Ophthalmic lens design with the optimization of the aspherical coefficients

Transcription

1 Ophthalmic lens design with the optimization of the aspherical coefficients Wen-Shing Sun Chuen-Lin Tien Ching-Cherng Sun, MEMBER SPIE National Central University Institute of Optical Sciences Chung-Li, 320, Taiwan Ming-Wen Chang, MEMBER SPIE Yuan-Ze University Department of Electrical Engineering Chung-Li, 320, Taiwan Horng Chang Chung Shan Institute of Science and Technology Material R&D Center Taoyuan, 325, Taiwan Abstract. An efficient approach for the designing of ophthalmic lenses, based on the optimization of the aspherical surface coefficients is presented. We show that five optimization constraints are enough to both reduce the aberrations and prevent the development of inflection points along the aspherical surface. In particular, we solve the inflection point problem by counting the second derivative of the sag with respect to the height of the aspherical surface as one of the constraints. The design data indicate that nonconicoidal aspheric lenses can be made thinner and flatter than the equivalent spherical or conicoidal lenses. The traditional damped least-squares methods are used and design results are given for both aspherical negative lenses and aspherical positive lenses Society of Photo-Optical Instrumentation Engineers. [S (00) ] Subject terms: ophthalmic lens; Coddington s equations; oblique astigmatism; inflection point; aspheric surface; optimization method. Paper received Dec. 18, 1998; revised manuscript received June 30, 1999, and Oct. 19, 1999; accepted manuscript received Oct. 28, Introduction In spherical ophthalmic lens design programs, three variables are generally used. These are the curvatures of the two surfaces and the thickness at the center. One surface curvature is used to eliminate oblique astigmatism error 1 OAE, and the other surface must be curved to provide the coincident lens power. As is commonly known, for ophthalmic lenses the center thickness should be as small as possible to reduce the weight. A series of designs have been produced, with related data listed in Table 1 and the related parameters described in Fig. 1. We find that the second surface radius R 2 is maintained within an approximate range at 50 to 60 mm for negative lenses from 1 to 8 D, and approximately 82 to 84 mm for positive lenses from 1 to 4 D. Since R 2 acts to eliminate the same OAE order, a larger R 2 will give a much larger aberration, as shown in Fig. 2, which means that we cannot get a flat lens solution with a reasonably corrected OAE. Jalie 2 used one spherical surface and one hyperbolic surface to obtain thinner and flatter spectacle lenses, but the higher order aspherical coefficient was not considered. Katz 3 studied high power 14 D ophthalmic lenses design with a well corrected OAE, using two aspherical surfaces. Atchison 4 used a single aspherical surface to design 6 D and 6 D lenses, getting a flatter result. However, its viewing angle for the eye was limited to only 27.5 deg. It is generally a requirement of ophthalmic lenses to have a thinner, flatter shape and a larger aperture size. To keep the lens as flat as possible, we made one of the surfaces nearly flat. That is to say, if we define s 1 and s 2 as the sag of the first and the second surface, respectively, s 1 should be equal to 0 for a negative lens and s 2 should be near 0 for a positive lens. To make the edge thickness e of a negative lens thinner, s 2 should be shorter. Because the larger aperture size of an ophthalmic lens usually causes an inflection point, we consider the second derivative value in the design of the aspherical surface to avoid the inflection point. In addition, for a positive lens, owing to cosmetic and visual-psychological purposes, we should make the power of the second surface to be slightly concave. In this paper, we demonstrate that this design goal can be achieved using only one aspherical surface, and the optimization process can be executed with just five variables and five constraints. 5 2 Theory 2.1 Aberrations Figure 3 shows the tangential and sagittal oblique vertex sphere powers of an ophthalmic lens with respect to the center of rotation of the human eye R. The distances are measured from the vertex sphere with a reference surface concentric with the eye s center of rotation. We can see that the radius of the vertex sphere is l 2 A 2 R 27 mm and the slope angle of the emergent ray on the second surface is u 2 30 deg. The distance v D 2 Z is given by v y 2 /sin u 2 l 2. The radius of curvature of the Petzval surface is therefore r p nf, where n is the refraction index of the lens and f is the focal length. The radius of curvature of the far point sphere FPS is l 2 f V, where f V is the back vertex focal length. We can determine the tangential and sagittal oblique vertex powers for each surface of the lens by using Coddington s equations: n s n n cos i n cos i, 2 s r s Opt. Eng. 39(4) (April 2000) /2000/$ Society of Photo-Optical Instrumentation Engineers

2 Table 1 Design results of spherical lenses. 75 mm Diameter Negative Lens Density 1.25 g/cm 3 Lens power Item 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 D First surface R 1 First surface sag s Second surface R 2 Second surface sag s Center thickness (mm) Edge thickness (mm) Axial height (mm) OAE(D) MOE (D) Distortion Mass (g) mm Diameter Positive lens Density 1.25 g/cm 3 Lens power Item 1 D 2 D 3 D 4 D First surface R 1 First surface sag s Second surface R 2 Second surface sag s Center thickness (mm) Edge thickness (mm) Axial height (mm) OAE(D) MOE (D) Distortion Mass (g) n cos 2 i n cos2 i n cos i n cos i t t r T 3 The symbols in the preceding equations are defined as follows: i is the incident angle; i is the refractive angle; n and n are the refractive indices of both sides, respectively; s is the distance from the object point of the sagittal surface to the vertex sphere; s is the distance from the image point of the sagittal surface to the vertex sphere; r s is the radius of the sagittal curvature; t is the distance from the object point of the tangential surface to the vertex sphere; t is the distance from the image point of the tangential surface to the vertex sphere; and r T is the radius of the tangential curvature. In ophthalmic lens design, the OAE Ref. 1, and the distortion are primarily considered. The object distance is set to be at infinity. First, free from the OAE, a point-focal lens has aberration properties as follows: Fig. 1 Structure of (a) negative and (b) positive lenses. OAE F T F S 0, 4 Optical Engineering, Vol. 39 No. 4, April

3 Fig. 4 Ophthalmic lens distortion. F T 1/f TV, F S f SV, F V 1/f V, f TV ZT and f SV ZS. Fig. 2 Initial optimization value: (a) lens shape for 6 D, (b) back vertex power versus oblique angle, (c) lens shape for 4 D, and (d) back vertex power versus oblique angle. Next, we consider the distortion, which is defined as the difference in height between the ideal image and the actual image. The distortion produced by a positive lens is shown in Fig. 4. Here H is the first principal point and H is the second principal point. The distortion is the distance Q Q MQ MQ. Using the geometry of the figure, we can obtain the ideal image height MQ and the actual image height MQ with MQ f tan u 1 1 F tan u tan u t/n F 1 F, v 9 MQ f v l 2 tan u 2 1 l 2 F v F v tan u 2, 10 where F is the lens equivalent power, f is the effective focal length, u 1 is the slope angle of the incident ray on the first surface, F V is the ophthalmic lens power, and F 1 is the ophthalmic lens power for the first surface. A normalized distortion is defined as distortion % 100 MQ MQ. 11 MQ Fig. 3 Tangential and sagittal oblique vertex sphere powers. where F T and F S are the tangential and sagittal oblique vertex sphere powers, respectively. This means that the oblique power error MOE is defined as MOE F T F S F 2 V, where Aspherical Surface without Inflection Points The equation for aspherical surfaces can be written as c v y 2 s 1 l Pc 2 v y 2 1/2 By4 Cy 6 Dy 8 Ey 10, 12 where s is the sag, c v is the curvature of lens surface, y is the vertical distance from any point on surface to the axis of revolution, P is the constant term of the conicoidal surface, and B, C, D and E are the high order coefficients of the aspherical surface. 980 Optical Engineering, Vol. 39 No. 4, April 2000

4 As shown in Fig. 3 a generally smooth surface with OAE 0 can be obtained only up to u 2 30 deg. When u 2 is over 30 deg, many high order coefficients for the aspherical surface easily result in surface inflection points, which increase manufacturing difficult. An inflection point is determined by the second derivative value of the sag s with respect to the height y given in Eq. 13. The inflection point exists if this is equal to zero. We can avoid inflection points by controlling the second derivative value d 2 s c v dy 2 1 Pc 2 v y 2 3/2 12By2 30Cy 4 56Dy 6 90Ey Optimization Method The damped least-squares method 6 is usually applied for the optimization of an optical design. A merit function is defined as the summation of the squared values of the weighting differences between the aberrations and their target values m i 1 w i e i t i 2, 14 where m is the total summation number, the w i s are the weighting factors, e i is the aberration and t i is the target value. We use P, B, C, D and E, ineq. 12 as variables. The merit function consists of five terms. The first three are the field of view values, 0.5, 07 and 1.0, which are relative to the OAE, the fourth is the 1.0 field distortion, and the last is the value of d 2 s/dy 2 for DA/2. We define a function f i (x 1,x 2,x 3,x 4,x 5 )as f i w i e i t i, 15 where the subscript i corresponds to the five terms x 1, x 2, x 3, x 4 and x 5, which are the aspherical surface variables corresponding to P, B, C, D and E, respectively. Before optimization, we denote the variables as x 10, x 20, x 30, x 40 and x 50, and the aberrations before the optimization are f 10, f 20, f 30, f 40 and f 50. After the optimization process, we denote the variables as x 1, x 2, x 3, x 4, and x 5 and the aberrations as f 1, f 2, f 3, f 4, and f 5. Here, we define matrix A, in which the elements are A ij f i x j. We then get the equation X A T A p I 1 A T f 0, where A T is the transpose matrix of A, I is a unit matrix, p is a damping factor, and f 0 is the matrix containing the five elements f 10, f 20, f 30, f 40 and f 50. If x and x 0 are the matrices containing the elements x 1, x 2, x 3, x 4 and x 5, and x 10, x 20, x 30, x 40 and x 50, respectively, we obtain x x 0 X. 18 Fig. 5 Spherical ophthalmic lens design for 6 D: (a) design parameters and the results, (b) lens shape, (c) back vertex power versus oblique angle, and (d) distortion versus oblique angle. 3 Examples 3.1 The 6 Diopter Lens Design Figure 5 shows the design data and the results for a 6 D spherical ophthalmic lens. It was made of plastic material polycarbonate PC with a refractive index n If we want to reduce the edge thickness and the axial height, we require the first curvature to be flat, but the oblique astigmatism is then very large, as shown in Fig. 2. The oblique astigmatism error must be completely eliminated by the aspherical surface variables P, B, C, D and E. The data for a negative lens, in Fig. 2, could be used as the initial value. We can optimize the second surface, i.e., the aspherical one, by modifying the variables P, B, C, D and E. The design procedures are as follows. First, to minimize the oblique astigmatic error, we first use the weighting factors, w 1 w 2 w 3 1 and w 4 w 5 0 and the target value t 1 t 2 t 3 0. We then obtain an optimized variable matrix x through Eqs , which contain a new conical constant P and the aspherical coefficients B, C, D and E. Based on the principles described in Section 2 and the optimized matrix, we further obtain the optimization terms e 1 e 2 e 3 0, e % and e The sag, its first derivative and its second derivative with respect to the radius of the lens of the optimized surface, are shown in Fig. 6. From Fig. 6 c, wefind that there is an inflection point at y 25 mm. From Fig. Optical Engineering, Vol. 39 No. 4, April

5 Fig. 6 Presence of the inflection point in the design procedure: (a) sag s versus height y, (b) first derivative of s with respect to y, and (c) second derivative of s with respect to y. 6 b, the variation of the slope ds/dy is anomalous at the same point. The slope changes from positive to negative at y 33.4 mm, where the sag s is minimal or maximal, as shown in Fig. 6 a. This makes it difficult to manufacture the aspherical surface under this condition, so the d 2 s/dy 2 value should be taken into consideration in the optimization design process. Second, we use the fifth term d 2 s/dy 2. Before optimization, we set w 1 w 2 w 3 1, w 4 0, and w 5 1 and the target values t 1 t 2 t 3 0 and t 5 0. After optimization, we obtain e 1 e 2 e 3 0, e % and e ; however, there is still an inflection point. Thus, we change the target value t 5 to Furthermore we obtain better results, e 1 e 2 e 3 0, e %, and e , after the optimization. Third, we consider distortion in the optimization process. We change the value of t 4 to obtain a minimum distortion value without changing the sign of e 5. Before optimization, we set w 1 w 2 w 3 w 4 w 5 1, and the target value is t 1 t 2 t 3 0, t 4 7.5% and t After optimization, we obtain e D, e Fig. 7 Absence of the inflection point after optimization: (a) sag s versus height y, (b) first derivative of s with respect to y, and (c) second derivative of s with respect to y D, e D, e 4 7.5% and e As a result, a smooth surface, as illustrated in Fig. 7, is obtained. The final result of the optimization design process is shown in the following. Iteration 0, Merit function , Iteration 1, Merit function , Iteration 2 Merit function , Iteration Optical Engineering, Vol. 39 No. 4, April 2000

6 Table 2 Comparisons between a spherical lens and an aspherical negative lens of the power 6 D. 75 mm Diameter 6 D Negative Lens ; Density 1.25 g/cm 3, Center Thickness 1mm Comparable items Spherical Surface Aspherical surface Edge thickness (e) mm mm 32 Axial height (ah) mm mm 59 Mass g g 20 The results for a 6 D lens are shown in Fig. 8. Table 2 gives a comparison between a spherical surface Fig. 5 and an aspherical surface Fig. 8. The axial height is reduced by 59%, the edge thickness is reduced by 32%, and the mass is reduced by 20%. Table 3 shows the data for 1 to 8 D aspherical negative lenses obtained through the process described above. For a 1 D aspherical negative lens, we find that if the first surface is a plane, the constant term of the conicoidal second surface is as large as To avoid this, the first surface cannot be flat, so we set the conicoidal surface constant P 10 and F D. Then, we obtain P In the same way, we can change the spherical lens power of the first surface from 2 to 4 D.The powers for the first surface are 1.2, 0.7 and 0.2 D, corresponding to 2, 3 and 4D, respectively. For the design of negative lenses from 5 to 8 D,the first curvature is flat because P 10. A comparison of the results of spherical and aspherical lens design is given in Table 4. The optimization of the aspherical lens enables us to reduce the edge thickness from 7.6 to 33%, while the axial height is obviously reduced from 50 to 75%, and mass is reduced from 16 to 20%, for the various diopters. Fig. 8 Aspherical ophthalmic lens design for 6 D: (a) design parameters and the results, (b) lens shape, (c) back vertex power versus oblique angle, and (d) distortion versus oblique angle. 3.2 The 4 Diopter Lens Design For a positive aspherical ophthalmic lens, the first surface should be aspherical and the second surface flat, so that it becomes thinner and flatter. The optimization process is similar to that for the negative lens, but the edge thickness changes each time. To keep it at 1 mm, the center thickness t s 1 s 2 e must be changed after each optimization. In addition, the lens power changes according to the center thickness. To keep the lens power and the optimized aspherical surface unchanged, the power of the second surface F 2 should be adjusted according to the following formula 1 Merit function , F 1 F 2 F v t/n F 1 19 Iteration 4 Merit function , Iteration 5 Merit function Changing t units in millimeters and F 2 does not cause much variation in the merit function. After a few iterations, we obtain the best optimization value. Figure 9 shows the results for a positive 4 D spherical lens. If we want to reduce the edge thickness and the axial height, the second curvature is chosen to be 0.5 D, as shown in Fig. 2 c, where the center thickness is t 5.44 mm and the edge thickness is e 1 mm. We use these two values as the initial values. In the optimization process, as shown in cycle 1 listed in the following, once the edge thickness is larger than 1 mm, the center thickness of the lens should be changed accordingly to t mm mm mm. In addition, F 2 must be corrected according to Eq. 19. Simultaneously, an edge thickness of e mm is obtained by e t s 2 s 1. Three cycles of the optimization processes are shown as follows. For cycle 1, Optical Engineering, Vol. 39 No. 4, April

7 Table 3 Design results of aspherical negative lenses ( 1 to 8 D). Aspherical Negative Lens (PC Density 1.25 g/cm 3, 75 mm Diameter) Lens power Item 1 D 2 D 3 D 4 D First surface (sphere) Second surface (asphere) Conic constant P Aspherical coefficient B C D Distortion Center thickness (mm) Edge thickness (mm) Axial height (mm) OAE (D) MOE (D) Distortion Mass (g) Lens power First surface (sphere) Item 5 D 6 D 7 D 8 D Second surface (asphere) Conic constant P Aspherical coefficient B C D E Center thickness (mm) Edge thickness (mm) Axial height (mm) OAE (D) MOE (D) Distortion Mass (g) t mm, e mm, Iteration 0, Merit function , Iteration 1, Merit function , Iteration 2, Merit function , Iteration 3, Merit function , t mm, e mm. For cycle 2, t mm, e mm, Iteration 0, Merit function , Iteration 1, Merit function , T mm, e mm. For cycle 3, t mm, e mm, Iteration 0, Merit function , Iteration 1, 984 Optical Engineering, Vol. 39 No. 4, April 2000

8 Table 4 Comparisons between a spherical and an aspherical negative lenses ( 1 Dto 8 D). Item Edge thickness (mm) Axial Height (mm) Mass (g) Lens power 1 D D D D D D D D Fig. 9 Spherical ophthalmic lens 4 D: (a) design parameters and the results, (b) lens shape, (c) back vertex power versus oblique angle, and (d) distortion versus oblique angle. Fig. 10 Aspherical ophthalmic lens design for 4 D: (a) design parameters and the results, (b) lens shape, (c) back vertex power versus oblique angle, and (d) distortion versus oblique angle. Optical Engineering, Vol. 39 No. 4, April

9 Merit function , t mm, e mm. Figures 9 and 10, respectively, show spherical and aspherical designs for a positive 4 D lens. In comparison with the results in Table 5, the center thickness is reduced by 26%, the axial height is reduced by 65%, and the mass is reduced by 30%. Table 6 shows the design results for aspherical positive lenses 1 to 4 D. A comparison between spherical and aspherical positive lenses is given in Table 7. In comparison with the spherical lens, the optimization of the aspherical lens enables the center thickness to be reduced from 15 to 26%, the axial height to be reduced from 63 to 65%, and the mass to be reduced from 15 to 30%, for the various diopters. 3.3 Comparison A comparison of the optimized data between a spherical and an aspherical lens for various powers is shown in Figs. 11 and 12. The results show that an aspherical lens is better than a spherical lens, in terms of the thickness, axial height and mass. It is interesting that we find that the thickness seems to be a linear function of the lens power in Figs. 11 a and 12 a. Therefore, once the lens power is known, the thickness can be obtained through a simple calculation of the linear relationship between the thickness and the lens power, as shown in the figures. Table 5 Comparisons between a spherical and an aspherical positive lens of the power 4 D. 72 mm Diameter 4 D Positive Lens Density 1.25 g/cm 3, Edge Thickness 1mm Lens Items Spherical Surface Aspherical Surface Center thickness (t) mm mm 26 Axial height (ah) mm mm 65 Mass g g 30 Table 6 Design results of aspherical positive lenses ( 1 to 4 D). Aspherically Positive lens (PC density 1.25 g/cm 3, 72 mm Diameter) Lens Power Item 1 D 2 D 3 D 4 D First surface (asphere) Second surface (sphere) Conic constant P Aspherical coefficient B C D E Center thickness (mm) Edge thickness (mm) Axial height (mm) OAE (D) MOE (D) Distortion Mass (g) Table 7 Comparisons between a spherical and an aspherical positive lens ( 1 to 4D). Item Center Thickness (mm) Axial Height (mm) Mass (g) Lens power 1 D D D D Optical Engineering, Vol. 39 No. 4, April 2000

10 Fig. 11 Comparisons between a negative aspherical lens and a spherical lens: (a) edge thickness versus lens power, (b) axial height versus lens power, and (c) mass versus lens power. Fig. 12 Comparisons between a positive aspherical lens and spherical lens: (a) edge thickness versus lens power, (b) axial height versus lens power, and (c) mass versus lens power. 4 Conclusion We proposed an efficient approach to the design of ophthalmic lenses with a thinner, flatter shape and a larger aperture size, using only one aspherical surface. We demonstrated that the design minimizes the oblique astigmatism error, avoids the reflection point and locally minimizes the distortion aberration. The optimization process is executed with just five variables and constraints. Acknowledgment This study is supported by the National Science Council of the Republic of China. References 1. M. Jalie, The Principles of Ophthalmic Lenses, The Association of Dispensing Opticians, London M. Jalie, Ophthalmic spectacle lenses having a hyperbolic surface, U.S. Patent No. 4,289, M. Katz, Aspherical surfaces used to minimize oblique astigmatic error, power error, and distortion of some high positive and negative power ophthalmic lenses, Appl. Opt , D. A. Atchison, Spectacle lens design: a review, Appl. Opt , M. W. Chang, W. S. Sun, and C. L. Tien, The design of ophthalmic lens by using optimized aspheric surface coefficients, Proc. SPIE 3482, D. P. Feder, Automatic optical design, Appl. Opt. 2 12, Wen-Shing Sun received his BS degree in physics from the Chung-Yuan University in 1984 and his MS degree in physics from the Fu Jen Catholic University, Taiwan, in From 1990 to 1992 he was a lens design engineer with the Industrial Technology Research Institute, Hsinchu, Taiwan. Currently he is a PhD candidate at the Institute of Optical Sciences, National Central University, Taiwan. His research interests include lens design and ophthalmic optics. Optical Engineering, Vol. 39 No. 4, April

11 Chuen-Lin Tien received his BS degree in physics from National Chang Hua University of Education in 1985 and his MS degree in physics from the Fu Jen Catholic University, Taiwan, in He was an assistant researcher with the Chung Shan Institute of Science and Technology from 1987 to Currently he is pursuing his PhD degree at the Institute of Optical Sciences, National Central University, Taiwan. His research interests include lens design, optical metrology and thin film. Ching-Cherng Sun received his BS in electrophysics from National Chiao Tung University in 1988 and his PhD in Optical Sciences, National Central University, in 1993 and was then a postdoctor with Optical Sciences Center of National Central University for a half year. He was in the Chinese Air Force beginning in July Immediately after, he became an associate professor in the Electronic Engineering Department of Chine Hsin College of Technology and Commerce for one year. In 1996, he joined the faculty of National Central University, Taiwan, and became an associate professor at the Institute of Optical Sciences. Professor Sun is currently a member of the Optical Engineering Society of ROC, SPIE and OSA. He has been the secretary of the SPIE Taiwan chapter since December His research interests are photorefractive devices, optical information processing, optical metrology and holography. from National Cheng Kung University in 1966, his MS degree in geophysics from the NCU in 1968, and this PhD degree in optics from the University of Arizona in He has been president of Optical Engineering Society of ROC, vice-president of International Commission for Optics (ICO), a member of International Activities Committee of Optical Society of America (OSA), chair of the Taiwan Chapter of SPIE, and a member of Engineering, Science and Technology Policy Committee of SPIE. His current interests include optical design and optical testing, holography and optical information processing. He is a fellow of SPIE and OSA. Horng Chang received his MSc in 1977 and his PhD in 1980 in applied optics from Imperial College, London, where he studied the lens optimization technique with some revisions on the Version 5 program. Since then he has been with the Chung Shan Institute of Science and Technology, conducting design, fabrication and testing of electro-optic systems. His current activities are the promotion of dual use optical technologies, including personal displays, fiber optic gyro, and IrDA. Ming-Wen Chang is professor emeritus with National Central University (NCU), and is a visiting professor at Yuan Ze University, Department of Electrical Engineering. He was a senior scientist in Chung Shan Institute of Science and Technology before coming to the universities. He was a professor with the National Chiao Tung University, Institute of Electro-Optical Engineering, and a professor and director of the Institute of Optical Sciences and of the Optical Sciences Center, NCU. He received his BS degree in physics 988 Optical Engineering, Vol. 39 No. 4, April 2000