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\nand localized transversal profile carrying finite power, that develop intensity\nsingularities when they are focused in a linear medium, such as vacuum. They\ninclude beams with orbital angular momentum and with radial polarization, in\nwhich case they develop punctual phase and polarization singularities\nsurrounded by infinitely bright rings, along with singular longitudinal fields.\nIn practice, these effects are manifested in focal intensities and spot sizes,\nvortex bright ring intensities and radii, and strengths of the longitudinal\nfield, that strongly change with the lens aperture radius. Continuous control\nof these focal properties is thus exercised without changing the light incident\non the lens, with substantially the same collected power, and while maintaining\nparaxial focusing conditions. As solutions of the Schr\\"odinger equation, these\nexploding beams have analogues in other areas of physics where this equation is\nthe fundamental dynamical model.\n