An effective medium theory for predicting the existence of surface states

We build an effective medium theory for two-dimensional photonic crystals comprising a rectangular lattice of dielectric cylinders with the incident electric field polarized along the axis of the cylinders. In particular, we discuss the feasibility of constructing an effective medium theory for the case where the Bloch wave vector is far away from the center of Brillouin zone, where the optical response of the photonic crystal is necessarily anisotropic and hence the effective medium description becomes inevitability angle dependent. We employ the scattering theory and treat the two-dimensional system as a stack of one-dimensional arrays. We consider only the zero-order interlayer diffraction and all the higher order diffraction terms of interlayer scattering are ignored. This approximation works well when the higher order diffraction terms are all evanescent waves and the interlayer distance is far enough for them to decay out. Scattering theory enables the calculation of transmission and reflection coefficients of a finite sized slab, and we extract the effective parameters such as the impedance ($Z_e$) and the refractive index ($n_e$) using a parameter retrieval method. We note that $n_e$ is uniquely defined only in a very limited region of the reciprocal space. ($n_e k_0 a<<1$, where $k_0$ is the wave vector inside the vacuum and a is thickness of the slab for retrieval), but $Z_e$ is uniquely defined and has a well-defined meaning inside a much larger domain in the reciprocal space. For a lossless system, the effective impedance $Z_e$ is purely real for the pass band and purely imaginary in the band gaps. Using the sign of the imaginary part of $Z_e$, we can classify the band gaps into two groups and this classification explains why there is usually no surface state on the boundary of typical fully gapped photonic crystals comprising of a lattice of dielectric cylinders.