## 2 Inverse Design of Silicon Photonics with Iterative Optimization Algorithms
### Optimization of Empirical Structures  
An empirical structure makes use of classical structures, where only a few parameters like radius, duty cycle, widths of waveguides, etc. are tuned for targeted optical response. These classical structures are proposed by researchers for different application scenarios. For example, micro-rings are preferred for wavelength selection due to their high wavelength-sensitivity [21; 27]. Directional couplers (DCs) are used for polarization beam splitters (PBS), mode-division multiplexers (MDM) and wavelength-division multiplexers (WDM) since DCs couple light from one waveguide to another in a reciprocating way [21; 24; 26]. Y-splitter with three ports is widely used for power splitting (PS) [59]. Mach-Zehnder interferometer (MZI) can separate light and finally combine with optical path difference, so that it could be utilized for optical switches or modulators [30; 60; 61]. Gratings which periodically arrange the silicon and air (or silica) have been applied to manipulate the effective refractive index of waveguides or act as fiber grating couplers (GC) [62; 63]. With known structures and underlying theories to design a device for targeted FOM, it still costs lots of time to tailor the key parameters, as each parameter set requires at least one EM simulation.  
Figure 1: Inverse design schemes for integrated silicon photonics.  
The relationship between the abstracted key parameters from an empirical structure and its optical response is a kind of NP hard problem, which can be solved by heuristic algorithms. GA, a kind of heuristic optimized algorithm, is inspired by Darwinian evolution theory of survival of the fittest [64]. After initial chromosomes, which represent different parameter sets, are randomly generated, the fitness values describing the FOM of each chromosome are computed as shown in Figure 2a. Chromosome with better fitness values will be chosen as elite and passed to the next generation directly. For the rest of chromosomes, some chromosomes experience crossover with others to create new chromosomes and some produce offspring by mutation. Through the above elite selection, crossover and mutation, a new generation is born. This procedure is repeated until the best FOM is found.  
We take reference [25] for example to illustrate utilizing GA to optimize an optical device with empirical structure. A broadband PBS based on directional coupler and sub-wavelength gratings (SWGs) structure is designed as shown in Figure 2b. Key parameters like pitch \(\Lambda\), duty cycle \(\alpha\) and coupling length \(L_{s}\) are abstracted to represent this structure and coded as a chromosome. Initial chromosomes (20 parameters sets) are generated randomly within parameter boundaries. The FOM of each chromosome is calculated with respect to the output of transverse electric (TE) mode at through waveguide and transverse magnetic (TM) mode at cross waveguide. After hundreds of iterations, the best parameter set is found as Figure 2c suggests, where the transmission efficiencies for both TE and TM modes at desired output ports are larger than 85% over wavelength range from 1250 nm to 1680 nm.  
Another popular heuristic algorithm is PSO [65]. Different form GA, where parts of parameter sets are updated at each iteration, PSO updates all the parameter sets towards the global optimized FOM. For PSO, the initial randomly generated parameter sets are called particles, each of which is assigned an initial velocity. PSO evaluates the FOM of each particle and finds the best global and local FOMs as well as the corresponding particles. According to the global best particle and the local best particle at their current iteration, the velocity of each particle is updated. With updated velocities, the new population are generated from original particles. This updating procedure is repeated until the global best FOM is converged to an acceptable value. PSO has also been widely used for the design of silicon photonic devices with empirical structures as listed in Table 1.