The laser second threshold: Its exact analytical dependence on detuning and relaxation rates

A. A. Bakasov, N. B. Abraham

An exact analysis has been carried out for general analytical expressions for the second threshold of a single-mode homogeneously broadened laser and for the initial pulsation frequency at the second threshold, for arbitary physical values of the relaxation rates, and at an arbitrary detuning between the cavity frequency and the atomic resonance frequency. These expressions also give correspondingly exact forms for asymptotic cases that have been previously studied with some approximations. Earlier approximate results are partly confirmed and partly improved by these more general expressions. The physical status of various expressions and approximations is reconsidered and specified more clearly, including an analysis of what reasonably can be attained in lasers or masers. A general analytical proof is given of the fact that, for a larger detuning of the laser cavity from resonance, a higher value of the laser excitation is required to destabilize the steady-state solution (the second threshold). We also present results for the minimum value of the second threshold at fixed detuning as a function of the other parameters of the system and for the dependence of the ratio of the second threshold to the first threshold as a function of detuning. Minima of the second threshold and of the threshold ratio occur only if the population relaxation rate is equal to zero. The minima of the threshold ratio are shown to be bounded from above as well as from below (as functions of the relaxation rates, so long as the second threshold exists). The upper bound on the minima ratio is equal to 17. The variation of the second threshold in the semi-infinite parameter space of the decay rates is shown at various detunings in plots with a finite domain by normalizing the material relaxation rates to the cavity decay rate.

А. Н. Ораевский

Vincenzo Gino Benza, E. Montaldi

Faisal H A Mathkoor, Chan Hin Kam, C. H. Raymond Ooi

We have studied the photon statistics of the non-degenerate two-photon laser composed of many noninteracting atoms with Kerr nonlinearity. Nonlinearity of the gain is accounted for by the Lamb–Scully approach. Under certain conditions, we obtain an approximate photon distribution function. In the region above lasing threshold, the first few moments of the photon numbers are calculated analytically, and the threshold condition for stable lasing that depends on the Kerr parameter and detuning are obtained. We analyze the properties of the cavity field through the Mandel Q parameter, second-order correlation functions, Cauchy–Schwarz inequality for a range of Kerr parameters, atomic injection rate, and cavity detuning. Nonclassicality of the field is found through sub-Poissonian statistics, particularly at a low injection rate and high Kerr coupling with negative detuning. These nonclassicality conditions correspond to low photon numbers and depend on the sign of the detuning. The violation of the Cauchy–Schwarz inequality also occurs at low photon numbers but over a wide range of parameters, especially at short transient times. Our results show that the two-photon laser with Kerr nonlinearity is a promising quantum nonlinear optical system for developing controllable nonclassical light sources.

B. K. Mohanty, N. Nayak

The results of the quantum theory of a laser given by Riska and Stenholm (see ibid., vol.3, p.536 (1970)) for the case of inhomogeneous broadening and detuning are re-derived by avoiding the Doppler limit approximation to show the exact influence of relaxation parameters on the theory. The threshold condition is seen to be dependent on gamma ab /ku where gamma ab is the relaxation parameter and ku the Doppler parameter. Its value is found to be higher than earlier results. It is noted that the photon distribution function at resonance is much closer to the corresponding distribution obtained from exact calculations for the tuned case. The photon number at the peak of the distribution curve is much lower than earlier values for comparable situations.

Vladimir A. Sychugov, G. P. Shipulo