deletions | additions       diff --git a/When_a_laser_beam_of__1.tex b/When_a_laser_beam_of__1.tex  index 221963c..7ba2ba5 100644  --- a/When_a_laser_beam_of__1.tex  +++ b/When_a_laser_beam_of__1.tex 
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In this study, we show that shear waves can be induced by a laser beam, with a model of the underlying physical phenomenon. We also applied the technique in a biological tissue to evaluate its potential application in shear wave elastography.  The Z axis is defined here as the laser beam axis, and the ultrasound probe imaging plane  is in the XZ YZ  plane, as illustrated by Figure \ref{Figure1}. In this experiment, we used first a 4x8x8 cm$^3$ tissue-mimicking phantom made of 5\% polyvinyl alcohol, 0.1 \% black graphite powder and 1\% salt. A freezing/thawing cycle was applied to stiffen the material to a value of 15$\pm$5 kPa \cite{17375819}.  The laser beam was emitted by a Nd:YAG laser (EverGreen 200, Quantel, Les Ulis, France), which produced a 200 mJ, 5 mm in diameter Q-switched pulse at a central wavelength of 532 nm during 10 ns. The laser is absorbed in the medium, and the optical intensity $I(x,y,z,t)$ along position $(x,y,z)$ and time $t$ $I$  decays exponentially along medium depth $z$\cite{scruby1990laser}: $r$\cite{scruby1990laser}:  \begin{equation}  I=I_0 \exp(- \gamma z) r)  \end{equation}  where $I_0$ is the incident intensity distribution at the surface and $\gamma$ is the absorption coefficient of the medium. In non-metallic solids, the absorption coefficient $\gamma$ is relatively small, so that the radiation is able to penetrate into the bulk of the material - contrary to metals where the radiation is absorbed within a few nanometres. The absorption of the laser beam by the medium gives then rise to an absorbed optical energy $q$ equal to $\gamma $q=\gamma  I$. Assuming that all the optical energy is converted to heat, a local increase of temperature appears. Temperature distribution $T(x,y,z,t)$ can be computed using heat equation:  \begin{equation}  \frac{\partial T}{\partial t} = \frac{k}{\rho C} \nabla ^2 T + \frac{q}{\rho C} 
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The propagation of the heat is slow compared to the duration of the heating (10 ns) and the thermal expansion duration (), so that the phenomenon can be considered as adiabatic.  %$\kappa$ is approximately equal to 10${^6}$ m$^2$.s$^{-1}$ for water, the main component of biological tissues; for a 10 ns laser pulse, the thermal diffusion path is then equal to 0.01 to 0.1 $\mu$m. $\gamma^{-1}$ of water is equal to 0.1 m, which is a million times higher; even for melanin and haemoglobin, highly absorbing at 532 nm, $\gamma^{-1}$ is respectively equal to 10 and 100 $\mu$m, far higher than the thermal diffusion path. The thermal conductivity effects are consequently negligible, and increase of temperature lies in laser absorption zone.  The local increase of temperature can lead to two main effects creating elastic waves: (1) Thermoelastic thermoelastic  expansion and (2) Ablation 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 expansion, 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, the solid is semi-infinite (the laser beam is absorbed on one side of the medium). The medium), and the  local expansion acts as a dipole force parallel to the surface. In the ablative regime, the local increase of temperature is so high that the surface of the medium melts and creates a point-force in the medium.The medium is then displaced locally inside the medium along Z axis mainly.  In both cases, the absorption of the laser by the phantom leads to a local displacement which can propagate as elastic wave waves  in the medium. To observe the elastic wave, 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 1000 1500  ultrasound frames 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). Y-displacement, depending on the position of the probe on the medium).  The laser beam was triggered 10 ms after the beginning of the first  ultrasound acquisition. acquisition, $t=0$ms being defined as the laser emission.