The signal-to-noise ratio (S/N) of the Brown–Twiss stellar intensity interferometer, in which the resolving time (τr) of the photodetectors is much longer than the coherence time of incident radiation (τc), is derived, especially for incident radiation with a very high effective temperature. The statistical properties of the wave amplitude of the incident radiation is assumed to be described by a gaussian distribution function. In order to evaluate the mean square error in the intensity-correlation measurement, the eighth-order moment of complex wave amplitudes is obtained by using the moment theorem for a gaussian random process. The mean square error of the high-intensity correlation experiment during a finite observation time (τ0) is determined chiefly by the random correlation of noise currents due to intensity fluctuations of the incident radiation. The S/N in the case of extremely high-intensity radiation, thus, is given by (τ0/τc)12∣γ12∣2, where γ12 is the complex degree of coherence between two observation points independent of the intensity of incident radiation. In the case of low-intensity radiation the mean square error of an intensity-correlation measurement is determined by the random correlation of shot noise in photoelectric currents. The S/N for low-intensity radiation, as originally discussed by Hanbury Brown and Twiss, is given by (τ0/τr)12∣γ12∣2ηnτc where n is the number of incident photons per second and η is the quantum efficiency of a photodetector. The S/N for the intermediate case is also discussed.
The signal-to-noise ratio (S/N) of the Brown–Twiss stellar intensity interferometer, in which the resolving time (τr) of the photodetectors is much longer than the coherence time of incident radiation (τc), is derived, especially for incident radiation with a very high effective temperature. The statistical properties of the wave amplitude of the incident radiation is assumed to be described by a gaussian distribution function. In order to evaluate the mean square error in the intensity-correlation measurement, the eighth-order moment of complex wave amplitudes is obtained by using the moment theorem for a gaussian random process. The mean square error of the high-intensity correlation experiment during a finite observation time (τ0) is determined chiefly by the random correlation of noise currents due to intensity fluctuations of the incident radiation. The S/N in the case of extremely high-intensity radiation, thus, is given by (τ0/τc)12∣γ12∣2, where γ12 is the complex degree of coherence between two observation points independent of the intensity of incident radiation. In the case of low-intensity radiation the mean square error of an intensity-correlation measurement is determined by the random correlation of shot noise in photoelectric currents. The S/N for low-intensity radiation, as originally discussed by Hanbury Brown and Twiss, is given by (τ0/τr)12∣γ12∣2ηnτc where n is the number of incident photons per second and η is the quantum efficiency of a photodetector. The S/N for the intermediate case is also discussed.
Introduction The degree of first-order temporal coherence, a function denoted by g (1) ( τ ), provides information about the coherence length and the power spectral density of a light source. However, without additional information, g (1) ( τ ) has no bearing on intensity fluctuations and higher-order statistics of the emitted light. A quasi-monochromatic laser beam and the beam of light from an incandescent light bulb, provided that the latter is properly filtered to match the spectral line-shape of the former, will have identical degrees of first-order coherence. Any interferometric experiment involving the splitting and superposition of amplitudes would yield identical results for the laser beam and the (properly filtered) thermal light. Therefore, on the basis of such experiments alone, there is no way to distinguish the two light sources. It turns out, however, that the intensity fluctuations of laser light are fundamentally different from those of thermal light. The two sources can, therefore, be distinguished based on their second-order coherence properties. An ideal photodetector produces an electrical signal proportional to the “cycle-averaged intensity” of the E -field (or B -field) of the light beam at the location of the detector. Assuming that the electrical bandwidth of the detector (including all associated circuitry) is greater than the bandwidth of the incident light wave by at least a factor of 2, the output of the detector should accurately represent the intensity fluctuations of the light beam as a function of time.
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]
Imaging is the estimation of the spatial distribution of a physical object by measuring the optical radiation it emits, or by making use of an optical wave that interacts with the object, via reflection or transmission, for example, before being measured by a detector. The resolution of an imaging system is limited by the inability to localize the optical field at points of the object. Under otherwise ideal conditions, resolution is limited by diffraction. The sensitivity of an imaging system is limited by the uncertainty in the measurement. Under ideal conditions, this is determined by photon noise, which depends on the statistical fluctuations of the light. In conventional imaging systems, an extended detector, such as a CCD camera or an array detector, measures the spatial distribution of the optical intensity, which is proportional to the photon flux density. In interferometric systems, the spatial distribution of the optical field is inferred from measurements of the optical intensity. With the emergence of coherence theory, imaging systems based on measurements of the second-order coherence function at pairs of points in the detection plane were developed. An example is the imaging of an incoherent object based on the van Cittert-Zernike theorem. Imaging systems based on measurement of intensity correlation, or the photon coincidence rate, at pairs of points, were developed in the 1960s. A classic example of the photon-correlation imaging of an object emitting thermal light is stellar imaging using a Hanbury-Brown-Twiss intensity-correlation interferometer. More recently, two-photon light, which may be generated via spontaneous parametric downconversion in a second-order nonlinear optical crystal, has been used for imaging. This type of two-photon (or biphoton) imaging, which has come to be called quantum imaging, is also based on the measurement of photon coincidence by the use of photon-counting array detectors or by scanning two photon-counting detectors at pairs of points. To compare the resolution and sensitivity of imaging systems based on the aforementioned types of measurements, it is necessary to derive expressions for the measured quantities in terms of the object distribution. The point-spread functions based on such expressions can be used to assess the resolution. One measure of the sensitivity of the imaging process is the signal-to-noise ratio (SNR) of the measured variables. The statistical nature of the light source must be known in order to determine the SNR. The purpose of this chapter is to compare the resolution and sensitivity of photon-correlation imaging systems that make use of thermal light and two-photon light. We will henceforth refer to these two imaging modalities as classical and quantum photon-correlation imaging, or simply classical and quantum imaging, respectively. Clearly, light in other quantum states can also be used for imaging.
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.
In the history of optical science, we were satisfied with the measurement of mean intensity of light, namely the measurement of first-order coherence of light, for optical studies and observations, until the middle of 20th century when two astrophysicists Hanbury Brown and Twiss (HBT) introduced their “intensity interferometer”. “Correlation” emphasizes its statistical nature, while “coherence” emphasizes its interference nature. The chapter extends the concept of second-order coherence to Nth-order coherence (N > 2) and analyzes a few simple measurements on the third-order temporal and spatial coherence of thermal field. It also calculates the second-order temporal coherence function and the second-order spatial coherence function of thermal field by assuming idealized photodetectors and correlation measurement electronics.
The shot noise which occurs in photoelectric detection is calculated utilizing fully quantum mechanical procedures. The results which are obtained in this paper include the effects of photon bunching and they differ considerably from classical results which are obtained on the assumption of Poisson statistics. In some cases the Poisson-based classical results lead to shot-noise powers that are in error by as much as an order of magnitude. The results reveal the existence of a relative minimum in the plot of shot noise vs input signal to noise ratio. This suggests the possibility of optimum input signal to noise ratios within certain restricted ranges of the input signal to noise ratio. A plot of the shot noise which is in excess of that predicted by Poisson-based statistics vs input signal to noise ratio reveals a maximum at an input signal to noise ratio of approximately 1.5. This suggests that the enhancement of the Hanbury-Brown and Twiss effect that is obtained when coherent radiation is superposed upon chaotic radiation will be maximized at a coherent-to-chaotic ratio of approximately 1.5.