Wavefronts and caustics associated with Mathieu beams

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 A(ϕ)=ce (ϕ,q)+ise (ϕ,q), with ν 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 A(ϕ)=ce (ϕ,q) and A(ϕ)=se (ϕ,q) are cones and their caustic is the z axis; thus, they are not structurally stable. However, in general, the Mathieu beam generated by A(ϕ)=ce (ϕ,q)+ise (ϕ,q) 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 a =0 and q=0,0.2,0.3,0.5. For q=0, we obtain the Bessel beam of order zero; in this case, the wavefronts are cones and the caustic coincides with the z axis. For q≠0, the wavefronts are deformations of conical ones, and the caustic surface, for some values of q, has singularities of the cusp ridge type. Furthermore, we remark that the set of Mathieu beams with characteristic value a =0 and 0≤q