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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Effect of cooperative atomic interactions on the linewidth of a single-mode laser

M. Mardale, Mariana Craiu, Paul E. Sterian

Published Jun 29, 2001• 0 citations
Publisher

Abstract

As the laser linewidth arises under the influence of spontaneous emission, it is of considerable interest to find effects that decrease it. In this paper it is considered the effect of cooperative atomic interactions on the linewidth of a single-mode laser. It can be shown by suitably generalizing the Scully-Lamb theory that the cooperative effects tend to decrease the natural linewidth of the laser at high intensities. Considering the interaction of two-two- level atoms with a single-mode quantized field and the effect of cooperative atomic interactions it was obtained an expression for the natural linewidth of the system. Based on the theoretical models we obtain the numerical results for the laser linewidth as a function of the cooperation parameter. We then present a set of graphs by suitably plotting these data and we discuss the results.

DOI
10.1117/12.432823

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As the laser linewidth arises under the influence of spontaneous emission, it is of considerable interest to find effects that decrease it. In this paper it is considered the effect of cooperative atomic interactions on the linewidth of a single-mode laser. It can be shown by suitably generalizing the Scully-Lamb theory that the cooperative effects tend to decrease the natural linewidth of the laser at high intensities. Considering the interaction of two-two- level atoms with a single-mode quantized field and the effect of cooperative atomic interactions it was obtained an expression for the natural linewidth of the system. Based on the theoretical models we obtain the numerical results for the laser linewidth as a function of the cooperation parameter. We then present a set of graphs by suitably plotting these data and we discuss the results.
Thomas et al. (1981, 1982) proposed the generation of cross correlation between two laser fields interacting with a three-level system as a means of reducing noise and subsequently exploited this property in the observation of very narrow Ramsey fringes. Cross correlation has been discussed theoretically by Dalton and Knight (1982, 1983) and shown to have interesting effects in population trapping. For such effects to be important, the cross correlation coefficient must be as large as possible. The authors discuss the degree of correlation between the two modes of a two-mode laser using the approach of Scully and Lamb (1967, 1968) and show it can be large. They also evaluate the linewidths of the two laser modes. They find that if the laser parameters for the two modes are equal, the two-mode linewidth is one half the value of the linewidth of the corresponding single-mode laser, well above threshold.
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.
We use standard quantum laser theory to examine the linewidth of lasers with a four-level atomic gain medium as a function of the spontaneous emission rate into the lasing mode and the decay rates from the upper and lower lasing levels. We find that the laser linewidth ceases to exhibit the customary decrease with increased photon number when a critical photon number, proportional to the ratio of the decay rate from the lower lasing level to the rate of spontaneous emission into the lasing mode, is reached. This critical photon number would decrease as the size of the optical cavity is decreased. In addition we show that, in general, the laser linewidth depends critically on the ratio of the overall decay rates from the upper and lower lasing levels.
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.
We propose a minimalistic model to account for the main properties of a continuous superradiant laser, in which a beam of atoms crosses the mode of a high-finesse Fabry-Perot cavity, and collectively emits light into the cavity mode. We focus on the case of weak single atom - cavity cooperativity, and highlight the relevant regime where decoherence due to the finite transit time dominates over spontaneous emission. We propose an original approach where the dynamics of atoms entering and leaving the cavity is described by a Hamiltonian process. This allows deriving the main dynamical equations for the superradiant laser, without the need for a stochastic approach. We derive analytical conditions for a sustained emission and show that the ultimate linewidth is set by the fundamental quantum fluctuations of the collective atomic dipole. We calculate steady-state values of the two-body correlators and show that the continuous superradiant regime is tied to the growth of atom-atom correlations, although these correlations only have a small impact on the laser linewidth.