Journal of China University of Science and Technology Vol.49-2011.10
Design the Compact Astigmatic Lens by even Aspheric
Toroidal Surface and Optimizing Mass of Lens
重量最佳化與偶次非球面複曲面的精巧型散光鏡片設計
陳德請 1 黃光榮 2 李世文 3 卓葆軒 4
Der-Chin Chen1, K-L Huang2, Shih-Wen Lee3, Bao-Xuan Zhuo4
1
逢甲大學電機工程系副教授
2
3
4
明道大學光電工程研究所
中華科技大學電機工程系助理教授
逢甲大學電機工程系大學四年級學生
Department of Electrical Engineering of Feng Chia University
Institute of Electro-Optical Engineering of Mingdau university
3
Department of Electrical Engineering, China University of Science and Technology
ABSTRACT
This paper studies the design of astigmatic eyeglass lens by using even aspheric toroidal
surface, multi-configuration and optimizing the mass of eyeglass lens. The innovation
method will provide the best solution of correcting astigmatism and minimum edge
thickness of eyeglass lens. Making a human eye modal with astigmatism and the
multi-configuration was defined three distances that are far, middle and near,
10.0e+09mm ,1000mm and 500mm respectively, according to the human eye’s adaptive
function with three different view angles. The designing result of 1.0 D astigmatic eyeglass
lens can largely correct astigmatism and the Modulation Transfer Function (MTF) is 0.3 at
30 lp/mm spatial frequency. And super thin and compact astigmatic eyeglass lenses are
designed. This innovation design is custom design solution.
Key Word :toroidal surface, eyeglass, MTF
摘 要
本文研究目的係使用重量最佳化與偶次非球面複曲面設計一精巧型散光眼鏡鏡
片。此種鏡片優點具超薄型重量輕，解像率又高。研究方法係採用 ZEMAX 光學設計
15
Design the Compact Astigmatic Lens by even Aspheric Toroidal Surface
and Optimizing Mass of Lens
程式的多重組態對無限遠、1000mm、500mm，3 種視場(View Field)進行設計，使具
散光眼患者在配戴精巧型散光眼鏡後也適合看遠、中、近距離。除了可矯正-1.00D 散
光之外，還可以使配戴精巧型散光眼鏡之後的視網膜解像率在 30 lp/mm 的 MTF 值達
0.3。
關鍵字：複曲面，眼鏡，解像率
I.INTRODUCTION
A corrective lens is a lens worn in front of the eye, usually used to treat myopia,
hyperopia, presbyopia, and astigmatism. Spectacles are worn on the face a short distance in
front of the eye. Myopia (near-sightedness) requires a divergent lens, whereas hyperopia
(far-sightedness) requires convergent lens [1][2]. Astigmatism eye caused by the cornea
that lacks a prefect spheric surface has two different diopters at vertical and horizontal axis.
Astigmatic eyes can be corrected via usage of astigmatic eyeglasses with two different
diopters. In general, astigmatic eyeglasses that utilize the cylinder surface enable the
process of production at a lower cost. The raw material of eyeglass has several diopters of
lens-front surface, and eyeglass manufacturers get the diopters of patient’s astigmatism
from glasses shops to compute the diopters of the lens-back surface and then produce
astigmatic eyeglasses. The spheric cylinder surface is unsatisfactory because people with
astigmatism cannot look far or near clearly. Lens manufacturers prefer to use aspheric
lenses improving vision instead of traditional spheric cylinder lenses. It is true that aspheric
lenses are used in cameras and binoculars. Therefore, the aspherics/toroidal are a sign of
good optics in eyewear. We had published about the design of astigmatic lenses by
ZEMAX optical software in the previous issue of this journal[3]. The designing result of
1.0D astigmatic lens can obviously correct astigmatism and the MTF is more than 0.3 at 30
lp/mm. This paper studies the design of toroidal eye-glasses for astigmatic eye by using
even aspheric toroidal surface and optimizing the mass of eyeglass lens. The designing
result of -1.0D astigmatic eyeglass lens can correct astigmatism and the total MTF is 0.3 at
30 lp/mm spatial frequency. The center thickness of the even aspheric toroidal astigmatic
lens can be reduced from 3 mm to 1.75mm, that is, 1.25mm thinner than that of the
spherical lens; the X-edge and the Y-edge thickness are 2.00mm and 0.99mm respectively.
Also, the accuracies of both design methods are fit for the requirements in the industry. An
aspheric design can reduce the amount of induced optical distortions. This study use even
16
Journal of China University of Science and Technology Vol.49-2011.10
aspheric surface to design an astigmatic eyeglass lens, and satisfy the human eye’s far,
middle and near view angles and distances.
II. Principle
The eyeglass is a single lens with negative or positive power to correct myopia and
hyperopia. Myopia is due to the parallel light entering the eye’s refraction system in a way
that the light is imaged in front of the retina, causing the image blurred. Therefore, we have
to add a negative lens in front of the eye to let the parallel light enter eyes, which move the
image to focus at the retina. Hyperopia is due to the parallel light that is imaged in back of
the retina. A positive lens lets the light be imaged at the retina. The focal length of a lens in
single refractive medium can be calculated from the lensmaker's equation:
n
1 1 t (n '  n )
 (n '  n )[  
]
f
r1 r2
n 'r1r2
(1)
Where f is the focal length of the lens, n is the refractive index of lens material, t is the
center thickness of the lens, r1 and r2 are the front and back radius of curvature of the lens.
If t is small compared to r1 and r2, then the thin lens approximation can be made. For a lens
in air, f is then given by
F  F1  F2
(2)
Where total power F = (n-1)/f, front surface power F1= (n-1)/r1 and back surface power
F2= (n-1)/r2. Its total power is equal to the sum of the front and back surface power.
The astigmatic eye has two different radiuses of curvature at the two orthogonal axes of
cornea [4]. We create the model of the astigmatic eye by the toroidal surface and Liou &
Brennan 1997 eye model and then design the compact even aspheric toroidal astigmatic
lens. The toroidal surface is defined an aspheric surface with two radiuses of curvature at
the orthogonal axes (yz and xz). The aspheric surface of vertical plane (yz) is defined:
z
cy 2
1  1  (1  k )c 2 y 2
 1 y 2   2 y 4  ...   7 y14
(3)
17
Design the Compact Astigmatic Lens by even Aspheric Toroidal Surface
and Optimizing Mass of Lens
where the z is the sag of surface, c is the curvature(c=1/r) (Shown in Fig.1), k is the
conic constant, the y is distance from surface to the optical axis and αi is the ith aspheric
coefficient.
The surface of horizontal plane (xz) is sphere with radius of curvature R as shown in
Fig.2.
Fig.1 The radius of curvature (r) at vertical plane (yz)
Fig.2 The radius of curvature (R) of at horizontal plane (xz)
If the index n, focal length f, shape of back surface radius of curvature (r2) and front
surface radius of curvature (r1) have known, the center thickness (t) is calculated by
equation (1) and (2). The size of astigmatic lens will optimize for ultra light and minimum
thickness at basic requirement of the structure, resolution, and power.
The volume of the eyeglasses lens calculated by the follow equation:

S
f ( x, y, z )dxdydz
  f ( X (u, v, w), Y (u, v, w), Z (u, v, w)) | J (u, v, w) | dudvdw
T
18
(4)
Journal of China University of Science and Technology Vol.49-2011.10
Where
 X
 u
 X
J (u, v, w)  
 v
 X
 w
Y
u
Y
v
Y
w
Z 
u 
Z 

v 
Z 
w 
In optics, Fermat's principle or the principle of least time is the principle that the path
taken between two points by a ray of light is the path that can be traversed in the least time.
We can use this principle and ray trace method to cancel the unnecessary light path to
reduce the lens material or the mass. The small size of eyeglasses lens is proportional to
small mass at same optical glass material of eyeglasses lens. Using TMAS subroutine in
ZEMAX, we can make the lens lighter and thinner by this optimization method.
III. DESIGN EXAMPLE
This research designed an optical power of Astigmatic eyeglass lens; the flow chart of
design is follow:
Step 1: Use ZEMAX optic design software and spheric eye data make a sphere eye
model. The parameter of human eye is consulting the Liou & Brennan 1997 eye model.
This normal eye model includes cornea, aqueous, pupil, crystalline lens, vitreous fluid, and
retina as shown in Fig.3.
Step 2:We choose a -1.0D astigmatic eye with X: -1.00D and Y: 0.00D, by changing the
cornea to the toroidal surface with two radiuses of curvature: 7.77mm (vertical plane (yz))
and 7.617mm (horizontal plane (xz))；other parameters are not changed in normal eye
model. Astigmatic eye model is created and the data was shown in Table 1. By the
Modulation Transfer Function (MTF) we know it horizontal MTF value is smaller than 0.3
at spatial frequency is 30 lp/mm.
Step 3: And then, add -1.0D corrective astigmatic lens and coordinate break (shown in
Table 2.), set the lens-front radius of curvature of corrective lens to 400 mm and set the
lens-back’s surface to toroidal surface with vertical plane’s radius of curvature 400 mm and
horizontal plane’s radius of curvature 250 mm.
Step 4: Considering human eye can adapt to the change from far to near object distances,
we set the object at 1.00e+09mm, 1000mm, and 500mm, and 0°, -10°, and -20° three field
of views let the light into the eye, there is three configurations( shown in Fig.4)[5][6].
19
Design the Compact Astigmatic Lens by even Aspheric Toroidal Surface
and Optimizing Mass of Lens
Step 5: Set the lens-back surface’s radiuses of curvature, conic constant and six aspheric
coefficient terms to variable. Then use MTFA, MTFT, and MTFS… to optimize the lens.
Fig.3 the eye model
Table.1 Astigmatic eye model Lens data
Surface Type
OBJ
1
2
Sphere
Sphere
Toroidal
Radius of curvature
(mm)
Infinity
Infinity
7.77⊥(7.617‖)*
Thickness
(mm)
1.00e+09
50.00
0.55
Index and abbe
number
Semi-diameter
(mm)
Comment
Nd=1.38,ν=50.2
5
Cornea
3
Sphere
6.40
3.16
Nd=1.34,ν=50.2
5
Aqueous
STO
Sphere
Infinity
0.00
Nd=1.34,ν=50.2
1.25
Pupil
5
Gradient 3
12.4
1.59
5
6
Gradient 3
Infinity
2.43
5
7
Sphere
-8.10
16.23883
Crystalline
lens-front
Crystalline
lens-back
Vitreous
Nd=1.34,ν=50.2
5
IMA Sphere
-12.00
5
Retina
* 7.77mm is the radius of curvature at vertical plane (yz); 7.617mm is the radius of curvature at horizontal
plane (xz)
20
Journal of China University of Science and Technology Vol.49-2011.10
Fig.4 Layout of the astigmatic lens at (a) far, (b) middle, and (c) near vision
IV. RESULT AND ANALYSIS
After optimized, the astigmatism eye was corrected by astigmatic lens. Table.2 shows
the lens data of astigmatic lens, and Fig 5, 6, and 7 shows the MTF value after optimized at
three visions respectively. The astigmatic eyeglass lens can correct astigmatism and the
MTF is over than 0.3 at 30 lp/mm spatial frequency. By optimizing mass of lens, the center
thickness of lens can be reduced to 1.75mm; the X-edge and Y-edge thickness are 2.00mm
and 0.99mm, respectively. In Fig.5 6,and 7, the MTF of tangential plane is superior to the
MTF of saggital plane, because the tangential plane of toroidal surface uses the asphereic
surface. The near vision has blur spot produced by defocusing at about 20 lp/mm spatial
frequency so the MTFS is bad as shown in Fig7. The total MTF is in excess of 0.3 so it still
fit performance requirement of vision.
Table.2 The lens data of the astigmatic lens (after optimized)
Surface Type
Radius of curvature (mm)
2
Sphere
400.00
3
Toroidal Surface
370.32 (Default 400)
Toroidal
Conic
α4
Surface’s aspheric constant
coefficient
-570.89
-21.9e-006
(tangential plane )
Thickness(mm)
1.75
28.00
α6
1.37e-009
21
α8
1.24e-011
Glass
Radius of curvature (mm)
Polycar.
0.00
267.55 (Default 250)
α12
α14
α10
4.43e-014
5.85e-017
-8.63e-019
Design the Compact Astigmatic Lens by even Aspheric Toroidal Surface
and Optimizing Mass of Lens
Fig.5 The MTF at the far vision
Fig.6 The MTF at the middle vision
Fig.7 The MTF at the near vision
22
Journal of China University of Science and Technology Vol.49-2011.10
V. CONCLUTION
This paper studies the design of toroidal eye-glasses for astigmatic eye by using even
aspheric toroidal surface , multi-configuration and optimizing the mass of eyeglass lens.
The aspheric toroidal lens-back surface of astigmatic lenses can not only correct the
astigmatic problem, but also get the best MTF. The designing result of -1.0D astigmatic
eyeglass lens can correct astigmatism and the total MTF is 0.3 at 30 lp/mm spatial
frequency. By the even aspheric toroidal surface and optimizing the mass of eyeglass lens,
the center thickness of the astigmatic lens can be reduced from 3 mm to 1.75mm, that is,
1.25mm thinner than that of the spherical lens; the X-edge and the Y-edge thickness are
2.00mm and 0.99mm respectively. This paper can offer innovative design method to
correct astigmatism eye. And super thin and compact astigmatic eyeglass lenses are
designed. Based on your statement of astigmatism eye, we develop a solution that meets
form, fit, and function specifications while satisfying your visual defect requirement.
Reference
[1] Yung-Feng Shih, Luke L-K Lin, Por-Tying Hung, “Studies of Ocular Biometry in
Taiwan”, Journal of Medical Ultrasound, Volume 15, Issue 1, pp.9-18, 2007.
[2] DarryI J Meister, Scott W Fisher, “Progress in the spectacle correction of presbyopia.
Part 1:Design and development of progressive lenses” , Clin. and Exp. Optom. Vol.91,
No. 3, pp. 240-250, May, 2008.
[3] Der-Chin Chen, Shang-Wei Hsieh ,Shih-Wen Lee, “ The Design of Astigmatic Lenses
by ZEMAX Optical Software” ,Journal of China University of Science and
Technology ,Vol.46, pp.77-86 ,April ,2011.
[4] Frank L. Pedrotti, S.J., Leno M. Pedrotti, Leno S. Pedrotti, “Introduction to Optics”,
Pearson Prentice Hall, pp. 433-434, 2007.
[5] Agnieszka Barcik, Damian Siedlecki, “Optical performance of the eye with progressive
addition lens correction”, International Journal for Light and Electron Optics , Vol.121,
Issue 21, pp.1937-1940, November, 2010
[6] Muhammad Nadeem Akram and Muhammad Hammad Asghar, “Wavefront aberrations
in the accommodated human eye based on individual eye model”, Journal of Applied
Optics, Vol. 42, Issue 13, pp.2312-2316, 2003.
23