# Terahertz Wideband Filter Based on Sub-Wavelength Binary Simple Periodic Structure
## 2 Design Principles and Results  
Figure 1 shows a binary sub-wavelength simple periodic structure whose refractive index is uniformly distributed in the \(y\) direction, periodically distributed in the \(x\) direction. \(d\) is the period of the grating, and \(a\) is the ridge width of the grating. In addition, \(b\) is the groove width of the grating, \(h\) is the groove depth of the grating, and \(f\) = \(a/d\) is the duty ratio of the grating--that is, the ratio of the minimum line width of the grating to the grating period. In the \(z\) direction, the structural region is divided into three regions--an incident region, a grating region, and a base region--within which \(n_{1}\) is the refractive index of the incident region, where \(z\) < 0; \(n_{2}\) is the refractive index of the grating region as a grating layer, where \(0\) < \(z\) < \(h\); \(n_{3}\) is the refractive index of the base region, where \(z\) > \(h\); and the incident angle of the terahertz wave is normally incident (i.e., \(\theta\) = 0). The thickness of the incident and base regions is considered to be infinite. By utilizing the characteristics of the sub-wavelength structure, the diffraction order intensity in each of the diffracted waves generated by the incident wave passing through the grating can be adjusted by adjusting the grating structure parameters, such as the duty ratio\(f\), the grating period \(d\), and the groove depth \(h\). Figure 1: Sub-wavelength binary simple periodic structure model. By inhibiting the intensity of the non-zero diffraction order, an evanescent wave on the grating surface is excited to increase the intensity (i.e., transmission efficiency) of the zero-order diffraction. However, due to the strong coupling between the sub-wavelength structure and the incident wave, there is no analytical relationship between each structural parameter and the diffraction efficiency of each level, and the diffraction efficiency of each level is very sensitive to the change of structural parameters. That is, a slight structural change would cause a sharp change in the filtering performance. It can be seen that it is very difficult to obtain a broadband filter with high transmission efficiency by manually adjusting the structural parameters. Therefore, in this paper, a method to combine the rigorous coupled wave method suitable to the periodic structure in the vector analysis method and the genetic optimization algorithm is introduced to establish the evaluation function of the wideband filter, and the best grating structural parameters of the binary simple periodic sub-wavelength structure are obtained to satisfy the broadband filtering effect. The goal of this paper is to design a terahertz wideband filter with a center frequency of around 4.07 THz, with a 3-dB bandwidth above 2.2 THz. The average transmission efficiency error and flatness error of the main and side lobes were selected as the factors of fitness function. The waveform was divided into three parts: the left-side lobe, the main lobe, and the right-side lobe. The definition of the transmission efficiency _Tr_ is the ratio of the intensity of the outgoing wave light field to the intensity of the incident wave field. The average transmission efficiency _Tra_ is defined in Equation (1):  
\[Tra=\frac{Tr_{1}+Tr_{2}+Tr_{3}+\ldots\ldots+Tr_{n}}{n}, \tag{1}\]  
which means discretely dividing a frequency band into \(n\) discrete points, dividing the sum of the transmission efficiencies of each discrete point. The average transmission efficiency error \(M_{i}\) is defined in Equation (2):  
\[M_{i}=|T_{i}-Tra|,\ i=1,2,3,4, \tag{2}\]  
which means the absolute value of the difference between the ideal transmission efficiency _Ti_ and the average transmission efficiency _Tra_, where _Ti_ is 1 for the main lobe and 0 for the side lobes. The average transmission efficiency errors of the three segments are defined as \(M\), _M_\({}_{2}\), and _M_\({}_{3}\), respectively, and the transmission efficiency error at the center frequency is _M_\({}_{4}\).