  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.
