2 Department of General Education, Dalian Polytechnic University, Dalian 116034, China 2  
[PERSON]  
1 School of Information Science and Engineering, Dalian Polytechnic University, Dalian 116034, China; [EMAIL_ADDRESS] (S.L. ); [EMAIL_ADDRESS] (B.L. ); [EMAIL_ADDRESS] (YL. ); [EMAIL_ADDRESS] (F.L. ); [EMAIL_ADDRESS] (F.L. ); [EMAIL_ADDRESS] (Y.C. ); 2305208540005579@features. These studies have made significant progress in the area of slot-waveguide multimode interference couplers. The investigation of silicon-based power splitters with a slot-waveguide structure has resulted in notable designs from various research teams. [PERSON]'s team proposed a \(1\times 8\) power splitter based on a GaN-SiO2 slot-waveguide structure [12], as well as a \(1\times 4\) power splitter based on a Si-GaN slot-waveguide structure [13]. Additionally, [PERSON]'s team developed a \(1\times 3\) polarization-insensitive power splitter based on a Si-SiNx slot-waveguide structure that can achieve polarization-insensitive function [17]. The various types of power splitters designed to date have demonstrated good performance, but further research is needed to improve their size reduction and reduce loss. With the increasing demand for device integration, the constraints of the traditional rectangular MMI coupler in terms of device structure size are gradually being highlighted. Therefore, reducing the structural size of MMI couplers has become a growing concern for researchers. In order to achieve small size and low loss, a multimode interference coupler with parabolic (including exponential and other quadratic curvilinear) coupling regions has been proposed [18]. The size of the device is reduced by designing the coupling region with a width that varies as a function of the parabola with length. In this article, we present the simulation and design of a \(1\times 3\) parabolic power splitter based on a Si-GaN slot-waveguide structure using the effective refractive index method and the eigenmode expansion method. Our objective is to effectively reduce the height and width of the device while minimizing excess losses. To achieve this, we employ a single-mode waveguide with a silicon-nitride-gallium-nitride (Si-GaN) structure, which is known for its excellent optical properties and compatibility with photonic integration. The coupling region of the device is designed with a parabolic-shaped structure. This parabolic design helps to reduce the width of the coupling region, thereby decreasing the overall size of the power splitter. Additionally, taper structures are integrated into the input/output waveguides and the connection sections of the coupling region. These taper structures play a critical role in reducing excess loss and improving the balance of the device. By gradually transitioning the width of the waveguide, the taper structures enhance the coupling efficiency from the single-mode input to the coupling region, thereby reducing waveguide crosstalk and improving overall device performance. The devices are analyzed for tolerances to reduce excess loss and achieve good imbalance. The designed device demonstrates small size and low losses compared to similar power splitters.