Ab initiomultimode linewidth theory for arbitrary inhomogeneous laser cavities

We present a multimode laser-linewidth theory for arbitrary cavity structures and geometries that contains nearly all previously known effects and also finds new nonlinear and multimode corrections, e.g., a correction to the $\alpha$ factor due to openness of the cavity and a multimode Schawlow-Townes relation (each linewidth is proportional to a sum of inverse powers of all lasing modes). Our theory produces a quantitatively accurate formula for the linewidth, with no free parameters, including the full spatial degrees of freedom of the system. Starting with the Maxwell-Bloch equations, we handle quantum and thermal noise by introducing random currents whose correlations are given by the fluctuation-dissipation theorem. We derive coupled-mode equations for the lasing-mode amplitudes and obtain a formula for the linewidths in terms of simple integrals over the steady-state lasing modes.