# Angle-Based Parametrization with Evolutionary Optimization for OESCL-Band Y-Junction Splitters
## 1 Introduction  
Photonic integrated circuits (PICs) enable the interconnection of different types of photonic devices into a single platform. Designing energy-efficient on-chip optical interconnects remains a challenge when scaling the number of devices per chip [1]. Many optical passive devices work traditionally in the SCL optical communications band, while the OESCL band has gained attention due to ultra-wideband applications [2]. Among the passive components on a chip, the 3 dB power splitters are present in the order of dozens [3; 4; 5]. A 3 dB power splitter distributes the signal across multiple optical interconnects and is the essential building block for switches [6; 7], modulators [8; 9], and multiplexers [10; 11]. Y-junctions based on multi-mode interferometers have become one of the essential building blocks in photonic integrated circuits [12]. Moreover, many inverse design methodologies for power splitters are based on multi-mode interferometer Y-junctions [10; 12; 13]. There are three main causes for excess loss in fabricated Y-junctions [12; 14]: (i) a low tolerance to fabrication errors, (ii) minimum fabricated resolution available, and (iii) deviations in the sharp angles (see \(\alpha_{i}\) in Section 2.1). Special mask techniques can to help overcome these problems, e.g., the electron beam technique [13; 14]. In this regard, any device geometry, regardless of the type of optimization, has to be adjusted to the fabrication tolerances of the machinery available to the designer. This is especially true for non-intuitive geometries generated by optimization methods. Post-processes are usually added [13; 15], in which the fabrication constraints are fed to a second algorithm to smooth and tweak the optimized geometry, making it fabrication-ready. Figure 1 shows the timeline of some popular optimization algorithms used in the photonic design and relates them to their adoption (by the number of citations). We observe that the most popular algorithms are particle swarm optimization (PSO) [16] and genetic algorithms (GAs) [17]. These algorithms are usually used with default parameters due to the computational complexity of performing a hyperparameter optimization, which can limit their performance [18]. The covariance matrix adaptation evolution strategy (CMA-ES) [19] is a global optimization algorithm that has not been widely adopted. However, it does not need to optimize its parameters--only its population size. This characteristic is important because increasing the population size usually leads to a more exhaustive search. Furthermore, CMA-ES has been demonstrated to be efficient, robust, and easy for antenna optimization [20]. This motivates us to compare the performance and convergence of CMA-ES, GA, and PSO. Figure 1 includes other optimization methods: (i) moving asymptotes (MMA) [21], (ii) limited memory Broyden-Fletcher-Goldfarb-Shanno bound constrained (L-BFGS-B) [22], and (iii) differential Evolution (DE) [23]. For a detailed review on optimization methods, please refer to [18; 24; 25]. Table 1 shows examples of photonic splitters designed using different algorithms to those presented in this work. Our CMA-ES-based optimized splitter presents a wider bandwidth along with a shorter footprint, and we also expect lower losses. Our work extends an _angle-based parametrization_[26]. It defines the device's internal angles and requires fewer parameters than conventional topological optimization because its number of parameters depends on one dimension of the design space. In contrast, conventional topology optimization relies on two design space dimensions. Then, the optimization of the device is reduced to a medium-dimensional problem, allowing us to use global optimization [18]. The usage of global optimization is a key difference compared to other approaches, such as level-set and density topology, which usually rely only on local optimization [9; 24; 27].