7
OPTICAL REVIEW Vol. 3, No. 2 (1996) 128-134 DeterminatiOn Of Initial Variable DeriVatiVe Increments Using Genetic Algorithm in Damped Least-Squares Automatic Lens Design Problem Hiroshi MATSUll and Kazuo TANAKA2 *Imaging Research Center, R~i:D Headquarters, CANON Inc', 3-30-2, Shimomaruko, Ohta-ku, Tokyo, 146 Japan, 'R~~:D Human Resource Development project, R~~:D Headquarters, CANON Inc', 2-7-1, Nishi-Shinjuku, Shinjuku-kz4 Tokyo, 163-07 Japan (Received December 13, 1995; Accepted February 5, 1996) An important factor in performing effective optimization with the damped least-squares method is to establish appropriate initial values for the variable derivative increments prior to starting the optimi2;ation process. It is shown first that the determination of these increments can be treated as a combinatorial problem. Then, a novel method of determining optimum variable derivative increments is developed using a genetic algorithm and the characteristics of the eigenvalues of the Jacobian matrix. Some numerical experiments to show the effectiveness 0L this method are also presented. The proposed method reduces the number of optimization reiterations required to reach a stationary point. Key words : automatic lens design, damped least-squares, genetic algorithm, aspheric lens, photographic lens 1. Introduction The so'called damped least-squares (DLS) methodl) has been widely used in automatic lens design. It requires nurnerical evaluation of the first order derivatives of performance functions with respect to lens parameter variables. Numerical derivative increments for these vari- ables must be determined prior to the automatic optimiza- tion process. The efEciency of optimization of the DLS method depends not only on the darnping factor but also on the variable derivative increments. In several reports such as those written by Wynne and Wormell2) and Matsui and Tanaka3) an effective damping factor has been discussed, and the authors' previous paper4) proposed a method for estimating numerical adequacy of variable derivative increments. We have known only the method proposed by Nash5) for the determination of numerical variable derivative increments. His method is a compromise between the machine precision of a computer used in the calculation and the expression of the function to be optimized, how- ever, it offers no explanations for the theoretical reasonability of the method. Although automatic lens design has been widely used, we have never had any systematic way to specify reasonble numerical derivative increments prior to automatic lens optimization process. Many experienced designers have used empirical methods to specify starting derivative incre- ments. Genetic algorithms6) are a type of combinatorial optimization7) tool. Betensky8) has used the genetic algor- ithm to change and/or select lens types during automatic lens design. There are other optimization techniques which are similar to the genetic algorithm; one is the cluster algorithm which was applied to lens design by Elsner9) and another method referred to as biological evolution was used to find a lens design starting point by Walk and Niklaus.10) The purpose of this paper is to present a method to set adequate variable derivative increments prior to the DLS automatic lens design optimization process using the genetic algorithm. Sorne numerical experiments are also given to show the effectiveness of this proposed method. 2. Analysis 2.1 Formulation of the Problem Here, the previous results3,4,11) on which this study is based are summarized and the characteristics of variable derivative increments are discussed. Lens design is formulated as a minimization problem of a merit function; ip (X)=F(X)TF(X) , (1) where F(X)~R" means performance functions and XE R" denotes variables. The superscript T means matrix transposition. When Eq. (1) is solved by the DLS method, the normal equation is obtained; (ATA+pl)AX= -ATF(XO) ' (2) where A indicates a Jacobian matrix, p signifies a damping factor, I is the unit matrix, and AX is the solution vector. In the evaluation of the zj-th element of Jacobian A, we use finite difference; aFi(X)/ axj ~{ Fi(X+ (5Xj)-Fi(X) } / (~j , (3) since the analytical expressions of derivatives are difficult to obtain in lens design. In Eq. (3), (~j is the j-th variable derivative increment and is given by (~:Xj=[O . . . O (~j O . . . O] . (4) The solution to Eq. (2) is expressed by AX= VAX' (5) with a normal orthogonal matrix V, and the i-th element of ZIX' is given by Ax'i=yi/(e +p) (6) where yi is the i-th element of Y=-STUTF(Xo)' U is a normal orthogonal matrix, S is the diagonal matrix whose 128

Determination of Initial Variable Derivative Increments Using Genetic Algorithm in Damped Least-Squares Automatic Lens Design Problem

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OPTICAL REVIEW Vol. 3, No. 2 (1996) 128-134

DeterminatiOn Of Initial Variable DeriVatiVe Increments Using Genetic Algorithm in Damped Least-Squares Automatic Lens Design Problem Hiroshi MATSUll and Kazuo TANAKA2 *Imaging Research Center, R~i:D Headquarters, CANON Inc', 3-30-2, Shimomaruko, Ohta-ku, Tokyo, 146 Japan, 'R~~:D Human Resource Development project, R~~:D Headquarters, CANON Inc', 2-7-1, Nishi-Shinjuku, Shinjuku-kz4 Tokyo, 163-07 Japan

(Received December 13, 1995; Accepted February 5, 1996)

An important factor in performing effective optimization with the damped least-squares method is to establish appropriate initial values for the variable derivative increments prior to starting the optimi2;ation process. It is

shown first that the determination of these increments can be treated as a combinatorial problem. Then, a novel

method of determining optimum variable derivative increments is developed using a genetic algorithm and the characteristics of the eigenvalues of the Jacobian matrix. Some numerical experiments to show the effectiveness

0L this method are also presented. The proposed method reduces the number of optimization reiterations required to reach a stationary point.

Key words : automatic lens design, damped least-squares, genetic algorithm, aspheric lens, photographic lens

1. Introduction

The so'called damped least-squares (DLS) methodl) has been widely used in automatic lens design. It requires nurnerical evaluation of the first order derivatives of performance functions with respect to lens parameter variables. Numerical derivative increments for these vari-ables must be determined prior to the automatic optimiza-tion process. The efEciency of optimization of the DLS method depends not only on the darnping factor but also on the variable derivative increments. In several reports

such as those written by Wynne and Wormell2) and Matsui and Tanaka3) an effective damping factor has been discussed, and the authors' previous paper4) proposed a method for estimating numerical adequacy of variable derivative increments.

We have known only the method proposed by Nash5) for the determination of numerical variable derivative increments. His method is a compromise between the machine precision of a computer used in the calculation and the expression of the function to be optimized, how-ever, it offers no explanations for the theoretical reasonability of the method.

Although automatic lens design has been widely used, we have never had any systematic way to specify reasonble numerical derivative increments prior to automatic lens optimization process. Many experienced designers have used empirical methods to specify starting derivative incre-

ments. Genetic algorithms6) are a type of combinatorial

optimization7) tool. Betensky8) has used the genetic algor-ithm to change and/or select lens types during automatic lens design. There are other optimization techniques which are similar to the genetic algorithm; one is the cluster algorithm which was applied to lens design by Elsner9) and another method referred to as biological evolution was used to find a lens design starting point by

Walk and Niklaus.10)

The purpose of this paper is to present a method to set adequate variable derivative increments prior to the DLS automatic lens design optimization process using the genetic algorithm. Sorne numerical experiments are also given to show the effectiveness of this proposed method.

2. Analysis

2.1 Formulation of the Problem Here, the previous results3,4,11) on which this study is

based are summarized and the characteristics of variable derivative increments are discussed.

Lens design is formulated as a minimization problem of a merit function;

ip (X)=F(X)TF(X) , (1) where F(X)~R" means performance functions and XE R" denotes variables. The superscript T means matrix transposition. When Eq. (1) is solved by the DLS method, the normal equation is obtained;

(ATA+pl)AX= -ATF(XO) ' (2) where A indicates a Jacobian matrix, p signifies a damping factor, I is the unit matrix, and AX is the solution vector.

In the evaluation of the zj-th element of Jacobian A, we use finite difference;

aFi(X)/ axj ~{ Fi(X+ (5Xj)-Fi(X) } / (~j , (3)

since the analytical expressions of derivatives are difficult

to obtain in lens design. In Eq. (3), (~j is the j-th variable

derivative increment and is given by

(~:Xj=[O . . . O (~j O . . . O] . (4)

The solution to Eq. (2) is expressed by

AX= VAX' (5) with a normal orthogonal matrix V, and the i-th element of ZIX' is given by

Ax'i=yi/(e +p) (6) where yi is the i-th element of Y=-STUTF(Xo)' U is a normal orthogonal matrix, S is the diagonal matrix whose

128

OPTICAL REVIEW Vol. 3, No. 2 (1996)

elements are the singular values of A and ei is the i-th eigenvalue of (A1'A).

By analyzing the relationship between eigenvalues ei, i E { 1,2, . . . ,n}, and the initial value for a damping factor

~), we can derive the relationships3) written by

efrlin<~)<ehaax (7) and

J~) ~~ ehled ' (8) where ehnln and ehlax are the minimum and maximum of eigenvalues, respectively, and ehled is the median of a series

of eigenvalues.

When the variable derivative increments are given, the elements of the Jacobian matrix are evaluated and the elements of eigenvalues are deterrnined. Using Eqs. (7) and (8) it is possible to estimate whether the given deriva-

tive increments are adequate to effective optimization or not. However, we have no way to analytically determine variable derivative increments which provide adequate distribution of the eigenvalues satisfying Eqs. (7) and (8).

The aim of this paper is to establish a method determin-ing (~j, j=1, . . . , n in Eq. (3) which satisfies Eq. (8).

Adequacy of variable derivative increments depends on the lens configuration. They are not required to be extremely exact. Lens designers generally specify these values by integers indicating exponential parts of the variables written in exponential forms. For example, when the variable derivative increment value is believed to lie

between 1><10-8 and lxl0-3, this exponent should be cho-

sen from among

8 -7,-6,-5,-4 3 (9) When there are n variables and each has a chance to be selected from k exponent integers, there are kn combina-tions in total. This is one of the combinatorial problems

and is called a non-deterministic polynomial complete (NP-complete) problem.7) Therefore, we recognize that the

problem to determine the adequate variable derivative increments is a NP-complete problem. 2.2 Determination of Variable Derivative Increments by the Genetic Algorithm

An astronomical period of time is generally required in the NP.complete problem for the computation to reach a strict solution. The genetic algorithm6) is one of the most

powerful tools to solve this kind of problem with high accuracy and within a reasonable time. Here, we propose use of the genetic algorithm to determine the adequate variable derivative increments prior to the lens optimiza-tion process.

The genetic algorithm requires the natural parameter set 0L the optimization problem to be coded as a finite-length string;

sl' s2' ' (10) . . , sn '

A string (chromosome) is composed of features (genes) si which take different values (alleles). Here, features are the

variable derivative increments in the DLS problem and values are the integers of exponents of those individual increments. When the i-th variable derivative increment is

H. MATSUI & K. TANAKA 129

given by

(i~i=ailOj(') (11) where i~{1,2, . . . , n}

j~E{pp+1p+2, . . . , p+q}

p and q; some integers , (12) then, ignoring the mantissa ai, a string is composed of exponents j(i) and is expressed by

j(1), j(2) . . , (13) , . j(n) .

To perform an effective search for a better structure the genetic algorithm needs objective function values associ-ated with individual strings. According to the previous analysis3,4) summarized in Sect. 2.1, the objective function

(OF) value should be higher for a string which satisfies Eqs. (7) and (8). Then, we define OF in the form:

OFV= ~ OF(ei) (14) i=1

where OF(ei)= C (e'-LL), LL<e'<(po

po~LL ' ' OF(ei)= C (e'-UL) p0<e'<UL /~)~UL ' ' '

OF(ei)=0, ei<LL or UL<ei . (15) In Eq. (15) C is a constant and LL and UL are the lower and upper limits of OF(ei), respectively. This objective function is graphically illustrated in Fig. I in which the abscissa is exponentially graduated.

The genetic algorithm starts with the making of a population of plural strings. This population of strings is

produced using a random number generator. An optimization process of the genetic algorithm is

composed of three operators, namely, reproduction/selec-tion, cross-over and mutation.

Reproduction/selection is a process in which individual strings are copied in the population or taken out from the population. According to the objective function value, we have obtained what we should expect; the strings which have the best objective function value are copied, the average ones stay even, and the worst ones die off.

After the reproduction/selection, a cross-over proceeds with the random selection of a mate, random selection of a cross-over site, and the exchange of sub-strings from the

beginning of the string to the cross-over site with the

Objective function

v

c

lue

Objective function

Eigenvalue

LL po UL Fig. 1. The objective function to evaluate individual strings.

130 OPTICAL REVIEW Vol. 3, No. 2 (1996)

corresponding sub-string of the chosen mate. The last process in the genetic algorithm is mutation.

This is the random alternation of the value of a feature in

a string with some probability.

A procedure in which we copy strings with some bias toward the best, mate and partially swap sub-strings and mutate an occasional feature value for good measure is one generation in the genetic algorithm.

Lastly, we should comment on the reusability of the variable derivative increments optirnized by the genetic algorithm. Once the algorithm is performed at the starting stage of lens design and the variable derivative increments

are refined, we can use them throughout the design procedure except when the lens configurations and/or its specification are changed on a large-scale.

Flow-charts of the entire procedure proposed in this paper are given in Fig. 2.

H. MATSUI & K. TANAKA

3. Numerical Experiments

To show the usefulness of the genetic algorithm in deterrnining the initial variable derivative increments prior

to the lens optimization process in the DLS method, we perform several numerical experiments.

After numerous of these experiments, we set the follow-ing conditions for the genetic algorithm. We start with a

random population of 10 strings. The lower and upper limits of the objective function are -30 and +15, respec-

tively, and the constant C is assumed to be two. The probability of mutation is set to be 0.05. The maximum number of generations is set at 30. The criterion to get out

of the genetic algorithm routine, narnely Eq. (8), is rewrit-

ten as

10-2.5p0<eined<102.5,~) . (16)

Both the mantissas and the exponents of initial individual

A

(a) initial damping factor p o

(b) setting of a string

generation the o

of ulation

initial variable derivative increments

6xi [ i=1 ,...,n l

Jacobian A

eigenval ues ei [ i=i,...,n l

no

no

emin< p 0<emax

yes

A refinement of 5xi by the genetic algorithm

B

emed~F p o

ye s

DLS optimization process

Jacobian A

eigenvalues ei [ i=1,. ,n l

min< p 0<emax no

yes

emed~~ p o

no

ye s

obj ec t ive

B

function val ues

reproduction/sel ection

c ro s s - over

mutation

Fig. 2. Flow-charts: (a) the whole process of the automatic lens design problem with the genetic algorithm, (b) the genetic algorithm to refine

variable derivative increments.

OPTICAL REVIEW Vol. 3, No. 2 (1996)

Fig. 3. Cross sectional view of the hrst numerical example. This system consists of an optical pick-up lens and a glass plate represent-

ing a cover-plate of CD.

derivative increment values are empirically given by an experienced designer. We determine the value of a damp-ing factor to be 1.0. The rate of convergence depends not only on variable derivative increments but also on the damping factor, and here we show how to determine adequate derivative increments after the value of the damping factor is arbitrarily given.

3. I Qptical Pick- Up Lens Figure 3 shows a cross sectional view of the first exam-

ple. This is a singlet having two aspheric surfaces together

with a glass plate and is designed as an optical pick-up lens.12) The focal length is 4.5 mm, the numerical aperture

is 0.5, the refractive index of the lens is 1.57645 and an object is positioned at infinity.

The mathematical expression of the asphericity is writ-

ten by

X- h2/r +A'Ihl3+Bh4+B'Ihl5+Ch6 ~ 1~ 1-(1+k)(hlr)2

+C' hl7+Dh8+D'Ih 9+Ehl0+E'I hl ll +Fhl2+F'Ihl 13+ Ghl4+ G' h 15+Hhl6 (17)

where X is the sagitta, h is the height measured perpendic-ularly from an optical axis, k is the conic constant, r is the

radius of curvature of a tangent sphere and A' through H are the 14-term aspheric coefficients.

Variables for the optimization are as follows: the radius

of curvature of the second surface, the lens thickness, conic constants of both surfaces and all the aspheric coefficients of both surfaces. Thus, there is a total of 32

adjustable variables. The radius of curvature of the first surface is used to adjust the focal length of the lens.

The targets for this design are 16 points of transversal

aberration of axial imaging, 16 points of meridional and 8 points of sagittal transversal aberration of the off-axis

imaging of a single-degree field angle, meridional and sagittal field curves of the off-axis imaging with a single-

degree angular field and the working distance. Thus, there are 43 targets in the problem.

We choose initial derivative increments for the optimi-zation whose values are listed in Table 1. Both the distribu-

tion and the median of a series of eigenvalues of (ATA) of the initial state are shown in Fig. 4(a). By applying the generic algorithm to the initial condition, the refined increments are obtained and are also listed in Table 1. The

time period to refine increments by the genetic algorithm routine was about 30 s by DEC3000/700 AXP computer. Both the distribution and the median of eigenvalues after the genetic algorithm routine are shown in Fig. 4(b).

H. MATSUI & K. TANAKA 131

Table 1. The initial and reflned derivative increments of variables (Ist numerical example).

Increment Initial value Refined value

(~(1lr')

(~(d )

(~(k, )

6(A',) (~(B,)

(~(B',)

(~(C,)

(~(C',)

(~(D,)

(~(D',)

(~(E, )

(~(E',)

(~(F,)

(~(F',)

(~( G, )

(~(G',)

(~(H, )

(~( k. )

(~(A'.)

(~(B.)

(~(B'.)

(~(C.)

(5'(C'.)

(~(D. )

(~(D'.)

6(E.) (~(E'.)

(~(F. )

(~(F'.)

~(G.) (~(G'.)

(~(H. )

1.0e-8

1.0e-3

1.0e-7

6.5e-8

3.1e-6

1.3e-8

5.5e-9

2.1e-9

8.0e-10

2.9e-10

1.1e-10

3.8e-11

1.4e-11

4.8e-12

1.7e-12

5.9e-13

2.0e-13

1.0e-8

2.1e-9

3.3e-10

4.7e-11

6.1e-12

7.8e-13

9.5e-14

1.1e-14

1.4e-15

1.6e-16

1.8e-17

2.1e-18 2 .4e-19

2.7e-20

3.1e-21

1.0e-3

1.0e-1

1.0e-4

6.5e-4

3.1e-4

1.3e-7

5.5e-5

2.1e-5

8.0e-7

2.9e-7

1.1e-7

3.8e-7

1.4e-9

4.8e-8

1.7e-8

5.9e-9

2.0e-9

1.0e-4

2.1e-5

3.3e-9

4.7e-7

6.1e-8

7.8e-9

9.5e-10

1.1e-11

1.4e-11

1.6e-15

1.8e-12

2.1e-14

2.4e-15

2.7e-18

3.1e-17

Figure 5 shows that the values of ip/ipo of the initial and

the refined states are plotted against the cumulative num-ber of iterations, where ipo is the value of merit function

for the starting stage. The optimization routine with 15-iterations took about 10 s in this example. In the refined

state which satisfies Eq. (16), better results have been obtained.

3.2 Photographic Lens 1 The second exarnple is a photographic objectivel3)

shown in Fig. 6 in which the object side surface of the last

lens is aspheric. The focal length is 51.6 mm, the f number

is 2.8, a diaphragm is situated in the space between the third and fourth lenses and an object is set at infinity.

The mathematical expression of an aspheric surface in this example is defined by

X=Ah2+Bh4+Ch6+Dh8+Ehlo (18) where X is the sagitta, h is the height measured perpendic-

ularly from an optical axis and A through E are the 5-term aspheric coefficients.

The variables for the optimization are as follows: 6 radii

of curvatures of the spherical surfaces, all the aspheric coefficients, 5 Iens thicknesses and 4 Iens separations. The

radius of curvature of the object side of the first lens and

that of the image side of the last lens are used to simultane-

ously adjust the focal length and the backfocus of the lens.

l 32 OPTICAL REVIEW Vol.

no. of samp[es

17

2

3, No.

1 ~ol ~o

A

emed.

2 (1996)

(a)

H. MATSUI & K TANAI(A

-1 o 10 100

~

po

value 10 10

Fig. 6. Cross sectional view of the second numerical example.

Table 2. The initial and refined derivative increments of variables (2nd numerical example).

no of samples

7

4

Increment Initial value Refined value

(b)

-;eO :20 -10 O IU iu I O I O value IdO ~

emed fb Fig. 4. The distribution of eigenvalues of (A~'A) of the hrst numeri-cal example. po means the initial value for a damping factor and e~*d indicates the median of a series of eigenvalues. (a) Initial state, (b)

Refined state.

Logl oipl~

~(1/•,) ~(1/•,) ~(1/•*) ~(1/•,) ~(1/•,) 6(1/•*) ~(d, )

~(d;=)

~(c~)

~(d,)

~(d,)

~(d,)

~(d;)

~(~) ~(d~)

~(A,) ~( J~,)

~(C,) ~(D, )

~(E* )

1.0e-6

1.0e-6

1.0e-6

1.0e-6

1.0e-6

1.0e-6

1.0e-3

1.0e-3

1.0e-3

1.0e-3

1.0e-3

1.0e-3

1.0e-3

1.0e-3

1.0e-3

1.0e-6

1.0e-12

1.0e-15

1.0e-20

1.0e-20

1 .Oe-6

1 .Oe-3

1 .Oe-2

1 .Oe-2

1.0e-3

1.0e-2

1.0e-1

1.0e-1

1.0e-1

1.0e-1

1.0e-1

1 .Oe- 1

1 .Oe- 1

1 .Oe- 1

1 .Oe-2

1 .Oe-3

1 .Oe-6

1.0e-11

1 .Oe- 12

1.0e-12

O

-1

-2

-3

-4

Initial state

Refined state

i O 20 No. of iterations for p

Fig. 5. Comparison of the rates of convergence of the optimization. (Ist numerical example).

This simultaneous operation can be analytically done by the generalized Gaussian constants.14)

The targets for this design are 4 points of transversal aberration of the axial imaging, 16 points 0L meridional and 4 points of sagittal transversal aberration of two off-axis irnagings (21.635 mm and 15 mm fleld height), meridional and sagittal fleld curves and distortion of two

off-axis imagings, Iongitudinal color aberration for the second spectrum, transversal color aberrations from off-axis imaging, and 9 mechanical constraints (4 Iens thick-nesses and 5 rims). Thus, there are 40 targets in this problem .

We choose initial derivative increments for the optim-ization and their values are shown in Table 2. Both the distribution and the median of a series of eigenvalues of

(ATA) of the initial state are shown in Fig. 7(a). By applying the generic algorithrn to the initial condition, the

refined increments are obtained and are also listed in Table

2. The computation tirne to refine increments by the genetic algorithm routine is about 15 s. Both the distribu-

tion and the median of eigenvalues after the genetic process are shown in Fig. 7(b), where it appears that the eigenvalues distribute closer to the damping factor when the refined increments are used. The values of ~/ipo of the

OPTICAL REVIEW Vol. 3, No. 2 (1996)

no. of samples

2

l~ o

(a)

-10 100 value 1010 10

i + A

emed. Po

H. MATSUI & K. TANAKA 133

Fig. 9. Cross sectronal vrew of the third numencal example

no. of samples

2

(b)

Table 3. The initial and refined derivative increments of variables (3rd numerical example).

Increment Initial value Reflned value

~O -10 100 value 1010 ~

emed . Po

Fig. 7. The distribution of eigenvalues of (ATA) of the second numerical example. (a) Initial state, (b) Refined state.

Logl o ~ I ~o

O

-1

-2

-3

~(1lr,)

~(1lr,)

~(1lr,)

~(1lr~)

~(1/•*) ~(1lr*)

~(1lr*)

~(1lrb)

~(1/•**) ~(1/•**) (~(1/r*,)

~(d, )

~(d;,)

~(~) ~(d,)

~(d ,)

~(d,)

~(d;)

~(d;*)

~(d~)

~(d**)

~(d* * )

1.0e-5

1.0e-7

1.0e-5

1 .Oe-5

1 .Oe-6

1 .Oe-5

1.0e-5

1 . Oe-7

1 .Oe-5

1.0e-7

1.0e-6

1.0e-3

1 . Oe-3

1 .Oe-3

1.0e-3

1 .Oe-3

1.0e-3

1.0e-3

1 .Oe-3

1 .Oe-3

1 .Oe-3

1.0e-3

1.0e-2

1 . Oe-2

1 .Oe-2

1.0e-4

1.0e-2

1.0e-2

1 .Oe-2

1 .Oe-2

1.0e-4

1.0e-2

1 .Oe-2

1 .Oe-3

1 .Oe-2

1.0e-2

1.0e-4

1 .Oe-2

1 .Oe-3

1 .Oe-3

1 .Oe-2

1.0e-2

1.0e-2

1 .Oe-2

Initial state

Refined state

20 No, of iterations for p

Fig. 8. Comparison of the rates of convergence of the optimization. (2nd numerical example).

initial and the refined states are plotted against the cumula'

tive number of the iterations in Fig. 8. The time required

for the 30-iterative-optimization was about 10 s. When refined increments which satisfy Eq. (16) are employed, faster convergence has been obtained.

3.3 Photographic Lens 2 The third example is the double-Gauss-type photo-

graphic objective shown in Fig. 9. The focal length is 50 mm. the f number is 1.8 and an object is set at inflnity.

The variables for the optimization are: 11 radii of curva-

tures, 6 Iens thicknesses and 4 Iens separations. The radius

of curvature of the last surface is used to adjust the focal

length of the objective.

The targets of this design are 4 points of transversal aberration of the axial imaging, 8 point of meridional and 4 points of sagittal transversal aberration of two off-axis imagings, Iongitudinal color aberration, transversal color aberration of the off-axis imaging, distortion of the off.axis

imaging and the rims of lenses. There are 45 targets in the

problem. The initial derivative increments for the optimization

are listed in Table 3. The distribution and the median of eigenvalues for the initial state are shown in Fig. 10(a).

By applying the genetic algorithm to the initial state, the

refined increments are obtained and are listed in Table 3.

The computation time to refine increments by the genetic algorithm was about 20 s. The distribution and the median of eigenvalues for the refined state are presented in Fig. 10(b).

134 OPTICAL REVIEW Vol. 3, No. 2 (1996)

no, of samples

2

(a)

l~ o

no, of samples

2

-1 o 10

~

emed.

100

+

po

value 10 10

(b)

~O -10 100 Value 1010 +. Aj

po emed. Fig. 10. The distribution of eigenvalues of (A1'A) of the numerical example. (a) Initial state, (b) Refined state.

third

Judging from these figures, the median of eigenvalues is

10cated closer to the damping factor when the refined increments are used. Figure 11 presents the values of clipo

of both the initial and the refined states plotted against the

cumulative number of iterations. The computation time for the 10-0ptimization-iteration was about 5 s. When refined increments are employed and Eq. (16) is satisfied, the convergence has been faster.

Since the Jacobian matrix must be computed to evaluate each performance function to obtain adequate variable derivative increments, the total time required for the design is increased. However, the method proposed here has an advantage in that it attains better lens configuration

data having lower merit function value than that obtained by the empirically determined increments which do not satisfy Eq. (8). This is dernonstrated in Figs. 5, 8 and 11,

where the optimization processes using the empirically determined initial increments have stagnated.

4. Conclusions

We first defined that the problem of determining the variable derivative increments in the DLS approach is a type of combinatorial optimization problem. By applying the genetic algorithm with the help of the relationship between the darnping factor and the eigenvalues of the Jacobian matrix of the performance functions, we have

H. MATSUI & K. TANAKA

Logl o ip / ~

O

-1

-2

-3

[nitial state

Refined state

Fig. 11.

tion. (3rd numerical example).

10

No. of iterations for p

Comparison of the rates of convergence of the optimiza-

proposed a method to determine adequate variable deriva-tive increments prior to the automatic lens optimization

process. The effectiveness of the method presented here has been

confirmed by the results of several numerical experiments.

Even though the entire time for design is longer, the method proposed here has given better lens configuration data with lower merit function value than the empirically assigned method of variable derivative increments.

Ref erences

1) G.C. Wynne: Proc. Phys. Soc. Lond. 73 (1959) 777. 2) C.G. Wynne and P.M.J.H. Wormell: Appl. Opt. 2 (1963) 1233. 3) H. Matsui and K. Tanaka: Appl. Opt. 33 (1994) 2411. 4) H. Matsui and K. Tanaka: Appl. Opt. 34 (1995) 642. 5) J.C. Nash: Compact Numerical Methods for Computers (Adam

Hilger, Bristol and New York, 1990) p. 218.

6) D.E. Goldberg: Genetic Algorithlns in Search. C~timization and Machine Learning (Addison-Wesley, Reading, MA., 1987) p. 1.

7) H. Konno and H. Suzuki: Integral Programming and Com-binatorial Q~,timization (Nikkagiren, Tokyo, 1982) p. 297 (in Japanese).

8) E. Betensky: Opt. Eng. 32 (1993) 1750. 9) G. Elsner: J. Opt. Theor. Appl. 59 (1988) 165.

10) M. Walk and J. Niklaus: J. Opt. Theor. Appl. 59 (1988) 173. 11) H. Matsui and K. Tanaka: Appl. Opt. 31 (1992) 2241. 12) K. Matsuoka: U.S. Patent No. 4,979,807 (25 Dec., 1990).

13) Y. Matsui: Jpn. Patent No. 44-559777 (21 Nov., 1969) (in Japanese).

14) K. Tanaka: Progress in C~tics XXIII, ed. E. Wolf (North-Holland, Amsterdam, 1986) p. 53.