Lasing spectra and thresholds of the whispering-gallery modes in a circular dielectric microcavity

Microlaser designs based on high-reflectivity whispering-gallery (WG) modes around the edge of a thin semiconductor microdisk have been studied since the early 1990´s. Optical pump is normally arranged with a wide external laser beam, hence the gain over the disk can be considered as uniform. The same, although less justified, holds for injection lasers. Mathematically, these modes are the source-free solutions of the 3-D Maxwell equations, however if the disk thickness is only a fraction of the wavelength the modes can be studied in 2-D formulation. Still surprisingly, it appears that accurate study of the circular-cavity lasing modes is absent, as the simplified analysis of Frateschi et al. (1995), was based on rough assumptions, and FDTD simulations of Fujita et al. (1999) did not address the modes directly. In this paper we study the lasing eigenvalue problem (LEP) for a circular resonator. We look for the non-attenuating time-harmonic electromagnetic field ∼e^-ikct, k=Rek>0 in and out of a dielectric circular cylinder of radius a. We assume that the field does not vary along the z axis and can be characterized by a scalar function U, which represents either E_z or H_z component depending on the polarization.