We consider the diffraction of time-harmonic electromagnetic waves by dielectric grating interfaces. It is known that the intensity of the electromagnetic wave can change significantly at certain frequencies. Such situations are known as Wood anomalies, and they may lead to the destruction of the grating. The strong changes in intensity are associated with the excitation of surface waves along the gratings. Therefore, a reliable method to predict the appearance of surface waves is of great importance. The diffraction of a TE or TM polarized electromagnetic plane wave by a periodic grating can be modelled with a quasiperiodic boundary-value problem for the Helmholtz equation. Mathematically, a surface wave is a function, which decays exponentially with distance from the grating and is a solution to the corresponding problem with no sources (homogeneous problem). Nazarov and Plamenevsky [2] formulated a general existence criterion for the surface waves based on the properties of the extended scattering matrix corresponding to a special asymptotics. In this work, we introduce a new computational method for the detection of surface waves. It is based on a finite element solver for the Helmholtz equation and on a gradient-based optimization method for finding the entries of the scattering matrix. In the previous work [1], the existence criterion was applied in the case of planar gratings, but the scattering matrix was computed using the rigorous coupled-waves approach. However, this approach can be applied only with very specific grating interfaces. The finite element method imposes only few restrictions on the geometry of the grating, and is thus able to deal with a much wider class of interfaces. We demonstrate by numerical tests the accuracy of the new method. Results with planar gratings coincide with the results given by the rigorous coupled-waves approach. We are able to present examples of nonplanar gratings which support the excitation of surface waves.

*Department of Mathematical Information Technology, University of Jyväskylä, Finland.
*Department of Mathematics and Mathematical Physics, St.Petersburg University, Russia.

[1] V. Grikurov, M. Lyalinov, P. Neittaanmäki, and B. Plamenevsky, On surface waves in diffraction gratings, Report B13/1999, Department of Mathematical Information Technology, University of Jyväskylä. To appear in Math. Methods Appl. Sci.
[2] S. Nazarov, B. Plamenevsky, Selfadjoint elliptic problems with radiation conditions on the edges of the boundary, Algebra i Analiz 4 (1992) 196-225. In Russian. English translation in St.Petersburg Math. Journal 4(3), 1993.