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Canadian Journal of Physics

Laser theory with natural inclusion of radiation loss: Density-operator approach

A. Liberato, B. Baseia

Published Sep 1, 1988• 3 citations
Publisher

Abstract

A recent result of a previous paper describing the field loss in an optical cavity in a natural way is applied to laser theory. The field loss, due to the transmitted beam light, and the field gain, now attributed to the presence of active atoms but previously attributed to two noninteracting mechanisms, are correctly added to provide the equation of motion for the density operator of the radiation field. The replacement of individual operators, which appear in the conventional treatment, with collective operators is a conceptual difference that emerges from the present approach.

DOI
10.1139/p88-126

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A recent result of a previous paper describing the field loss in an optical cavity in a natural way is applied to laser theory. The field loss, due to the transmitted beam light, and the field gain, now attributed to the presence of active atoms but previously attributed to two noninteracting mechanisms, are correctly added to provide the equation of motion for the density operator of the radiation field. The replacement of individual operators, which appear in the conventional treatment, with collective operators is a conceptual difference that emerges from the present approach.
A semiclassical treatment of the radiation loss in a laser, previously investigated by Baseia and Nussenzveig [Opt. Acta 31, 39 (1984)], is extended to the quantum-theory level. The model employs an optical cavity having an output coupling that includes the radiation loss due to the beam extraction through appropriate boundary conditions, instead of the unrealistic loss reservoir of the conventional treatment. This allows one to work entirely within the continuous spectrum and obtain the equation for the radiation field both inside and outside the cavity. Inside the cavity the results agree, in the coherent representation and absence of quantum noise, with those of the previous semiclassical theory. Also, the results of both treatments for the field inside the cavity coincide with those of the conventional theory, since we reinterpret its ``mode amplitude'' as a collective variable. Outside the cavity (an inaccessible region for a standard approach) the explicit solution for the laser field exhibits a natural relationship between the intensities inside and outside the cavity. Further considerations, such as those concerning the role played by the noise term in the field buildup, are also discussed.
Conventional laser theory deals with discrete ‘cavity modes’ and introduces artificial mechanisms to simulate radiation field losses caused by beam extraction. A more realistic model of a laser with a transmitting window, previously investigated by Lang, Scully and Lamb, is reanalysed semiclassically. In the previous work, the continuous radiation field spectrum was treated as the limit of a discrete but very dense spectrum of modes, and the narrowness of the laser linewidth was attributed to the locking together of these modes. Working entirely within the continuous spectrum, we find that the treatment is considerably simplified, leading to an explicit solution for single-quasimode operation within the linear approximation which exhibits the role of the excitation conditions in the build-up of the laser field. As was pointed out by Lang and Scully, the transmission loss plays the role of an effective noise source, satisfying the fluctuation-dissipation theorem. However, the continuum treatment does not support the mode-locking mechanism, but rather the usual explanation of the laser linewidth in terms of gain narrowing. The nonlinear analysis agrees with the main conclusion of Lang, Scully and Lamb that the ‘mode amplitude’ of the conventional theory should be interpreted as a collective variable that satisfies the well-known rotating-wave Van der Pol oscillator equation. The domain of validity of this result and of the laser quasimode concept depends on an approximation, analogous to the Weisskopf-Wigner approximation, which is well satisfied for the usual case of very long-lived quasimodes.
The inclusion of damping in the quantized radiation field in pure and mixed states is accomplished in a natural way, without using traditional loss reservoirs, through the application of appropriate boundary conditions in a simple model of an optical cavity coupled to the outside world. By using collective operators (which are suitable weighted superpositions of the usual creation and annihilation operators for modes of the continuous spectrum, extended over each Fox–Li band) and convenient initial conditions, we are able to describe the radiation loss from an optical cavity to the outside world. The physical situations corresponding to initial conditions leading to the exponential damping are also investigated.
"We propose to study the random laser problem. As a lasing system one considers an active gain medium interacting with a light field where the damping of the latter is treated in the spirit of open systems theory, considering a multimode case with the spectral overlap of the modes in the losses term [11]. We approach the problem by using the density matrix formalism, deriving the master equation for the light field in order to find out the detailed description of the laser like emission. The specific character of the field damping and gain effects essentially modifies the master equation in comparison to that of the ordinary laser and thus contributes to the properties of the resulting laser emission. The derivation of the density operator for the field is the main ingredient for describing the quantum statistical properties of the lasing and such a progress represents a great interest for the study of random laser systems."
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.