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Annual Meeting Optical Society of America

Laser theory without the rotating wave approximation

Reeta Vyas, Surendra Singh

Published Jan 1, 1985• 0 citations
Publisher

Abstract

In laser theory and other branches of quantum electronics certain terms oscillating at twice the resonance frequency are dropped from the interaction Hamiltonian. The effect of these counterrotating terms on the operation of a single-mode laser is considered using the model of Scully and Lamb. It is found that these terms lead to a shift of the laser threshold and a broadening of the photon number distribution.

DOI
10.1364/oam.1985.tuw6

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Journal of physics. A, Proceedings of the Physical Society. General•Mar 1, 1970•14 citations
In laser theory and other branches of quantum electronics certain terms oscillating at twice the resonance frequency are dropped from the interaction Hamiltonian. The effect of these counterrotating terms on the operation of a single-mode laser is considered using the model of Scully and Lamb. It is found that these terms lead to a shift of the laser threshold and a broadening of the photon number distribution.
The effect of counter-rotating terms in the interaction Hamiltonian on the photon statistics is investigated for a single-mode laser. The treatment is based on the Scully-Lamb model. A perturbation expansion is used to derive an equation of motion for the field-density matrix. The counter-rotating terms are found to lead to both transient and secular effects. With some reasonable approximations the steady-state photon-number distribution is derived and it is shown that the counter-rotating terms tend to raise the threshold of laser action and broaden the photon-number distribution. The relative size of these effects is found to be very small and depends on the square of the ratio of the atomic linewidth and the transition frequency.
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.
The question of how the light output of a laser and its fluctuations depend on the optical-cavity length at a given frequency is investigated theoretically for a laser oscillating in a single Gaussian mode. The treatment is quantum mechanical, and is based on the Scully-Lamb laser model, except that a perturbation expansion is used and possible cooperative atomic effects are included. It is shown, with the help of some reasonable approximations, that the probability distribution of the photon-occupation number can be cast into a form that is similar to the Scully-Lamb formula, but with coefficients that depend on cavity length in a more complicated way. Curves are presented that illustrate the behavi\`or and should lend themselves to experimental test.
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.