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

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

1999-03-15

arxiv.org/abs/physics/9903022v2

J. Opt. A: Pure Appl. Opt. 1 (1999) 471-475

Physics

Classical Physics

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

Scientific paper

10.1088/1464-4258/1/4/310

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 $\epsilon$ 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 $\triangle_L$ 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 $\delta\geq 20$. On the interval $\delta\in(1,100)$, 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.

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