A study of laser rate equations by Liapunov's second method

An investigation is made of a system of coupled nonlinear differential equations (Statz-DeMars equations), describing the time variation of photon density and inversion in a laser or maser, without solving these equations explicitly. The method applied is based onLiapunov's stability theory. The results are rigorous and imply no approximation; they are, therefore, valid for arbitrarily large nonlinear terms. In the physically meaningful halfspace of the phase plane, i.e. where the photon density is not negative, the Statz-DeMars equations admit only damped periodic and damped aperiodic solutions. The transition between the aperiodic and the periodic mode is achieved, when the pumping rate exceeds a critical value. It is proven that the whole halfspace considered belongs to the domain of asymptotic stability of the equilibrium state and, therefore, no limit cycles and no diverging solutions exist.