Unraveling the vector nature of generalized space-fractional Bessel beams

We introduce an exact analytical solution of the homogeneous space-fractional Helmholtz equation in cylindrical coordinates. This solution, called vector Space-Fractional Bessel Beam (SFBB), has been established from the Lorenz' gauge condition and Hertz vector transformations. We perform scalar and vector wave analysis focusing on electromagnetics applications, especially in cases where the dimensions of the beam are comparable to its wavelength $(k_r \approx k)$. The propagation characteristics such as the diffraction and self-healing properties have been explored with particular emphasis on the polarization states and transverse propagation modes. Due to continuous order orbital angular momentum dependence, this beam can serve as a bridge between the ordinary integer Bessel beam and the fractional Bessel beam and, thus, can be considered as a generalized solution of the space-fractional wave equation that is applicable in both integer and fractional dimensional spaces. The proposed SFBBs provide better control over the beam characteristics and can be readily generated using Digital Micromirror Devices (DMDs), Spatial Light Modulators (SLMs), metasurfaces, or spiral phase plates. Our findings offer new insights on electromagnetic wave propagation, thus paving a new route towards novel applications in optical tweezers, refractive index sensing, optical trapping, and optical communications.