JPH11242690A - Method and device for designing optical system and medium recorded with program for implementing the method - Google Patents

Abstract

PROBLEM TO BE SOLVED: To obtain an optical system as a global optimal solution in a practicable time independent of initial data given in advance by selecting at least two parent individuals from a population of an n-th generation and newly generating a population consisting of plural child individuals and generating a population of the next generation. SOLUTION: A parent 1 and a parent 2 out of selected parents are made to be one's parents to generate two children in accordance with a normal distribution set around them. The standard deviation of the normal distribution is so set that the component σ1 in the direction of the major axis connecting the parents is proportional to the distance between the parents (σ1=αd1, d1: the distance between parents 1 and 2), and the component σ2 in the directions of the minor axis is proportional to the distance between the major axis and a parent 3 (σ2=βd2, d2: the distance from the parent 3 to the axis connecting parents 1 and 2). The crossing operator is effective for functions, where there is a strong dependence between variables, or a multi-peak function. Thus, unnecessary retrieval is eliminated, and an optimal solution or its approximate value can be obtained in a shorter time.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an optical system including at least one optical element (lens element, reflecting mirror, etc.), for example, a photographic lens, a microscope lens, a batch exposure system or a scanning exposure system. (Step and repeat type stepper, step and scan type stepper, etc.), such as a design method and an apparatus such as a projection optical system, particularly, each lens element belonging to the optical system to be designed is strictly a design value Optimal rearrangement in the case where processing is not performed in the same way (that is, correction of the arrangement), and the combination of each lens element in the case where a plurality of optical systems including lens elements that are not processed in accordance with the design values are manufactured. The present invention relates to a method and apparatus for designing an optical system that enables optimization.

[0002]

2. Description of the Related Art Conventionally, designing an optical system including a lens element has been known as a very difficult problem. This is because factors such as multidimensions, hypermultimodality, strong dependencies between variables, and complex constraints make the problem difficult. In addition, there are many evaluation criteria for an optical system to be designed, such as Seidel's five aberrations, such as size and cost.

In a conventional optical system design method, a local search around the initial solution is fundamental. If the initial solution is not appropriate, the search falls into a local solution and the search fails. Therefore, in order to find an optical system having a target performance, a method of changing an initial solution by trial and error has been adopted. In addition, since the conventional search can basically optimize only one evaluation criterion,
Despite the fact that there are many evaluation criteria, the trade-off ratio has been set and made simpler (Yoshiya Matsui: Lens Design Method, Kyoritsu Publishing (1986); Nakagawa Jihei: Lens Design Engineering, Tokai University Press (1986) Toru Kusakawa: Lens Optics, Tokai University Press (1988)).

A trade-off ratio for obtaining an optical system having a target performance cannot be known in advance. As described above, at present, the burden on the expert is extremely heavy in the search for the initial solution for the local search and the search for the trade-off ratio between the evaluation criteria. Further, in the conventional optical system correction method, the curvature of the boundary surface of each optical element (lens element, reflecting mirror, etc.) belonging to this optical system is defined as one optical system data given in advance by a designer as initial data. , The distance between boundary surfaces, the refractive index of the space located between the boundary surfaces (lens elements and the air lens between lens elements), etc. To the index.

[0005] Then, the optical system data represented by the plurality of changed parameters is set as a new solution (that is, an optical system to be improved), and a similar improvement procedure is repeated. For example, when the quality of the optical performance is reflected in the increase or decrease of the evaluation function, a plurality of parameters are changed so that the evaluation function increases, and each is updated as a new parameter value.

On the other hand, a genetic algorithm (Genetic Algor
Ithm: GA) is known as one of the optimization methods that imitate the evolutionary process of living organisms by engineering. This genetic algorithm (hereinafter, referred to as GA) is a generative test method, and has a feature that it is only necessary to evaluate the superiority or inferiority between two solution candidates. Therefore, conditions such as differentiability of the evaluation function are not required, and the present invention is effective for problems having complicated constraints. Further, GA has a feature that a search is performed using a group of a plurality of solution candidates, and is attracting attention as a global search method. Furthermore, taking advantage of the above two characteristics, a plurality of evaluation criteria are explicitly handled, and attention is being paid to a multi-objective optimization method for finding a Pareto optimal solution set by a single search.

[0007] For example, M.WALK AND J.NIKLAUS, "So
me Remarks on Computer-Aided Design of Optical Len
s System "(JOURNAL OF OPTIMIZATION THEORY AND APPLI
CATION: Vol.59, No.2, pp.173-181, NOVEMBER 1988)
Ya, X.CHEN AND K.YAMAMOTO, "Genetic algorism and it
s application in lens design "(SPIE, Vol.2863, PP.2
16-221) shows a technique in which the above GA is applied to the design of an optical system.

[0008]

However, in the conventional optical system design method and correction method, once the evaluation function locally reaches a local maximum value, even if the improvement procedure is repeated, further improvement is not achieved. I can't. The optical system corresponding to a plurality of parameters obtained in such a situation is a local optimal solution, that is, a local optimal optical system, and is very unlikely to be a true optimal solution, that is, a global optimal solution. Therefore, the performance of the optical system as a solution obtained by the conventional optical system design method and correction method depends on the optical system initially given as an initial solution by the designer. That is, when the optical system of the optimal solution is significantly different from the optical system initially given as the initial solution by the designer, it is highly likely that the optimal solution does not appear.

In the prior art for designing an optical system using a genetic algorithm, the genetic operator used in the GA can handle only a discrete number. It was necessary to discretize the parameters, and it was difficult to obtain a solution with sufficient accuracy in a practical time. That is, in the conventional technology for discretely executing GA, as shown in FIG. 24, each of a plurality of parameters constituting a parent individual and a child individual is represented by a binary code as a coding / crossover method, and the binary A combination of a code and a one-point crossover, a combination of the binary code and a two-point crossover, or a combination of the binary code and a uniform crossover is often used. FIG. 24 (a)
FIG. 24B is a diagram showing a step of generating one of the parameters of the child individual by one-point crossover from one of the parameters of the parent individual represented by the binary code, and FIG. FIG. 24C is a diagram showing a step of generating one of the parameters of the child individual from one of the parameters of the parent individual by two-point crossover, and FIG. FIG. 4 is a diagram illustrating a step of generating one of parameters.

As described above, in the discrete GA, the topological structure (binary code) of the genotype space and the topological structure (actual value) of the phenotype space are significantly different.
This means that the traits (characteristics) of the parent individual are less likely to be inherited by newly generated child individuals. Therefore, even if a parent of a certain generation approaches the optimal solution,
The next-generation offspring individual has a problem that the probability of deviating from the optimal solution becomes extremely higher than that of the parent individual, resulting in a remarkable increase in unnecessary searches.

In other words, in the case of a discrete GA, in order to obtain sufficient accuracy, it is necessary to make the degree of discretization of each parameter fine, and the amount of genetic information inevitably increases, resulting in a significant increase in calculation time. Will be invited. Conversely, if each parameter is roughly discretized in order to obtain a solution in a practical calculation time, there is a risk that an optimal solution may be overlooked if it exists between the minimum steps of discretization.

Further, when considering the evaluation criteria of the designed optical system, there are a plurality of evaluation criteria (for example, resolution and distortion) that often conflict with each other in terms of actual performance. In each case, the reciprocity of the plurality of evaluation criteria is uniquely weighted in advance, so that the plurality of evaluation criteria are integrated and represented as one evaluation function. However, whether the weighting set in advance is appropriate can be confirmed only after obtaining various optimum solutions while changing the weight of each of the plurality of evaluation criteria. A solution that meets each evaluation criterion is generally called a Pareto optimal solution, and there are a plurality of Pareto optimal solutions corresponding to various weightings.

Therefore, when there are a plurality of evaluation criteria for the designed optical system, it is necessary to perform multi-objective optimization so as to meet each evaluation criteria.
With the conventional optical system design method and correction method, the obtained solution cannot always be determined to be the optimal solution. SUMMARY OF THE INVENTION The present invention has been made to solve the above-described problems, and has been designed to obtain an optical system as a global optimum solution within a practical time without depending on initial data given in advance. It is a main object to provide a method, an apparatus, and a medium on which a program for implementing the design method is recorded.

[0014]

The method for designing an optical system according to the present invention includes at least one optical element (lens element,
This is a method of designing an optical system including a reflecting mirror using an evolutionary calculation method (a genetic algorithm), and is characterized by optimizing the optical system using a genetic operator that directly handles continuous values. . In other words, the design method of the optical system is such that n (≧
A selection step of selecting at least two parent individuals from a generation population, and applying at least one of a crossover operator and a mutation operator to the selected parent individuals By doing so, at least a child generation step of newly generating a group consisting of a plurality of child individuals and a selection step of generating a next-generation group are provided.

In particular, an individual which is a candidate for an optical system to be designed includes, for example, the curvature of a boundary surface defining a lens element, a reflecting mirror, and the like included in the optical system to be designed, a distance between the boundary surfaces, Given as a real number vector having a component such as a refractive index of a space located between the boundary surfaces, mapping from the optical system data (phenotype) of the individual to a genotype is performed. The optimization of the optical system need not be performed for all components of the real number vector, but is performed individually for specific vector components (at least one of a plurality of parameters characterizing the optical system to be designed). It is possible.

In the method of designing an optical system according to the present invention, in the child generation step, a child individual is generated by at least one of a crossover operator (crossover step) and a mutation operator (mutation step). That is, the crossover operator sets a real vector having a value that appears in accordance with the occurrence probability from a subspace defined by a continuous predetermined occurrence probability distribution set based on the real vector component of the selected parent individual. , As child individuals. In addition, the mutation operator, according to the occurrence probability, from within a subspace defined by a continuous predetermined occurrence probability distribution in which the occurrence probability increases as approaching at least one parent individual among the selected parent individuals. A real number vector having the appearing value as a component is generated as a child individual.

Therefore, the selecting step, the child generating step,
And in the search process in which the selection process is repeatedly executed,
The child individual newly generated from the selected parent individual inherits the traits (characteristics) of the parent individual, thereby avoiding unnecessary searching. That is, in the initial stage of the search process, the parent individuals are separated from each other, and each individual is scattered in various spaces according to the distance between the parent individuals. (As the number of executions of each of the above steps increases), the distance between parent individuals also approaches,
According to the distance between the parent individuals, more child individuals are generated in the subspace where the optimal solution exists.

The crossover operator is a genetic operator that directly handles continuous values, and is, for example, UNDX.
(Ono, I. and Kobayashi, S: A Real-coded Genetic A
lgorithm for Function Optimization Using Unimodal
Normal Disttribution Crossover, Prpoceeding of 7th
International Conference on Genetic Algorithms, p
p.246-253 (1997)) and BLX-α (NJRadcliffe:
Formal Analysis andRandom Respectful Recombinatio
n, Proceeding of the Fourth InternationalConferenc
e on Genetic Algorithms, pp. 222-229, 1991); ND
X (I. Ono, M. Yamamura and S. Kobayashi: A Genetic
Algorithm with Characteristic Preservation for Fun
ction Optimization, Proceedings of IIZUKA '96, pp.
511-514, 1996); or UNDX (Ono, I. and Kobayas).
hi, S: A Real-coded GeneticAlgorithm for Function
Optimization Using Unimodal Normal DisttributionC
rossover, Prpoceeding of 7th International Confere
nce on Genetic Algorithms, pp.246-253 (1997)).

Further, in the selection step, an individual to be an individual of the next generation population is selected from an n generation population including a parent individual and a newly generated child individual population. Here, in the selection step, in order to realize multi-objective optimization, an individual that satisfies at least one of one or two or more evaluation criteria is selected from a population including the generated child individuals. In this individual selection, it is preferable to sequentially select individuals to be the next-generation individuals in proportion to the fitness of each individual, starting with the individual that best meets the predetermined evaluation criteria.

Further, in the method of designing an optical system according to the present invention, in the selection step, the selected individuals are replaced with the unselected individuals in the n generation population.
Preferably, a next generation population is generated. The method of designing an optical system according to the present invention can be realized by a program described in a predetermined language. This program includes, for example, an arithmetic circuit having a memory (RAM, ROM), an input / output It is executed by a computer having a basic configuration including a unit, a calculation unit, a storage unit, and a control unit. In particular, when the program is executed by a computer, a program for realizing the method of designing the optical system includes a CD, an MO, an FD, a hard disk,
It is preferable that the information is optically or magnetically recorded on a predetermined recording medium such as a magnetic tape or a ROM.

With the above arrangement, the present invention makes it possible to automatically generate an optical system as a global optimal solution that does not depend on a given initial solution. In other words, it is possible to obtain an optimal solution of a plurality of parameters (each component of a real number vector) characterizing the optimal optical system or an approximate solution of the optimal solution from an arbitrarily given initial solution within a practical time.

In the conventional optical system design method and correction method, if a local optimal solution is once found and there is hardly any change in the increase or decrease of the evaluation function even when the improvement procedure is repeated, it is obtained at this time. The local optimal solution of the obtained optical system is temporarily stored in a memory as a single solution candidate, and one or more of the plurality of parameters of the solution that are arbitrarily changed is newly obtained as an optical object to be improved. A method of obtaining a plurality of local optimal solutions by repeating the improvement procedure again as a system and repeating the operation of obtaining another local optimal solution is also conceivable. However, as the optical system becomes more complicated due to an increase in the number of optical elements and the like, the local optimal solution generally becomes enormous, and it is difficult to find a global optimal solution by this method.

In addition, since a plurality of optical systems as multi-objective optimal solutions which should exist in correspondence with a plurality of conflicting evaluation criteria can be simultaneously selected in the selection step, a plurality of evaluation criteria for evaluating the optical system are required. When given, by performing multi-objective optimization that simultaneously handles a plurality of evaluation criteria, it becomes possible to obtain a plurality of Pareto optimal solutions or an approximate solution of a plurality of Pareto optimal solutions within a practical time.

Further, according to the present invention, the optimal solution of the parameter characterizing the optimal optical system or the approximate solution of the optimal solution is satisfied even within the range satisfying the constraint on the specific parameter (s) arbitrarily designated by the user. Can be obtained within a practical calculation time. For example, the constraint that the first surface of the first lens must be convex, the refractive index of the glass material of the second lens must be x (eg, 1.5266), y (eg, 1.6010), or z (eg, 1.7294). Even if restrictions are given,
Multi-objective optimization is possible within a range satisfying such restrictions.

[0025]

DESCRIPTION OF THE PREFERRED EMBODIMENTS Embodiments of an optical system designing method and apparatus according to the present invention and a recording medium on which a program for realizing the designing method is recorded will be described below with reference to FIGS. . In the drawings, the same portions are denoted by the same reference numerals, and description thereof is omitted. The method for designing an optical system according to the present invention includes:
It can also be realized by a program described in a predetermined language. This program is executed by, for example, an arithmetic circuit having a memory (RAM, ROM) or a computer 1 as shown in FIG. In particular, FIG.
The computer 1 shown in FIG.
It includes a control unit 101, a calculation unit 102, and a storage unit 103 (memory).
6 is provided. The output unit 105 includes a display device such as a CRT 115 and a liquid crystal display. When the method for designing an optical system according to the present invention is executed by the computer 1 as shown in FIG. 1, a program for implementing the method for designing an optical system may be stored in the external storage device 106 or the CD 115 , MO, FD, magnetic tape, ROM, etc., it is preferable that the information is optically or magnetically recorded.

Next, in order to facilitate understanding, the definitions of the terms used in this specification are listed below. Individuals: autonomous individuals characterized by genes; populations: collections of individuals; population size: the number of individuals allowed to exist in the population; genes: basic components that represent genetic information; alleles: each gene can take Chromosome: a gene expressed as a string (character string), vector (numerical string), or a mixture thereof; locus: location of the gene on the chromosome; phenotype: external expression of a trait expressed by the chromosome; gene Type: Gene expression that defines the trait (internal expression); Encoding: Mapping from phenotype to genotype; Decoding: Inverse mapping from genotype to phenotype; Fitness: Indicates the degree of adaptation of the individual to the environment Evaluation value of scalar (when targeting an optimization problem, the value of the objective function itself or the value of the penalty function considering the constraint conditions is set); Genetic operation: crossover, Mutation and selection operations Selection: Selection of individuals by fitness (creating a pair of individuals for mating considering differences in fitness); Crossover: Genetic recombination between individuals (pair of individuals) Mutation of both chromosomes between them to create a new individual); Mutation: Replacement with an allele (replacement of the value of one locus with another allele); Generation: Genetic manipulation 1 cycle; Diversity: Degree of preserving gene diversity; Parallelization: Model in a decentralized population; Immigration: Exchange of individuals between multiple populations; First, the design method of the optical system according to the present invention is applied to the design of a three-lens lens system and a four-lens lens system in which two evaluation criteria are made single-purpose, and the global search capability of the designing method is improved. Confirm (first embodiment). Next, treat the two evaluation criteria as multipurpose,
Regarding the design of a triplet lens system, the excellent multi-objective optimization ability of the method for designing an optical system according to the present invention will be shown (second embodiment).

In the embodiment of the optical system designing method according to the present invention, the design specifications of the lens system are such that the focal length f, the brightness F, and the angle of view 2w are set by the user, and each lens surface ( Consider optimizing the radius of curvature ri of each boundary surface of the optical elements constituting the optical system, the lens thickness, and the inter-lens distance di. The refractive index of glass and the number N of lenses are given in advance. The radius of curvature of the lens surface closest to the film surface (image surface) and the distance from the lens surface to the film surface) are corrected to satisfy the given focal length using paraxial tracking. Therefore,
An N-piece lens system design is equivalent to 4N-2 dimensional function optimization.

The designed lens system converts a total of eleven rays centered on the principal ray into three rays of 0 °, 0.65 w °, and w °.
It is evaluated based on three spot diagrams created by passing through the lens system at different angles of incidence. Note that the two evaluation criteria are distortion (Distortion: D) and resolution (Resolution: R). In this embodiment,
The distortion D is given by the distance between the ideal image forming position (ftanw) and the image point of the principal ray, as shown in Expression 1 below.
In this embodiment, the resolution R is given by the standard deviation of the degree of dispersion of the image points of the other ten rays from the image point of the principal ray, as shown in the following Expression 2.

[0029]

(Equation 1)

[0030]

(Equation 2)

Where (xideal, yideal) is an ideal image forming position, (x0, y0) is the position of the image point of the principal ray, and (xk, yk)
Represents the positions of the image points of the other ten rays.
Next, a genetic algorithm (GA) will be described.
GA is an engineering model that mimics the evolutionary process of living things,
The general algorithm is configured as follows. That is, (1) Generation of Initial Populat
ion) Generate a plurality of individuals (solution candidates) and use them as the initial population. (2) Selection for Reproduction A parent individual (parent individual) is selected from a population. (3) Generation of Children A crossover operator or a mutation operator is applied to a parent individual to generate a child individual (new solution candidate). (4) Selection for Survival From among individuals in the current generation population and the population of offspring individuals generated in the above step (3), individuals constituting the next generation population are selected. (5) Steps (2) to (5) until a predetermined termination condition is satisfied.
Repeat (4).

GA design items are broadly classified into coding / crossover / mutation design and generation model design. Coding /
Crossover / mutation design is a design of an individual expression method and a new individual generation method and depends on the problem domain. However, since the performance of GA is greatly influenced by the coding / crossover / mutation design, it is very This is an important design item. On the other hand, generational alternation model design is a method of selecting a parent to generate offspring and a method of selecting individuals to be left in the next generation population. By devising this, the relationship between multiple trade-offs in multi-objective optimization is improved. At a time.

Next, the coding / crossover design will be described. In the embodiment of the method for designing an optical system according to the present invention, a real vector expression is adopted as coding. Individuals are represented by real vectors (r1, r2, ..., rn, d1, d2, ... dn) whose components are the radius of curvature and the distance between surfaces (parameters that characterize each candidate of the optical system to be designed). Is done.

In this embodiment, UNDX (Ono, I. and Kobayashi, S: A Real-c
oded Genetic Algorithm for Function Optimization U
singUnimodal Normal Disttribution Crossover, Prpoc
eeding of 7th International Conference on Genetic
Algorithms, pp.246-253 (1997)).
UNDX generates two children according to a normal distribution set around parents 1 and 2 among the selected parents, as shown in FIG. The standard deviation of the normal distribution is such that the component σ1 in the main axis direction connecting the parents is proportional to the distance between the parents (σ1 = αd1, d1: the distance between the parent 1 and the parent 2), and the component σ2 of the other axes is the main axis and the parent. 3 (σ2 = βd3, d3: distance from parent 3 to axis connecting parent 1 and parent 2). FIG.
The case of variables is shown.

The crossover operator is effective for a function having a strong dependency between variables and a multimodal function. This is because UNDX can search globally at the beginning of the search where the parent is at the same distance as the parent, and locally at the end of the period when the parents are close to each other, and can search independently of the coordinate system. it is conceivable that. (First Embodiment) In an optical system designing method according to a first embodiment of the present invention, a scalar evaluation value is set by giving a trade-off ratio of resolution R and distortion D to form a linear image, and an experiment is performed. To confirm the effectiveness of GA for global search ability.

As the generational alternation model of the GA of the first embodiment, an MGG model (Satoh, H., Yamamur) shown in FIG.
a, M. and Kobayashi, S .: Minimal Generation Gap M
odelfor GAs Considering Both Exoplortion and Explo
itation, Proceedings of IIZUKA '96, pp.494-497 (19
96)) is used. In this generation alternation model, two individuals are randomly crossed n times from the population as parents. Then, when the best one individual among the parent and child individuals and the one individual selected by roulette selection are returned to the group, generation change is performed.

In order to confirm the effectiveness of the design method in the first embodiment, the optimization of the three-lens lens system and the four-lens lens system was performed. With respect to the four-lens lens system, three types of experiments were performed with different design specifications. In all experiments, the population was 100 individuals and the number of crossings n was 10
0, α of UNDX is 0.5, β is 0.35. The trade-off ratio between the distortion D and the resolution R is 1: 1. (Experiment 1) As Experiment 1 of the first embodiment, optimization of a three-unit lens system was performed. Design specification is focal length f = 100
mm, brightness F = 3.0, and angle of view 2w = 38.0 °. In this experiment, starting from an initial group of random lenses L1 to L3 as shown in FIGS. 4A and 4B, the evaluation was terminated when 4 million lenses were evaluated. In FIG. 4, S indicates an aperture and I indicates an image plane.
FIG. 5 shows the lens shape and the spot diaphragm of the obtained optical system, and FIG. 6 shows the aberration diagram. The spot diagram is displayed in a range of ± 0.5 mm. (Experiment 2) As Experiment 2 of the first embodiment, optimization of a four-lens set was performed. The design specifications are f = 50 mm, F = 3.0, 2w = 46 ° as standard lens specifications, f = 135 mm, F = 2.8, 2w = 1 as telephoto lens specifications.
8.2 °, f = 20 mm as wide-angle lens specification, F =
5.6, 2w = 92 °. Similar to the three-lens lens system, the search was started from randomly shaped lenses L1 to L4, and the number of evaluations was 1,000,000 and the search was terminated. The lens shapes and spot diagrams of the obtained optical system are shown in FIGS. 7 to 9, respectively.
7 shows an optical system obtained in the case of the standard lens specification, FIG. 8 shows an optical system obtained in the case of the telephoto lens specification, and FIG. 9 shows an optical system obtained in the case of the wide-angle lens specification.

In Experiment 1 of the first embodiment, 1
In all zero trials, a lens system having a lens shape as shown in FIG. 5, ie, a triplet, was selected. The search was started from a completely random lens system without depending on knowledge, and the empirically-optimized triplet was obtained.It can be said that the global search of the lens system by GA was effective. .

Also, in Experiment 2, a relatively high-performance lens system was obtained with each design specification. In the telephoto lens and the wide-angle lens, almost the same shape was obtained in each trial as shown in FIGS. 8 and 9 (two patterns each), but in the standard lens, it was shown in FIG. Thus, lenses of various shapes (six patterns) were obtained. (Second Embodiment) Next, a multi-objective optimization by GA will be described as a second embodiment according to the present invention.

In the second embodiment, the distortion and the resolution
The objectives are set explicitly to confirm the effectiveness of the GA with regard to its multi-objective optimization capabilities. In the second embodiment, as a generation alternation model, a non-Pareto solution selection strategy (Shigenobu Kobayashi, Koji Yoshida, Masayuki Yamamura: Generation of Pareto optimal setting tree set by GA, Journal of the Japanese Society for Artificial Intelligence, vol. 11, No. .5, pp.778-785
(1996)) (see FIG. 10).

Crossover is performed by selecting n sets at random from the group. The generated offspring are combined with the current population to eliminate individuals that are not Pareto solutions, and the Pareto solution alone is used as the next generation population. A Pareto solution is a solution that is superior to all other solutions in at least one criterion. This allows explicit multi-objective optimization without setting a trade-off ratio between resolution and distortion.

In the second embodiment, the effectiveness of the multi-objective optimization by GA is confirmed by experiments on a three-lens lens system. The prepared initial population is 5 Pareto solutions obtained by the search with the single-purpose GA, and the number of crossovers n is 4
0, and the number of evaluations is 1.6 million (the search is 1
Up to 600,000 times). The design specifications of the lens system are the same as those in Experiment 1 of the first embodiment described above. FIG.
In, the resulting Pareto solution set is plotted on the distortion-resolution plane. The number of Pareto solutions obtained is 750.

In the second embodiment, when the search is further continued from the Pareto solution set shown in FIG. 11, the Pareto solution surface can be evolved in the lower left direction. However, the speed of evolution has slowed, and it is presumed that it is sufficiently close to the true Pareto optimal solution surface. In addition, up to a certain value, the resolution is improved without sacrificing the distortion, but it is found that it is difficult to improve the resolution after that, even if the distortion is sacrificed. All the shapes of the obtained solutions are triplets, and in this design specification, it is considered that the triplet is optimal in all regions.

FIG. 12 shows a state in which the best solution P (the lens system shown in FIG. 5) obtained in Experiment 1 of the first embodiment described above is plotted on the enlarged view of FIG. . It should be noted that S in the figure is a lens system that governs a solution found based on a single-purpose evaluation criterion. As can be seen from FIG. 12, in the second embodiment (multi-objective optimization), it is apparent that many better solutions than the solution obtained by the single-purpose GA are obtained. This suggests that forcing a multipurpose problem into a single purpose may make the problem more difficult.

Rather than finding a single lens by searching after linearly imaging a plurality of evaluation criteria to obtain a single objective, a plurality of lenses having a trade-off relationship are found by a single search. It seems natural to select an appropriate lens according to the preference of the person. As described above, the effectiveness of the embodiment of the optical system designing method according to the present invention is apparent from the above-described experiments. In particular, in a multipurpose GA that explicitly handles each evaluation criterion, a plurality of lenses having different performances can be searched at once.

The method for designing an optical system according to the present invention sequentially selects an approximate optimal solution from the following two features of the principle of the genetic algorithm (1) or (1a) and (2). Finally, since the principle of outputting the optimal solution is adopted, an approximate solution or an optimal solution according to the required calculation time can be efficiently generated. (1) Simultaneous progress of local search and global search of problem space only by crossover operator, (1a) Simultaneous progression of local search and global search of problem space by combination of crossover operator and mutation operator, ( 2) A series of operations of a genetic operator is defined as one generation, and generational alternation is performed by repeating a series of operations on a finite number of genes including the best gene of each generation. The gene at the position (locus) where each parameter appears is represented by a string (variable string with the gene as a variable) of the corresponding length, and corresponds to each parameter characterizing the lens system to be designed.

The crossover operator determines, for each parameter, a value that appears at a predetermined probability in a subspace defined by the parameters of the two parents from the one-to-two parent genes. It works as follows. Further, when the mutation operator is used together, for each parameter, a value that appears according to a continuous predetermined occurrence probability distribution in which the occurrence probability increases as the value approaches the parameter of the arbitrarily selected parent is set as a parameter value. Genes of offspring individuals are generated.

As described above, the internally generated solution candidates can always be directly evaluated. Therefore, if the arbitrarily given constraints are not satisfied, the candidate may be killed. In this case, multi-purpose evaluation is possible. Furthermore, during GA execution, the best solution (or Pareto-optimal solution) among a plurality of memorized (satisfying any constraints) genes is alternately changed so as to be always maintained and excellent. Is selected so as to be less likely to die, and is left for the next generation, so that efficient multipoint simultaneous search is possible. (Third Embodiment) A third embodiment of the method for designing an optical system according to the present invention will be described below with reference to FIGS. In the third embodiment, an optimal solution of a photographic lens system, which is a three-unit lens system having a known refractive index, is obtained. That is, the refractive index of each of the three lenses is set by the user and is a constraint.

FIG. 13 is a schematic view of the configuration of a photographic lens system. G in this figure is the image plane. Also, since the photographic lens system in this figure is an example of a three-lens configuration, there are six boundary surfaces a to g, each having a curvature, and six boundary surfaces d1 to d6 (distance between AB is d1). , BC between d2 and C
D3 between D, d4 between DE, d5 between EF, d between FG
6) exists.

In the photographic lens system shown in FIG. 13, the refractive index between the lens boundary surfaces is given in advance. If the refractive index between the lens boundary surfaces has a degree of freedom, the number of parameters increases by that amount. Also, the angle of view and the combined focal length of the lens optical system are usually constraints given by the designer,
6 of a to g (for example, a to f), and d1 to d6
When five of the parameters (for example, d1 to d5) are determined, the remaining two parameters (for example, g and d6) are naturally determined. Therefore, the third embodiment is a problem of simultaneous optimization of 10 parameters.

For d1 to d6, the distance between the boundary surfaces is kept as it is (that is, in order not to physically interfere,
1-d6> 0 is a necessary condition), and the reciprocal of the radius of curvature is used to indicate each of the curvatures a to g. The reason for using the reciprocal is that when the plane (that is, the radius of curvature is infinite) changes slightly, the correspondence to the transition of the surface shape from convex to concave is treated continuously in the parameter space. .

There are various evaluation criteria for the lens system.
Accordingly, various evaluation functions can be defined. Five evaluation criteria represented by Seidel's five aberrations are well-known and can be applied to this embodiment. However, in this embodiment, for simplicity, an example using ray tracing by a spot diagram and two opposite evaluation criteria is used. Are listed. FIG. 14 is a genetic expression characterizing a photographic lens system commonly referred to as a triplet type. The left side of the figure is the object side, and the right side is the image side. Assuming light from infinity on the object side as a parallel light flux, this light flux should converge to one point on the image plane g. Also, from the left side of the figure, a parallel light beam parallel to the optical axis of the lens system (a light beam shown by a solid line in the figure), and a parallel light beam whose angle with the optical axis is the specified maximum angle of view of the lens system. Representative examples are a light beam (a light beam indicated by a broken line in the figure) and a parallel light beam (a light beam indicated by a dashed line in the figure) whose angle with the optical axis is about half of the maximum specification angle of view of the lens system. From a viewpoint, it is possible to evaluate an image formed on the entire image plane.

For example, as one of the evaluation criteria, the degree of convergence is determined by the paraxial ray of the light beam, the light beam on the maximum outer diameter of the light beam (usually using the diameter corresponding to the entrance pupil diameter),
And the intermediate diameter of the two rays (for example, the maximum outer diameter of 70).
It is considered that three types of light rays, that is, light rays on the image plane having a diameter of about% and about 80%) are evaluated based on variations from the image plane arrival positions of paraxial rays. This criterion is
If sagittal, coma, and flare are ignored, respectively, they roughly correspond to the resolution of the image formed on the image plane. Considering this variation with respect to the above-mentioned three types of parallel light flux, it is possible to roughly evaluate the resolution of the formed image.

Further, for example, as another evaluation criterion, for each of the above three types of parallel light beams, the deviation between the position at which the principal ray reaches the image plane and the position at which the image is theoretically formed is measured. Think about it. An evaluation value obtained by comprehensively evaluating deviations from the theoretical values of three principal rays obtained for each of the principal rays (for example, measuring a standard deviation from an average value of the deviations) is an imaging value. This roughly corresponds to the distortion of the image.

As described above, when two different evaluation criteria are used and multi-objective optimization is performed to minimize both of the two objective functions corresponding to the two evaluations (resolution and distortion), 2 Many multi-objective optimal solutions (Pareto optimal solutions), such as a solution in which one of the objective function values is extremely small and the other is large, and both are fairly small
Will be present.

In the embodiment of the method of designing an optical system according to the present invention, as a genetic operator in a genetic algorithm which is one of evolutionary calculation methods, a crossover operator for directly operating a continuous value or a continuous operator Both crossover operators, which operate directly on values, and mutation operators, which operate directly on continuous values, are used. FIG.
Is a genetic representation of ten continuous value parameters characterizing the three-lens lens system shown in FIG. In each of a to g and d1 to d5 in the drawing, corresponding ones of the parameters of the lens system are stored as continuous values. Among such genes, n genes (n> 1) that satisfies the minimum constraint are arbitrarily generated.

Then, a crossover operator which directly handles continuous values for one to two genes appropriately selected is applied.
The crossover operator determines that the corresponding parameters in the two genes (for example, the value of the parameter a of one gene is a
In v1, the value of parameter a of the other gene is assumed to be av2), and the occurrence probability distribution of the new gene is set on the parameter (a) space.

The occurrence probability distribution generally has a shape in which the occurrence probability increases as approaching av1 and av2. Then, from within the subspace defined by the occurrence probability distribution, two values of the parameter a that the new gene should have, which appear according to the occurrence probability, (for example, av3
And av4) are selected. All of these tasks (see FIG. 1)
(In the example of 5, 10 parameters).

In the third embodiment, a method of treating individual parameters independently or a method of simultaneously selecting a plurality of parameters or all parameters in consideration of the correlation between parameters, such as NDX, can be applied. is there. Thus, from the n gene populations, the preset crossover rate P
According to c, Pc × n times, the one-to-two genes are more easily selected as the evaluation results are better, and are modified and extracted. The crossover operator is applied again to the extracted gene group.

Thus, the number of genes in the gene increases from n to n × (1 + 2Pc). When there is a single evaluation function (that is, when there is a single evaluation criterion, or when multiple evaluation criteria are put together and represented by one evaluation function)
Are always left unreconstructed and extracted so that genes with better evaluation values are more easily selected until the number of genes in the gene population reaches n. The other 2 × n × Pc genes are deleted.
Thus, a new n gene population is generated. The gene population generated in this way is called one generation of a genetic algorithm. When such generational changes are repeated,
As long as sufficient calculation time is allowed, an optimal solution will be eventually derived, and even when sufficient calculation time cannot be obtained, a good solution corresponding to the calculation time is efficiently generated.

When there are a plurality of evaluation functions and multi-objective optimization is performed, all (n
Of the × (1 + 2Pc) gene population, only the multi-objective optimal gene is left (or the multi-objective optimal gene is always left, and k-fold (k> 1) non-multi-objective optimal genes are also included). Other genes are killed to form a new generation of gene populations. However, in this case, care must be taken because the number n of genes in the gene population varies depending on the generation.

As a genetic operator that directly handles continuous values, not only a crossover operator that directly handles continuous values, but also a mutation operator that directly handles continuous values can be used. In this case, the gene is randomly reconstructed and extracted Pm × n × (1 + 2Pc) times from the (n × (1 + 2Pc)) gene group after the crossover operator is applied, according to a preset mutation rate Pm. Is done. Then, any (one or more) parameters of the gene are arbitrarily (or
A mutant nascent gene is generated that is altered (with a probability distribution biased near the original value). That is, the number of genes in the gene population immediately after the use of the mutation operator is n × (1 + Pm) × (1 + 2Pc).
Then, in the same manner as described above, a gene to be left in the next generation is selected from this gene population, and a next-generation gene population is generated.

In the above embodiment, the radius of curvature of the boundary surface,
Although the distance between the boundary surfaces has been described as a typical parameter, an optimum solution may be obtained by adding parameters of continuous values related to lens design such as the refractive index of each of the three lenses and the air pressure between the lenses. Needless to say. As described above, the method for designing an optical system according to the present invention employs the principle of sequentially selecting an approximate optimal solution from the two features of the principle of the genetic algorithm and finally presenting the optimal solution. Therefore, an approximate solution or an optimal solution according to the required calculation time can be efficiently generated. (Fourth Embodiment) Next, a fourth embodiment of the method for designing an optical system according to the present invention will be described with reference to FIGS.

In the third embodiment, FIG. 13 described above is used as a diagram for defining each parameter. FIG. 16 is a flowchart for explaining the procedure of the method for designing an optical system according to the fourth embodiment, and FIG. 17 is a diagram schematically showing operations in the fourth embodiment. The optical system shown in FIG. 13 includes the curvatures a to f of the lens surfaces (boundary surfaces) of the lenses L1 to L3, the distance between lens surfaces (distance between boundary surfaces) AB = d1, BC = d2, CD
= D3, DE = d4, EF = d5, FG = d6 and the refractive index of the medium between the lens surfaces nab = n1, nbc = n2, ncd =
n3, nde = n4, nef = n5, nfg = n6, Abbe number νab = ν1, νbc = ν2, νcd = ν3, νde = ν4, ν
It has parameters of ef = ν5 and νfg = ν6.

The chromosome of this optical system is -abcde
-f-d1-d2-d3-d4-d5-d6-n1-n2-n3-n4-n5-n6-ν1-ν2-ν3
It is represented by the gene string -v4-v5-v6-.
Here, the curvatures a to f of the lens surfaces are continuous values, and the lens surface intervals d1 to d6 are also continuous values. In the fourth embodiment, in the chromosome, the values (alleles) of the genes corresponding to the curvature and the surface interval are continuous values.

Next, the operation of the fourth embodiment will be described with reference to FIGS. 16 and 17. Here, first, the curvature a to f of each lens surface and the lens surface distances d1 to d6 are treated in the same dimension. Therefore, each of these parameters is -2.048
An example of normalization within the range of -2.048 will be described. In this case, the influence on each light beam is continuously equalized. For example, consider a lens system characterized by the following parameters:

[0067] In the above-mentioned lens system, when a desired focal length is obtained in a state where the lens surface (final surface) located closest to the image plane is pending, the above-mentioned lens system is expressed as a vector having a maximum of four dimensions or less. When the final surface is not pending, the above-described lens system is expressed as a vector having a maximum of five dimensions or less.

If the above-mentioned lens system is expressed without standardization using the five-dimensional vector (the refractive index is a fixed value), (0.0100,-
0.0083, -0.0077,8,5). In order to normalize each parameter (each vector component), it is necessary to set a maximum value and a minimum value with respect to at least the curvature and the lens surface interval (glass, air). The setting of the maximum value and the minimum value is performed by a user (designer). For example, -0.5 to +0.5 is set for the curvature, and 0.1 to 5 is set for the lens surface distance (corresponding to the lens thickness) sandwiching glass as a medium.
When 0 is set, and when the distance between the lens surfaces (air distance between lenses) sandwiching air as a medium is set to 0.1 to 100, the normalized value of each parameter is defined as follows. .

(Normalized value) = 4.096 × (parameter value) / (parameter maximum value−parameter minimum value) Therefore, the normalized vector is (0.04096, -0.03
39968, -0.0315392, 0.6567, 0.41042). In the fourth embodiment, first, a parameter t indicating a generation is set as an initial value.
Is set to 0 (step ST0).

Subsequently, the parameters a to f, d1 to d
6, a plurality of lens data having n1 to n6 and ν1 to ν6 as components are generated and set as an initial group (step ST
1). Here, a random number is used to generate a plurality of parameters that serve as an initial group. At this time, each parameter generated by the random number is standardized as described above. Then, when an optical system (candidate of an optical system to be designed) characterized by each of these parameters satisfies a predetermined constraint condition and all light rays incident on this optical system can reach the image plane, The parameters corresponding to the optical system are set as an initial group. In addition, as the constraint conditions,
For example, it is preferable to exclude a physically nonexistent optical system (such as an optical system having a negative lens surface interval) (the search becomes efficient). Further, such a constraint condition can be set by the designer himself when setting the maximum value and the minimum value such as the curvature.

In the fourth embodiment, since the angle of view and the combined focal length of the optical system are restricted by the designer, the curvature f of the lens surface closest to the image side of the optical system, As the distance d6 from the lens surface to the image plane, a value that satisfies the above-described constraint obtained by paraxial ray tracing for the optical system is used. In the fourth embodiment, the refractive indexes of the lenses L1 to L3 are given in advance, and the medium between the lenses L1 to L3 is air.
The chromosome of the optical system in this embodiment is -abcde-d1-
d2-d3-d4-d5-.

At this time, a plurality of individuals having different chromosomes exist in the initial population. continue,
After adding 1 to the generation parameter t (step ST2),
As shown in FIG. 17, a set of three parents Pa1, Pa2, and Pa3 are selected from the initial group (step ST).
3). If t> 2, a set of three parents Pa1, Pa2, and Pa3 are selected not from the initial group but from the group of the generation (t-1) immediately before the current generation (t). You.

In step ST4, a new gene is generated from the parents Pa1, Pa2, Pa3 selected in step 3 above. That is, crossover is performed. The crossover operator in the fourth embodiment includes UNDX (Uni-modal Normal Diode) among the genetic operators that directly handle continuous values.
distribution crossover) is used. Therefore, in the following description, step ST when UNDX is applied will be described.
4 will be described with reference to FIG.

First, an n-dimensional space corresponding to the number of genes in the chromosome of an individual is considered, and each individual is represented by this n-dimensional vector. In the fourth embodiment, as described above,
Since the chromosome is composed of 10 genes -abcde-d1-d2-d3-d4-d5-, n = 10 and each individual has 10
It is represented by a dimensional vector (a, b, c, d, e, d1, d2, d3, d4, d5). FIG. 18 is represented by a three-dimensional vector for illustration. Sub-step ST4-1 First, as shown in FIG.
In T4-1, the parent Pa1 selected in step ST3,
Points VC1, VC1 on the space respectively corresponding to Pa2, Pa3,
VC2 and VC3 are set. Here, the points VC1 and V
M is the middle point of C2, and a segment VC1VC2 from the point VC3.
Let H be the foot of the perpendicular that has been lowered. Sub-step ST4-2 Next, as shown in FIG.
At T4-2, a normal random number is generated with the midpoint M as the expected value and σa as the standard deviation. In accordance with the normal random numbers, a point P1 is generated on the line segment VC1VC2 (the point P1 is generated on the line segment VC1VC2 with a probability according to a normal distribution such that the midpoint M is a vertex). Substep ST4-3 Further, as shown in FIG. 18C, in substep ST4-3, an n-dimensional normal distribution according to n normal random numbers with the point P1 as an expected value and σb as a standard deviation is used. Is generated. In this embodiment, the probability space has a 10-dimensional normal distribution. Sub-step ST4-4 As shown in FIG.
In 4-4, the n generated in sub-step S4-3 is used.
The point P2 is generated according to an n-dimensional normal random number having a three-dimensional normal distribution probability space Ex. That is, the point P2 is generated with a probability according to the n-dimensional normal distribution. Sub-step ST4-5 As shown in FIG.
In 4-5, a hyperplane π including the point P1 and perpendicular to the vector VC1VC2 is set. The hyperplane π is a surface of dimension (n-1 order) obtained by subtracting 1 from the number of dimensions of the space, and has a ninth order in this embodiment. The foot dropped from the point P2 to the hyperplane π is defined as a point P3. Sub-step ST4-6 As shown in FIG.
In 4-6, a vector P1P4 having a component parallel to the vector P1P3 starting from the point P1 and having a length according to a normal random number with the point P1 as an expected value and σc as a standard deviation.
Are generated such that the point P4 is obtained. Where σc
Is proportional to the nth root of the distance HVC3 between the point H and the point VC3 in step S4-1.

The n-dimensional coordinates of the point P4 generated in the above steps ST4-1 to ST4-6 are the new genes, that is, the n parameters a, b, c, d, e, f, d1 of the child chromosome. , d
Corresponds to 2, d3, d4, d5. Step S of the fourth embodiment
In T4, the above-mentioned sub-steps ST4-1 to ST4
-6 is repeated m times, and three parents Pa1, Pa2, Pa
Three to m new genes are generated.

When step ST4 is completed, the evaluation value of the set of m new genes and two parents Pa1 and Pa2 (family set) generated in step S4 is calculated. In the fourth embodiment, the evaluation value (fitness) is represented by a single evaluation criterion or a single evaluation function φ by combining a plurality of evaluation criteria. For example, k evaluation criteria abe (1), a
If be (2), ... abe (k) exists, it is sufficient to weight each evaluation criterion abe (1), abe (2), ... abe (k) and linearly combine them . That is, the evaluation function φ is φ = W1 * abe (1)
+ W2 * abe (2) +... + Wk * abe (k). Then, the individual with the best evaluation value and one individual selected by roulette selection are selected from the family set (step S).
T5).

The roulette selection is a technique for selecting individuals at a predetermined probability in proportion to the fitness (evaluation value) of each individual or the rank of the fitness. For example, a case where four individuals A, B, C, and D exist will be described.
In the former fitness proportional method, the fitness of individual A is 40, the fitness of individual B is 60, the fitness of individual C is 100, and the fitness of individual D is
Is 200, the probability that individual A is selected is 10%, the probability that individual B is selected is 15%, the probability that individual C is selected is 25%, and the probability that individual D is selected is 50
% (A: B: C: D = 10: 15: 25: 50). Also, in the latter fitness rank proportional method, the probability that individual A is selected is 10%, and the probability that individual B is selected is 20%.
%, The probability that individual C is selected is 30%, and the probability that individual D is selected is 40% (A: B: C: D = 1: 2: 3: 4)
Becomes In the fourth embodiment, the latter fitness rank proportional method (ranking method) is used, but the present invention is not limited to this method.

Further, in step S6, individuals other than the two individuals selected in step S5 are killed (selected). Then, the selected individual is converted to the parent Pa in the initial population.
1, replacement with Pa2, and replacement of the selected individual with a parent in the group is called generational change. Subsequently, in step ST7, the individual having the best evaluation value is output from the individuals in the population whose generation has been changed (best solution presentation), and the operation of step ST2 is repeated again.

Here, the above steps ST2 to ST7 are as follows:
Called the first generation of genetic algorithms. In the second and subsequent generations, three parent Pas
1, Pa2 and Pa3 are selected. By repeating such generation alternation, an optical system as a global optimal solution that does not depend on a given initial solution can be automatically generated.

In the fourth embodiment, only the crossover operator is used. However, a mutation operator may be used in addition to the crossover operator. First, a case where a mutation operator is applied in addition to a crossover operator will be described. In the fourth embodiment, the above-described step ST
In 5 and ST6, a gene to be left to the next generation is selected from a set (family set) of m new genes and two parents Pa1 and Pa2.

When applying the mutation operator, m
A preset mutation rate Pm from a set of two new genes and two parents Pa1 and Pa2 (family set)
, A new chromosome is randomly extracted only Pm × (m + 2) times. Then, any (one or more) chromosomes among the genes of the extracted chromosomes are optionally
(Or with a probability distribution biased near the original value) produces a mutant, nascent gene. Thereafter, a gene to be left to the next generation is selected from a set of m new genes, a new gene generated as a mutant, and two parents Pa1 and Pa2. That is,
The individuals with the best evaluation value and the individuals selected by roulette selection from this set are the two parents Pa1, Pa
Replaced with 2.

In the fourth embodiment, only the mutation operator can be applied instead of the crossover operator. In this case, instead of steps ST3 to ST6, a new chromosome is randomly extracted from the population of chromosomes corresponding to the plurality of lens data by the number of times according to the mutation rate Pm. Then, a mutant is generated by changing any (one or more) genes of the extracted chromosome with a probability distribution biased near the original value. Then, chromosomes to be left for the next generation are selected from the set of mutants.

Next, a description will be given of a case where multi-objective optimization is performed in the fourth embodiment. As described above, in the fourth embodiment, in steps ST5 and ST6, when an individual having the best evaluation value is selected from a set (family set) of m new genes and two parents Pa1 and Pa2, and roulette When an individual is selected by selection, one evaluation function φ is used. Then, these two selected individuals are replaced with the two parents Pa1 and Pa2 in the group. That is, even when there are a plurality of evaluation criteria, these plurality of evaluation criteria are treated as a single evaluation function by being linearly combined.

When performing multi-objective optimization, the following steps S11 and S15 to S17 are executed in place of steps ST1 and ST5 to ST7 described above. Hereinafter, FIG.
The operation of the multi-objective optimization will be described with reference to the flowchart shown in FIG. 20 and the schematic diagram of the operation shown in FIG. The difference between step ST11 and step ST1 is that a plurality (k) of a plurality of individuals (lens data) in the initial group is different.
The evaluation values of a plurality of evaluation functions φ1 to φk corresponding to the evaluation criteria are calculated.

Next, in step ST15, the m new genes and the two parent Ps generated in step ST4 are generated.
The evaluation values of the plurality of evaluation functions φ1 to φk included in each set (family set) of a1 and Pa2 are calculated. Then, an individual that is not Pareto-optimal (non-Pareto solution) is selected from these evaluation values. Here, the Pareto-optimal solution means a solution in which, when there are a plurality of evaluation criteria, at least one of the plurality of evaluation criteria is superior to the other solutions.

Subsequently, in step ST16, individuals that are not Pareto-optimal are eliminated. That is, among the evaluation values of the plurality of evaluation functions φ1 to φk, an individual that is inferior to other individuals in any of the evaluation functions is deleted. Here, the remaining Pareto-optimal individuals become the next-generation group, and in the next generation, a parent for the crossover operator is selected from this group.

After the end of step ST16, step ST
In step 17, the Pareto-optimal individual obtained in step ST16 is output, and each step is repeatedly executed in order from step ST2 again. Hereinafter, similarly, step S
By repeating T2 to ST4 and steps S15 to S16, multiobjective optimization of individuals in the population can be performed.

Even when such multi-objective optimization is performed, it is possible to apply the above-described mutation parameter in addition to the crossover operator. Also, a mutation parameter can be applied instead of the crossover operator. The above steps (steps ST11, ST2 to ST2)
By performing ST4 and ST15 to ST17), even if there are a plurality of evaluation criteria of the optical system, it is possible to execute multi-objective optimization without applying a trade-off ratio (weight) to the plurality of evaluation criteria. it can. As a result, the best individual among the plurality of chromosomes in the population is always retained, and the more excellent individuals are selected with a predetermined probability so as to be less likely to be selected and are left for the next generation, thereby improving efficiency. A good multipoint simultaneous search becomes possible.

In such Pareto optimization, the number of individuals in a group differs depending on the number of generations. However, the Pareto optimization as described above has an advantage that the group size does not become extremely large because non-Pareto solutions are eliminated. In the above-described fourth embodiment, UNDX is applied as the crossover operator, but BLX-α or NDX may be applied instead.

Further, for example, some of the plurality of evaluation criteria may be linearly combined to form a plurality of evaluation functions, and the plurality of evaluation functions may be multi-objectively optimized. Hereinafter, an application example applied to an optical system including 15 lenses by the method for designing an optical system according to the fourth embodiment will be described. (Application Example 1) In Application Example 1 of the fourth embodiment, a projection optical system for transferring a circuit pattern on a mask onto a wafer at a reduced magnification is designed.

In the first application example, the criteria (1) to (16) listed below are applied as evaluation criteria of the optical system, and an evaluation function obtained by linearly combining a plurality of evaluation criteria is applied. (1) Spherical aberration at maximum numerical aperture (2) Spherical aberration at 75% of maximum numerical aperture (3) Sagittal image plane at maximum object height (4) Meridional coma of upper ray at 70% of maximum numerical aperture at maximum object height ( 5) Meridional coma of lower ray at 70% of maximum numerical aperture at maximum object height (6) Meridional coma of upper ray at 50% of maximum numerical aperture at maximum object height (7) 50% of maximum numerical aperture at maximum object height Meridional coma of lower ray (8) Meridional coma of upper ray of 70% of maximum numerical aperture at 75% of maximum object height (9) Meridional coma of lower ray of 70% of maximum numerical aperture at 75% of maximum object height (10 ) Meridional coma of upper ray of 50% of maximum numerical aperture at 75% of maximum object height (11) Meridional coma of lower ray of 50% of maximum numerical aperture at 75% of maximum object height (12) 50% of maximum object height Maximum in Meridional coma of upper ray at 70% of the numerical aperture (13) Meridional coma of lower ray at 70% of maximum numerical aperture at 50% of maximum object height (14) Upper ray of 50% of maximum numerical aperture at 50% of maximum object height (15) Meridional coma of the lower ray of 50% of the maximum numerical aperture at 50% of the maximum object height (16) Distortion aberration at the maximum object height In this application example 1, (17) to (17) are listed below. (19)
Are added. (17) Distance WD from the lens surface closest to the reticle side to the reticle> 25 (18) Magnification β of the entire system = -3.0 (when ray tracing from the wafer side to the reticle side) (19) Distance d < 25 In the first application example, the parameters that are changed when designing the optical system are the curvature and the surface interval of each lens surface. The glass material refractive index n of each lens element is given in advance as n = 1.56384.

Under the above assumptions, the parameters in the method for designing an optical system according to the fourth embodiment are set as listed below. Initial population size: 50 Number of intersections: 300,000 Number of children generated by the intersection operator: 20 σa of UNDX: 0.5 × VC1VC σb of UNDX: σc of UNDX: 0.35 × (VC1VC2) 1 / n From the adaptation example 1 described above, the projection optical system shown in Table 1 below was generated. In Table 1, β is the magnification from the reduction side (wafer side) to the enlargement side (reticle side), L is the conjugate length (distance between object images), NA is the numerical aperture on the reduction side, and WD is the most reticle side. The distance from the lens surface to the reticle is shown. In Table 1 below, the numerical value at the left end represents the surface number from the wafer surface, r is the radius of curvature, d is the surface interval,
n represents a refractive index, respectively.

[0093]

[Table 1] β = -3.0 L = 380.52758 NA = 0.30 WD = 24.96409 r dn 0 ∞ 12.49751 1.000000 (conjugated surface on the wafer side) 1 -120.85562 18.23479 1.563840 2 -70.82037 12.67890 1.000000 3 -92.19251 7.16914 1.563840 4 -59.21449 17.55595 1.000000 5 -2708.99510 18.44782 1.563840 6 -132.78761 6.51286 1.000000 7 ∞ 8.63269 1.000000 (aperture stop) 8 173.72702 17.51225 1.563840 9 -238.22683 12.06856 1.000000 10 -101.43114 17.11040 1.563840 11 -203.60368 15.33085 1.000000 12 166.15887 5.69882 1.56322788 -13. 6.11086 1.563840 15 -78.07944 6.05381 1.000000 16 74.48074 12.50264 1.563840 17 -220.48073 11.91135 1.000000 18 -82.31072 8.08050 1.563840 19 275.42052 19.38991 1.000000 20 -67.68594 9.24998 1.563840 21 -210.48779 7.57457 1.000000 22 -244.62903 15.02358 1.563840 23. 97.21932 10.30374 1.000000 26 -28.90898 14.32626 1.563840 27 102.56872 7.92284 1.000000 28 43.32668 8.22930 1.563840 29 2828.72964 10.81079 1.000000 30 243.23452 16.81079 1.563840 31 39.78134 24.96409 1.000000 32 共 役 (Conjugate plane on the reticle side) FIG. 21 is a cross-sectional view of the projection optical system shown in Table 1 above, and FIG. Here, FIG.
22A is a spherical aberration diagram, FIG. 22B is an astigmatism diagram, FIG. 22C is a meridional coma at the maximum object height, FIG. 22D is a meridional coma at 75% of the maximum object height, and FIG. (E) is a meridional coma at 50% of the maximum object height, FIG. 22 (f) is a meridional coma on the optical axis, FIG. 22 (g) is a sagittal coma at the maximum object height, and FIG. 22 (h) is a 75% of the maximum object height. 22 (i) is a sagittal coma at 50% of the maximum object height, FIG. 22 (j) is a sagittal coma on the optical axis, and FIG. 22 (k) is a distortion diagram. In the astigmatism diagram of FIG. 22B, a broken line indicates a meridional image plane,
The solid line represents the sagittal image plane. In FIG. 22, NA indicates the numerical aperture, and Y indicates the object height.

FIG. 23 shows that the size of the initial group is 4
00, the cross-sectional view of the projection optical system when 24,000 times of crossover are performed so as to generate 50 children is shown in an adaptive example 2.
It is shown as As described above, in the method of designing the optical system according to each of the application examples 1 and 2, a good solution (individual) can be obtained despite the large number of lens elements.

Note that in the above-mentioned adaptation examples 1 and 2,
Although the glass material of each lens is given, it goes without saying that this may also be included in the parameters. At this time, since the glass material often exists discretely, a genetic parameter that handles discrete values may be used.

[0096]

As described above, according to the present invention, it is possible to generate an optical system as a global optimal solution that does not depend on a given initial solution. Further, in the present invention, since the genetic parameter that explicitly handles continuous values is used, it is possible to repeatedly inherit the traits (characteristics) of the parent individual to the next-generation offspring individual generated from the parent individual. Since there is no need to perform useless search, there is an effect that an optimal solution or an approximate solution of a matching optimal solution can be obtained in a shorter time. Further, according to the present invention, there is an effect that a plurality of optical systems can be simultaneously generated as a multi-objective optimal solution that should exist in correspondence with a plurality of conflicting evaluation criteria.

[Brief description of the drawings]

FIG. 1 is a diagram showing a medium on which a program for realizing an optical system designing apparatus and an optical system designing method according to the present invention is recorded.

FIG. 2 is a diagram for explaining a crossover operator in the optical system designing method according to the present invention.

FIG. 3 is a diagram for explaining a generation alternation model according to the first embodiment of the present invention.

FIG. 4 is a diagram illustrating an example of an initial group according to the first embodiment of the present invention.

FIG. 5 is an optical path diagram of an optical system (lens system) obtained in Experiment 1 of the first embodiment according to the present invention.

FIG. 6 is an aberration diagram of the lens system shown in FIG. 5;

FIG. 7 shows a lens system (focal length 50 mm, brightness F2.0, focal length 50 mm) obtained in Experiment 2 of the first embodiment according to the present invention.
It is a figure which shows the optical path diagram of 46 degree of view, and a spot diagram.

FIG. 8 shows a lens system (focal length 135 mm, brightness F2.
8 shows an optical path diagram and a spot diagram at an angle of view of 18.2 °).

FIG. 9 shows a lens system (focal length: 20 mm, brightness: F5.6, obtained by Experiment 2 of the first embodiment according to the present invention).
It is a figure which shows the optical-path figure of 92 degree of view, and a spot diagram.

FIG. 10 is a diagram illustrating a generation alternation model according to a second embodiment of the present invention.

FIG. 11 is a diagram showing a Pareto solution set obtained by the second embodiment according to the present invention.

FIG. 12 is a diagram showing a state in which the optimum solution obtained in Experiment 1 of the first embodiment according to the present invention is plotted on FIG. 11;

FIG. 13 is a diagram for specifically explaining parameters in a third embodiment according to the present invention.

FIG. 14 is a diagram showing how various light beams converge on an image plane when a spot diagram is used as an example of evaluating the photographic lens of FIG. 13;

FIG. 15 shows a case where a plurality of continuous value parameters representing a lens optical system are set to continuous values when an angle of view, a focal length, and a refractive index of each lens glass material are given in advance and each of them is a constraint. This is an example (real number vector) in which genes are represented as they are.

FIG. 16 is a flowchart illustrating a method for designing an optical system according to a fourth embodiment of the present invention.

FIG. 17 is a view schematically showing the operation of the fourth embodiment according to the present invention.

FIG. 18 is a diagram illustrating a crossover operator to which UNDX is applied.

FIG. 19 is a flowchart for explaining an operation when performing multi-objective optimization in the optical system designing method according to the fourth embodiment of the present invention.

FIG. 20 is a diagram schematically illustrating an operation when performing multi-objective optimization in the optical system designing method according to the fourth embodiment of the present invention.

FIG. 21 is an optical path diagram of a projection optical system generated according to a fourth embodiment of the present invention (part 1).

22 is an aberration diagram of the projection optical system shown in FIG.

FIG. 23 is an optical path diagram of a projection optical system generated according to a fourth embodiment of the present invention (part 2).

FIG. 24 is a diagram for describing a coding / crossover method in a conventional optical system design method using discrete GAs.

[Explanation of symbols]

DESCRIPTION OF SYMBOLS 1 ... Computer, 100 ... Main control system, 101 ... Control part, 102 ... Calculation part, 103 ... Memory, 104 ... Input part, 105 ... Output part, 114 ... Recording medium (CD), 11
5. Display device (CRT).

 ────────────────────────────────────────────────── ─── Continuing from the front page There is an application for the application of Article 30, Paragraph 1 of the Patent Law June 26-June 27, 1997 The 22nd Optical Symposium (Japan Optical Society) hosted by the Japan Society of Applied Physics ( (72) Inventor Shigenobu Kobayashi 4259 Nagatsuda-cho, Midori-ku, Yokohama-shi, Kanagawa-ken, Tokyo Institute of Technology

Claims (23)

[Claims] At least two parent individuals are selected from a population of n (≧ 1) generations, each of which is a real number vector representing a candidate for an optical system to be designed. A selection step, by applying at least one of a crossover operator and a mutation operator as a genetic operator to the selected parent individual, a child generation step of newly generating a population of a plurality of child individuals. A selection step of selecting an individual to be an individual of a next generation population from the population of the n generations and the population of the child individuals. 2. The method according to claim 2, wherein, in the selection step, individuals satisfying at least one of one or two or more evaluation criteria are selected from the n generation population and the generated offspring population. The method for designing an optical system according to claim 1, wherein the optical system is selected as an individual. 3. In the child generation step, the crossover operator includes: a subspace defined by a continuous predetermined occurrence probability distribution set based on a real number vector component of each of the selected parent individuals; 3. The method of designing an optical system according to claim 1, wherein a real number vector having a component appearing according to the occurrence probability as a component is generated as a child individual. 4. In the child generation step, the mutation operator is defined as a continuous predetermined occurrence probability distribution in which the occurrence probability increases as approaching at least one parent individual among the selected parent individuals. From within the subspace
2. The method according to claim 1, wherein a real number vector having a component appearing according to the occurrence probability as a component is generated as a child individual.
Or the method for designing an optical system according to 2. 5. The method for designing an optical system according to claim 1, wherein the selection step, the child generation step, and the selection step are performed a plurality of times in order. 6. A generation of a population comprising a plurality of individuals, wherein each individual has a plurality of parameters representing candidates for an optical system to be designed including at least one optical element. In a method for designing an optical system that optimizes an optical system to be designed while repeatedly selecting an individual to be an individual in a group, the method for designing at least one of a plurality of parameters of the individual The optimization includes selecting a plurality of parent individuals from the generated individuals, setting a continuous predetermined occurrence probability distribution based on the selected parameters of each of the plurality of parent individuals, A light which newly generates a child individual having a value which appears according to the occurrence probability as a value of the selected parameter from a subspace defined by a distribution. Academic design method. 7. An individual having, as a value of the selected parameter, a value matching any one or more of two or more evaluation criteria from a group including at least the parent individual and the generated child individual. 7. The method for designing an optical system according to claim 6, wherein the method is selected as an individual of a next generation population. 8. The method according to claim 8, wherein the selected parameter of the individual is at least one of a radius of curvature of a boundary surface in the optical element, a distance between the boundary surfaces, and a refractive index of a space located between the boundary surfaces. 7. The method for designing an optical system according to claim 5, wherein 9. A group of a plurality of individuals corresponding to each candidate for an optical system to be designed, each individual being a real vector n having one or more predetermined parameters characterizing the optical system as components. (≧ 1) a parent selecting step of selecting at least two parent real vectors from the generation population, and a continuous predetermined occurrence probability distribution set based on the real vector components of each of the selected parent individuals. A crossing step of generating, as a child individual, a real vector having a value that appears according to the occurrence probability as a component from within the defined and expressed subspace, and approaching at least one parent individual among the selected parent individuals. From a subspace defined by a continuous predetermined occurrence probability distribution in which the occurrence probability increases with the real number vector having a value that appears according to the occurrence probability as a component, A child generation step of performing at least one of a mutation step of generating as an individual, and a selection step of selecting an individual to be an individual of a next generation group from the n generation population and the generated child individuals, How to design an equipped optical system. 10. The optical system according to claim 9, wherein, in the selection step, the selected individual is replaced with an unselected individual in the n-generation population to generate a next-generation population. Design method. 11. The selection step includes, in a group of the parent individual and the generated offspring individual, an individual that most closely matches a predetermined evaluation criterion, and the next generation in succession in proportion to the fitness of each individual. 11. The method for designing an optical system according to claim 9, wherein an individual to be an individual is selected. 12. In the selection step, an individual satisfying at least one of one or two or more evaluation criteria is selected from a population of the parent individual and the generated offspring individual. The method for designing an optical system according to claim 9, wherein the method is selected as: 13. The real vector component of the individual is at least one of a radius of curvature of a boundary surface of the optical element, a distance between the boundary surfaces, and a refractive index of a space located between the boundary surfaces. The method for designing an optical system according to any one of claims 9 to 12. 14. Generating a plurality of parameters each representing a candidate for an optical system to be designed including at least one optical element, and selecting a parameter to be left out of the generated plurality of parameters. While iteratively executing, the optical system design device comprising an arithmetic unit that optimizes the optical system to be designed, and a memory that temporarily stores the generated parameters, wherein the arithmetic unit is From an n (≧ 1) generation group consisting of a plurality of real vectors given as the plurality of parameters,
A parent selection step of selecting at least two parent real vectors, from within a subspace defined by a continuous predetermined occurrence probability distribution set based on the real vector components of each of the selected parent individuals, A crossover step of generating, as a child individual, a real vector having a value that appears according to the occurrence probability as a child individual; and a continuous generation in which the occurrence probability increases as approaching at least one parent individual among the selected parent individuals. A child generation step of executing at least one of a mutation step of generating, as a child individual, a real vector having a component appearing in accordance with the occurrence probability as a component from a subspace defined by a predetermined occurrence probability distribution, at least a selection step of selecting an individual to be an individual of the next generation population from the n generation population and the generated offspring individuals. That the optical system design device. 15. The operation unit according to claim 14, wherein in the selection step, the selected individual is replaced with an unselected individual in the n-generation population to generate a next-generation population. Optical system design equipment. 16. The computing unit according to claim 1, wherein, in the selection step, among the population of the parent individual and the generated offspring individual, the individual that is most suitable for a predetermined evaluation criterion is in proportion to the fitness of each individual. 16. The optical system designing apparatus according to claim 14, wherein an individual to be a next-generation individual is selected in turn. 17. The computing unit, in the selection step, comprising: a group of the parent individual and the generated offspring individual
The optical system design according to any one of claims 14 to 16, wherein an individual who satisfies at least one of one or two or more evaluation criteria is selected as an individual of a next-generation population. apparatus. 18. The real vector component of the individual handled by the arithmetic unit is at least one of a radius of curvature of a boundary surface of the optical element, a distance between the boundary surfaces, and a refractive index of a space located between the boundary surfaces. The optical system designing apparatus according to any one of claims 14 to 17, wherein: 19. A group of a plurality of individuals each corresponding to a candidate optical system to be designed, each individual being a real vector n having one or more predetermined parameters characterizing the optical system as components. (≧ 1) a parent selecting step of selecting at least two parent real vectors from the generation population, and a continuous predetermined occurrence probability distribution set based on the real vector components of each of the selected parent individuals. From within the defined subspace, a crossover step of generating, as a child individual, a real vector having a component that appears according to the occurrence probability as a component, and as the selected parent individual approaches at least one parent individual, From a subspace defined by a continuous predetermined occurrence probability distribution in which the occurrence probability is high, a real vector having a component appearing according to the occurrence probability as a component is defined as a child individual. A child generation step of performing at least one of a mutation step of generating, and a selection step of selecting, from the n generation population and the generated child individuals, individuals to be individuals of a next generation population. Media on which the recorded program is recorded. 20. The program according to claim 19, wherein in the selection step, the selected individual is replaced with an unselected individual in the n-generation population to generate a next-generation population. The described medium. 21. The selecting step, wherein, in the population of the parent individuals and the generated offspring individuals, the individuals that best meet a predetermined evaluation criterion are sequentially ordered in proportion to the fitness of each individual. 21. The medium according to claim 19, wherein a program for selecting an individual to be an individual is recorded. 22. In the selection step, an individual who satisfies at least one criterion among one or two or more types of evaluation criteria is selected from a population of the parent individual and the generated offspring individual. 22. The medium according to any one of claims 19 to 21, wherein the program is recorded as selected as: 23. The real vector component of the individual is at least one of a radius of curvature of a boundary surface of the optical element, a distance between the boundary surfaces, and a refractive index of a space located between the boundary surfaces. The medium according to any one of claims 19 to 22, wherein a program to be executed is recorded.