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Japanese Journal of Applied Physics

Convergence and Asymptotic Behavior of the Solution for a High Intensity Single-Mode Gas Laser in the Form of Continued Fractions

Kazuo Kuroda, Iwao Ogura

Published Nov 1, 1973• 4 citations
Publisher

Abstract

It is proved that the continued fractions as the solution of Lamb's semiclassical laser equation are convergent and satisfy the boundary condition required for the convergence of the Fourier expansion of the density matrix. Using the content of the proof the asymptotic behavior of a high intensity single-mode gas laser is investigated. The intensity increases proportionally to the relative excitation asymptotically and the rate of increase is independent of the detuning.

DOI
10.1143/jjap.12.1758

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It is proved that the continued fractions as the solution of Lamb's semiclassical laser equation are convergent and satisfy the boundary condition required for the convergence of the Fourier expansion of the density matrix. Using the content of the proof the asymptotic behavior of a high intensity single-mode gas laser is investigated. The intensity increases proportionally to the relative excitation asymptotically and the rate of increase is independent of the detuning.
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.
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.
This paper extends the calculations of the semiclassical Lamb theory of a Doppler-broadened gas laser (optical maser) to arbitrary intensities for the case of single-mode operation. The coupled equations of the classical electromagnetic field and an ensemble of two-level atoms are set up, and a solution is obtained in the form of a continued fraction. This is used to compute intensity-detuning curves and atomic population inversion densities as functions of both the velocity and the position in the laser. When the laser is tuned to resonance, the velocity dependence of the inversion density shows a previously unknown fine structure consisting of a "bump" in the bottom of the hole. The calculations are compared to results obtained from a perturbation expansion in powers of the field and exact results known for atoms with zero velocity. The response of the laser output to a slow modulation and the buildup of oscillations are also discussed.
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.
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.