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Wavefront aberrations in the accommodated human eye based on individual eye model
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Optik
Optics
Optik 118 (2007) 271–277 www.elsevier.de/ijleo
Wavefront aberrations in the accommodated human eye based on individual eye model Yang Wanga, Zhao-Qi Wanga,, Huan-Qing Guoa, Yan Wangb, Tong Zuob a Institute of Modern Optics, Nankai University, Key Laboratory of Opto-electronic Information Science and Technology, Ministry of Education, Tianjin 300071, PR China b Tianjin Eye Hospital, Refractive Surgery Center, Tianjin 300020, PR China
Received 9 November 2005; accepted 10 March 2006
Abstract In this research, we firstly construct individual eye models based on the wavefront and the measured cornea structure of the eyes. Then we analyze the influence of accommodation on the wavefront aberrations based on individual eye model. The individual eye model has the same wavefront aberration as that measured from Hartmann–Shack wavefront sensor. The optical design software ZEMAX is used to construct the individual eye models for 20 normal eyes. Accommodative conditions are from 0 to –4 diopter in steps of one diopter. The variations of the total, the spherical, the coma and the higher-order root-mean-square wavefront aberrations, as accommodations, are illustrated. Influence of accommodation on wavefront aberration varies from individual to individual, and the variation magnitude is independent of the magnitude of the wavefront aberration of the eye. r 2006 Elsevier GmbH. All rights reserved. Keywords: Accommodation; Individual eye model; Wavefront aberration; ZEMAX
1. Introduction Wavefront aberrations of human eyes are expected to change when eyes are forced to see objects at various distances clearly. Accommodation that includes changes of refractive power and position of crystalline lens is thought to have an effect on wavefront aberrations of the human eyes [1] and then on the retinal image quality. So it is significant to analyze the wavefront aberrations in the accommodated normal human eye. Impact of accommodation on wavefront aberrations has been reported in some previous researches. Atchison et al. [2] made the first attempt to characterize the Corresponding author.
E-mail address: [email protected] (Z.-Q. Wang). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.03.019
wavefront aberrations of eyes for 15 subjects in detail by using the aberroscope technique. They found no clear trend, on the wavefront aberrations, up to the forth order in amount or direction of change with 0D, 1.5D and –3D accommodative conditions. Lopez-Gil et al. [3] studied the change in retinal image quality with accommodation by using near-infrared double-pass technique and reported that the double-pass image for the accommodated eye tended to be more symmetric than that of the unaccommodated eye. He et al. [4] found a consistent trend in the change of aberrations over the accommodative range by subjective ray-tracing technique [4], and only one eye from the eight eyes studied changed very little. With regard to the particular aberrations, authors in previous studies suggested that the amount of spherical aberration decreases with
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accommodation, although most of those results were subject dependent. The change in coma in He et al. did not show a variation trend, and the average over eight eyes gave almost no change with accommodation. Influence of accommodation in previous studies was all based on actual measurements of normal human eyes. In this article, we analyze the influence of accommodation based on an individual eye model in principle. Some early work has been done on the eye model for optical performance. Gullstrand–Le Grand scheme eye model offered a powerful tool in the early 20th century. Many later researchers paid more interests on it and continued to modify this model. Those eye models in previous researches [5–7] were all based on anatomy and were summarized from the average results by statistics. However, actually, each eye corresponds to its own individual physiological characteristics and eye models should reflect the individual ocular aberrations. To better describe the optical performance of human eye, we firstly construct the individual eye model by optical design software ZEMAX. As a result, the wavefront aberrations calculated from the individual eye model are equal to those measured from the real eye by Harmann–Shack wavefront sensor. We then investigate the influence of the accommodation on the ocular aberrations with the constructed individual eye model. The data from 20 eyes are reported in this study. For all subjects whose ages range from 18 to 32, the pupil sizes are greater than 5 mm for cornea and aberration measurements. No cyclopentolate hydrochloride, for dilating pupils, is adopted in this study because of its influence on aberrations measurements. There are emmetropes, myopes and hyperopias in these subjects whose defocus range from –6D to +3D. No subject in this experiment has a record of ocular disease.
2. Construction of individual eye model Hartmann–Shack is used to measure the wavefront aberrations of the whole eye. The principle of the Hartmann–Shack wavefront sensor is as follows [8]. A narrow near-infrared beam projects onto the subject’s retina acts as a beacon source. On the way out, the light propagates through the eye’s optics, suffering local phase shifts in the wavefront, before reaching a lenslet that samples the local average of the wavefront tilt over the eye pupil. This sampling generates a distribution of spots that is captured by a CCD camera in the focal plane of the lenslet. When a perfect plane wave is measured, the lenslet array forms a regular array of focus spots. If a deformed wavefront is measured, the image spot focused by each lenslet is displaced in
proportion to the local wavefront slopes in x and ydirections. In the Hartmann–Shack image, the relative displacement of each sample spot is proportional to the wavefront slope within the corresponding sub-aperture. The wavefront aberration is expressed as Zernike polynomial expansion. The corneal topographic system used to measure the corneal surfaces is Orbscan II. With a Placido-based videokeratographic device, the discrete set of corneal elevation data in radial distribution over the pupil plane can be obtained for the anterior and the posterior corneal surface. The elevation data of the corneal surfaces in vertical distribution that is defined as along the optical axis are calculated from the elevation data in radial distribution. Differences in path length between the ray passing the pupil center and the rays traveling through the other pupil area are calculated with software MatLab. Then, the corneal surfaces including the anterior and posterior surfaces are established. The least-square procedure is used to decompose the corneal surfaces to 35 terms of Zernike polynomials [9]. Zernike coefficients of the cornea with the unit of micrometers are all calculated. The detailed corneal structural parameters of 20 eyes are listed in Table 1, which are input into optical design software ZEMAX. Only three Zernike coefficients have been showed because of the limited space. The accuracy of this procedure was previously evaluated with the use of reference surfaces. It can be seen from the table that different eye carries different radius and thickness of cornea. Radius of anterior cornea is larger than that of posterior cornea for all eyes. The seventh eye has the largest anterior and posterior corneal radii. While the 12th eye has the smallest anterior corneal radius, the 14th eye has the smallest posterior corneal radius. We use the medical BMF-200 A/B Ultrasonic Diagnostic Instrument to measure the eye’s axial lengths, including the depth of cornea, anterior chamber, crystalline lens and vitreous body. According to ultrasonic spreading, at different times in different types of medium, we can get the depth of the very part of the eye by measuring the spread time accurately. Each eye is measured ten times to get an average. The parameters of the crystalline lens are difficult to be measured. To do this, we choose Gullstrand–Le Grand eye model as the initial configuration with the individual corneal and eye’s axial length parameters. In order to make the total aberrations of eye model correspond to the aberrations of actual eye, the operands ZERN in ZEMAX is added to the merit function with the coefficients defining the aberrations of the actual eye. Zernike Fringe Sag surface helps to optimize the lens surface for getting the Zernike coefficients, which can fit the lens well. After optimizing, the eye model shows similar characters with the actual
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Table 1. Eye no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Table 2. Eye no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
273
Detailed corneal structural parameters of 20 eye models Anterior corneal radius
Posterior corneal radius
Corneal thickness (mm)
8.32 8.26 7.88 7.93 8.30 7.98 8.38 8.14 8.06 8.03 8.13 7.0 7.87 7.46 8.25 7.54 7.90 7.85 8.13 8.23
6.64 6.52 6.50 6.38 6.55 6.46 6.87 6.63 6.64 6.50 6.52 6.38 6.17 5.88 6.75 5.96 6.38 6.40 6.65 6.59
0.562 0.546 0.543 0.531 0.544 0.527 0.552 0.504 0.501 0.537 0.523 0.596 0.484 0.511 0.531 0.535 0.596 0.597 0.496 0.587
Zernike coefficients of anterior/posterior cornea (mm) Z3 0.0016/0.0058 0.0032/ 0.0069 0.0037/ 0.0002 0.0027/ 0.0092 0.0024/0.0061 0.0058/0.0087 0.0043/ 0.0032 0.0095/ 0.0036 0.0054/0.0078 0.0047/ 0.0033 0.0002/0.0025 0.0003/ 0.0011 0.0051/0.0006 0.0010/0.0003 0.0095/0.0076 0.0090/0.0051 0.0012/0.0012 0.0005/ 0.0050 0.0060/0.0081 0.0031/0.0006
Z4 0.0008/ 0.0087 0.0015/ 0.0126 0.0017/ 0.0018 0.0004/ 0.0034 0.0015/ 0.014 0.0046/ 0.0115 0.0024/ 0.0083 0.0022/0.0019 0.0009/0.0042 0.0037/ 0.0033 0.0018/ 0.0082 0.0002/ 0.0005 0.0004/0.0012 0.0058/ 0.0088 0.0010/ 0.0013 0.0061/ 0.0088 0.0026/ 0.0113 0.0024/ 0.0075 0.0005/ 0.0048 0.00216/ 0.0053
Z5 0.0007/0.0047 0.0007/ 0.0024 0.0018/ 0.0021 0.0008/0.0102 0.0001/0.0060 0.0005/0.0095 0.0011/0.0026 0.0001/0.0018 0.0003/0.0043 0.0009/0.0013 0.0021/ 0.0022 0.0013/ 0.0021 0.0015/ 0.0032 0.0004/0.0082 0.0021/0.0012 0.0008/ 0.0029 0.0011/ 0.0065 0.0003/ 0.0007 0.0048/ 0.0081 0.0018/0.0018
Detailed structural parameters of crystalline lens of 20 eye models Anterior crystalline lens radius 10.22 10.90 11.02 9.96 11.11 10.17 12.08 11.33 9.98 10.62 13.05 11.07 10.25 11.23 10.27 12.66 13.14 11.07 9.76 10.35
Posterior crystalline lens radius 6.09 6.33 5.98 5.77 6.21 6.25 6.35 6.41 5.67 6.01 6.11 6.07 6.23 6.27 6.05 6.15 6.66 6.08 6.22 6.00
Thickness of crystalline lens (mm) 3.46 3.75 3.81 3.44 3.78 3.36 3.51 3.46 3.42 3.73 3.92 3.99 4.05 3.37 3.54 3.61 3.50 3.45 3.54 3.51
eye. Table 2 lists the detailed crystalline lens structural parameters of 20 eye models. We give the thickness of the lens, the radius of the surfaces and three Zernike
Zernike coefficients of anterior/posterior crystalline lens (mm) Z3 0.0008/0.0009 0.0122/0.0036 0.0096/0.0082 0.0151/0.0132 0.0031/ 0.0001 0.0029/ 0.0006 0.0004/ 0.0015 0.0002/0.0020 0.0025/0.0026 0.0033/ 0.0017 0.0055/ 0.0005 0.0016/0.0007 0.0015/0.0025 0.0005/0.0008 0.0009/0.0001 0.0010/0.0002 0.0011/ 0.0002 0.0007/ 0.0003 0.0003/0.0007 0.0022/0.0002
Z4 0.1182/0.1759 0.0074/0.1240 0.1009/0.1624 0.0085/0.1385 0.0073/0.1214 0.0423/0.0754 0.0672/0.1195 0.0860/0.1396 0.0891/0.1457 0.0977/0.1509 0.1031/0.2747 0.0026/0.1008 0.0273/0.0932 0.0770/0.1449 0.2662/0.4660 0.0013/0.1792 0.0558/0.0918 0.0965/0.1537 0.0435/0.0597 0.0827/ 0.1197
Z5 0.0009/ 0.0015 0.0029/ 0.0005 0.0016/0.0028 0.0003/0.0004 0.0006/ 0.0009 0.0010/0.0023 0.0011/0.0018 0.0003/ 0.0005 0.0010/ 0.0016 0.0015/ 0.0024 0.0000/ 0.0030 0.0001/ 0.0005 0.0023/ 0.0002 0.0002/ 0.0020 0.0013/0.0032 0.0007/0.0051 0.0008/ 0.0013 0.0023/0.0035 0.0031/ 0.0051 0.0033/0.0055
coefficients of the surfaces here. We can see from this table that the sixth eye has the smallest thickness and the 13th eye has the largest thickness. The 17th eye has the
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largest anterior and posterior surface radii. The 19th eye has the smallest anterior surface radius and the ninth eye has the smallest posterior surface radius. In the accommodation study, we optimize the radius of the crystalline lens’s surface with different object distances until the incident light focuses to the image plane (retinal plane for the human eye). The accommodation diopter can be input to the eye by changing the object distance. In this experiment, we estimate Zernike coefficients in five accommodative conditions, ranging from 0 to 4 diopter in step of one diopter. Diopter corresponds to the target distance, for example, one diopter corresponds to the distance of 1000 mm, and two diopters corresponds to the distance of 500 mm. When the merit function is optimized to zero, the Zernike coefficients are just what we want. We calculate the Zernike coefficients in ZEMAX after finishing every optimization process [10].
3. Influence of accommodation on wavefront Fig. 1 illustrates the effect of accommodation on rootmean-square (RMS) wavefront value for all 20 eyes. The abscissa indicates the eye number, which is sequenced by the size of the RMS value of wavefront aberration of the whole eye (the defocus term is excluded) for 0D accommodative condition, and the ordinate is the RMS value of the wavefront aberrations in micrometers. The symbol of , J, K, n and & represents 0D, 1D, 2D, 3D and 4D, respectively. The RMS wavefront value provides a general estimate of the variation of the wavefront from the ideal one. In general, different eyes have different RMS wavefront values as the accommodation varies. From this figure, we can see that the RMS
Fig. 1. RMS values of total wavefront aberrations excluding defocus for 20 eyes with four accommodative conditions ( 1D, 2D, 3D and 4D).
wavefront values of most eyes decrease as the accommodation varies from 0D to –2D, and then gradually increase as the accommodation varies from 2D to –4D. The RMS wavefront values of some eyes decrease as the accommodation varies from 0D to –1D, and then gradually increase as the accommodation varies from –1D to –4D. Only the RMS wavefront values of the eighth and the 16th eye decrease from 0D to –3D and increase from –3D to –4D. Changes in RMS wavefront values with accommodation for all eyes are found, but the amplitude of the changes varies from individual to individual. Some RMS values of the wavefront have a little variation, such as the third and the fifth eyes. It is only 0.092 mm from the maximum RMS value of 0.419 mm at –4D to the minimum RMS value of 0.327 mm at –2D for the third eye, and it is 0.077 mm from the maximum of 0.478 mm at 0D to the minimum of 0.401 mm at –2D for the fifth eye, which is 22.0% and 16.3% in the corresponding proportion to the maximum RMS values. However, changes of some RMS values cover a wide range, such as the 17th eye and the 20th eye. It is about 0.689 mm from the maximum RMS value of 1.019 mm at 0D to the minimum RMS value of 0.33 mm at –2D for the 17th eye, and it is 1.058 mm from the maximum RMS value of 1.358 mm at 0D to the minimum RMS value of 0.30 mm at –2D for the 20th eye, which is 67.7% and 77.9% in the corresponding proportion to the maximum RMS values, respectively. In addition, larger magnitude variation seems to be independent of wavefront aberration value. For instance, the first eye has the smaller wavefront aberration, but it has the larger change of 84.8% in RMS wavefront value, which is from the maximum value of 0.329 mm at 0D to the minimum value of 0.05 mm at –2D. The 12th eye has the maximum RMS wavefront value of 0.756 mm at 0D, while the maximum RMS difference is about 0.556 mm, which is 74% in proportion to the maximum RMS value. However, the 18th eye has a larger RMS value of 1.038 mm at 0D, and has only smaller change of 13.0% from 1.082 mm (maximum) at –4D to 0.941 mm (minimum) at –2D. This seems to be similar with the results of He et al. In their study, subject KK has very little wavefront aberrations but has the largest relative change in RMS wavefront value. Fig. 2 shows the change of RMS in primary spherical aberration with accommodation for all 20 eyes. The abscissa indicates the accommodative stimulus from 0D to –4D and the ordinate is the spherical RMS wavefront value with the unit in micrometers. There are four subfigures in Fig. 2. Among them, the top-left figure illustrates the spherical RMS values from the first to the fifth eyes, and the top-right figure shows those from the sixth to the 10th eyes. The down-left figure illustrates the spherical RMS values from the 11th to the 15th eyes and those from the 16th to the 20th eyes are showed on
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Fig. 2. Spherical RMS wavefront value as a function of the accommodative stimulus for 20 eyes.
Fig. 3. x-Axis coma RMS as a function of the accommodative stimulus for 20 eyes.
the down-right figure. In each sub-figure, five curves are differently represented by the symbols of n, J, &, B and *, which are labeled on the figure. Changes of spherical RMS can be shown clearly in the four subfigures. We can see that the spherical RMS value of wavefront aberrations of almost every eye has a graduate decreasing trend from 0D to –4D accommodative conditions, but the variation pattern does not hold the same for all eyes. For example, the fourth eye has the minimum RMS value of 0.07 mm at 0D, and the RMS value increases to the maximum of 0.15 mm at –1D, and then decreases gradually to 0.116 mm at –4D. The difference between maximum and minimum RMS values is about 53.3% in proportion to the maximum RMS wavefront value. Among 20 eyes, the 12th eye has the larger change of 94.4% in spherical RMS value from the maximum RMS value of 0.5 mm at 0D to the minimum RMS value of 0.028 mm at –4D. While the spherical RMS value of the 18th eye changes gently, which has the smallest variation of 14.3% from the maximum of 0.35 mm at –1D to the minimum of 0.3 mm at –4D. The variation magnitude is independent of the spherical RMS value. The 20th eye can be taken as an example. It has the largest spherical RMS among 20 eyes, but only 19.4% variation from the maximum of 0.62 mm at 0D to the minimum of 0.5 mm at –2D. Fig. 3 describes the change of x-axis coma RMS wavefront values with accommodation for all eyes. The abscissa indicates the accommodative stimulus from 0D to –4D and the ordinate is the x-axis coma RMS wavefront value with the unit in micrometers. There are also four sub-figures in Fig. 3, in which the symbols are labeled. We can also see the individual variation in this figure. The second and the 19th eye have considerable
variations in the x-axis coma RMS wavefront value. There is a variation magnitude of 0.15 mm of the second eye from the maximum of 0.3 mm at 0D to the minimum of 0.15 mm at –1D, which is 50% in proportion to the maximum value. The x-axis coma RMS value of the 19th eye decreases most among the 20 eyes as the accommodation varies and the difference in RMS wavefront value is 0.232 mm from the maximum of 0.432 mm at 0D to the minimum of 0.2 mm at –3D, which is 53.7% in proportion to the maximum value. But the variance trend over the accommodative range is not so significant for most eyes. For instance, the 20th eye has the largest x-axis coma RMS value among the 20 eyes, while there is only little change of 19.4% in proportion to the maximum RMS value from the maximum of 0.62 mm at 0D to the minimum of 0.5 mm at –2D. The yaxis coma RMS wavefront value has similar changes with the x-axis coma RMS wavefront value as the accommodation varies, so we do not present the corresponding graph here. Fig. 4 shows the total contribution of wavefront aberrations from the fifth to the seventh Zernike order as a function of accommodation. The abscissa indicates the accommodative stimulus from 0D to –4D and the ordinate is the higher-order RMS wavefront value with the unit in micrometers. We also give four sub-figures, and the symbols are also labeled. From the graph, we can see that as the accommodation varies from 0D, the higher-order RMS values of most eyes decrease firstly, and then increase gradually. Different eyes have different decreasing as well as increasing values. The RMS value of the ninth eye decreases from 0.28 mm at 0D to 0.11 mm at –2D, which is 56.7% in proportion to the maximum RMS value, and then increases to 0.3 mm at –4D, which is 63.3% in proportion to the maximum
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Fig. 4. Higher-order RMS as a function of the accommodative stimulus for 20 eyes.
RMS value. The RMS value of the 13th eye decreases from 0.13 mm at 0D to 0.036 mm at –1D, which is 26.1% in proportion to the maximum RMS wavefront value, and then increases to 0.36 mm at –4D, which is 90% in proportion to the maximum RMS wavefront value. In comparison, the higher-order RMS wavefront value of the seventh eye varies gently, which decreases to 0.018 mm between 0D and –3D and increases to 0.038 mm between –3D and –4D. It is only 23.8% variation of the RMS wavefront value in proportion to the maximum RMS value. The situation of the variance trend for few eyes is totally different. For instance, the RMS wavefront value of the second eye decreases from 0.13 mm at 0D to 0.09 mm at –1D, increases to 0.12 mm at –2D, decreases to 0.002 mm at –3D, and increases to 0.152 mm at –4D.
4. Conclusions We have constructed 20 individual eye models to analyze the impact of accommodation in this study. The variations of overall wavefront aberrations and different Zernike term aberrations as different accommodation conditions (range from 0D to –4D) have been presented. In conclusion, accommodation influences wavefront aberrations of the whole eye, and the situation varies substantially from individual to individual. The RMS wavefront values of most eyes decrease firstly, and then increase as the accommodation varies from 0D to –4D. Spherical RMS wavefront value has a graduate decreasing trend as the accommodation varies from 0D to –4D. Most higher-order RMS values decrease firstly, and then increase as the accommodation varies from 0D to –4D. These results are in good agreement with those actually measured by He et al.
Coma does not how large variation as accommodation, like other kinds of aberrations, except for a few eyes in our study. No matter how large the wavefront aberration of the un-accommodated eye, all eyes have their own aberration variations as the accommodation varies. Generally speaking, accommodation plays an important role in contributing wavefront aberrations to the eyes, and has an impact on overall optical imaging quality. Therefore, we have to consider not only the wavefront aberration at the un-accommodative condition but also that at the accommodative condition during the clinical ocular therapy. Individual eye model is firstly adopted to analyze the influence of accommodation on wavefront aberration in this paper. As a new way, it can be adopted widely for ocular performance diagnosis of human eye and for clinical vision therapy.
Acknowledgements This research is supported by the National Nature Science Foundation of China (Key Research Project, no. 60438030), the Key Research Foundation of Scientific and Technical Committee of Tianjin City of China (no. 033183711) and the Chinese Ministry of Education’s Nankai University and Tianjin University Cooperation Foundation.
Reference [1] H. Hofer, P. Artal, B. Singer, J.L. Aragon, D.R. Williams, Dynamics of the eye’s wave aberration, J. Opt. Soc. Am. A. 18 (2001) 497–506. [2] D.A. Atchison, M.J. Collins, C.F. Wildsoet, J. Christensen, M.D. Water worth, Measurement of monochromatic ocular aberrations of human eyes as a function of accommodation by the Howland aberroscope technique, Vis. Res. 35 (1995) 313–323. [3] N. Lopez-Gil, I. Iglesias, P. Artal, Retinal image quality in the human eye as a function of the accommodation, Vis. Res. 38 (1998) 2897–2907. [4] J.C. He, S.A. Burns, S. Marcos, Monochromatic aberrations in the accommodated human eye, Vis. Res. 40 (2000) 41–48. [5] 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. [6] I. Escudero-Sanz, R. Navarro, Off-axis aberrations of a wide-angle schematic eye model, J. Opt. Soc. Am. A. 16 (1999) 1881–1890. [7] H.-L. Liou, N.A. Brennan, Anatomically accurate, finite model eye for optical modeling, J. Opt. Soc. Am. A. 14 (1997) 1684–1695.
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[8] J. Liang, D.R. Williams, Aberrations and retinal image quality of the normal human eye, J. Opt. Soc. Am. A. 14 (1997) 2873–2883. [9] P.M. Prieto, F. Vargas-Martin, S. Goelz, P. Artal, Analysis of the performance of the Hartmann–
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Optik
Optics
Optik 118 (2007) 271–277 www.elsevier.de/ijleo
Wavefront aberrations in the accommodated human eye based on individual eye model Yang Wanga, Zhao-Qi Wanga,, Huan-Qing Guoa, Yan Wangb, Tong Zuob a Institute of Modern Optics, Nankai University, Key Laboratory of Opto-electronic Information Science and Technology, Ministry of Education, Tianjin 300071, PR China b Tianjin Eye Hospital, Refractive Surgery Center, Tianjin 300020, PR China
Received 9 November 2005; accepted 10 March 2006
Abstract In this research, we firstly construct individual eye models based on the wavefront and the measured cornea structure of the eyes. Then we analyze the influence of accommodation on the wavefront aberrations based on individual eye model. The individual eye model has the same wavefront aberration as that measured from Hartmann–Shack wavefront sensor. The optical design software ZEMAX is used to construct the individual eye models for 20 normal eyes. Accommodative conditions are from 0 to –4 diopter in steps of one diopter. The variations of the total, the spherical, the coma and the higher-order root-mean-square wavefront aberrations, as accommodations, are illustrated. Influence of accommodation on wavefront aberration varies from individual to individual, and the variation magnitude is independent of the magnitude of the wavefront aberration of the eye. r 2006 Elsevier GmbH. All rights reserved. Keywords: Accommodation; Individual eye model; Wavefront aberration; ZEMAX
1. Introduction Wavefront aberrations of human eyes are expected to change when eyes are forced to see objects at various distances clearly. Accommodation that includes changes of refractive power and position of crystalline lens is thought to have an effect on wavefront aberrations of the human eyes [1] and then on the retinal image quality. So it is significant to analyze the wavefront aberrations in the accommodated normal human eye. Impact of accommodation on wavefront aberrations has been reported in some previous researches. Atchison et al. [2] made the first attempt to characterize the Corresponding author.
E-mail address: [email protected] (Z.-Q. Wang). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.03.019
wavefront aberrations of eyes for 15 subjects in detail by using the aberroscope technique. They found no clear trend, on the wavefront aberrations, up to the forth order in amount or direction of change with 0D, 1.5D and –3D accommodative conditions. Lopez-Gil et al. [3] studied the change in retinal image quality with accommodation by using near-infrared double-pass technique and reported that the double-pass image for the accommodated eye tended to be more symmetric than that of the unaccommodated eye. He et al. [4] found a consistent trend in the change of aberrations over the accommodative range by subjective ray-tracing technique [4], and only one eye from the eight eyes studied changed very little. With regard to the particular aberrations, authors in previous studies suggested that the amount of spherical aberration decreases with
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accommodation, although most of those results were subject dependent. The change in coma in He et al. did not show a variation trend, and the average over eight eyes gave almost no change with accommodation. Influence of accommodation in previous studies was all based on actual measurements of normal human eyes. In this article, we analyze the influence of accommodation based on an individual eye model in principle. Some early work has been done on the eye model for optical performance. Gullstrand–Le Grand scheme eye model offered a powerful tool in the early 20th century. Many later researchers paid more interests on it and continued to modify this model. Those eye models in previous researches [5–7] were all based on anatomy and were summarized from the average results by statistics. However, actually, each eye corresponds to its own individual physiological characteristics and eye models should reflect the individual ocular aberrations. To better describe the optical performance of human eye, we firstly construct the individual eye model by optical design software ZEMAX. As a result, the wavefront aberrations calculated from the individual eye model are equal to those measured from the real eye by Harmann–Shack wavefront sensor. We then investigate the influence of the accommodation on the ocular aberrations with the constructed individual eye model. The data from 20 eyes are reported in this study. For all subjects whose ages range from 18 to 32, the pupil sizes are greater than 5 mm for cornea and aberration measurements. No cyclopentolate hydrochloride, for dilating pupils, is adopted in this study because of its influence on aberrations measurements. There are emmetropes, myopes and hyperopias in these subjects whose defocus range from –6D to +3D. No subject in this experiment has a record of ocular disease.
2. Construction of individual eye model Hartmann–Shack is used to measure the wavefront aberrations of the whole eye. The principle of the Hartmann–Shack wavefront sensor is as follows [8]. A narrow near-infrared beam projects onto the subject’s retina acts as a beacon source. On the way out, the light propagates through the eye’s optics, suffering local phase shifts in the wavefront, before reaching a lenslet that samples the local average of the wavefront tilt over the eye pupil. This sampling generates a distribution of spots that is captured by a CCD camera in the focal plane of the lenslet. When a perfect plane wave is measured, the lenslet array forms a regular array of focus spots. If a deformed wavefront is measured, the image spot focused by each lenslet is displaced in
proportion to the local wavefront slopes in x and ydirections. In the Hartmann–Shack image, the relative displacement of each sample spot is proportional to the wavefront slope within the corresponding sub-aperture. The wavefront aberration is expressed as Zernike polynomial expansion. The corneal topographic system used to measure the corneal surfaces is Orbscan II. With a Placido-based videokeratographic device, the discrete set of corneal elevation data in radial distribution over the pupil plane can be obtained for the anterior and the posterior corneal surface. The elevation data of the corneal surfaces in vertical distribution that is defined as along the optical axis are calculated from the elevation data in radial distribution. Differences in path length between the ray passing the pupil center and the rays traveling through the other pupil area are calculated with software MatLab. Then, the corneal surfaces including the anterior and posterior surfaces are established. The least-square procedure is used to decompose the corneal surfaces to 35 terms of Zernike polynomials [9]. Zernike coefficients of the cornea with the unit of micrometers are all calculated. The detailed corneal structural parameters of 20 eyes are listed in Table 1, which are input into optical design software ZEMAX. Only three Zernike coefficients have been showed because of the limited space. The accuracy of this procedure was previously evaluated with the use of reference surfaces. It can be seen from the table that different eye carries different radius and thickness of cornea. Radius of anterior cornea is larger than that of posterior cornea for all eyes. The seventh eye has the largest anterior and posterior corneal radii. While the 12th eye has the smallest anterior corneal radius, the 14th eye has the smallest posterior corneal radius. We use the medical BMF-200 A/B Ultrasonic Diagnostic Instrument to measure the eye’s axial lengths, including the depth of cornea, anterior chamber, crystalline lens and vitreous body. According to ultrasonic spreading, at different times in different types of medium, we can get the depth of the very part of the eye by measuring the spread time accurately. Each eye is measured ten times to get an average. The parameters of the crystalline lens are difficult to be measured. To do this, we choose Gullstrand–Le Grand eye model as the initial configuration with the individual corneal and eye’s axial length parameters. In order to make the total aberrations of eye model correspond to the aberrations of actual eye, the operands ZERN in ZEMAX is added to the merit function with the coefficients defining the aberrations of the actual eye. Zernike Fringe Sag surface helps to optimize the lens surface for getting the Zernike coefficients, which can fit the lens well. After optimizing, the eye model shows similar characters with the actual
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Table 1. Eye no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Table 2. Eye no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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Detailed corneal structural parameters of 20 eye models Anterior corneal radius
Posterior corneal radius
Corneal thickness (mm)
8.32 8.26 7.88 7.93 8.30 7.98 8.38 8.14 8.06 8.03 8.13 7.0 7.87 7.46 8.25 7.54 7.90 7.85 8.13 8.23
6.64 6.52 6.50 6.38 6.55 6.46 6.87 6.63 6.64 6.50 6.52 6.38 6.17 5.88 6.75 5.96 6.38 6.40 6.65 6.59
0.562 0.546 0.543 0.531 0.544 0.527 0.552 0.504 0.501 0.537 0.523 0.596 0.484 0.511 0.531 0.535 0.596 0.597 0.496 0.587
Zernike coefficients of anterior/posterior cornea (mm) Z3 0.0016/0.0058 0.0032/ 0.0069 0.0037/ 0.0002 0.0027/ 0.0092 0.0024/0.0061 0.0058/0.0087 0.0043/ 0.0032 0.0095/ 0.0036 0.0054/0.0078 0.0047/ 0.0033 0.0002/0.0025 0.0003/ 0.0011 0.0051/0.0006 0.0010/0.0003 0.0095/0.0076 0.0090/0.0051 0.0012/0.0012 0.0005/ 0.0050 0.0060/0.0081 0.0031/0.0006
Z4 0.0008/ 0.0087 0.0015/ 0.0126 0.0017/ 0.0018 0.0004/ 0.0034 0.0015/ 0.014 0.0046/ 0.0115 0.0024/ 0.0083 0.0022/0.0019 0.0009/0.0042 0.0037/ 0.0033 0.0018/ 0.0082 0.0002/ 0.0005 0.0004/0.0012 0.0058/ 0.0088 0.0010/ 0.0013 0.0061/ 0.0088 0.0026/ 0.0113 0.0024/ 0.0075 0.0005/ 0.0048 0.00216/ 0.0053
Z5 0.0007/0.0047 0.0007/ 0.0024 0.0018/ 0.0021 0.0008/0.0102 0.0001/0.0060 0.0005/0.0095 0.0011/0.0026 0.0001/0.0018 0.0003/0.0043 0.0009/0.0013 0.0021/ 0.0022 0.0013/ 0.0021 0.0015/ 0.0032 0.0004/0.0082 0.0021/0.0012 0.0008/ 0.0029 0.0011/ 0.0065 0.0003/ 0.0007 0.0048/ 0.0081 0.0018/0.0018
Detailed structural parameters of crystalline lens of 20 eye models Anterior crystalline lens radius 10.22 10.90 11.02 9.96 11.11 10.17 12.08 11.33 9.98 10.62 13.05 11.07 10.25 11.23 10.27 12.66 13.14 11.07 9.76 10.35
Posterior crystalline lens radius 6.09 6.33 5.98 5.77 6.21 6.25 6.35 6.41 5.67 6.01 6.11 6.07 6.23 6.27 6.05 6.15 6.66 6.08 6.22 6.00
Thickness of crystalline lens (mm) 3.46 3.75 3.81 3.44 3.78 3.36 3.51 3.46 3.42 3.73 3.92 3.99 4.05 3.37 3.54 3.61 3.50 3.45 3.54 3.51
eye. Table 2 lists the detailed crystalline lens structural parameters of 20 eye models. We give the thickness of the lens, the radius of the surfaces and three Zernike
Zernike coefficients of anterior/posterior crystalline lens (mm) Z3 0.0008/0.0009 0.0122/0.0036 0.0096/0.0082 0.0151/0.0132 0.0031/ 0.0001 0.0029/ 0.0006 0.0004/ 0.0015 0.0002/0.0020 0.0025/0.0026 0.0033/ 0.0017 0.0055/ 0.0005 0.0016/0.0007 0.0015/0.0025 0.0005/0.0008 0.0009/0.0001 0.0010/0.0002 0.0011/ 0.0002 0.0007/ 0.0003 0.0003/0.0007 0.0022/0.0002
Z4 0.1182/0.1759 0.0074/0.1240 0.1009/0.1624 0.0085/0.1385 0.0073/0.1214 0.0423/0.0754 0.0672/0.1195 0.0860/0.1396 0.0891/0.1457 0.0977/0.1509 0.1031/0.2747 0.0026/0.1008 0.0273/0.0932 0.0770/0.1449 0.2662/0.4660 0.0013/0.1792 0.0558/0.0918 0.0965/0.1537 0.0435/0.0597 0.0827/ 0.1197
Z5 0.0009/ 0.0015 0.0029/ 0.0005 0.0016/0.0028 0.0003/0.0004 0.0006/ 0.0009 0.0010/0.0023 0.0011/0.0018 0.0003/ 0.0005 0.0010/ 0.0016 0.0015/ 0.0024 0.0000/ 0.0030 0.0001/ 0.0005 0.0023/ 0.0002 0.0002/ 0.0020 0.0013/0.0032 0.0007/0.0051 0.0008/ 0.0013 0.0023/0.0035 0.0031/ 0.0051 0.0033/0.0055
coefficients of the surfaces here. We can see from this table that the sixth eye has the smallest thickness and the 13th eye has the largest thickness. The 17th eye has the
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largest anterior and posterior surface radii. The 19th eye has the smallest anterior surface radius and the ninth eye has the smallest posterior surface radius. In the accommodation study, we optimize the radius of the crystalline lens’s surface with different object distances until the incident light focuses to the image plane (retinal plane for the human eye). The accommodation diopter can be input to the eye by changing the object distance. In this experiment, we estimate Zernike coefficients in five accommodative conditions, ranging from 0 to 4 diopter in step of one diopter. Diopter corresponds to the target distance, for example, one diopter corresponds to the distance of 1000 mm, and two diopters corresponds to the distance of 500 mm. When the merit function is optimized to zero, the Zernike coefficients are just what we want. We calculate the Zernike coefficients in ZEMAX after finishing every optimization process [10].
3. Influence of accommodation on wavefront Fig. 1 illustrates the effect of accommodation on rootmean-square (RMS) wavefront value for all 20 eyes. The abscissa indicates the eye number, which is sequenced by the size of the RMS value of wavefront aberration of the whole eye (the defocus term is excluded) for 0D accommodative condition, and the ordinate is the RMS value of the wavefront aberrations in micrometers. The symbol of , J, K, n and & represents 0D, 1D, 2D, 3D and 4D, respectively. The RMS wavefront value provides a general estimate of the variation of the wavefront from the ideal one. In general, different eyes have different RMS wavefront values as the accommodation varies. From this figure, we can see that the RMS
Fig. 1. RMS values of total wavefront aberrations excluding defocus for 20 eyes with four accommodative conditions ( 1D, 2D, 3D and 4D).
wavefront values of most eyes decrease as the accommodation varies from 0D to –2D, and then gradually increase as the accommodation varies from 2D to –4D. The RMS wavefront values of some eyes decrease as the accommodation varies from 0D to –1D, and then gradually increase as the accommodation varies from –1D to –4D. Only the RMS wavefront values of the eighth and the 16th eye decrease from 0D to –3D and increase from –3D to –4D. Changes in RMS wavefront values with accommodation for all eyes are found, but the amplitude of the changes varies from individual to individual. Some RMS values of the wavefront have a little variation, such as the third and the fifth eyes. It is only 0.092 mm from the maximum RMS value of 0.419 mm at –4D to the minimum RMS value of 0.327 mm at –2D for the third eye, and it is 0.077 mm from the maximum of 0.478 mm at 0D to the minimum of 0.401 mm at –2D for the fifth eye, which is 22.0% and 16.3% in the corresponding proportion to the maximum RMS values. However, changes of some RMS values cover a wide range, such as the 17th eye and the 20th eye. It is about 0.689 mm from the maximum RMS value of 1.019 mm at 0D to the minimum RMS value of 0.33 mm at –2D for the 17th eye, and it is 1.058 mm from the maximum RMS value of 1.358 mm at 0D to the minimum RMS value of 0.30 mm at –2D for the 20th eye, which is 67.7% and 77.9% in the corresponding proportion to the maximum RMS values, respectively. In addition, larger magnitude variation seems to be independent of wavefront aberration value. For instance, the first eye has the smaller wavefront aberration, but it has the larger change of 84.8% in RMS wavefront value, which is from the maximum value of 0.329 mm at 0D to the minimum value of 0.05 mm at –2D. The 12th eye has the maximum RMS wavefront value of 0.756 mm at 0D, while the maximum RMS difference is about 0.556 mm, which is 74% in proportion to the maximum RMS value. However, the 18th eye has a larger RMS value of 1.038 mm at 0D, and has only smaller change of 13.0% from 1.082 mm (maximum) at –4D to 0.941 mm (minimum) at –2D. This seems to be similar with the results of He et al. In their study, subject KK has very little wavefront aberrations but has the largest relative change in RMS wavefront value. Fig. 2 shows the change of RMS in primary spherical aberration with accommodation for all 20 eyes. The abscissa indicates the accommodative stimulus from 0D to –4D and the ordinate is the spherical RMS wavefront value with the unit in micrometers. There are four subfigures in Fig. 2. Among them, the top-left figure illustrates the spherical RMS values from the first to the fifth eyes, and the top-right figure shows those from the sixth to the 10th eyes. The down-left figure illustrates the spherical RMS values from the 11th to the 15th eyes and those from the 16th to the 20th eyes are showed on
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Fig. 2. Spherical RMS wavefront value as a function of the accommodative stimulus for 20 eyes.
Fig. 3. x-Axis coma RMS as a function of the accommodative stimulus for 20 eyes.
the down-right figure. In each sub-figure, five curves are differently represented by the symbols of n, J, &, B and *, which are labeled on the figure. Changes of spherical RMS can be shown clearly in the four subfigures. We can see that the spherical RMS value of wavefront aberrations of almost every eye has a graduate decreasing trend from 0D to –4D accommodative conditions, but the variation pattern does not hold the same for all eyes. For example, the fourth eye has the minimum RMS value of 0.07 mm at 0D, and the RMS value increases to the maximum of 0.15 mm at –1D, and then decreases gradually to 0.116 mm at –4D. The difference between maximum and minimum RMS values is about 53.3% in proportion to the maximum RMS wavefront value. Among 20 eyes, the 12th eye has the larger change of 94.4% in spherical RMS value from the maximum RMS value of 0.5 mm at 0D to the minimum RMS value of 0.028 mm at –4D. While the spherical RMS value of the 18th eye changes gently, which has the smallest variation of 14.3% from the maximum of 0.35 mm at –1D to the minimum of 0.3 mm at –4D. The variation magnitude is independent of the spherical RMS value. The 20th eye can be taken as an example. It has the largest spherical RMS among 20 eyes, but only 19.4% variation from the maximum of 0.62 mm at 0D to the minimum of 0.5 mm at –2D. Fig. 3 describes the change of x-axis coma RMS wavefront values with accommodation for all eyes. The abscissa indicates the accommodative stimulus from 0D to –4D and the ordinate is the x-axis coma RMS wavefront value with the unit in micrometers. There are also four sub-figures in Fig. 3, in which the symbols are labeled. We can also see the individual variation in this figure. The second and the 19th eye have considerable
variations in the x-axis coma RMS wavefront value. There is a variation magnitude of 0.15 mm of the second eye from the maximum of 0.3 mm at 0D to the minimum of 0.15 mm at –1D, which is 50% in proportion to the maximum value. The x-axis coma RMS value of the 19th eye decreases most among the 20 eyes as the accommodation varies and the difference in RMS wavefront value is 0.232 mm from the maximum of 0.432 mm at 0D to the minimum of 0.2 mm at –3D, which is 53.7% in proportion to the maximum value. But the variance trend over the accommodative range is not so significant for most eyes. For instance, the 20th eye has the largest x-axis coma RMS value among the 20 eyes, while there is only little change of 19.4% in proportion to the maximum RMS value from the maximum of 0.62 mm at 0D to the minimum of 0.5 mm at –2D. The yaxis coma RMS wavefront value has similar changes with the x-axis coma RMS wavefront value as the accommodation varies, so we do not present the corresponding graph here. Fig. 4 shows the total contribution of wavefront aberrations from the fifth to the seventh Zernike order as a function of accommodation. The abscissa indicates the accommodative stimulus from 0D to –4D and the ordinate is the higher-order RMS wavefront value with the unit in micrometers. We also give four sub-figures, and the symbols are also labeled. From the graph, we can see that as the accommodation varies from 0D, the higher-order RMS values of most eyes decrease firstly, and then increase gradually. Different eyes have different decreasing as well as increasing values. The RMS value of the ninth eye decreases from 0.28 mm at 0D to 0.11 mm at –2D, which is 56.7% in proportion to the maximum RMS value, and then increases to 0.3 mm at –4D, which is 63.3% in proportion to the maximum
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Fig. 4. Higher-order RMS as a function of the accommodative stimulus for 20 eyes.
RMS value. The RMS value of the 13th eye decreases from 0.13 mm at 0D to 0.036 mm at –1D, which is 26.1% in proportion to the maximum RMS wavefront value, and then increases to 0.36 mm at –4D, which is 90% in proportion to the maximum RMS wavefront value. In comparison, the higher-order RMS wavefront value of the seventh eye varies gently, which decreases to 0.018 mm between 0D and –3D and increases to 0.038 mm between –3D and –4D. It is only 23.8% variation of the RMS wavefront value in proportion to the maximum RMS value. The situation of the variance trend for few eyes is totally different. For instance, the RMS wavefront value of the second eye decreases from 0.13 mm at 0D to 0.09 mm at –1D, increases to 0.12 mm at –2D, decreases to 0.002 mm at –3D, and increases to 0.152 mm at –4D.
4. Conclusions We have constructed 20 individual eye models to analyze the impact of accommodation in this study. The variations of overall wavefront aberrations and different Zernike term aberrations as different accommodation conditions (range from 0D to –4D) have been presented. In conclusion, accommodation influences wavefront aberrations of the whole eye, and the situation varies substantially from individual to individual. The RMS wavefront values of most eyes decrease firstly, and then increase as the accommodation varies from 0D to –4D. Spherical RMS wavefront value has a graduate decreasing trend as the accommodation varies from 0D to –4D. Most higher-order RMS values decrease firstly, and then increase as the accommodation varies from 0D to –4D. These results are in good agreement with those actually measured by He et al.
Coma does not how large variation as accommodation, like other kinds of aberrations, except for a few eyes in our study. No matter how large the wavefront aberration of the un-accommodated eye, all eyes have their own aberration variations as the accommodation varies. Generally speaking, accommodation plays an important role in contributing wavefront aberrations to the eyes, and has an impact on overall optical imaging quality. Therefore, we have to consider not only the wavefront aberration at the un-accommodative condition but also that at the accommodative condition during the clinical ocular therapy. Individual eye model is firstly adopted to analyze the influence of accommodation on wavefront aberration in this paper. As a new way, it can be adopted widely for ocular performance diagnosis of human eye and for clinical vision therapy.
Acknowledgements This research is supported by the National Nature Science Foundation of China (Key Research Project, no. 60438030), the Key Research Foundation of Scientific and Technical Committee of Tianjin City of China (no. 033183711) and the Chinese Ministry of Education’s Nankai University and Tianjin University Cooperation Foundation.
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