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Optica Acta International Journal of Optics

Semiclassical Theory of Laser Transmission Loss

B. Baseia, H. M. Nussenzveig

Published Jan 1, 1984• 31 citations
Publisher

Abstract

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.

DOI
10.1080/713821390

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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.
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.
Previous theories of laser radiation have described the electromagnetic field in terms of a discrete set of quasimodes (Fox and Li "modes") of the laser cavity. Each quasimode was assumed to have a finite quality factor Q. In steady state (when gain due to lasing atoms cancels the losses) and in the absence of noise sources, the semiclassical theory predicts a $\ensuremath{\delta}$-function line shape. In the present semiclassical analysis (fluctuations are ignored), this theory is generalized for a maser with a cavity having a semitransparent wall as one of the mirrors so that there are now many modes of the universe corresponding to each Fox-Li-type quasimode. It is demonstrated that the normal modes of the universe associated with a single Fox-Li "mode" may, under proper conditions, lock together and the $\ensuremath{\delta}$-function laser line shape may be regained. Specifically, we investigate the conditions under which the coupled multimode equations associated with laser oscillation can be reduced to an equation of the form found in the usual single-quasimode theory.
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.
We summarize the basic problems in laser coherence and the concepts and techniques needed to solve them. Special techniques needed to handle partition noise in semiconductor lasers will be touched on. We address the problem of how a self-sustained oscillator works. Why is the laser line so narrow? Why does a small amount of noise produce a broadening of the line as opposed to additive side bands? What is the role of zero point oscillations and spontaneous emission in determining the line width? In what sense can the transition to a lasing state be described as a phase transition? The concepts used to deal with the response of a laser system involve a quasi-Markoffian description. This in turn validates a quantum regression theorem which relates the spectrum of the noise to the transport response even in a system far from equilibrium. A significant simplification is made by introducing non-commuting (quantum) noise sources. The techniques required in the solution of laser noise problems involve a multi-time correspondence between quantum variables and c-number random variables that reduces a quantum stochastic problem to a classical stochastic problem. Adiabatic elimination is used to eliminate the fast variables to obtain a reduced problem for the fields alone or the fields and some total atomic variables (total inversion and polarization). Reduction to a rotating wave version of the van der Pol oscillator and numerical solution of the Fokker-Planck equation yields (a) the line shape (b) the spectrum of intensity fluctuations and (c) the distribution of photocounts observed over some time interval. Comparison of theoretical results with later experiments will be presented.
Abstract This present paper is an attempt to describe the operation of a laser by a method patterned closely after the classical theory of dispersion. There, the electromagnetic field is treated as a classical system interacting with a collection of classical harmonic oscillators which are at rest. In the present case the radiation field is still treated classically, but, in accordance with the correspondence principle, the oscillators are now virtual oscillators associated with the up-and-down transitions in effective two-state atoms which are described quantum mechanically. The atoms carrying these virtual oscillators are not at rest but move with different velocities. We show how the Lorentz averaging can be performed for such systems, and derive a closed set of equations linking the average electric field E, with the average polarization density P„ and the average population inversion density Mp associated with atoms moving with the velocity v. Next, we investigate the conditions for the existence of a steady state if pumping is present, and if the electric field is represented by a standing wave, or a travelling wave of a single frequency (single-mode case). It turns out that, notwithstanding the nonlinear nature of the equations, the steady-state conditions give a simple complex dispersion relation; moreover, the real and imaginary part of this dispersion relation are equivalent to the usual heuristic expressions which specify the operating frequency in terms of the index of refraction and cavity length, and which balance the gain against the losses. The single-mode case is analysed in detail without any smallness assumptions for the resultant intensity. The small-intensity case gives the usual results exhibiting the tuning dip. For high intensities the relative depth of the dip tends to zero. At the end of the paper we discuss proposed extensions and additional applications.
The problem of the quantum-limited or intrinsic linewidth of a good-cavity laser is revisited. Starting from the Scully-Lamb master equation, we present a fully analytical treatment to determine the correlation function and the spectrum of the cavity field at steady state. For this purpose, we develop an analytical approximation method that implicitly incorporates the microscopic fluctuations of both the phase and intensity of the field, and, in addition, takes full account of the saturation of the nonlinear gain. Our main result is a simple formula for the quantum-limited linewidth that is valid from near to far above threshold and also includes the presence of thermal photons. Close to the threshold, the linewidth is twice as large as predicted by the standard phase-diffusion treatment neglecting intensity fluctuations, and even 50% above threshold the increase is still considerable. In general, quantum fluctuations of the intensity are present and continue to influence the linewidth as long as the photon-number distribution is not strictly Poissonian. This inherent relationship is displayed by a formula relating the linewidth and the Mandel Q parameter. More than 100% above treshold the linewidth is found to be smaller than predicted by the standard treatment, since the simple phase-diffusion model increasingly overestimates the rate of phase fluctuations by neglecting gain saturation. In the limit of a very large mean photon number the expected perfectly coherent classical field is obtained.