Ab-initio theory of quantum fluctuations and relaxation oscillations in multimode lasers

We present an \emph{ab-initio} semi-analytical solution for the noise spectrum of complex-cavity micro-structured lasers, including central Lorentzian peaks at the multimode lasing frequencies and additional sidepeaks due to relaxation-oscillation (RO) dynamics. In~Ref.~1, we computed the central-peak linewidths by solving generalized laser rate equations, which we derived from the Maxwell--Bloch equations by invoking the fluctuation--dissipation theorem to relate the noise correlations to the steady-state lasing properties; Here, we generalize this approach and obtain the entire laser spectrum, focusing on the RO sidepeaks. Our formulation treats inhomogeneity, cavity openness, nonlinearity, and multimode effects accurately. We find a number of new effects, including new multimode RO sidepeaks and three generalized $\alpha$ factors. Last, we apply our formulas to compute the noise spectrum of single- and multimode photonic-crystal lasers.