Polarization flowers

Two types of singularities of the polarization vector E of paraxial laser beams are discussed: vector singularities and Stokes singularities. Vector singularities are time independent points at which the direction of E becomes undefined; Stokes singularities are time independent points at which the normalized Stokes parameters become undefined. We show how to construct both types of singularities in the form of polarization "flowers" with arbitrarily large positive Poincaré-Hopf indices, and in the form of "hyperbolic webs" with arbitrary negative indices. We also show how to construct arbitrary arrays of these singularities. We extend the sign rule for phase singularities to vector singularities, and define additional, general properties of singular vector fields not considered previously. A given singularity structure embedded in the waist of a Gaussian laser beam will generally change upon propagation away from the waist, and we briefly study these changes, describing them in terms of singularity trajectories in space and in time.