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Physical Review

Coherence Properties and Photon Correlation

Hans Morawitz

Published Aug 16, 1965• 33 citations
Publisher

Abstract

An intensity-correlation experiment is proposed to test the second-order coherence of laser light from a cw-operated laser based on the Glauber formalism. The calculation utilizes the known form of the density matrix for radiation from a randomly excited source such as a discharge tube which is equivalent to filtered black-body radiation, and represents the laser radiation field by a coherent state. The intensity correlation calculated exhibits the standard Hanbury Brown-Twiss term and an extra term due to intensity interference between the Fourier components of the thermal field and the laser mode.

DOI
10.1103/physrev.139.a1072

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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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An intensity-correlation experiment is proposed to test the second-order coherence of laser light from a cw-operated laser based on the Glauber formalism. The calculation utilizes the known form of the density matrix for radiation from a randomly excited source such as a discharge tube which is equivalent to filtered black-body radiation, and represents the laser radiation field by a coherent state. The intensity correlation calculated exhibits the standard Hanbury Brown-Twiss term and an extra term due to intensity interference between the Fourier components of the thermal field and the laser mode.
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.
In 1956, Hanbury Brown and Twiss performed their famous intensity-correlation experiment with two 'coherent' beams of light from a thermal source. They argued that such intensity-correlation must entail an excess of photon arrival-time coincidences over chance ones for the two beams, and went on (with Little) to demonstrate this in an experiment (Twiss, Little and Hanbury Brown 1957). About the same time, Brannen and Ferguson (1956) performed a photon-coincidence experiment with coherent thermal light, and obtained a null result (due to lack of sensitivity).
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.
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)}g(2) g ( 2 ) which reflects spatial coherence properties of the emitted light. The measurement of g(2)g^{(2)}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)}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.
In this experiment, the observation by Hanbury Brown and Twiss that light intensity fluctuations from an extended broadband source are spatially correlated is demonstrated with a quasi-thermal light source. Laser light is used to illuminate a ground glass screen and generate a diffraction pattern, also known as a speckle pattern. The speckle pattern is recorded using an electronic camera and analyzed. Calculated auto-correlations of the intensity fluctuations agree with theoretical results originating from speckle analysis.