Singular perturbation analysis of the mean-field limit of semiclassical optics.

Abstract

A linear stability analysis of a plane-wave model of a unidirectional ring cavity, which allows the slowly varying field amplitudes to vary along the length of the atomic medium, yields a characteristic equation having singularities coincident with estimates of the atomic-dominant eigenvalues derived under the mean-field limit. This result calls into question the validity of previous estimates of these eigenvalues. Using this general characteristic equation, we estimate the cavity- and atom-dominant eigenvalues by applying a perturbation expansion in sequential powers of the uniform-field parameter \ensuremath{\eta}=2 ln\ensuremath{\Vert}scr\ifmmode \bar{E}\else \={E}\fi{}(L)/scr\ifmmode \bar{E}\else \={E}\fi{}(0)\ensuremath{\Vert}. The resulting eigenvalue estimates appear as asymptotic expansions with limited domains of validity. The cavity-dominant eigenvalues, which are responsible for the well-known ``self-pulsing'' instability in unidirectional ring lasers and optically bistable resonators, are shown to be consistent with estimates derived in previous studies. However, the new expressions for the atom-dominant eigenvalues exhibit a more intricate distribution in the complex plane.