# 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, SiO_{2} and TiO_{2} 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.

- 1.
Priority of mutation to cross-over.

- 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

### 2.2 Mutation

### 2.3 Recombination

### 2.4 Selection

## 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.

_{2}\((n_{L} = 1.46)\) and TiO

_{2}\((n_{H} = 2.25)\), 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.

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 | SiO | 1.42157 | 111.70 | 111.70 | 111.70 |

2 | TiO | 2.18750 | 189.05 | 146.98 | 145.14 |

3 | SiO | 1.42157 | 206.99 | 282.24 | 268.01 |

4 | TiO | 2.18750 | 162.72 | 164.76 | 174.17 |

5 | SiO | 1.42157 | 183.07 | 241.75 | 223.34 |

6 | TiO | 2.18750 | 213.02 | 159.74 | 174.17 |

7 | SiO | 1.42157 | 260.24 | 274.58 | 223.34 |

8 | TiO | 2.18750 | 192.58 | 184.31 | 203.20 |

9 | SiO | 1.42157 | 255.51 | 315.82 | 223.34 |

10 | TiO | 2.18750 | 180.76 | 156.72 | 203.20 |

11 | SiO | 1.42157 | 253.84 | 284.51 | 268.01 |

12 | TiO | 2.18750 | 186.73 | 160.84 | 203.20 |

13 | SiO | 1.42157 | 250.98 | 272.46 | 268.01 |

14 | TiO | 2.18750 | 182.50 | 148.69 | 174.17 |

15 | SiO | 1.42157 | 233.68 | 270.23 | 223.34 |

16 | TiO | 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 |

## 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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