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Physical Review A

Pressure Effects in a High-Intensity Gas Laser

Stig Stenholm

Published Nov 1, 1970• 16 citations
Publisher

Abstract

The present paper includes the influence of collisions in the high-intensity laser theory by Stenholm and Lamb. The history of an atom is divided into field- and collision-dominated intervals. The collisions are treated as instantaneous independent events (impact approximation). The equation of motion for the density matrix of an ensemble of atoms undergoing collisions is derived. The collisions manifest themselves as an integral operator in the equation of motion. As a model, collisions which do not change the populations of the atomic levels are treated. Physical conditions on the kernel matrix of the collision integral are used to construct a simple nontrivial model. To simplify the treatment, the phase and velocity changes are assumed to take place in different collisions. This model is used to solve the problem of a high-intensity laser with collisions. Only the lowest nonvanishing terms in the Fourier expansion of the density matrix elements are used. Typical numerical results are evaluated and shown in the figures. Connections with other papers and generalizations are discussed.

DOI
10.1103/physreva.2.2089

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The present paper includes the influence of collisions in the high-intensity laser theory by Stenholm and Lamb. The history of an atom is divided into field- and collision-dominated intervals. The collisions are treated as instantaneous independent events (impact approximation). The equation of motion for the density matrix of an ensemble of atoms undergoing collisions is derived. The collisions manifest themselves as an integral operator in the equation of motion. As a model, collisions which do not change the populations of the atomic levels are treated. Physical conditions on the kernel matrix of the collision integral are used to construct a simple nontrivial model. To simplify the treatment, the phase and velocity changes are assumed to take place in different collisions. This model is used to solve the problem of a high-intensity laser with collisions. Only the lowest nonvanishing terms in the Fourier expansion of the density matrix elements are used. Typical numerical results are evaluated and shown in the figures. Connections with other papers and generalizations are discussed.
A theoretical model for the pressure dependence of the intensity of a gas laser is presented in which only velocity-changing collisions with foreign-gas atoms are included. This is a special case where the phase shifts are the same for the two atomic-laser levels or are so small that deflections are the dominant effect of collisions. A collision model for hard-sphere repulsive interactions is derived and the collision parameters, persistence of velocity and collision frequency, are assumed to be independent of velocity. The collision theory is applied to a third-order expansion of the polarization in powers of the cavity electric field (weak-signal theory). The resulting expression for the intensity shows strong pressure dependence. The collisions reduce the amount of saturation and the laser intensity increases with pressure in a characteristic fashion. It is recommended that the best way to look for this effect is to make the measurements under conditions of constant relative excitation.
The dynamics of a multimode high-intensity gas laser including some velocity-changing-collision effects is investigated theoretically. Using a spatial Fourier expansion method for the density matrix of the active medium, one obtains the system of equations which contain only the Fourier components of the population inversion. To obtain the approximate solutions in the high-intensity case, the saturation terms due to hole burning and the population-pulsation terms are treated separately. The former are calculated exactly and the result is equivalent to the rate-equation approximation. The latter are treated by the perturbation method. The equations of motion for mode amplitudes have a similar form to those of Lamb's perturbation theory but can be applied to a high-intensity multimode laser. Experimental results of three-mode oscillation of a He-Ne laser are compared with the theory.
This paper presents a semiclassical theory of a single-mode gas laser valid for essentially arbitrary values of field intensity. The theory is formulated in terms of an ensemble-averaged form of the density-matrix equations of motion involving a single set of coordinates. The polarization and the population-inversion density are expanded in spatial Fourier components. One obtains a set of coupled difference equations in the Fourier coefficients yn{y}_{n}yn​, which may be solved subject to appropriate boundary conditions on yn{y}_{n}yn​ for large nnn. The solutions may be expressed in terms of continued fractions in the general case, and in closed form in important special cases. Quantities of interest are evaluated by computer. The steady-state laser intensity and frequency are obtained from the coupled electromagnetic field equations as functions of the cavity parameters and the linear gain of the amplifying medium. At moderate values of field intensity, the velocity distribution of the space-averaged population inversion density exhibits a fine structure for low-velocity atoms superimposed on the usual broader depletion. This structure is a manifestation of the coherent ringing of the amplifying medium. The results are compared with those obtained in the rate-equation approximation, and with the recent work of Stenholm and Lamb. Discrepancies with the latter paper are explained.
We present a theory of pressure effects in which the atomic center-of-mass motion is treated quantum mechanically. The quantum-mechanical calculation treats the perturber-induced energy-level variations and the velocity changes of the emitter caused by collisions on an equal basis. Specifically, we shall treat the problem of a laser to first order in the laser field, allowing for the fact that the laser atoms are undergoing collisions, but our results will also be applicable to the cases of stimulated emission or absorption. It will be sufficient to carry out a perturbation solution of the problem assuming that the laser atoms undergo at most one collision in their lifetime, since such a restricted calculation reveals the salient features of the theory. We shall find that, in general, there is no classical limit for our results. Thus, previous treatments employing a classical Boltzmann-equation approach for the atomic center-of-mass motion are invalid, and a quantum-mechanical description is necessary to correctly treat cases where both the modified Doppler effect (modified by collisions) and perturber-induced energy-level variations are present.
The effects of phase-interrupting and velocity-changing collisions between active and perturber atoms on the spectral profile of a gas laser are studied. Exact analytical results for the density matrix are presented for the case of strong collisions in a unidirectional ring laser, and the laser intensity is then evaluated in the Doppler limit under the condition of self-consistency. The washout of the hole burning in the steady-state population difference and the broadening of the laser line shape are discussed. The results of the third-order perturbation theory are compared with the exact calculations valid for an arbitrarily intense pump.
The authors consider a system of two-level atoms coupled to a radiation field and immersed in a buffer gas. The corresponding Bloch-like equations are augmented by collision operators stemming from the linear Boltzmann equation. Collision operators describe velocity-changing collisions that give rise to various kinetic effects of light. They study the general properties of collision operators, the main one consisting of the fact that such operators are Hermitian in a suitably defined space of velocity distribution functions They introduce a set of functions that serve as approximations to the eigenfunctions. The eigenvalues are connected with transport coefficients (they give an example of such a relation), but they keep eigenvalues as free parameters of the approach. In this way the authors can describe the whole class of collision operators, the Keilson-Storer model being a particular example. Employing the introduced eigenfunctions they derive and formally solve the equations of motion for the expansion coefficients of the elements of the density matrix of active atoms.