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The local increase of temperature can lead to two main effects creating elastic waves: (1) thermoelastic expansion and (2) ablation of medium.
In
metals, transition from first to second regime occurs approximately about 10$^7$ W.cm$^{-2}$. This is equal to the energy of the laser we used, so the predominant regime in our experiment cannot be determined yet.
In the thermoelastic regime, a local dilatation of the medium occurs. In
an unbounded solid, this would lead to a
curl-free displacement, so no shear wave would occur. However, in the case presented, unidimensional analysis along $z$ (as the
solid depth of absorption is
semi-infinite (the laser small compared to the beam
diameter), the stress $\sigma_{zz}$ can be written \cite{scruby1990laser}:
\begin{equation}
\sigma_{zz} = (\lambda + \mu) \frac{\partial u_z}{\partial z} - 3 (\lambda + \frac{2}{3}\mu) \frac{\alpha E}{\rho C S \delta}
\label{eq:stressUnidim}
\end{equation}
where $\lambda$ and $\mu$ are respectively the first and second Lamé's coefficient, $\alpha$ is
absorbed on one side of the
medium), thermal dilatation coefficient and
$\delta$ the
local expansion acts as dipolar forces parallel to average depth of laser beam absorption. In the
surface. A unidimension analysis lead absence of external constraints normal to
a local displacement $u_z$ along $z$ \cite{scruby1990laser}: the surface, the stress across the surface must be zero, i.e. $\sigma_{zz} (z=0) = 0$, so that equation \ref{eq:stressUnidim} can be integrated:
\begin{equation}
u_z =
\frac{(3 \lambda \frac{3(\lambda +
2 \frac{2}{3} \mu)}{(\lambda + 2\mu)} \frac{\alpha
E \delta}{\rho C S \delta} \approx 3 \alpha E}{\rho C S}
\label{eq:deplUnidim}
\end{equation}
where $\lambda$, $\mu$ are respectively the first and second Lamé's coefficient and $\alpha$ is the thermal dilatation coefficient. In as in a biological soft tissues, $\mu \gg
\lambda$, so that \ref{eq:deplUnidim} becomes $u_z = 3\alpha E/(\rho C S)$. \lambda$. Taking as an order of magnitude $\alpha$ =
69.10$^{-6}$ 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 still smaller than the typical displacement resolution with ultrasound, of a few micrometers.
This unidimensional analysis cannot explain the induction of shear waves, as displacement is curl-free. However,
in a tridimensional model, the local expansion acts as dipolar forces parallel to the surface, so displacements along X and Y axis
are higher, as $\delta V = u_x u_y u_z = \frac{3\alpha E}{\rho C}$ where displacement along $z$ is known, so that displacement along $x$ (or, equivalently, along $z$) is equal to ??? $\mu$m. can be higher.
In the ablative regime, the local increase of temperature is so high that the surface of the medium
melts. is vaporized. This phenomenon creates a
point-like force $f$ stress $\sigma$ in the medium, given by \cite{scruby1990laser}:
\begin{equation}
f \sigma =
\frac{S}{\rho}\frac{I^2}{(L+C(T_V-T_0))^2} \frac{1}{\rho}\frac{I^2}{(L+C(T_V-T_0))^2}
\label{eq:ablation}
\end{equation}
where $L$ is the latent heat required to vaporize the solid, $T_0$ and $T_V$ the initial and vaporization temperatures.
3
In both cases, absorption of the laser by the phantom leads to a local displacement which can propagate as elastic waves in the medium. To observe the shear waves, the medium was scanned with a 5 MHz ultrasonic probe made of 128 elements linked 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 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.
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