Band-Gap Analysis Of One-Dimensional Photonic Crystals And Conditions For Gap Closing

In this paper an analytical theory for arbitrary one-dimensional periodic media is presented. The analysis relies on the mathematical properties of Hill's equation. It is shown that the position of the band gaps can be obtained by quite simple expressions. As a special case, a one-dimensional multilayered medium (conventional photonic crystal) is studied. An exact formula for the location of the gap edges is derived for an infinite number of gaps, for both polarizations, at arbitrary angle of incidence. The gap closing conditions and the difference between the even- and the odd-numbered gaps are obtained. An extension for periodic structures with an arbitrary number of different layers is also presented. This method can be useful for the design of photonic crystal devices.