This application solves Maxwell’s equations in the frequency domain in parallel for multiple instances of a parametrized geometry using finite difference frequency domain numerical (FDFD) scheme, with the goal of

1. solving for metasurfaces using domain decomposition,
2. train surrogate models for offline Maxwell’s solve and large-scale optimization.

The design of metasurfaces — large-area ultrathin nanopatterned surfaces designed to mimic bulky lenses — is computationally challenging because of the enormous range of scales they involve. Indeed, they present two very different length scales, namely their feature size (of the order of 10 nm) and their diameter (of the order of the centimeter). Recent work from Johnson group @MIT Mathematics [1] has made the design possible using decomposition methods which breaks the computational domain of the metasurface into multiple smaller domains. A first approach is to solve Maxwell's equations for a multitude of subdomains in parallel and online. However, during large scale optimization, we need to simulate many different metasurfaces and this online approach remains slow. A second approach is to use a surrogate model, which is trained using solutions to Maxwell’s equation on the subdomain but is much faster to evaluate. A surrogate model dramatically increases the speed of simulation for metasurfaces and makes large scale optimization possible, at the cost of a tradeoff between training time and accuracy if the surrogate model. Using embarrassingly parallel solves for the training subdomains simulations alleviates this trade-off.

Julia enables us to write and solve Maxwell’s equations in 2D very elegantly and efficiently. Julia also gives us more freedom and control than its commercial software counterpart.

[1] R. Pestourie, C. Pérez-Arancibia, Z. Lin, W. Shin, F. Capasso, and S. G. Johnson, “Inverse design of large-area metasurfaces,” Opt. Express, vol. 26, no. 26, pp. 33732–33747, Dec. 2018, doi: 10.1364/OE.26.033732.