Tight-focusing properties of propagable fractional-order vector vortex beams

We investigate the tight-focusing properties of propagable fractional-order vector vortex beams (VVBs). The superposition of the phase singularity from optical vortices (OVs) and polarization singularity from VVBs plays a significant role in creating propagable fractional-order VVBs. Applying this unique characteristic, we derive an integrated analytical model to calculate the tight-focused electromagnetic fields and Poynting vector distributions based on the Richards–Wolf vector diffraction integral. The results show that the tight-focusing intensity of fractional-order VVBs exhibits uniaxial symmetric distribution, in which an odd number of petals with non-uniform amplitude occurs as the beams pass through the polarizer. In contrast to classical integer-order OVs with homogeneously polarized states, propagable fractional-order VVBs show semi-circular patterns with different directions after the polarizer, owing to their intriguing polarization distributions. Besides, to deeply understand the focusing characteristics of the calculated fractional-order VVBs, the influence of different-order carrier VVBs and OVs is studied in detail. These results might provide a valuable reference for particle manipulation and optical trapping.