Abstract: An apparatus for spin polarizing a particle beam is adapted to process an input particle beam in such a way as to generate an at least partially spin polarized output particle beam. A vortex beam generator for imparting orbital angular momentum to the input particle beam. An electromagnetic field generator generates a transverse magnetic field, space-variant and symmetric with respect to the axis of the input particle beam, in such a way as to change the spin of the particles and attach thereto different values of orbital angular momentum in dependence on their input spin values. A beam component separating group spatially separates the particles in dependence on their orbital angular momentum values, in such a way as to obtain the at least partially spin polarized output particle beam.

Abstract: Geometrical phase optical elements comprising, as a birefringent means, a liquid crystal material with a spatially inhomogeneous orientation of the molecular director in a plane orthogonal to the direction of propagation of an input radiation beam, ordered according to a predetermined geometrical pattern. Also, an optical system for generating helical modes of propagation of an optical radiation beam, including a geometrical phase optical element orientated transversely to the direction of propagation of the beam, whose optical axis is orientated according to a predetermined geometrical relation adapted to transform a circular polarized input radiation beam into a helical mode with a wavefront having a helical surface whose handedness is a function of the handedness of the input polarization, in which the switching between different orders of helicity of the radiation beam emerging from the geometrical phase optical element is carried out by switching the circular polarization state of the input radiation.

Abstract: Geometrical phase optical elements comprising, as a birefringent means, a liquid crystal material with a spatially inhomogeneous orientation of the molecular director in a plane orthogonal to the direction of propagation of an input radiation beam, ordered according to a predetermined geometrical pattern. Also, an optical system for generating helical modes of propagation of an optical radiation beam, including a geometrical phase optical element orientated transversely to the direction of propagation of the beam, whose optical axis is orientated according to a predetermined geometrical relation adapted to transform a circular polarized input radiation beam into a helical mode with a wavefront having a helical surface whose handedness is a function of the handedness of the input polarization, in which the switching between different orders of helicity of the radiation beam emerging from the geometrical phase optical element is carried out by switching the circular polarization state of the input radiation.