# O-Band Grating Couplers Using Silicon Nitride Structures
## 3 Simulation Results  
The numerical studies of the GCs of Si\({}_{3}\)N\({}_{4}\) structures were performed using RSoft-CAD software by utilizing a 2D FDTD method. The mode solution of the waveguide and SMF was contacted by the Finite Element Method (FEM), which is a generalized mode solver. The FEM solver enabled us to gather valuable insights into the mode profiles and propagation characteristics of the waveguide, which is crucial in the optimization process. The data obtained from the FDTD results were analyzed using MATLAB codes and, in our simulations, the grid size was set to 10 nm for both axes. To comprehend the issue of the mismatch size between the Si\({}_{3}\)N\({}_{4}\) waveguide mode and the standard SMF mode with dimensions of a core diameter of 8.2 \(\upmu\)m and cladding diameter of 125 \(\upmu\)m, an illustration of the mode solution profile at an operating wavelength of 1310 nm was simulated. Figure 2 displays the outcomes of the TE fundamental mode of the standard SMF depicted in red color, and the Si\({}_{3}\)N\({}_{4}\) waveguide with dimensions of 500 \(\times\) 760 nm is depicted in blue color. The width size was selected to ensure a single TE mode solution, and the Si\({}_{3}\)N\({}_{4}\) thickness was optimized. It is evident from the figure that a significant mode mismatch is obtained, resulting in a minimum energy loss of 80%. Thus, to improve the light coupling between the Si\({}_{3}\)N\({}_{4}\) waveguide chip and the SMF, a GC is employed to increase the Si\({}_{3}\)N\({}_{4}\) waveguide mode so that it is more suitable to the SMF mode, and as a result, the overlap area between the two modes is increased, which leads to more energy efficiency. Figure 3 showcases the optimal grating period \(\Lambda\) in the GC waveguide, which is determined to be 0.84 \(\upmu\)m. This carefully chosen value guarantees a robust configuration that can tolerate shifts of up to \(\pm\)30 nm from the optimal value. With this grating period, light can propagate through the GC waveguide to the SMF in an exceptional manner, enabling efficient and reliable transmission. However, fabricating smaller grating periods demands the utilization of high-resolution lithography techniques, and achieving the desired resolution can pose significant challenges. This optimization is crucial for achieving high efficiency and ensuring good fabrication quality. Figure 4 presents an essential analysis of the influence of etch depth on the coupling efficiency performance of the GC. Etch depth is a key fabrication parameter that governs the periodic corrugations etched into the GC structure. This parameter significantly impacts the interaction of light with the GC, thereby affecting the coupling efficiency and overall device performance. The graph showcases high coupling efficiency at an optimal value of 0.05 \(\upmu\)m, yielding a coupling efficiency of 28%, with a tolerance range of \(\pm\)0.05 \(\upmu\)m from  
Figure 3: Coupling efficiency as a function of grating period \(\Lambda\). Figure 2: Gaussian profile of the TE fundamental mode for SMF (red color) and Si\({}_{3}\)N\({}_{4}\) waveguide (blue color). the optimal value. As a result, this range can be useful when handling fabrication errors that can happen at the fab faculty. Figure 5a,b provide a visualization of the thickness of the SiO\({}_{2}\) layer (T\({}_{\text{SiO}_{2}}\)) and the Si\({}_{3}\)N\({}_{4}\) thickness (T\({}_{\text{Si}_{3}\text{N}_{4}}\)) versus coupling efficiency to the SMF. In Figure 5a, it can be observed that a T\({}_{\text{SiO}_{2}}\) thickness of 1 \(\upmu\)m demonstrates a high coupling efficiency of 28%, with a tolerance shift of \(\pm\)25 nm. Figure 5b demonstrates the optimized value for T\({}_{\text{Si}_{3}\text{N}_{4}}\), which is determined to be 0.76 \(\upmu\)m, and the tolerance shift of \(\pm\)30 nm ensures the allowable variation within this range. At this thickness, alongside the operation wavelength of 1310 nm, there is a reliable operation in the O-band spectrum.