A simple formula for the Lgap width of a facecentred cubic photonic crystal
Abstract
The width icons/Journals/Common/triangle" ALT="triangle" ALIGN="TOP"/>_{L} of the first Bragg scattering peak in the (111) direction of a facecentred cubic lattice of air spheres can be well approximated by a simple formula which only involves the volume averaged icons/Journals/Common/varepsilon" ALT="varepsilon" ALIGN="TOP"/> and icons/Journals/Common/varepsilon" ALT="varepsilon" ALIGN="TOP"/>^{2} over the lattice unit cell, icons/Journals/Common/varepsilon" ALT="varepsilon" ALIGN="TOP"/> being the (position dependent) dielectric constant of the medium, and the effective dielectric constant icons/Journals/Common/varepsilon" ALT="varepsilon" ALIGN="TOP"/>_{eff} in the longwavelength limit approximated by MaxwellGarnett's formula. Apparently, our formula describes the asymptotic behaviour of the absolute gap width icons/Journals/Common/triangle" ALT="triangle" ALIGN="TOP"/>_{L} for high dielectric contrast icons/Journals/Common/delta" ALT="delta" ALIGN="TOP"/> exactly. The standard deviation icons/Journals/Common/sigma" ALT="sigma" ALIGN="TOP"/> steadily decreases well below 1% as icons/Journals/Common/delta" ALT="delta" ALIGN="TOP"/> increases: for example, icons/Journals/Common/sigma" ALT="sigma" ALIGN="TOP"/><0.1% for the sphere filling fraction f = 0.2 and icons/Journals/Common/delta" ALT="delta" ALIGN="TOP"/>icons/Journals/Common/geq" ALT="geq" ALIGN="TOP"/>20. In the interval icons/Journals/Common/delta" ALT="delta" ALIGN="TOP"/>icons/Journals/Common/in" ALT="in" ALIGN="TOP"/>(1,100), our formula still approximates gap widths with a reasonable precision, namely the absolute gap width icons/Journals/Common/triangle" ALT="triangle" ALIGN="TOP"/>_{L} with a standard deviation of 3% for low filling fractions up to 6.5% for the closepacked case and the relative gap width icons/Journals/Common/triangle" ALT="triangle" ALIGN="TOP"/>_{L}^{r}, from 4.2% to 8%. Differences between the case of air spheres in a dielectric and dielectric spheres in air are briefly discussed.
 Publication:

Journal of Optics A: Pure and Applied Optics
 Pub Date:
 July 1999
 DOI:
 10.1088/14644258/1/4/310
 arXiv:
 arXiv:physics/9903022
 Bibcode:
 1999JOptA...1..471M
 Keywords:

 Physics  Classical Physics;
 Condensed Matter;
 Mathematical Physics;
 Physics  Optics
 EPrint:
 13 pages, 4 figs., RevTex, two references added. For more info see http://www.amolf.nl/external/wwwlab/atoms/theory/index.html