The dielectric materials play an important role in optical devices. The permittivity values of all dielectric materials are frequency-dependent. In order to simulate real materials, it is required to incorporate the frequency dependent properties of the materials in a well-defined way whenever optical structures are expected to operate over a wide range of frequency. The well-known parametric models such as Debye model, Lorentz model, Lorentz–Drude model, modified-Debye model, Brendel–Bormann model, etc. can be used to generate such optical properties by fitting them to experimental data using nonlinear optimizer. Electromagnetic simulations with arbitrary geometries and inhomogeneous materials are often accomplished by using computational electromagnetic methods such as the finite difference time-domain (FDTD)^[1] method in which the parametric models are required in order to integrate the dispersive properties of different materials. However, an efficient optimization technique is needed to extract the parameters of the different models to fit both the real and imaginary parts of the complex permittivity in order to reduce the computational cost and obtain accurate results.

Several optimization techniques have been used to model the frequency dependent properties of the materials in optical range. Researchers have been working on different novel materials with aiming to find out the precise modeling parameters using different models. Rakic et al.^[2] determined the dielectric parameters for different metals by using Brendel–Bormann and Lorentz–Drude models through using simulated annealing algorithm technique.^[2] Pernice et al. used an adaptive nonlinear least square algorithm to extract the parameters for metal.^[3] Particle swarm least squares optimization technique was utilized by Kelley et al. to fit the Debye model function.^[4] Saber et al.^[5] used a genetic algorithm based technique for extracting optical parameters. Sagor et al.^[6] presented a comparative study of the performances of different mathematical models for some materials. Other traditional and bio-inspired algorithms have also been used in this field by several researchers.^[4,7–10]

Most of the efforts have been made to compare different material models by using single optimization methods. However, the comparison of different optimization techniques for a particular model to reduce the computational cost while using the material in different simulators is yet to be done. Recently, particle swarm optimization (PSO) and invasive weed optimization (IWO) have received increased interest in the field of optimization. These techniques have shown great ability in numerous real-world applications.^[11–13] Researchers made a lot of efforts to compare these algorithms to validate that one algorithm outperforms another in a specified set of problems. However, to the best of our knowledge, a detail comparative study of different optimization techniques on complex material modeling has never been made before.

In this paper, we present a comparative study of PSO and IWO to extract the parameters for dielectric materials in optical range by using Debye model. The dielectric material used here is graphene oxide in a spectral range of 350 nm–800 nm. It is observed that the performances of IWO and PSO depend on the mathematical definition of the objective function used and also on the chosen boundary conditions. We work with two different objective functions and three different boundary conditions for each objective function. The observations show the relative suitability of the objective functions and the boundary conditions under different circumstances.

Several studies in the literature can be found where Debye function expansions of complex permittivity are considered.^[4] The Debye model refers to dielectrics in terms of electric dipoles. The applied electric field causes the dipoles to respond accordingly. Oscillating frequency governs the polarization of the material.

The general equation for Debye model can be written as

Graphene oxide is chosen for evaluating PSO and IWO by using Debye model. The experimental data of the complex permittivity of graphene oxide are obtained from Ref. [18] in a spectral range of 350 nm–800 nm. The performances of the optimization methods are evaluated by using two objective functions as follows:

Boundary 1:

The experimental values of refractive index, n and extinction coefficient, k for graphene oxide at different frequencies are obtained from Jung et al.’s work.^[18] From these values the real and imaginary parts of complex permittivity are calculated as follows:

For different population sizes, the curves of fitness versus iteration number for different boundary conditions and objective functions are shown in Figs. 2 and 3 for IWO and Figs. 4 and 5 for PSO. The optimized permittivity curves in comparison to the experimental curves are shown in Figs. 6–9.

Fig. 2.

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	Fig. 2. Variations of fitness with the number of iterations for different population sizes for IWO working with objective function 1 and boundary conditions (a) 1, (b) 2, and (c) 3.

Fig. 3.

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	Fig. 3. Variations of fitness with the number of iterations for different population sizes for IWO working with objective function 2 and boundary conditions (a) 1, (b) 2, and (c) 3.

Fig. 4.

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	Fig. 4. Variations of fitness with the number of iterations for different population sizes for PSO working with objective function 1 and boundary conditions (a) 1, (b) 2, and (c) 3.

Fig. 5.

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	Fig. 5. Variations of fitness with the number of iterations for different population sizes for PSO working with objective function 2 and boundary conditions (a) 1, (b) 2, and (c) 3.

Fig. 6.

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	Fig. 6. Comparisons between the experimental and fitted permittivities of graphene oxide for IWO working with objective function 1 and boundary conditions (a) 1, (b) 2, and (c) 3.

Fig. 7.

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	Fig. 7. Comparisons between the experimental and fitted permittivities of graphene oxide for IWO working with objective function 2 and boundary conditions (a) 1, (b) 2, and (c) 3.

Fig. 8.

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	Fig. 8. Comparisons between the experimental and fitted permittivities of graphene oxide for PSO working with objective function 1 and boundary conditions (a) 1, (b) 2, and (c) 3.

Fig. 9.

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	Fig. 9. Comparisons between the experimental and fitted permittivities of graphene oxide for PSO working with objective function 2 and boundary conditions (a) 1, (b) 2, and (c) 3.

From Table 1 and 2, it can be observed that for both the algorithms and both the objective functions, the Root mean square (RMS) deviation and objective function value are almost immune to the variation in population size, irrespective of the chosen boundary condition.

Table 1.

Table 1.

Table 1. Performance parameters of the PSO and IWO algorithms in optical parameter extraction for first objective function. . Pop Size Boundary 1 Boundary 2 Boundary 3 RMS deviation Elapsed time (seconds) No. of iterations required to reach convergence Objective function value, F1 Iteration stopped at RMS deviation Elapsed time (seconds) No. of iterations required to reach convergence Objective function value, F1 Iteration stopped at RMS deviation Elapsed time (seconds) No. of iterations required to reach convergence Objective function value, F1 Iteration stopped at PSO 50 0.0631 2.4986 49 3.7416 180 0.0631 2.3679 82 3.7437 175 0.1548 1.9424 16 0.0965 65 75 0.0631 3.0046 44 3.7416 178 0.0631 3.2043 80 3.7404 188 0.1548 1.9899 14 0.0949 63 100 0.0631 3.3119 44 3.7416 167 0.0631 3.9159 80 3.7437 197 0.1548 2.3279 9 0.0948 58 125 0.0631 6.0751 39 3.7416 272 0.0631 6.9226 82 3.7309 316 0.1548 2.8176 14 0.0947 63 150 0.0631 4.8958 33 3.7416 184 0.0631 4.6872 80 3.7365 179 0.1548 6.1259 65 0.0959 114 IWO 50 0.0916 2.3683 72 3.7487 121 0.1308 1.9887 41 3.7818 90 0.0274 3.9433 180 0.0935 290 75 0.0916 2.7525 64 3.7422 113 0.1308 3.9389 96 3.7397 147 0.0281 3.0369 75 0.0935 191 100 0.0916 2.5305 39 3.7455 88 0.1308 2.0764 1 4.22 51 0.0293 4.0849 80 0.0937 212 125 0.0916 3.2992 46 3.7469 95 0.1308 2.6071 1 4.22 51 0.0286 4.0101 80 0.0935 175 150 0.0916 5.2117 91 3.7416 140 0.1308 2.9129 1 4.22 51 0.0294 4.4484 80 0.0936 172

Table 1. Performance parameters of the PSO and IWO algorithms in optical parameter extraction for first objective function. .

Table 2.

Table 2.

Table 2. Performance parameters of the PSO and IWO algorithms in optical parameter extraction for second objective function. . Pop Size Boundary 1 Boundary 2 Boundary 3 RMS deviation Elapsed time (seconds) No. of iterations required to reach convergence Objective function value, F1 Iteration stopped at RMS deviation Elapsed time (seconds) No. of iterations required to reach convergence Objective function value, F1 Iteration stopped at RMS deviation Elapsed time (seconds) No. of iterations required to reach convergence Objective function value, F1 Iteration stopped at PSO 50 0.073 2.5278 50 688.036 173 0.073 2.3301 81 689.627 173 0.073 2.3301 81 689.627 173 75 0.073 2.9009 46 688.036 175 0.073 2.8795 83 690.299 173 0.073 2.8795 83 690.299 173 100 0.073 3.827 40 688.036 189 0.073 3.9845 85 689.869 204 0.073 3.9845 85 689.869 204 125 0.073 3.9376 37 688.036 175 0.073 7.3575 101 685.711 324 0.073 7.3575 101 685.711 324 150 0.073 5.2092 46 688.036 193 0.073 4.3616 81 689.298 173 0.073 4.3616 81 689.298 173 IWO 50 0.0916 2.3683 72 3.7487 121 0.1308 1.9887 41 3.7818 90 0.1308 1.9887 41 3.7818 90 75 0.0916 2.7525 64 3.7422 113 0.1308 3.9389 96 3.7397 147 0.1308 3.9389 96 3.7397 147 100 0.0916 2.5305 39 3.7455 88 0.1308 2.0764 1 4.22 51 0.1308 2.0764 1 4.22 51 125 0.0916 3.2992 46 3.7469 95 0.1308 2.6071 1 4.22 51 0.1308 2.6071 1 4.22 51 150 0.0916 5.2117 91 3.7416 140 0.1308 2.9129 1 4.22 51 0.1308 2.9129 1 4.22 51

Table 2. Performance parameters of the PSO and IWO algorithms in optical parameter extraction for second objective function. .

From Table 1, we observe that for the first objective function, in the case of PSO, the RMS deviation remains constant at 0.0631 for the first two boundary conditions. But for the third boundary condition i.e. for the smallest boundary, the deviation increases to 0.1548. But the objective function value is observed to have an opposite behavior with respect to change in boundary size. For boundary conditions 1 and 2, we obtain the objective function value of approximately 3.74 which decreases to approximately 0.0965 for the third boundary condition.

For IWO working with the first objective function, the change in RMS deviation is very minute between boundary conditions 1 and 2. For the first boundary condition the RMS deviation is 0.0916 and for the second boundary condition it is 0.1308. But for the third boundary condition, it comes down to around 0.029. The objective function value remains within the range of 3.74–4.22 for the first two boundary conditions but decreases to around 0.0935 for the third boundary condition. So, for the first objective function, IWO performs better for boundary condition 3 which is the smallest boundary. This agrees with the common intuition that the smaller the search area, the easier it is to find the solution.

For the second objective function, a change in boundary condition in PSO does not cause any significant variation in the RMS deviation (it changes slightly around 0.073) and the value of the objective function (it remains within 685–690). So, for this objective function, the choice of boundary condition has no significant effect on the performance of PSO.

For IWO using the second objective function, it is observed that the RMS deviation (0.0916) for the first boundary condition is very close to those for the other two boundary conditions (for both these condition, the deviations are both 0.1308). Besides, the objective function value for the three boundary conditions remain within a range of 3.74–4.22. So, for IWO working with the second objective function, the choice of boundary condition has little effect on the performance of the algorithm, an observation which is similar to that for PSO working with the same objective function.

So, the use of the second objective function makes the performances of IWO and PSO less dependent on the chosen boundary conditions.

Moreover, we observe that for PSO, the function value for the first objective function is smaller (around 0.0965–3.74) than that for the second objective function (around 690) while the RMS deviations for the two are comparable, irrespective of the boundary condition. So, the performance of PSO is better if the first objective function is used.

For IWO, the two objective functions have comparable outputs for the first two boundary conditions. But as the boundary size in condition 3 decreases, the first objective function gives a much lower objective function value (around 0.0935) than the second one (around 3.74–4.22). So, for smaller boundaries, IWO performs better with the first objective function.

Furthermore, for the second objective function, it can be observed that there are no differences in RMS deviation and objective function value obtained for boundaries 2 and 3 for both PSO and IWO. This gives an indication that reducing the boundary size beyond a certain extent may have no effect on the optimization performance.

In this work, we choose single pole-pair Debye model over multi-pole pair models since it is found to be adequate to model the variation in permittivity of graphene oxide with reasonable accuracy. Besides, the use of a single pole-pair model would result in a smaller computational time. However, multi-pole pair models have to be used in the cases where a single-pole pair model is not able to provide reasonable optimization.

To the best of our knowledge, a comparative study of PSO and IWO in extracting optical parameters has not been published before. In this work we analyze the performances of these two algorithms for extracting the parameters required to model the frequency dependent permittivity of graphene oxide using single pole-pair Debye model for two different objective functions and three boundary conditions, thus working with six different sets of choices. After comparing the performance of the two algorithms, we find relative advantages of the objective functions for different choices. This study shows that both PSO and IWO are robust in extracting mathematical parameters required to model the experimental variation of permittivity of graphene oxide. Optical properties of other materials can also be replicated based on other models using these algorithms.