ANALYTIC EXPRESSIONS FOR THE REFLECTION DELAY, PENETRATION DEPTH, AND ABSORPTANCE OF QUARTER-WAVE DIELECTRIC MIRRORS

In this paper we analyze the operation of high-reflectivity quarter-wave (QW) dielectric mirrors at the band-stop center (Bragg) frequency, relevant for the design of small-cavity optoelectronic structures. We discuss the mirror penetration depth from the standpoints of reflection delay and energy storage in the quarter-wave mirrors and derive exact analytic expressions for the penetration depth at the mirror center frequency based on both definitions. We show that, in general, the two models yield different expressions for the penetration depth, but for certain practical cases the difference is negligible. We also present the mathematical analysis and expressions for the absorptance and the peak reflectivity of a dielectric mirror with weak material absorption. The quarter-wave mirrors have typically been analyzed using coupled-wave theory, which applies to mirrors with small index differences, or by exact numerical calculation using the transmission matrix approach. In this paper we derive expressions for the penetration depth and mirror reflection delay that are valid for arbitrary material refractive index combinations and any number of layers, and are therefore applicable both for semiconductor and amorphous dielectric mirrors. The use of our results is illustrated for a typical vertical cavity surface-emitting laser structure.