Elliptic propagation invariant optical fields:: Mathieu beams

The interest in propagation invariant optical fields (PIOF's) is due to the fact that, under optimal conditions, they propagate long distances without significant change of their transverse intensity distribution. These kind of wavefields were first identified and described in terms of the nonsingular Bessel function. Based on the separability of the Helmholtz equation in elliptic cylindrical coordinates we have demonstrated that there exist another class of PIOF's. The lowest order mode may have a highly localised distribution along one of the transverse directions and a sharply peaked quasi-periodic structure along the other. Higher order modes are composed Of elliptical vortices and the corresponding intensity profiles are formed by propagation-invariant confocal elliptical rings. These fields are described by the Mathieu-Hankel functions which are exact solutions of Helmholtz equation and for this reason we have called them Mathieu beams. We have also showed that Bessel beams are a particular case of Mathieu beams, which have a broaded fan of interesting features. Since the Mathieu-Hankel functions form a whole set of exact travelling wave solutions of the Helmholtz wave equation they can be used to describe a class of PIOF's. Using the McCutchen theorem obtained the relation between the general class of PIOF's and these new beams.