- Email: [email protected]
Individual eye model based on wavefront aberration
ARTICLE IN PRESS
Optik
Optics
Optik 116 (2005) 80–85 www.elsevier.de/ijleo
Individual eye model based on wavefront aberration Huanqing Guoa, Zhaoqi Wanga,, Qiuling Zhaob, Wei Quanc, Yan Wangd a
The Key Laboratory photoelectron Information Science & Technique, Modern Optical Institute of Nankai University, Nankai University, Tianjin University, Tianjin, 300071, PR China b Precision Instrument and Photoelectron Engineering Institute, Tianjin University, Tianjin, 300071, PR China c Communication Engineering Institute, Jilin University, Jilin, 130022, PR China d Tianjin Eye hospital, Refractive Surgery Center, Tianjin, 300020, PR China Received 17 August 2004; accepted 5 December 2004
Abstract Based on the widely used Gullstrand-Le Grand eye model, the individual human eye model has been established here, which has individual corneal data, anterior chamber depth and the eyeball depth. Furthermore, the foremost thing is that the wavefront aberration calculated from the individual eye model is equal to the eye’s wavefront aberration measured with the Hartmann-shack wavefront sensor. There are four main steps to build the model. Firstly, the corneal topography instrument was used to measure the corneal surfaces and depth. And in order to input cornea into the optical model, high-order aspheric surface-Zernike Fringe Sag surface was chosen to fit the corneal surfaces. Secondly, the Hartmann-shack wavefront sensor, which can offer the Zernike polynomials to describe the wavefront aberration, was built to measure the wavefront aberration of the eye. Thirdly, the eye’s axial lengths among every part were measured with A-ultrasonic technology. Then the data were input into the optical design software–ZEMAX and the crystalline lens’s shapes were optimized with the aberration as the merit function. The individual eye model, which has the same wavefront aberrations with the real eye, is established. r 2005 Elsevier GmbH. All rights reserved. Keywords: Wavefront aberration; Eye model; Corneal topography; Zernike polynomials; ZEMAX
1. Introduction Many experiments in optical design, digital image processing, robotics etc., must take into account the behavior and contribution of the optical system of the eye [1]. Le Grand had modified the eye model with six refraction surfaces, proposed by Sweden’s ophthalmology expert Gullstrand in the early 20th century, with four spherical refractions [2]. Gullstrand-Le Grand eye model offered a powerful tool for the ophthalmologist Corresponding author. Fax: +86 22 23508332.
E-mail address: [email protected] (Z. Wang). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2004.12.005
and the ophthalmological instrument engineers. Many later researchers continued to modify this model by importing aspherical surface and lens grads refractive index etc. [3–5]. These eye models were all based on the anthropotomy and biological experiments and were summed up from the average results with statistics. As a complex and precise optical system, each person’s eye has its own individual physiological characteristics. In recent years, wavefront technology of the human eye is being used as an effective tool for measuring ocular wavefront aberration and diagnosing the vision [6–9]. The wavefront-optometer, such as the ray-tracing optometer, the Hartmann-Shack sensor, the OPD optometer
ARTICLE IN PRESS H. Guo et al. / Optik 116 (2005) 80–85
etc., can give the accurate wavefront aberration of the individual eye and describe it by a mathematic expression (exp. Zernike polynomials) [10,11]. Different eyes have different wavefront aberrations, and individual eye model made for individual people has its own optical characteristic and ocular quality. Instead of the conventional anthropotomy and biological experimental methods, individual eye model has been established here through the process of measuring scathelessly, numerical calculation and optical modeling. High-order aspheric surface was used to fit the cornea measured with the topography meter. The axial lengths of the eye were fixed values given by Aultrasonic meter. As for the lens, its shape was solved with the wavefront aberration of the individual eye obtained from the Hartman-Shack wavefront sensor. The software ZEMAX was chosen to construct the model with the Gullstrand-Le Grand eye model as the initial configuration. As a result, the individual eye model we built has the same wavefront aberration as that of the real eye.
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to describe the wavefront aberration W(x,y). The expression is as follows: W ðx; yÞ ¼
N X
C i Z i ðx; yÞ ðx2 þ y2 p1Þ:
i¼1
Described by Zernike polynomials, wavefront aberrations are very different from one to person to another person. And the pattern of aberrations varies greatly among different individual eyes, such as emmetropia, myopia and hyperopia. High-order aberrations even exist in the emmetropia. The eye model constructed for individual people must show its own special wavefront aberrations. That is, each Zernike coefficients calculated from individual eye model should be equal to the real eye’s coefficient measured by the wavefront sensor. The Hartmann-Shack wavefront sensor was chosen to measure the wavefront aberration of the eye. Fig. 1 shows the experimental setup we built. The HartmannShack sensor can measure wavefront emerging from eye in reference to a perfect plane wave at the eye’s entrance pupil. And this is equivalent to measuring the wavefront error of the eye at the exit pupil in reference to a perfect reference sphere [6].
2. Methods and processes Table 1 shows the detailed structural parameters of the Gullstrand-Le Grand eye model. R1 and R2 stand for the curvature radius of corneal anterior surface and posterior surface, respectively, R3 and R4 stand for the curvature radius of the lens’s two surfaces, and R5 stands for the retinal curvature radius. The new glass catalog with corresponding refraction indices for the eye was created. All these data were then input into ZEMAX and became the initial configuration of our following work.
2.2. High-order aspheric corneal surface Small changes of the corneal surface will bring great effect on the eye’s optical quality because about 2/3 diopter of the whole eye’s dioptric system lies on its
2.1. Wavefront aberration of the eye In addition to exhibiting some defocus and astigmatism, normal human eyes are known to suffer from higher order wavefront aberrations. All these aberrations degrade the retinal image quality. To provide a more complete description of all the aberrations of the eye, Zernike polynomials Zi(x,y), which can quantify every individual aberration components, are used
Table 1.
Fig. 1. Experimental setup of Hartmann-Shack wavefront sensor for measuring the eye.
Structural parameters of Gullstrand-Le grand eye model
Radius (mm)
Depth (mm)
Refractive index
Medium
R1 R2 R3 R4 R5
D1 D2 D3 D4
1.3771 1.3374 1.420 1.336
Cornea Anterior chamber Lens Vitreous body
¼ 7:8 ¼ 6:5 ¼ 10:2 ¼ 6:0 ¼ 12:3
¼ 0:55 ¼ 3:05 ¼ 4:0 ¼ 16:60
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2.3. Measurement with ultrasonic technology
K
x
R
z
Corneal surface Reference sphere
Fig. 2. Orbscan II gives relative altitude from the corneal surface to the reference sphere.
cornea. Ophthalmologists used to measure the corneal surface with Placido Disc. In order that the measured data are imported to the eye model, which is constructed with ZEMAX, proper polynomials should be considered to fit the corneal surface accurately. Here, the following high-order aspheric functions are selected: z ¼ z0 þ
N X x 2 þ y2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ai Zi ðx; yÞ: R þ R2 ð1 þ cÞðx2 þ y2 Þ i¼1
(1) This expression is the Zernike Fringe Sag surface with a rectangular coordinate system. Here, coordinate axis z points to the optical axis, z0 is constant. And R, c are the curvature radius and conic constant, respectively, and C ¼ 0 is for spheres. In the next sum item, Zi(x,y) are Zernike polynomials normalized on unit circle and ai are their respective coefficients. N is the number of item. During the practical operation, most of the topography meters with Placido Disc will give the relative altitude K(x,y) between the measured corneal surface and a reference sphere in the radial direction. Fig. 2 shows the truth. There is a coordinate transformation formula obtained from Fig. 2 as follows: Dðx; yÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 ðx2 þ y2 Þ ðR Kðx; yÞÞ2 ðx2 þ y2 Þ
(2) and then let Dðx; yÞ equal to the sum item of formula (1): Dðx; yÞ ¼
N X
ai Z i ðx; yÞ ðx2 þ y2 p1Þ:
3. Experiments 3.1. Construct HQ’s right eye model Testee HQ’s right eye model has been constructed and his right eye has about 3.0D myopia and about 0.75D astigmatism measured subjectively. Measured with the experimental setup that is shown in Fig. 1, the wavefront aberrations of HQ’s right eye were expressed by 14 Zernike items in our research, and we found that it is enough to get perfect result for the individual eye model. The detailed data are expressed in Fig. 3. We use the prepositional standard of Optical Society of America to be the marshaling sequence of the Zernike items. Here, the pupil size was 5.2 mm. Orbscan II corneal topography meter was used to measure HQ’s corneal shape and its thickness every 5 min. And the average value of six measurements was calculated. Eq. (3) was solved by the least square method and the coefficients were obtained. Table 2 shows these Zernike coefficients of HQ’s cornea. The Root Mean Square (RMS) value at the last row shows the fitting error. The medical BMF-200 A/B Ultrasonic Diagnostic Instrument was used to measure the eye’s axial lengths. Fig. 4 shows one result as the map of ultrasonic reflective intensity, and the length between two peak values is taken as the measuring result. The average value of eight measurements was calculated and Table 3 shows the measurement result. In Table 3, the corneal
(3)
i¼1
In formula (3), every measuring point (x,y) will give its equation. So firstly, we can decide the curvature radius R in formula (1), and then we solve Eq. (3) by Gauss Least Square Method to get ai : Until now, a set of continuous function, which ZEMAX can recognize, has been used to fit the corneal surface. The processes are similar to solve both the surfaces of the cornea.
Zernike Coefficient (µm)
y
The eye’s axial lengths could be measured with ultrasonic technology. Ultrasonic can reflect on interface between two media and transmit different distance with different time. A-type-ultrasonic technology, which has the advantage of exact orientation and high-axial resolving power, had been used to measure the anterior chamber depth, the lens depth and the depth of the vitreous body. Then, the corresponding data in Table 1 were replaced with these measured depths for individual eye model.
6 4 2 0 -2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 Zernike Order
Fig. 3. Zernike description of HQ’s right eye’s aberrations.
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depth was measured by Orbscan II and the pupil size was used in wavefront calculation. Next, all the above data were input into ZEMAX. The crystalline lens of the eye has complementary effect with the cornea and it offers about 1/3 diopter of the eye. However, it is difficult to measure the intravital eye lens surface. In order to make every Zernike coefficient equal to that of measurement, we optimize the lens surfaces. The operands ZERN was used and the Zernike Fringe Sag surfaces were selected for both the surfaces of the lens. It is easy to optimize and the merit function converges quickly to reach zero. The lens was just what
Table 2. Corneal anterior and posterior Zernike fringe sag surface coefficients for HQ’s right eye (units: mm) Zernike coefficients
Anterior
Posterior
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 RMS
0.97429 0.55382 0.27326 0.87593 0.25229 0.019822 0.085622 0.13672 0.10259 0.0030071 0.011180 0.0027173 0.031730 0.0022907 1.1235
9.0498 7.2686 3.3932 11.327 2.8665 0.012261 0.35361 0.22671 0.16375 0.046279 0.11893 0.85623 0.094030 0.20292 4.8872
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we wanted now because the wavefront aberrations given by the model were equal to the actual eye’s aberrations. By now, the HQ’s individual model with his individual corneal surface, eye depths and exact wavefront aberration have been established. Fig. 6(a) is the layout of the model. To sum up the whole process discussed above, Fig. 5 gives a work flow chart.
3.2. Calculations from the model After completing the individual eye model, we can easily calculate many optical parameters from it. These parameters can be used to evaluate the whole optical system and the retinal image quality of the eye. Fig. 6 shows the standard spot diagram, diffraction encircled energy plots and the modulation transfer function (MTF) calculated from the eye model. Only the central visual field is considered here. MTF is a very important tool to evaluate retinal image quality. Because of plentiful aberrations besides the defocus and astigmatism, HQ’s right eye has a very poor MTF value (the lowest two plots in Fig. 6(d)). However, when we devise a ‘‘spectacle’’ for HQ and get rid of the 4th (defocus), 3rd and 5th (astigmatism) Zernike item, the MTF plot will be better than the original one (the middle two plots of Fig. 6(d)). But it is still much lower than the MTF of diffraction limit (the highest plots of Fig. 6(d)) because the remanent high-order aberrations still exist. Traditional optical lens cannot rectify most of the high-order wavefront aberrations.
Corneal shape measured by Orbscan II
Corneal surface fitting with Zernike polynomials
Eye’s axial lengths measured by BMF-200
Wavefront aberration measured by HS sensor
Individual Eye Model
Definition of Merit function
Lens’s surfaces
00
10
20 (MM)
30
40
Fig. 4. Ultrasonic reflective intensity of BMF-200 A/B Ultrasonic Diagnostic Instrument.
Table 3.
Interested calculations, such as PSF, MTF etc.
Fig. 5. Work flow chart for building the individual eye model.
Each several part of the eye’s axial lengths for HQ’s right eye (units: mm)
Eye
Corneal depth
Depth of anterior chamber
Depth of lens
Depth of vitreous body
Pupil size
Value
0.57
3.03
3.51
18.62
5.2
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model we built is a static model and cannot show the eye’s accommodation which is a very complex process. If we can measure the corresponding data with some relative static accommodations, it will be possible to build models for the eye’s different accommodations. Though they can show different status of the eye, these models will be still static. We also built some individual models for the people whose vision was normal or only ametropic. We did not do experiments for pathological eyes. That is, we just care much about the optical system of the eye and do not care about neural or psychological effects. For this kind of subject, good optical system of the eye perhaps shows excellent retinal image quality but does not mean good vision. For ophthalmologists, the individual eye model can help devise appliances to improve subjects’ vision. And the most important application is in the field of the popular corneal ablation surgery. The ablation profile of the corneal ablation surgery guided by the wavefront and topography can be calculated accurately from the model by changing the corneal anterior surface, and this is the succeeding work.
100.00
(a)
FRACTION OF ENCLOSED ENERGY
(b)
1.0 0.9 0.8 0.7 0.6 0.5
Acknowledgements
0.4 0.3 0.2 0.1 0.0 0.000
25.000
(c)
50.000
RADIUS FROM CENTROID IN MICRONS
1.0 MODULUS OF THE OTF
0.9 0.8 0.7 0.6 0.5
References
0.4 0.3 0.2 0.1 0.0 0.00
(d)
This work is supported by the National Natural Science Foundation of China (Key research project, No. 60438030), the Chinese Ministry of Education’s Nankai University, Tianjin University Cooperation Foundation and the Key Research Foundation of Scientific and Technical Committee of Tianjin City of China (No. 033183711).
30.00
60.00
SPATIAL FREQUENCY IN CYCLES MILLIMETER
Fig. 6. The optical system’s quality calculated from individual eye model. (a) Layout of HQ’s individual model. (b) Spot diagram on the retina (units: mm). (c) diffraction encircled energy plots. (d) The tangential sagittal MTF plots.
the eye The and
4. Discussion The individual eye model is custom-built for individual people. It has its own corneal surfaces, axial lengths and the individual wavefront aberrations. Here, the
[1] R. Navarro, J. Santamaria, J. Bescos, Accommodationdependent model of the human eye with aspherics, J. Opt. Soc. Am. A 2 (1985) 1273–1281. [2] J. Warren Blaker, Toward an adaptive model of the human eye, J. Opt. Soc. Am. 70 (1980) 220–223. [3] W. Lotmar, Theoretical eye model with aspherics, J. Opt. Soc. Am. 61 (1971) 1522–1529. [4] A.C. Kooijman, Light distribution on the retina of a wide-angle theoretical eye, J. Opt. Soc. Am. 73 (1983) 1544–1550. [5] H.-L. Liou, N.A. Brennan, Anatomically accurate, finite model eye for optical modeling, J. Opt. Soc. Am. A 14 (1997) 1684–1695. [6] J. Liang, B. Grimm, S. Goelz, J.F. Bille, Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor, J. Opt. Soc. Am. A 11 (1994) 1949–1957. [7] W. Quan, Z.Q. Wang, G.G. Mu, L. Ning, Correction of the aberrations in the human eyes with SVAG1 thin-film
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transistor liquid-crystal display, OPTIK 114 (2003) 467–471. [8] J. Liang, D.R. Willams, Aberrations and retinal image quality of the normal human eye, J. Opt. Soc. Am. A 14 (1997) 2873–2883. [9] E. Moreno-Barriuso, R. Navarro, Laser Ray-Tracing versus Hartmann-Shack sensor for measuring optical aberrations in the human eye, J. Opt. Soc. Am. A 17 (2000) 974–984.
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[10] P.M. Prieto, F. Vargas-Martin, S. Goelz, P. Artal, Analysis of the performance of the Hartmann-Shack sensor in the human eye, J. Opt. Soc. Am. A 17 (2000) 1388–1398. [11] J.F. Bille, U. von Page, The development of wavefront technology and its application to ophthalmology, In: J.F. Bille, C.F.H. Harner, F.H. Loesel (Eds.), Aberration-Free Refractive Surgery, second enlarged ed., Springer, Berlin, 2003, 2004, pp.1–23.
Optik
Optics
Optik 116 (2005) 80–85 www.elsevier.de/ijleo
Individual eye model based on wavefront aberration Huanqing Guoa, Zhaoqi Wanga,, Qiuling Zhaob, Wei Quanc, Yan Wangd a
The Key Laboratory photoelectron Information Science & Technique, Modern Optical Institute of Nankai University, Nankai University, Tianjin University, Tianjin, 300071, PR China b Precision Instrument and Photoelectron Engineering Institute, Tianjin University, Tianjin, 300071, PR China c Communication Engineering Institute, Jilin University, Jilin, 130022, PR China d Tianjin Eye hospital, Refractive Surgery Center, Tianjin, 300020, PR China Received 17 August 2004; accepted 5 December 2004
Abstract Based on the widely used Gullstrand-Le Grand eye model, the individual human eye model has been established here, which has individual corneal data, anterior chamber depth and the eyeball depth. Furthermore, the foremost thing is that the wavefront aberration calculated from the individual eye model is equal to the eye’s wavefront aberration measured with the Hartmann-shack wavefront sensor. There are four main steps to build the model. Firstly, the corneal topography instrument was used to measure the corneal surfaces and depth. And in order to input cornea into the optical model, high-order aspheric surface-Zernike Fringe Sag surface was chosen to fit the corneal surfaces. Secondly, the Hartmann-shack wavefront sensor, which can offer the Zernike polynomials to describe the wavefront aberration, was built to measure the wavefront aberration of the eye. Thirdly, the eye’s axial lengths among every part were measured with A-ultrasonic technology. Then the data were input into the optical design software–ZEMAX and the crystalline lens’s shapes were optimized with the aberration as the merit function. The individual eye model, which has the same wavefront aberrations with the real eye, is established. r 2005 Elsevier GmbH. All rights reserved. Keywords: Wavefront aberration; Eye model; Corneal topography; Zernike polynomials; ZEMAX
1. Introduction Many experiments in optical design, digital image processing, robotics etc., must take into account the behavior and contribution of the optical system of the eye [1]. Le Grand had modified the eye model with six refraction surfaces, proposed by Sweden’s ophthalmology expert Gullstrand in the early 20th century, with four spherical refractions [2]. Gullstrand-Le Grand eye model offered a powerful tool for the ophthalmologist Corresponding author. Fax: +86 22 23508332.
E-mail address: [email protected] (Z. Wang). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2004.12.005
and the ophthalmological instrument engineers. Many later researchers continued to modify this model by importing aspherical surface and lens grads refractive index etc. [3–5]. These eye models were all based on the anthropotomy and biological experiments and were summed up from the average results with statistics. As a complex and precise optical system, each person’s eye has its own individual physiological characteristics. In recent years, wavefront technology of the human eye is being used as an effective tool for measuring ocular wavefront aberration and diagnosing the vision [6–9]. The wavefront-optometer, such as the ray-tracing optometer, the Hartmann-Shack sensor, the OPD optometer
ARTICLE IN PRESS H. Guo et al. / Optik 116 (2005) 80–85
etc., can give the accurate wavefront aberration of the individual eye and describe it by a mathematic expression (exp. Zernike polynomials) [10,11]. Different eyes have different wavefront aberrations, and individual eye model made for individual people has its own optical characteristic and ocular quality. Instead of the conventional anthropotomy and biological experimental methods, individual eye model has been established here through the process of measuring scathelessly, numerical calculation and optical modeling. High-order aspheric surface was used to fit the cornea measured with the topography meter. The axial lengths of the eye were fixed values given by Aultrasonic meter. As for the lens, its shape was solved with the wavefront aberration of the individual eye obtained from the Hartman-Shack wavefront sensor. The software ZEMAX was chosen to construct the model with the Gullstrand-Le Grand eye model as the initial configuration. As a result, the individual eye model we built has the same wavefront aberration as that of the real eye.
81
to describe the wavefront aberration W(x,y). The expression is as follows: W ðx; yÞ ¼
N X
C i Z i ðx; yÞ ðx2 þ y2 p1Þ:
i¼1
Described by Zernike polynomials, wavefront aberrations are very different from one to person to another person. And the pattern of aberrations varies greatly among different individual eyes, such as emmetropia, myopia and hyperopia. High-order aberrations even exist in the emmetropia. The eye model constructed for individual people must show its own special wavefront aberrations. That is, each Zernike coefficients calculated from individual eye model should be equal to the real eye’s coefficient measured by the wavefront sensor. The Hartmann-Shack wavefront sensor was chosen to measure the wavefront aberration of the eye. Fig. 1 shows the experimental setup we built. The HartmannShack sensor can measure wavefront emerging from eye in reference to a perfect plane wave at the eye’s entrance pupil. And this is equivalent to measuring the wavefront error of the eye at the exit pupil in reference to a perfect reference sphere [6].
2. Methods and processes Table 1 shows the detailed structural parameters of the Gullstrand-Le Grand eye model. R1 and R2 stand for the curvature radius of corneal anterior surface and posterior surface, respectively, R3 and R4 stand for the curvature radius of the lens’s two surfaces, and R5 stands for the retinal curvature radius. The new glass catalog with corresponding refraction indices for the eye was created. All these data were then input into ZEMAX and became the initial configuration of our following work.
2.2. High-order aspheric corneal surface Small changes of the corneal surface will bring great effect on the eye’s optical quality because about 2/3 diopter of the whole eye’s dioptric system lies on its
2.1. Wavefront aberration of the eye In addition to exhibiting some defocus and astigmatism, normal human eyes are known to suffer from higher order wavefront aberrations. All these aberrations degrade the retinal image quality. To provide a more complete description of all the aberrations of the eye, Zernike polynomials Zi(x,y), which can quantify every individual aberration components, are used
Table 1.
Fig. 1. Experimental setup of Hartmann-Shack wavefront sensor for measuring the eye.
Structural parameters of Gullstrand-Le grand eye model
Radius (mm)
Depth (mm)
Refractive index
Medium
R1 R2 R3 R4 R5
D1 D2 D3 D4
1.3771 1.3374 1.420 1.336
Cornea Anterior chamber Lens Vitreous body
¼ 7:8 ¼ 6:5 ¼ 10:2 ¼ 6:0 ¼ 12:3
¼ 0:55 ¼ 3:05 ¼ 4:0 ¼ 16:60
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2.3. Measurement with ultrasonic technology
K
x
R
z
Corneal surface Reference sphere
Fig. 2. Orbscan II gives relative altitude from the corneal surface to the reference sphere.
cornea. Ophthalmologists used to measure the corneal surface with Placido Disc. In order that the measured data are imported to the eye model, which is constructed with ZEMAX, proper polynomials should be considered to fit the corneal surface accurately. Here, the following high-order aspheric functions are selected: z ¼ z0 þ
N X x 2 þ y2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ai Zi ðx; yÞ: R þ R2 ð1 þ cÞðx2 þ y2 Þ i¼1
(1) This expression is the Zernike Fringe Sag surface with a rectangular coordinate system. Here, coordinate axis z points to the optical axis, z0 is constant. And R, c are the curvature radius and conic constant, respectively, and C ¼ 0 is for spheres. In the next sum item, Zi(x,y) are Zernike polynomials normalized on unit circle and ai are their respective coefficients. N is the number of item. During the practical operation, most of the topography meters with Placido Disc will give the relative altitude K(x,y) between the measured corneal surface and a reference sphere in the radial direction. Fig. 2 shows the truth. There is a coordinate transformation formula obtained from Fig. 2 as follows: Dðx; yÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 ðx2 þ y2 Þ ðR Kðx; yÞÞ2 ðx2 þ y2 Þ
(2) and then let Dðx; yÞ equal to the sum item of formula (1): Dðx; yÞ ¼
N X
ai Z i ðx; yÞ ðx2 þ y2 p1Þ:
3. Experiments 3.1. Construct HQ’s right eye model Testee HQ’s right eye model has been constructed and his right eye has about 3.0D myopia and about 0.75D astigmatism measured subjectively. Measured with the experimental setup that is shown in Fig. 1, the wavefront aberrations of HQ’s right eye were expressed by 14 Zernike items in our research, and we found that it is enough to get perfect result for the individual eye model. The detailed data are expressed in Fig. 3. We use the prepositional standard of Optical Society of America to be the marshaling sequence of the Zernike items. Here, the pupil size was 5.2 mm. Orbscan II corneal topography meter was used to measure HQ’s corneal shape and its thickness every 5 min. And the average value of six measurements was calculated. Eq. (3) was solved by the least square method and the coefficients were obtained. Table 2 shows these Zernike coefficients of HQ’s cornea. The Root Mean Square (RMS) value at the last row shows the fitting error. The medical BMF-200 A/B Ultrasonic Diagnostic Instrument was used to measure the eye’s axial lengths. Fig. 4 shows one result as the map of ultrasonic reflective intensity, and the length between two peak values is taken as the measuring result. The average value of eight measurements was calculated and Table 3 shows the measurement result. In Table 3, the corneal
(3)
i¼1
In formula (3), every measuring point (x,y) will give its equation. So firstly, we can decide the curvature radius R in formula (1), and then we solve Eq. (3) by Gauss Least Square Method to get ai : Until now, a set of continuous function, which ZEMAX can recognize, has been used to fit the corneal surface. The processes are similar to solve both the surfaces of the cornea.
Zernike Coefficient (µm)
y
The eye’s axial lengths could be measured with ultrasonic technology. Ultrasonic can reflect on interface between two media and transmit different distance with different time. A-type-ultrasonic technology, which has the advantage of exact orientation and high-axial resolving power, had been used to measure the anterior chamber depth, the lens depth and the depth of the vitreous body. Then, the corresponding data in Table 1 were replaced with these measured depths for individual eye model.
6 4 2 0 -2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 Zernike Order
Fig. 3. Zernike description of HQ’s right eye’s aberrations.
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depth was measured by Orbscan II and the pupil size was used in wavefront calculation. Next, all the above data were input into ZEMAX. The crystalline lens of the eye has complementary effect with the cornea and it offers about 1/3 diopter of the eye. However, it is difficult to measure the intravital eye lens surface. In order to make every Zernike coefficient equal to that of measurement, we optimize the lens surfaces. The operands ZERN was used and the Zernike Fringe Sag surfaces were selected for both the surfaces of the lens. It is easy to optimize and the merit function converges quickly to reach zero. The lens was just what
Table 2. Corneal anterior and posterior Zernike fringe sag surface coefficients for HQ’s right eye (units: mm) Zernike coefficients
Anterior
Posterior
a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 RMS
0.97429 0.55382 0.27326 0.87593 0.25229 0.019822 0.085622 0.13672 0.10259 0.0030071 0.011180 0.0027173 0.031730 0.0022907 1.1235
9.0498 7.2686 3.3932 11.327 2.8665 0.012261 0.35361 0.22671 0.16375 0.046279 0.11893 0.85623 0.094030 0.20292 4.8872
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we wanted now because the wavefront aberrations given by the model were equal to the actual eye’s aberrations. By now, the HQ’s individual model with his individual corneal surface, eye depths and exact wavefront aberration have been established. Fig. 6(a) is the layout of the model. To sum up the whole process discussed above, Fig. 5 gives a work flow chart.
3.2. Calculations from the model After completing the individual eye model, we can easily calculate many optical parameters from it. These parameters can be used to evaluate the whole optical system and the retinal image quality of the eye. Fig. 6 shows the standard spot diagram, diffraction encircled energy plots and the modulation transfer function (MTF) calculated from the eye model. Only the central visual field is considered here. MTF is a very important tool to evaluate retinal image quality. Because of plentiful aberrations besides the defocus and astigmatism, HQ’s right eye has a very poor MTF value (the lowest two plots in Fig. 6(d)). However, when we devise a ‘‘spectacle’’ for HQ and get rid of the 4th (defocus), 3rd and 5th (astigmatism) Zernike item, the MTF plot will be better than the original one (the middle two plots of Fig. 6(d)). But it is still much lower than the MTF of diffraction limit (the highest plots of Fig. 6(d)) because the remanent high-order aberrations still exist. Traditional optical lens cannot rectify most of the high-order wavefront aberrations.
Corneal shape measured by Orbscan II
Corneal surface fitting with Zernike polynomials
Eye’s axial lengths measured by BMF-200
Wavefront aberration measured by HS sensor
Individual Eye Model
Definition of Merit function
Lens’s surfaces
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10
20 (MM)
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Fig. 4. Ultrasonic reflective intensity of BMF-200 A/B Ultrasonic Diagnostic Instrument.
Table 3.
Interested calculations, such as PSF, MTF etc.
Fig. 5. Work flow chart for building the individual eye model.
Each several part of the eye’s axial lengths for HQ’s right eye (units: mm)
Eye
Corneal depth
Depth of anterior chamber
Depth of lens
Depth of vitreous body
Pupil size
Value
0.57
3.03
3.51
18.62
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model we built is a static model and cannot show the eye’s accommodation which is a very complex process. If we can measure the corresponding data with some relative static accommodations, it will be possible to build models for the eye’s different accommodations. Though they can show different status of the eye, these models will be still static. We also built some individual models for the people whose vision was normal or only ametropic. We did not do experiments for pathological eyes. That is, we just care much about the optical system of the eye and do not care about neural or psychological effects. For this kind of subject, good optical system of the eye perhaps shows excellent retinal image quality but does not mean good vision. For ophthalmologists, the individual eye model can help devise appliances to improve subjects’ vision. And the most important application is in the field of the popular corneal ablation surgery. The ablation profile of the corneal ablation surgery guided by the wavefront and topography can be calculated accurately from the model by changing the corneal anterior surface, and this is the succeeding work.
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0.4 0.3 0.2 0.1 0.0 0.000
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References
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This work is supported by the National Natural Science Foundation of China (Key research project, No. 60438030), the Chinese Ministry of Education’s Nankai University, Tianjin University Cooperation Foundation and the Key Research Foundation of Scientific and Technical Committee of Tianjin City of China (No. 033183711).
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Fig. 6. The optical system’s quality calculated from individual eye model. (a) Layout of HQ’s individual model. (b) Spot diagram on the retina (units: mm). (c) diffraction encircled energy plots. (d) The tangential sagittal MTF plots.
the eye The and
4. Discussion The individual eye model is custom-built for individual people. It has its own corneal surfaces, axial lengths and the individual wavefront aberrations. Here, the
[1] R. Navarro, J. Santamaria, J. Bescos, Accommodationdependent model of the human eye with aspherics, J. Opt. Soc. Am. A 2 (1985) 1273–1281. [2] J. Warren Blaker, Toward an adaptive model of the human eye, J. Opt. Soc. Am. 70 (1980) 220–223. [3] W. Lotmar, Theoretical eye model with aspherics, J. Opt. Soc. Am. 61 (1971) 1522–1529. [4] A.C. Kooijman, Light distribution on the retina of a wide-angle theoretical eye, J. Opt. Soc. Am. 73 (1983) 1544–1550. [5] H.-L. Liou, N.A. Brennan, Anatomically accurate, finite model eye for optical modeling, J. Opt. Soc. Am. A 14 (1997) 1684–1695. [6] J. Liang, B. Grimm, S. Goelz, J.F. Bille, Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor, J. Opt. Soc. Am. A 11 (1994) 1949–1957. [7] W. Quan, Z.Q. Wang, G.G. Mu, L. Ning, Correction of the aberrations in the human eyes with SVAG1 thin-film
ARTICLE IN PRESS H. Guo et al. / Optik 116 (2005) 80–85
transistor liquid-crystal display, OPTIK 114 (2003) 467–471. [8] J. Liang, D.R. Willams, Aberrations and retinal image quality of the normal human eye, J. Opt. Soc. Am. A 14 (1997) 2873–2883. [9] E. Moreno-Barriuso, R. Navarro, Laser Ray-Tracing versus Hartmann-Shack sensor for measuring optical aberrations in the human eye, J. Opt. Soc. Am. A 17 (2000) 974–984.
85
[10] P.M. Prieto, F. Vargas-Martin, S. Goelz, P. Artal, Analysis of the performance of the Hartmann-Shack sensor in the human eye, J. Opt. Soc. Am. A 17 (2000) 1388–1398. [11] J.F. Bille, U. von Page, The development of wavefront technology and its application to ophthalmology, In: J.F. Bille, C.F.H. Harner, F.H. Loesel (Eds.), Aberration-Free Refractive Surgery, second enlarged ed., Springer, Berlin, 2003, 2004, pp.1–23.