deletions | additions       diff --git a/When_a_laser_beam_of__1.tex b/When_a_laser_beam_of__1.tex  index bdc8567..7135cee 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 first 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 \% 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.  We assume assumed  that the reflection of the laser is negligible, as the on this black  medium was black, 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.In non-metallic solids, the absorption coefficient  $\gamma$ is relatively small, 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 g.L$^{-1}$, the order of magnitude of $\gamma$ is 1000 m$^{-1}$, meaning  that most of  the radiation is able to penetrate into absorbed in  the bulk first millimeter  of the material - contrary to 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}  \rho C \frac{\partial T}{\partial t} = k \nabla ^2 T + \gamma I  \end{equation}  where $\rho$ is the density, $\kappa$ the thermal diffusivity and $C$ the heat capacity.  The thermal diffusion path, equal to $\sqrt{4kt}$, where $k$ is ..., is equal to 0.01 $\mu$m.   The propagation of the heat is slow compared to the duration of the heating (10 ns) and the thermal expansion duration (), so that the phenomenon can be considered as adiabatic.  %$\kappa$ is approximately equal to 10${^6}$ m$^2$.s$^{-1}$ for water, $\sqrt{4\kappa t}$, with $t$  themain component of biological tissues; for a 10 ns  laser pulse, the thermal diffusion path emission duration,  isthen  equal to0.01 to  0.1 $\mu$m. $\gamma^{-1}$ As $\gamma^{-1} \gg \sqrt{4\kappa t}$, propagation  of water is equal to 0.1 m, which is a million times higher; even for melanin and haemoglobin, highly absorbing at 532 nm, $\gamma^{-1}$ heat  is respectively equal to 10 and 100 $\mu$m, far higher than the thermal diffusion path. The thermal conductivity effects are consequently negligible, and increase of temperature lies in negligible during  laser absorption zone. emission.  The local increase of temperature can lead to two main effects creating elastic waves: (1) thermoelastic expansion and (2) ablation of medium. 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. 
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