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Physical Review A

High-intensity single-mode laser theory

Jehuda Ziegler, P. R. Berman

Published Aug 1, 1977• 11 citations

Abstract

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.

DOI: 10.1103/physreva.16.681

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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 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

{y}_{n}

, which may be solved subject to appropriate boundary conditions on

{y}_{n}

for large

n

. 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.

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.

An exact analysis has been carried out for general analytical expressions for the second threshold of a single-mode homogeneously broadened laser and for the initial pulsation frequency at the second threshold, for arbitary physical values of the relaxation rates, and at an arbitrary detuning between the cavity frequency and the atomic resonance frequency. These expressions also give correspondingly exact forms for asymptotic cases that have been previously studied with some approximations. Earlier approximate results are partly confirmed and partly improved by these more general expressions. The physical status of various expressions and approximations is reconsidered and specified more clearly, including an analysis of what reasonably can be attained in lasers or masers. A general analytical proof is given of the fact that, for a larger detuning of the laser cavity from resonance, a higher value of the laser excitation is required to destabilize the steady-state solution (the second threshold). We also present results for the minimum value of the second threshold at fixed detuning as a function of the other parameters of the system and for the dependence of the ratio of the second threshold to the first threshold as a function of detuning. Minima of the second threshold and of the threshold ratio occur only if the population relaxation rate is equal to zero. The minima of the threshold ratio are shown to be bounded from above as well as from below (as functions of the relaxation rates, so long as the second threshold exists). The upper bound on the minima ratio is equal to 17. The variation of the second threshold in the semi-infinite parameter space of the decay rates is shown at various detunings in plots with a finite domain by normalizing the material relaxation rates to the cavity decay rate.

This paper discusses the case of a Doppler-broadened gas laser with a saturable intracavity absorber gas. The semiclassical equations governing the field intensities and the atomic density-matrix elements are set up. The latter are written as Fourier series, the components of which are shown to obey coupled difference equations. In this paper the exact steady-state solution in single-mode operation is obtained. The lowest approximation (rate-equation approximation) is used to compute detailed characteristics of the system. In particular, the influence of the pumping rates in the two cells are investigated and displayed in the figures. The regions of bistable operation are determined and conditions for the occurrence of an inverted Lamb dip are given. The effect of a slow modulation of the pumping rates is discussed. The range of validity of the lowest approximation is investigated both analytically and numerically. Finally, there is a brief discussion of the influence of pressure on the parameters of the theory.