## 3 Design and Simulation  
### Design of the Single-Mode Waveguide  
Optical waveguides are designed using slot waveguides based on a silicon-gallium nitride (Si-GaN) structure. Gallium nitride (GaN) has been widely used for integrating optically active nanodevices and non-photonic devices due to its superior electrical properties, temperature resistance, and potential to cover a broad-spectrum range [13]. GaN components can be grown on epitaxial substrates or directly on silicon substrates. It is recommended to use materials with the smallest refractive index difference to incorporate slot waveguides into the construction of multimode interference devices. Recent research has highlighted the superiority of GaN compared to materials such as silicon-alumina and silicon-silica (SiO2) for this application [13]. By utilizing GaN as the material for slot waveguides, the performance of the devices is significantly enhanced. GaN's high refractive index contrast and its compatibility with silicon technology make it an excellent choice for high-performance integrated optical devices. The use of GaN in slot waveguides not only improves the optical confinement but also enhances the overall efficiency and functionality of the photonic devices. In the fabrication of silicon gallium nitride (Si-GaN) structure, it needs to be prepared through various processes such as photolithography and etching. In the preparation of silicon-based slot waveguides, \(0.2\ \upmu\mathrm{m}\) is easier to achieve single-mode propagation than \(0.22\ \upmu\mathrm{m}\), which helps to reduce mode interference and optical loss, and it has stronger mode constraints, which makes it more suitable for high-density integrated photonic devices. In this paper, we use a mode solution to simulate the design of the device in the 1550 nm optical wavelength range. Figure 1a depicts a three-layered slot waveguide comprising two Si layers with a refractive index of 3.47 and a GaN layer that has a refractive index of 2.305. This waveguide is encircled by a SiO\({}_{2}\) cladding layer, which has a refractive index of 1.45. \(H_{\text{Si}}\) and \(H_{\text{GaN}}\) are the heights of Si and GaN, respectively. \(W\) is the width of the slot waveguide, which operates at a wavelength of 1550 nm. Figure 1b shows an example of the structure of the power splitter, which comprises three components, i.e., the input waveguide, the multimode interference coupling region, and the output waveguide. \(L_{\text{MMI}}\) is the length of the coupling region. \(W_{0}\) is the width of the narrow end of the coupling region, whereas \(W_{1}\) is the width of the broad end of the coupling region. \(L_{\text{Taper}}\) and \(W_{\text{Taper}}\) are the length and width of the taper structures, respectively. All the waveguides, including the input and output ones, use a slot-waveguide configuration. The multimode interference coupling region is designed to achieve efficient power splitting. Stable transmission of optical signals was achieved through a waveguide under single-mode transmission conditions by using an effective refractive index method to determine the optimal height and width of the slot waveguide. Waveguide configurations with varying heights for silicon and gallium nitride as well as different widths for the slot waveguide were scanned. The single-mode conditions are analyzed for the Si waveguide height, and the variation of the mode refractive index with the waveguide height is analyzed in the range of 0.16-0.26 \(\upmu\)m for the Si waveguide height. From Figure 2a, it can be observed that the criteria for achieving single-mode transmission are fulfilled only when the height of Si is less than and not equal to 0.24 \(\upmu\)m. It is noteworthy that the standardized preparation process usually assumes the height of Si to be around 0.2 or 0.22 \(\upmu\)m. Therefore, in our study, the height of silicon was set to 0.2 \(\upmu\)m to align with the standardized preparation process.