Time-Reversal Mirror inside a granular suspension: a way of measuring the ultrasound diffusion coefficient

We demonstrate that the diffusion coefficient, $D$, for ultrasound propagating in a multiple scattering medium, such as a dense granular suspension, can be measured using a time reversal experiment. This requires an unprecedented experimental setup in which a piezoelectric transducer, acting as a Time-Reversal Mirror (TRM), is embedded within the granular suspension at a depth much larger than the scattering mean free path, while an array of transducers is placed in the far field of the scattering sample. A single element of the array emits a short pulse and the TRM records the resulting ultrasonic field, which consists of a ballistic coherent wave followed by a coda wave. When the entire coda wave is time-reversed and re-emitted from the TRM, it is observed to refocus on the original source, with a focal spot size that decreases with the inverse depth of the TRM, characteristic of a diffusive regime. Time-reversal of short windows selected at different times $t$ in the coda wave reveals a focal spot size that decreases as the inverse square root of time, i.e., $1 / \sqrt{Dt}$. By fitting the predictions of a microscopic diffusion theory to our experimental data, we are able to accurately measure the diffusion coefficient in the granular suspension. Remarkably, this method does not require ensemble averaging due to stability of time-reversal against statistical fluctuations.