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We introduce a simple, single beam method for determination of the topological charge of polychromatic optical vortices. It is based on astigmatic transformation of singular optical beams, where the intensity pattern of a vortex beam acquires a form of dark stripes in the focal plane of a cylindrical lens. The number of the dark stripes is equal to the modulus of the vortex topological charge, while the stripe tilt indicates the charge sign. We demonstrate experimentally the effectiveness of this technique by revealing complex topological structure of polychromatic singular beams.

We consider the propagation of a tilted high order paraxial vortex-beam through a homogeneous anisotropic medium of a uniaxial crystal. We found that the initially circularly polarized beam bearing the lorder optical vortex splits into ordinary and extraordinary beams with a complex vortex structure. After a series of dislocation reactions the vortices gather together at the axis of the partial beam with the initial circular polarization shaping the l-order optical vortex. However, only l-1 vortices gather together on the axis of the partial beam with the orthogonal circular polarization. One optical vortex is shifted along the direction perpendicular to the inclination plane of the beam. Such a vortex displacement induces the transverse shift of the partial beam. In fact, we deal with the beam quadrefringence in a uniaxial, homogeneous anisotropic medium. The first two beams is a result of the splitting of the initial tilted beam into the ordinary and extraordinary ones. The appearance of the second two beams is caused by the transverse shift of one of the circularly polarized components in the initially circularly polarized vortex-beam or both circularly polarized components in the initially linearly polarized beam .We consider this effect both in terms of the conservation law of the angular momentum flux along the crystal optical axis and on the base of the solutions to the paraxial wave equation for the initially circularly and linearly polarized beams. We revealed that the transverse shift of the crystal traveling beam depends on neither a magnitude nor a sign of the vortex topological charge being defined only by a handedness of the initial circular polarization and a sign of the inclination angle of the beam. Also we analyze the deformation of the cross-section of the shifted vortex-beams and its evolution as the beam propagates along the crystal.

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