Angle-resolved photon-coincidence measurements in a multiple-scattering medium

We present angle-resolved correlation measurements between photons after propagation through a three-dimensional disordered medium. The multiple-scattering process induces photon correlations that are directly measured for light sources with different photon statistics. We find that multiple-scattered photons between different angular directions with angles much larger than the average speckle width are strongly correlated. The time dependence of the angular photon correlation function is investigated, and the coherence time of the light source is determined. Our results are found to be in excellent agreement with the continuous mode quantum theory of multiple scattering of light. The presented experimental technique is essential in order to study quantum phenomena in multiple-scattering random media, such as quantum interference and quantum entanglement of photons.

Figure 1

(a) Sketch of the experimental setup. L: lens; G: rotating ground glass plate; I: iris; P: polarizer; F: 10-nm interference filter; $\Delta \Theta$: angle between detectors $D$${}_1$ and $D$${}_2$; $a$: incident light channel; and $b_0$, $b_1$: exit light channels. Optionally, a neutral density filter (ND) is inserted in front of detector $D$${}_1$, attenuating the light beam. The relative angle between $D$${}_2$ and the optical axis of the incident light beam is 4.5${}^{°}$. To probe different realizations of disorder, the sample is displaced. (b) Recorded photon counting distribution $P(\mathrm{n̂}(\Delta t))$ of the light source on detector $D$${}_1$ after the neutral density filter and without the sample using a time interval of $\Delta t=266\hspace{0.5em}\mu$s to bin the measured photons. The photon counts $\mathrm{n̂}(\Delta t)$ have been measured $N=11\hspace{0.5em}000$ times. (c) Same measurement as in (b), but using a time interval of $\Delta t=100\hspace{0.5em}\mu$s.

Figure 2

Classical intensity correlation function $\stackrel{̲}{C_{b_0{b}_1}^{(C)}(\Delta \Theta )}$ vs $\Delta \Theta$ for the speckle pattern shown in the inset. The full width at half maximum represents the average width of the speckles $\alpha$. The speckle pattern is obtained by rotating detector $D$${}_1$ behind the sample. $\Theta =0^{°}$ corresponds to the configuration where $D$${}_1$ is positioned on the optical axis of the incident light source. The uncertainty in $\stackrel{̲}{C_{b_0{b}_1}^{(C)}(\Delta \Theta )}$ does not depend on the angle and yields $\pm 0.4$.

Figure 3

Photon correlation function $\stackrel{̲}{C_{b_0{b}_1}^Q(0,\Delta t)}$ (gray circles) depending on $\langle \mathrm{n̂}(\Delta t)\rangle$ and $F(\Delta t)$ (top axis) of the light source that is obtained by varying the time interval $\Delta t$ while slowly rotating the ground glass plate [see Fig. 1a]. For a coherent state, $\stackrel{̲}{C_{b_0{b}_1}^Q(0,\Delta t)}$ is independent of $\langle \mathrm{n̂}(\Delta t)\rangle$ (black squares). The angle between the detectors is chosen to be $\Delta \Theta =9^{°}$, and the theoretical predictions contain no free fitting parameters (solid curves).

Figure 4

(a) Angular dependence of $\stackrel{̲}{C_{b_0{b}_1}^Q(0,\Delta t)}$ for a coherent light source (black squares) and a Gaussian radiation source (gray circles). Detector $D$${}_1$ is rotated behind the sample to change $\Delta \Theta$. The straight lines are the theoretical predictions assuming that the classical intensity correlations equal unity. (b) Measured temporal dependence of the angular photon correlation function for a Gaussian radiation source (gray circles) and a coherent light source (black squares). Each data point represents an individual measurement using $\Delta \Theta =9^{°}$ and $\Delta t=266\hspace{0.5em}\mu$s. For visibility, the uncertainties of the data points are not plotted, which are $\pm 0.03$ for the coherent light source and $\pm 0.07$ for the Gaussian radiation source. The gray curve is a fit to the data with the coherence time as a free parameter.