Nonlinear equation method for band structure calculations of photonic crystal slabs

In this letter a nonlinear equation method based on effective-medium approximation is presented to calculate the band structure of photonic crystal slabs. At a definite wave vector, the normalized frequency is inversely proportional to the effective refractive index of the photonic crystals. Effective-medium approximation is used to calculate the effective refractive indices of the two dimensional photonic crystals and photonic crystal slab. The corresponding relation between the band structure of the photonic crystal slab and that of two dimensional photonic crystals is given by a nonlinear equation. Through the nonlinear equation, the three dimensional photonic crystal slabs problems are simplified as two dimensional simulations. The results by this method agree very well with those of three dimensional finite-difference time-domain method and the calculation is much faster.