The classical differential method, the rigorous coupled wave theory, and the modal method: comparative analysis of convergence properties in staircase approximation

The diffraction by periodic structures using a representation of the field in some functional basis leads to a set of ordinary differential equations, which can be solved by numerical integration. When the basic functions are the exponential harmonics (Fourier decomposition) one arrives at the well-known classical differential method. In the case of simple lamellar profiles, the numerical integration can be substituted by eigenvalue-eigenvector technique, known in the field of diffraction by periodic systems under the name of rigorous coupled-wave analysis or method of Moharam and Gaylord. When the basis functions are searched as the rigorous solutions of the diffraction problem inside the lamellar grooves, the theory is known under the name of modal method. A comparative analysis of the three methods is made to reveal the convergence rate for an arbitrary shaped grating using the staircase approximation. It is shown that in TM polarization this approximation leads to sharp peaks of the electric field near the edges. A higher number of fourir harmonics is then required to describe the field, compared with the case of a smooth profile, and a poor convergence is observed. The classical differential method, which does not use the staircase approximation does not suffer from this problem.