# General analytical solution for the electromagnetic grating diffraction problem

## TL;DR

This paper presents a comprehensive analytical solution for electromagnetic grating diffraction using modal methods and coordinate transformations, clarifying the conditions under which Rayleigh expansion is valid.

## Contribution

It introduces a general analytical T-matrix solution for 1D grating diffraction problems via curvilinear coordinate transformation, linking modal and Rayleigh expansions.

## Key findings

- Derived a general analytical T-matrix solution for 1D gratings
- Established the validity conditions for Rayleigh expansion based on modal expansion
- Unified modal and Rayleigh methods through coordinate transformation

## Abstract

Implementing the modal method in the electromagnetic grating diffraction problem delivered by the curvilinear coordinate transformation yields a general analytical solution to the 1D grating diffraction problem in a form of a T-matrix. Simultaneously it is shown that the validity of the Rayleigh expansion is defined by the validity of the modal expansion in a transformed medium delivered by the coordinate transformation.