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The Z axis is defined here as the laser beam axis, and the ultrasound imaging plane is in the YZ plane, as illustrated by Figure \ref{Figure1}.
In this experiment, we used a laser beam emitted by a Nd:YAG laser (EverGreen 200, Quantel, Les Ulis, France), which produced a 200 mJ,
5 mm in diameter Q-switched pulse at a central wavelength of 532 nm during 10
ns. ns in a beam of section $S$=20 mm$^2$. The laser beam was absorbed in a 4x8x8 cm$^3$ tissue-mimicking phantom made of water and of 5\% polyvinyl alcohol, 1 \% black graphite powder and 1\% salt. A freezing/thawing cycle was applied to stiffen the material to a value of 15$\pm$5 kPa \cite{17375819}.
We assumed that the reflection of the laser on this black medium was negligible,% and we also suppose a one-dimensional heating - radial heat flow is consequently neglected, as the laser size (5 mm in diameter) is quite large compared to the characteristic heat flow distance () or the thermal diffusion length in the material (a few tens of micrometers).
The laser was absorbed in the medium with an exponential decay of the optical intensity $I$ along medium depth $r$\cite{scruby1990laser}:
\begin{equation}
I=I_0 I=(1-R)I_0 \exp(- \gamma r)
\label{eq:opticalIntensity}
\end{equation}
where
$I_0$ $R$ is the
reflexion coefficient of the medium (typically less than a few pourcents for a black mat medium such as the one used here, so can be neglected), $I_0=\frac{d E}{Sdt}$ the incident intensity distribution at the surface and $\gamma$
is the absorption coefficient of the medium.
%A measurement of the fraction of light which go through different thicknesses of the medium, cut by a microtome, indicates that $\gamma \approx$ 10$^4$ m$^{-1}$ in our sample, meaning that most of the radiation is absorbed in the first hundred of micrometers.
%Even if the sample is mainlyeFor low concentration medium, $\gamma$ is hard to calculate in our case, as the sample is composed of different materials, but the graphite, even in low concentration, absorbate much more than other components, so we can approximate $\gamma \approx \gamma_{graphite}$. For graphite particles of 1.85 $\mu$m at a concentration of 10 g.L$^{-1}$, the order of magnitude of $\gamma$ is 10$^4$ m$^{-1}$, meaning that most of the radiation is absorbed in the first hundred of micrometers of sample.
This is quite higher than metals where the radiation is absorbed within a few nanometres.
The absorption of the laser beam by the medium gives then rise to an absorbed optical energy $\gamma I$.
The absorption of the laser beam by the medium gives then rise to an absorbed optical energy $q = \gamma I$. Assuming that all the optical energy is converted to heat, a local increase of temperature appears. Temperature distribution $T$ can be computed using heat equation:
\begin{equation}
\rho C \frac{\partial T}{\partial t} = k \nabla ^2 T +
\gamma I q
\label{eq:eqChaleur}
\end{equation}
where $\rho$ is the density, $\kappa$ the thermal diffusivity and $C$ the heat capacity.
The thermal diffusion path, equal to $\sqrt{4\kappa t}$, with $t$ the laser emission duration, is equal to 0.1 $\mu$m. As $\gamma^{-1} \gg \sqrt{4\kappa t}$, propagation of heat is negligible during laser
emission. emission, and term $k \nabla ^2 T$ can be neglected in equation \ref{eq:eqChaleur}. Combination with equation \ref{eq:opticalIntensity} and integration over time lead then to a temperature $T$ at the end of the laser emission:
\begin{equation}
T = T_0 + \frac{\gamma}{S \rho C}E \exp(-\gamma r)
\ref{eq:eqTemperature}
\end{equation}
The local increase of temperature can lead to two main effects creating elastic waves: (1) thermoelastic expansion and (2) ablation of medium. In metals, transition from first to second regime occurs approximately about 10$^7$ W.cm$^{-2}$. This is equal to the energy of the laser we used, so the predominant regime in our experiment cannot be determined yet.
In the thermoelastic expansion, a local dilatation of the medium occurs. In an unbounded solid, this would lead to a curl-free displacement, so no shear wave would occur. However, in the case presented, the solid is semi-infinite (the laser beam is absorbed on one side of the medium), and the local expansion acts as
a dipole force dipolar forces parallel to the surface. In the ablative regime, the local increase of temperature is so high that the surface of the medium melts and creates a point-force in the medium.
In both cases,
the absorption of the laser by the phantom leads to a local displacement which can propagate as elastic waves in the medium. To observe the
elastic shear waves, the medium was scanned with a 5 MHz ultrasonic probe made of 128 elements linked to a Verasonics scanner (Verasonics V-1, Redmond, WA, USA). The probe was used in ultrafast mode \cite{bercoff2004supersonic}, acquiring 1500 ultrasound images per second. Due to the presence of graphite particles, the medium presented a speckle pattern on the ultrasound image. Tracking the speckle spots with an optical flow technique (Lucas-Kanade method) allowed to compute one component of the displacement in the medium (Z-displacement or Y-displacement, depending on the position of the probe on the medium). The laser beam was triggered 10 ms after the first ultrasound acquisition, $t=0$ms being defined as the laser emission.
