deletions | additions       diff --git a/When_a_laser_beam_of__1.tex b/When_a_laser_beam_of__1.tex  index 7ba2ba5..079bf4a 100644  --- a/When_a_laser_beam_of__1.tex  +++ b/When_a_laser_beam_of__1.tex 
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In this experiment, we used first a 4x8x8 cm$^3$ tissue-mimicking phantom made of 5\% polyvinyl alcohol, 0.1 \% 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 beam was emitted by a Nd:YAG laser (EverGreen 200, Quantel, Les Ulis, France), which produced a 200 mJ, 5 mm in diameter Q-switched pulse at a central wavelength of 532 nm during 10 ns. The We assume that the reflection of the  laser is negligible, as the medium was black. The laser was  absorbed in the medium, and medium with an exponential decay of  the optical intensity $I$decays exponentially  along medium depth $r$\cite{scruby1990laser}: \begin{equation}  I=I_0 \exp(- \gamma r)  \end{equation}  where $I_0$ is the incident intensity distribution at the surface and $\gamma$ is the absorption coefficient of the medium. In non-metallic solids, the absorption coefficient $\gamma$ is relatively small, so that the radiation is able to penetrate into the bulk of the material - contrary to 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 $\gamma  I$. Assuming that all the optical energy is converted to heat, a local increase of temperature appears. Temperature distribution $T(x,y,z,t)$ can be computed using heat equation:  \begin{equation}  \rho C  \frac{\partial T}{\partial t} = \frac{k}{\rho C} k  \nabla ^2 T + \frac{q}{\rho C} \gamma I  \end{equation}  where $c_k$ the thermal wave speed (usually taken as equal to the compression wave speed), $\rho$ the density, $\kappa$ the thermal diffusivity and $C$ the heat capacity. 
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