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Let's describe now the other regime. Solving equation \ref{eq:eqChaleurApprox} with same experimental parameters as before but a laser energy of 200 mJ, we find a maximum increase of temperature of 60 K, giving a maximum
medium temperature of about 298+60
$\approx$ = 360 K. While slightly below to the vaporization point of our medium, supposed close to 373K (water vaporization temperature), it can be sufficient to vaporize the medium, as it has been demonstrated that the presence of small particles like the graphite particles acts as nucleation sites and facilitate thus the vaporization of the medium at lower temperature \cite{Alimpiev_1995}. Vaporization of the particles leads by reaction to displacements inside the medium, which generate then shear waves: this constitutes the \textit{ablative regime}.
To describe
physically this regime, we
suppose supposed the medium as homogeneous and isotropic, and we
adopt adopted again a 1-D model.
The stress $\sigma_{zz}$ is, similarly Similarly to the previous section,
we calculated the stress $\sigma_{zz}$, which is the sum of the axial strain component and a term given by the second law of motion
representing due to the reaction to the
ejection of particles
ejected outside the medium when they
each reach vaporization point \cite{ready1971effects}:
\begin{equation}
\sigma_{zz} = (\lambda + 2 \mu) \frac{\partial u_z}{\partial z} - \frac{1}{\rho}\frac{I^2}{(L+C(T_V-T_0))^2}
\label{eq:stressAbla}
\end{equation} where $L$ is the latent heat required to vaporize the solid, $T_0$ and $T_V$ the initial and vaporization temperatures.
By assuming again that $\mu \ll
\lambda$, the \lambda$ and a zero stress
at the medium surface, equation \ref{eq:stressAbla} leads to a displacement $u_z$:
\begin{equation}
u_z = \frac{\zeta}{\rho \lambda}\frac{I^2}{(L+C(T_V-T_0))^2}
\label{eq:deplAblaApprox}
\end{equation}
Using high-energy experimental parameter, $\zeta \approx \gamma^{-1} = 40 \mu$m (average depth of absorption), $\lambda$ = 2 GPa (first Lamé's coefficient of water), $L$ = 2.2 MJ.kg$^{-1}$ (vaporization latent heat of water) and $T_V-T_0$ = 373-298 = 75 K, we obtain a displacement $u_z$ approximately equal to 2.9 $\mu$m.
This While slightly higher, this value is
quite close to the in good agreement with experimental
measured displacement (2.5 $\mu$m).
To calculate the propagation of the displacement along space and time, we modeled the ablation regime as a point force directed along Z direction with a depth of
50 40 $\mu$m
during 100 $\mu$s and
increasing linearly of constant value from -2.5 to
0 mm and decreasing symmetrically from 0 to 2.5 mm. The magnitude of the force is stored in a matrix $H_z^{abla}(x,y,z,t)$. Propagation as a shear wave was calculated using Green operators $G_{zz}$ \cite{aki1980quantitative}:
\begin{equation}
G_{zz}(r,\theta,z,t) = \frac{\cos^2 \theta}{4\pi \rho c_p^2 r}
f(t-\frac{r}{c_p}) \delta(t-\frac{r}{c_p}) + \frac{\sin^2 \theta}{4\pi \rho c_s^2 r}
f(t-\frac{r}{c_s}) \delta(t-\frac{r}{c_s}) + \frac{3\cos^2 \theta-1}{4\pi \rho r^3} \int_{r/c_p}^{r/c_s}{\tau f_{NF}(t)dt}
\label{eq:Gzz}
\end{equation}
with same notations as in equation \ref{eq:Gyz}.
Displacement Displacements can again be computed by convoluting the applied force 4-D matrix
$H_z(x,y,z,t)$ $H_z^{abla}(x,y,z,t)$ with $G_{zz}$.
Using same values for the physical quantities as previously, results are illustrated in Figure \ref{figGreen}-(B) which represents displacement maps along Z axis, 1.0, 1.5, 2.0, 2.5 and 3.0 ms after force application. The displacement maps present many similarities with the experimental results of the Figure \ref{Figure2}: initial displacement is directed inside the medium and only two half cycles are propagating.
Ablation regime has also confirmed visually: at high power, a disk of paler color of the same size as the beam diameter appears at the impact location of the laser on the phantom, which is consistent with
a vaporization the theory of a
fraction partial vaporization of the
material. medium. This was not observed in the low-energy experiments.
