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IEEE Journal of Quantum Electronics

1A1 - Photoelectron statistics produced by a laser operating below and above the threshold of oscillation

Charles A. Freed, H. A. Haus

Published Aug 1, 1966• 43 citations
Publisher

Abstract

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.

DOI
10.1109/jqe.1966.1074028

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Photoelectron statistics produced by a laser operating below and above the threshold of oscillation

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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 of n photoelectrons 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.

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Preliminary observations of a new effect in photoelectron statistics is reported. By utilizing an AND-gate system, the joint probability of emission of two photoelectrons from a photocathode at times ${t}_{1}$ and ${t}_{2}$ has been measured over a range of intervals, $\ensuremath{\tau}={t}_{2}\ensuremath{-}{t}_{1}$, lying between 4.0 and 50 nsec with a resolution $\ensuremath{\Delta}\ensuremath{\tau}$ at 1.0 nsec, under illumination with a single-mode He-Ne laser and a tungsten lamp. The joint probability of emission as a function of $\ensuremath{\tau}$, under both conditions of illumination, rises to a large peak in the neighborhood of $\ensuremath{\tau}=15$ nsec, and when $\ensuremath{\tau}$ exceeds 25 nsec, it drops rapidly to the constant value predicted by the classical theory of Mandel. In addition, there is a distinct fine structure which becomes more pronounced in the "dark" emission. Significantly, the amplitude of the peak varies linearly with the mean intensity of the incident light, while the constant value obtained at long intervals varies quadratically. To explain the observed phenomena, a generalization of the Mandel theory is formulated on the basis of the axiomatic theory of probability and the following hypotheses: (1) The emission of two photoelectrons separated by a time interval are statistically dependent events owing to intrinsic processes within the cathode as well as extrinsic processes of the radiation field; (2) If the instantaneous intensity of the radiation field is proportional to the square of the field amplitude, then the probability of emission is a function of this amplitude. Both hypotheses are found to be necessary to provide a base for interpreting the linear dependence of the joint probability peak on field intensity. Moreover, the generalized formulation reduces to Mandel's expressions when the following restrictions are imposed: The emission process is statistically independent; the probability of emission is proportional to the intensity of radiation, and the radiation field is a Gaussian random process.

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A relation is derived between the probability distribution of the integrated intensity of light incident on a photoelectric detector and that of the emission of n photoelectrons in a time interval (t, t + T) from the properties of a quantized electromagnetic field. Both the direct and the inverse relations are derived, thereby enabling the statistical properties of light to be determined from photoelectric measurements. An explicit expression is given for the coefficients for calculating the moments ( W k) by means of the moments (n J) (] = 1, 2 ..... k). The correctness of the operations in which generalized functions occur is confirmed by some direct calculations. The significance of photoelectric detectors has been increasing of late. Since their response is proportional to the intensity of incident light (to the square of the amplitude of light) (more exactly, to the local mean time value of the intensity), one speaks of quadratic detectors. These detectors are of basic importance in the theory of the coherence of the electromagnetic field, since if we place N such detectors into N different points in space and if we lead the outputs of these detectors to the correlator we can measure the correlation (coherence) effects of the 2N-th order in the given field [1]. Such correlation measurements provide information on the energy distribution in the radiation spectrum [2], the angular dimensions of stars [3], and the state of polarization [4] and also enable the phase of physically important functions (e.g. the scattering amplitudes) [5] to be determined. Although in such experiments the properties of photoelectrons are studied rather than of photons, they reflect the basic properties of photons. Let the probability distribution Of the emission of n photoelectrons in a time interval (t, t + T) be p(n, T, t) - p(n) and let the probability distribution of the quantity W(T, t) = W = J'tt+r I(t') dr', where I(t) is the intensity of linearly polarized light incident on the photodetector (for simplicity we assume that the sensitivity of the photocathode is a = 1), be P(W). The quantity W can be regarded as a classical characteristic of the field, the quantity n represents the quantum characteristic of the field. Then, on the basis of such a semi-classical description, in which the field is described classically, but its interaction is described quantum-wise, one can derive the following relation [6]

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Correlations in Light from a Laser at Threshold

Ren Fang Chang, Victor Korenman, C. O. Alley +1 more

In this study, we measured the temporal correlations in the electromagnetic field radiated by a laser in the threshold region of oscillation, from $\frac{1}{10}$ 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 $\frac{1}{10}$ 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.

Physical Review•Feb 10, 1969•79 citations
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.
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 of n photoelectrons 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.
Preliminary observations of a new effect in photoelectron statistics is reported. By utilizing an AND-gate system, the joint probability of emission of two photoelectrons from a photocathode at times t1{t}_{1}t1​ and t2{t}_{2}t2​ has been measured over a range of intervals, $\ensuremath{\tau}={t}_{2}\ensuremath{-}{t}_{1}$, lying between 4.0 and 50 nsec with a resolution $\ensuremath{\Delta}\ensuremath{\tau}$ at 1.0 nsec, under illumination with a single-mode He-Ne laser and a tungsten lamp. The joint probability of emission as a function of $\ensuremath{\tau}$, under both conditions of illumination, rises to a large peak in the neighborhood of $\ensuremath{\tau}=15$ nsec, and when $\ensuremath{\tau}$ exceeds 25 nsec, it drops rapidly to the constant value predicted by the classical theory of Mandel. In addition, there is a distinct fine structure which becomes more pronounced in the "dark" emission. Significantly, the amplitude of the peak varies linearly with the mean intensity of the incident light, while the constant value obtained at long intervals varies quadratically. To explain the observed phenomena, a generalization of the Mandel theory is formulated on the basis of the axiomatic theory of probability and the following hypotheses: (1) The emission of two photoelectrons separated by a time interval are statistically dependent events owing to intrinsic processes within the cathode as well as extrinsic processes of the radiation field; (2) If the instantaneous intensity of the radiation field is proportional to the square of the field amplitude, then the probability of emission is a function of this amplitude. Both hypotheses are found to be necessary to provide a base for interpreting the linear dependence of the joint probability peak on field intensity. Moreover, the generalized formulation reduces to Mandel's expressions when the following restrictions are imposed: The emission process is statistically independent; the probability of emission is proportional to the intensity of radiation, and the radiation field is a Gaussian random process.
A relation is derived between the probability distribution of the integrated intensity of light incident on a photoelectric detector and that of the emission of n photoelectrons in a time interval (t, t + T) from the properties of a quantized electromagnetic field. Both the direct and the inverse relations are derived, thereby enabling the statistical properties of light to be determined from photoelectric measurements. An explicit expression is given for the coefficients for calculating the moments ( W k) by means of the moments (n J) (] = 1, 2 ..... k). The correctness of the operations in which generalized functions occur is confirmed by some direct calculations. The significance of photoelectric detectors has been increasing of late. Since their response is proportional to the intensity of incident light (to the square of the amplitude of light) (more exactly, to the local mean time value of the intensity), one speaks of quadratic detectors. These detectors are of basic importance in the theory of the coherence of the electromagnetic field, since if we place N such detectors into N different points in space and if we lead the outputs of these detectors to the correlator we can measure the correlation (coherence) effects of the 2N-th order in the given field [1]. Such correlation measurements provide information on the energy distribution in the radiation spectrum [2], the angular dimensions of stars [3], and the state of polarization [4] and also enable the phase of physically important functions (e.g. the scattering amplitudes) [5] to be determined. Although in such experiments the properties of photoelectrons are studied rather than of photons, they reflect the basic properties of photons. Let the probability distribution Of the emission of n photoelectrons in a time interval (t, t + T) be p(n, T, t) - p(n) and let the probability distribution of the quantity W(T, t) = W = J'tt+r I(t') dr', where I(t) is the intensity of linearly polarized light incident on the photodetector (for simplicity we assume that the sensitivity of the photocathode is a = 1), be P(W). The quantity W can be regarded as a classical characteristic of the field, the quantity n represents the quantum characteristic of the field. Then, on the basis of such a semi-classical description, in which the field is described classically, but its interaction is described quantum-wise, one can derive the following relation [6]
Nonclassical light beams generated by the degenerate optical parametric oscillator operating below threshold are analyzed in terms of photoelectron-counting sequences. The positive-P representation is used to calculate the generating function for photoelectron statistics in a closed form. This generating function is used to derive expressions for the photoelectron-counting and waiting-time distributions. The dependence of these distributions on mean photon number inside the cavity and efficiency of detection is studied. The relationship between photoelectron-counting sequence and the photon emission sequence is used to present a simple physical picture of light beams produced by the degenerate parametric oscillator.
The statistical behavior of photoelectrons emitted at L points of a photocathode in partially coherent Gaussian light is considered. The exact formula for the cumulants and the factorial moment generating function is derived. An exact expression for the photocount distribution and for the factorial moments is obtained in the case when the detection time is much smaller than the coherence time. Two cases of interest which were treated earlier by the same authors are also considered. In particular, the case of two points of detection is of special interest because a physical meaning can be given to the difference of the normalized eigenvalues. Finally, both the exact and approximate theories are compared with the experimental photocount distribution for four different geometrical arrangements.
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.