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Physical Review A

Theory of a high-intensity multimode laser

Kazuo Kuroda

Published Jan 1, 1979• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.19.231

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Physical Review A•Oct 1, 1973•83 citations
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.
The semiclassical interaction of a two-level system with a multimode electromagnetic field is discussed in detail. The equations of motion for the macroscopic density matrix are transformed into a single integral equation for the population density, with a complex kernel which can take into account both running- and standing-wave configurations. The iterative solution to this integral equation, which corresponds to an expansion in terms of powers of the electromagnetic field amplitudes, turns out to be divergent when the field amplitudes are greater than some critical value. In the case in which the kernel is a periodic or quasiperiodic function of time, a general solution to the integral equation is found, and this solution turns out to be unique and convergent for every value of the electromagnetic (e.m.) field amplitude. This solution is expressed through a continued-fraction expansion, whose terms are matrices, which can be readily obtained from the kernel. This solution is a generalization of the continued-fraction expansion usually met in the strong-signal semiclassical theory of lasers. Some particular cases are then treated in detail; among them, the solution to the multimode operation in laser devices is given in which the excited level lifetime is long compared with the period of the beating term between two adjacent modes. Although the theory presented is directed towards the steady-state regime in multimode lasers, it may also treat the transient regime of laser operations, or it may be extended to include phenomena where a strong e.m. field, made up of several modes equally spaced in frequency, interacts with a two-level system.
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.
A high-intensity single-mode gas laser is analyzed by several mathematical methods. Exact numerical results are best obtained by solving for the coefficients of a Fourier series solution by using a backward recurrence scheme or a successive convergents method for evaluating a continued fraction. Numerical integration techniques can also be used to convert the periodic boundary conditions to initial conditions and then solve the resulting initial-value problem. Direct analytic integration leads to exact solutions in terms of Bessel functions for the special case of zero detuning, equal decay rates, and neglecting of collision broadening. For general single-mode operation, an improved analytic approximation [the second recursion relation approximation (2RCRA)] is obtained by taking an additional term in the recursion relations for the Fourier coefficients. This approximation predicts the existence of one of the secondary resonances in graphs of the spatially averaged population inversion and quadrature polarization coefficient densities and matches the exact solution from the position of the first resonance and outward independent of the intensity. The 2RCRA also matches the exact detuning curve except at small detunings. The first-order ("rate equation") approximation, by comparison, does not predict any secondary resonances and does not match the exact solution at higher intensities. A physical interpretation for the appearance of the secondary resonances is presented in terms of multiphoton interactions and a prediction for the positions of the resonances for the special case discussed above is obtained. As predicted by Greenstein the exact detuning curves show quantitatively that the Lamb dip decreases and disappears as the intensity increases for high excitations.
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.
The optical pumping of atoms by a multimode gas laser, in the presence of a magnetic field, is studied theoretically. The atoms are described by their density matrix, which is expanded on an irreducible tensorial set. The atomic relaxation is assumed to be isotropic. In order to avoid the usual perturbation theory, we use the so-called "Broad-Line approximation" (BLA). BLA is valid when the width of the "Bennett hole" created by one laser mode in the velocity distribution of the excited atoms is comparable to the spacing between modes; under these conditions, the atomic response does not depend on the velocity and the multimode laser irradiation is equivalent to a broad-line excitation. For the atomic system, BLA leads to a set of coupled equations which is valid at arbitrary intensities of the laser field and which depends on this field through a unique parameter, $\ensuremath{\gamma}$. This parameter can be understood as a laser-induced transition probability. In this article, we analyze the nonlinear effects which can be deduced from this theory, for any value of the angular momenta involved in the laser transition: Hanle-effect broadening, alignment coupling, saturation resonances on the populations are shown. The exact calculations, valid at arbitrary laser intensities for $J=1\ensuremath{\leftrightarrow}J=0$, $J=1\ensuremath{\leftrightarrow}J=1$, and $J=1\ensuremath{\leftrightarrow}J=2$ transitions, and the corresponding experimental results will be presented in forthcoming papers.