Longitudinal polynomial expansion method for arbitrary-shaped gratings

The multilayer Fourier Modal Method (or Rigorous Coupled Wave Analysis) is one of the simplest and most popular method for analyzing arbitrary-shaped surface-relief gratings. This technique consists in replacing the grating by a staircase approximation. Hence the actual structure becomes a stack of lamellar gratings in which Maxwell´s equations may be written as an eigenvalue problem. Recently, the Legendre polynomial expansion method 2 has been introduced. In each layer, Maxwell´s equations in Fourier space are analytically projected onto the Hilbert space spanned by the Legendre polynomial basis functions. In principle, the above method avoids the staircase approximation of the actual grating. However this nice property is lost when the integrals defining the inner product are calculated by using the rectangle method. In our presentation, we will describe a new formulation for analyzing arbitrary-shaped gratings based on Fourier expansions in the direction of periodicity and Legendre polynomial expansions in the longitudinal direction. We will compare the numerical performance of our approach with that of other well established methods.