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

Abstract

The width ${△}_L$ 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 $\epsilon$ and ${\epsilon }^2$ over the lattice unit cell, $\epsilon$ being the (position dependent) dielectric constant of the medium, and the effective dielectric constant ${\epsilon }_{eff}$ in the long-wavelength limit approximated by Maxwell-Garnett's formula. Apparently, our formula describes the asymptotic behaviour of the absolute gap width ${△}_L$ for high dielectric contrast $\delta$ exactly. The standard deviation $\sigma$ steadily decreases well below 1% as $\delta$ increases. For example $\sigma <0.1$ for the sphere filling fraction $f=0.2$ and $\delta \ge 20$. On the interval $\delta \in (1,100)$, our formula still approximates the absolute gap width ${△}_L$ (the relative gap width ${△}_L^r$) 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.

13 pages, 4 figs., RevTex, two references added. For more info see http://www.amolf.nl/external/wwwlab/atoms/theory/index.html

References in corpus (3)

• Photonic Band Gaps of Three-Dimensional Face-Centered Cubic Lattices
• Density-of-States Calculations and Multiple-Scattering Theory for Photons
• Resonance-Induced Effects in Photonic Crystals

Cited by in corpus (1)

• Metallo-dielectric diamond and zinc-blende photonic crystals