Accelerating Bessel-Coulomb optical beams: planar, spherical and parabolic traveling structured wavefields

For centuries, parabolic mirrors have been used like magnifying makeup mirrors, parabolic antennas and parallel beam reflectors. These mirrors have the property that light rays coming out from their focal point after reflection on the surface of the mirror will travel parallel to the axis of the paraboloid and vice versa; light rays parallel to the axis incident on the mirror will converge at the focus after reflection. The Greek mathematician Diocles (fl. ca. 190 B.C.) is credited with the geometrical optics solution to the latter situation of "<i>how to find a mirror surface such that when it is placed facing the sun the rays reflected from it meet a (single) point and thus cause burning</i>." However, to the best of our knowledge, a complete wave theory has not been presented to date and only the stationary case has been reported in the literature. Here, using an alternative approach to that used for stationary fields, the possibility of creating paraboloidal traveling wavefields of the Helmholtz equation is demonstrated. Moreover, their relation to spherical and planar wavefields associated with parabolic mirrors is unveiled.