LETTERS TO THE EDITOR Relation between Radiation Statistics and Two-Quantum Photocurrent Spectra*

Two methods of investigating the statistical properties of radiation fields in single beams have been found useful, particularly to obtain information about laser sources. One is the measurement of photocounting statistics 1 ; the other is the analysis of the more easily obtained photocurrent spectrum.' The information most readily extracted from these measurements, such as moments of the counting distribution or bandwidths, is ultimately related to certain statistical parameters of the fluctuations in the radiation field. Correlation properties, such as photon-bunching effects, can be revealed by detailed processing of ordinary (singlequantum) photocounting measurements, but can be made more apparent by using double-quantum detectors.' 5 The relation between two-quantum photocounting statistics and the distribution of irradiance fluctuations has been analyzed in an earlier paper. In this paper, we complement that analysis with a determination of the kind of information about the stochastic properties of the radiation field that can be obtained from the twoquantum photocurrent spectrum. The results generalize those of a similar analysis made for the single-quantum detector by Freed and Haus. In both cases, the photocurrent spectrum is related to a moment of the two-time joint-probability density of the radiation, but the one-quantum spectrum corresponds to a second moment of that distribution, whereas the two-quantum spectrum relates to a particular fourth-order moment. Thus, from a single measurement, the twoquantum photocurrent spectrum provides a determination of a fourth-order correlation function of the incident radiation. What is referred to in the spectral measurement is the excess photocurrent noise spectrum, above the shot-noise level. In the case of single-photon detection, that excess noise is directly related to the spectral noise power of the radiation. The proper generalization to double-quantum detection (or to any order) is obtained by recognizing that the relevant statistics measured by the excessnoise spectrum are those of the probability of emission of a photoelectron in the detector. For the one-quantum detector, this is indeed proportional to the irradiance I(t), but for the twoquantum detector, the emission probability w(t) is proportional to the square of the fluctuating irradiance, 12(t). For sufficiently narrow photocurrent pulses, the generalized relation between the photocurrent spectrum ,s (X) and the irradiance is given by