Characterizing Individual Scattering Events by Measuring the Amplitude and Phase of the Electric Field Diffusing through a Random Medium

Zhongping Jian, Jeremy Pearce, and Daniel M. Mittleman

Phys. Rev. Lett. 91, 033903 – Published 18 July 2003

Abstract

We describe observations of the amplitude and phase of an electric field diffusing through a three-dimensional random medium, using terahertz time-domain spectroscopy. These measurements are spatially resolved with a resolution smaller than the speckle spot size and temporally resolved with a resolution better than one optical cycle. By computing correlation functions between fields measured at different positions and with different temporal delays, it is possible to obtain information about individual scattering events experienced by the diffusing field. This represents a new method for characterizing a multiply scattered wave.

Received 24 March 2003

DOI:https://doi.org/10.1103/PhysRevLett.91.033903

Authors & Affiliations

Zhongping Jian, Jeremy Pearce, and Daniel M. Mittleman

Department of Electrical and Computer Engineering, Rice University, MS 366, Houston, Texas 77251-1892, USA

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Vol. 91, Iss. 3 — 18 July 2003

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Images

Figure 1
Schematic of the experimental setup, showing the fiber-coupled THz detector in two different positions (labeled $θ_{1}$ and $θ_{2}$ ). A spherical wave produced by a hypothetical scattering event, centered to the right of the axes of both of the displayed detector locations, is also shown. This partial spherical wave produces a correlated signal at the two detectors, with a temporal shift resulting from the geometrical path length difference. The detector at $θ_{2}$ receives the signal earlier.Reuse & Permissions
Figure 2
(a) A typical input waveform. (b) A typical multiply scattered waveform. These data have been spectrally filtered to remove noise at both low and high frequencies. In these data, the zero of the time axis is determined by the length of the optical delay line, and is therefore arbitrary with respect to the arrival of the incident pulse at the input facet of the random medium. However, this arbitrary $t = 0$ is fixed for all measurements. (c) Photon time-of-flight distribution, obtained by averaging the squares of all measured waveforms. This is plotted on the same delay axis as in (b).Reuse & Permissions
Figure 3
Correlation function $C_{δ θ} (τ, T)$ , plotted versus the correlation offset $τ$ and the delay of the temporal window $T$ . The $T$ axes (vertical axes) are the same as the time axes in Figs. 2b, 2c. These data represent an ensemble average of all pairs of measured fields with the indicated angular separation for one particular realization of the random configuration of scatterers. In this example, an extended oscillatory correlation is observed at a window delay of $T \sim 730 p s$ . This correlation persists even for large angular separations. The dashed lines indicate cuts through these data sets displayed in Fig. 4.Reuse & Permissions
Figure 4
Correlation functions $C_{δ θ} (τ, T)$ as a function of the cross-correlation offset at two delays corresponding to the two dashed lines in Fig. 3a. The upper curve has a pronounced oscillatory correlation which is absent at other delays. The lower curve has been shifted downwards by 0.5 for clarity. The inset shows the evolution of a portion of the oscillatory correlation with increasing angular separation (corresponding to the three dashed lines at $T = 730 p s$ in Fig. 3; dashed line: $δ θ = 2 °$ , solid line: $δ θ = 5 °$ , dotted line: $δ θ = 8 °$ ). As $δ θ$ increases, the correlation signal shifts towards negative offsets. This shift is related to the tilt of the correlated wave front emerging from the random medium as illustrated in Fig. 1.Reuse & Permissions
Figure 5
Histograms of the correlation phase shift, for all of the observed scattering events in measurements of four distinct random configurations. These are displayed for both the cases of forward ( $θ = 65 °, \dots, 75 °$ ) and backward ( $θ = 105 °, \dots, 115 °$ ) scattering. The forward scattering events exhibit an asymmetric correlation phase, indicating that the scattering events are more likely to originate from the side of the detector axis closer to the input beam. In contrast, the backward scattering events are nearly symmetric.Reuse & Permissions

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