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Journal of physics. A, Proceedings of the Physical Society. General

The effects of detuning on the quantum theory of an inhomogeneously broadened laser

D.O. Riska, Stig Stenholm

Published Sep 1, 1970• 14 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/3/5/011

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Physical Review A•Aug 1, 1976•28 citations
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.
The results of the semi-classical theory for the laser are compared with the results of the quantum theory given by Scully and Lamb (1967). It is found that the spatial structure of the cavity mode leads to important corrections. For moving atoms the cavity mode is smeared, and a generalized form of the Scully-Lamb theory (for resonance between the atomic transitions and the cavity mode) leads to agreement with the semi-classical theory. The photon distribution for the case with moving atoms is obtained and compared with the results of the Scully-Lamb theory.
A quantum-statistical analysis of multi-photon lasers including the effect of detuning and atomic motion is presented. The equations are solved to fourth order in the atom-field coupling constant. Systems in detailed balance (with multi-photon loss mechanism) as well as systems without detailed balance (with single-photon loss mechanism) are considered separately. Treatments for the single-mode and multi-mode processes are given. Expressions for threshold conditions, peak photon number and the half width of photon distribution are obtained. The variation of mean output photon number, the effect of detuning, the coherence characteristics, etc. are studied and results are illustrated for two- and three-photon lasers as examples.
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.
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 single-mode laser model as formulated by Haken and co-workers is considered. Within the approximation which takes into account only the interaction (at all orders in the coupling constant) between the mode and the single atoms, this model leads to a linear time-evolution equation for the Glauber quasiprobability distribution $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ of the field mode, containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. This equation is solved exactly in the stationary situation. Thus the quantum effects due to the terms with derivatives of order higher than the second one are fully taken into account. It is shown that these terms have a shrinking effect on the stationary distribution above threshold, counteracting the broadening effect due to the second-derivative term. The photon and photoelectron distributions are given. Exact relations are deduced, which allow the calculation of all the moments of the steady-state distribution from the first one. This distribution is compared with the stationary solutions of Risken's equation, of P. Mandel's equation, and with the stationary distribution deduced by Weidlich, Risken, and Haken. The connection with the single-mode laser theory of Scully and Lamb is analyzed in the high-intensity region. The complete Scully-Lamb master equation is translated into the Glauber representation, yielding a linear equation for $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. Suitably connecting the parameters AAA, BBB, and CCC of the Scully-Lamb equation with the parameters of the model of Haken and co-workers, it is shown that there is perfect agreement between the strong-signal Scully-Lamb steady-state distribution and that of Weidlich, Risken, and Haken for all values of the pump parameter.