# Terahertz Wideband Filter Based on Sub-Wavelength Binary Simple Periodic Structure
## 2 Design Principles and Results
The flatness error \(N\) is defined as Equation (3):  
\[N_{i}=\frac{Tr_{\max}-Tr_{\min}}{Tr_{\max}+Tr_{\min}},\ (i=1,2,3), \tag{3}\]  
where each frequency domain, the difference between the highest transmission efficiency and the lowest transmission efficiency is divided by the sum of the highest transmission efficiency and the lowest transmission efficiency. If the value of \(N_{i}\) is smaller, it indicates that the waveform is flatter. The flatness errors of each segment are denoted as \(N_{1}\), \(N_{2}\), and \(N_{3}\), respectively. According to the design goal requirements, in order to ensure the filter characteristics to approach the idea curve as shown in Figure 2, the following objective functions are established:  
\[G(X)=\sum_{i=1}^{3}\left(P_{i}M_{i}+Z_{i}N_{i}\right)+P_{4}M_{4}, \tag{4}\]  
where \(X\) is a collection of structural parameters, including (_f_, \(d\), _h_). The first six terms consider the transmission efficiency and flatness of each frequency segment. The seventh term considers the transmission efficiency of the center operating frequency. _Pi_ and _Zi_ are weighting factors, which take values in the range [0,1] with their sum equal to 1, which can be adjusted according to different design requirements. The essence of the design is to find a set of binary simple periodic sub-wavelength structure parameters (_f_, \(d\), _h_) to make sure the function \(G\) is the smallest. The frequency range of an incident terahertz wave is from 1.66 to 8.40 THz, and a wave of TE polarization is perpendicularly incident--an ordinary terahertz source is polarized light,and sub-wavelength devices are related to the polarization of incident light. In this paper, for simplicity, we take TE incident light as an example, where the incident medium is air of refractive index \(n_{0}=1\), and the grating dielectric material is silicon of refractive index \(nr=3.42\), which is almost transparent. Combined with genetic algorithm optimization, the optimal filtering effect of the filter is shown in Figure 2.  
Through actual optimization, we set the stall generation, coding method, crossover function, scaling function, generation, hybrid function, etc. Before the optimization started, the fitness of the corresponding target environment of each individual in the initial population was obtained. By finding the individual with the highest fitness in the population as the optimal individual in the population, the optimal design result searched by the genetic optimization algorithm was obtained. At the same time, the genetic optimization algorithm was very simple, adaptable, robust, and could process optimization results in parallel. Additionally, the convergence speed was quite fast. Consequently, we obtained the structural parameters of the filter \(d=129.817\)\(\upmu\)m, the duty ratio \(f=0.432\), and the groove depth \(h=30.379\)\(\upmu\)m, with the highest transmission efficiency in the main lobe reaching close to 50%. Compared with DCF, the highest transmittance was increased to 47% from 37%, and 3-dB bandwidth was increased to 2.33 THz from 2.20 THz, indicating that this work uses a simple structure to achieve a high-transmission efficiency terahertz wideband filter that breaks through the design limitations of the abovementioned terahertz filtering.