Simulated-annealing-based geneticalgorithm for modeling the optical constants of solids

Aleksandra B. Djurisic, Jovan M. Elazar, and Aleksandar D. Rakic

We propose a simulated-annealing-based genetic algorithm for solving model parameter estimationproblems. The algorithm incorporates advantages of both genetic algorithms and simulated annealing.Tests on computer-generated synthetic data that closely resemble optical constants of a metal wereperformed to compare the efficiency of plain genetic algorithms against the simulated-annealing-basedgenetic algorithms. These tests assess the ability of the algorithms to find the global minimum and theaccuracy of values obtained for model parameters. Finally, the algorithm with the best performance isused to fit the model dielectric function to data for platinum and aluminum. © 1997 Optical Society ofAmerica

Key words: Optical constants, simulated annealing, genetic algorithms.

1. Introduction

Optical properties of solids are often described interms of the complex optical dielectric function er~v!5 er1~v! 1 ier2~v!. Interpretation of optical spectrais usually accomplished by fitting of the suitablemodel to experimental data. Successful determina-tion of the model parameter values permits not onlyextrapolation of the data to wider spectral range, butalso further inclusion of temperature, pressure, orstrain dependence in the model.

Models that are closely related to the electronicband structure are often exceedingly complicated forpractical use ~e.g., in real-time spectroscopic ellip-sometry or thin-film filter design!. Therefore in thispaper we employ one of the simple phenomenologicalmodels, the Lorentz–Drude ~LD! oscillator model.The LD model employs oscillators at major criticalpoints in the joint density of states, corresponding tointerband transition energies \vj, with some addi-tional oscillators modeling absorption between criti-cal points. Each oscillator is characterized by itsoscillator strength, damping constant, and frequency.

A. B. Djurisic and J. M. Elazar are with the Faculty of ElectricalEngineering, University of Belgrade, P.O. Box 816, Belgrade, Yu-goslavia. A. D. Rakic is with the University of Queensland, De-partment of Electrical and Computer Engineering, St. Lucia QLD4072, Brisbane, Australia.

Received 24 September 1996; revised manuscript received 19February 1997.

0003-6935y97y287097-07$10.00y0© 1997 Optical Society of America

The number of oscillators employed for modeling theoptical constants of metals usually ranges from 4 to 6.Consequently, the number of the adjustable modelparameters is 14–20. Initial estimates of oscillatorfrequencies are available from band-structure calcu-lations, but for the oscillator strength values and thedamping constants even an order of magnitude isunknown, leaving 10–14 parameters whose initialvalues are extremely difficult to guess.

Owing to both the lack of good initial estimates andthe existence of the multiple minima in objectivefunction, conventional downhill methods are incon-venient for the model parameter estimation.Clearly, an optimizing algorithm capable of findingthe global minimum without external supervision isneeded to fulfill this task.

Some robust methods capable of locating the globalminimum without good initial guesses, such as thesimulated-annealing ~SA! algorithm and genetic al-gorithms ~GA’s! have been recently proposed for solv-ing the complex optimization problems.1,2

GA’s search for optimal solutions by use of mech-anisms of natural survival: selection, mating ~selec-tion of parents and crossover!, and mutation, asillustrated in Fig. 1. The first step in implementa-tion of a GA is the generation of a population, whichrepresents a set of strings called chromosomes, thatare possible solutions of the problem. In the case ofmodel parameter estimation, elements of the strings,referred to as genes, are values for the model param-eters. The strings are characterized by performancewith respect to some objective function, termed fit-ness.

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Fig. 1. Illustration of GA. Shades of gray illustrate the fitness of a gene: the lighter the shade, the better the gene. After a numberof generations, the final population that has mostly good genes is obtained.

The population of the next generation is formed byapplication of selection, crossing over, and mutationoperations to the chromosomes in the mating popu-lation. Strings with high fitness may survive in themating population or enter the next generation bymeans of selection, while strings with low fitness arekilled off. Mutation is performed by random alter-ation of parameter values in strings, i.e., genes, tointroduce a certain diversity in the population andthus prevent quick convergence to the local mini-mum. The mutation probability, Pmute, has to bechosen very carefully, because values of Pmute thatare too large cause poor convergence owing to thepossible loss of important genetic information.

GA’s have one important feature: to find theglobal minimum, they do not need initial estimatesfor the values of model parameters. However, it isdifficult to obtain the necessary number of significantdigits in model parameter values with GA’s. To ob-tain even two significant digits in the solution, onewill have to ~a! use the population with a very largenumber of strings or to ~b! increase the mutationprobability. The first strategy is computationallyintensive and requires substantial memory, and withthe second strategy large values of the mutationprobability can cause poor convergence. The prob-lem of obtaining the required number of significantdigits in model parameters values with GA’s was al-ready addressed in Ref. 3.

The SA technique was introduced by Kirkpatrick etal.2 in 1983. The foundations for this algorithm aregiven by Metropolis et al.4 The SA algorithm isbased on an analogy with the annealing of solids:The function to be minimized i.e., the objective func-tion, is analogous to the energy of solids, and thecontrol parameter, called temperature, is analogousto the temperature of solids. A flowchart of the al-gorithm is shown in Fig. 2. The algorithm startsfrom arbitrary initial state and generates a sequenceof random changes of model parameter values,termed moves. Downhill moves are always ac-cepted, whereas uphill moves are accepted with prob-ability that is a function of temperature. Theacceptance probability is usually given by the Boltz-mann distribution p 5 exp~2DEyT!, where DE is thechange in the objective function and T is the temper-ature. The possibility of accepting the uphill movepermits escaping from the local minima. However,convergence to the global minimum depends on the

7098 APPLIED OPTICS y Vol. 36, No. 28 y 1 October 1997

cooling schedule and the move-step size. Small val-ues of move-step size, as well as too large a coolingrate, could cause final-solution dependence of the ini-tial parameter values.

Theoretically, the final solution obtained by the SAalgorithm is insensitive to the choice of the initialparameter values. However, in solving practicalproblems, an inadequate choice of the move-generation procedure or the cooling schedule couldcause the final-result dependence of the initial val-ues. On the other hand, there are no difficulties inobtaining the necessary number of significant digitsin model parameter values like those encounteredwith GA’s, since in the SA algorithm the number ofobtained significant digits depends on the move-stepsize.

Fig. 2. Flowchart of SA algorithm.

Algorithms that combine SA and GA techniques toyield an algorithm capable of more reliably locatingthe global minimum and that is also capable of locat-ing it with the desired accuracy have beenreported.5–9 There are two main strategies: Thefirst is performing a specified number of the Metrop-olis steps on each string in the population and thenapplying some simple selection mechanisms.5,6 Thesecond strategy is introducing the temperature-controlled amount of change in mutations9 or accep-tance of a child string as new parent by means of SA,with the number of generated children for each par-ent depending on the success of the family.7 Thefirst strategy is usually more SA oriented, and thesecond is dominated by the characteristics of GA’s.In both strategies the nondominant algorithm is sim-plified to permit easier implementation and to re-quire less CPU time for computation. However, thealgorithm proposed in Ref. 6 enables one to determineby setting certain initial parameters, whether SA orselection mechanisms will dominate. In Ref. 8 acombination of SA and a GA is suggested that in-cludes a move-generation strategy of SA and geneticoperators in forming the new population.

Since with the simplification of the nondominantalgorithm both strategies produced unsatisfactory re-sults when tried on our test problems, we developeda number of SA-based genetic algorithms ~SAGA’s!that preserve the strengths of both algorithms.That is, our algorithms retain the final-solution in-dependence of the initial estimates ~which is a char-acteristics of GA’s! and also have the higher accuracyinherent in SA techniques.

In SAGA’s, as described in Section 2, the matingpopulation is formed through performance of a spec-ified number of Metropolis steps on each string in thecurrent population. A new generation is obtained byapplication of selection, mating, and mutation oper-ations to the strings in the mating population. Insuch a manner we obtain the necessary diversity inparameter values that are undoubtedly provided bySA and correctly locate the valley containing theglobal minimum that is provided by the use of mech-anisms of natural selection.

Section 2 gives the description of the two SAGA’sthat differ in the selection, mating, and crossovermechanisms employed. In Section 3 these algo-rithms are compared with plain GA’s with the sameselection, mating, and mutation mechanisms for twosets of computer-generated synthetic data ~with andwithout noise! resembling the optical constants ofmetals. After the effectiveness of the SAGA’s hasbeen verified for problems with known solutions~where these algorithms showed better performancethan the corresponding GA’s!, the algorithm with thebest performance, SAGA2, was applied in Section 4 tofit the model dielectric function to data for platinumand aluminum.

2. Description of the Algorithm

Chromosomes in the population are represented bystrings of finite length. Because of the continuous

nature of the model parameter values, we adoptedfloating-point representations of the parameter val-ues in chromosomes, as suggested in Refs. 3 and 9.As a result of the floating-point number coding, thelength of the chromosome is given by the number ofmodel parameters. Parameter values p~k! in initialpopulation strings are generated according to for-mula

p~k! 5 pl~k! 1 @pu~k! 2 pl~k!#r, (1)

where r is a random number r [ @0, 1#. This confinesall the parameters within the specified boundariespl~k! and pu~k! to ensure that obtained values havephysical significance.

For every string in the population, a specified num-ber g of Metropolis iterations is performed. Startingfrom the initial string, the new string is generatedaccording to

pj~k! 5 pi~k! 1 r@D~k!#, (2)

where r is an integer chosen randomly from the set$21, 1%, and D~k! is the move-step size for parameterpi~k!. If the new parameter value pj~k! is outside thespecified boundaries, the value of pj~k! is set to thevalue of the nearest boundary. The new string isalways accepted when the change in the objectivefunction DE is negative. If the change is positive,the new string could be accepted with the probabilityp 5 exp~2DEyT!, where T is the temperature. Theinitial step size has to be large enough to providesufficient mobility of the algorithm at the beginning,but it has also to be small enough in the final stage toreduce the fluctuations. We reduced the next gen-eration step size in a nearly inverse quadratic man-ner, as suggested by Catthoor et al.10 When theratio D~k!yp~k! is less than 0.005, further reduction ofthe step size for that parameter is stopped. Afterexecuting g Metropolis iterations for every string inthe population, we perform selection, mating, andmutation to produce the next generation.

Two different mechanisms of selection and matingare investigated. In the first case we employ thetournament selection, as suggested in Ref. 11 for thecase of a binary tournament. The tournament pop-ulation is formed taking into account the n1 stringswith the best fitness in Metropolis iterations per-formed for each string in the current population.Therefore the tournament population consists of Nn1strings, where N is the number of strings in the ini-tial population. To form the mating population of Nstrings, the tournament population is divided ran-

Table 1. Target Values of the Semiquantum Model Parameter ValuesEmployed for the Generation of the Synthetic Data Sets with and

without Noise

j 0 1 2 3 4

fj 0.700 0.200 0.300 0.200 0.050Gj 0.060 0.300 0.300 1.000 3.000vj 0 0.400 1.500 2.000 4.500

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Table 2. Final Fitness Values and Values of the Model Parameters Obtained for Synthetic Data Set without Noise

Algorithm SAGA1 GA1 GA1 SAGA2 GA2 GA2

N 500 500 2000 500 500 2000fitness 1.32 3 1024 2.107 1.240 8.85 3 1025 9.402 0.044f0 0.700 0.685 0.691 0.700 0.618 0.701G0 0.060 0.060 0.060 0.060 0.055 0.060f1 0.200 0.177 0.182 0.200 0.186 0.195G1 0.300 0.281 0.268 0.300 0.284 0.291v1 0.400 0.393 0.396 0.400 0.354 0.402f2 0.300 0.198 0.381 0.300 0.503 0.305G2 0.300 0.274 0.379 0.300 0.504 0.306v2 1.500 1.490 1.521 1.500 1.566 1.503f3 0.200 0.361 0.151 0.201 0.013 0.209G3 0.999 0.951 1.071 1.007 3.219 1.112v3 2.001 1.801 2.141 1.999 2.604 1.991f4 0.050 0.017 0.045 0.049 0.053 0.040G4 2.999 3.462 3.097 2.994 2.935 2.936v4 4.505 5.619 4.709 4.515 4.442 4.739

domly in N groups of n1 strings. The best string ofthe n1 strings in each group enters the mating pop-ulation. The crossover between two randomly cho-sen strings in the mating population is performedwith probability Pcross. If the crossover does nottake place, strings survive in the next generation.We performed crossover at random points by choos-ing a random integer N2 [ @nmin, npar#, where npar isa number of model parameters, i.e., number of ele-ments in strings, and nmin is the minimal number ofelements exchanged in the crossover. Then we gen-erate N2 random integers ni [ @1, npar#, i 5 1, N2,where N2 is the number of elements to be swapped,and swap the elements at the positions ni.

In the second case we form the mating populationby replacing each string with the chromosome withthe best fitness in g Metropolis iterations performedfor that string. A new generation is produced byselection and crossover.12 The strings with the bestfitness ~Ps percents of the mating population! directlyenter the new generation. The remaining number ofstrings in the new population is generated by cross-

over between the parent strings. Parent strings arerandomly chosen from among all the strings in themating population. The probability of choosing astring as a parent string is inversly proportional to itsnormalized objective function,

F~i! 5f ~i!

(j51

N

f ~ j!, (3)

where f ~i! is the objective function value of the ithstring. In this case we employed the two-pointcrossover. Points of the crossover are randomly cho-sen, and the elements of the strings between thesetwo points are swapped.

In both cases we perform random mutations of thestring elements with probability Pmute. Mutation isperformed with the new value of the parameter gen-erated in the same way as in population generation.After the mutation is executed, temperature is re-duced according to T~m 1 1! 5 a@T~m!#, where m isthe counter of generations. The algorithm termi-

Table 3. Final Fitness Values and Values of the Model Parameters Obtained for Synthetic Data Set with Noise

Algorithm SAGA1 GA1 GA1 SAGA2 GA2 GA2

N 500 500 2000 500 500 2000Fitness 0.746 3.549 2.653 0.746 10.63 1.824f0 0.699 0.670 0.694 0.699 0.692 0.696G0 0.061 0.060 0.060 0.061 0.062 0.061f1 0.198 0.181 0.178 0.198 0.137 0.203G1 0.302 0.269 0.277 0.303 0.213 0.317v1 0.399 0.385 0.390 0.398 0.389 0.397f2 0.303 0.242 0.415 0.305 0.560 0.358G2 0.301 0.326 0.395 0.303 0.551 0.337v2 1.500 1.494 1.530 1.501 1.587 1.525f3 0.192 0.296 0.116 0.189 0.044 0.118G3 0.961 0.837 1.179 0.951 3.633 0.782v3 2.002 1.804 2.291 2.009 2.672 2.137f4 0.054 0.048 0.041 0.055 0.008 0.075G4 2.934 2.710 3.156 2.939 1.406 2.705v4 4.430 4.174 4.773 4.412 4.667 4.112

7100 APPLIED OPTICS y Vol. 36, No. 28 y 1 October 1997

nates when the specified number of generations isreached.

We compared these two SAGA’s with the corre-sponding plain GA’s that do not have the step ofperforming g Metropolis iterations for each string.Therefore in the first case we have a slightly differentselection mechanism that employs a binary tourna-ment, as suggested in Ref. 11, where the tournamentpopulation consists of two copies of every string in thepopulation and the tournament is held for randomlychosen pairs of strings.

3. Test of the Performance on the Synthetic Data

To investigate the performance of the algorithms, wegenerated two sets of synthetic data that closely re-semble the optical constants of metals. The targetparameter values used to generate er~v! in both datasets are given in Table 1. The first data set is with-out noise, while the other set is with Monte Carlogenerated additive Gaussian noise with frequency-dependent variance, determined in such a manner asto give uncertainties of 0.5% in the reflectance calcu-lated from the er

noise~v! values, providing greatersimilarity with the real experimental data. Valuesof the dielectric constants are generated in the range

Fig. 3. Fitness versus number of generations for SAGA1 andGA1.

Fig. 4. Fitness versus number of generations for SAGA2 andGA2.

from 6.3 meV to 15 eV. These tests gave useful in-formation about the ability of the algorithm to locatemodel parameter values correctly.13,14 We brieflydiscuss the applied model. The dielectric constanter~v! can be expressed in the form that separatesintraband effects from interband effects, as wasshown in Refs. 15–18:

er~v! 5 er~ f !~v! 1 er

~b!~v!. (4)

The intraband part er~ f !~v! of the dielectric constant

is given by the well-known free electron or Drudemodel,19

er~ f !~v! 5 1 2

Vp2

v~v 1 iG0!, (5)

and the interband part of the dielectric constanter

~b!~v! is given by the simple semiquantum modelresembling the Lorentz result for insulators:

er~b!~v! 5 2 (

j51

k fjvp2

~v2 2 vj2! 1 ivGj

, (6)

where vp is the plasma frequency, k is the number ofinterband transitions with frequency vj, oscillatorstrength fj, and damping constant Gj, while Vp 5=f0vp is the plasma frequency associated with intra-band transitions with oscillator strength f0 anddamping constant G0.

For determining the fitness of strings the followingobjective function was used:

E~p! 5 (i51

i5N FUer1~vi! 2 er1exp~vi!

er1exp~vi!

U1 Uer2~vi! 2 er2

exp~vi!

er2exp~vi!

UG2

. (7)

We investigated the performance of the algorithmsSAGA1, an algorithm with tournament selection;GA1, a corresponding plain GA; SAGA2, an algo-rithm with the selection of the Ps percent of best-performing strings; and GA2, a plain GA with thesame selection and mating mechanism. The resultsobtained, including fitness, final parameters values,and the number of strings in the population em-ployed, are given in the Table 2 for the data set with-out noise and in Table 3 for the data set with noise.Figure 3 and shows fitness for the data set with noiseas a function of the number of generations for SAGA1and GA1, and Fig. 4 for SAGA2 and GA2.

In both cases SAGA’s showed better performance,having the lower fitness and greater number of sig-nificant digits obtained even for considerably smaller

Table 4. Model Parameter Values for Aluminum

j 0 1 2 3 4

fj 0.468 0.283 0.078 0.161 0.006Gj 0.042 0.293 0.403 1.809 2.157vj 0 0.106 1.562 1.981 5.994

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number of strings in the population. Besides, SA-GA’s are less dependent on the selection, mating, andcrossover mechanisms employed. Both SAGA’syielded similar results with computer-generated syn-thetic data sets, but SAGA2 gave a slightly lowerfitness value with the test data without noise.Therefore in the following section SAGA2 is employedto estimate the values of the parameters of the modelfor optical constants of aluminum and platinum.

4. Application to Aluminum and Platinum

As a final test of our technique, we fit the model forthe optical constants to real data. We have chosentwo metals having qualitatively different opticalspectrum: aluminum ~exhibiting nearly free elec-tron behavior! and platinum ~as a typical transitionmetal!. Aluminum is chosen as a well-known mate-rial and the semiquantum model was employed forparameterization of optical constants of aluminumseveral times.19–22 Expected interband transitionsare at 0.4, 1.5, 2.1, and 4.5 eV. Obtained modelparameters values are presented in Table 4. Valuesof the oscillator strengths correspond to the plasmafrequency15 \vp 5 14.98 eV. Real and imaginaryparts of the dielectric constant of aluminum ~solidcurve, model; dashed curve, experimental data! ver-sus energy are presented in Fig. 5.

To fit the semiquantum model to the data for plat-inum, we used experimental results given in Ref. 23,based on the work of Weaver.24 Weaver usedreflectance25–27 and transmittance28 data to obtain nand k by the Kramers–Kronig technique. Expectedtransitions for platinum are, according to Ref. 29,near 6.3, 7.8, 10.5, and, 10.8 eV. Like other transi-

Fig. 5. Real and imaginary parts of the optical dielectric functionof aluminum versus energy ~solid curve, model; dashed curve, tab-ulated data!.

Table 5. Model Parameter Values for Platinum

j 0 1 2 3 4 5 6

fj 1.090 1.490 4.631 0.026 3.091 6.909 8.969Gj 0.078 0.838 5.194 0.403 5.467 11.539 9.274vj 0 0.861 2.464 6.168 9.293 14.264 20.012

7102 APPLIED OPTICS y Vol. 36, No. 28 y 1 October 1997

tion metals, platinum has the characteristic mini-mum in er2 ~in the case of platinum located near 13eV! with additional structure at higher energies.There is also the strong absorption at 0.7 eV and theevident structure in optical constants24 at the 7.4, 9.8,and 19.9 eV. Therefore we had to use six oscillators,where the oscillator strength values correspond to theplasma frequency30 \vp 5 5.15 eV. Obtained modelparameter values for platinum are presented in Ta-ble 5. Figure 6 shows er1, er2 versus energy for plat-inum ~solid curve, semiquantum model; dashedcurve, experimental data!.

5. Conclusion

Our principal aim has been to develop an algorithmcapable of determining parameters of the model foroptical constants of metals without good initialguesses or external supervision. We investigatedperformance of the plain genetic algorithms ~GA’s!and the algorithms that employ both simulated-annealing and genetic algorithm operators ~SAGA’s!.Tests on the computer-generated data sets, with andwithout noise, proved that SAGA’s obtained lowerfitness values for fewer strings in the population.Moreover, final fitness values obtained by SAGA’s areindependent of the selection and the mating mecha-nisms involved. We employed the algorithm withthe best performance ~SAGA2! to fit the model to realdata ~for aluminum and platinum!. Excellent agree-ment with experiment was achieved for both metals.

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