SiliconDOI 10.1007/s12633-014-9267-2

ORIGINAL PAPER

A Genetic Algorithm Based Approach for the Extractionof Optical Parameters

Md. Ghulam Saber ·Rakibul Hasan Sagor ·Ashik Ahmed

Received: 4 February 2014 / Accepted: 26 November 2014© Springer Science+Business Media Dordrecht 2015

Abstract We present the Debye model parameters of fourmaterials which are extensively used in the field of electron-ics and photonics. The parameters have been extracted usinga Genetic Algorithm (GA) based technique. We have deter-mined the complex relative permittivity using the extractedmodeling parameters and compared with the experimentallyobtained ones. A very good agreement has been found ineach case which validates our extracted parameters. Theassociated root-mean-square (RMS) deviations have beenfound to be 0.3894, 0.026, 0.8163 and 0.4370 for graphene,graphene oxide, aluminum zinc oxide (AZO) and galliumzinc oxide (GZO) respectively.

Keywords Genetic algorithm · Debye model · Opticalproperties · Material optics

1 Introduction

Material’s interaction with electromagnetic (EM) wavesis described by polarization which can be both electricaland magnetic. However, in the case of naturally occur-ring materials, the magnetic polarization can be neglectedfor frequencies higher than several hundred terahertz. Thecomplex relative permittivity or dielectric function is usedto describe the electrical polarization of a material. Manyparametric models have been developed to determine thedielectric properties of the materials over a broad frequency

M. G. Saber (�) · R. H. Sagor · A. AhmedDepartment of Electrical and Electronic Engineering,Islamic University of Technology (IUT),Board Bazar, Gazipur, 1704, Bangladeshe-mail: gsaber@iut-dhaka.edu

range. However, the modeling parameters need to be opti-mized in order to obtain the best fit to the experimentallymeasured data.

Simulation of EM problems involving arbitrary geome-tries and inhomogeneous materials are often performedusing computational electromagnetics methods such as thefinite-difference time-domain (FDTD) method [1] whichrequires the parametric models in order to integrate thedispersive properties of different materials. Since FDTDsolves the EM fields as a function of time, the parametricmodel we choose needs to be expressed in the time domain.One such model is the Debye model which can be appliedconveniently in both time and frequency domain.

Optimization techniques are required to fit such materialmodels. Rakic et al. [2] used the acceptance-probability-controlled simulated annealing algorithm in order to solvethe parameter estimation problem. Pernice et al. [3] utilizedan adaptive nonlinear least square algorithm to fit the data.Recently some researchers have focused on non-traditionaloptimization methods like particle swarm optimization(PSO), ant colony optimization and genetic algorithm (GA).Kelley et al. [4] used a hybrid particle swarm-least squaresoptimization approach to fit the Debye model function.Clegg et al. [5] fitted the tissue dielectric properties for theDebye model using GA. We utilized the GA to minimize thedifference between the experimentally obtained values andthe Debye model values. The GA has been chosen becauseof its robustness in finding solutions from a large searchspace and its ability to overcome the problem of gettingtrapped in local minima.

The modeling parameters for several materials using dif-ferent parametric models have been reported. Deinega et al.[6] proposed a simple effective model in which the dielec-tric polarization depends both on the electric field and itsfirst time derivative and they extracted the parameters for

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silicon in the spectral range of 300–1000 nm. Sagor etal. [7] reported the extracted parameters for five metalsusing the modified Debye model. Rakic et al. [2] presentedthe modeling parameters for eleven materials using boththe Lorentz-Drude model and the Brendel-Bormann model.Saber et al. [8] determined the parameters of Si0.6Ge0.4

using both modified Debye and Lorentz models.In this paper, we report the extracted parameters of

graphene, graphene oxide, AZO and GZO for the Debyemodel. The modeling parameters have been optimized usingthe GA. In order to validate our obtained results, we havecompared them with the experimental ones. We have deter-mined the root-mean-square (RMS) deviation for each caseand an excellent agreement has been observed. The obtainedRMS deviations are 0.3894, 0.026, 0.8163 and 0.4370respectively for graphene, graphene oxide, AZO and GZO.The frequency range for which we have determined themodeling parameters focuses on the plasmonic applications.A lot of research work is going on in the field of plasmon-ics due to its ability to overcome the diffraction limit [9].To the best of our knowledge, this is the first time one hasextracted the modeling parameters for these materials usingthe Debye model. It is expected that the extracted parame-ters will be useful for efficiently defining materials in thesimulation model which will give birth to new types ofplasmonic devices.

2 Material Models

The Debye model describes dielectrics in terms of electricdipoles. The dipoles respond to an applied electric field andfollow the behavior of the field applied with some relaxationtime. The polarization of the material depends on the oscil-lating frequency. If the frequency is fast, polarization willbe low and vice-versa.

The frequency dependent complex permittivity functionfor the single pole-pair Debye model [10] is given by thefollowing equation

εr(ω) = ε∞ + εs − ε∞(1 + jωτ)

(1)

where, ε∞ is the infinite frequency relative permittivity, εs

is the zero frequency relative permittivity or the static per-mittivity, j is the imaginary unit and τ is the relaxationtime.

From (1), it can be observed that the single pole-pairDebye model can be described by three parameters whichare ε∞, εs and τ . However, in the case of metal another term,σ , related to conductivity is added and (1) becomes,

εr(ω) = ε∞ + εs − ε∞(1 + jωτ)

− jσ

ωεo

(2)

In this case, we have four variables; however, a relationshipcan be obtained among them by comparing with the Drudemodel equation as,

σ = εo(ε∞ − εs)/τ (3)

Thus, there are three parameters that need to be extractedfor fitting the materials and the other can be obtainedfrom equation (3).

3 Optimization Method

The GA is a very robust and powerful non-traditional opti-mization technique when the problem has a very largeparameter search space [11]. The general procedure ofthe GA requires a randomly picked initial population ofprobable solutions and for our case we choose an initial pop-ulation of 500. Roulette selection (also known as stochasticsampling) is then used to select the parents from this initialpopulation. In roulette selection, the chance of an individ-ual getting selected is proportional to its fitness. The fitnessfunction for the roulette selection is given in equation (4).After selecting the parents, crossover is done in order toproduce offspring. The final step is to perform mutation onthe new offspring and this process is continued until thestopping criteria are met.

In this study we have utilized the optimization toolbox ofMATLAB� in order to optimize the modeling parametersof the materials. We have used a double vector popula-tion type and constraint dependent creation function. Theproportional scaling function has been chosen for fitnessscaling. For mutation, an adaptive feasible function has beenused and for crossover, we have used the heuristic func-tion. We have run the simulation for 150 generations andfound it to be sufficient for the convergence of the mini-mization function. For other parameters we have used thedefault values (Fig. 1).

Inside the GA, we have allowed ε∞, εs and τ to vary forthe convergence of the minimization function. In the caseof τ the search space was very large in comparison to thesearch space of ε∞ and εs . We have used logarithmic valuesof τ in the GA in order to avoid this problem. The fitnessfunction that we employed:

ff =∑

(εrexp − εropt )2 + (εimexp − εimopt )

2 (4)

where, εrexp is the real part of the experimentally obtainedpermittivity function, εropt is the real part of the permittivityfunction obtained using optimized parameters, εimexp is theimaginary part of the experimentally obtained permittivityfunction and εimopt is the imaginary part of the permittivityfunction obtained using optimized parameters.

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Fig. 1 Flow chart of the GA

The boundary conditions that need to be maintainedfor the Debye parameters to be integrated in the FDTDalgorithm are ε∞ > 1, εs < ε∞and σ ≥ εo(ε∞ − εs)

/τ .

4 Results and Discussion

4.1 Graphene

Graphene has been the subject of extensive research sincethe first experimental demonstration by Novoselov et al.[12]. It has unique band structure and high mobility[13] which have attracted the attention of the researchers.Recently Bruna et al. [14] reported that a dielectric functioncan be simply derived by imposing the consistence with var-ious experiments. The optical constants they obtained can

be used to model the optical behavior of graphene in the vis-ible range. The complex refractive index that they reportedis in the visible range of electromagnetic spectrum:

n = 3.0; k = C1

3λ (5)

The approximation that the refractive index n is constantover the visible spectral range is reasonable because thereis no sharp resonance within this range. We have used thismethod of obtaining n and k and determined the complexrelative permittivity using the obtained n and k values withinthe visible and near-infrared spectral range. Then we havefitted the Debye function for the obtained complex relativepermittivity using the GA.

4.2 Graphene Oxide

Chemical exfoliation technique yields a large number ofsingle layers of graphene. However, this alternative methodof producing single layer graphene also produces heav-ily oxidized layers of graphene oxide. Since the opticalproperties of a material are sensitive to its oxide as well,they are useful in estimating the optical behavior of thatparticular material. Jung et al. [15] reported the disper-sion functions for the refractive index, n and the extinc-tion coefficient, k of single and multiple-layer grapheneoxide measured using the imaging spectroscopic ellipsome-try technique. They measured the dispersion parameters ofa 100 nm thick graphene oxide stack and determined thedispersion parameters of individual graphene oxide sheetsusing an effective medium approximation. In order to opti-mize the parameters for fitting the Debye function usingthe GA for graphene oxide, we have used this data as thereference.

4.3 AZO and GZO

AZO and GZO are high performance conductive zinc oxideswhich have potential applications in transparent electron-ics and photonics. From the reported optical characteristicsof these conductive films, it has been found that at opti-cal communication wavelength AZO and GZO have lower

Table 1 Optimized Debye model parameters for the materials

Parameters Graphene Graphene Oxide Aluminum zinc oxide Gallium zinc oxide

ε∞ 8.00 5.0771 1.0011 2.1011

εs −77.00 3.327 −40.5 −51.5

τ (sec) 4.118e-15 4.7143e-17 2.863e-15 2.863e-15

σ (S/m) N/A N/A 1.2834e5 1.6576 e5

Range of wavelength (nm) 300–1200 370–790 1300–2100 1400–2200

RMS Deviation 0.3894 0.026 0.8163 0.4370

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Table 2 Covariance of the optimized parameters for different materials

Graphene Graphene oxide Aluminum zinc oxide Gallium zinc oxide

Covariance of the complex permittivity 8.8065 ± 2.0881i 0.0112 ± 0.00021i 1.6284 ± 1.3392i 2.0921 ± 2.4363i

Covariance of the real part 0.7303 0.0012 1.8407 2.6437

Covariance of the imaginary part 8.0762 0.0099 0.2124 0.5515

(i) (ii)

(iii) (iv)

300 400 500 600 700 800 900 1000 1100 12002

4

6

8

10

12

14

Wavelength (nm)

Rel

ativ

e pe

rmitt

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Curve fitting for graphene

Re[Ref.]Imag[Ref.]Re[Simulated]Imag[Simulated]

350 400 450 500 550 600 650 700 750 8000

0.5

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1.5

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Curve fitting for Graphene Oxide

Re[Ref.]Imag[Ref.]Re[Simulated]Imag[Simulated]

1400 1500 1600 1700 1800 1900 2000-8

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Re[Ref.]Imag[Ref.]Re[Simulated]Imag[Simulated]

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Curve fitting for Gallium Zinc Oxide

Re[Ref.]Imag[Ref.]Re[Simulated]Imag[Simulated]

Fig. 2 Comparison of relative permittivity between our results andexperimental values for (i) Graphene, (ii) Graphene Oxide, (iii) AZOand (iv) GZO, obtained using the single pole-pair Debye model. The

red color indicates the imaginary part and the black color indicates thereal part of the complex relative permittivity

Fig. 3 Plot of number ofiterations vs time elapsed

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

500

1000

1500

Number of Iterations

Tim

e E

laps

ed (

seco

nds)

Six-pole Lorentz-DrudeSingle Pole-pair Debye

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loss than silver [16–18]. This characteristic makes AZO andGZO potential candidates in the field of nanophotonics. Wehave obtained the experimental values of the real and imag-inary part of the complex relative permittivity for AZO andGZO from West et al. [19]. These materials offer lowerlosses at telecommunication wavelength which is of par-ticular interest in the field of optical communication. Theextracted parameters using the GA for the Debye functionare presented in Table 1. We have determined the RMSdeviation of the complex permittivity in order to find outthe difference between the experimental results and ourextracted parameters. A maximum RMS deviation of 0.8163has been observed in the case of AZO and for other mate-rials it is less than that. The graphical representation ofour obtained results and experimental results are given inFig. 2. The solid lines indicate the results obtained using ourextracted parameters and the dotted lines indicate the resultsobtained from the references. It can be easily understoodfrom the figures that our obtained results agree well withthe results provided in the references for the range of thewavelengths we have used. The RMS deviation for nickelusing the parameters reported by Rakic et al. [2] has beendetermined as 4.38 in the wavelength range 600–1100nmand for palladium it is 3.1251 in the wavelength range of300–700nm while we have obtained a maximum RMS devi-ation of 0.8163 for AZO which indicates robustness of ourapproach in determining the modeling parameters. In thecase of graphene and AZO, the simulated real part of thecomplex relative permittivity exhibits slight deviation fromthe experimental ones. This is the best we could obtain usingthe single pole-pair model. It can be improved by increas-ing the number of poles, however, that will result in morecomputational time.

Covariance is another method that estimates the errorof a fitted parameter. We have determined the covarianceof the fitted parameters which are presented in Table 2.The covariance of the experimental and optimized complexpermittivity as well as the real and imaginary part of theexperimental and optimized complex permittivity has beendetermined.

In our study, we have used the single pole-pair Debyemodel to fit the dielectric functions of the materials. Thisprovides us with an advantage of less computation timeover multiple-pole models. If we would have used multiple-pole models, then the computation time would have beenmuch higher. In order to make an estimate of how muchlonger it would take to simulate using multiple-pole mod-els, we have developed a one-dimensional FDTD modeland simulated optical signal propagation through grapheneand silver. For defining graphene in the simulation model,we have used our extracted parameters for the single pole-pair Debye model and for silver, we have used the six-poleLorentz-Drude (LD) model parameters reported by Rakic

et al. [2]. We have run the simulation for different numberof iterations and recorded the corresponding time elapsed.The simulation has been done using a first generation Intel�

CoreTM i5 processor based computer and the simulationmodel has been developed in MATLAB� version R2012a(7.14.0.739). The obtained data are presented graphically inFig. 3.

From the figure, it is clearly visible that the compu-tation time required for simulation using the six-pole LDmodel is much higher than the single pole-pair Debye caseand the difference between the two increases drasticallyas the number of iterations are increased. When the num-ber of iterations is 10000, the six-pole LD model requiresabout 3.12 times higher computation time than the singlepole-pair Debye model. Therefore, it is evident that ourextracted parameters exhibit negligible deviations and arecomputationally efficient.

5 Conclusion

In summary, we have extracted the Debye model param-eters for four materials using the GA. The accuracy ofthe obtained parameters has been determined by compar-ing them with the results provided in the literature. Ourextracted parameters for the single pole-pair Debye modelalso have an advantage of computational efficiency overmultiple-pole models and it enables us to run simulationsin the time domain. We believe that these material mod-eling parameters will help researchers in simulating newnano-structures and devising new plasmonic applications.

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