Rigorous coupled-wave analysis

Definition

Consider a structure that is periodic in $x$ with period $\Lambda$, invariant in $y$, and made of layers $z_{l-1} < z < z_l$. In each layer the relative permittivity $\varepsilon(x)$ does not depend on $z$. Let a $y$-polarized (TE) plane wave of vacuum wave number $k_0$ arrive from above with in-plane wave number $\kappa_0$, and let $\kappa_m = \kappa_0 + 2\pi m / \Lambda$. Fix a truncation order $M$. Rigorous coupled-wave analysis of order $M$ computes the diffracted field as follows.

1. In each layer, write $E_y = \sum_{\abs{m} \le M} S_m(z) e^{i \kappa_m x}$, and expand $\varepsilon(x) = \sum_p \varepsilon_p e^{2\pi i p x / \Lambda}$ in a Fourier series.
2. Substitute into $\partial_x^2 E_y + \partial_z^2 E_y + k_0^2 \varepsilon E_y = 0$, which follows from the Maxwell equations, and keep the orders $\abs{m} \le M$; this gives $S'' = A S$ with the matrix $A = K^2 - k_0^2 E$.
3. Here $K$ is diagonal with entries $\kappa_m$, and $E$ is the Toeplitz matrix with entries $E_{mp} = \varepsilon_{m - p}$.
4. Solve each layer by the eigenvalues and eigenvectors of $A$. Match the tangential fields at the interfaces, and with the plane-wave orders above and below, as in the transfer-matrix method.

The output is the amplitude, hence the diffraction efficiency, of each order $m$.