deletions | additions       diff --git a/When_a_laser_beam_of__1.tex b/When_a_laser_beam_of__1.tex  index 7135cee..c46f929 100644  --- a/When_a_laser_beam_of__1.tex  +++ b/When_a_laser_beam_of__1.tex 
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The Z axis is defined here as the laser beam axis, and the ultrasound imaging plane is in the YZ plane, as illustrated by Figure \ref{Figure1}.  In this experiment, we used a laser beam 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 laser beam was absorbed in a 4x8x8 cm$^3$ tissue-mimicking phantom made of water and of 5\% polyvinyl alcohol, 0.1 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}. We assumed that the reflection of the laser on this black medium was negligible,% and we also suppose a one-dimensional heating - radial heat flow is consequently neglected, as the laser size (5 mm in diameter) is quite large compared to the characteristic heat flow distance () or the thermal diffusion length in the material (a few tens of micrometers). The laser was absorbed in the medium with an exponential decay of the optical intensity $I$ 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. $\gamma$ 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 1 10  g.L$^{-1}$, the order of magnitude of $\gamma$ is 1000 10$^4$  m$^{-1}$, meaning that most of the radiation is absorbed in the first millimeter 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 $\gamma I$. Assuming that all the optical energy is converted to heat, a local increase of temperature appears. Temperature distribution $T$ can be computed using heat equation:  \begin{equation} 
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