# 16-Channel Wavelength Division Multiplexers Based on Subwavelength Grating
## 2 Structure and Method
The bandwidth of a single channel is given by:  
\[\Delta\lambda=\frac{2\lambda_{0}^{2}}{\pi\Big{(}n_{eff1}+n_{eff2}\Big{)}}\sqrt {\kappa^{2}+\Big{(}\frac{\pi}{L}\Big{)}^{2}} \tag{4}\]  
It can be seen from Equation (4) that the bandwidth of single-channel filtering is mainly related to the sum of the mode-effective refractive index of the two waveguides, the coupling coefficient, and the coupling length. Among them, the coupling coefficient is mainly realized by the grating width and waveguide spacing, and the coupling length (the number of cycles) is determined. In this work, we demonstrate a WDM structure with a channel spacing of 5 nm, and this structure can be designed according to actual needs in specific applications. The structure of a single filter is shown in Figure 2a. The WDM waveguide for reverse transmission can be either an SWG waveguide or a strip waveguide. Figure 2b simulates the effective refraction of a 220 nm thick Si waveguide at 1310 nm and 1550 nm. The curve of the rate is a function of the waveguide width. According to the waveguide width, after simulating the respective effective refractive index, the grating perturbation structure can be designed according to the needs to meet the corresponding filtering function. As shown in the figure, when Wb = 600 nm, Ws = 400 nm, \(n_{eff1}\) = 2.570, \(n_{eff2}\) = 2.229, at this time \(\lambda_{0}\) = 1.55 \(\upmu\)m, \(p\) = 322.9 nm can be calculated from Equation (3), there will be some gaps between actual simulation and calculation, as appropriate. After adjustment, \(p\) = 319.4 nm, the duty cycle is 0.5, and Wa is 100 nm. The light field diagram at 1548-1552 nm is shown in Figure 2c shows the optical field diagram at 1548-1552 nm. It can be seen that Gaussian light with a bandwidth of 5 nm is input from the input port of Ws, almost completely back-coupled to the Wb waveguide, and output from the output port under phase-matching conditions. Figure 2d shows the transmission spectral lines of the through and output ports. the light output from the through port is transmitted to the next WDM channel with an insertion loss of less than 0.05 dB. A low-loss C-band transmission should be obtained in the O-band WDM area. As shown in Figure 3a, the spectral lines of three different structures of SWG are demonstrated when transmitting O-band and C-band, and through simulation, we selected three SWG structures with good wavelength division performance at 1310 nm and their corresponding C-band simulation results. Since the structure parameters of each type are different, the Bragg wavelength is different. The Bragg wavelength of type B is around 1300 nm, which cause a decrease in the loss around 1300 nm for type B but not for types A and C. As shown in Figure 3b, for the same waveguide width (Wb or Ws), different wavelengths (\(\lambda_{1}\) = 1310 nm and \(\lambda_{2}\)=1550 nm) will have different mode refractive indices, which will lead to different wave mismatch conditions. It is the coupling coefficient that determines whether the forward coupling of C-band will occur. The coupling coefficient is determined by the width of the grating, the waveguide interval, and the coupling length, by adjusting the width of Wb and Ws and the width of the grating disturbance Wa, so that when the O-band is reverse-coupled, the forward coupling coefficient of the C-band is low, and the output from through is almost lossless. As shown in Figure 3a, we optimized the loss in the C-band from 0.6 dB of type A to 0.05 dB of type B, while type A, B, and C represented different dimensional parameter setting of Ws, Wb, gap and p in our optimization process. The optimized O-band main line waveguide is a bar waveguide, and the back-coupled filter is an SWG waveguide. In contrast, the C-band is the SWG waveguide, and the back-filtered waveguide is the bar waveguide. At this time, the through and output port transmission lines of O and C are shown in Figure 3b, and the transmission light field diagram is embedded in the figure.