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OpticsOptikOptikOptik 119 (2008) 383–387

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doi:10.1016/j.ijl

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The study of wavelength-dependent wavefront aberrations

based on individual eye model

Ming Liua, Zhao-Qi Wanga,�, Yan Wangb, Tong Zuob, Yang Wanga

aKey Laboratory of Opto-electronic Information Science and Technology, Ministry of Education,

Institute of Modern Optics, Nankai University, Tianjin 300071, PR ChinabTianjin Eye Hospital, Refractive Surgery Center, Tianjin 300020, PR China

Received 28 July 2006; accepted 6 January 2007

Abstract

Theoretical calculations of the wavelength dependence of the ocular wavefront aberrations are performed withindividual eye model. Individual eye model, based on the traditional Gullstrand–Le Grand eye model, has beenestablished with measured individual cornea data, eyeball depth and wavefront aberrations. We analyze thewavelength-dependent wavefront aberrations at 12 different visible wavelengths (between 400 and 750 nm) for eighteyes. The change of defocus with wavelength (longitudinal chromatic aberration, LCA) is noticeable, and in goodagreement with the results from references. In most cases, the primary spherical aberration changes significantly withwavelength. In comparison with the primary spherical aberration, the other higher-order wavefront aberrations have asmaller change with wavelength. These results have potential applications in those situations where defocus or higher-order wavefront aberrations information in arbitrary wavelength is required.r 2007 Elsevier GmbH. All rights reserved.

Keywords: Chromatic wavefront aberration; Visible spectrum; Individual eye model; ZEMAX

1. Introduction

The human eye exhibits a diverse range of wavefrontaberrations, which significantly limit the retinal imagequality. The two main contributors to the overallwavefront aberrations are cornea and crystalline lens.Their surfaces and their internal refractive indexdistribution are mainly responsible for the ocularwavefront aberrations. The refractive indices are wave-length dependent and, as a consequence, the wavefrontaberrations of the human eye vary over the visiblespectrum.

e front matter r 2007 Elsevier GmbH. All rights reserved.

eo.2007.01.002

ing author.

ess: wangzq@nankai.edu.cn (Z.-Q. Wang).

Longitudinal chromatic aberration (LCA), chromaticfocus shift, causing shorter wavelengths to focus closerto the lens than longer wavelengths, has been measuredin the human eyes [1–4]. It has been found to be abouttwo diopters (D) across the visible spectrum, with somedifferences across studies. Because the higher-orderaberrations are small compared to the defocus, thechange of higher-order aberrations with wavelength isexpected to be modest in comparison with defocus. Thechange of the wavefront aberrations across the visiblespectrum was directly measured, using a spatiallyresolved refractometer [5]. Measured LCA was in goodagreement with that of the references, and the higher-order wavefront aberrations increased slightly withwavelength.

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Eye models that can reproduce optical properties ofthe human eye are especially useful to simulateexperiments. Since the beginning of 20th century, manymodels have been proposed, progressively developedand improved. The famous Gullstrand eye, later revisedby Le Grand, has been highly successful and widely usedfor first-order aberration calculations. The ocularchromatic aberration has been modeled using the LeGrand’s theoretical eye model [7] and the Indianachromatic reduced eye model [11]. The predictions ofthese models were in good agreement with the averagevalues of experimental results over the visible range.However, the prediction on the ocular aberrations wasbasically limited to the LCA. Higher-order aberrations,even if astigmatism, could not be predicted with thesemodels.

To obtain the predictions of higher-order aberrations,individual eye model has been established based onGullstrand–Le Grand eye model using optical designsoftware ZEMAX, which has the same wavefrontaberrations as that of real eye [6]. In this paper, weinvestigate the wavelength-dependent wavefront aberra-tions, especially the higher-order aberrations, at differ-ent visible wavelengths with the constructed individualeye model.

2. Individual eye model

Individual optical elements, including cornea, anteriorchamber, crystalline lens and vitreous body, areincorporated into the traditional Gullstrand–Le Grandeye model to construct the individual eye model byoptical design software ZEMAX. As a result, thewavefront aberrations calculated from the individualeye model are equal to those measured from the real eye.Theoretical calculations of the wavelength dependenceof wavefront aberrations are performed with theindividual eye model.

The wavefront aberrations of the human eyes aremeasured with a Hartmann–Shack wavefront sensorwith a natural pupil [8]. No cyclopentolate hydrochlor-ide for dilating pupils is adopted in this study because ofthe influence on aberrations measurements. The mea-sured wavefront aberrations are expressed as Zernikepolynomial expansion, and the coefficient ordering andnormalization follow the Optical Society of Americastandardization committee recommendations [9].

Corneal elevation maps are obtained using a Bausch& Lomb ORBSCAN II corneal topographer. Theanalysis of corneal topography involves fitting the rawdata to a parametric geometric model that includes aregular basis surface plus irregular residual component.The regular basis surface accounts for the overall shapeof the cornea, which is spherical model in this paper.

The irregular residual component is fitted with Zernikepolynomial expansion using a least-squares procedure.

The medical BMF-200 A/B Ultrasonic DiagnosticInstrument is used to measure the eye’s axial lengths,including the depth of cornea, anterior chamber,crystalline lens and vitreous body. Each eye is measured10 times for average.

To obtain the geometric parameters of the crystallinelens, including a regular basis surface and irregularresidual component, we firstly modify the traditionalGullstrand–Le Grand eye model with the measuredindividual elements about the geometric parameters ofthe cornea, and the depth of cornea, anterior chamber,crystalline lens and vitreous body in the softwareZEMAX. We then make optimization with the geo-metric parameters of the crystalline lens as variables andthe measured wavefront aberrations as the target of themerit function [6]. After optimizing, the individual eyemodel has the similar character with actual eye.

The refractive indices of the cornea, the aqueoushumor, the crystalline lens and the vitreous humor alldepend on the wavelength being used. The refractiveindices (n(l)) as a function of wavelength (l) areobtained by reference [7], with l expressed in nan-ometers. For the limited space, we only list the refractiveindices of the cornea and the crystalline lens:

Refractive index for the cornea:

nðlÞ ¼ 1:51167� 0:000636054lþ 1:17� 10�6l2

� 1:01� 10�9l3 þ 3:31� 10�13l4, ð1Þ

Refractive index for the crystalline lens:

nðlÞ ¼ 1:53808� 0:000448268l

þ 5:74� 10�7l2 � 2:61� 10�10l3. ð2Þ

3. Results

We have established eight individual eye models, andanalyzed the wavefront aberrations of the eight eyemodels at 12 different wavelengths in the visible range,which are 400, 450, 490, 500, 530, 555, 570, 600, 620,650, 700 and 750 nm.

The change of defocus of all the eight eyes over thevisible spectrum is studied separately. Data from each ofthe eight individual eye models are shown in Fig. 1. Theabscissa is the wavelength studied with, and the ordinateindicates the defocus in diopters, which is estimateddirectly from the Zernike polynomial expansion of thewavefront aberrations [10]. Due to the individual levelof refractive error of each eye, there is a bodily shift ofthe curves up or down the dioptric axis. Nonetheless, theevolution of defocus with wavelength among eyes is verysimilar. In order to compare eyes, we first need to factorout any overall refractive error that an individual eye

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Fig. 1. Change of defocus for all the eight eyes as a function of

wavelength in the visible range.Fig. 2. Average defocus from all the eight eyes as a function of

wavelength in the visible range. Dashed curve is third-order

polynomial fit to the average data. Error bars represent the

standard deviation of the results.

Fig. 3. Change of astigmatism for all the eight eyes as a

function of wavelength in the visible range.

M. Liu et al. / Optik 119 (2008) 383–387 385

may have had. This is done by adding or subtracting aconstant amount at each series of data points, aprocedure that affects the vertical position of anindividual’s function but leaves the change in powerwith wavelength unaffected. Here we make the functioncross zero at the wavelength of 570 nm. The averagedefocus values from the eight eyes as a function ofwavelength are shown in Fig. 2. Error bars represent thestandard deviation of the eight eyes. It can be seen fromthe figure that the error bars are very small. Using aleast-squares procedure, the average data are fitted to athird-order polynomial, described by

DðlÞ ¼ � 12:27þ 0:04590l� 5:74� 10�5l2

þ 2:57� 10�8l3, ð3Þ

where D is defocus in diopters and l is wavelength innanometers. The mean LCA from the eight eyes is 1.26D

between 450 and 650 nm. This value agrees well with theexperimental data of Marcos [5]. And the mean LCA is2.15D between 400 and 750 nm, which is very similar tothe LCA of 2.23D predicted by the Indiana chromaticreduced eye model [11].

Fig. 3 shows the change of astigmatism in diopterswith wavelength for all the eight eyes, calculating fromthe second-order astigmatism terms in the Zernikepolynomial expansion [10]. The difference between themaximum and the minimum of the same eye is from 0.01to 0.06D. The biggest relative variation of astigmatismoccurs in the curve with symbol x, and the value is from1.73D at wavelength 400 nm to 1.67D at wavelength750 nm. The smallest relative variation occurs in thecurve with symbol �, and the value is from 0.64D atwavelength 400 nm to 0.63D at wavelength 700 nm.These are compared with the measured results fromreference [5], where the biggest variation is from �0.69D

at wavelength 450 nm to �0.81D at wavelength 650 nm,and the smallest variation is from �0.41D at wavelength450 nm to �0.34D at wavelength 650 nm.

Fig. 4 shows the wavelength dependence of the rootmean square (RMS) of higher-order wavefront aberra-tions for all the eight eyes, leaving out tilt, defocus andastigmatism terms. The difference between the max-imum and the minimum of the same eye is from 0.0060to 0.0336 mm. The biggest relative variation occurs inthe curve with symbol +, and the value is from 0.6028mm at wavelength 400 nm to 0.5692 mm at wavelength750 nm. The smallest variation occurs in the curve withsymbol J, and the value is from 0.5166 mm at wave-length 400 nm to 0.5106 mm at wavelength 600 nm. Theseare compared with the measured results from reference[5], where the biggest variation is from �1.09 mm atwavelength 450 nm to �1.27 mm at wavelength 650 nm,and the smallest variation is from �1.28 mm atwavelength 450 nm to �1.27 mm at wavelength 650 nm.In the study we also find that for the same eye, theRMS values of some Zernike terms increase withwavelength, while that of the others decrease withwavelength.

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Fig. 4. Change in the root-mean-square wavefront error with

wavelength for all the eight eyes.

Fig. 5. Change of the primary spherical aberration as a

function of wavelength for all the eight eyes.

Fig. 6. Change of x-axis coma as a function of wavelength for

all the eight eyes.

Fig. 7. Change of trefoil as a function of wavelength for all the

eight eyes.

M. Liu et al. / Optik 119 (2008) 383–387386

Fig. 5 shows the change of RMS in primary sphericalaberration as a function of wavelength for all the eighteyes. Two of eight eyes’ spherical aberration increasewith wavelength, while the rest decreases with wave-length. They all change monotonously with wavelength,without undulation. In most cases, the primary sphericalaberration changes significantly with wavelength ac-cording to the Marechal criterion. The differencebetween the maximum and the minimum of the sameeye is from 0.0117 to 0.0908 mm. The spherical RMS oftwo eyes with the symbol D and +, which have largervalue of 0.3203 and 0.3592 mm at wavelength 400 nmand reach the minimum 0.3086 and 0.3451 mm atwavelength 750 nm, change slightly with wavelength.The eye with symbol x has the biggest variation, which isfrom 0.1190 mm at wavelength 400 nm to 0.2098 mm atwavelength 750 nm. These are compared with themeasured result from reference [5], where the variationis from �0.12 mm at wavelength 450 nm to �0.38 mm atwavelength 650 nm.

Fig. 6 describes the change of the third-order x-axiscoma RMS wavefront values with wavelength. Thedifference of x-axis coma RMS between the maximumand the minimum of the same eye is from 0.0013 to

0.0249 mm. The biggest variation occurs in the curvewith symbol +, and the value is from 0.3965 mm atwavelength 400 nm to 0.3716 mm at wavelength 750 nm.The smallest variation occurs in the curve with symbol&, and the value is from 0.0045 mm at wavelength400 nm to 0.0058 mm at wavelength 750 nm. These arecompared with the measured results from reference [5],where the variation is from �2.20 mm at wavelength450 nm to �2.19 mm at wavelength 650 nm. The RMSwavefront value of the y-axis coma has the similarchange with that of the x-axis coma as wavelengthvaries, so we do not present the correspondent graphhere.

Fig. 7 illustrates the change of the third-order trefoilRMS wavefront values as a function of wavelength. Thedifference of trefoil RMS between the maximum and theminimum of the same eye is from 0.0026 to 0.0152 mm.The biggest variation occurs in the curve with symbolJ, and the value is from 0.1846 mm at wavelength400 nm to 0.1694 mm at wavelength 750 nm. The smallestvariation occurs in the curve with symbol D, and thevalue is from 0.0367 mm at wavelength 400 nm to0.0393 mm at wavelength 750 nm. These are comparedwith the measured results from reference [5], where the

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variation is from �1.20 mm at wavelength 450 nm to�1.40 mm at wavelength 650 nm.

4. Conclusions

We have demonstrated the use of individual eyemodel for analyzing the variation of wavefront aberra-tions of human eye across visible spectrum (range from400 to 750 nm). The results are compared with the dataprovided by references.

The most noticeable change across visible wave-lengths occurs in defocus for all the eight eyes, and theevolution of defocus with wavelength among eyes is verysimilar. The mean LCA is 2.15D between 400 and750 nm. Astigmatism is nearly constant over visiblespectrum in all eight eyes.

The RMS of higher-order aberrations changes gentlywith wavelength. For the same eye, the RMS values ofsome Zernike terms increase with wavelength, while thatof the others decrease with wavelength. Especially, theprimary spherical aberration changes significantly withwavelength. In comparison with the primary sphericalaberration, the other higher-order wavefront aberra-tions, such as coma and trefoil, have smaller changewith wavelength.

In conclusion to this simulation exercise, the indivi-dual eye model accurately predicts the chromaticdifference in wavefront aberrations of the human eye.Individual eye model allow us to estimate the wavefrontaberrations at each wavelength, not only defocus andastigmatism, but also higher-order aberrations. Theresults reveal the individual eye model to be valuabletool for chromatic aberrations analysis.

Acknowledgments

This research is supported by the National NatureScience Foundation of China (Key Research Project,No. 60438030), the Key Research Foundation ofScientific and Technical Committee of Tianjin City ofChina (No. 033183711), the Chinese Ministry of

Education’s Nankai University and Tianjin UniversityCooperation Foundation, and the Specialized ResearchFund for the Doctoral Program of Higher Education(No. 20050055002).

References

[1] G. Wald, D.R. Griffin, The change in refractive power of

the human eye in dim and bright light, J. Opt. Soc. Am.

37 (1947) 321–336.

[2] R.E. Bedford, G. Wyszecki, Axial chromatic aberration

of the human eye, J. Opt. Soc. Am. 47 (1957) 564–565.

[3] W.N. Charman, J.A. Jennings, Objective measurements

of the longitudinal chromatic aberration of the human

eye, Vision Res. 16 (1976) 999–1005.

[4] P.A. Howarth, A. Bradley, The longitudinal chromatic

aberration of the human eye and its correction, Vision

Res. 26 (1986) 361–366.

[5] S. Marcos, S.A. Burns, E. Moreno-Barriuso, R. Navarro,

A new approach to study ocular chromatic aberrations,

Vision Res. 39 (1999) 4309–4323.

[6] H. Guo, Z. Wang, Q. Zhao, W. Quan, Y. Wang,

Individual eye model based on wavefront aberration,

Optik 116 (2005) 80–85.

[7] E.R. Villegas, L. Carretero, A. Fimia, Le Grand eye for

the study of ocular chromatic aberration, Ophthalmol.

Physiol. Opt. 16 (1996) 528–531.

[8] J. Liang, B. Grimm, S. Goelz, J.F. Bille, Objective

measurement of wave aberration of the human eye with

the use of a Hartmann–Shack wave-front sensor, J. Opt.

Soc. Am. A 11 (1994) 1949–1957.

[9] L.N. Thibos, R.A. Applegate, J.T. Schwiegerling, R.

Webb, Standards for reporting the optical aberrations of

eyes, in: V. Lakshminarayanan (Ed.), Vision Science and

Its Applications of Trends in Optics and Photonics Series,

vol. 35, Optical Society of America, Washington, DC,

2000, pp. 232–244.

[10] L.N. Thibos, X. Hong, A. Bradley, X. Cheng, Statistical

variation of aberration structure and image quality in a

normal population of healthy eyes, J. Opt. Soc. Am. A 19

(2002) 2329–2348.

[11] L.N. Thibos, M. Ye, X. Zhang, A. Bradley, The

chromatic eye: a new reduced-eye model of ocular

chromatic aberration in humans, Appl. Opt. 31 (1992)

3594–3600.