
  When a laser beam of sufficient energy is incident on a medium, the absorption of the electromagnetic radiation leads to an increase of the local temperature. Due to thermal effects, displacements occur in the medium which can propagate as elastic waves. Elastic waves can be separated in two components in a bulk: compression waves, corresponding to a curl-free propagation; and shear waves, corresponding to a divergence-free propagation \cite{aki2002quantitative}. This phenomenon has been notably observed in metals. Measures of the compression and shear waves is notably used as a method of inspection to reveal potential cracks in the solid.


In a medical context, induction of compression waves by laser has been studied for the last ten years, with the development of photoacoustic imaging \cite{Xu_2006}. In this technique, a laser beam is absorbed by the tissue, which induces by thermal expansion compression waves, which are themselves acquired by acoustic transducers. Time of flight measurements allows then to find the source of the waves and thus, to map optical absorption of the tissues \cite{22442475}. As the optical absorption coefficient of the tissue depends on the optical wavelength, different structures can be observed by tuning properly the laser wavelength. For example, oxygenated and de-oxygenated haemoglobin can be discriminated in this way. Noninvasive imaging of hemoglobin concentration and oxygenation in the rat brain using high-resolution photoacoustic tomography\cite{16674205}. The frequency of the elastic waves used in photoacoustic imaging are typically of a few megahertz. At this frequency, shear waves are quickly attenuated, typically over a few microns in soft tissues, so only compression waves can propagate over a few centimeters.

We hypothesized in this study that a laser beam in a soft tissue can nevertheless induce shear waves in addition to compression waves. Shear waves have drawn an increasing interest in medical imaging, with the development for the last two decades of shear wave elastography techniques \cite{muthupillai1995magnetic}, \cite{sandrin2002shear}. As its names indicates, this term covers the techniques used to measure or map the elastic properties of biological tissues using shear wave propagation. The shear modulus, directly proportional to Young's modulus in soft tissues, varies indeed of several orders of magnitude in human body and potentially offers an excellent contrast. As a shear wave propagates in an organ at a speed proportional to the square root of the shear modulus, measuring its speed throughout the organ allows to compute the shear modulus of the tissue. Shear waves can be induced by an external vibrator \cite{muthupillai1995magnetic}, a focused acoustic beam \cite{sarvazyan1998shear}, \cite{11937286}, the Lorentz force\cite{grasland2014elastoEMarticle}, or natural body displacements \cite{gallot2011passive}. Shear wave elastography techniques have been successfully applied for the detection of various pathologies in organs such as the liver \cite{sandrin2003transient}, the breast \cite{goddi2012breast}, \cite{sinkus2005viscoelastic}, the prostate \cite{cochlin2002elastography}, \cite{souchon2003visualisation}, the bladder \cite{25574440} and the eye cornea \cite{tanter2009high}, \cite{22627517}.

In this study, we show that shear waves can be induced in soft tissues 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.

In the first experiment, illustrated by Figure \ref{Figure1}, we used a laser beam emitted by a Nd:YAG laser (EverGreen 200, Quantel, Les Ulis, France), which produced a Q-switched pulse of energy $E$ = 200 mJ at a central wavelength of 532 nm during 10 ns in a beam of section $S$=20 mm$^2$. The laser beam was absorbed in a 4x8x8 cm$^3$ tissue-mimicking phantom made of water and of 5\% polyvinyl alcohol, 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 was absorbed in the medium with an exponential decay of the optical intensity $I$ along medium depth $r$\cite{scruby1990laser}:
\begin{equation}
I=(1-R)I_0 \exp(- \gamma r)
\label{eq:opticalIntensity}
\end{equation}
where $R$ is the reflexion coefficient of the medium (typically less than a few pourcents for a black mat medium such as the one used here, so can be neglected thereafter), $I_0=\frac{1}{S}\frac{d E}{dt}$ the incident intensity distribution at the surface and $\gamma$ the absorption coefficient of the medium.

\textcolor{red}{We measured the fraction of light which go through different thicknesses of the medium with a laser beam power measurement device (): it indicated that $\gamma \approx$ ??? m$^{-1}$ in our sample,} meaning that most of the radiation is absorbed in the first hundred of micrometers.
%Even if the sample is mainlyeFor low concentration medium, $\gamma$ is hard to calculate in our case, as the sample is composed of different materials, but the graphite, even in low concentration, absorbate much more than other components, so we can approximate $\gamma \approx \gamma_{graphite}$. For graphite particles of 1.85 $\mu$m at a concentration of 10 g.L$^{-1}$, the order of magnitude of $\gamma$ is 10$^4$ m$^{-1}$, meaning that most of the radiation is absorbed in the first hundred of micrometers of sample.
This is quite higher than 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 $q = \gamma I$. Assuming that all the optical energy is converted to heat, a local increase of temperature appears. Temperature distribution $T$ can be computed using heat equation:
\begin{equation}
    \rho C \frac{\partial T}{\partial t} = k \nabla ^2 T + q
    \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 $\sqrt{4\kappa t}$, with $t$ the laser emission duration (20 ns) and $\kappa$ = 1.43 10$^{-7}$ m$^2$.s$^{-1}$ for water \cite{Blumm_2003}, is equal to 0.1 $\mu$m. As $\gamma^{-1} \gg \sqrt{4\kappa t}$, propagation of heat is negligible during laser emission, and term $k \nabla ^2 T$ can be neglected in equation \ref{eq:eqChaleur}. Combination with equation \ref{eq:opticalIntensity} and integration over time lead then to a temperature $T$ at the end of the laser emission:
\begin{equation}
    T = T_0 + \frac{\gamma}{S \rho C}  E \exp(-\gamma r)
    \label{eq:eqTemperature}
\end{equation}
where $T_0$ is the initial temperature of the medium.

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 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), and the local expansion acts as dipolar forces 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.

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.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  