A simple formula for the L-gap width of a face-centered-cubic photonic crystal

Abstract

The width $\triangle_L$ of the first Bragg's scattering peak in the (111) direction of a face-centered-cubic lattice of air spheres can be well approximated by a simple formula which only involves the volume averaged $ε$ and $ε^2$ over the lattice unit cell, $ε$ being the (position dependent) dielectric constant of the medium, and the effective dielectric constant $ε_{eff}$ in the long-wavelength limit approximated by Maxwell-Garnett's formula. Apparently, our formula describes the asymptotic behaviour of the absolute gap width $\triangle_L$ for high dielectric contrast $δ$ exactly. The standard deviation $σ$ steadily decreases well below 1% as $δ$ increases. For example $σ< 0.1%$ for the sphere filling fraction $f=0.2$ and $δ\geq 20$. On the interval $δ\in(1,100)$, our formula still approximates the absolute gap width $\triangle_L$ (the relative gap width $\triangle_L^r$) with a reasonable precision, namely with a standard deviation 3% (4.2%) for low filling fractions up to 6.5% (8%) for the close-packed case. Differences between the case of air spheres in a dielectric and dielectric spheres in air are briefly discussed.