  An azimuthally electric-polarized vector beam (APB), with a polarization
vortex, has a salient feature that it contains a magnetic-dominant region
within which electric field ideally has a null while longitudinal magnetic
field is maximum. Fresnel diffraction theory and plane-wave spectral (PWS)
calculations are applied to quantify field features of such a beam upon
focusing through a lens. The diffraction-limited full width at half maximum
(FWHM) of the beam's longitudinal magnetic field intensity profile and
complementary FWHM (CFWHM) of the beam's annular-shaped total electric field
intensity profile are examined at the lens's focal plane as a function of the
lens's paraxial focal distance. Then, we place a subwavelength dense dielectric
Mie scatterer in the minimum-waist plane of a self-standing converging APB and
demonstrate for the first time that a very high resolution magnetic field at
optical frequency is achieved with total magnetic field FWHM of 0.23{\lambda}
(i.e., magnetic field spot area of 0.04{\lambda}^2) within a magnetic-dominant
region. The theory shown here is valuable for development of optical microscopy
and spectroscopy systems based on magnetic dipolar transitions which are in
general much weaker than their electric counterparts.
