Statistics of Multiply Scattered Broadband Terahertz Pulses

We describe the first measurements of the diffusion of broadband single-cycle optical pulses through a highly scattering medium. Using terahertz time-domain spectroscopy, we measure the electric field of a multiply scattered wave with a time resolution shorter than one optical cycle. This time-domain measurement provides information on the statistics of both the amplitude and phase distributions of the diffusive wave. We develop a theoretical description, suitable for broadband radiation, which adequately describes the experimental results.

Figure 1

A schematic of the diffusion experiment. The detector is situated at 90° to the input beam direction, in order to avoid measuring ballistic terahertz photons.

Figure 2

Several typical terahertz waveforms measured using the setup shown in Fig. 1. Each waveform represents a unique configuration of the random medium. For reference, the upper curve shows the single-cycle pulse incident on the random medium. This curve has been scaled by a factor of 600 and is shown on a shifted time axis. The bottom curve shows a measurement with no scatterers in the sample cell, and is therefore an indication of the noise level. Only the first 200 psec of each waveform is shown, although the measurable signals extend beyond 600 psec.

Figure 3

The probability distribution of the normalized real (triangles) and imaginary (open squares) parts of the complex scattered electric field, $P(r/\sigma )$ and $P(i/\sigma )$, plotted on a log scale. The dashed line shows the Gaussian distribution, which is the result expected for monochromatic radiation [28]. The solid curve is the prediction of Eq. (3), using an experimentally determined estimated for the mean spectral intensity.

Figure 4

The probability distribution of the normalized spectral phase derivative ${\widehat{\varphi }}^{\prime }$, plotted on a log scale. The solid line is the probability distribution given in Eq. (4), equivalent to the monochromatic case [30], with $Q=0.234$.