Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures

TLDR

The authors derive an exact expression for the electromagnetic mode density and group velocity of a finite, N‑period, one‑dimensional photonic band‑gap structure. They first derive a general formula for mode density using the complex transmission coefficient of any index profile, then specialize it to an N‑period structure via the unit‑cell coefficient, and illustrate the approach with a quarter‑wave stack and its applications to 3‑D structures, spontaneous‑emission control, delay lines, band‑edge lasers, and superluminal tunneling times. The analysis reveals mode‑density enhancement and suppression at the photonic band edge and midgap, respectively, and demonstrates these effects in a quarter‑wave stack with implications for 3‑D structures, spontaneous‑emission control, delay lines, band‑edge lasers, and superluminal tunneling times. © 1996 The American Physical Society.

Abstract

We derive an exact expression for the electromagnetic mode density, and hence the group velocity, for a finite, N-period, one-dimensional, photonic band-gap structure. We begin by deriving a general formula for the mode density in terms of the complex transmission coefficient of an arbitrary index profile. Then we develop a specific formula that gives the N-period mode density in terms of the complex transmission coefficient of the unit cell. The special cases of mode-density enhancement and suppression at the photonic band edge and also at midgap, respectively, are derived. The specific example of a quarter-wave stack is analyzed, and applications to three-dimensional structures, spontaneous emission control, delay lines, band-edge lasers, and superluminal tunneling times are discussed. \textcopyright{} 1996 The American Physical Society.