We study the convergence properties of the two-dimensional Rigorous Coupled Wave Approach (RCWA) for $p$-polarized monochromatic incident light. The RCWA is a semi-analytical numerical method that is widely used to solve the boundary-value problem of scattering by a grating. The approach requires the expansion of all electromagnetic field phasors and the relative permittivity as Fourier series in the spatial variable along the direction of the periodicity of the grating. In the direction perpendicular to the grating periodicity, the domain is discretized into thin slices and the actual relative permittivity is replaced by an approximation. The approximate relative permittivity is chosen so that the solution of the Maxwell equations in each slice can be computed without further approximation. Thus, there is error due to the approximate relative permittivity as well as the truncation of the Fourier series. We show that the RCWA embodies a Galerkin scheme for a perturbed problem, and then we use tools from the Finite Element Method to show that the method converges with increasing number of retained Fourier modes and finer approximations of the relative permittivity. Numerical examples illustrate our analysis, and suggest further work.

我们研究了二维严格耦合波方法（RCWA）的收敛特性 $p$-偏振单色入射光。RCWA是一种半解析数值方法，广泛用于解决光栅散射的边值问题。该方法需要扩展所有电磁场相量，并沿光栅的周期性方向在空间变量中将其相对介电常数作为傅立叶级数。在垂直于光栅周期性的方向上，将畴离散成薄片，并用近似值代替实际的相对介电常数。选择近似的相对介电常数，以便无需进一步近似即可计算每个切片中的麦克斯韦方程组的解。因此，由于近似的相对介电常数以及傅立叶级数的截断而存在误差。我们证明了RCWA体现了一个针对扰动问题的Galerkin方案，然后我们使用了有限元方法的工具来证明该方法随着越来越多的保留傅立叶模态和相对介电常数的近似值收敛。数值示例说明了我们的分析，并提出了进一步的建议。