Time-reversal inside a granular suspension to probe ultrasound diffusion

We demonstrate that ultrasound diffusion—typically associated with the transport of average wave energy and the breaking of time-reversal symmetry—can nonetheless be revealed through a time-reversal experiment. This is achieved using an unprecedented configuration: A single piezoelectric transducer, acting as a time-reversal mirror (TRM), is buried deep inside a strongly scattering medium (a dense granular suspension), while an array of transducers is positioned at a distance, outside the scattering region. A short pulse is emitted by a single array element and the TRM records the resulting ultrasonic field, composed of a coherent ballistic wave followed by a diffuse coda wave. When the entire coda is time-reversed and re-emitted from the TRM, the wave refocuses at the original source with a focal spot size that decreases with the inverse of the TRM depth, consistent with diffusive transport. By time-reversing short coda segments at increasing times <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>t</a:mi></a:math>, we observe a focal spot size scaling as <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mn>1</b:mn><b:mo>/</b:mo><b:msqrt><b:mrow><b:mi>D</b:mi><b:mi>t</b:mi></b:mrow></b:msqrt></b:mrow></b:math>, where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>D</c:mi></c:math> is the ultrasound diffusion coefficient. Fitting this evolution with a microscopic diffusion model allows us to extract <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>D</d:mi></d:math>. Remarkably, this measurement does not require ensemble averaging, because of the inherent stability of time-reversal against statistical fluctuations.