deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 6ac3026..2b19638 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. If melting temperature is reached, a part of the absorbed heat will be used to melt the solid without increase of temperature.  The thermal diffusion path, equal to $\sqrt{4\kappa t}$, with $t$ the laser emission duration (20 ns) and $\kappa$ = 1.43 10$^{-7}$ m$^2$.s$^{-1}$ for water \cite{Blumm_2003}, is equal to 0.1 $\mu$m. As $\gamma^{-1} \gg \sqrt{4\kappa t}$, propagation of heat is negligible during laser emission, so term $k \nabla ^2 T$ can be neglected in equation \ref{eq:eqChaleur}.Combination with equation \ref{eq:opticalIntensity} and integration over time allows to calculate the temperature $T$ at the end of the laser emission (discarding any potential melting effects).  The local increase of temperature can lead to two main effects creating elastic waves: (1) thermoelastic expansion and (2) ablation of medium. Inmetals, transition from first to second regime occurs approximately about 10$^7$ W.cm$^{-2}$. This is equal to  the energy thermoelastic expansion, a local dilatation  of the laser we used, medium occurs. In an unbounded solid, this would lead to a curl-free displacement,  so the predominant regime no shear wave would occur. However,  in our experiment cannot be determined yet. 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 dipolar forces parallel to the surface.  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 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 this case, the net stress $\sigma$ is given by Newton's second law of motion and is equal to \cite{scruby1990laser}:  \begin{equation}  \sigma = \frac{1}{\rho}\frac{I^2}{(L+C(T_V-T_0))^2}  \label{eq:ablation}  \end{equation}  where $L$ is the lastent heat required to vaporize the solid, $T_V$ and $T_0$ the initial and vaporization temperatures.  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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