Anisotropic beam spreading in uniaxial crystals

The anisotropic spreading of a paraxial beam propagating orthogonally to the optical axis inside a uniaxial crystal is investigated. The lack of rotational invariance around the propagation direction suggests to consider the mean square width (MSW) of the beam in any transverse direction, which turn out to be a quadratic form of the unit vector associated with the considered direction and to depend quadratically on the propagation distance z. The z-evolution of the MSW is compared with its standard evolution in vacuum. A simple connection is establish, for cylindrically symmetric input fields, between the anisotropic and the free space MSW, which in particular allows us to easily picture the spreading of the beam by means of a simple geometrical construction. The particular case of an astigmatic Gaussian beam is analytically carried out