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Journal of Physics A Mathematical and General

Effect of relaxation parameters on the quantum theory of a detuned laser with inhomogeneous broadening

B. K. Mohanty, N. Nayak

Published Apr 1, 1976• 6 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/9/4/020

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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.
The results of the Riska-Stenholm quantum theory for an inhomogeneously broadened laser are re-derived without using the Doppler limit approximation which restricts the utility of this theory. The photon distribution peak is seen to occur at a significantly lower photon number and the threshold condition is dependent on Yablku, where Yab is the relaxation parameter and ku the Doppler parameter. The dimensionless intensity parameter is obtained and the results are compared with the existing theories of Riska and Stenholm and of Scully and Lamb.
A quantum statistical analysis of a two-photon laser is presented in a generalization of the one-photon laser. A two-photon loss mechanism is assumed by taking detailed-balance considerations into account. The expressions for the photon statistical distribution exhibit the exact influence of the Doppler parameter and other relaxation parameters. In addition, the photon statistics can be studied as a function of the detuning. Threshold conditions for the laser action are noticed. The peak photon number and half-width of the photon distributions are obtained. It is found that the photon distribution can be written as a ratio of the one-photon laser-photon distribution to the Poisson distribution for photons.
The quantum theory of a laser developed by Scully and Lamb (1967) is reformulated to include the effects of atomic motion and detuning. The equations are solved to fourth order in the atom-field coupling constant. An approximate treatment of the mode structure is shown to reproduce the results of Lamb's semi-classical theory in the Doppler limit. In addition the photon distribution may be studied as a function of the detuning. The width of the photon distribution is found to increase monotonically with increasing detuning.
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.
We generalize Lamb's quantum theory of lasers to treat the off-resonance operation of two-transition lasers in inhomogeneously broadened media. The master equation is derived from the density-matrix equation of motion in the Doppler limit. We first find the stationary-state solution from which we then obtain such operation characteristics as the threshold condition, the mean photon number, and the half-width of photon distribution. Compared with the one-transition laser, it is found that in the two-transition laser it is much easier to start oscillation and to obtain higher gain. Furthermore, we find that the two-transition laser is less sensitive to the detuning effect and its deviation from the coherent state is smaller than that of the one-transition laser. The remarkably different hole-burning phenomenon is also discussed.
We investigate the effect of collisional broadening, of population trapping and repumping, and of detuning of the atomic levels on the quantum-statistical properties of Raman laser radiation. We find that under suitably chosen operating conditions the effect of such additional incoherent relaxation mechanisms for the atomic polarization and populations can be made very small. One obtains highly nonclassical values for the Mandel Q parameter and the output-intensity fluctuation spectrum S(\ensuremath{\omega}) for experimentally reasonable parameters. Including detuning of the laser and the pump light from the atomic levels in certain cases can lead to further narrowing of the intracavity photon number distribution, as compared to the resonant case, down to a Mandel Q parameter close to the optimum possible value of Q=-1. A subsequent preliminary analysis prevails that inhomogeneous Doppler broadening of the atomic lines, both for collinear and for counterpropagating pump and laser light, is not detrimental to the nonclassical intracavity state.