Radially polarized hyperbolic sine non-uniform coherent beams and their propagation properties.

We introduce a radially polarized partially coherent beam characterized by a tailored non-uniform correlation function, termed the radially polarized hyperbolic sine non-uniform coherent (RPHSNUC) beam. We establish the validity conditions for generating this physical optical field. Employing the ordinary Huygens-Fresnel principle, we derive analytical expressions for the spectral intensity and spectral degree of polarization in free space and explore the beam's propagation properties through numerical simulations. Our results demonstrate that RPHSNUC beams preserve their dark hollow core and radial polarization during propagation in free space, while exhibiting a distinctive self-focusing behavior. These findings suggest potential applications in free-space optical communications, polarization-sensitive imaging, and optical trapping.