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Physical Review A

Photoelectric detection of polychromatic light

Harry J. Kimble, L. Mandeļ

Published Aug 1, 1984• 38 citations
Publisher

Abstract

The problem of photoelectric detection of light is reexamined when the light is polychromatic and not strictly stationary, and when the frequency response of the detector has to be taken into account. It is shown that, whereas the one-photoelectron detection probability can always be expressed in a familiar form involving the expected light intensity, the same is not true of the twofold detection probability. In general, the photodetector does not count photons in any precise sense.

DOI
10.1103/physreva.30.844

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The problem of photoelectric detection of light is reexamined when the light is polychromatic and not strictly stationary, and when the frequency response of the detector has to be taken into account. It is shown that, whereas the one-photoelectron detection probability can always be expressed in a familiar form involving the expected light intensity, the same is not true of the twofold detection probability. In general, the photodetector does not count photons in any precise sense.
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]
It is well known that photoelectric measurements yield information about the statistical behavior of fluctuating light beams. The probability of n photoelectrons being ejected, in a fixed time interval, from the photosensitive surface of the photoelectric detector, upon which the light beam is normally incident, is a linear (Poisson) transform of the probability density for the intensity of the beam. With the help of some plausible assumptions, the present analysis provides solutions to the problem of inverting the Poisson transform, thus determining the probability density for the intensity from experimentally obtained photocounting distributions. The effectiveness of the method is demonstrated by an actual inversion of a typical experimental counting distribution. The technique is of particular interest in connection with efforts to understand the statistical behavior of optical fields, especially of laser fields.
The basic formulae governing the fluctuations of counts registered by photoelectric detectors in an optical field are derived. The treatment, which has its origin in Purcell's explanation of the Hanbury Brown-Twiss effect, is shown to apply to any quasi-monochromatic light, whether stationary or not, and whether of thermal origin or not. The representation of the classical wave amplitude, of the light by Gabor's complex analytic signal appears naturally in this treatment. It is shown that the correlation of counts registered by N separate photodetectors at N points in space is determined by a 2 N th order correlation function of the complex classical field. The variance of the individual counts is shown to be expressible as the sum of terms representing the effects of classical particles and classical waves, in analogy to a well-known result of Einstein relating to black-body radiation. Since the theory applies to correlation effects obtained with any type of light it applies, in particular, to the output of an optical maser, although, for a maser operating on one mode, correlation effects are likely to be very small.
This chapter begins with a brief description of methods of primary photon detection: the photoelectric effect, rectifying detection, photothermal effects, photon beam amplifiers, single-microwave-photon counting, and quantum nondemolition detection. The theory of photoelectric detection is applied to derive the connection between the statistics of the incident light field and the ejected photoelectrons. This is followed by a description of several electron multiplication schemes that convert the microscopic detection event into a classical, irreversible process that is registered as a count. The remaining topics are coincidence counting, and homodyne and heterodyne detection schemes.
Wait-time distributions for the nthn\mathrm{th}nth photodetection at a detector illuminated by a stationary light beam are studied. Both unconditional measurements, initiated at an arbitrary instant, and conditional measurements, initiated upon a photodetection, are considered. Simple analytic expressions are presented for several classical and quantum sources of light and are used to quantify and compare photon sequences generated by them. These distributions can be measured in photon-counting experiments and are useful in characterizing and generating photon sequences with prescribed statistics. The effects of nonunit detection efficiency are also discussed, and curves are presented to illustrate the behavior.