LASER 2ND THRESHOLD - ITS EXACT ANALYTICAL DEPENDENCE ON DETUNING ANDRELAXATION RATES

An exact analysis has been carried out for general analytical expressi ons for the second threshold of a single-mode homogeneously broadened laser and for the initial pulsation frequency at the second threshold, for arbitrary physical values of the relaxation rates, and at an arbi trary detuning between the cavity frequency and the atomic resonance f requency. These expressions also give correspondingly exact forms for asymptotic cases that have been previously studied with some approxima tions. Earlier approximate results are partly confirmed and partly imp roved by these more general expressions. The physical status of variou s expressions and approximations is reconsidered and specified more cl early, including an analysis of what reasonably can be attained in las ers or masers. A general analytical proof is given of the fact that, f or a larger detuning of the laser cavity from resonance, a higher valu e of the laser excitation is required to destabilize the steady-state solution (the second threshold). We also present results for the minim um value of the second threshold at fixed detuning as a function of th e other parameters of the system and for the dependence of the ratio o f the second threshold to the first threshold as a function of detunin g. Minima of the second threshold and of the threshold ratio occur onl y if the population relaxation rate is equal to zero. The minima of th e threshold ratio are shown to be bounded from above as well as from b elow (as functions of the relaxation rates, so long as the second thre shold exists). The upper bound on the minima is equal to 17. The varia tion of the second threshold in the semi-infinite parameter space of t he decay rates is shown at various detunings in plots with a finite do main by normalizing the material relaxation rates to the cavity decay rate.