Polarization singularities in nondiffracting Mathieu–Poincaré beams

We introduce a new family of nondiffracting full Poincaré beams based on a superposition of nondiffracting Mathieu beams, which we call the Mathieu–Poincaré beams (MPBs). We studied the polarization structure of the MPBs and how it is traced on the Poincaré sphere, and found that the first region mapping the Poincaré sphere is contained within an ellipse of circular polarization of constant size for all beam orders m for a given semi-focal distance and as expected a higher order beam covers the Poincaré sphere m-fold in a nonuniform way given the noncircular symmetry of the Mathieu beams. Finally, we looked into the polarization singularities along the inter-focal line and observed that the all -points have a star (lemon) morphology for even (odd) beam order m when we used positive helical Mathieu beams to synthesize the MPBs, and that this relationship is reversed when we switched to a negative helical Mathieu beam.