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In the thermoelastic regime, a local dilatation of the medium occurs. We suppose later on that the medium is homogenous and isotropic, and we adopt a 1D model, as the depth of absorption is small compared to the beam diameter. The stress $\sigma_{zz}$ is the sum between the axial strain component and the thermal expansion component \cite{scruby1990laser}:
\begin{equation}
\sigma_{zz} = (\lambda + 2 \mu) \frac{\partial u_z}{\partial z} - 3(\lambda + \frac{2}{3}\mu) \frac{\alpha E}{\rho C S \zeta}
\label{eq:stressUnidim} \label{eq:stressThermo}
\end{equation}
where $\lambda + 2 \mu$ is the P-wave modulus and $\lambda + \frac{2}{3}\mu$ the bulk modulus, with $\lambda$ and $\mu$ respectively the first and second Lamé's coefficient, $\alpha$ is the thermal dilatation coefficient and $\zeta$ 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, giving a displacement $u_z$ from the surface:
\begin{equation}
u_z = \frac{(3\lambda + 2\mu)}{(\lambda + 2\mu)} \frac{\alpha E \zeta}{\rho C S \zeta}
\label{eq:deplUnidim} \label{eq:deplThermo}
\end{equation}
As in a biological soft tissues, $\mu \gg \lambda$, the displacement $u_z$ can be approximated as
$3 $\frac{3 \alpha
\frac{E}{\rho E}{\rho C S}$. 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 slightly smaller than the typical ultrasound resolution of displacement, typically of a few micrometers. In a tridimensional model, displacements along X and Y axis would also occurs, as the local expansion acts as dipolar forces parallel to the surface.
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
$\sigma$ $\sigma_{zz}$ in the medium,
as the sum of the P-wave modulus and a term given by
the second law of motion \cite{scruby1990laser}:
\begin{equation}
\sigma \sigma_{zz} =
(\lambda + 2 \mu) \frac{\partial u_z}{\partial z} - \frac{1}{\rho}\frac{I^2}{(L+C(T_V-T_0))^2}
%\label{eq:ablation} \label{eq:stressAbla}
\end{equation}
where $L$ is the latent heat required to vaporize the solid, $T_0$ and $T_V$ the initial and vaporization temperatures.
Similarly to the thermoelastic regime, this leads to a displacement $u_z$:
\begin{equation}
u_z = \frac{\zeta}{\rho (\lambda + 2 \mu)}\frac{I^2}{(L+C(T_V-T_0))^2} \approx \frac{\zeta}{\rho \lambda}\frac{E^2}{S^2 \tau^2 (L+C(T_V-T_0))^2}
\label{eq:deplAbla}
\end{equation}
Estimating $\zeta$ equal to 100 $\mu$m (average depth of absorption), $\lambda$ = 2 GPa (first Lamé's coefficient of water), $L$ = 2.2 MJ.kg$^{-1}$ (vaporization latent heat of water), $C$ = 4180 J.kg$^{-1}$.K$^{-1}$ (water heat capacity), $T_V-T_0$ = 373-293 = 80 K (water vaporization temperature minus laboratory temperature), $\rho$ = 1000 kg.m$^{-3}$ (water density), $E$ = 200 mJ, $S$ = 20 mm$^2$, $\tau$ = 20 ns, we obtain a displacement $u_z$ approximately equal to 2.0 $\mu$m. This value is four times higher than the one obtained with thermoelastic expansion.
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.
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