Ultralow Raman lasing threshold and enhanced gain of whispering gallery modes in silica microspheres

Remarkable demonstrations of a cavity-induced giant reduction of the Raman lasing threshold in silica microspheres have been reported in Spillane et al. [Nature (London) 415, 621 (2002)], yet a complete quantum electrodynamics (QED) treatment of this process including the rigorous expressions for the electromagnetic modes of the sphere is lacking. In this paper, we employ a cavity QED approach to describe Raman lasing in microscopic spherical dielectric resonators with third-order nonlinearity to derive explicit expressions for the Raman gain and lasing threshold in terms of mode-overlapping coefficients determined from full analytic expressions of the electromagnetic eigenmodes of a dielectric sphere. We present dependencies of the Raman lasing threshold on the the order n of overlapping whispering gallery modes for microspheres with diameter in the range 20-35 mu m, demonstrating ultralow thresholds of between 70 and 90 mu W, consistent with recent experimental results. We explain the reduction of the lasing threshold as due to an increase in the overlapping coefficients as the mode order is increased toward the regime where the modal energy is mostly confined to the surface region of the resonator. The presented theory can be easily generalized to any microcavity of regular morphology and for any combination of interacting whispering gallery modes.