面心立方光子晶体L型带隙宽度的一个简单公式
A simple formula for the L-gap width of a face-centered-cubic photonic crystal
摘要：
空气球的面心立方晶格的（111）方向上的第一布拉格散射峰的宽度$\triangle_L$可以通过简单的公式很好地近似，该公式仅涉及晶格晶胞上的体积平均$ε$和$ε^2$，$ε$是介质的（位置相关的）介电常数，以及通过Maxwell-Garnett公式近似的长波长极限中的有效介电常数$ε_{eff}$。显然，我们的公式准确地描述了高介电对比度$δ$的绝对间隙宽度$\triangle_L$的渐近行为。随着$δ$的增加，标准差$σ$稳定地下降到远低于1%。例如，球体填充分数$f=0.2$和$δ\geq 20$的$σ< 0.1%$。在间隔$δ\in(1,100)$上，我们的公式仍然以合理的精度近似绝对间隙宽度$\triangle_L$（相对间隙宽度$\triangle_L^r$），即对于低填充分数具有3%（4.2%）的标准偏差，对于密堆积情况具有6.5%（8%）的标准偏差。简要讨论了空气中的电介质球和空气中的电介质球的情况之间的差异。
英文摘要：
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.
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