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diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex
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+++ b/The_absorption_of_the_laser__.tex
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Substituting low-energy experimental parameters ($\gamma^{-1} \approx$ 40 $\mu$m$^{-1}$, $S$ = 20 mm$^{2}$, $E$ = 10 mJ, $\rho$ = 1000 kg.m$^{-3}$, $C$ = 4180 J.kg$^{-1}$.K$^{-1}$) lead to a maximum increase of temperature of 3 K. This local increase of temperature gives rise to a local dilatation of the medium. The induced displacements can then lead to shear waves: this constitutes the \textit{thermoelastic regime}.
To describe this
dilatation, regime, we
take a homogeneous and isotropic suppose the medium
and as
homogeneous and isotropic; As the depth of absorption (about 50 $\mu$m) is hundred times smaller than the beam diameter (5 mm), we
have adopted also 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 \alpha \frac{ E}{\rho C S \zeta}
\label{eq:stressThermo}
\end{equation}
where $\lambda$ and $\mu$
are respectively the first and second Lamé's coefficient, $\alpha$ the thermal dilatation coefficient and $\zeta$ the
average depth
over which there is a temperature rise. In of absorption. This equation can be simplified by remarking that in most soft media, including biological tissues, $\mu \ll \lambda$. Moreover, in the absence of external constraints normal to the surface, the stress across the surface must be zero, i.e. $\sigma_{zz} (z=0) = 0$, so that equation \ref{eq:stressThermo} can be integrated, giving a displacement $u_z$ from the surface:
\begin{equation}
u_z = \frac{(3\lambda + 2\mu)}{(\lambda + 2\mu)} \frac{\alpha E \zeta}{\rho C S \zeta}
\label{eq:deplThermo}
\end{equation}
As in a biological soft tissues, $\mu \ll \lambda$, the displacement can be approximated as:
\begin{equation}
u_z = \frac{3 \alpha E}{\rho C S}
\label{eq:deplThermoApprox}
\end{equation}
Substituting same experimental parameters as previously and $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient), we obtain a displacement $u_z$= 0.025 $\mu$m. This value is slightly higher than the experimental displacement (about 0.02 $\mu$m). Note that as shown theoretically, the displacement is directed outside the medium, which is confirmed in the experimental images.
To calculate the propagation of the displacement along space and time, we have to take into account the dilatation along X and Y axis which lead to stronger displacements than along Z. We modeled thus the thermoelastic regime as two opposite forces during 100 $\mu$s directed along Y axis with a depth of 50 $\mu$m and decreasing linearly from 2.5 to 0 mm (respectively -2.5 to 0 mm) \cite{Davies_1993}.
The magnitude of the force along space and time is stored in a matrix $H_y^{thermo}(x,y,z,t)$. Propagation as a shear wave along Z axis was calculated using Green operators $G_{yz}$ as calculated by Aki Richards \cite{aki1980quantitative}:
\begin{equation}
G_{yz} (r,\theta,z,t)= \frac{\cos \theta \sin \theta}{4\pi \rho c_p^2 r}
f(t-\frac{r}{c_p}) \delta(t-\frac{r}{c_p}) - \frac{\sin \theta \cos \theta}{4\pi \rho c_s^2 r}
f(t-\frac{r}{c_s}) \delta(t-\frac{r}{c_s}) + \frac{3\cos \theta \sin \theta}{4\pi \rho r^3} \int_{r/c_p}^{r/c_s}{\tau f_{NF}(t) dt}
\label{eq:Gyz}
\end{equation}
where $\theta$ is the angle between the applied force and the considered point (r,$\theta$,z), $c_p$ and $c_s$ the compression and shear wave speed respectively, $\tau$ the
time time, $\delta$ a Dirac distribution and $f_{NF}(t)$ representing near-field effects. The three terms correspond respectively to the far-field compression wave, the far-field shear wave and the near-field component.
Displacements along Z are then equal to the convolution between
the time and spatial extent of the force $H_y $H_y^{thermo} (x,y,z,t)$ with $G_{yz}$.
Using a medium density $\rho$ of 1000 kg.m$^{-3}$, a compression wave speed of 1500 m.s$^{-1}$ and a shear wave speed of 5.5 m.s$^{-1}$, we calculated the displacements along space and time. Figure \ref{figGreen}-(A) represents resulting displacement maps along Z axis, 1.0, 1.5, 2.0, 2.5 and 3.0 ms after force application. The normalized displacement maps present many similarities with the low-energy experimental results of the Figure \ref{Figure2}, with a initial central displacement directed outside the medium and the propagation of three half cycles.
