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In this work we present a new and wide class of scalar, rectangular symmetrical optical fields, the free-space propagation of which can be given in a closed-form in the paraxial approximation. In particular it is shown how such fields can be expressed as a finite linear combination of the recently introduced Lorentz beams [O. El Gawhary, S. Severini, J. Opt. A: Pure Appl. Opt., 8 (2006) 409.] that, in this way, act as a basis for the newly introduced class. Because of their mathematical form, we call such fields super-Lorentzian beams. Some common features of the class are pointed out and the concept of order of the beam introduced. Moreover, by using these results, we demonstrate the existence of a new family of mutually orthogonal paraxial fields with a related new class of orthogonal polynomials.

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.

In this work we present an extensive study of the scaling symmetry typical of a paraxial wave theory. In particular, by means of a Lagrangian approach we derive the conservation law and the corresponding generalized charge associated with the scale invariance symmetry. In general, such a conserved charge, qs say, can take any value that remains constant during propagation. However, it is explicitly proven that for the whole class of physically realizable shape-invariant fields, that is, fields whose intensity distribution maintains its shape on propagation, qs must necessarily vanish. Finally, an interesting relation between such charge qs and the effective radius of a beam, as introduced by Siegman some years ago, is derived.

The nonparaxial corrections for Bessel–Gauss beams were derived recently using two different approaches [Borghi et al., J. Opt. Soc. Am. A 18, 1618 (2001) and Vaveliuk et al., J. Opt. Soc. Am. A 24, 3297 (2007)]. However, the two obtained results do not agree, so it is necessary to determine which method is correct. In the most recent of those papers, Vaveliuk et al. claimed that their method is correct while the method described by Borghi et al. is incorrect. In the present work, just by solving the rigorous propagation problem, we show that exactly the converse is true.

We introduce the degree of paraxiality as a paraxiality measure of a monochromatic light beam. Computation of this parameter is possible once the plane-wave spectrum of the field being analyzed is given for a starting source plane. On the basis of this definition, quantitative comparisons of the paraxiality of different types of fields are possible. In addition, the recently introduced paraxial estimator [Opt. Lett.32, 927 (2007)] is identified as the limiting case of the degree of paraxiality.

We comment on a recent paper by Seshadri [Appl. Opt.45, 5335 (2006)]APOPAI0003-693510.1364/AO.45.005335. In the cited work the author utilized the so-named Felsen method in order to derive the full wave expression of a scalar Bessel-Gauss field, propagating in free space, originated from an assigned input distribution on the source plane z = 0. As we show, the author's obtained results do not correctly reproduce the aforesaid input field distribution, as, on the contrary, they have to do.