The classical differential method, the rigorous coupled wave theory, and
the modal method: comparative analysis of convergence properties in staircase
Evgueni K. Popov, Michel Neviere, Boris Gralak, Gerard Tayeb.
Physics, Theory, and Applications of Periodic Structures in Optics, Ph. Lalanne,
Editor, Proc. of SPIE, Vol.4438, p.12-18, 2001.
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 an 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 Fourier 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.