Poincar\'e sphere representation for classical inseparable Bell-like states of the electromagnetic field

Classical beams of light with non-uniform polarization patterns (e.g. radially and azimuthally polarized doughnut beams) may exhibit quantum-like features as, for instance, inseparability. We establish an exact correspondence between radially and azimuthally polarized classical modes of the electromagnetic field and the two-qubit quantum Bell states. We demonstrate the existence of a special representation for such classical modes by means of a pair of Poincar\'e spheres. Points on these spheres are described by Stokes parameters associated with such modes, and their explicit expressions are given.