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Exploding or concentrating beams, vortex beams, and cylindrical vector beams, have a precisely shaped transversal amplitude profile such that they produce a continuously concentrating and intensifying focal spot upon focusing as the lens aperture is opened. This effect is the physical manifestation of the mathematical fact that Fresnel diffraction integral predicts an infinite intensity at the focus when the aperture effects are ignored. Here, using a full electromagnetic, nonparaxial focusing model, we show that the singularity in exploding cylindrical vector beams is an artifact of the paraxial approximation. Nevertheless, the exploding or concentrating effect, alien to any other light beam with finite power, keeps going on up to unit numerical aperture, equivalent to infinite aperture radius. This unique feature enables a dynamic control of the focal intensity and spot size down to the sub-wavelength scale using a single light beam, imitating similar control when focusing an ideal plane wave, but requiring a finite amount of power.