  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.
