Optical thin-films today have many applications in various high-tech areas such as lasers, simulation technology, guidance technology, and so on. One of the main concerns in designing a filter is to develop a method of creating a filter with the best performance and the lowest number of layers and thickness.^[1] Different methods of this design are divided into three general categories: analytical, graphical, and numerical methods.^[2–4]

Numerical methods include refinement methods and synthesis methods. In designing filters, the starting point is an important issue that should be considered in the use of refinement methods, but in synthesis methods, by creating a preliminary filter design, optimization steps can be done. Synthesis methods such as the evolutionary algorithm and needle method^[5–7] are much less sensitive to the point and can evolve the design for suitable results automatically.

In this paper, we propose an evolutionary algorithm as a new method of designing optical thin films. First, we explain how this algorithm works, then how it is adapted to the optimization problem of a filter, and finally, we compare its results with those from other methods. To illustrate the ability of the algorithm to optimize the filters, we consider two high-performance examples and optimize them by two efficient and powerful algorithms, the genetic algorithm (GA) and differential evolution (DE), which are used to optimize optical thin films with various applications,^[8–11] and then we compare the results with those from the new algorithm. The results obtained from optimization of the polarizer at the wavelength of 1540 nm by the new method indicate a thin-film design with a higher ratio of P-polarization transmittance to S-polarization transmittance (T_p/T_s) than other methods. This polarizer is used in a laser system to obtain linear polarization for the Q switching process. On the other hand, in the second example, the results obtained in the new method represent a plan with fewer layers and less thickness.

The imperialist competitive algorithm (ICA) is a type of evolutionary algorithm. It is a synthesis optimization method inspired by political, social, and cultural competitions between countries. For more theoretical details, see Ref. [12].

2.2. Assimilation

Assimilation of the colonies is done in order to approach the character of their imperialists.

According to Fig. 1 the colony country moves as far as x towards the imperialist but with angle of deviation (θ) and moves into a new position. The number of x is random with uniform distribution as

where β is a number between one and two and d is the distance between colony and imperialist. Also, θ is considered randomly and with uniform distribution:

Fig. 1.

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	Fig. 1. (color online) General form of colony movement towards its imperialist.

2.4. Imperialist competition

The cost of empire is equal to the cost of the imperialist country plus a percentage of the total cost of its colonies and is expressed as

where T · C_n is the total cost of the n-th empire and ξ is a positive number between zero and one. Over time, the weakest imperialist loses its power and stronger empires will take over its colonies in iteration of the algorithm. For modeling this process first we introduce the normalized total cost of the n-th empire (N · T · C_n) as follows:

Here, {T · C} is the maximum total cost between empires. According to Eq. (4), an empire with the least total cost has the most power. Therefore, the probability of achieving the colony by each imperialist is

from which by using the roulette wheel mechanism the colony of competition proportional to can be achieved. Empires that lose their colonies in competition with the imperialists, will be eliminated. There are several ways to terminate the algorithm. Here, the number of iteration is the end condition. The diagram of the ICA algorithm is shown in Fig. 2.

Fig. 2.

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	Fig. 2. (color online) General diagram of ICA algorithm.

In this part, we investigate, for example, two multilayer filters using ICA, genetic algorithm, and differential evolution, and compared their results. To match the algorithm with the optimization process, each country with its total characteristics corresponds to a filter with its thickness of layers. Each characteristic of a country (p_i’s) such as politics, culture, economy, etc., corresponds to the thickness of a layer of filter.

Several parameters such as thickness, materials, and number of layers are important for designing these examples. On the other hand, founding the global optimal point by selecting appropriate effective parameters of the algorithm is important. To design such filters, two materials with the highest difference between their refractive indices are required. Therefore, we consider a material with a low refractive index SiO₂, which has a high laser-induced damage threshold against laser radiation for high refractive index material TiO₂ was selected. It has transparent layers in the visible and near infrared region and also has a high laser-induced damage threshold.^[13–16] These multilayer filters consist of several pairs of layers with high and low refractive index materials, which are TiO₂ and SiO₂ with refractive indices n_H = 2.25 and n_L = 1.46 respectively which are deposited on BK₇ (n_s = 1.51) substrate. Design is carried out by MATLAB software.

The merit function is one of the important elements in optimizing the thin-film filters. The main goal is to minimize the merit function to achieve a more satisfactory result, which is obtained by changing the layer thickness.^[17,18] The merit function is the root-mean-square error between the calculated transmittance and the target value of transmittance in each iteration. A suitable merit function is defined as

3.1. Polarizer

The first example is a Brewster polarizer^[19] at a wavelength of 1540 nm, which was designed and optimized by using the ICA method. The incident ray angle is the Brewster angle of the BK₇ substrate (56°). The selection of the Brewster angle as the incident angle reduces the reflection of P-polarization and improves the performance of the polarizer.^[20–22]

Our target is designed to have high P-polarization (T_p and low S-polarization (T_s) for transmittance lightwave at 1540-nm wavelength. On the other hand, our main effort is to achieve a greater value for the T_p/T_s ratio in optimizing this polarizer. The initial number of layers was chosen to be 20 and the thickness of layers was selected to be in a range from 10 nm to 300 nm. According to Ref. ^[23], a thin layer less than 10 nm does not have much effect on the quality of target approximation. Table 1 shows the differences among the results caused by changing the effective parameters of the algorithm. Based on it, despite changing parameters, the results are located around a global optimal point. It could be an important advantage in quickly achieving the global optimal point of the algorithm. While other algorithms have a great deal of sensitivity in determining the proper values for reaching the global optimal point.^[24] This is a time consuming process because of the repetition of the algorithm to find the appropriate values of the parameters. Figure 3 demonstrates how the procedure has reached convergence. In this figure, the values of the merit function are shown in terms of the iteration. Clearly, with increasing iteration, the optimization process approaches to the lower values of the merit function. Table 2 gives some characteristics of filters, which were optimized by different methods and figures 4 and 5 show the values of S and P transmittances of the three methods.

Fig. 3.

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	Fig. 3. (color online) Convergence curve of the algorithm.

Fig. 4.

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	Fig. 4. (color online) P-polarization transmittance spectra from different methods.

Fig. 5.

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	Fig. 5. (color online) S-polarization transmittance spectra from different methods.

Table 1.

Table 1.

Table 1. Some differences among results obtained by changing algorithm parameters. . Effective parameters Parameter value Iteration 800 Npop a 300 Prevolution b 0.1 0.3 0.5 Mu c 0.03 0.05 0.07 0.03 0.05 0.07 0.03 0.05 0.07 Zeta d 0.2 0.4 0.2 0.4 0.2 0.4 0.2 0.4 0.2 0.4 0.2 0.4 0.2 0.4 0.2 0.4 0.2 0.4 TP/% 96.20 96.19 96.20 96.19 96.19 96.20 96.17 96.18 96.19 96.18 96.18 96.21 96.20 96.21 96.20 96.20 96.17 TS/% 0.3010 0.3004 0.3010 0.3004 0.3007 0.3017 0.3001 0.3011 0.3010 0.3011 0.3008 0.3007 0.3004 0.3019 0.3004 0.3019 0.3016 0.3012 TP/TS 319.60 320.20 319.60 320.20 319.89 318.86 320.46 319.43 319.57 319.43 319.75 319.85 320.27 318.65 320.27 318.65 318.96 319.29 Physical thickness/nm 3271 3279 3271 3279 3279 3278 3270 3279 3275 3279 3277 3276 3275 3270 3275 3270 3274 3270 a Number of population, b Revolution probability of countries, c Revolution rate of countries, d the coefficient determining the percentage of colonies participation in imperialist competition

Table 1. Some differences among results obtained by changing algorithm parameters. .

Table 2.

Table 2.

Table 2. Characteristics of optimized polarizers. . Number of layer Material Physical thickness/nm GA DE ICA 1 TiO2 a 181.28 146.98 170.22 2 SiO2 b 236.63 282.24 285.86 3 TiO2 163.81 164.76 157.23 4 SiO2 230.91 241.75 265.68 5 TiO2 201.18 159.74 161.25 6 SiO2 194.73 274.58 277.21 7 TiO2 232.57 184.31 170.31 8 SiO2 213.50 315.82 291.09 9 TiO2 187.35 156.72 167.13 10 SiO2 258.49 284.51 284.09 11 TiO2 171.43 160.84 159.70 12 SiO2 212.85 272.46 263.52 13 TiO2 196.05 148.69 157.03 14 SiO2 220.24 270.23 285.13 15 TiO2 190.59 193.76 175.15 /nm 3202.31 3260.39 3270.06 Tp/% 99.56 91.20 96.17 Ts/% 0.68 0.34 0.30 Tp/Ts 146 268 320 a nH=2.18750; b nL=1.42157.

Table 2. Characteristics of optimized polarizers. .

According to Table 2 it seems that using the ICA method, a more appropriate value is obtained for the T_p/T_s ratio. In addition, it is noticeable in comparison with the reported results in other previous designs.^[25]

3.2. Edge filter

The second example is an edge filter in the visible range. It is also composed of two materials TiO₂ and SiO₂ on BK₇ substrate. Our target has a high transmittance value between 400 nm and 550 nm but a low transmittance value between 550 nm and 700 nm. Table 3 shows the comparison among some parameters in three different algorithms. Figure 5 shows the optimal transmittance spectra from three methods in the visible range. Based on Figs. 6 and 7, it seems that the optimized diagrams of the ICA method are more effective than those of the previous design models, which is more acceptable due to the number of layers and the edge width of the spectrum.^[26]

Fig. 6.

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	Fig. 6. (color online) Transmittance spectra of edge filter from different methods.

Fig. 7.

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	Fig. 7. (color online) Reflectance spectra of edge filter from different methods.

Table 3.

Table 3.

Table 3. Characteristics of optimized edge filters. . Number of layer Material Physical thickness/nm GA DE ICA 1 TiO2 a 148.90 283.61 227.32 2 SiO2 b 172.99 200.64 85.40 3 TiO2 113.87 113.13 203.10 4 SiO2 188.01 200.70 287.85 5 TiO2 201.94 119.26 94.42 6 SiO2 132.91 18.73 48.29 7 TiO2 182.61 77.09 90.48 8 SiO2 130.74 136.11 114.74 9 TiO2 178.82 182.58 69.87 10 SiO2 132.45 124.22 299.91 11 TiO2 177.01 191.54 46.26 12 SiO2 134.77 83.51 152.17 13 TiO2 173.77 97.22 58.42 14 SiO2 141.21 226.37 292.96 15 TiO2 170.08 92.22 74.26 16 SiO2 150.36 134.25 137.75 17 TiO2 156.30 29.31 287.83 18 SiO2 150.64 161.02 71.93 19 TiO2 61.68 182.39 20 SiO2 70.39 234.34 /nm 2888.24 2969.44 2642.96 a nH=2.18750; b nL=1.42157.

Table 3. Characteristics of optimized edge filters. .

Figure 6 shows the transmittance spectra of the optimized edge filter by using the ICA method, which has a more acceptable edge than that by using the DE method, and on the other hand, it can be said that there are fewer ripples in the band pass than those by using the GA method. In addition, Figure 7 displays the reflectance spectra of the edge filter by using the three methods. It is shown that the ripples in the ICA diagram are less than the others.

The ICA method of optimizing the optical thin-films is proposed. Using the ICA method, a polarization with a high T_p/T_s ratio at the desired wavelength can be obtained. In addition, an edge filter with a wide passband, close to 100%, a wide stopband, and fewer layers is available from this method. As examples indicate, the design of a polarizer and an edge filter shows that the ICA method is a very robust method. Also, due to the low sensitivity of this algorithm to the change of its parameters, it will be easier to reach a global optimal point than by other algorithms. So this evolutionary algorithm could be a suitable alternative for some algorithms such as GA and DE in the optimization of optical thin-film filters.