deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index a487ec7..abf9fe3 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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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. We supposelater on  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}  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. To observe the shear waves, the medium was scanned with a 5 MHz ultrasonic probe made of 128 elements linked 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.