Analytical expression for the optimized stop bands of fcc photonic crystals in the scalar-wave approximation

We present an analytical model based on scalar-wave approximation to study the stop bands of face-centered-cubic photonic crystals with dielectric spheres at lattice sites. An analytical expression is obtained for the normalized stop band widths along the [111] and [100] directions as a function of volume fraction and relative dielectric contrast. These calculations enable us to map out the dielectric constants and volume fractions which results in overlapping L and X stop bands. From this map, it is possible to extract the optimum volume fraction that results in the widest overlapping stop band for a given dielectric contrast. We find that the relation between the optimum volume fraction and the dielectric contrast can be approximated by a simple power-law behavior for fee photonic crystals.