Theory of quarter-wave-stack dielectric mirrors used in a thin fabry-perot filter

I present a new derivation of the analytic form for the phase shift near resonance and the optical penetration length upon reflection from a distributed dielectric mirror consisting of a quarter-wave stack. The requirement of proper termination to achieve high reflectivity is suspended to investigate large optical penetration depths. Separate equations, derived for N and N + 1/2 layer pairs, are convenient for the design of tunable Fabry-Perot filters with a specified tuning range. The analysis is also applicable to distributed Bragg reflectors, vertical-cavity surface-emitting lasers, and resonant photodiodes. I show that the penetration length can sharply reduce the overly broad free spectral range of an ultrathin Fabry-Perot filter that might be useful in applications such as tunable wavelength filters for wavelength division multiplexing applications. The results also demonstrate regimes of zero dispersion and of superluminal reflection in the dielectric mirrors, which are of particular interest in photonic bandgap structures.