## Maxwell's Equations

Each layer of the grating region may consist of a single, or arbitrary number of index of refraction regions. The full vector wave equation may be solved within a single layer by representing the periodic index of refraction as a Fourier series. This results in a set of linear coupled differential equations which are equivalent to Maxwell’s equations within a grating layer. The only approximation is in the truncation order the user sets for the Fourier expansion. Solution of the truncated set of differential equations is found by algebraic eigensystem methods. This results in a basis set of inhomogeneous plane waves which satisfy Maxwell’s equations within the layer. The full grating solution then follows from matching boundary conditions across each layer. This is done by employing S-matrix methods, which are much more stable than Gaussian elimination with pivoting.

For further information on the solution methods employed in GSolver refer to the Users Manual.