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.
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.
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.
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 second-order autocorrelation function, g(2)(0) , measured in the Hanbury Brown–Twiss scheme is the strong indicator separating the thermal (incoherent) from the coherent state of light emitted by lasers or photon or polariton condensates. Consequently, g(2)(0) can be used to determine if the Bose–Einstein condensate forms in the system. The fast dynamics of non-equilibrium light–matter Bose–Einstein condensate make implementing the Hanbury Brown–Twiss scheme challenging and expensive. A good time resolution requires single-photon detectors operating at cryogenic temperatures. An alternative approach to measuring g(2)(0) , taken in many experiments, is to rely on the scheme with a single detector and measure the intensity fluctuations to directly obtain g(2)(0) . In this paper, we show that the Hanbury Brown–Twiss scheme and the single detector scheme are not equivalent in the general case because we cannot directly use the intensity fluctuations measured in the single detector scheme to obtain g(2)(0) of light emitted by a multi-mode system. Nevertheless, we develop a method allowing for the reconstruction of g(2)(0) from the measurements in the single detector scheme for a multi-mode system with fast thermalization and illustrate it for a special case of non-equilibrium light–matter Bose–Einstein condensate.
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.
We investigated the properties of various light fields based on the Hanbury-Brown-Twiss(HBT)experiment consisted by two single-photon-counting modules(SPCMs),one 50/50beam splitter and the Fast-Com P7888card.Having experimentally investigated the second-order degree of coherence g2(τ)for the continuous-wave and the pulse lasers and the single-photon source generated by single atom,we confirmed the experiment results by theoretical analysis.Specially,we have achieved a high quality of single-photon source based on single atom trapped in the magnetic-optical trap and the far-off-resonance trap with g(2)(τ=0)=0.09and g(2)(τ=0)=0.08,respectively.Using the pulse laser and the single atom,we can perform the triggered single photon experiment.
The researches on higher-order coherence and quantum statistics of light field are the important researching issues in quantum optics. In 1956, Hanbury-Brown and Twiss (HBT) (Hanbury-Brown R, Twiss R Q 1956 Nature 177 27) revolutionized optical coherence and demonstrated a new form of photon correlation. The landmark experiment has far-reaching influenced and even inspired the quantum theory of optical coherence that Glauber developed to account for the conclusive observation by HBT. Ever since then, the HBT effect has motivated extensive studies of higher-order coherence and quantum statistics in quantum optics, as well as in quantum information science and cryptography. Based on the HBT scheme, the degree of coherence and photon number distribution of light field can be derived from correlation measurement and photon counting technique. With the rapid development of the photoelectric detection technology, single-photon detection, which is the most sensitive and very widespread method of optical measurement, is used to characterize the natures of light sources and indicate their differences. More recently, HBT scheme combined with single-photon detection was used to study spatial interference, ghost imaging, azimuthal interference effect, deterministic manipulation and detection of single-photon source, etc. Due to broadband RF spectrum, noiselike feature, hypersensitivity to the initial conditions and long-term unpredictability, chaotic laser meets the essential requirements for information security and cryptography, and has been developed in many applications such as chaos-based secure communications and physical random number generation, as well as public-channel secure key distribution. But the research mainly focused on macroscopic dynamics of the chaotic laser. Moreover, the precision of measurement has reached a quantum level at present. Quantum statistcs of light field can also uncover profoundly the physical nature of the light. Thus, it is important to exploit the higher-order degree of coherence and photon statistics of chaotic field, which contribute to characterizing the field and distinguishing it from others. In this paper, photon number distribution and second-order degree of coherence of a chaotic laser are analyzed and measured based on HBT scheme. The chaotic laser is composed of a distributed feedback laser diode with optical feedback in fiber external cavity configuration. The bandwidth of the chaotic laser that we obtain experimentally is 6.7 GHz. The photon number distribution of chaotic laser is fitted by Gaussian random distribution, Possionian distribution and Bose-Einstein distribution. With the increase of the mean photon number, the photon number distribution changes from Bose-Einstein distribution into Poissonian distribution and always accords with Gaussian random distribution well. The second-order coherence g(2)(0) drops gradually from 2 to 1. By changing the bias current (I = 1.0Ith-2.0Ith) and feedback strength (010%), we compare and illustrate different chaotic dynamics and g(2)(0). From low frequency fluctuation to coherence collapse, the chaotic laser shows bunching effect and fully chaotic field can be obtained at the broadest bandwidth. Furthermore, the physical explanation for sub-chaotic or weakening of bunching effect is provided. It is concluded that this method can well reveal photon statistics of chaotic laser and will open up an avenue to the research of chaos with quantum optics, which merges two important fields of modern physics and is extremely helpful for the high-speed remote chaotic communication.