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We present a set of paraxial light beams with cylindrical symmetry, smoothand localized transversal profile carrying finite power, that develop intensitysingularities when they are focused in a linear medium, such as vacuum. Theyinclude beams with orbital angular momentum and with radial polarization, inwhich case they develop punctual phase and polarization singularitiessurrounded 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 longitudinalfield, that strongly change with the lens aperture radius. Continuous controlof these focal properties is thus exercised without changing the light incidenton the lens, with substantially the same collected power, and while maintainingparaxial focusing conditions. As solutions of the Schr\"odinger equation, theseexploding beams have analogues in other areas of physics where this equation isthe fundamental dynamical model.

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