Electromagnetic modes of corrugated thin films and surfaces with a transition layer: Basic formalism

We study the electromagnetic modes of a system composed of three media characterized by complex dielectric functions ${\epsilon }_j$(ω) (j=1, 2, 3), one interface being corrugated, while the other interface, parallel to the first, is planar. This configuration may correspond to a corrugated thin film, or to a corrugated surface with a transition layer. The corrugation profile f(x) has an arbitrary periodicity and we consider the cases of p- (TM) and s- (TE) polarized waves propagating in a direction perpendicular to the grooves. Applying the Rayleigh hypothesis we calculate the transverse electromagnetic fields and we derive an implicit dispersion relation for the modes. Then a perturbational approach leads to explicit solutions for the propagation constant α as a function of the frequency ω and for ω as a function of α. It is assumed that the height of the corrugation is very small compared to its period d and compared to the wavelength 2π/α, that Im${\epsilon }_j$≪‖Re${\epsilon }_j$‖, and that α≃oslnπ/d (n=±1,±2,. . .). The solutions for α(ω) and for ω(α) are found to be linear in ‖f^(m)${\Vert }^2$, where f^(m) is the Fourier transform of f(x).