## Optimization of the Cooke triplet with the various evolution strategies and the damped least squares

Darko Vasiljeviĉ
darkov@diginaut.com

Keywords: evolution strategies, damped least squares, optical design, optimization, Cooke triplet

ABSTRACT

Optimization of the Cooke triplet with various evolution strategies and the damped least squares is presented. All algorithms are described and their advantages and shortcomings are presented. After detailed presentation of the evolution strategies their adaptation to the optimization of optical systems are discussed. Analysis of the Cooke triplet optimizations is given and optimum optimized optical system is chosen.

1.Introduction

Problem of the automatic lens design and optimization of optical systems is very old. Many researchers proposed various methods or their improvement in order to solve this problem, which belongs to a class of highly nonlinear optimization problems. All optimization problems can be classified in two broad groups:

• Classical optimization methods which are based on the computations of the first and the second partial derivatives. In the optimization of optical systems some kind of the damped least squares (DLS) is mostly used.
• Modern optimization methods which are based on analogies in nature like the simulated annealing (SA) and the evolutionary algorithms like the genetic algorithms (GA), the evolution strategies (ES) and the evolutionary programming (EP).

Currently many automatic lens design programs are commercially available or developed for proprietary use and most of them employ some variant of the damped least squares. Author’s intention is to try to implement modern optimization techniques in the optimization of optical systems. Following evolutionary algorithms, based on the evolution strategies which are applied in the optimization of optical systems, are presented in this paper:

• two membered evolution strategy – (1+1) ES – EVOL;
• multimembered evolution strategy in three variants:
• with mutation as only genetic operator –(μ, λ) ES – GRUP;
• with mutation and recombination as genetic operators – (μ, λ) ES – REKO;
• the most general method with various genetic operators – (μ, λ) ES – KORR.

All those algorithms are compared among themselves and with the classical damped least squares optimization and the best evolutionary algorithm is selected.

Research in applying those algorithms in optimization of various types of objectives (doublets cemented and separated, Cooke triplets, Tessars, Petzvals) are conducted for several years and it is subject of author’s Ph.D. Thesis 1. First results in applying evolutionary algorithms in optimization of the cemented doublets are published in 2, 3. All optimizations are done on proprietary optical design program called ADOS (Automatic Design of Optical Systems) developed as a part of the author’s Master of Science Thesis 4.

2.Definition of the problem

Cooke triplet is one of very popular objectives, which is used mainly for normal and narrow angle applications. Maximum field angle for this objective is 50˚ - 60˚ and maximum relative aperture is f/2.8. As a rule one can say that the higher the aperture of a Cooke triplet, the smaller the field that it covers. Standard Cooke triplet found in Cox 5 (catalog of the optical systems at the end of the book) is used for testing various optimization methods.

Optical systems are defined by the parameters associated with the individual surfaces of the system. These parameters can be divided in two main groups:

• basic parameters which must be defined for every optical system;
• optional parameters which are defined only if this parameter exists, for example if the surface is aspheric one must define aspherical constant.

Basic parameters are following parameters:

• radius of curvature for each surface of optical system;
• separation between the surfaces of optical system;
• glasses from which are made components of optical system;
• clear radius for each surface of optical system.

To completely define and analyze optical system, one must, beside basic parameters, define ray data, which describes rays that are to be traced through the optical system.

The optimization concept is fairly simple but its implementation can be quite complex. One can define it as given an initial system definition and set of target performance goals determine the set of system configuration variables which minimizes the deviation of actual performance from targeted goals without violating boundary limitations. This simply stated problem becomes a complex problem when the number of variables and targets are large, errors are nonlinear functions of the variables, and the errors are not orthogonal to each other with respect to the variables. In the typical optical system optimization case, all of these conditions are true to varying degrees.

When working with various types of optimization methods one usually define single number, called a merit function value, to characterize actual system performance compliance with the targeted system performance. In other words the merit function value is the measure of the effectiveness of the optimization method as it is the goal of the optimization to reduce the merit function value.

The selection of an appropriate merit function for the optimization is fundamental to the successful outcome of the process. From the mathematical point of view most appropriate merit function is function in quadratic form. This type of the merit function is used in all optimization methods (classical damped least squares and various evolution strategies). It can be defined as a sum of squares of the aberrations:

where is:

ψ - the merit function value;

m - the number of parameters of the optimization;

ωi - waiting factor for each calculated aberration;

fi - each aberration that is calculated by raytrace through the optical system.

Waiting factor for each calculated aberration is necessary because one calculate different types of aberrations (transverse ray, angular, waveform) that may differ very much. In order to be able to compare aberrations and to reduce their values one have to bring them to similar values.

3. Classical damped least squares optimization

Least squares optimization is a modification of the Newton - Raphson method first developed by Levenberg [6]. It was introduced into optics by Rosen and Eldert [7], Merion [8,9], Wynne [10] and others. Damped least squares optimization evolved after a period of intensive research and experimentation in late 1950s and mid 1960s. Now almost every optical design program has some kind of the DLS optimization. Experience gained with this method seams to confirm that the DLS optimization is probably very efficient and general method available to the optical designer. Author’s implementation of the DLS optimization is based upon works of the researchers from the Imperial College in London (Wynne, Wormell and Kidger [11-14]). Mathematical theory of the DLS optimization is well known and will not be presented here.

The DLS optimization belongs to a broader group of linear optimization models, which does not take any explicit account of the fact that there may be many local minima of the merit function in the space of all variables. The number of local minima depends on merit function and number of variables. The damped least squares generally drives the merit function to the local minima nearest the starting point (optical system). Optical designer has several tools besides optical design program, which allows him possibility to find satisfactory optical system. These tools are:

• The selection of a promising starting optical system. If one optical system doesn’t give required quality optical designer might try with another entirely different optical system.
• The choice of optimization variables. A poor choice of variables may even prevent a linear convergence.
• The weights used in construction of the merit function. Good selection of weights is essential in balancing the aberrations.
• The choice of damping factor (adaptive or multiplicative damping) or choice of modification the whole merit function.

These tools are highly dependent on the skill and experience of the designer. Even an experienced designer may have difficulty in searching out a satisfactory solution for a "state-of-art" design requirement.

4.Evolution strategies

Evolution strategies are algorithms, which imitate the principles of natural evolution, such as mutation, recombination and selection as a method to solve parameter optimization problems. Bienert, Rechenberg and Schwefel developed them during 1960s at the Technical University of Berlin in Germany. From that time they evolved from relatively simple (1+1) ES to a powerful, robust and self adapting tool for mathematical and technical optimization. The main application domain of ES is optimization of high dimensional continuous problems. The strategy performs well in domains where it is impossible, difficult or expensive to find a precise mathematical description of the problem at hand.

The most general algorithmic description of ES is following:

1.The problem is defined as finding the real-valued n dimensional vector x that is associated with the minimum of the function F(x).

2.An initial population of parent vectors, is selected at random from feasible range in each dimension. The distribution of initial trials is typically uniform.

3. An offspring vector is created from each parent by adding a Gaussian random variable with zero mean and preselected standard deviation to each component of x.

4. Selection then determines which of those vectors to maintain by ranking the errors  and The N vectors that possess the least error become the new parents for the next generation.

The process of generating new trials and selecting those with least error continues until an optimum solution is reached or the available computation is exhausted.

Schwefel gives detail mathematical theory of the evolution strategies in 15. First he developed two membered ES as a minimal concept for organic evolution. Principles of mutation and selection are used for change of variable parameters and choosing of the individuals during optimization. To completely specify algorithm one must define following things:

• define algorithm;
• define optimization’s step length (in this case standard deviation for the Gaussian normal distribution);
• define convergence criterion.

Schwefel in 15 first gives description of the algorithm using only terms from biology and after that using terms from mathematics.

For the control of the optimization’s step length Schwefel in 15 uses 1/5 success rule which reflects theoretical result that, on average, one out of five mutation should cause an improvement of the merit function. The 1/5 success rule states that the ratio of successful mutations to all mutations should be 1/5. If it is greater then 1/5, optimization method should increase the step length, otherwise decrease the step length.

For the convergence criterion Schwefel in 15 uses two functions:

• the optimization step length must be smaller then predefined small value;
• the difference in two adjacent merit functions values must be smaller than predefined small value.

Fogel in 16 discusses two major drawbacks of the two membered ES:

• constant step length in each dimension made the optimization slow to converge on optimal solution;
• brittle nature of point-to-point search made the optimization susceptible to stagnation and local minima, although the optimization can be shown to asymptotically converge to the global optimum.

The two membered ES offers only essential imitation of the evolutionary process. In order to overcome these difficulties Schewefel decided to reach higher level of imitation of the evolutionary process by increasing the number of the individuals in the population and including set of the genetic operators instead of only one genetic operator (mutation). He developed several multimembered ES. First come method called GRUP. It has μ parents and λ offsprings (λ ≥ 6μ). It is generational type of ES, which means that λ offsprings chose among themselves μ best offsprings to form new generation. In Schwefel notation one can write (μ, λ) ES – GRUP. It uses only one genetic operator – mutation. Further research in this field took Schwefel to the new method of the multimem-bered ES which he called REKO. It has everything that has method GRUP and has two genetic operators:

• mutation of variable parameters according to Gaussian normal distribution law;
• recombination of the optimization step lengths.

The final and most advanced method of the multimembered ES developed by Schwefel is method KORR. It has everything that have methods GRUP and REKO and adds many more features. KORR strategy has following characteristics:

• Two possibilities of the parent selection for the next generation:
• Plus strategy – parents and offsprings form population from, which are chosen best individuals to, become parents in the next generation. Theoretically it is possible that one individual lives for several generations.
• Coma strategy – parents for new generation are chosen only from the current generation offsprings. Here every individual lives only for one generation.
• Five different genetic operators:mutation of the variable parameters according to Gaussian normal distribution law;
• discrete recombination of pairs of parents;
• intermediary recombination of pairs of parents;
• discrete recombination of all parents
• intermediary recombination of all parents in pairs.

All genetic operators can be applied to:

• object variables;
• optimization step lengths;
• rotation angles of the mutation hyperellipsoid.

KORR is only ES method that has linearly correlated mutations.

Schwefel in 15 gives all necessary definitions for the multimembered ES. He first gives description of the algorithm using only terms from the biology and after that using terms from the mathematics.

In the multimembered ES there is no fixed rule to control step lengths. There step lengths become variable parameters and can be changed along with other variables in the optimization. In this way nature is simulated more precisely.

Convergence criterion for the multimembered ES is the same as for the two membered ES.

5.Application of the evolution strategies to the optimization of optical systems

Optimization of optical systems is very specific and evolution strategies must be adapted to it. Flow chart diagram is given on the figure 1. Each type of the evolution strategy needs different input data to be defined. Simple rule is the more complex type of the evolution strategy the more input data needs to be defined. To start optimization with the evolution strategies one needs a starting point i.e. initial optical system. This system must be valid i.e. it has to fulfill all necessary geometric boundary conditions. If this initial optical system is valid, then one can calculate aberrations and merit function and proceed with search for improved optical system. If this initial optical system don't fulfill all conditions one have to formulate auxiliary merit function which represents deviation measure from the geometric boundary conditions. This auxiliary merit function is optimized by the same evolution strategy. Process of the optimization is stopped when all conditions are fulfilled i.e. auxiliary merit function is equal to zero.

After detail testing of the starting optical system one have to make initial population by random changes of the starting optical system according to Gaussian normal distribution law. Each new optical system must be completely tested by:

• calculating all necessary paraxial values;
• calculating ray data;
• testing boundary geometric conditions;
• calculating aberrations and merit function.

If the new optical system is valid it is accepted in the population, otherwise it is rejected. When the whole initial population completed the evolution process can start. First thing is forming new optical system by application of the genetic operators. Various ES methods have different genetic operators, from only point mutation in EVOL to quite complex and numerous genetic operators in the KORR. When genetic operators form new optical system, optimization method must decide whether optimization is possible or not. If the optimization is not possible and search method is looking for the new starting point (valid optical system) then the geometrical boundary conditions of the new optical system is tested. If all conditions are fulfilled then new valid initial optical system is found and optimization is stopped. It has to be restarted with this optical system as an initial optical system.

Figure 1: General flow chart for the evolution strategies

If the optimization is possible then new optical system is tested. If it is OK, optical system is accepted in the population. Process of forming, testing and accepting new optical systems is repeated until whole new population is not fulfilled. After forming new population, the best optical systems with minimum merit functions are selected to become parents for next generation. It is important to notice that the optimization method knows in every moment which optical system is the best with minimal merit function.

At the end of the each generation optimization method is testing fulfillment of the all convergence criterions. If any of the criterions is fulfilled then optimization is finished and currently best optical system becomes optimal optical system. If the convergence criterions are not fulfilled then new generation is started and new optical systems are made.

6. Analysis of the Cooke triplet optimizations

Cooke triplet, one relatively simple optical system, is chosen to be starting point for optimization. This is because the evolutionary algorithms are rather time consuming and if there are lot of variable design parameters they may take time before they come with results. They also have stochastic nature so they ought to be run several times. Each evolutionary algorithm is run five times which is minimum value for calculating necessary average values. Optimization by each algorithm is done when:

• radiuses and separations are variable.

Glasses aren't varied because they are discrete variables that are taken from glass database. Radiuses and separations are continuous variables and the evolutionary strategies are optimization algorithms that use only continuous variables for the optimization.

Principal optical data for the chosen Cooke triplet are given in table 1:

Table 1: Principal optical data for the Cooke triplet

 focal length [mm] f = 25 relative aperture (f-number) f / 4 aperture stop at the fifth surface field angle [°]

Results of the optimizations are characterized by the merit function of the optical system. Results from the optimizations when only radiuses were variable are presented in table 2:

Table 2: Optimization of the Cooke triplet with only radiuses variable

 (1+1) ES - EVOL (m, l) ES - GRUP (m, l) ES - REKO (m, l) ES - KORR 1st run 4.462 4.466 4.464 4.462 2nd  run 4.462 4.467 4.465 4.463 3rd  run 4.462 4.464 4.463 4.464 4th  run 4.462 4.470 4.462 4.465 5th  run 4.462 4.466 4.463 4.465 average 4.462 4.466 4.463 4.464 standard deviation 0.000 0.002 0.001 0.002 DLS optimization 4.544

One can see very small variation of the merit function with change of the optimization algorithm. It is important to notice that initial Cooke triplet is very good optical system and optimization methods can improve only little this optical system. When the radiuses of the optical system are only variables DLS method didn’t find any improvement. Optimized and initial optical systems are identical. Various evolutionary strategies found little improvement and optical system with the smallest merit function is found by (1+1) ES – EVOL method. All optical systems found by ES methods are very similar and one can say that they represent global minimum when the radiuses are only variables.

Results from the optimizations when radiuses and distances were variables are presented in table 3:

Table 3: Optimization of the Cooke triplet with radiuses and distances variable

 (1+1) ES - EVOL (m, l) ES – GRUP (m, l) ES - REKO (m, l) ES - KORR 1st  run 0.517 0.565 0.522 0.490 2nd  run 0.511 0.509 0.501 0.500 3rd  run 0.521 0.512 0.472 0.464 4th  run 0.506 0.514 0.502 0.418 5th  run 0.505 0.529 0.502 0.457 average 0.512 0.526 0.500 0.466 standard deviation 0.007 0.023 0.017 0.032 DLS optimization 0.398

One can see that the merit function is reduced from the value 4.544 to the values from 0.565 to 0.398. This is rather significant reduction of the merit function. From the table 3 it is obvious that the differences in the merit function values are greater then when the radiuses were only variables. The smallest value of the merit function (0.398) is found by classical DLS method. Methods EVOL, GRUP and REKO found very similar values of merit functions and method KORR found values of merit function in between from methods EVOL, GRUP, REKO and DLS. From the table 3 can be seen that every ES method found almost the same local minimum all five runs of the optimization.

It is very important to notice that the merit function is only one indicator of the quality of the optical systems. Optical designer has to take in account other indicators like aberrations and geometrical properties of the optical systems if he wants to make complete picture of the each optimized optical system and possibly choose the optimal one.

In the table 4 are shown aberrations of the initial and optimized optical systems. Optical systems with the smallest merit function are chosen for the each ES method.

Table 4: Aberrations of the optimized Cooke triplets

 Initial DLS EVOL GRUP REKO KORR Longitudinal spherical aberration [mm] H=Hmax 0.0169 0.0305 0.0353 0.0303 0.0296 0.0325 H = 0.7·Hmax -0.0766 -0.0348 -0.0399 -0.0363 -0.0399 -0.0407 Transvere spherical aberration [mm]    H=Hmax 0.0021 0.0038 0.0044 0.0038 0.00373 0.0041 H = 0.7·Hmax -0.0068 -0.0031 -0.0034 -0.0032 -0.0035 -0.0036 Astigmatism [mm] -0.1620 -0.1222 -0.1207 -0.1134 -0.0917 -0.0860 -0.0787 -0.0879 -0.0802 -0.0763 -0.0743 -0.0688 Field curvature [mm] -0.1312 -0.0477 -0.0806 -0.1134 -0.0850 -0.0702 -0.0778 -0.0250 -0.0450 -0.0493 -0.0421 -0.0368 Distortion [%] -0.4552 -0.2703 -0.6734 -0.7026 -0.5592 -0.7959 -0.1978 -0.1016 -0.2937 -0.3020 -0.2428 -0.3373 Comma [mm] -0.0006 0.0145 0.0048 0.0071 0.0059 0.0068 0.0060 0.0094 0.0024 0.0028 0.0034 0.0007 -0.0099 -0.0092 0.0008 0.0014 -0.0007 0.0025 0.0011 -0.0019 0.0008 0.0012 0.0011 0.0002

In the table 4 are shown five main aberrations:

• spherical aberration (longitudinal and transvere) for the two ray heights (maximal and 0.7 · maximal);
• astigmatism for the two field angles (maximal and 0.7 · maximal);
• field curvature for the two field angles (maximal and 0.7 · maximal);
• comma for the two field angles (maximal and 0.7 · maximal) and for the two ray heights (maximal and 0.7 · maximal).

The initial optical system is very good optical system with spherical aberration (longitudinal and transvere) and comma corrected and other aberrations small enough. All optimized optical systems have corrected spherical aberration (longitudinal and transvere) and reduced astigmatism and field curvature. Comma is small but not corrected. Comma and distortion of the optical systems are in connection: if comma is smaller then distortion is greater and vice versa.

Initial and optimized optical systems are shown on the figure 2.

 Initial optical system Classical DLS optimization (1+1) ES – EVOL optimization (m, l) ES – GRUP optimization (m, l) ES – REKO optimization (m, l) ES – KORR optimization

From the figure 2 it is obvious that the optical system found by the classical DLS method has very thick first lens and very thin second lens. It is not rational to make such optical system. Optical systems found by ES methods are, in general, better systems. The EVOL method found optical system that has only second lens non-adequate (too thin). The GRUP and REKO methods found optical systems that have all lenses adequate. The KORR method found optical system that is similar to the optical system found by the classical DLS. It has first lens too thick and second lens too thin. After detailed analysis of the all data one can decide that the optimal optimized optical system is the system found by the (m, l) ES – REKO method. It has merit function value 0.472 which is in the middle of the merit function values (0.565 – 0.398). It has corrected spherical aberration (longitudinal and transvere) and reduced astigmatism and field curvature. Distortion is little bigger then the distortion of the initial optical system. Comma is corrected for the  and small enough for the . It has all three lenses that are adequate (neither too thick nor too thin).

7.Conclusions

In this paper Cooke triplet, one of very popular objectives, is optimized by various evolution strategies (twomembered EVOL and multimembered GRUP, REKO and KORR) and classical damped least squares method. All optimized optical systems are compared among themselves and with initial optical system, which is very good optical system.

When radiuses are only variable parameters of optical system, all optimization methods found similar results. Classical DLS didn’t manage to find better optical system then initial optical system. Various ES found optical systems with merit function little different from merit function value of the initial optical system.

When radiuses and distances are variable parameters of optical system all optimization methods found optical systems reduced merit function values. To completely analyze any optical system, optical designer can’t rely only on merit function value instead he has to analyze also aberrational balance and geometrical properties of optical system. When all factors are taken in consideration optimum optical system is found by the multimembered ES – REKO.

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