Electromagnetic modes of corrugated thin films and surfaces with a transition layer. II. Minigaps.

Abstract

Our recent calculation [Phys. Rev. B 36 , 1007 (1987)] of electromagnetic modes that may propagate parallel to a thin corrugated film is applied to the study of the ``minigap'' regions. Two polariton branches of the planar film, when displaced by a reciprocal vector of the corrugation period, may intersect at values of the propagation constant (quasimomentum) that do not coincide with a Brillouin-zone boundary. In the vicinity of an intersection a coupling of modes occurs, and our first-order perturbational calculation shows that the behavior is determined by the signs of two parameters B and G, giving rise to four cases. B is a coupling constant, proportional to ${h}^{2}$ (where h is the height of the corrugation), and G is positive (negative) if the slopes of the unperturbed modes at the intersection have opposite (equal) signs. For B>0 and G>0, an energy gap occurs; in general, the minigap is ``indirect;'' that is, the minimum of the upper band and the maximum of the lower band are not located at the same value of the momentum. Within the gap we find solutions with complex values of the momentum. In the case B>0 and G0, a ``repulsion'' of modes takes place; there is no gap. If B0 and G>0, then we get a momentum gap that is also indirect, in general. Within this gap there exist solutions with a complex value of the frequency. In the last case, B0 and G0, a frequency gap and a momentum gap are obtained simultaneously. The widths of the minigaps are proportional to \ensuremath{\Vert}f^(i-j)\ensuremath{\Vert} (hence to h), the Fourier coefficient of the corrugation profile corresponding to unperturbed branches i and j. Special attention is given to the case that two branches of the smooth film intersect symmetrically at a Brillouin-zone boundary. In this case, B>0 and G>0 are applicable; the energy gap then becomes direct. We have also investigated the amplitudes of the partial, diffracted waves in the minigap regions. All our results are directly applied to a surface with a transition layer, as well as to a sharp surface, and to guided waves with either TE or TM polarization.