Forward-Mode Differentiation of Maxwell's Equations

TL;DR

This paper introduces a forward-mode differentiation method for Maxwell's equations that offers exact gradients, improved memory and speed efficiency, and serves as a precise alternative to finite-difference methods, demonstrated through sensitivity analyses.

Contribution

The paper presents a novel forward-mode differentiation technique for Maxwell's equations, enhancing efficiency and accuracy over existing methods in electromagnetic sensitivity analysis.

Findings

Exact gradients for Maxwell's equations computed efficiently.

Significant memory and speed improvements in FDTD simulations.

Validated sensitivity analysis on scatterers and surface gratings.

Abstract

We present a previously unexplored forward-mode differentiation method for Maxwell's equations, with applications in the field of sensitivity analysis. This approach yields exact gradients and is similar to the popular adjoint variable method, but provides a significant improvement in both memory and speed scaling for problems involving several output parameters, as we analyze in the context of finite-difference time-domain (FDTD) simulations. Furthermore, it provides an exact alternative to numerical derivative methods, based on finite-difference approximations. To demonstrate the usefulness of the method, we perform sensitivity analysis of two problems. First we compute how the spatial near-field intensity distribution of a scatterer changes with respect to its dielectric constant. Then, we compute how the spectral power and coupling efficiency of a surface grating coupler changes with respect to its fill factor.