Lorentz beams and symmetry properties in paraxial optics

A new kind of tridimensional scalar optical beams is introduced. These beams are called Lorentz beams because the form of their transverse pattern in the source plane is the product of two independent Lorentz functions. A closed-form expression of free-space propagation under the paraxial limit is derived. Moreover, as the slowly varying part of these fields fulfils the scalar paraxial wave equation, it follows that there also exist Lorentz-Gauss beams, i.e. beams obtained by multiplying the original Lorentz beam by a Gaussian apodization function. Although the existence of Lorentz-Gauss beams can be shown by using two different and independent ways obtained recently by Kiselev (2004 Opt. Spectrosc. 96 497-81) and Gutierrez-Vega and Bandres (2005 J. Opt. Soc. Am. 22 289-98), here we have followed a third different approach, which makes use of Lie's group theory, and which possesses the merit to put into evidence the symmetries present in paraxial optics.