A new approach to measure the second order correlation function g(2) and the coherence time was investigated. The g(2) was calculated from the photon pair time interval distribution by direct numerical self-convolution with the high order correction. The accuracy of this method was examined using an optical fiber based Hanbury-Brown-Twiss interferometer with a pseudo-thermal light source. We found that the significance of the high order correction is related to the factor Īτc, which is the overlapping of the photon wave packets. A novel technique was also demonstrated to measure the coherence time τc of a light source using the random phase modulation. This method is more suitable for a weak light source with a long coherence time using a simple experimental setup.
A new approach to measure the second order correlation function g(2) and the coherence time was investigated. The g(2) was calculated from the photon pair time interval distribution by direct numerical self-convolution with the high order correction. The accuracy of this method was examined using an optical fiber based Hanbury-Brown-Twiss interferometer with a pseudo-thermal light source. We found that the significance of the high order correction is related to the factor Īτc, which is the overlapping of the photon wave packets. A novel technique was also demonstrated to measure the coherence time τc of a light source using the random phase modulation. This method is more suitable for a weak light source with a long coherence time using a simple experimental setup.
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.
Abstract We present an experimental technique that can accurately measure the high-order photon correlations of various light fields based on double Hanbury Brown–Twiss scheme. The system consists of four conventional binary response single-photon counting modules which are non-photon-number-resolving detectors. We theoretically analyze the effects of the experimental conditions on the high-order photon correlations of coherent, chaotic thermal, Fock and squeezed vacuum states with considering the overall efficiency and background noise. In experiment, we determine the proper resolution time from 2 9 = 512 ns to 2 11 = 2048 ns and counting rate from 10 kc s −1 to 100 kc s −1 for characterizing the exact second-, third- and fourth-order photon correlations of chaotic thermal state. At complete time delays, we also observe the third- and fourth-order photon correlations of chaotic thermal state, with the maximum high-order correlation values of 5.99 ± 0.02 and 23.9 ± 0.2 respectively, which are in good agreement with the theoretical values of 6 and 24. The demonstrated experimental technique with high accuracy is a promising tool for investigating quantum optical coherence and ghost imaging.
Two-photon correlation measurement provides a promising way to experimentally demonstrate the statistical nature of a light source, which is very significant for the deep understanding of the photon-generating process and the diagnosing of the coherence property. Quantitatively two- photon correlation is described by second-order coherence. Usually the behavior of the second-order coherence against any of the parameters defining the phase volume is different for different photon statistics. The Poisson photon statistics for coherent light gives its second-order coherence as a flat response; The Bose-Einstein photon statistics for chaotic light gives its second-order coherence as a bunching effect; While the Sub-Poisson photon statistics for non-classical light gives its second-order coherence as an anti-bunching effect. Therefore the measurement of two-photon correlation is proved to be a good finger print to check whether light is in coherent state or incoherent state such as thermal state or non-classical state. Historically the measurement of two-photon correlation was first performed by Hanbury-Brown and Twiss (HBT) in 1956. They used a linear mixer to realize the correlation of the two currents from the photoelectric detectors illuminated by a stationary thermal light souece, a mercury arc, and the photo-bunching effect was first successfully observed in the visible region of 435.8 nm. HBT method is no doubt a good way to extract the small excess two-photon correlation for a stationary light because the background, that is the DC components, has been cut off automatically by the broad band amplifiers, which is in fact the key of the success of HBT experiment. However there exists a general problem, to which no attention has ever been paid, in measuring the two-photon correlation of non-stationary light such as synchrotron radiation (SR) by the HBT method. Here the "non-stationary" means a sense of classical mechanics that the observed intensity has some deterministic time structure. The systematic time structure of SR decided by the bunch distribution of the electric current in a storage ring will give rise to a large amount of unexpected accidental correlation, which in fact has nothing to do with the inherent photon statistics of light source and usually l000～l0000 times larger than the true two-photon correlation due to the short bunch separation length (2ns) and the short coherence time (～0.1ps) which is not comparable to the time resolution (1ns) of the measuring system. The existence of the accidental correlation would severely prevent us from observing the bunching effect of the true two-photon correlation. Therefore to suppress the much larger accidental correlation and to extract the small true two-photon correlation, a novel intensity interferometer has been developed for soft X-ray synchrotron radiation. This intensity interferometer consists of an optical vacuum chamber and an electric correlator. All the essential optical elements which includes a wire scanner, a precise diffraction slit, a grating monochromator with a coherence time modulator, a beam divider and two fast-response photon detectors (microchannel plates) are mounted in this high vacuum chamber. The electric correlator completes the multiplication of the two broad band electric currents coming from the photoelectric detectors. The basic idea to suppress the much larger accidental correlation is to modulate the coherence time by modulating the entrance slit width of the monochromator by a piezoclectric translator. The two sets of light intensity are simultaneously modulated too. When the frequency of modulation is f, the third harmonics 3f is detected with a loch-in amplifier because the 3f components include only the true two-photon correlation. Practically it is difficult to modulate with frequency f without any higher order harmonics distortion which might add some false 3f components. To overcome this difficulty we have used a sharp bandpass filter of l00～350 MHz in each branch of the correlator, which is lower than the RF frequency 500 MHz and much higher than 1.6 MHz, the revolution frequency of the stored beam of the 2.5 GeV storage ring. This new apparatus has been operated successfully in the measurement of the horizontal two-photon correlation for the first harmonic of undulator radiation with photon energy of 70 eV at the Photon Factory, KEK. By narrowing the precise slit width which correspondingly changes the spatial coherence of the incident SR, a bunching effect of the normalized excess two-photon correlation has been clearly observed. This explicit bunching effect implies that synchrotron radiation is chaotic radiation. Further investigation shows that although second-order coherence is completely determined by the first-order coherence for the case of chaotic light, the measured information from the light source is essentially different. The two-photon correlation of synchrotron radiation does not depend on the response time of the detectors but gives the information of instantaneous emittance of the stored beam with the time scale of coherence time τC. By fitting the experimental data, the horizontal instantaneous emittance of the stored beam is estimated to be 40nmrad. This intensity interferometer can be utilized to characterize the coherence properties of incomplete FELs, such as SASE, because if they are fully coherent light sources the normalized excess two-photon correlation would have a flat response, but not showing a photon-bunching effect.
Non-photon-number-resolving single-photon-counting modules (SPCM's), also called on-off photon detectors, have been used in quantum optics for investigating the properties of various light fields based on the Hanbury-Brown-Twiss (HBT) experiment. However, for different incident light fields the experimental conditions can affect the measured photon statistics. In this paper, the second-order degree of coherence g(2), known as the important factor for quantitying a single-photon source, is experimentally investigated by means of the HBT scheme consisting of two SPCM's. By comparing the results of coherent field with that of the thermal field, we show that the measured g(2) is affected by the photon-counting rate and the resolution time from pulsed to continuous wave fields. The proper counting rate and resolution time for characterizing the exact photon statistical properties of input fields are determined.
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 Hanbury Brown–Twiss (HBT) interferometer, which is used to measure the correlation of two photons or two light beams, has played a crucial role in quantum optics and quantum information. We propose a generalized HBT interferometer, which is able to measure the correlation of photons or light beams up to eight in a single experimental run. With an improved time correlated single-photon counting (TCSPC) algorithm, two-, three-, and four-photon coincidence counts, and the degree of fifth-order coherence of pseudothermal light are measured with a data collection time of 60 s. The generalized HBT interferometer can be used to measure the correlation of two multi-photon detection events by simulating photon number resolving detectors with multiple single-photon detectors. The generalized HBT interferometer and improved TCSPC algorithm provide a versatile framework for studying multi-photon correlations and interference, which is helpful to understand the physics of multi-photon interference and may find applications in fluorescence lifetime imaging and other techniques using TCSPC.