Bogoliubov theory of the laser linewidth and application to polariton condensates

For a generic semi-classical laser dynamics in the complex Ginzburg-Landau form, we develop the Bogoliubov approach for the computation of the linewidth. Our method provides a unifying perspective of the treatments by Henry and Petermann: both broadening mechanisms are ascribed to the non-orthogonality of the Bogoliubov modes, which live in a space with doubled degrees of freedom. As an application, the method allows to study the interplay of interactions and spatial inhomogeneity typical of polariton condensates: the concept of gain guided versus conservative modes is illustrated by varying the width of the pump spot and the frequency of the trap; it is shown that the traditional theory of the Henry and Petermann factors fails dramatically in the presence of a nonlinear refractive index (i.e. polariton-polariton interactions). In particular, we point out that the Henry's single mode approach performs poorly for an inhomogeneous condensate because the exact density-density correlator has to be considered in order to describe quantitatively phase diffusion as driven by density fluctuations.