# O-Band Grating Couplers Using Silicon Nitride Structures
## 2 Grating Coupler Design and Theoretical Aspects  
Figure 1 provides a detailed representation of the x-z cross-section structure of the GC waveguide with the SMF, highlighting its key geometrical parameters. In our model, we selected the GC based on a uniformly grooved teeth configuration for its ease of fabrication and compact footprint, which leads to low-cost GC devices. It is important to emphasize that while complex configurations, such as using a reflector [35; 36], a GC with non-uniform grooved teeth [37; 38; 39; 40; 41; 42], or edge couplers [38], can improve coupling efficiency, they also introduce fabrication difficulties and increased costs. For example, the reflectors used in GCs are sensitive to fabrication variations, such as dimensional errors or imperfections in the grating structure. Similarly, non-uniform grooved teeth increase the complexity of the fabrication process, making precise control over the grating structure and dimensions more challenging. Even small fabrication variations can significantly impact device performance. Due to these considerations, the design of a GC with a uniformly grooved teeth configuration is a reliable, easy-to-fabricate, and cost-effective choice. Our proposed design consists of three layers with different thicknesses that are carefully chosen to achieve maximum coupling efficiency between the GC and SMF at a wavelength of 1310 nm. The sub-ground layer is made of Si (T\({}_{\text{Si}}\)) with a thickness of 4 \(\upmu\)m, followed by a middle layer of SiO\({}_{2}\) with a thickness of 1 \(\upmu\)m. The top layer is composed of Si\({}_{3}\)N\({}_{4}\) (T\({}_{\text{Si}_{3}\text{N}_{4}}\)) with a thickness of 0.76 \(\upmu\)m. The top layer features grooved teeth made of SiO\({}_{2}\) material. The grating period between the grooved teeth, denoted as \(\Lambda\), has been set to 0.84 \(\upmu\)m, where the SiO\({}_{2}\) width of each grooved tooth is 0.34 \(\upmu\)m, represented by W, and the Si\({}_{3}\)N\({}_{4}\) width of each grooved tooth is 0.5 \(\upmu\)m. The angle of the SMF, defined as \(\theta\), is 7.5 degrees, and the distance between the SMF and GC, denoted as h, is 2.6 \(\upmu\)m. The etch depth corresponds to the thickness of the grooved teeth, and the optimal value for this parameter is 0.38 \(\upmu\)m. By optimizing these key values, the light can propagate through the  
Figure 1: 2D schematic diagram of the GC configuration and key parameters. GC more effectively, and as a result, achieve higher light coupling efficiency between the GC and SMF. To analyze the propagation characteristics of light within the GC waveguide, we employed the FDTD method. This numerical technique provides valuable insights into the behavior of electromagnetic fields and their propagation through the GC waveguide. The principal work of GC is to change the mode size of the Si\({}_{3}\)N\({}_{4}\) waveguide to be a better match to the SMF mode. Mode matching refers to ensuring that the guided mode in the Si\({}_{3}\)N\({}_{4}\) waveguide is appropriately coupled to the desired SMF mode for efficient light transmission. To achieve this goal, the Bragg condition [1] must be fulfilled, which can be achieved by carefully optimizing the grating parameters, such as the grating period, the coupling angle from the SMF, and etch depth, and the GC can enhance the coupling efficiency, enabling the effective transmission of light into and out of the Si\({}_{3}\)N\({}_{4}\) waveguide. The grating period \(\Lambda\) is the key parameter that refers to the distance between the grooves in the grating structure. The grating period \(\Lambda\) of the GC is given by [21]:  
\[\Lambda=\frac{\lambda}{n_{eff}-n_{c}sin(\theta)} \tag{1}\]  
Within this mathematical expression, \(\lambda\) symbolizes the wavelength, \(n_{eff}\) is the refractive index, \(n_{c}\) is the refractive index of the cladding material, and \(\theta\) is the coupling angle from the SMF. The diffraction behavior of a GC can be described using the Bragg condition, which is also referred to as the phase matching condition.