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Physical Review A

Semiclassical study of the laser transition

Guy Michel Stéphan

Published Feb 1, 1997• 24 citations
Publisher

Abstract

A semiclassical, nonlinear theory of a single-mode laser is developed so that the transition around threshold can be studied. The field is expressed in the frequency domain which allows us to emphasize the role of the Fabry-P\'erot cavity. A generalized Airy function is obtained for the laser line shape: it contains the source line shape and the empty cavity line shape. It describes the laser both above and below threshold, including the transition region where there is an abrupt increase of intensity and decrease of the laser linewidth. The general arguments are illustrated by detailed numerical calculations for the 3.39-\ensuremath{\mu}m line of the He-Ne laser. The intensity of the spontaneous emission source which plays a central role is computed from first principles.

DOI
10.1103/physreva.55.1371

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A semiclassical, nonlinear theory of a single-mode laser is developed so that the transition around threshold can be studied. The field is expressed in the frequency domain which allows us to emphasize the role of the Fabry-P'erot cavity. A generalized Airy function is obtained for the laser line shape: it contains the source line shape and the empty cavity line shape. It describes the laser both above and below threshold, including the transition region where there is an abrupt increase of intensity and decrease of the laser linewidth. The general arguments are illustrated by detailed numerical calculations for the 3.39-\ensuremath{\mu}m line of the He-Ne laser. The intensity of the spontaneous emission source which plays a central role is computed from first principles.
We give a general equation which allows the description of the evolution of the laser properties when the gain is varied from below to above threshold. The method is general in the context of the semi-classical theory of lasers. It is numerically illustrated in the simplest case of a single mode laser with a source term characterized by a very narrow spectrum. The transformation of the Airy line shape of the passive Fabry-Perot cavity into the laser line shape is described. A classical linewidth which complements the usual quantum limit can thus be introduced, together with its control parameters.
A quantum theory for noise in the steady state of the laser oscillator above threshold is presented. The concept of operator noise source is introduced. A simple laser model operating in a single mode with resonant frequency tuned to the center of the material line is considered. The resonance linewidth of the inverted material is collision-broadened, and assumed to be much broader than the cold-cavity width. We restrict ourselves to one type of collision. The quantum description of this laser model leads to nonlinear operator equations. These are linearized, but this linearization is justified only for operation "sufficiently" above threshold. The first- and second-order Glauber functions and the expectation value of the field commutator are derived for the field inside the laser cavity. These results are compared with the predictions of a "semiclassical" theory in which the classical equations contain noise sources such that properties of the field below threshold are correctly predicted. Our results differ by certain terms from the results of this semiclassical theory. These terms are too small to have been detected experimentally.
We study the properties of the laser field in the case where the cavity contains two cells, one which amplifies and one which absorbs the radiation. We adopt the simplest possible model, in which there is only one running mode in the cavity and all atoms are fixed and homogeneously distributed in space. We first set up a semiclassical description. In the frame of the stationary semiclassical (SC) theory the most striking feature is the occurrence of a domain in which three solutions coexist. A linear-stability analysis shows that two of these solutions are stable. This gives rise to an hysteresis cycle. Most of the results in this section are analytic. In the frame of a purely quantum description, we derive a generalized Fokker-Planck equation which is exactly solved in the stationary case. This yields a stationary quasiprobability distribution function for the field. In the domain where the SC theory predicts three solutions, the field-distribution function has two maxima and one minimum whose positions are precisely given by the SC intensities. With this distribution function we study the intensity and the intensity fluctuations: the intensity shows a rather abrupt increase when crossing the narrow transition region; in this region the intensity fluctuations are extremely high. By means of an approximate method to solve the time-dependent Fokker-Planck equation we study the linewidth and show that it drastically decreases above threshold. Finally we consider the linearized and quasilinearized Fokker-Planck equations which can be solved analytically. We show that the domain in which neither of these equations fits the results of the generalized Fokker-Planck equation is extremely small. The properties of the metastable states are analyzed in terms of the generalized and asymptotic Fokker-Planck equations.
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.
We develop a first-principles description of the mechanical effects of light on a single two-level atom in a Fabry-Perot interferometer. Our analysis is based on the plane-wave and semiclassical approximations, but within this context it yields general equations that include propagation effects and allow arbitrary parameters of the resonator. We then focus on the conditions under which the uniform field limit holds and derive an explicit expression for the cavity-induced force on the atom in the low saturation limit. We also provide a physical explanation of the role played by the resonator in slowing down or accelerating the atoms.