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Physical Review

Photostatistics of Light Scattered by a Liquid

M. Bertolotti, B. Crosignani, Paolo Di Porto, D. Sette

Published May 5, 1967• 9 citations
Publisher

Abstract

The structure of a liquid can be investigated through the photostatistics of scattered light. A theoretical approach is given in which the deviation from the Poisson distribution, followed by the photons of a single-mode laser beam, is connected with the two- and four-particle distribution functions of the scattering medium. In particular, the presence of the Mandelstam-Brillouin doublet is put into evidence through the time behavior of a non-Poissonian statistical term, by using an explicit form of the two-particle correlation function.

DOI
10.1103/physrev.157.146

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The structure of a liquid can be investigated through the photostatistics of scattered light. A theoretical approach is given in which the deviation from the Poisson distribution, followed by the photons of a single-mode laser beam, is connected with the two- and four-particle distribution functions of the scattering medium. In particular, the presence of the Mandelstam-Brillouin doublet is put into evidence through the time behavior of a non-Poissonian statistical term, by using an explicit form of the two-particle correlation function.
Microscopic scattering processes are discussed for two structures of light incident on thermal particles. One is composed of two collinear light beams originating from different oscillating modes of a laser, and another is composed of two counter propagating beams originating from the same oscillating mode. A comparison of the cumulant expansions for the above two cases has allowed discerning which terms in the scattering structures led to the same statistical properties of the scattered photons. It is shown that, for the scattering of multimode-laser light, the statistical properties of scattered photons are not affected by the noncorrelating random phases of the incident laser fields but are the same as those in the case of single-mode laser light under practical scattering experiments. The factorial moments ${F}_{k}[=〈n(n\ensuremath{-}1)\ensuremath{\cdots}(n\ensuremath{-}k+1)〉]$ up to sixth order of the scattered photons were measured for various scattering situations including the above two cases, and they were found to support the theoretical results.
Brillouin–Mandelstam scattering is one of the most accessible nonlinear optical phenomena and has been widely studied since its theoretical discovery one hundred years ago. The scattering mechanism is a three-wave-mixing process between two optical fields and one acoustic field and has found a broad range of applications spanning microscopy to ultra-narrow-linewidth lasers. Building on the success of utilizing this nonlinearity at a classical level, a rich avenue is now being opened to explore Brillouin scattering within the paradigm of quantum optics. Here, we take a key step in this direction by employing quantum optical techniques yet to be utilized for Brillouin scattering to characterize the second-order coherence of Stokes scattering across the Brillouin lasing threshold. We use a silica microsphere resonator and single-photon counters to observe the expected transition from bunched statistics of thermal light below the lasing threshold to Poissonian statistics of coherent light above the threshold. Notably, at powers approaching the lasing threshold, we also observe super-thermal statistics, which arise due to instability and a “flickering” in and out of lasing as the pump field is transiently depleted. The statistics observed across the transition, including the “flickering,” are a result of the full nonlinear three-wave-mixing process and cannot be captured by a linearized model. These measurements are in good agreement with numerical solutions of the three-wave Langevin equations and are well demarcated by analytical expressions for the instability and the lasing thresholds. These results demonstrate that applying second-order-coherence and photon-counting measurements to Brillouin scattering provides new methods to advance our understanding of Brillouin scattering itself and progress toward quantum-state preparation and characterization of acoustic modes.
We analyze the quantum statistical properties of light scattered by an acousto-optic modulator. From the perspective of quantum mechanics, the acousto-optic effect arises from a nonlinear interaction between three bosonic fields: the laser light, the Brillouin-scattered light, and an acoustic wave. The Hamiltonian coincides with that of sum-frequency generation, where the sum-frequency mode is known to exhibit sub-Poissonian statistics. We calculate the free temporal evolution of the Fano factor for Brillouin backscattered light and determine that sub-Poissonian statistics are obtained when faint laser light is scattered by a counter-propagating coherent acoustic wave with low average phonon number. The use of sub-Poissonian light would lead to an increase of the security for quantum key distribution.
The quantum statistical properties of independent light beams are considered, forming interference fringes, which are in interaction with a particle (electron) beam treated as a two-level system. Using the master equation and the generalized Fokker-Planck equation techniques, the photocount statistics are derived and it is demonstrated that the whole, initially coherent, radiation field remains coherent for all times while single modes exhibit two-peak bistable behaviour, varying their photon statistics from Poisson statistics to some intermediate statistics for coherent and chaotic radiation.
If a beam of light is coherent, then the photons constituting the beam arrive at random moments in time. Consequently, the photon counting statistics in a given time interval below the coherence time, should in theory follow a Poissonian distribution. When the measurement interval exceeds the coherence time, the distribution should broaden and become a super-Poissonian distribution. The aim of this study is to experimentally demonstrate the Poissonian nature of light, as well as quantifying the transition to a super-Poissonian distribution. In the experiment, a low-intensity laser beam was measured with single-photon detectors, able to capture individual photons. The arrival time of the photons was detected with an oscilloscope, and the data further analysed with our own Python code. The probability distribution of the arrival time of photons was constructed and fitted with a Poissonian distribution for different time intervals. The results showed a clear Poissonian distribution for all intervals from 0.2 μs up to 2000 μs. We did however, observe tendencies of randomness for the longest measured intervals. A Poissonian distribution is defined by the variance being equal to the mean, whilst for a super-Poissonian distribution the variance is higher. The biggest measured ratio between the variance and the mean in our experiment was 1.06 for interval time 2000 μs, which is too small to infer super-Poissonian. Due to limited data memory, our dataset was restricted in length and resolution. Therefore, longer intervals could not be studied without statistical uncertainty. To cross-check the result we also calculated the probability distribution for the arrival time itself. The result perfectly fitted an exponential distribution which was in accordance with the theory of coherent light. Hence, within the given measurement accuracy and statistical uncertainty, it is safe to conclude that the laser beam is coherent for all tested time intervals. However, this does not allow us to conclude that it is a single-mode laser.