README.md

APF_inverse_design

APF (Augmented Partial Factorization) is an efficienct method proposed in Nature Computational Science 2, 815–822 (2022) that can solve multi-input electromagnetic forward problems in one shot, offering substantial speed-up and memory usage reduction compared to existing methods. Here, we generalize APF to enable efficient gradient computation for and inverse design of multi-input nanophotonic devices.

This repository includes codes used to inverse design a metasurface beam splitter using APF method, which splits the incident light equally into the $\pm$ 1 diffraction orders for any incident angle within a wide angular range. Detailed documentation is given in the comments of each file.

• inverse_design_beam_splitter.m: Build the system and perform the optimization
• FoM_and_grad.m: Compute the figure of merit and its gradient with respect to optimization variables.
• constraint_and_grad.m: Build the inequality constraint and its gradient with respect to optimization variables.
• gradient_descent_BLS.m: Perform the gradient-descent optimization with the learning rate determined by backtracking line search.
• build_epsilon_pos.m: Build the permittivity profile.

To take fully advantage of the APF method and efficienctly perform optimizations, one needs to

• Install MESTI.m [download it and add the MESTI.m/src folder to the search path using the addpath command in MATLAB], the serial version of MUMPS and its MATLAB interface.
• Install NLopt to use various well-developed algorithms for optimizations. If NLopt is not installed, one can specify other methods in inverse_design_beam_splitter.m.
• Run inverse_design_beam_splitter.m [one can customise the optimization using options; check the documentation of this file for more details].

An animation on the evolution of the metasurface and its transmission matrix is shown below.

See the corresponding paper Fast Multichannel Inverse Design through Augmented Partial Factorization for more details.