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This article describes the generation of elastic shear waves in a soft medium using a laser beam. Our experiments show two different regimes depending on laser energy. Physical modeling of the underlying phenomena reveals a thermoelastic regime caused by a local dilatation resulting from temperature increase, and an ablative regime caused by a partial vaporization of the medium by the laser. Computed theoretical displacements are close to experimental measurements. A numerical study based on the physical modeling gives propagation patterns comparable to those generated experimentally. These results provide a physical basis for the feasibility of a shear wave elastography technique (a technique which measures a soft solid elasticity stiffness  from shear wave propagation) by using a laser beam.