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Design of Spherical Aberration Free Aspherical Lens by use of Ray Reverse Tracing Method Han Seob KIM([email protected]), Kyu Yeol PARK([email protected]), Won Kyu LEE([email protected]), Jeong Up JEON ([email protected]), Sung Tae PARK([email protected]) ABSTRACT In this study, aberration free aspherical lens design method named ray reverse tracing method is introduced. Differently from the traditional design method, the ray reverse tracing method traces the shape and location of a real object by use of its virtual image. From the result, especially spherical aberration free aspherical lens could be designed by use of the ray reverse tracing method. Furthermore, it could reduce the degree of dependence of optical characteristics on designer’s ability, because deformation terms and optimization can be eliminated, which has been performed in conventional lens design process. Key Words : Spherical Aberration, Aspherical Lens, Ray Reverse Tracing Method, Deformation term 1. Introduction Various studies are going on in the field of material technology and design technology etc. for lenses which ranges from small-sized one that is used in endoscope for surgical operation of internal treatment to large-sized one that is used in astronomical telescope, in order to get more accurate, clear and better image. The most important function required from optical lense is their optical role that embodies high precision image exactly same as real object. But there is an undesirable factor in lens characteristics so called aberration that hinders lens features. When defining an element that transforms a point in an object space to the corresponding one in an image space as a lens or an optical system, the idealistic optical system is one that transforms exactly 1:1. And the aberration is the phenomenon (1) that the real optical system deviates from the idealistic optical system. Thus, in order to improve the optical system performance, many studies on removal or reduction

Design of Spherical Aberration Free Aspherical Lens

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In this study, aberration free aspherical lens design method named ray reverse tracingmethod is introduced. Differently from the traditional design method, the ray reversetracing method traces the shape and location of a real object by use of its virtual image.From the result, especially spherical aberration free aspherical lens could be designed byuse of the ray reverse tracing method. Furthermore, it could reduce the degree ofdependence of optical characteristics on designer’s ability, because deformation terms andoptimization can be eliminated, which has been performed in conventional lens designprocess.

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Page 1: Design of Spherical Aberration Free Aspherical Lens

Design of Spherical Aberration Free Aspherical Lens

by use of Ray Reverse Tracing Method

Han Seob KIM([email protected]), Kyu Yeol PARK([email protected]),

Won Kyu LEE([email protected]), Jeong Up JEON ([email protected]),

Sung Tae PARK([email protected])

ABSTRACT

In this study, aberration free aspherical lens design method named ray reverse tracing

method is introduced. Differently from the traditional design method, the ray reverse

tracing method traces the shape and location of a real object by use of its virtual image.

From the result, especially spherical aberration free aspherical lens could be designed by

use of the ray reverse tracing method. Furthermore, it could reduce the degree of

dependence of optical characteristics on designer’s ability, because deformation terms and

optimization can be eliminated, which has been performed in conventional lens design

process.

Key Words : Spherical Aberration, Aspherical Lens, Ray Reverse Tracing Method,

Deformation term

1. Introduction

Various studies are going on in the field of material technology and design technology

etc. for lenses which ranges from small-sized one that is used in endoscope for surgical

operation of internal treatment to large-sized one that is used in astronomical telescope, in

order to get more accurate, clear and better image. The most important function required

from optical lense is their optical role that embodies high precision image exactly same as

real object. But there is an undesirable factor in lens characteristics so called aberration

that hinders lens features. When defining an element that transforms a point in an object

space to the corresponding one in an image space as a lens or an optical system, the

idealistic optical system is one that transforms exactly 1:1. And the aberration is the

phenomenon(1) that the real optical system deviates from the idealistic optical system. Thus,

in order to improve the optical system performance, many studies on removal or reduction

Page 2: Design of Spherical Aberration Free Aspherical Lens

of the aberration effect are going on. As a typical example, various methods such as an

analytic method(2) that applies higher degree of order in the mathematical expression of the

aberration and materialistic method(3) that uses new material for one lens to have various

refractive indices are presented. The most widely used method is the one that uses lens for

aberration correction. But, because this method using multiple lenses has some demerit that

the whole system becomes complicated and large-sized, it has its limitation today when

speculating the trend that micronization and light-weightization of the optical system is

rapidly realized.

At present, most optical system changes from spherical lens optical system to aspherical

lens optical system(4)(5), due to the trend of system micronization and necessity of high-

function and high quality including aberration correction. Aspherical lens optical system

has many advantages that it removes spherical aberration by improving focusing

performance and does not need complicated lens combination but still has high

permeability. However, because it has many difficulties in designing and manufacturing,

use of spherical lens that has approximate solution has been preferred(6). In recent years,

instead of the difficulties in designing and manufacturing, study on the application of the

shape of the aspherical lens and its optical merits in many other fields is actively being

proceeded. As a typical example of special use of an aspherical lens, in order to improve

performance and function of high-tech military equipment, DARPA (Defense Advanced

Research Projects Agency) is carrying out a study on the accuracy enhancement in

information by maintaining broader visual field and reducing the aberration from changing

the traditional shape of canopy part of an airplane or head part of a missile which usually

obtains forward information into aspherical shape(7).

Because, differently from spherical lens, aspherical lens is not defined in one curvature,

it has design difficulties. Even though it is an axis-symmetric aspherical lens that is in

general use, it is inevitable to apply design equation for the complicated aspherical surface,

and individual coefficients used in the equation are very important factors that affect the

lens characteristics. And optimization procedure required when designing lens using

commercial software also has great affects on lens characteristics. But, instead this

optimization procedure and selection of individual coefficients play as very important

factors in lens design, these factors are arbitrarily determined by the designer relying on

the designer’s experiences. Thus, eventually, this has the problem that the lens

Page 3: Design of Spherical Aberration Free Aspherical Lens

characteristics have to be determined differently depending on designers’ experience and

capability though the initial values for lens design are identical to each other.

Based on this background, in this study, in order to avoid the traditional design method that

mainly depends on the designer’s experience, a new design method named ‘ray reverse

tracing’ method was devised. By using the ray reverse tracing method, it was confirmed

that it was possible to remove the spherical aberration of a spherical lens, and it was also

possible to design a spherical aberration free aspherical lens.

2. Spherical Aberration and Ray Reverse Tracing Method

2.1 Spherical Aberration

Generally, there are various kinds of aberration in spherical lens of optical devices.

Among the various aberrations that could be found in an optical system, spherical

aberration is especially regarded as the most important factor that determines the

performance of the optical device. Fig. 1 shows the effects of spherical aberration. Rays of

light parallel incident along the optical axis of a lens pass the lens and gather on one point,

which is called a focus. But if the spherical aberration of the lens exists, the lens focus can

not form geometrically a single point in a space and has a specific region called focal

region as shown in the figure 1/(b). This phenomenon is caused by the fact that the incident

angle of the ray incident onto the spherical lens is getting larger and larger, as the incident

ray parallel to the optical axis gets far away from the axis, comparing with that of the

incident ray nearer to the axis. In this case, the relation between the incident angle and the

refractive angle can be explained by the Snell’s law(8).

Snell’s law is an equation consisting of incident angle of a ray incident onto a lens

surface, its refractive angle and index of refraction of lens material. The bigger the

diameter of a spherical lens is, the bigger the spherical aberration is. And the size of the

spherical aberration directly influences the size of the focus. Therefore, in a design of a

Spot image

Fig. 1 Illustration of spherical aberration

(a) Spherical aberration free

Spot image

(b) Spherical aberration Focal region

Page 4: Design of Spherical Aberration Free Aspherical Lens

high-function and high-quality lens, the spherical aberration that has great influences on

the image forming function of the lens must indispensably be corrected.

2.2 Ray Reverse Tracing Method

In geometrical optics, as a traditional method to draw lens multiplying factor and the

location of an image, ray tracing method has been used(8)(9). Fig. 2 shows the basic

principle of the ray tracing method. This method is a simple one to determine the size and

the location of an image made through the lens using the lens focal length that is known

when the size and the location of an object vary, and has rules as follows.

1.Ray of light ? starting from the object, parallel incident upon a lens, passes F2 .

2.Ray of light ? starting from the object and passing F1 proceeds parallel to optical

axis after passing through the lens.

3.Ray of light ? starting from the object and passing the center of lens proceeds

straight without refraction.

Rays of light starting from the object pass the lens and form the image of the object

depending on a focal length, the distance between the object and the lens, and the size of

the object. In ray tracing method, all the rays of light starting from the object are used to

Fig. 2 Principle of ray (reverse) tracing method

(b) Ray reverse tracing method

Object

Image Optical axis

lens

Ray

Ray

A

α`

F

Ray

nSinα=n`Sinβ

nSinα`=n`Sinβ

α

(a) Ray tracing method

F1

F2

Object

Image

Optical axial

Lens

Focal point

Ray ?

?

?

? ’

? ’

? ’

Page 5: Design of Spherical Aberration Free Aspherical Lens

determine the location and the size of an image formed by the lens.

But, in this study, ray reverse tracing method was devised for correction of the

spherical aberration that is inherent to the spherical lens and for generation of aspherical

surface. In this method, an intended virtual image is first made depending on the optical

characteristics of the lens used, and then all the rays of light start not from the object but

from the virtual image and form the object after passing the lens. The principles of the ray

reverse tracing method can be explained as follows. When making an image, at a point on

F, of an object at infinite distance using an arbitrary lens, an arbitrary line is drawn from

the point F to the lens surface. When the end point of the line on the lens surface is denoted

as a point A, line AF is a refractive ray of the lens and the angle between the normal line

at A and the line AF is a refractive angle. By substituting the refractive angle derived

from this and the lens index of refraction into the Snell’s expression, the incident angle

between the lens and the object and the incident ray can be found. But, in this case, if the

lens has aberration, the incident ray with the incident angle that was found from the

refractive angle and the refractive ray that connects the virtual image and a point on the

lens surface will not pass the corresponding point on the object. Therefore, in order for the

incident ray with the incident angle found form Snell’s law to pass exactly the

corresponding point on the object, the curvature of the lens surface must inevitably be

changed. In this study, on purpose of removing the spherical aberration of a lens and

designing the shape of the aspherical lens surface by use of these optical characteristics,

the ray reverse tracing method was devised and utilized.

F

A1′ A1

p1’ P p1

Fig. 3 Spherical aberration removal method(Ray reverse tracing)

O F’

h

R

B

Page 6: Design of Spherical Aberration Free Aspherical Lens

3. Removal of spherical aberration and generation of an aspherical surface

3.1 Removal of spherical aberration

As described above, the spherical aberration in an optical system, due to which parallel

rays having passed through a lens do not converge into one point called a focus of the lens,

can be the cause of deterioration in an optical characteristic such as resolution. But, in the

case of the lens whose aberration is completely removed, all the parallel rays having

passed through the lens converge into one single point. In ray tracing of an object at a finite

distance from a lens, various rays with various incident angles from the object passing

through the lens finally forms an image exactly same as the shape of the object. On the

other hand, if the object is at infinite distance from the lens, all incident rays from the

object onto the lens will be regarded as parallel and form a point image that is same as the

focus of the lens. Therefore, in this study, by using these principles and applying the ray

reverse tracing method, an aberration removal method was suggested.

Fig. 3 shows spherical aberration removal principles using the ray reverse tracing

method. In the case of a spherical lens, its focal length near its radical axis zone can easily

be found from mathematical expression with its radius of curvature and index of refraction

only. The method of spherical aberration removal of an arbitrary lens whose focal length is

already known is as follows. First, a line FA1 is drawn from the lens focus(F) lying on the

optical axis to an arbitrary point(A) on the lens surface where the spherical aberration

removal is required. The line FA1 is a refractive ray between the lens and an image.

Because the point is located on the lens surface, when a line RA1 is drawn from the point

to the center of curvature of the lens, it becomes normal to the lens surface at A1. Thus, the

( )

refraction ofIndex :

(3) tan)(tan90sintan90sin

(2) tan9090

(1) tan)(tan90 90

90180

)(tan

111

1

11

1

n

ph

pOFhnp

h

phAORORA

ph

pOFhAOROFAFAR

AORORAARF

pOFhOFA

−−°=

−=∠−°=∠

−−°=∠−∠−°=∠

∠+°=∠−°=∠

−=∠

−−−

−−

Page 7: Design of Spherical Aberration Free Aspherical Lens

angle ∠FA1R between the line and the refractive ray becomes a refractive angle of the lens

at A1. An incident angle can be found using a index of refraction of the lens and the

refractive angle shown above. But, because the lens has spherical aberration, it is

impossible to draw an incident ray between the lens and the object using the incident angle

found above. To make it possible, the radius of curvature of the lens must be varied

depending on the point on the lens surface.

In the concrete, in order to correct the spherical aberration using the ray reverse tracing

method, the spherical aberration of the lens must be defined first.

To do this, all incident rays must be parallel to an optical axis, and all refractive rays must

converge into one single point. And for the point F to be the focus, a ray starting from an

object and passing through an arbitrary point A1 must be parallel to the optical axis, and the

angle ∠ORA becomes an incident angle at an arbitrary point A1. Here, the incident angle

and the refractive angle at an arbitrary point A1 on the lens surface must satisfy the Snell’s

law. Accordingly, substituting equation (1) and (2) into the Snell’s law will yield the

equation (3). Because the index of refraction (n), the height (h) of an arbitrary point A1 on

the lens surface and the focal length are constant in the equation (3), the spherical

aberration of a lens becomes a function only of p1 the distance from the origin (O) to an

arbitrary point A1 on the optical axis. Therefore, finding out the correct p1’ makes A1

change to A1’ where eventually the spherical aberration is completely removed.

In the case of real practice, very small value is assigned to h, and A1 is let be the origin

of the next point A2 in the removal process of the spherical aberration. By repeating this

process until the accumulated value of h becomes the same as the radius of a lens, the

spherical aberration all over the lens surface can be removed. When this process is finished,

a new lens surface called an aspherical surface that is different from the original spherical

surface is generated.

3.2 Design of an aspherical surface

As described before, in order to remove the spherical aberration of a lens by use of the

ray reverse tracing method and Snell’s law, focal length and curvature corresponding to

basic data of a lens used are required. But by application of this process, shape of an

aspherical surface that has no spherical aberration can possibly be designed using the focal

length of a lens only, even though the basic shape of the lens is not yet determined. Here,

Page 8: Design of Spherical Aberration Free Aspherical Lens

the focal length is the design data given basically in design of all lenses. The principle of

shape design of an aspherical lens without spherical aberration using the ray reverse

tracing method and a focal length only is as shown in Fig. 4. F is a focus. A focus (F) is

marked at a distance of a focal length given for design from the origin (O). The line

passing the origin and the focus is an optical axis of the lens for design. As described in the

previous section, in order to apply the ray reverse tracing method with the given condition

on the basic shape of a lens, an arbitrary point on the lens surface is picked at the distance

of h from the origin and Snell’s law is applied to this point. But, when the basic shape of a

lens is not given and the ray reverse tracing method is to be applied using the focal length

only, an arbitrary point A1 is picked at the distance of h vertically from the origin. Here, the

location of A1 has a same coordinate as that of the origin in the direction of the optical axis

means that the surface formed from revolution of line OA1 around the optical axis is

normal to the optical axis. In this case, because the normal ray incident onto this surface

does not refract and proceeds straight forward and plays a role like the optical axis while

A1 changes its location in the course of design, the problem on selection of an initial

location of the point A1 can be neglected. Thus, the line connecting the focus F and an

arbitrary point A1 becomes refractive ray, and the incident ray is produced by drawing a

line passing through the point A1 and parallel to the optical axis. When drawing a line

connecting an arbitrary point A1 and the origin, and drawing another line 11RA passing A1

and normal to the line OA1 , the angle between the line 11RA and the refractive ray becomes

a refractive angle, and the angle between the line 11RA and the incident ray becomes an

incident angle. After this process, in the same manner as in the spherical aberration

removal method described in the previous section, substituting the refractive angle, the

Fig. 4 Illustration of aspherical lens design using ray reverse tracing method

F

h

A1

A2

A3

A4

A5

p1 p2 p3 p4 p5

h

h

h

h

B

R1 R2 O

Page 9: Design of Spherical Aberration Free Aspherical Lens

incident angle, and the index of refraction of

a lens material into the Snell’ law and

expanding it result in an equation (3). It is a

function of the distance between the origin

and the coordinate of A1 in the direction of

optical axis. Therefore, the value of p1 that

satisfies the equation (3) becomes the

coordinate of A1 in the direction of the

optical axis. In the Fig. 4, A1 is a temporary

origin to find out the location of A2, and in

turn A2 is for A3 and so on. By repeating

these process until the accumulated value of

h becomes the same as the design radius of a

lens, the shape of an aspherical lens is designed. It can be decided that applying this

method has advantage in that design process can be standardized in comparison with the

traditional design method which uses complicated and high order equation for aspherical

lens design.

3.3 Design module for aspherical lens

Fig. 5 shows a design module for an aspherical lens to implement on a PC (personal

Computer) the design process of an aspherical lens using the ray reverse tracing method

suggested in this study. In this module, by entering the index of refraction of a lens, lens

diameter and the curvature of a spherical lens, it is possible to design an idealistic shape of

aspherical lens surface free of spherical aberration through the aberration removal process

of a spherical lens.

And, this module is so structured that even in the case of entering the focal length only

without entering the shape data of spherical lens, it is possible to design aspherical lens that

has required optical performance.

3.4 Generation of an aspherical curvature and coefficients

Generally, in the design of an aspherical lens, following design equation for the

aspherical lens has been used. In this equation, k is defined from the basic shape of a lens,

Fig. 5 Aspherical lens design Module

Page 10: Design of Spherical Aberration Free Aspherical Lens

and design variables are vertex curvature and aspherical coefficients. But, these variables

are floating during the course of designing, and finally decided by optimization. In this

paper, design method for an aspherical lens using the ray reverse tracing method was

suggested, and in consideration of the its correlation with a general design method using

the equation mentioned above. An example of the process for derivation of vertex

curvature and deformation term from the aspherical lens shape data was shown in Fig. 6.

Here, an arbitrary aspherical surface of 15mm radius was used.

Fig. 6/(a) shows the process to find out the curvature of the spherical shape closest to a

given aspherical shape. When the curvature of a spherical shape close to an aspherical

lens shape is plotted, the spherical shape that is the closest to the aspherical shape

corresponds to the one that makes the least area by crossing the aspherical shape, and its

curvature is selected as that of the aspherical shape. Here, the selected curve has a radius of

17.82mm, hence the vertex curvature is 0.0561167.

Fig. 6/(b) shows the process of evaluating the error occurred when the aspherical

surface does not perfectly coincide with the spherical surface. The error shows the

difference in y coordinates at the same x coordinate. And Fig. 6/(c) shows the curve type of

the error when drawn in different y scale. It is possible to generate deformation term by use

of the error data obtained in this procedure. That is, if the error data obtained above are

expressed in the aspherical surface design equation aforementioned, the coefficient for

each term in the equation will become the corresponding coefficient in the aspherical

design equation. But, because the aspherical design equation consists only of even order

terms, using only the coefficients of even order terms as they were in the polynomial

obtained from the error curve is inadequate. Therefore, when a polynomial is made using

the error curve, curve fitting is performed with the polynomial consisting only of even

order terms except odd order terms and constants. Then, the coefficient of each term

n termDeformatio : DC,B,A,constant ConiccurvatureVertex

222

(4) ....1086422)1(11

2)(

==

+=

+++++−+

=

kc

yxr

DrCrBrArrck

crrz

Page 11: Design of Spherical Aberration Free Aspherical Lens

obtained can be used as deformation term. By applying this process, it was possible to

evaluate the deformation terms as follows.

A= 0.0000399112 B= -0.0000003442 C= 0.0000000006

Fig.6/(d)is the result of comparison between aspherical shape generated from this

process using the deformation terms and the initial basic aspherical shape. In Fig. 6/(d) as

compared with Fig.6/(a) which shows the shape error of spherical shape that has a radius

closest to that of the initial basic aspherical shape :17.82mm, vertex curvature : 0.0561167,

it can be confirmed that an aspherical shape in which almost no shape error exist is

obtained.

It can be concluded that this result shows that it is possible to calculate the deformation

terms for lens design from the shape data of an aspherical lens, and by use of these terms it

is possible to trace design information on an aspherical lens in an arbitrary shape.

4. Conclusion

In this study, as a removal method of spherical aberration that hinders optical lens

performance, ray reverse tracing method that is new and different from the traditional lens

Aspheric

Sphere

Sphere

Sphere(15.82

Fig. 6 Generation of aspherical curvature and deformation terms

(a) Calculation of basic curvature for aspherical shape

(b) Calculation of error curve

(c) Curve fitting (d) Comparison with aspherical curves

Page 12: Design of Spherical Aberration Free Aspherical Lens

design method was suggested. And it was confirmed that by use of the ray reverse tracing

method, not only removal of spherical aberration that could be found in a spherical lens but

also design of aspherical lens is easy

The conclusion of this study is arranged as follows.

1. It is possible to design an aspherical shape free of spherical aberration by use of the

ray reverse tracing method.

2. In the case of designing an aspherical lens by use of the ray reverse tracing method,

vertex curvature and the deformation terms which have been used as the initial

design data in the traditional design method for an aspherical lens is unnecessary,

and it is possible to design an aspherical shape with a given focal length only.

3. In the design of an aspherical lens using the ray reverse tracing method,

optimization process that is required in the traditional design method can be

omitted, hence the ray reverse tracing method has advantage that the design process

can be standardized.

4. It is possible to calculate the deformation terms used in the lens design from the

shape data of a aspherical lens, and by use of these, it is possible to trace back the

design information about an arbitrary aspherical shape.

Acknowledgement

This study was carried out as a project on commission “development of an intelligent

grinding machine“ managed by KITECH and operated by Ministry of Commerce, Industry

& Energy of the government of Korea.

References

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3. Roman Ilinsky, "Gradient-index meniscus lens free of spherical aberration", J. Opt. A:

Pure Appl. Opt. 2, pp. 449 - 451, 2000.

Page 13: Design of Spherical Aberration Free Aspherical Lens

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8. Hecht, Eugene., "Optics", Addison-Wesley Pub. Co., 2nd ed., pp. 211 - 239, 1987.

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recording with non-spherical waves", J. Opt. A: Pure Appl. Opt. 3 pp. 387 - 397, 2001.