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Physical Review

Theory of Electromagnetic Field Measurement and Photoelectron Counting

Paul L. Kelley, Walter H. Kleiner

Published Oct 19, 1964• 540 citations
Publisher

Abstract

A theory of electromagnetic field measurement by means of photoionization is developed and applied to photoelectron counting. A probability theory involving multitime joint probability functions for a sequence of photoionizations is formulated. A general quantum-theory definition is proposed for the nonexclusive probability function which occurs in the probability theory. Approximations are then introduced to derive expressions for this probability function which involve correlation functions of the photoionization detector and the electromagnetic field-plus-source. The general theory is used to derive quantities of interest in photo-ionization counting experiments. Expressions are derived for (1) the probability ${P}_{n}(t, t+T)$ that $n$ photo-ionizations are observed in the time interval $t$ to $t+T$, and (2) quantities related to ${P}_{n}(t, t+T)$, such as its generating function and various moments. ${P}_{n}(t, t+T)$ is found to be a compound Poisson distribution determined by the density operator of the field when the latter is expressed in Glauber's $P$ representation. Using this result, the character of ${P}_{n}(t, t+T)$ is examined for several specific density operators. These correspond to a coherent state, various fields with the mode phases distributed independently of the mode amplitudes, and a "spread-out" coherent state.

DOI
10.1103/physrev.136.a316

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A theory of electromagnetic field measurement by means of photoionization is developed and applied to photoelectron counting. A probability theory involving multitime joint probability functions for a sequence of photoionizations is formulated. A general quantum-theory definition is proposed for the nonexclusive probability function which occurs in the probability theory. Approximations are then introduced to derive expressions for this probability function which involve correlation functions of the photoionization detector and the electromagnetic field-plus-source. The general theory is used to derive quantities of interest in photo-ionization counting experiments. Expressions are derived for (1) the probability Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) that nnn photo-ionizations are observed in the time interval ttt to t+Tt+Tt+T, and (2) quantities related to Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T), such as its generating function and various moments. Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) is found to be a compound Poisson distribution determined by the density operator of the field when the latter is expressed in Glauber's PPP representation. Using this result, the character of Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) is examined for several specific density operators. These correspond to a coherent state, various fields with the mode phases distributed independently of the mode amplitudes, and a "spread-out" coherent state.
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]
We consider the probability P(n)(t0,t){P}^{(n)}({t}_{0}, t)P(n)(t0​,t) of counting nnn photoelectrons during any time interval [t0,t{t}_{0}, tt0​,t] in a photodetection model built in the general framework of quantum field theory. Owing to an equilibrium assumption on the detector state, P(n)(t0,t){P}^{(n)}({t}_{0}, t)P(n)(t0​,t) is evaluated from the incident field and some electron-correlation functions. The conditions for P(n)(t0,t){P}^{(n)}({t}_{0}, t)P(n)(t0​,t) to be a compound Poisson distribution as expected are discussed. This leads to the definition of coherence properties for the electrons of the detector in connection with the reliability of the detection operation in counting experiments.
We show that time distributions of photoelectrons can best be described by a generalized probability-density function which we call the probability-density functional. The probability-density functional gives complete information about the statistical properties of any random-point process, and provides the most natural definitions of the ensemble average and generating functional of a quantity distributed in time. Using this functional, we develop a new approach to photoelectron statistics that systematically relates various joint probability distributions of photoelectrons to one another and to the corresponding generating functionals. We demonstrate that this method can be used to derive previously obtained results and we use it to obtain new results. In particular, the amount of information about the statistical properties of photoelectrons that is contained in a probability distribution can be obtained by comparing the corresponding generating functional with the probability-density functional. Further, we examine time derivatives of the probability distribution for the number of counts and develop new relationships for probability distributions of triggered counting. Finally, we apply our results to obtain an ``inverse'' relationship between photon bunching (antibunching) and super-Poisson (sub-Poisson) statistics.
The functional formalism, providing a unified treatment of the optical coherence theory is developed for arbitrarily ordered field operators. Multimode ordered expansions of operators in terms of boson amplitude operators as well as integral expansions are discussed. They form a basis of associating a class of s-ordered probability functionals with the density operator of the field. As an application, the distributions of photons and photoelectrons are studied. The generalized photo-detection equation of Perina and Horak (1970) is rederived by using the functional approach. A formula expressing the photocount generating function through the characteristic functional of the field is derived.
In the theoretical study of photoionization, certain properties of the photoelectrons are not defined in terms of any (even idealized) model for the electron-detection process, in contrast to the well-known Glauber theory of photon detection. Thus, for instance, the energy spectrum of the electron produced in atomic ionization is usually defined simply as the long-time probability for the electron to have a given energy. In this paper a formal model for an electron detector localized to a small region in space, similar to the DeWitt monopole detector, is set up in terms of which results analogous to those obtained for photon detection in the Glauber theory can be obtained. In addition, it is possible to define correlation functions describing the joint probability of detecting an electron and a photon. This correlation function is calculated here in the case of two-photon ionization of an atom modeled as having two discrete energy levels and an ionization continuum. When the atom is subjected to an incident driving field, both a photoelectron and scattered photons arising from the spontaneous decay of the upper of the two discrete levels are produced. Correlations between these photons and the photoelectron are studied here by considering the joint probability of detection of a photoelectron and a scattered photon for a monochromatic driving field.
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