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!!!We measured the fraction of light which go through different thicknesses of the medium with a laser beam power measurement device (): it indicated that $\gamma \approx$ ??? 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 $q = \gamma I$. Assuming that all the optical energy is converted to heat, a local increase of temperature appears. Temperature distribution $T$ in absence of melting can be computed using heat equation:
\begin{equation}
\frac{\partial T}{\partial t} = \kappa \nabla ^2 T + \frac{q}{\rho C}
%\label{eq:eqChaleur}
\end{equation}
where $\rho$ is the density, $C$ the heat capacity and $\kappa$ the thermal diffusivity. If melting temperature is reached, a part of the absorbed heat will melt the solid without increase of temperature. The thermal diffusion path, equal to $\sqrt{4\kappa \tau}$, with $\tau$ = 20 ns the laser emission duration and $\kappa$ = 1.43 10$^{-7}$ m$^2$.s$^{-1}$ for water \cite{Blumm_2003}, is approximately equal here to 0.1 $\mu$m. As $\gamma^{-1} \gg \sqrt{4\kappa t}$, propagation of heat is negligible during laser emission, and term $k \nabla ^2 T$ can be neglected during this time.
The local increase of temperature can lead to two main effects creating elastic waves: (1) thermoelastic expansion and (2) ablation of medium.
In the thermoelastic regime, a local dilatation of the medium occurs. In a unidimensional analysis along $z$ (as the depth of absorption is small compared to the beam diameter), the stress $\sigma_{zz}$ can be written \cite{scruby1990laser}:
\begin{equation}
\sigma_{zz} = (\lambda + \mu) \frac{\partial u_z}{\partial z} - (3\lambda + 2\mu) \frac{\alpha E}{\rho C S
\delta z} \zeta}
\label{eq:stressUnidim}
\end{equation}
where $\lambda$ and $\mu$ are respectively the first and second Lamé's coefficient, $\alpha$ is the thermal dilatation coefficient and
$\delta z$ $\zeta$ the average depth of laser beam absorption. In the absence of external constraints normal to the surface, the stress across the surface must be zero, i.e. $\sigma_{zz} (z=0) = 0$, so that equation \ref{eq:stressUnidim} can be
integrated: integrated, and there is a displacement $u_z$ from the surface:
\begin{equation}
u_z = \frac{(3\lambda + 2\mu)}{(\lambda + 2\mu)} \frac{\alpha E
\delta z}{\rho \zeta}{\rho C S
\delta z} \zeta} \approx 3 \alpha \frac{E}{\rho C S}
% \label{eq:deplUnidim}
\end{equation}
as in a biological soft tissues, $\mu \gg \lambda$. Taking as an order of magnitude $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient), $E$ = 200 mJ, $\rho$ = 1000 kg.m$^{-3}$ (water density), $C$ = 4180 kg.m$^{-3}$ (water calorific capacity) and $S$ = 20 mm$^2$, we obtain a displacement $u_z$= 0.5 $\mu$m. This value is still smaller than the typical displacement resolution with ultrasound, of a few micrometers. This unidimensional analysis cannot explain the induction of shear
waves, waves in a thermoelastic expansion, as
this displacement is curl-free. However, in a tridimensional model, the local expansion acts as dipolar forces parallel to the surface, so displacements along X and Y axis
can be higher. could also lead to shear waves.
In the ablative regime, the local increase of temperature is so high that the surface of the medium is vaporized. This phenomenon creates a stress $\sigma$ in the medium, given by \cite{scruby1990laser}:
\begin{equation}
...
\end{equation}
where $L$ is the latent heat required to vaporize the solid, $T_0$ and $T_V$ the initial and vaporization temperatures.
3
In both cases, absorption of the laser by the phantom leads to a local displacement which can propagate as elastic waves in the medium. To observe the 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.
...
