Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media

TLDR

The paper establishes the most general differential equations governing the Fourier components of electromagnetic fields diffracted by arbitrary periodic anisotropic media. These equations are derived using the fast‑Fourier‑factorization method, enabling rapid convergence, and can be solved via classic differential theory, modal methods for specific groove profiles, or extended to nonperiodic media with a Fourier transform. The resulting framework recovers diffraction by classic isotropic gratings, improves both differential and modal approaches, and includes crossed gratings, arbitrary 2‑D periodic surfaces, and 3‑D photonic crystals as special cases.

Abstract

We establish the most general differential equations that are satisfied by the Fourier components of the electromagnetic field diffracted by an arbitrary periodic anisotropic medium. The equations are derived by use of the recently published fast-Fourier-factorization (FFF) method, which ensures fast convergence of the Fourier series of the field. The diffraction by classic isotropic gratings arises as a particular case of the derived equations; the case of anisotropic classic gratings was published elsewhere. The equations can be resolved either through classic differential theory or through the modal method for particular groove profiles. The new equations improve both methods in the same way. Crossed gratings, among which are grids and two-dimensional arbitrarily shaped periodic surfaces, appear as particular cases of the theory, as do three-dimensional photonic crystals. The method can be extended to nonperiodic media through the use of a Fourier transform.