# Angle-Based Parametrization with Evolutionary Optimization for OESCL-Band Y-Junction Splitters
## 2 Angle-Based Parametrization
### Materials and Methods  
We used a silicon-on-insulator (SOI) platform, which is extremely attractive because of its compatibility with microelectronics, introducing now-popular scalable photonic integrated circuits.  
In the inverse design, we need to specify the permittivity distribution via parametrization. The most used methods are (i) the density topology and (ii) level-set. In both methods, the design region is divided into pixels, and a permittivity is associated to each pixel [24]. For more methods, please refer to [34].  
A _drawback_ of these methods is that they usually require a high number of parameters to have the flexibility of exploring desired fabricable devices. The higher the degrees of freedom, the better we can explore the designs. However, this process can require considerable computing resources (i.e., CPU- or memory) [18], leading to significant execution times.  
We propose the _angle-based parametrization_ depicted in Figure 2. The pink region depicts the silicon core and the white background color represents silicon oxide. We divided a rectangular MMI splitter design region into \(z\) equally spaced vertical-aligned segments. The segments have different widths. Therefore, the division gives us a set of angles between consecutive segments, where \(\alpha_{i}|_{1}^{p}\) are the angles and optimization parameters, and \(p=z-1\).  
The waveguide width \(W_{g}\), MMI body length, \(L_{MMI}\), and gap \(\delta\) can be adjusted to fit multiple outputs or operation frequencies. For simplicity, we demonstrate a 1 \(\times\) 2 splitter working at 1550 nm.  
A 1 \(\times\) 2 MMI splitter parametrization can be summarized in three main steps:  
1. First, we divide the initial topology into \(W_{i}[i=1\dots z]\) longitudinal equally spaced segments 1 based on [12; 35; 36], and we set symmetry on the transverse direction of the propagation. We also set the values of \(\delta\) = 0.2 \(\upmu\)m and \(L_{MMI}\) = 2 \(\upmu\)m. This allows us to reduce the number of angles \(\alpha_{i}\) to optimize because the angles on the upper edge of the device match those at the bottom edge.
2. Second, we modify the dimensions of the current width \(W_{i}\) 2 based on the previous width and angle, as shown in Equation (1). In particular, we can set \(W_{1}=W_{g}=0.5\)\(\upmu\)m for the 1550 nm working window. \[W_{i}=W_{i-1}+2d\cdot\tan(\alpha_{i-1})\quad\text{for $i>1$}\] (1)  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Work** & **Device** & **Method** & **Bandwidth** & **Loss** & **Size** \\ \hline Mak 2016 [28] & 2 \(\times\) 2 splitter & PSO & 1530–1570 nm & S: 4.11 dB E: 6 dB & 4.8 \(\times\) 4.8 \(\upmu\)m\({}^{2}\) \\ \hline Lu 2018 [29] & Beam splitter & PSO & 1500–1600 nm & S: no losses & 2 \(\times\) 2 \(\upmu\)m\({}^{2}\) \\ \hline Chang 2018 [10] & 3 dB Y-splitter &
\begin{tabular}{c} Subwavelength \\ inverse design \\ \end{tabular} & 1520–1580 nm & E: 1.5 dB & 2.88 \(\times\) 2.88 \(\upmu\)m\({}^{2}\) \\ \hline Xu 2022 [30] & 3 dB Y-splitter & ADTO & 1588–2033 nm & S: 0.83 dB & 5.4 \(\times\) 2.88 \(\upmu\)m\({}^{2}\) \\ \hline Xu 2017 [31] & 3 dB Y-splitter & Nonlinear fast search & 1530–1560 nm & E: 0.96 dB & 3.6 \(\times\) 3.6 \(\upmu\)m\({}^{2}\) \\ \hline [PERSON] 2017 [13] & 1 \(\times\) 3 splitter & Inverse design & 1400–1700 nm & E: 0.64 dB & 3.8 \(\times\) 2.5 \(\upmu\)m\({}^{2}\) \\ \hline [PERSON] 2016 [32] & 3 dB Y-splitter & Taper design & 1530–1600 nm & E: \textless{}0.19 dB & 5 \(\upmu\)m length taper \\ \hline [PERSON] 2019 [33] & 3 dB Y-splitter & DNN inverse design & 1450–1650 nm & S: 0.45 dB & 2.6 \(\times\) 2.6 \(\upmu\)m\({}^{2}\) \\ \hline Our work & 3 dB Y-splitter & CMA-ES & 1260–1625 nm & S: 0.15 dB & 2 \(\times\) 1.2 \(\upmu\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of photonic devices designed using iterative optimization algorithms.  
3. Third, we place a power monitor 3 at one waveguide output to measure the transmittance of the fundamental TE (transverse electric) mode in order to establish the figure of merit.