Design and optimization of thin film polarizer at the wavelength of 1540 nm using differential evolution algorithm

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Abstract

In this paper, a thin film polarizer at the wavelength of 1540 nm in infrared region was designed and optimized using differential evolution method. It is shown how the algorithm’s parameters can change the output result to obtain the best consequence of optimization. This polarizer consists of a few pairs of high and low refractive index dielectric materials, titanium dioxide and silicon dioxide, respectively, with \(BK_{7}\) glass substrate and the angle of incident light was supposed 56° that is the Brewster angle for \(BK_{7}\) glass. Our final optimized polarizer has 91.20 and 0.336% transmittance for P and S polarization, respectively, and a 271 ratio of \(\frac{{T_{P} }}{{T_{S} }}\) which has high significance for this polarizer. It consists of eight pairs of layers with low and high refractive index materials and 3369.1 nm physical thickness which is used to separate S and P polarized light for Q-switching process.

Keywords

Polarizer Optimization Differential evolution Dielectric materials Infrared region Brewster angle 

1 Introduction

Polarizers are optical filters that separate two orthogonal polarization components (S and P) of light into different directions. Polarizers are the main components in optical elements such as multiplexers, tunable filters and Q-switching (Perla and Azzam 2007; Tilma et al. 2011; Yang et al. 2014). Plate and cube are the two main types of thin film polarizers. In the plate type, the layers are deposited on a flat substrate (Macleod 2010). The researchers considered that flat plate Brewster angle polarizers consist of multilayer dielectric thin films from repeated pairs of low and high refractive index layers. If Brewster angle is used as the incident angle, it will help to decrease the reflection of P polarization as well as increase the efficiency of the polarizer (Macleod 2010; Dobrowolski and Waldorf 1981; Gavrilov et al. 1983). The performance of polarizers may be varied by changing the characteristics of the layers, i.e. refractive index and optical thickness (Macleod 2010; Thelen 1989). Many different attitudes for designing optical filters exist to reach the solutions of different problems, these approaches can be split into three types: graphical vector methods, analytical synthesis methods and numerical design methods (Yang and Kao 2001a, b; Dobrowolski 1995). Numerical methods are of interest for designing optical filters because they are especially robust for the solution of complicated spectral issues that can not be solved with other methods. They can be used for the design of optical coatings with very complicated specifications which require a large number of layers for their solution (Abed et al. 2014). There are different methods for optimization of filters such as genetic algorithm (GA), particle swarm optimization (PSO), and differential evolution (DE). According to the properties of the filters, performance of the methods is different. It seems that determining the effective parameters of DE algorithm leads to better performance than other methods (Sahraee et al. 2014).

Refinement methods (Willey 2014; Tikhonravov and Trubetskov 2012; Lie and Dobrowolski 1992; Tikhonravov 1993), and synthesis methods (Baedi et al. 2010; Back et al. 1995) are two basic approaches for designing of numerical optical coatings. An initial design has been required as a starting solution in the refinement methods, and the problem includes better improvement of this solution (Willy 2002). Since synthesis methods generate their own starting designs automatically, they are combined with refinement methods as an effective synthesis method to improve the quality of the solutions. Evolutionary algorithms (Greiner 1996) have been applied to some issues encountered in optical designs and coatings that are computationally complex. These approaches have been adopted by ideas borrowed from genetics and natural selections. In this paper, the authors applied a method called DE algorithm to optimize optical thin film systems with different number of layers. The main purpose of this research is the design and optimization of a polarizer at a wavelength in infrared region, 1540 nm using the minimum number of layers and highest ratio of P to S-polarization transmittance \(\left( {\frac{{T_{P} }}{{T_{S} }}} \right),\) the value criterion of which should be more than 200 (Sahraee et al. 2014). We applied MATLAB software for calculating optimization and obtaining high \(\frac{{T_{P} }}{{T_{S} }}\) ratio with eight pairs of low and high refractive index, SiO2 and TiO2 materials, respectively.

2 DE algorithm procedure

Differential evolution (DE) is an efficient heuristic for the global optimization of algorithm introduced by Storn and Price (1997) and was developed to optimize real parameter and real valued functions. DE is an evolutionary algorithm that can be used to find approximate solutions for problems that are not analytically solved. We could consider a global diagram for evolutionary algorithm.

There exist two differences between DE and other optimization algorithms:
  1. 1.

    Priority of mutation to cross-over.

     
  2. 2.

    Different mutation procedure: mutations are generated with differences between the existing members in population while they follow from a probability distribution in the other algorithms.

     

2.1 Initialization

It is supposed a function with n real parameters will be optimized, n being the number of layers and N the number of population in this paper. Here, we consider a parameter vector as a filter. The parameter vectors have the form:
$$x_{i,G} = [x_{1,i,G} ,x_{2,i,G} ,x_{3,i,G} , \ldots ,x_{n,i,G} ]\quad i = 1,2,3, \ldots ,N$$
where \(x_{i,G}\) is ith member of population and G is the generation number. They are considered the upper and lower bounds of parameters:
$$x_{j}^{L} \le x_{j,i,1} \le \,x_{j}^{U}$$
which selects initial values on the intervals \(\left[ {x_{j}^{L} ,x_{j}^{U} } \right]\).

2.2 Mutation

Mutation operator randomly selects three vectors \(x_{m1,G} ,x_{m2,G}\) and \(x_{m3,G}\) for each given parameter vector \(x_{i,G}\) that generates \(v_{i,G + 1}\) as:
$$v_{i,G + 1} = x_{m1,G} + \beta (x_{m2,G} - x_{m3,G} )$$
\(v_{i,G + 1}\) is called donor vector and scaling factor, Beta, is a constant which is selected on interval [0.2, 0.5]. It seems that if the population increases, the beta should be decreased due to the increasing algorithm’s accuracy. According to the experimental results, high values for population size get premature convergence.

2.3 Recombination

Cross-over operator incorporates solutions from the previous generation. The trial vector \(u_{i,G + 1}\) is generated from the elements of the parameter vector \(x_{i,G}\), and the elements of the donor vector, \(v_{i,G + 1}\). Elements of the donor vector enter the trial vector with cross-over probability, PCR as:
$$u_{j,i,G} = \left\{ {\begin{array}{*{20}l} {v_{j,i,G + 1} } \hfill & {if\quad rand_{i,j} \le PCR\quad or\quad j = I_{rand} \quad i = 1,2,3, \ldots ,N} \hfill \\ {x_{j,i,G} } \hfill & {if\quad rand_{i,j} > PCR\quad and\quad j_{rand} \quad j = 1,2,3, \ldots ,n} \hfill \\ \end{array} } \right.$$
where \(rand_{j,i} \sim U[0,1],\) \(I_{rand}\) is a random integer from [1,2,3,…,n] and \(I_{rand}\) ensures that \(v_{i,G + 1} \ne x_{i,G}\).

2.4 Selection

The parameter vector \(x_{i,G}\) is compared with the trial vector \(v_{i,G}\) and the one with the lowest function value is admitted to the next generation:
$$x_{i,G + 1} = \left\{ {\begin{array}{*{20}l} {u_{i,G + 1} } \hfill & {if\quad f(u_{i,G + 1} ) \le f(x_{i,G} )\quad or\quad j = I_{rand} } \hfill \\ {x_{i,G} } \hfill & {otherwise} \hfill \\ \end{array} } \right.$$
This process continues until stopping criterion is reached. There are various criterions to stop the process. Here, it is number of iterations which is 600.

3 Design and discussion

According to Ref. Kaiser and Pulker (2003), if the total number of films is relatively small (up to 10–20) then the effective design approach is to begin the refinement process many times with arbitrarily setting beginning designs, and then for obtaining the best final design. Therefore, the authors decided to design a thin film polarizer at the wavelength of 1540 nm with 16 layers by the DE method. In this optimization problem, DE modifies the initial design parameters for a better filter that can be obtained through different steps. Members of population are equivalent to a filter and each one consists of eight pairs of layers with high and low refractive index.

The merit function is one of the important elements in optimization of thin-film filters. The merit function is root-mean-square error between the calculated transmittance and the target value of transmittance in every iteration. A suitable merit function is defined as:
$$f(\eta ,d,\lambda_{k} ) = \left\{ {\frac{1}{w}\sum\limits_{k = 1}^{w} {\frac{{[T(\eta ,d,\lambda_{k} ) - T(\lambda_{k} )]^{2} }}{{\delta T_{k} }}} } \right\}^{{\frac{1}{2}}}$$
(1)
where \(T(\eta ,d,\lambda_{k} )\) and \(T(\lambda_{k} )\) are desired and the target transmittance is at wavelength \(\lambda_{k}\). \(\eta\) and d parameters are refractive index and thickness vectors of a coating system, respectively and \(\delta T_{k}\) is tolerance at the wavelength \(\lambda_{k}\). Generally, \(\delta T_{k}\) is set to 0.01 and W is the number of points where the merit function is evaluated which is considered 400 points.
The actual calculations of transmittance and reflectance were done based on the characteristic matrix method (Macleod 2010). At oblique incident light, the effective refractive index of a layer with a refractive index must be defined for each polarization as (Dobrowolski 1995):
$$\eta^{s} = (n^{2} - n_{m}^{2} \sin^{2} \theta_{0} )^{{\frac{1}{2}}}$$
(2)
$$\eta^{p} = \frac{{n^{2} }}{{(n^{2} - n_{m}^{2} \sin^{2} \theta_{0} )^{{\frac{1}{2}}} }}$$
(3)
The value of refractive index of the medium is \(n_{m}\) = 1 and the incident angle is \(\theta_{0}\) = 56°. The parameters of \(\eta^{s}\) and \(\eta^{p}\) are effective refractive indexes of the S and P polarizations, respectively. The substrate is \(BK_{7}\) glass \((n = 1.51)\) and the materials with low and high refractive indexes are SiO2 \((n_{L} = 1.46)\) and TiO2 \((n_{H} = 2.25)\), respectively. Compared to the work of others (Sahraee et al. 2014), to design and optimize such a filter with DE algorithm, we obtained the design parameters with less number of layers and lower physical thickness. Table 1 shows comparison of some parameters of two different filters with different number of layers as initial conditions. \(T_{p}\) and \(T_{s}\) are P and S-polarization for transmittance light and \(\sum d\) is total thickness.
Table 1

Comparison of some parameters of different optimized filters

DE algorithm

This work

Sahraee et al. (2014)

Layer

Material

Refractive index

Physical thickness

Physical thickness

Physical thickness

1

SiO2

1.42157

111.70

111.70

111.70

2

TiO2

2.18750

189.05

146.98

145.14

3

SiO2

1.42157

206.99

282.24

268.01

4

TiO2

2.18750

162.72

164.76

174.17

5

SiO2

1.42157

183.07

241.75

223.34

6

TiO2

2.18750

213.02

159.74

174.17

7

SiO2

1.42157

260.24

274.58

223.34

8

TiO2

2.18750

192.58

184.31

203.20

9

SiO2

1.42157

255.51

315.82

223.34

10

TiO2

2.18750

180.76

156.72

203.20

11

SiO2

1.42157

253.84

284.51

268.01

12

TiO2

2.18750

186.73

160.84

203.20

13

SiO2

1.42157

250.98

272.46

268.01

14

TiO2

2.18750

182.50

148.69

174.17

15

SiO2

1.42157

233.68

270.23

223.34

16

TiO2

2.18750

181.22

193.76

145.14

17

  

244.08

 

268.01

18

  

194.54

 

145.14

\(\sum {d(nm)}\)

  

3683.21

3369.09

3644.63

\(T_{p}\)

  

91.52%

91.2%

87.82%

\(T_{s}\)

  

0.23%

0.336%

0.43%

\(\frac{{T_{p} }}{{T_{s} }}\)

  

397

271

204

Our results have been obtained after 600 iterations. The proper selection of population in this algorithm is important and is one of the factors that influences reaching the optimal point, so that excessive increase in it will cause loss of process, time, and speed. In this design, by repeating the algorithm we conclude that the population should be nearly ten times the number of decision variables. Therefore, we found an optimal population of about 240. Figure 1 shows the profile of the filter which has 16 layers. We considered the initial thicknesses of layers from 100 to 200 nm (see Fig. 1). Then, we obtained the determinant parameters in achieving the final optimal point; one of these coefficients is the scaling factor (\(\beta\)). Inappropriate selection of this factor leads to premature convergence and eventually reaches local optimum points. In this algorithm, it has been proven that the value for the scale parameter is about 0.5, but in this optimization problem, the sensitivity of work is high in the values around this number. However, by repeating the runs, we were able to get the best value for this parameter. In addition to specifying this factor, determining the cross-section probability (PCR) parameter is also important. This factor also takes values between zero and one. The excessive increase in this factor increases the convergence speed of the algorithm and eventually reaches the local optimum points. Thus, we obtained an appropriate amount for this factor simultaneously with the beta factor. Therefore, the determination of the factors that improved the search and optimization process with the DE algorithm was well done. By finding them, we can obtain the Global Optimum point for values of 0.5 and 0.4 for beta and PCR, respectively.
Fig. 1

Index profile of optimized filter by DE method

Figure 2 demonstrates the DE method parameters, PCR and Beta change to achieve a global point with maximum value for \(\frac{{T_{p} }}{{T_{s} }}\). We found that if β = 0.5 and PCR = 0.4, the \(\frac{{T_{p} }}{{T_{s} }}\) will be 271. The comparison of transmittance is shown in Figs. 3 and 4. It seems that as the number of layers increases, \(T_{s}\) decreases and \(T_{p}\) increases, and therefore, the ratio of \(\frac{{T_{p} }}{{T_{s} }}\) is increased.
Fig. 2

\(\frac{{T_{p} }}{{T_{s} }}\) oscillations by Beta and PCR parameters of DE method in 240 population and 600 iterations

Fig. 3

S and P-polarization optimized transmittance spectra for 16-layers filter

Fig. 4

S and P-polarization optimized transmittance spectra for 18-layers filter

4 Conclusions

This study demonstrates that DE is a robust tool for multilayer thin film designs. It seems that by selecting suitable Beta and PCR parameters, the DE could be an effective method for designing filters with less thickness and less number of layers with better desired optical effects. The results demonstrate the selection of suitable range for parameters, Beta and PCR are efficient to find global point of optimization which has been considered in ranges of [0.2–05] and [0.1–0.9], respectively. In addition, comfortable choices for the number of population and the number of iterations could be significant in discovery of the best point of optimization with less time of calculation, selected as 240 and 600, respectively. The best results for the ratio of \(\frac{{T_{p} }}{{T_{s} }}\) can be achieved at β = 0.5 and PCR = 0.4, which is the value of 271.5. Also, the total physical thickness of the DE-designed is 3369.1 nm that can be an acceptable thickness for this polarizer. It was observed that the DE-designed filter could be useful in the desired optical Q-switching and other applications of polarizer at infrared region.

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Authors and Affiliations

  1. 1.Department of PhysicsPayame Noor UniversityTehranIran
  2. 2.Department of Physics, Faculty of SciencesShahrekord UniversityShahrekordIran
  3. 3.Nanotechnology Research CenterShahrekord UniversityShahrekordIran

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