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If enough energy is deposited, the local increase of temperature can also vaporize a part of Let's describe now the
surface of the medium \cite{scruby1990laser}. 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,
which leads to giving a maximum 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
be sufficient to vaporize the
medium; Moreover, medium, as it has been demonstrated that the presence of small particles like the graphite particles
in our medium acts as nucleation sites
for vaporization, which 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
can create generate then shear waves: this constitutes the \textit{ablative regime}.
To describe this
vaporization, regime, we suppose the medium as homogeneous and
isotropic. isotropic, and we adopt again a 1-D model. The stress $\sigma_{zz}$ is, similarly to the previous section, the sum of the
P-wave modulus axial strain component and a term given by the second law of motion
\cite{scruby1990laser}: representing the ejection of particles outside the medium when they each 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.
The By assuming again that $\mu \ll \lambda$, the stress leads to a displacement $u_z$:
\begin{equation}
u_z = \frac{\zeta}{\rho (\lambda + 2 \mu)}\frac{I^2}{(L+C(T_V-T_0))^2}
\label{eq:deplAbla}
\end{equation}
As in a biological soft tissues, $\mu \ll \lambda$, the displacement can be approximated as:
\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 $T_V-T_0$ =
373-293 373-298 =
80 75 K, we obtain a displacement $u_z$ approximately equal to
3.1 2.9 $\mu$m. This is quite close to the 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 $\mu$m and increasing linearly 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}) + \frac{\sin^2 \theta}{4\pi \rho c_s^2 r} f(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}
...
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
coherent to consistent with a vaporization of a fraction of the material. This was not observed in the low-energy experiments.
