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

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.