# Compact SOI Dual-Mode (De)multiplexer Based on the Level Set Method
## 1 Introduction  
Silicon photonics, with a highly rapid development, is considered as a promising technical field in optical communications and sensing due to its utilization of CMOS compatibility, small footprint, and lower power consumption [1, 2]. Recently, an increasing number of optical devices have been reported based on silicon-on-insulator (SOI) platforms, further enhancing the compactness and integration of photoelectric systems [3]. The growing demand for increased communication capacity has become an inevitable challenge, particularly with the advancement of technologies such as cloud computing [4] and big data [5]. Traditionally, parallel data are distinguished by wavelength during data processing and transmission, using a technology known as wavelength division multiplexing (WDM) [6, 7]. In response to the need for enhanced signal parallelism, researchers have explored mode division multiplexing (MDM) technology to expand channel capacity. Within the optical waveguide, multiple modes with different orders propagate simultaneously due to the orthogonality between different optical waveguide modes. In MDM systems, different modes are employed for parallel signal transmission, with the mode (de)multiplexer, denoted as mode (de)MUX, serving as a vital device for individual signal detection and processing. Various methods have been employed to implement mode (de)MUXs, with reported structures including the asymmetric directional coupler (ADC) [8, 9], micro ring resonators(MRR) [10], Y-junctions [11], multimode interference couplers (MMI) [12; 13], and subwavelength gratings (SWG) [14]. Among these, the ADC structure, based on phase matching conditions for mode converting, is the most commonly used for (de)MUX devices due to its short coupling length and simple manufacturing process, although its sensitivity to the manufacturing process has been a limitation. The sensitivity can be mitigated by employing tapered structures, as reported by [PERSON] et al. [9]. An MRR-type device offers the advantage of low crosstalk and compatibility with WDM systems, but they have a relatively narrow operating bandwidth [10]. Y-junction devices, on the other hand, have a broader bandwidth, as demonstrated by [PERSON] et al. in the wavelength range of 1450 nm to 1630 nm [11]. However, Y-junction devices typically have a footprint exceeding 50 um in length. MMI-based mode (de)MUXs provide low insertion loss and low crosstalk, but they require a larger length, exceeding 400 um [12; 13], which poses challenges for achieving high integration. Moreover, (de)MUX based on SWG structure exhibits relatively low insertion loss (!0.5 dB) and a broad bandwidth (?120 nm) [14]. However, the manufacturing process required for SWG structure production is complex. The inverse design method has emerged as an innovative approach for designing mode (de)MUX devices, providing an automated and optimized process for searching the optimal solution within a designed region [15; 16]. Several algorithms have been proposed for mode (de)MUXs, including particle swarm optimization (PSO) [17], direct binary search (DBS) [18; 19; 20], Bayesian DBS [21], density method [22; 23; 24], gradient-probability-driven search algorithm (GPDS) [25], and the digitized adjoint method [26]. Based on the PSO algorithm, [PERSON] et al. proposed a four-channel mode (de)MUX that exhibits good scalability [17]. However, due to the issue of premature convergence, PSO may struggle to converge on a suboptimal solution, which can impact the final results. The DBS method, which is powerful and easy to implement, optimizes device performance by dividing the design region into cells and controlling the materials in each cell. However, as a brute force search method, it requires a significant amount of computation, which hampers the optimization speed. To address this challenge, [PERSON] et al. proposed an improved DBS method called Bayesian DBS, which reduces the number of iterations needed for optimal structure search.