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Physical Review A

Statistical Properties of Laser Radiation During a Transient Buildup

F. T. Arecchi, Vittorio Degiorgio

Published Mar 1, 1971• 152 citations
Publisher

Abstract

The transient statistical properties of a single-mode laser field are investigated both experimentally and theoretically. The physical system is a $Q$-switched He-Ne gas laser oscillating on the 6328-\AA{} transition. The measurements are performed by photon-counting techniques, using the so-called linear method which enables resolving times to be in the nsec region. The effects of the photomultiplier and of the attenuators on the statistics are taken into account. The measurements are interpreted in terms of the available laser theories as well as in terms of a simple phenomenological model which attributes the statistical spread to the initial photon distribution, neglecting the stochastic forces in the course of the buildup. Finally, some of the approximations of the theories are discussed, and the sensitivity of a transient measurement is compared with that of a stationary-intensity-correlation measurement.

DOI
10.1103/physreva.3.1108

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The transient statistical properties of a single-mode laser field are investigated both experimentally and theoretically. The physical system is a QQQ-switched He-Ne gas laser oscillating on the 6328-\AA{} transition. The measurements are performed by photon-counting techniques, using the so-called linear method which enables resolving times to be in the nsec region. The effects of the photomultiplier and of the attenuators on the statistics are taken into account. The measurements are interpreted in terms of the available laser theories as well as in terms of a simple phenomenological model which attributes the statistical spread to the initial photon distribution, neglecting the stochastic forces in the course of the buildup. Finally, some of the approximations of the theories are discussed, and the sensitivity of a transient measurement is compared with that of a stationary-intensity-correlation measurement.
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.
We measure the statistical distribution of buildup times for the field of a single-mode CO2{\mathrm{CO}}_{2}CO2​ laser when cavity losses have rapidly changed from a large to a small value. The ratio between the average buildup time and its statistical spread is shown to be a sensitive test of the quantum fluctuations in the off state. This method operates as a ``statistical microscope,'' permitting the accurate determination of a small mean photon number in the infrared region by optical amplification, and a discrimination of the initial mean photon number from noise contributions along the amplification process. Following the buildup, the laser pulse decays on two time scales. The faster one, due to the radiation coupling, can be fitted by a Toda oscillator model; the slower one, due to the collisional dynamics of the excited population, provides a quantitative evaluation of two distinct population decay rates, one within the rotational manifold and the other one out of the vibrational band.
The paper contains a quantum theoretical analysis of laser beam fluctuations and of the light beat experiments with two lasers. With the help of experimental results on photon counting fluctuations in a single-mode laser field, some correlation properties of the field are derived. It is shown that the correlation equations are satisfied by states of the field which are much more general than "coherent" states. The equations lead directly to the spectral density of the intensity operator in the light beat experiments, which can be obtained from photoelectric measurements. The resulting expression is practically identical to that found by Forrester for light having thermal statistical properties. The reasons for this are discussed by a comparison of the corresponding probability distributions of photon counts and of the classical wave amplitude.
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Laser fluctuations are measured and characterized with a new technique based on the concept of first-passage-time distributions. This first-passage-time technique makes it possible to discern quantum noise in a dye laser even in the presence of pump fluctuations that are many orders of magnitude larger. We report new analytic and experimental results for the mean, variance, and skewness of the first-passage-time distributions. The analytic results agree well with numerical computations and experimental measurements. Our technique is particularly suited to the quantitative description of laser fluctuations above threshold; it complements the conventional photoelectric counting and correlation methods which are valuable for lasers below and near threshold but are difficult to implement for lasers far above threshold. The limits of validity of the theory containing a cubic nonlinearity are examined in the context of passage-time measurements. The effects of multiplicative noise and saturation on the variance of the intensity fluctuations are also discussed. A new signature of multiplicative noise is predicted.
The single-mode laser model as formulated by Haken and co-workers is considered. Within the approximation which takes into account only the interaction (at all orders in the coupling constant) between the mode and the single atoms, this model leads to a linear time-evolution equation for the Glauber quasiprobability distribution $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ of the field mode, containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. This equation is solved exactly in the stationary situation. Thus the quantum effects due to the terms with derivatives of order higher than the second one are fully taken into account. It is shown that these terms have a shrinking effect on the stationary distribution above threshold, counteracting the broadening effect due to the second-derivative term. The photon and photoelectron distributions are given. Exact relations are deduced, which allow the calculation of all the moments of the steady-state distribution from the first one. This distribution is compared with the stationary solutions of Risken's equation, of P. Mandel's equation, and with the stationary distribution deduced by Weidlich, Risken, and Haken. The connection with the single-mode laser theory of Scully and Lamb is analyzed in the high-intensity region. The complete Scully-Lamb master equation is translated into the Glauber representation, yielding a linear equation for $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. Suitably connecting the parameters AAA, BBB, and CCC of the Scully-Lamb equation with the parameters of the model of Haken and co-workers, it is shown that there is perfect agreement between the strong-signal Scully-Lamb steady-state distribution and that of Weidlich, Risken, and Haken for all values of the pump parameter.