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Physical Review A

Statistical properties of a many-mode laser

F. T. Arecchi, A. M. Ricca

Published Jan 1, 1977• 17 citations
Publisher

Abstract

Using a generalized version of the Volterra-Wiener functional technique, we study a model for a many-mode laser action in which the coupling among modes takes place through the intensities. It is shown that the coupled Langevin equations for the field amplitudes can be solved avoiding the infinities at the mode thresholds. In particular, mean photon numbers, variances, and correlation and cross-correlation functions of the modes can be computed. In order actually to carry out the computations, it is necessary to evaluate the solution of a set of coupled deterministic first-order differential equations and to diagonalize a matrix whose dimension is given by the number of modes envisaged. As an application, the two-mode case is treated in detail and, besides a derivation of the stationary properties, in part already known, the transient photon statistics, during the buildup of the two modes, is thoroughly studied. The advantage of this approach is due to its simplicity and to the possibility of dealing with a large number of interacting modes.

DOI
10.1103/physreva.15.308

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Using a generalized version of the Volterra-Wiener functional technique, we study a model for a many-mode laser action in which the coupling among modes takes place through the intensities. It is shown that the coupled Langevin equations for the field amplitudes can be solved avoiding the infinities at the mode thresholds. In particular, mean photon numbers, variances, and correlation and cross-correlation functions of the modes can be computed. In order actually to carry out the computations, it is necessary to evaluate the solution of a set of coupled deterministic first-order differential equations and to diagonalize a matrix whose dimension is given by the number of modes envisaged. As an application, the two-mode case is treated in detail and, besides a derivation of the stationary properties, in part already known, the transient photon statistics, during the buildup of the two modes, is thoroughly studied. The advantage of this approach is due to its simplicity and to the possibility of dealing with a large number of interacting modes.
A theory of a quantum amplifier based on two-photon transitions is presented. The model consists of an assemblage of atoms interacting with a single mode of the light field via two-photon absorption and emission. The treatment is quantum mechanical, and the emphasis is on the photon-statistical aspects of the problem. Difference-differential equations for the density matrix of the field mode are derived. The equations are solved for a special case. Differential equations for the expectation vaues of the number of photons and of the field amplitudes, and for their higher statistical moments, are also obtained. Approximate solutions for the average number of photons are found, and the effect of the amplification process on the statistical fluctuations of the number of photons and the field amplitudes is discussed. The results of this study suggest that such an amplifier does not have the coherence properties of a laser amplifier because it tends to increase the statistical fluctuations of the field.
The noise characteristics of a single-mode laser with a low-Q cavity are investigated theoretically. After the electric field is adiabatically eliminated from the Maxwell-Bloch equations, coupled Langevin equations with both additive and multiplicative white noises are examined. The equations are solved using Rice's method in a framework of quasilinear Fourier analysis. Noise spectral densities are calculated analytically to study the dependence of the relative intensity noise (RIN) on the pumping. Through an investigation of the auto- and cross-correlations of the light intensity and population noises, their variances are obtained in order to compare their properties to the good-cavity case. The stationary intensity cumulants, the photon counting coefficient, and the photon counting probability are explicitly derived and compared with the results of a Fokker-Planck analysis previously carried out for both the good- and bad-cavity cases.>
Analytic expressions for the correlations of the light emitted by an N-mode laser are given. We study in particular a special case of strongest couplings and show, for example, that at steady state, the second order cross intensity correlation tends toward a negative value, −1/(N−1), for large pump parameters. We also study the time dependent solutions, and express the second order amplitude and intensity correlations of the emitted light in terms of the eigenvalues and eigenfunctions of a Schrödinger-type differential equation which reduces to that studied by Risken for N=1 and to that studied by M-Tehrani and Mandel for N=2. Analytic expressions which approximate the eigenvalues of this differential equation for a general N are given.
A quantum statistical analysis of phase-locked coupled light beams is presented with the coupling mechanism between the individual modes provided by the time-modulated dielectric constant of the crystal inserted in the cavity. The intense electric field that drives the modulator is treated classically. A Hamiltonian including the modulator interaction is formulated and the Heisenberg equations of motion of the mode operators are solved utilizing a generating function of the mode operators. The evolution in time of the phase-space distribution functions for single and multiple modes is derived under various initial conditions. Initially coherent oscillating modes are shown to conserve their coherent character. Below threshold, the modes are assumed to be chaotic Gaussian. It is shown that when all modes have the same average number of photons, they remain uncorrelated Gaussian. When the average photon numbers are nonidentical, the field remains a pure noise field, but the various modes are correlated. The analysis of the resonant-case mode of operation, properly interpreted, leads directly to the case of nonresonant modulation.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.