Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings

TLDR

The rigorous coupled‑wave analysis technique for describing the diffraction of electromagnetic waves by periodic grating structures is reviewed. The study presents formulations for a stable and efficient numerical implementation of rigorous coupled‑wave analysis for one‑dimensional binary gratings (TE/TM and conical diffraction) and investigates how the number of harmonics affects convergence of diffraction efficiencies. The authors formulate a stable, efficient numerical implementation that anticipates and preempts underflow/overflow instability, and analyze how the number of harmonics influences convergence. The efficient formulation achieves up to an order‑of‑magnitude improvement in numerical efficiency, the method is inherently stable, and diffraction efficiencies converge correctly with more harmonics, especially for larger periods, deeper gratings, TM polarization, and conical diffraction.

Abstract

The rigorous coupled-wave analysis technique for describing the diffraction of electromagnetic waves by periodic grating structures is reviewed. Formulations for a stable and efficient numerical implementation of the analysis technique are presented for one-dimensional binary gratings for both TE and TM polarization and for the general case of conical diffraction. It is shown that by exploitation of the symmetry of the diffraction problem a very efficient formulation, with up to an order-of-magnitude improvement in the numerical efficiency, is produced. The rigorous coupled-wave analysis is shown to be inherently stable. The sources of potential numerical problems associated with underflow and overflow, inherent in digital calculations, are presented. A formulation that anticipates and preempts these instability problems is presented. The calculated diffraction efficiencies for dielectric gratings are shown to converge to the correct value with an increasing number of space harmonics over a wide range of parameters, including very deep gratings. The effect of the number of harmonics on the convergence of the diffraction efficiencies is investigated. More field harmonics are shown to be required for the convergence of gratings with larger grating periods, deeper gratings, TM polarization, and conical diffraction.