The Jones vector as a spinor and its representation on the Poincar\'e sphere

It is shown that the two complex Cartesian components of the electric field of a monochromatic electromagnetic plane wave, with a temporal and spatial dependence of the form ${\rm e}^{{\rm i} (kz - \omega t)}$, form a SU(2) spinor that corresponds to a tangent vector to the Poincar\'e sphere representing the state of polarization and phase of the wave. The geometrical representation on the Poincar\'e sphere of the effect of some optical filters is reviewed. It is also shown that in the case of a partially polarized beam, the coherency matrix defines two diametrically opposite points of the Poincar\'e sphere.