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The laser is absorbed in the medium with an exponential decay of the optical intensity $I(z)=I_0 \exp(- \gamma z)$ along medium depth $z$, where $I_0=\frac{1}{S}\frac{d E}{dt}$ is the incident intensity distribution at Let's describe the
surface (the reflection on the black mat material being neglected) and $\gamma$ the absorption coefficient phenomenon in a physical point of
the medium. 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 $\delta depth:
\begin{equation}
\delta = (\pi \sigma \mu_r \mu_0
\nu)^{\frac{1}{2}}$, \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
$\sigma$ = 1 $\sigma \approx$ 0.2 S.m$^{-1}$, $\mu_r \mu_0 \approx 4 \pi \times 10^{-7} H.m^{-1}$ and $\nu$ =
3 10$^8$ / 532
nm, 10$^{-9}$ = 5.6 10$^{14}$ Hz, the skin depth for our medium is about 47
$\mu$m. $\mu$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\'e bec, 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,
meaning that 63\% of the radiation is absorbed as illustrated in the
first 50 micrometers of the sample. 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 for maximum laser energy (200 mJ). 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
$u_z$ from the
surface: 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\lambda + 2\mu)}{(\lambda + 2\mu)} \frac{\alpha E \zeta}{\rho \frac{3 \alpha E}{\rho C
S \zeta}
\label{eq:deplThermo} S}
\end{equation}
As in a biological soft tissues, $\mu \ll \lambda$, the displacement $u_z$ can be approximated as $\frac{3 \alpha E}{\rho C S}$. 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
typical ultrasound elastography resolution (5--10 $\mu$m \cite{22545033}). 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.
If the laser beam surface, but calculus is
focused, beyond the
local increase scope of
temperature can vaporize a part of the surface of the medium. this article.
In both cases, absorption of If the laser
by beam is focused, the
phantom leads to a local
displacement which can propagate as elastic waves in the medium. In both cases, shear wave can occur because of the limited size increase of
the source. To observe the shear waves, the medium was scanned with temperature could also vaporize a
5 MHz ultrasonic probe made part of
128 elements connected to a Verasonics scanner (Verasonics V-1, Redmond, WA, USA). The probe was used in ultrafast mode \cite{bercoff2004supersonic}, acquiring 1500 ultrasound images per second. Due to the
presence surface of
graphite particles, the medium presented a speckle pattern on the ultrasound image. Tracking the
speckle spots with an optical flow technique (Lucas-Kanade method) allowed to compute one component of the displacement in the medium (Z-displacement or Y-displacement, depending on the position of the probe on the medium). The laser beam was triggered 10 ms after the first ultrasound acquisition, $t$ = 0 ms being defined as the laser emission. medium.
