Semianalytical formulations for the surface modes of photonic woodpiles

We have developed semianalytical methods that allow us to perform a comprehensive analysis of the surface modes of photonic woodpiles. The surface modes of both finite and semi-infinite woodpiles are characterized using transfer matrix and plane-wave matrix formulations, and, in the case of finite structures, we give a general analytical description of the ''double-interface'' modes, which propagate simultaneously along the top and bottom surfaces. We show that if the number of layers is even, then such modes will only exist for specific directions of the Brillouin zone. However, if the number of layers is odd, then every surface mode is a double-interface mode, and, in this case, the direction of propagation plays an important role in determining the coupling strength between the two surfaces: for certain directions, the coupling is negligible even when the number of layers is small. The dispersion curves of two different double-interface modes can anticross or be interwoven, depending on the symmetry of the modes. We also describe the conditions under which coupled surface modes will exist when two woodpiles are used to create a Fabry-Perot cavity.