deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index b8a6104..6922b6e 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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In this study, we show that shear waves can be induced in soft tissues by a laser beam. We also propose a model for the underlying physics. We finally applied the technique in a biological tissue to evaluate its potential application in shear wave elastography.  In the first experiment, illustrated by Figure \ref{Figure1}, we used a Q-switch Nd:YAG laser (EverGreen 200, Quantel, Les Ulis, France), which produced a pulse of energy $E$ = 200 mJ at a central wavelength of 532 nm during 20 ns in a beam of section $S$=20 mm$^2$. We defined Z as the laser beam axis, and the laser beam impact on the medium is the origin of coordinates (0,0,0). The laser beam was absorbed in a 4x8x8 cm$^3$ tissue-mimicking phantom made of water and of 5\% polyvinyl alcohol, 1 \% black graphite powder and 1\% salt. A freezing/thawing cycle was applied to stiffen the material to a value of 15$\pm$5 kPa \cite{17375819}.  The laser is absorbed in the medium with an exponential decay of the optical intensity $I(z)=I_0 \exp(- \gamma z)$ along medium depth $z$, where $I_0=\frac{1}{S}\frac{d E}{dt}$ is the incident intensity distribution at the surface (the reflection on the black mat medium material  being neglected) and $\gamma$ the absorption coefficient of the medium. !!!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 convection and of phase transition can be computed using the  heat equation: \begin{equation}  \frac{\partial T}{\partial t} = \kappa \nabla ^2 T + \frac{q}{\rho C}  \label{eq:eqChaleur} 
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