deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 29d90d3..2155345 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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\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. The thermal diffusion path, equal to $\sqrt{4\kappa \tau}$, with $\tau$ = 10 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 80 nm. As $\gamma^{-1} \gg \sqrt{4\kappa t}$, propagation of heat is negligible during laser emission, so that equation \ref{eq:eqChaleur} can be simplified as $\frac{\partial T}{\partial t} = \frac{q}{\rho C}$. Substituting the experimental parameters lead to a maximum increase of temperature of 12 K for maximum laser energy (200 mJ).  The This  local increase of temperature can leadto 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. We suppose that the medium is homogeneous and isotropic, and as the depth of absorption is small compared to the beam diameter, we adopt a 1D model. 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:stressThermo} 
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\end{equation}  As in a biological soft tissues, $\mu \ll \lambda$, the displacement $u_z$ can be approximated as $\frac{3 \alpha E}{\rho C S}$. 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 elastography resolution (5--10 $\mu$m \cite{22545033}). In a 3D model, displacements along X and Y axis would also occurs, as the local expansion acts as dipolar forces parallel to the surface.  At high If the  laser energy, beam is focused,  the local increase of temperature can vaporize a part of the surface of the medium.This phenomenon creates a stress $\sigma_{zz}$ 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_{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: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}  \label{eq:deplAbla}  \end{equation}  As in a biological soft tissues, $\mu \ll \lambda$, the displacement $u_z$ can be approximated as $\frac{\zeta}{\rho \lambda}\frac{E^2/S^2 \tau^2}{(L+C(T_V-T_0))^2}$. Estimating $\zeta$ equal to 50 $\mu$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$^2$ and $\tau$ = 10 ns, we obtain a displacement $u_z$ approximately equal to 3.9 $\mu$m. This value is eight 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. In both cases, shear wave can occur because of the limited size of the source. To observe the shear waves, the medium was scanned with a 5 MHz ultrasonic probe made of 128 elements connected 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.