We study resonance patterns of a spiral-shaped dielectric microcavity with chaotic ray dynamics. Many resonance patterns of this microcavity, with refractive indices n=2 and 3, exhibit strong localization of simple geometric shape, and we call them quasiscarred resonances in the sense that there is, unlike conventional scarring, no underlying periodic orbits. It is shown that the formation of a quasiscarred pattern can be understood in terms of ray dynamical probability distributions and wave properties like uncertainty and interference.

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We report that unidirectional lasing emission can be generated from a rounded isosceles triangular microcavity within a low nkD range, where n is the refractive index, k is the vacuum wave number, and D is the characteristic size of the microcavity. It is shown that unidirectional resonance modes have relatively high-Q values and in a nonlinear dynamic model appear as stationary lasing solutions with a low threshold. The formation of a whispering-gallery-type pattern along the rounded part on the symmetry axis is responsible for the unidirectionality of the resonances.

In this paper, we consider several special polynomials related to associated sequences of polynomials. Finally, we give some new and interesting identities of those polynomials arising from transfer formula for the associated sequences.1991 Mathematics Subject Classification. 05A10, 05A19.