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.
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.
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.
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Introduction In this chapter we will put the concept of coherence on a more mathematical basis. This will involve the formal definition of a number of functions that are used to describe the coherence properties of optical fields. These include the analytic signal, various correlation functions, and the degree of coherence for describing both temporal and spatial coherence. We shall see that the degree of temporal coherence is quantitatively related to the lineshape function and that the degree of spatial coherence between two points is determined by the size, intensity distribution, and location of an illuminating source. We will use the wave equation, and special solutions to the wave equation called Green's functions, to show how spatial coherence varies from point to point. The chapter will conclude with a discussion of how the intensity fluctuations of a source depend on its coherence properties, and we will examine a specific scheme, namely the Hanbury Brown–Twiss experiment, by means of which this relationship is studied. This discussion will involve a discussion of “photon statistics,” namely the time variation of the “detection” of photons from a source. In a quantum-mechanical context, square-law detectors respond to these quantized excitations of the optical field, which we call photons. In classical coherence theory it is advantageous to represent the real electromagnetic field by a complex quantity, both for its mathematical simplicity and because it serves to emphasize that coherence theory deals with phenomena that are sensitive to the “envelope” or average intensity of the field.
Abstract Investigations of photon correlations similar to the Hanbury Brown and Twiss experiment are since the advent of quantum optics in the center of both fundamental and application-oriented research. We derive a factorial moment-generating function of the probability distribution of electromagnetic radiation emitted by a source and absorbed by a photodetector from which we calculate an analytical expression for the second-order correlation coefficient g(2) g ( 2 ) which reflects spatial coherence properties of the emitted light. The measurement of g(2) g ( 2 ) thus allows to retrieve the spatial coherence parameter of the light source in terms of the combination of source area, source-detector distance, wavelength and detector area. The validity of the concept is proven by investigating g(2) g ( 2 ) of true thermal light from a Xenon arc lamp and from the sun for various detector areas, in excellent agreement with the theory. Finally, we suggest a novel scheme for determining light source diameters by exploiting the spatially dependent statistics and confirm its validity by exemplary calculations. These results based on radiation thermodynamics and photon statistics give fresh insight into quantum optical properties of classical light sources, photon correlations and photodetection with promising applications perspectives, more than 68 years after the original Hanbury Brown and Twiss experiment.
High-order photon correlations of light fields are important for characterizing the quantum nature. Since Hanbury Brown and Twiss conducted the pioneering experiments in the 1950s, the HBT effect has inspired extensive research on high-order photon correlation in quantum optics, quantum information, and quantum imaging. The Single-photon counting module is one of the most widely used single-photon detectors. Due to its high detection efficiency and low dark counts in the visible and near-infrared region, it is reasonably chosen for basic research on quantum mechanics. Many researches have demonstrated that the maximum value of second-order photon correlation g(2)(τ) at zero delay (τ = 0) can be used to distinguish different light fields. Therefore, the HBT scheme containing two single photon detectors have been widely used in many advanced studies, such as space interference, ghost imaging, single photon detection with high efficiency, etc. However, higher-order photon correlations g(n) (n > 2) can reveal more measurable characteristics of light fields, such as information about the non-Gaussian scattering process, the skewness and kurtosis of photon number distribution, etc. When the extended HBT scheme is used to measure higher-order photon correlations, the experimental conditions including quantum efficiency and background noise greatly affect the photon correlation measurement. The influences of the counting rate and resolution time of the detection system on the measurements are also very important and cannot be ignored. Therefore, the comprehensive considering of various influence factors is necessary for accurately measuring the high-order photon correlations and also a challenge. In this paper, we present a method based on double Hanbury Brown-Twiss scheme for the accurate measuring of high-order photon correlations g(n) (n > 2). The system consists of four single photon counting modules and is used to detect and analyze the joint distribution probability of temporal photon correlation. Considering the effects of the background noise and overall efficiency, theoretically, we analyze the correlations of the third- and fourth-order photon with the incident light intensity, squeezing parameter and photon number respectively for thermal state, coherent state, squeezed vacuum state, and Fock state. Meanwhile, experimentally we study the influences of resolution time and counting rate on correlations of the coherent state and thermal state with third- and fourth-order photon. On condition that the resolution time is 210 ns and the counting rate is 80 kc/s, the correlations of third and fourth-order photon with the thermal state at zero time delay are accurately measured, and the relative statistical deviations of the measured vales from the theoretical values are 0.3% and 0.8%, respectively. In addition, the third- and fourth-order photon correlations of the thermal state at different delay times are also observed. It is demonstrated that the high-order photon correlations of light fields are measured accurately by comprehensively analyzing various influencing factors. This technique provides a promising and useful tool to investigate quantum correlated imaging and quantum coherence of light fields.
Hanbury-Brown and Twiss (HBT) effect is the foundation for stellar intensity interferometry. However, it is a phase insensitive two-photon interference effect. Here we extend the HBT interferometer by mixing intensity-matched reference fields with the input fields before intensity correlation measurement. With the freely available coherent state serving as the reference field, we experimentally demonstrate the phase sensitive two-photon interference effect when the input fields are thermal fields in either continuous wave or non-stationary pulsed wave and measure the complete complex second-order coherence function of the input fields without bringing them together from separate locations. Moreover, we discuss how to improve the signal level by using the more realistic continuous wave broadband anti-bunched light fields as the reference field. Our investigations pave the way for developing new technology of remote sensing and interferometric imaging with applications in long baseline high-resolution astronomy.