers. The center-to-center spacing of lenslets in the pupil was 0.42 mm. d.–f. are the corresponding wave aberrations of the threeeyes from measurements of the wave-front slopes. The contour interval in the wave-aberration plots is 0.15 m for OP and 0.3 m forJL and ML. Defocus and astigmatism have been removed from the wave aberrations, thus showing the presence of substantial irregularaberrations. The peak-to-valley wave-front error for the 7.3-mm pupil is approximately 7 m, 4 m, and 5 m for JL, OP, and ML,respectively.Fig. 4. Repeatability of measurements with the wave-front sensor.a. Measurements of the wave aberration along one crosssection of a 3-mm pupil for a real eye (RNB) and an artificial eye.b. Measurement of the wave aberration along one cross sectionof a 7.3-mm pupil for observer JL. The error bars are 1 standarddeviation.the light that would exit the eye’s pupil is shown with thedashed curve in Fig. 4. The mean of the standard deviationof these measurements is only 0.0013 m or/487,35 times smaller than the standard deviation when a realeye was measured. This shows that the variability dueto the instrument is considerably smaller than that introducedby the eye. Possible sources of variability includefluctuations in focus and possibly other aberrations (despitethe use of cyclopentolate hydrochloride), smallmovements of the eye and head, and variations in thethickness of the tear film. Eye movements probably havea very small effect on the wave-front measurement, becausethe shift of pupil position is small for the fixatingeye. For example, a relatively large fixational eye movementof 10 arc min produces a pupillary displacement ofless than 35 m, which is a small fraction of the spacingbetween samples taken in the pupil. Our results withthe wave-front sensor show that the important aberrationscorrespond to gradual enough variations in phaseacross the pupil that normal fixational eye movementsare not a problem. In addition, the images acquired bythe CCD camera with real eyes contain some laserspeckle that is not present when a plane wave is introducedinto the system. Laser speckle, which changesfrom exposure to exposure, adds variability to the measurements.Figure 4b. shows with circular symbols a vertical crosssection through the wave aberration for one of the observers(JL) measured for a large (7.3-mm) pupil. The standarddeviation averages 0.052 m or /12, againshowing the high repeatability of the wave-front measurementfor the large (7.3-mm) pupil.C. Accuracy of the Wave-Front SensorWe assessed the accuracy with which the wave-front sensorcan measure known aberrations by introducing defocusof known amounts into the system. We passed aplane wave through a spherical trial lens positionedwhere the eye’s pupil would otherwise have been and theninto the wave-front sensor. The measured dioptric powerof the trial lens was calculated from the correspondingmode of the Zernike expansion of the wave aberration.Figure 5 shows the relationship between the measuredpower and the nominal dioptric power of the trial lensover an 8-D range. The discrepancy between the nominaland the measured power never exceeded 0.17 D and

J. Liang and D. R. Williams Vol. 14, No. 11/November 1997/J. Opt. Soc. Am. A 2877was generally much smaller for small powers. Similaraccuracy was obtained when cylindrical lenses were usedinstead of spherical lenses. The error should be consideredan upper bound on the error of the wave-front sensor,since some of these discrepancies may be attributableto manufacturing errors in the trial lenses. As expected,all the Zernike terms corresponding to aberrations otherthan either defocus or astigmatism, depending on thetype of trial lens used, were very close to zero. Thoughwe have not explicitly assessed the accuracy with whichthe wave-front sensor measures higher-order aberrationsother than defocus and astigmatism, this shows that thewave-front sensor does not introduce spurious higherorderaberrations that are not present in the wave aberration.Since higher-order aberrations displace the imageson the CCD, much like defocus and astigmatism butsimply in a different pattern, it is reasonable to expectthat the sensor produces accurate measurement of theseaberrations as well.Fig. 5. Measurement of trial lenses with the wave-front sensor.The curve shows the power in diopters derived from the wavefrontsensor as a function of the nominal power of trial lenses insertedinto the system.D. Comparison of the Wave-Front-Sensor MTF withDouble-Pass and Interferometric Estimates ofthe Eye’s MTFIn this section we compare estimates of the eye’s MTF obtainedwith the wave-front sensor with results reportedby Williams et al. 12 obtained with the double-pass and interferometrictechniques. The wave-front-sensor measurementswere carried out under conditions chosen tomatch those used in the double-pass and interferometricmeasurements. We used the same three observers, thesame refractive state, the same alignment procedure, thesame pupil size (3 mm), the same wavelength (632.8 nm),and the same retinal location (fovea). The wave-frontsensormeasurements were made approximately 2 yearsafter the interferometric and double-pass measurements.Figures 6a.–c. show the MTF’s of the three subjects obtainedwith the wave-front sensor compared with theMTF’s obtained with the interferometric and double-passtechniques. The eye’s MTF from the wave-front sensorwas taken as the autocorrelation of the eye’s pupil function.The MTF’s from the wave-front sensor do not includethe Stiles–Crawford effect. The Stiles–Crawfordeffect 15,16 is small at the foveal center, 17 and MTF’s calculatedfor this pupil size (3 mm) incorporating the Stiles–Crawford effect were negligibly different from those calculatedwithout it. Each MTF in Fig. 6 displays the crosssection of the two-dimensional MTF corresponding to thatfor horizontal gratings. The double-pass technique producesthe lowest MTF because of choroidal scatter, whichgrows in red light. Williams et al. 12 showed that whengreen light is used in the double-pass technique, thedouble-pass MTF was raised slightly into agreement withthe interferometric MTF. This is probably because redFig. 6. Comparison of MTF’s obtained with wave-front sensing, the double-pass method, and the interferometric technique. a.–c. comparethe MTF’s for each of the three observers, and d. shows the mean for the three observers. The interferometric and the double-passdata are from Williams et al., 12 who studied the same observers measured here. The curves show the eye’s MTF for horizontal gratings.The error bars for the interferometric MTF’s are 1 standard error of the measurements.

2878 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 J. Liang and D. R. Williamslight is more subject to multiple scattering in the choroidthan is green light.The within-subject variability of the MTF obtainedfrom wave-front sensing was very small; the standard errorof the individual MTF’s is smaller than the datapoints in Figs. 6a.–c. Nonetheless, as can be seen fromthe size of the error bars in Fig. 6d., the intersubject variabilityis greater for the wave-front-sensor MTF’s than foreither the double-pass or the interferometric MTF’s. Thewave-front-sensor data were collected in a single sessionon a single day, whereas the interferometric and doublepassmethods involved measurements over several days.It is possible that day-to-day variation in the eye’s MTF,perhaps as a result of shifts of refractive state and alignment,can explain this difference.In all three subjects the MTF from the wave-frontsensorlies somewhat above the interferometric data atlower spatial frequencies but agrees well at the highestspatial frequencies. One hypothesis is that forward lightscatter may account for the discrepancy. The wave-frontsensor discretely samples the pupil and does not captureimperfections in the eye’s optics at a very fine spatialscale such as those that would cause forward light scatter.To explain the low-frequency discrepancy, forwardscatter would have to produce a relatively broad skirt inthe eye’s point-spread function that affected the interferometricand the double-pass but not the wave-front-sensormeasurements. However, such a skirt would reducemodulation transfer at high as well as at low spatial frequencies,yet the discrepancy is confined to low frequencies.Therefore it does not appear that forward scatter byitself will explain the higher wave-front-sensor MTF’s atlow frequencies.Another possibility is that the refractive state of the observersin the different experiments was different. Althoughthe same grating target and procedure were usedin each case, the wave-front-sensor data were collectedwithin a single experimental session, whereas the interferometricdata were obtained in several sessions each ofgreater length and were run on different days. It is conceivablethat the refractive error was somewhat larger inthe longer, interferometric experiments. The amount ofdefocus required to bring the mean MTF’s into approximateregister is only 0.15 diopters. Another factor thatcould contribute to the discrepancy is differences in pupilalignment. Although the same subjective criterion, optimumimage quality of a horizontal grating, was used forall three methods, we do not know whether the subjectsselected the same pupil location with each method.Despite these small differences among the MTF’s obtainedwith different techniques, the three techniquesyield reasonably similar MTF’s. The Hartmann–Shackwave-front sensor captures the most important sources ofimage blur, a conclusion that is also supported by evidencethat wave-front-sensor data can be used to correcthigher-order aberrations in the eye. 18 Although wavefront-sensingmethods are insensitive to light scattercaused by structures on a spatial scale in the pupil finerthan that of the sampling array, the agreement in Fig. 6shows that in the normal eye, light scatter is a minorsource of image blur compared with the aberrations thatthe wave-front sensor can measure.E. Comparison of the Wave-Front-Sensor MTF with theMTF from the Objective AberroscopeFigure 7 compares the wave-front-sensor MTF with thatfrom the aberroscope, 19 both for a 3-mm pupil. Defocusand astigmatism have been removed from the wave-frontsensorMTF’s, as they were from the data of Walsh andCharman. 19 The wave-front sensor produces an MTFthat lies below that of the aberroscope. This differencecould be due to individual differences among the subjectsin the two studies, which is known to be large. 3,4 Toevaluate this possibility, we computed the MTF’s from the12 observers measured with the wave-front sensor for a3.4-mm pupil. The shaded area in Fig. 7 indicates therange of MTF’s obtained with the wave-front sensor,showing that the Walsh and Charman data lie at the upperedge of our sample. Walsh and Charman’s MTF isfor a wavelength of 590 rather than 633 nm, but we expectthis difference to be relatively unimportant. Althoughit is true that short wavelengths will generallyproduce a higher MTF in an optical system that has no orlittle aberrations, the aberrations in real eyes tend tomask this effect. The reason is that a fixed wave aberrationproduces a larger reduction in the MTF for shorterwavelengths, roughly compensating for the reduced effectof diffraction.Another possibility is that the difference is related tothe finer spatial scale with which we analyzed the waveaberration. The wave-front sensor had a fourfold increasein pupil sampling density relative to the aberroscope,and the wave aberration was described with basisfunctions up to tenth order instead of only fourth order,which corresponds to a greater than fourfold increase inthe number of modes characterized in each eye.The 65 polynomials in the wave-front fit include allZernike modes with radial power less than or equal to 10,Fig. 7. Comparison of MTF’s obtained with the wave-front sensorand with the aberroscope for a 3-mm pupil. Circles show theMTF of the eye from the objective aberroscope. 19 Squares show,for three observers, the mean MTF’s with both defocus and astigmatismremoved. Triangles show, for the same observers, themean MTF’s derived from wave aberrations that included onlythird- and fourth-order Zernike aberrations, with higher andlower orders removed. The shaded area shows the range of theMTF’s for 12 eyes for a 3-mm pupil measured with our wavefrontsensor.

J. Liang and D. R. Williams Vol. 14, No. 11 /November 1997/J. Opt. Soc. Am. A 2879Fig. 8. Zernike description of the eye’s aberrations. a. The wave aberration in YL’s eye for a 7.3-mm pupil, shown at the top, isdecomposed into Zernike polynomials up to tenth order. Contour line spacing is 0.15 m. Modes shown include classical aberrationssuch as defocus (0.1 D), astigmatism (0.8 D at 15 deg), coma, and spherical aberration. The decomposition reveals higher-order, irregularaberrations in addition to classical aberrations. b. The upper curve (squares) shows the RMS wave-front error of each Zernikeorder for a 7.3-mm pupil averaged across 14 human eyes. Error bars indicate the standard deviation among eyes. For the second-orderZernike modes, only astigmatism is shown. The average amount of astigmatism in these observers was 0.6 D, corresponding to a meanRMS value of 0.77 m. The middle curve (triangles) shows the data for a 3.4-mm pupil averaged across 12 tested eyes. The lowercurve (circles) is for an artificial eye. The error bars show the standard deviation of ten repeated measurements.except for the piston term. The first-order Zernike modesare the linear terms (corresponding to tilt, which we ignorehere). The second-order modes are the quadraticterms corresponding to the familiar aberrations—defocusand astigmatism. The third-order modes represent comaand comalike aberrations. The fourth order containsspherical aberration as well as other modes. The fifth totenth orders are the higher-order, irregular aberrations.Local irregularities in the wave front within the pupil arerepresented by these higher-order Zernike modes. Forthis pupil size, the wave-front-sensor data show that aberrationshigher than fourth order play a very small role.The square symbols in Fig. 7 show the MTF for the originalthree observers when defocus and astigmatism havebeen corrected. The triangles show the MTF when theirregular aberrations corresponding to orders 5–10 havebeen removed, leaving only diffraction and third- andfourth-order aberrations to determine the MTF. There islittle difference between these two MTF’s, indicating thataberrations corresponding to orders 5–10 do not play animportant role in image quality for 3-mm pupils.F. Irregular AberrationsIrregular aberrations play a more significant role whenthe pupil is large. In that case, the aberrations of the eyeare not well described by the classic aberrations of conventionaloptical systems. Figure 8a. shows the Zernikedecomposition of the wave aberration of subject LY’s eyefor a 7.3-mm pupil. The contribution of the classic aberrationssuch as defocus, astigmatism, coma, and sphericalaberration, as well as irregular aberrations, are shown.The root-mean-square (RMS) error of an individualZernike order is a measure of that order’s role in degradingoptical quality. Figure 8b. shows the RMS wavefronterror contributed by each Zernike order for a3.4-mm pupil (triangles), a 7.3-mm pupil (squares), andan artificial eye with a 6.7-mm pupil (circles). The averageRMS wave-front error decreases monotonically as theZernike order increases for both pupil sizes in the humaneye, though the pattern varies somewhat among individualobservers. The RMS error for the small pupil lies3–4 times lower than that for the large pupil of real eyes.This illustrates the well-known fact that aberrations growwith increasing pupil size. An RMS error of /14 (0.045m at 0.633 m) is a common tolerance for diffractionlimitedperformance in an optical system. 14 For the3.4-mm pupil, only Zernike orders up to third order exceedthis tolerance. At the larger pupil size, however,the mean RMS value of each Zernike order from 2 to 8 isgreater than /14.Figure 8b. also shows the RMS wave-front error for anartificial eye (circles) measured with the same instrument.The artificial eye consisted of an achromatic dou-

2880 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 J. Liang and D. R. Williamsblet ( f 16 mm) and a diffuser to mimic the retina.Though our measurements reveal some expected sphericalaberration in the artificial eye, the mean RMS value ofeach order averages approximately one order of magnitudelower than in human eyes for the large (7.3-mm) pupil.This indicates that the measured higher-order, irregularaberrations are true defects of the human eyesand not aberrations in the apparatus.G. Similarity of Left and Right EyesWe confirmed previous reports 3,4 that aberrations differgreatly from observer to observer. This leads to thespeculation that aberrations in the eye may not be systematicbut rather reflect random errors in the structureof the lens and cornea. However, we found that aberrationsare relatively similar between the left and right eyesof the same observer. This can be seen in Fig. 9a. whichshows three-dimensional plots of the wave aberrations oftwo observers. Figure 9b. shows the Zernike coefficientsFig. 9. Similarity of the eye’s aberrations in the left and righteyes. a. Three-dimensional surface plots of the wave aberrationof the left and right eyes of two observers, showing the mirrorsymmetry of left and right eyes. Defocus and astigmatismhave been removed from the wave aberration. b. Coefficientsof individual Zernike modes in the right eye plotted against thecorresponding coefficients of the left eye, showing the correlationbetween left and right eyes. Data of four subjects are combined.of the left eye plotted against the same coefficient of theright eye. The data points lie close to a straight line,with a slope of 1 indicating the correlation between theaberrations of the left and right eyes.H. Modulation Transfer Functions and Pupil SizeFrom our measurements of the eye’s wave aberration fora small (3.4-mm) and a large (7.3-mm) pupil, we show inFig. 10 the mean MTF for pupil sizes from 2 to 7.3 mm.The eye’s MTF’s were computed from the wave aberration,assuming uniform transmittance across thepupil. 20,21 Our results are in qualitative agreement withthose from other methods such as the interferometrictechnique 22 and the double-pass technique, 23 and frommeasurements of the eye’s wave aberration. 24 Specifically,at low frequencies the MTF is highest for pupils between2 and 3 mm, with aberrations reducing modulationtransfer for larger pupils. At high frequencies as shownin Fig. 10b., however, larger pupil sizes provide improvedmodulation transfer despite the presence of aberrations.I. Influence of the Higher-Order Irregular Aberrationson the Eye’s MTF for Small and Large PupilsIn this section we show the importance of higher-orderaberrations for retinal image quality. Figure 11a. showsthe MTF (squares) when only defocus and astigmatismhave been corrected for the 3-mm pupil. This curve lieswell below the uppermost curve, which is the MTF for anaberration-free 3-mm pupil. This discrepancy shows thedegradation of the eye’s optical quality by the eye’s monochromaticaberrations. Triangular symbols show theMTF if second-, third-, and fourth-order aberrations werecorrected, leaving only the higher-order, irregular aberrationsto produce image blur. There is little difference betweenthis curve and diffraction alone. As the uppercurves in Fig. 11a. show, the ratio of the diffraction MTFto the MTF with only defocus and astigmatism correctedgrows to more than a factor of 2 at high spatial frequencies.The higher-order (fifth–tenth) irregular aberrationsby themselves reduce image contrast by less than30%.In contrast, Fig. 11b. shows the MTF (squares) whenonly defocus and astigmatism have been corrected for the7.3-mm pupil. This curve lies far below the uppermostcurve, which is the MTF for an aberration-free 7.3-mmpupil. This discrepancy reveals the deleterious effect ofthe eye’s monochromatic aberrations on retinal imagequality. Triangles show the MTF if second-, third-, andfourth-order aberrations were corrected. Even in thiscase, the higher-order, irregular aberrations produce asubstantial loss in retinal image quality compared withan eye suffering only from diffraction. As shown by theupper curves, the ratio of the diffraction MTF to the MTFwith only defocus and astigmatism corrected grows tomore than a factor of 10 at high spatial frequencies.Even the higher-order (fifth–tenth) irregular aberrationsby themselves reduce image contrast by up to three tofour times at high frequencies.J. Strehl Ratio and Higher-Order AberrationsOne important reason to characterize higher-order aberrationsis to determine how much improvement in retinal

J. Liang and D. R. Williams Vol. 14, No. 11/November 1997/J. Opt. Soc. Am. A 2881Fig. 10. Mean of the radially averaged MTF’s of the eye for different pupil sizes. The numbers on the curves indicate the pupil size ofthe eye. The mean MTF’s for a 2- and a 3-mm pupil are derived from the wave aberration of 12 eyes measured across a 3.4-mm pupil,and the mean MTF’s for 4-, 5-, 6-, and 7.3-mm pupils are derived from the wave aberration of 14 eyes measured across a 7.3-mm pupil.Defocus and astigmatism were removed for each eye and each pupil size. Plots a. (linear) and b. (semilog) are from the same data.Fig. 11. Mean of the radially averaged MTF’s for two pupil sizes: a. 3 mm and b. 7.3 mm. The top curve in the lower part of the figureis the MTF for an aberration-free eye, in which diffraction is the sole source of image blur. The lowest curve is for eyes corrected toremove defocus and astigmatism entirely. The middle curve is for eyes with second-, third-, and fourth-order Zernike aberrations correctedbut with the higher-order, irregular aberrations (orders 5–10) uncorrected. The error bars are the standard deviation of 12 testedeyes for the small (3-mm) pupil and of 14 tested eyes for the large (7.3-mm) pupil. The upper part of the figure plots the ratio of thediffraction MTF to the mean MTF of real eyes if only defocus and astigmatism are corrected (squares) and the ratio of the diffractionMTF to the mean MTF of real eyes if the higher-order (fifth to tenth) irregular aberrations remain uncorrected (triangles).image quality could be achieved by correcting various aberrations.For this purpose we use the Strehl ratio as ametric for retinal image quality. The Strehl ratio, whichcan range from 0 to 1, is the ratio of the peak intensity ofthe eye’s PSF to that of a PSF for an aberration-free eyewith the same pupil size, in which diffraction is the onlysource of blur. Strehl ratios greater than 0.8 are generallyconsidered to correspond to diffraction-limitedimaging. 14Figure 12 shows how the Strehl ratio for 3- and 7.3-mmpupils improves as we remove successively higher Zernikeorders from the wave aberration. For example, the valueof 4 in the abscissa corresponds to the case in which thesecond-, third-, and fourth-order Zernike aberrationshave been removed from the wave aberration. One canthink of the removal of Zernike orders as corresponding tothe effect of a hypothetical optical element akin to a deformablemirror that was capable of correcting lowerorderaberrations while leaving the higher-order aberrationsunaffected. For 3-mm pupils, one need correct onlysecond-, third-, and fourth-order aberrations to achieve adiffraction-limited Strehl ratio. However, for 7.3-mm pupils,Zernike orders up to and including at least eighth ordermust be corrected to achieve diffraction-limited per-

2882 J. Opt. Soc. Am. A/Vol. 14, No. 11/November 1997 J. Liang and D. R. WilliamsFig. 12. Strehl ratio of the eye’s PSF for a 3-mm pupil (squares)and for a 7.3-mm pupil (triangles). Error bars show the standarddeviation of 12 tested eyes for the 3-mm pupil and of 14tested eyes for the 7.3-mm pupil. Each data point shows theStrehl ratio that would have been obtained with lower Zernikeorders removed to provide a measure of optical quality werelower orders corrected. The abscissa indicates the highest lowerorder removed in each case. For example, a value of 2 on theabscissa means that only second-order aberrations, correspondingto defocus and astigmatism, have been removed. A value of3 indicates that both second- and third-order aberrations havebeen removed.formance. This should be considered a lower boundbecause it assumes that all the eye’s aberrations are describedwith a tenth-order fit. Although the trend illustratedin Fig. 8 would suggest that still-higher-order aberrationsthan tenth are relatively unimportant, theypresumably also reduce retinal image quality to some extent.4. DISCUSSIONUnlike other techniques that measure only the eye’sMTF 12,22,23,25–27 techniques such as the aberroscopicmethod and wave-front sensing also measure the phasetransfer function of the eye, which plays an importantrole in retinal image quality when the pupil is large. 28 Inaddition, these techniques measure the wave aberrationsimultaneously at many locations throughout the pupil,unlike psychophysical methods, 7,29,2,30 which must sequentiallymap the pupil point by point. Rapid measurementcan reduce the possibility that temporal fluctuationsof refractive state will distort the estimate of the wave aberration.The knife-edge technique 6 allows simultaneousmeasurement of the entire pupil, but the phase error ateach location is estimated from the intensity of the lightat each point of the pupil, which can be confounded withother factors such as the Stiles–Crawford effect. Thewave-front sensor’s measurements are independent of intensityvariations in the light returning from the retina.The objective aberroscope 4 provides simultaneouswave-aberration measurements of the entire pupil butcannot sample the pupil with a spacing finer than0.9 mm. 31 Our wave-front sensor shows that it is possibleto sample the pupil at a much finer spatial scale.For example, for the 3.4-mm pupil measurements, asample spacing of 0.2 mm was used, which is more thanfour times smaller. In principle, even finer samplingcould be employed. Finer sampling helps to characterizethe more abrupt changes in phase error associated withirregular aberrations. A preliminary comparison of thewave aberration obtained with fine (0.2 mm) and coarse(0.42 mm) sampling for the same pupil size suggestedthat the finer sampling produced somewhat higher estimatesof the eye’s aberrations. Our wave-front sensorhas a number of potential applications because of its abilityto reliably capture irregularities at small as well aslarge spatial scales in the pupil. Because the PSF of theeye (or alternatively the MTF and the phase transferfunction) can be computed from the wave aberration,wave-front sensing reveals the impact of these irregularitieson retinal image quality. Conventional measures ofvisual performance, such as Snellen acuity or contrastsensitivity, do not link visual performance to specific defectsin the eye’s optics. Wave-front sensing can be usedto measure the wave aberration before and after laser refractivesurgery, 32,33 to quantitatively describe the opticalconsequences of corneal ablation and perhaps ultimatelyto provide a superior visual outcome. Moreover, it can aidin the design and fitting of contact lenses, so that onecould develop techniques to correct additional aberrationsbesides defocus and astigmatism. The wave-front sensorcan also be used in conjunction with a deformable mirrorto compensate for the eye’s wave aberration, providing supernormalvision and unprecedented resolution for retinalimaging. 18Although it is well known that defocus and astigmatismtend to be correlated in left and right eyes, we findthat this principle can be extended to other aberrations aswell. The genetic and environmental factors that controlall the aberrations of the eye’s dioptrics must operate inapproximate mirror symmetry on the left and right eyes.The biological optics of the human eye reveal local irregularitiesthat are not present in man-made optics.The wave-front measurements provide the most completedescription of the eye’s aberrations, showing conclusivelythat for large pupils the eye suffers from higher-order, irregularaberrations. These irregular aberrations do notreduce visual performance when the pupil is small(3 mm), such as in very-high-light-level conditions.However, they do reduce retinal image contrast in the visiblerange of spatial frequencies when the pupil is large.For example, for a 7.3-mm pupil at 20 c/deg, aberrationsbeyond defocus and astigmatism reduce retinal imagecontrast by a factor of approximately 7. The effect ofthese aberrations is especially deleterious in attempts toimage the retina at very high resolution. 34 To obtaindiffraction-limited retinal imaging with large pupils,methods such as adaptive optics must correct these irregularaberrations as well as the classical, lower-orderdefects in the eye.ACKNOWLEDGMENTSThis research is supported by National Institutes ofHealth grants EY04367 and EY01319, a Rochester Eyeand Human Parts Bank fellowship, and an ophthalmologydevelopment grant from Research to Prevent Blindness,Inc. The authors thank Don Miller and G. MichaelMorris for their contributions to this work.

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