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Physical Review A

Generalized-moment-expansion method for intensity correlation functions of the single-mode laser

J. M. Noriega, Luis Pesquera, M. A. Rodrguez

Published Nov 1, 1991• 3 citations

Abstract

The steady-state intensity correlation functions C(t) of the single-mode laser are analyzed with the generalized-moment-expansion method proposed by Nadler and Schulten [Z. Phys. B 59, 53 (1985); J. Chem. Phys. 82, 151 (1985)]. This method approximates C(t) by a superposition of N exponentials that exactly reproduces n derivatives of C(t) at t=0 and m relaxation moments (n+m+1=2N). In this way, the short- and long-time regimes of C(t) are well described even near threshold. With the use of relaxation moments we develop a method to obtain the eigenvalues of the Fokker-Planck operator. This method is applied to the single-mode laser. We show that the knowledge of only two relaxation moments is enough to obtain good results for the two first eigenvalues. The method gives excellent results when the eigenvalues are well separated (below threshold).

DOI: 10.1103/physreva.44.6087

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The steady-state intensity correlation functions C(t) of the single-mode laser are analyzed with the generalized-moment-expansion method proposed by Nadler and Schulten [Z. Phys. B 59, 53 (1985); J. Chem. Phys. 82, 151 (1985)]. This method approximates C(t) by a superposition of N exponentials that exactly reproduces n derivatives of C(t) at t=0 and m relaxation moments (n+m+1=2N). In this way, the short- and long-time regimes of C(t) are well described even near threshold. With the use of relaxation moments we develop a method to obtain the eigenvalues of the Fokker-Planck operator. This method is applied to the single-mode laser. We show that the knowledge of only two relaxation moments is enough to obtain good results for the two first eigenvalues. The method gives excellent results when the eigenvalues are well separated (below threshold).

Intensity fluctuations for a single-mode laser near threshold are calculated from the Scully-Lamb equation of motion for the reduced density matrix of the radiation field. The magnitude and correlation time of the intensity fluctuations are contrasted with the results obtained from the usual Fokker-Planck analysis of the rotating-wave, Van-der-Pol (RWVP) model. When the photon number at threshold is not large, these calculations predict measurable differences from the RWVP Fokker-Planck results. Two methods are used to calculate the dynamics. One is a direct eigenfunction expansion of the Green's function of the equation of motion. The other is based on the calculation of the initial derivatives of the correlation function.

We consider a single-mode laser model driven by correlated quantum and pump noise and study the effects of the cross correlation time τ between noises on the steady-state intensity correlation function and the associated relaxation time. Based on an approximated Fokker–Planck description of the laser model and the Stratonovich-like ansatz, we find the steady-state probability distribution function P st ( I ), the steady-state intensity correlation function K ( s ) and the associated relaxation time T . From numerical computations we found the following: (1) τ suppresses the fluctuation of the laser intensity for the case of positively correlated noises (i.e. the correlation strength between noises λ > 0) and increases the intensity fluctuation for the case of negatively correlated noises (λ > 0); (2) τ slows down the decay of the intensity correlation for the case of λ > 0 but speeds up the decay for the case of λ < 0. The effects of τ on P st ( I ), K ( s ) and T are entirely opposite for λ < 0 and for λ > 0.

Intensity correlations in a two-mode laser oscillating on two coupled transitions have been studied by means of the Fokker-Planck equation. It is shown that they can be expressed in terms of the eigenvalues and eigenfunctions of a one-dimensional Schr\"odinger-type equation. It is found that near threshold intensity correlations cannot be approximated by a single exponential. For moderately large excitations, however, a single exponential dominates. Approximate analytic expressions for large excitations have been obtained which show that the correlation time increases with increasing excitation.

Based on the Fokker-Planck description of a single-mode laser model with correlations between the real and imaginary parts of the quantum noise as well as the pump noise,the dynamical properties of the laser system were studied by using of the mean relaxation time approximation method.The intensity correlation function and the associated mean relaxation time were calculated.It is found that the presence of correlations between the real and imaginary parts of the quantum noise always slows down the relaxation of the light intensity fluctuation when the laser is operated over the threshold.

The form and the decay time of the intensity correlation function of the light from a single-mode He:Ne laser have been measured in the neighborhood of threshold. The experiments were performed at several predetermined values of the pump parametera, ranging froma = -1toa = 4. By combination of the experimental results with the theoretically predicted eigenvalues appearing in the expansion of the correlation function, it was possible also to determine the amplitudes of the first two terms in the expansion over the same range. The experimental results are found to be substantially in agreement with the nonlinear oscillator theory of the laser, without adjustment of any arbitrary scaling parameters.