Abstract
In this work we compute the wavefronts and the caustics associated with the solutions to the scalar wave equation introduced by Durnin in elliptical cylindrical coordinates generated by the function , with being an integral or nonintegral number. We show that the wavefronts and the caustic are invariant under translations along the direction of evolution of the beam. We remark that the wavefronts of the separable Mathieu beams generated by and are cones and their caustic is the axis; thus, they are not structurally stable. However, in general, the Mathieu beam generated by is stable because locally its caustic has singularities of the fold and cusp types. To show this property, we present the wavefronts and the caustics for the Mathieu beams with characteristic value and . For , we obtain the Bessel beam of order zero; in this case, the wavefronts are cones and the caustic coincides with the axis. For , the wavefronts are deformations of conical ones, and the caustic surface, for some values of , has singularities of the cusp ridge type. Furthermore, we remark that the set of Mathieu beams with characteristic value and has associated a caustic with singularities of the swallowtail type, which is structurally stable. Therefore, we conclude that this type of Mathieu beam is more stable than plane waves, Bessel beams, parabolic beams, and those generated by and . To support this conclusion, we present experimental results showing the pattern obtained after obstructing a plane wave, the Bessel beam of order , and the Mathieu beam of order and with complex transversal amplitude given by , where (, ) are the elliptical coordinates on the plane.
© 2018 Optical Society of America
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