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

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