Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings

Abstract

A new numerical mathematical method is presented for stack of lamellar gratings. It is based on expansion of the electromagnetic field in terms of the eigenfunctions of the Helmholtz equation. These eigenfunctions are in turn expanded in terms of sets of polynomial basis functions. It is shown that, for arbitrary polarization and for both dielectrics and lossy metals, the eigenvalues and the eigenvectors and, consequently, the spectral location of resonances converge at an exponential rate with increasing dimension of the polynomial basis. For the solution of the boundary-value problem physical arguments are used to derive a new algorithm that is of high numerical accuracy and is inherently stable. Single-precision arithmetic is sufficient, even for the calculation of strong resonances.