# 16-Channel Wavelength Division Multiplexers Based on Subwavelength Grating
## 2 Structure and Method  
The 16-channel O- and C-band WDM we demonstrate are composed of SWG inverse couplers so that the two waveguide modes in opposite directions satisfy the phase matching condition to achieve coupling. The overall structure of the device is shown in Figure 1. There is the main waveguide and 16 coupling branches. The main waveguide is bent back and forth in an \"s\" shape to reduce the size of the device. There are eight output channels for each of the O-band and C-band, which are sequentially distributed on both sides of the device. The width of the waveguide and the period of the SWG determine the coupling conditions of the bus waveguide and the branch waveguide: the O-band adopts the strip waveguide as the bus waveguide and the SWG as the output channel; the C-band adopts the SWG as the bus waveguide and the strip waveguide as the output channel, such a structure The design can meet dual-band low-loss transmission. Filtering by SWG inverse coupling can achieve a flat-topped waveform, which is preferred in WDM. Broad-spectrum light is coupled into the device from the input port, and then the O-band WDM is performed. In our design, the O-band WDM can ensure low-loss transmission in the C-band. C-band WDM is performed following the O-band WDM to complete the O + C WDM. Theoretically, the bandwidth of this structure can be expanded arbitrarily, and here we demonstrate the WDM of 1280-1340 nm and 1520-1580 nm by simulation. We use a grating perturbation inverse coupler for wavelength division. A single coupling structure is shown in Figure 2a. The waveguide with grating perturbation has a width of Ws, and the width of the strip waveguide is WB. The internal reduction is reduced to form a perturbation structure with a modulation width of Wa. Such a structure can ensure that the average effective refractive index in one period of the grating perturbation region is not much different from that of a straight waveguide with a width of Ws, so that the wavelength drift is small, and results closer to the theoretical value can be obtained. Figure 2b shows the variation of the effective refractive index of the strip waveguide with the O-band and C-band thickness of 220 nm with the width of the waveguide. When there is no auxiliary grating, due to the different widths of Ws and Wb, the phase mismatch causes the co-coupling deadline. At this time, if a grating perturbation structure is added to one of the waveguides when the following conditions in Equation (1) are met, the matching of the propagation constants of the strip waveguide and the SWG-based perturbation waveguide can be achieved:  
\[\Delta\beta=\beta_{1}+\beta_{2}-\frac{2\pi m}{p}=0 \tag{1}\]Figure 1: Single-channel grating structure diagram. Broad-spectrum light is input from the input port, the light is transmitted along the main waveguide, and is coupled to the output waveguide at a position that satisfies the wave vector matching condition. Figure 2: (a) Structure of a single reverse—coupled SWG wavelength division device. (b) Variation of effective refractive index with width for 220 nm thick strip waveguides in O—band and C—band. (c) Optical field map of the grating with a central wavelength of 1550 nm. (d) Losses at the through and output ends of the 1550 nm center wavelength grating. Here, \(\beta_{1}\) and \(\beta_{2}\) are the propagation constants of the two waveguides, respectively, \(p\) is the grating period, m is an integer, and \(2\pi/p\) is the grating vector. In addition, there is:  
\[\beta=\frac{2\pi n_{eff}}{\lambda_{0}} \tag{2}\]  
where \(\lambda_{0}\) is our target wavelength. When \(m\) = 1, put Equation (2) into Equation (1) to get  
\[n_{eff1}+n_{eff2}=\frac{\lambda_{0}}{p} \tag{3}\]  
where \(n_{eff1}\) and \(n_{eff2}\) are the mode refractive index of the strip waveguide and the SWG-based perturbation waveguide, respectively. It can be seen that the period of the perturbation grating is related to the sum of the effective refractive index of the strip waveguides, and the grating period can be calculated from this.