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Physical Review A

Frequency dependence of laser light fluctuations

Surendra Singh, Stephen R. Friberg, L. Mandeļ

Published Jan 1, 1983• 13 citations
Publisher

Abstract

The equations of motion for an inhomogeneously broadened laser operating off resonance in the neighborhood of threshold are solved. It is shown that the effects of detuning cannot be described just by a frequency-dependent laser pump parameter, but that the laser characteristics are modified in other ways also. The relative intensity fluctuations and the correlation time of the laser are calculated as a function of mean intensity and frequency, and it is shown that both are affected by detuning. These theoretical conclusions are then confirmed quantitatively by direct photoelectric counting and two-time correlation measurements of a He:Ne laser.

DOI
10.1103/physreva.27.381

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The equations of motion for an inhomogeneously broadened laser operating off resonance in the neighborhood of threshold are solved. It is shown that the effects of detuning cannot be described just by a frequency-dependent laser pump parameter, but that the laser characteristics are modified in other ways also. The relative intensity fluctuations and the correlation time of the laser are calculated as a function of mean intensity and frequency, and it is shown that both are affected by detuning. These theoretical conclusions are then confirmed quantitatively by direct photoelectric counting and two-time correlation measurements of a He:Ne laser.
The light intensities of the two counter-rotating modes of a ring laser have been investigated by photoelectric counting measurements near the laser threshold and for various detunings. The results are found to be in good quantitative agreement with the theory. They confirm that, at the line center, competition effects prevent the light output in one mode from growing and becoming coherent as the pump parameter is increased. Instead the relative intensity fluctuations of this mode become unity, as for thermal light. Negative cross correlations between the intensity fluctuations of the two modes are also measured. The competition effects virtually disappear when the laser is detuned three natural linewidths from line center, and an intermediate situation is found for smaller detuning.
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.
Expressions for the fluctuations and correlations of the light emitted by a ring laser with counter-rotating traveling wave modes are derived theoretically. It is shown that the competition of the two modes for the excited atomic population leads to negative correlations between their intensity fluctuations, whose magnitude depends on the detuning of the laser cavity from the atomic line center. As a result of the mode competition, the relative intensity fluctuations do not die out in general as the working point of the laser is raised above threshold at line center, and the emitted light does not become fully coherent as in a conventional laser. The two-time amplitude and intensity correlation functions of the light are calculated at line center, and are shown to be expressible in terms of the eigenfunctions and eigenvalues of a certain Schr\"odinger equation, in complete analogy with the theory of the single-mode laser. However, the intensity correlation time increases with pump parameter above threshold, unlike that for the single-mode laser.
The steady-state fluctuations of an inhomogeneously broadened two-mode laser with both additive and multiplicative white noise are investigated theoretically. A Fokker-Planck equation of the intensity-distribution function is presented. Analytic expressions of the autocorrelation and cross correlation of the steady-state two-mode intensities are derived for equal pump parameters of the two modes. Large intensity fluctuations occur when the laser is operated below threshold. However, when the laser is operated far above threshold, there is almost no difference between the laser models with and without multiplicative noise.
The correlations of the fluctuations of the electric field are studied for a simple model of the buildup from spontaneous-emission noise of the output of a single mode of a laser cavity. Results for when the laser cavity is detuned with respect to the center frequency of the gain medium are compared with the case in which the cavity is resonantly tuned. Resonantly tuned lasers display transient phase and amplitude correlations with a peak at the time that the evolving amplitude departs from the neighborhood of the origin in the complex plane of the amplitude where phase diffusion dominates. Deterministic frequency chirps during the switch-on of a detuned laser delay and significantly strengthen the transient correlation of phase and intensity fluctuations when the detuning is sufficiently large compared with the strength of the noise. In this case the peak in the correlations is related to the anomalous intensity fluctuations characteristic of transient switching.
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.