  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.
