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IEEE Journal of Quantum Electronics

Intensity correlation function of a laser beam near threshold

S. Chopra, L. Mandeļ

Published Mar 1, 1972• 18 citations
Publisher

Abstract

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.

DOI
10.1109/jqe.1972.1076973

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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.
We present an experimental investigation of the third-order intensity correlations in the light from a single-mode intensity-stabilized He-Ne laser in the threshold region. Accurate results were obtained by utilizing a fast digital correlator operating in real time. The computation of the intrinsic third-order correlations requires also a knowledge of the second-order intensity correlation which was measured under the same operating conditions. The experimental data are in agreement with recent theoretical computations based on the usual Van der Pol model of the laser oscillator. A new definition of the intrinsic correlations, which seems more appropriate for the interpretation of the experimental results, is discussed.
A procedure for the measurement of intensity correlations of the type $〈:\stackrel{^}{I}(\mathrm{x}, t)\stackrel{^}{I}(\mathrm{x}, t+\ensuremath{\tau}):〉$ and $〈:\stackrel{^}{I}(\mathrm{x}, t)\ifmmode\times\else\texttimes\fi{}\stackrel{^}{I}(\mathrm{x}, t+{\ensuremath{\tau}}_{1})\stackrel{^}{I}(\mathrm{x}, t+{\ensuremath{\tau}}_{2}):〉$ with the use of a time-to-amplitude converter is developed. The procedure is used to measure second- and third-order intensity fluctuation correlation functions of the optical field produced by a gas laser operated at various levels near its threshold of oscillation. The results are shown to be in agreement with the predictions of laser theories.
In this study, we measured the temporal correlations in the electromagnetic field radiated by a laser in the threshold region of oscillation, from 110\frac{1}{10}101​ to 10 times threshold intensity. The experimental results are compared with theoretical predictions based on solutions of a Fokker-Planck equation. We stabilized the intensity of a He-Ne cw gas laser by means of a long-time-constant servo system which controlled the cavity length. Using a fast photomultiplier as detector, we recorded the photoelectron count distribution within a short counting time (3 \ensuremath{\mu}sec) while the photomultiplier was exposed to the laser light. From the photoelectron count distribution measurement, we calculated second, third, and fourth normalized cumulants of the intensity probability density function of the light field. The normalized cumulant is a measure of "pure" correlations among photons because the contributions from lower-order correlations are removed. The statistics of the photoelectron count distribution shows that the intensity fluctuations at about 110\frac{1}{10}101​ threshold are nearly those of a Gaussian field, and continuously approach those of a constant-amplitude field as the intensity is raised to about 10 times threshold. The normalized second, third, and fourth cumulants of the intensity probability density function of the laser light were also measured at 17 and 42% of threshold intensity as the counting time was increased from 3 to 1000 \ensuremath{\mu}sec. The results agree with predictions computed under the assumption that the dependence of the correlation functions on the time variables is the same as for Gaussian light.
In the case of the cross-correlation intensity of noises being modulated by time period,the intensity correlation function of the single-mode laser system was presented in terms of the linear approximation method.The evolution of intensity correlation function with the time and find the periodic oscillation of the intensity correlation function was investigated.The results show that oscillation amplitude and period of the intensity correlation function do not vary with the quantum noise intensity,pump noise intensity and cross-correlation intensity.The input signal frequency can change the oscillation amplitude and period of the intensity correlation function,but not change the equilibrium site of the oscillation.The quantum noise intensity can change the position of the oscillation extremum.The parameters can all change the initial decay value of the intensity correlation function,while the intensity correlation function is not sensitive to the time period modulation frequency.
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).