Study of eigenmodes in periodic waveguides using the Lorentz reciprocity theorem

Photonic crystals, frequency-selective surfaces, gratings, and many so-called metamaterials are composed of periodic arrangements of objects. Such an arrangement of objects quickly forms a periodic waveguide. In this paper, we investigate a number of properties of the eigenmodes in general periodic waveguides using the Lorentz reciprocity theorem. Such an analysis seems to be missing in the literature. We present an original proof for the intimate relation between bidirectionality of a periodic waveguide and reciprocity. We also derive compact expressions for the excitation coefficients of the eigenmodes when the waveguide is excited by a source density or an incident field. The analysis is generalized to include periodic waveguides composed of anisotropic and bianisotropic materials.