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Applied Physics Letters

MEASUREMENTS OF PHOTOELECTRIC COUNTING DISTRIBUTIONS FOR A LASER NEAR THRESHOLD

David W. Meltzer, W. Davis, L. Mandeļ

Published Sep 15, 1970• 24 citations
Publisher

Abstract

Photoelectric counting probabilities have been measured for a laser operating near the threshold of oscillation, for various values of the pump parameter and various counting time intervals. The results are compared with values derived from the recent calculations of Lax and Zwanziger (based on the rotating-wave van der Pol oscillator model of a laser) and show very good agreement.

DOI
10.1063/1.1653383

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Photoelectric counting probabilities have been measured for a laser operating near the threshold of oscillation, for various values of the pump parameter and various counting time intervals. The results are compared with values derived from the recent calculations of Lax and Zwanziger (based on the rotating-wave van der Pol oscillator model of a laser) and show very good agreement.
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A previously constructed laser model with quantum (noncommuting) noise sources was shown to lead near threshold to a quantum rotating-wave Van der Pol oscillator. A full dynamical correspondence between quantum and classical random processes allows one to compute the average of any time-ordered, normal-ordered operator function by averaging the associated function of classical random variables. Numerical calculations for the associated classical Van der Pol oscillator of the steady-state distribution, the total intensity fluctuations, and the linewidth versus operating point were amply confirmed experimentally. Measurements (and calculations) of higher than two-time correlations were sparse and contradictory. Photocount distributions, at times short compared to the intensity correlation time, confirm only the steady state of the laser. Photocount distributions at intermediate and longer times are difficult to compute because they involve multitime correlations of high ($\ensuremath{\infty}$) order. By providing an exact solution for photocount distributions and their moments for all times, we expected to stimulate measurements near threshold which would provide an adequate test of the Van der Pol laser model. Comparison of the results reported here with recent photocount experiments of Meltzer, Davis, and Mandel and of Jakeman, Oliver, and Pike provides gratifying agreement and confirmation of our statistical understanding of laser fluctuations near threshold.
With the help of a fast digitizer, the probability distribution of the light intensity of a single-mode dye laser has been measured photoelectrically, for various working points of the laser, from well-below to well-above threshold. In the region below threshold the laser exhibits short spiking and a double-peaked probability distribution with large value at zero intensity. A plot of most probable vs mean light intensity shows a discontinuity suggestive of a first-order phase transition. The experimental results are compared with Monte Carlo solutions of the equations of motion, with both additive spontaneous emission noise and multiplicative Ornstein-Uhlenbeck pump noise, and reasonable agreement is found.
Two techniques are described for obtaining information about the intensity correlation function of an optical field, from measurements of photoelectric counting distributions. The first method is based on double differentiation of the second factorial moment with respect to the counting time. It requires very accurate data, but is shown to be quite feasible, and yields the intensity-correlation function explicitly. The second procedure is less sensitive to inaccuracies and allows the correlation time and certain other parameters to be determined from the factorial moments. In addition, it yields some information about the general form of the correlation function, but no explicit function. The procedures are illustrated and compared by application to photoelectric counting measurements of a laser far below the threshold of oscillation.
A simple semiclassical derivation is given of the statistics of the photoelectrons emitted from a photosurface illuminated by light with statistically time dependent intensity. Comparison is made with the quantum mechanical expressions. Experiments are described that yield the probability of emission ofnphotoelectrons within a fixed time interval from a photocathode illuminated by a laser below and above threshold. The experimental results are compared with the theory developed in this paper and expressions derived by Glauber.