Effective mode volume of a natural mode of an open dielectric resonator with an active region

Abstract

We study the lasing eigenvalue problem for a generic open dielectric resonator with gain material. The gain is introduced within the active region via the “active” imaginary part of the refractive index. Each eigenvalue is constituted of two positive numbers, namely, the lasing wavenumber and the threshold value of material gain. This approach yields clear insight into the lasing thresholds of individual modes. The Optical Theorem, if applied to the lasing-mode field, puts familiar “gain=loss” condition on firm footing. It enables us to rigorously introduce the conception of the volume of an open resonator and then the effective-mode volume, both for the passive cavities and cavities with active regions.