Purcell effect in sub-wavelength semiconductor lasers

We present a formal treatment of the modification of spontaneous emission rate by a cavity (Purcell effect) in sub-wavelength semiconductor lasers. To explicitly express the assumptions upon which our formalism builds, we summarize the results of non-relativistic quantum electrodynamics (QED) and the emitter-field-reservoir model in the quantum theory of damping. Within this model, the emitter-field interaction is modified to the extent that the field mode is modified by its environment. We show that the Purcell factor expressions frequently encountered in the literature are recovered only in the hypothetical condition when the gain medium is replaced by a transparent medium. Further, we argue that to accurately evaluate the Purcell effect, both the passive cavity boundary and the collective effect of all emitters must be included as part of the mode environment.