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...
The absorption Absorption of the laser beam by the medium
subsequently gives
then rise to an absorbed optical
energy energy, $\gamma I$. Assuming that all the optical energy is converted to heat, a local increase
of in temperature
appears. occurs. Temperature
distribution $T$ distribution, $T$, in
the absence of convection and
of phase
transition transition, can be computed using the
following heat equation:
\begin{equation}
k \nabla ^2 T = \rho C \frac{\partial T}{\partial t} - \gamma I
\label{eq:eqChaleur}
\end{equation}
where $\rho$ is the density, $C$
is the heat capacity and $k$
is the thermal conductivity. Calculating the exact solution
of to this equation is
behind beyond the scope of this article, but
during laser emission, we can roughly approximate the first
term by and second terms to be $k T / \gamma^2$ and
the second by $\rho C T /
\tau$. Taking \tau$, respectively, during laser emission. Given that $k$ = 0.6
W.m$^{-1}$.K$^{-1}$, W.m$^{-1}$.K$^{-1}$ (water thermal conductivity), $\rho$ = 1000
kg.m$^{-3}$, kg.m$^{-3}$ (water density), $C$ = 4180 J.kg$^{-1}$.m$^{-3}$ (water
thermal conductivity, density and heat
capacity respectively), capacity), $\gamma^{-1} \approx$ 40 $\mu$m and $\tau$ = 10 ns, the first term is negligible compared to the second
one, so that one; thus, the equation \ref{eq:eqChaleur} can be simplified as:
\begin{equation}
\frac{\partial T}{\partial t} = \frac{\gamma I}{\rho C} = \frac{\gamma}{\rho C S} \frac{dE}{dt}
\label{eq:eqChaleurApprox}
\end{equation}
Substituting low-energy experimental parameters ($E$ = 10 mJ, $S$ = 20 mm$^{2}$) leads to a maximum increase
of in temperature of 3
K. This increase of temperature gives rise to K, which produces a local dilatation of the medium. The induced displacements can then generate
to shear
waves: this waves, which constitutes the \textit{thermoelastic regime}.
To estimate the initial displacement amplitude in this regime, we
supposed assumed the medium
as was homogeneous and isotropic. As the depth of absorption (about 40 $\mu$m) is
hundred 100 times smaller than the beam diameter (5 mm), we discarded any
boundaries effect. boundary effects. The
stress $\sigma_{zz}$ stress, $\sigma_{zz}$, is the sum
between of 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) \alpha \frac{ E}{\rho C S \zeta}
\label{eq:stressThermo}
\end{equation}
where $\lambda$ and $\mu$ are respectively the first and second Lamé's
coefficient, coefficients, $\alpha$
is the thermal dilatation
coefficient coefficient, and $\zeta$
is the average depth of absorption. This equation can be simplified by
remarking the fact that in most soft media, including biological tissues, $\mu \ll \lambda$. Moreover, 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 0$. This allows the equation \ref{eq:stressThermo}
can to be integrated, giving
a displacement $u_z$ the following displacement, $u_z$, from the surface:
\begin{equation}
u_z = \frac{3 \alpha E}{\rho C S}
\label{eq:deplThermoApprox}
\end{equation}
Substituting
the same experimental parameters
as used previously
and along with $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient), we obtain a displacement $u_z$
= of 0.025 $\mu$m. This value is very close to the measured experimental displacement (about 0.02 $\mu$m). Note that both
the theoretical and experimental central displacements are directed
towards the outside
of the medium (see white circle arrow in the Figure \ref{figElastoPVA}-(A)).
To calculate the propagation of the displacements as shear waves, we
have to take into account must first consider the transverse
dilatation dilatation, which leads to stronger displacements than
those occurring along
Z. the Z axis. We
modeled thus
modeled the thermoelastic regime in 2D as two opposite forces directed along
the Y axis with a depth of 40 $\mu$m and
with an amplitude decreasing linearly
respectively from 2.5 to 0 mm
(respectively and from -2.5 to 0
mm) mm \cite{Davies_1993}. The magnitude of the force along space and time is stored in a
matrix matrix, $H_y^{thermo}(y,z,t)$. Displacements along
the Z
axis are then equal to the convolution between $H_y^{thermo} (y,z,t)$
with and $G_{yz}$ \cite{aki1980quantitative}:
\begin{equation}
G_{yz}(r,\theta,t) = \frac{\cos \theta \sin \theta}{4\pi \rho c_p^2 r} \delta(t-\frac{r}{c_p}) - \frac{\sin \theta \cos \theta}{4\pi \rho c_s^2 r} \delta(t-\frac{r}{c_s})
\label{eq:Gyz}
...
+\frac{3\cos \theta \sin \theta}{4\pi \rho r^3} \int\limits_{r/c_p}^{r/c_s}{\tau \delta(t-\tau) d\tau}
\label{eq:Gyz2}
\end{equation}
where (r,$\theta$) are the coordinates of the considered point with regards to the force location and direction, $c_p$ and $c_s$
are the compression and shear wave speed respectively, $\tau$
is the
time time, and $\delta$
is a Dirac
distributions. distribution. The three terms
of the equation correspond respectively to the far-field compression wave, the far-field shear
wave wave, and the near-field component.
Using
$\rho$ constants \rho$ = 1000 kg.m$^{-3}$, $c_p$ = 1500 m.s$^{-1}$ and $c_s$ = 5.5 m.s$^{-1}$,
results are illustrated in Figure \ref{figGreen}-(A) which represents displacement maps along
the Z
axis, axis were calculated 1.0, 1.5, 2.0,
2.5 2.5, and 3.0 ms after force
application. application, as illustrated in Figure \ref{figGreen}-(A). The normalized displacement maps present many similarities
with to the experimental results
displayed in
the Figure \ref{figElastoPVA}-(A), with
a an initial central displacement directed
towards the outside
of the
medium medium, and
the a propagation of three half cycles.
