Simple Application of the Envelope-Function Approximation for Photonic Crystals

A useful technique is presented for calculating the approximate behavior of electromagnetic wavepackets in a periodic dielectric medium. The technique produces an operator equation which describes the time evolution of the envelope of the wave packet. The operator equation is shown to be equivalent to the Helmholtz equation in linear, isotropic, homogeneous media and to a Schrödinger-like equation in a photonic band-gap material. A reflection experiment is used to demonstrate the technique. In this experiment, a photonic-crystal cavity is excited at a frequency above its band edge. It is found that the reflection coefficient has Fabry-Perot-like resonances which increase with the square of the frequency. Both the frequency and curvature of the band edge are accurately predicted by fitting the envelope-function theory to the reflection data.