Exploding paraxial beams, vortex beams, and cylindrical beams of light with finite power in linear media, and their enhanced longitudinal field

We present a set of paraxial light beams with cylindrical symmetry, smooth and localized transversal profile carrying finite power, that develop intensity singularities when they are focused in a linear medium, such as vacuum. They include beams with orbital angular momentum and with radial polarization, in which case they develop punctual phase and polarization singularities surrounded by infinitely bright rings, along with singular longitudinal fields. In practice, these effects are manifested in focal intensities and spot sizes, vortex bright ring intensities and radii, and strengths of the longitudinal field, that strongly change with the lens aperture radius. Continuous control of these focal properties is thus exercised without changing the light incident on the lens, with substantially the same collected power, and while maintaining paraxial focusing conditions. As solutions of the Schr\"odinger equation, these exploding beams have analogues in other areas of physics where this equation is the fundamental dynamical model.