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

Abstract

We study the electromagnetic modes of a system composed of three media characterized by complex dielectric functions ${\ensuremath{\epsilon}}_{j}$(\ensuremath{\omega}) (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 \ensuremath{\alpha} as a function of the frequency \ensuremath{\omega} and for \ensuremath{\omega} as a function of \ensuremath{\alpha}. It is assumed that the height of the corrugation is very small compared to its period d and compared to the wavelength 2\ensuremath{\pi}/\ensuremath{\alpha}, that Im${\ensuremath{\epsilon}}_{j}$\ensuremath{\ll}\ensuremath{\Vert}Re${\ensuremath{\epsilon}}_{j}$\ensuremath{\Vert}, and that \ensuremath{\alpha}\ensuremath{\simeq}osln\ensuremath{\pi}/d (n=\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,. . .). The solutions for \ensuremath{\alpha}(\ensuremath{\omega}) and for \ensuremath{\omega}(\ensuremath{\alpha}) are found to be linear in \ensuremath{\Vert}f^(m)${\ensuremath{\Vert}}^{2}$, where f^(m) is the Fourier transform of f(x).