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 a Bogoliubov approach for the computation of the laser emission 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 example of application, the method allows to study the interplay of driven-dissipation, interactions and spatial inhomogeneity typical of polariton condensates. The traditional theory of the Henry and Petermann factors is found to fail dramatically in the presence of sizable polariton-polariton interactions. In particular, also in a strong confining potential, the intrinsically multi-mode nature of the density fluctuations has to be considered in order to describe quantitatively phase diffusion {\em \`a la} Henry.