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Pol Grasland-Mongrain added Let_s_describe_the_phenomenon__.tex
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Let's describe the phenomenon in a physical point of view.
The optical intensity $I_0$ of the laser beam is defined as $I_0=\frac{1}{S}\frac{d E}{dt}$, where $E$ is the beam energy and $S$ the beam surface. When the laser beam is emitted on the medium, it absorbed with an exponential decay along medium depth $z$: $I(z)=(1-R) I_0 \exp(- \gamma z)$, where $R$ is the reflection coefficient of the material (supposed negligible on a black mat material as the one used here) and $\gamma$ the absorption coefficient of the medium. The absorption coefficient $\gamma$ can be estimated by calculating the skin depth:
\begin{equation}
\delta = (\pi \sigma \mu_r \mu_0 \nu)^{-\frac{1}{2}}
\label{eq:skinDepth}
\end{equation}
where $\sigma$ is the electrical conductivity of the medium, $\mu_r \mu_0$ its permeability and $\nu$ the frequency of the radiation. Substituting $\sigma \approx$ 0.2 S.m$^{-1}$, $\mu_r \mu_0 \approx 4 \pi \times 10^{-7} H.m^{-1}$ and $\nu$ = 3 10$^8$ / 532 10$^{-9}$ = 5.6 10$^{14}$ Hz, the skin depth for our medium is about 47 $\mu$m: it means that about 63\% of the radiation is absorbed in the first 47 micrometers of the sample. We have validated experimentally this value by measuring the fraction of light which go through different thicknesses of the medium (respectively 0, 30, 50 and 100 $\mu$m) with a laser beam power measurement device (QE50LP-S-MB-D0 energy detector, Gentec, Qu\'ebec, QC, Canada). We found respective power of 100\%, 88\%, 71\% and 57\%. An exponential fit indicated that $\gamma^{-1} \approx$ 50 $\mu m$ in our sample, as illustrated in the little graph in Figure 1.
The absorption of the laser beam by the medium gives then rise to an absorbed optical energy $q = \gamma I$. Assuming that all the optical energy is converted to heat, a local increase of temperature appears. Temperature distribution $T$ in absence of convection and of phase transition can be computed using the heat equation:
\begin{equation}
\frac{\partial T}{\partial t} = \kappa \nabla ^2 T + \frac{q}{\rho C}
\label{eq:eqChaleur}
\end{equation}
where $\rho$ is the density, $C$ the heat capacity and $\kappa$ the thermal diffusivity. The thermal diffusion path, equal to $\sqrt{4\kappa \tau}$, with $\tau$ = 10 ns the laser emission duration and $\kappa$ = 1.43 10$^{-7}$ m$^2$.s$^{-1}$ for water \cite{Blumm_2003}, is approximately equal here to 80 nm. As $\gamma^{-1} \gg \sqrt{4\kappa t}$, propagation of heat is negligible during laser emission, so that equation \ref{eq:eqChaleur} can be simplified as $\frac{\partial T}{\partial t} = \frac{q}{\rho C}$. Substituting the experimental parameters lead to a maximum increase of temperature of 12 K.
This local increase of temperature can lead a local dilatation of the medium occurs. We suppose that the medium is homogeneous and isotropic, and as the depth of absorption is small compared to the beam diameter, we 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 E}{\rho C S \zeta}
\label{eq:stressThermo}
\end{equation}
where $\lambda + 2 \mu$ is the P-wave modulus and $\lambda + \frac{2}{3}\mu$ the bulk modulus with $\lambda$ and $\mu$ respectively the first and second Lamé's coefficient, $\alpha$ is the thermal dilatation coefficient and $\zeta$ the average depth of laser beam absorption. 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 from the surface $u_z = \frac{(3\lambda + 2\mu)}{(\lambda + 2\mu)} \frac{\alpha E \zeta}{\rho C S \zeta$. 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}
\end{equation}
Taking as an order of magnitude $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient), $E$ = 200 mJ, $\rho$ = 1000 kg.m$^{-3}$ (water density), $C$ = 4180 kg.m$^{-3}$ (water calorific capacity) and $S$ = 20 mm$^2$, we obtain a displacement $u_z$= 0.5 $\mu$m. This value is slightly smaller than the experimental displacement (about 3 $\mu$m). This local displacement can lead to shear waves because of the limited size of the source. In a 3D model, displacements along X and Y axis would also occurs, as the local expansion acts as dipolar forces parallel to the surface, but calculus is beyond the scope of this article.
If the laser beam is focused, the local increase of temperature could also vaporize a part of the surface of the medium.
