dorsal/arxiv
View SchemaA simple formula for the L-gap width of a face-centered-cubic photonic crystal
| Authors | Alexander Moroz |
|---|---|
| Categories | |
| ArXiv ID | physics/9903022 |
| URL | https://arxiv.org/abs/physics/9903022 |
| DOI | 10.1088/1464-4258/1/4/310 |
| Journal | J. Opt. A: Pure Appl. Opt. 1 (1999) 471-475 |
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 $\epsilon$ 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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"abstract": "The width $\\triangle_L$ of the first Bragg\u0027s scattering peak in the (111)\ndirection of a face-centered-cubic lattice of air spheres can be well\napproximated by a simple formula which only involves the volume averaged\n$\\epsilon$ and $\\epsilon^2$ over the lattice unit cell, $\\epsilon$ being the\n(position dependent) dielectric constant of the medium, and the effective\ndielectric constant $\\epsilon_{eff}$ in the long-wavelength limit approximated\nby Maxwell-Garnett\u0027s formula. Apparently, our formula describes the asymptotic\nbehaviour of the absolute gap width $\\triangle_L$ for high dielectric contrast\n$\\delta$ exactly. The standard deviation $\\sigma$ steadily decreases well below\n1% as $\\delta$ increases. For example $\\sigma\u003c 0.1%$ for the sphere filling\nfraction $f=0.2$ and $\\delta\\geq 20$. On the interval $\\delta\\in(1,100)$, our\nformula still approximates the absolute gap width $\\triangle_L$ (the relative\ngap width $\\triangle_L^r$) with a reasonable precision, namely with a standard\ndeviation 3% (4.2%) for low filling fractions up to 6.5% (8%) for the\nclose-packed case. Differences between the case of air spheres in a dielectric\nand dielectric spheres in air are briefly discussed.",
"arxiv_id": "physics/9903022",
"authors": [
"Alexander Moroz"
],
"categories": [
"physics.class-ph",
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"math.MP",
"physics.optics"
],
"doi": "10.1088/1464-4258/1/4/310",
"journal_ref": "J. Opt. A: Pure Appl. Opt. 1 (1999) 471-475",
"title": "A simple formula for the L-gap width of a face-centered-cubic photonic crystal",
"url": "https://arxiv.org/abs/physics/9903022"
},
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