Global rate equation description of a laser

The global integro-differential rate equations describing a multimode laser are analyzed. Expressions for the relaxation oscillation frequencies and their damping rates in the single-mode and two-mode regimes are obtained without specifying either the cavity geometry or the longitudinal pump profile. On the same level of generality, we prove the existence of universal relations relating the peaks of the power spectra in the two-mode regime. For a Fabry-Perot with arbitrary longitudinal pump profile, series expansions of all the physical functions are derived in powers of the pump moments. These moments are averages of the pump profile over cavity modes at linear combinations of the lasing frequencies and their harmonics. These results apply to end-pumped and/or partially filled lasers. For a single mode Fabry-Perot laser, we prove that the contribution to the steady state intensity from the lasing mode varies from 75% close to the lasing threshold to zero at high intensity. The remainder comes from the harmonics of the lasing mode. Analyzing the steady state single mode intensity equation in terms of the pump gratings, we prove that close to the lasing threshold only the space average of the pump and its grating oscillating at twice the lasing wave number do not vanish. This provides a hint towards the justification of the usual modal rate equations which retain only these two functions in the dynamical evolution of a laser. For a Fabry-Perot with constant pump profile, an exact expression for the upper boundary of the stable single mode regime is derived. In that two-mode regime, we prove that there is a critical value of the pump at which the ratio of the two relaxation oscillation frequencies is 2, leading to an internal resonance