## 2 Inverse Design of Silicon Photonics with Iterative Optimization Algorithms
### Comparison of Iterative Optimization Algorithms for Silicon Photonics Design  
We have discussed three examples of inverse designed PBSs via different initial structures with corresponding iterative optimization algorithms in the above three subsections. The PBS based on empirical structure has high transmission efficiency (over 85% within 420 nm wavelength range) for both TE and TM modes and the critical size is under the control of designers. However, it also has the largest footprint (approximately 7 \(\times\) 4 \(\upmu\)m\({}^{2}\)) and the lowest DOF (only 3). It takes about one week to optimize this device on two 12-core central processing units (CPUs). Profound prior knowledge about the coupling theory is also required and a skillful initial structure has to be chosen, otherwise, the device may not work at all. The PBS based on QR-code like structure has ultra-compact size (2.4 \(\times\) 2.4 \(\upmu\)m\({}^{2}\)) as well as high DOF (i.e., 400). Its design process does not need much prior knowledge but the optimization time is long (\(\sim\)140 design hours) as each QR-code has to be verified during DBS. This time-consuming optimization process limits both the size of footprint and the number of DOF, so the transmission efficiency of DBS optimized PBS is not very high. The peak efficiency of optimized PBS is \(\sim\)80% and its 1 dB bandwidth is 83 nm. The irregular PBS optimized via adjoint method is also ultra-compact (2.8 \(\times\) 2.4 \(\upmu\)m\({}^{2}\)), and it has 16,800 DOF due to its ultra-high resolutions (i.e., 20 nm). After 64 h of optimization on two 12-core CPUs, the final irregular PBS has transmission efficiency over 90% covering 420-nm wavelength range. The irregular PBS has a time-efficient optimization process, where all the parameters can be updated in one round via only two simulations (one forward simulation and one adjoint simulation). However, profound knowledge about light-material interaction process, as well as advanced mathematics, is required for gradient calculations. Furthermore, gradient-based algorithms are easy to fall into local optimal. For example, the resolution of an irregular structure is usually 20 nm, which may result in minimal features too small to be fabricated. Apart from the examples of PBS discussed above, inverse design based on these methods has been widely used for other silicon photonic devices as shown in Table 1. Empirical structure-based devices are large but they have high performance and are easy to process since their critical sizes are easy to control. QR-code like structures have compact footprints and controllable minimal feature sizes at the cost of some decrease in optical performance. Irregular devices have the most appealing performance, but minimal feature size control is in great need for the convenience of device fabrication. Some researchers also combined different structures and algorithms to solve the limitations of certain methods for the design of a single device. For example, [PERSON] et al. combined QR-code like structure into an empirical waveguide crossing structure to work as a wavelength filter [35]. [PERSON] et al. proposed a global optimization method combining GA and annealing algorithm for optimizing on-chip twisted light emitter [81].