Julián-Macías Israel, Rickenstorff-Parrao Carolina, Cabrera-Rosas Omar de Jesús, Espíndola-Ramos Ernesto, Juárez-Reyes Salvador Alejandro, Ortega-Vidals Paula, Silva-Ortigoza Gilberto, Sosa-Sánchez Citlalli Teresa
J Opt Soc Am A Opt Image Sci Vis. 2018 Feb 1;35(2):267-274. doi: 10.1364/JOSAA.35.000267.
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<1 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 A(ϕ)=ce(ϕ,q) and A(ϕ)=se(ϕ,q). To support this conclusion, we present experimental results showing the pattern obtained after obstructing a plane wave, the Bessel beam of order m=5, and the Mathieu beam of order m=5 and q=50 with complex transversal amplitude given by Ce(ξ,50)ce(η,50)+iSe(ξ,50)se(η,50), where (ξ, η) are the elliptical coordinates on the plane.
在这项工作中，我们计算了与杜宁在椭圆圆柱坐标系中引入的标量波动方程的解相关的波前和焦散，该坐标系由函数(A(\phi)=ce(\phi,q)+ise(\phi,q))生成，其中(\nu)为整数或非整数。我们表明，波前和焦散在沿光束传播方向的平移下是不变的。我们注意到，由(A(\phi)=ce(\phi,q))和(A(\phi)=se(\phi,q))生成的可分离马蒂厄光束的波前是圆锥面，其焦散是(z)轴；因此，它们在结构上是不稳定的。然而，一般来说，由(A(\phi)=ce(\phi,q)+ise(\phi,q))生成的马蒂厄光束是稳定的，因为在局部其焦散具有折叠和尖点类型的奇点。为了展示这一特性，我们给出了特征值(a = 0)且(q = 0、0.2、0.3、0.5)的马蒂厄光束的波前和焦散。对于(q = 0)，我们得到零阶贝塞尔光束；在这种情况下，波前是圆锥面，焦散与(z)轴重合。对于(q≠0)，波前是圆锥面的变形，并且对于某些(q)值，焦散面具有尖点脊类型的奇点。此外，我们注意到，特征值(a = 0)且(0≤q＜1)的马蒂厄光束集合具有与燕尾形奇点相关的焦散，这在结构上是稳定的。因此，我们得出结论，这种类型的马蒂厄光束比平面波、贝塞尔光束、抛物线光束以及由(A(\phi)=ce(\phi,q))和(A(\phi)=se(\phi,q))生成的光束更稳定。为了支持这一结论，我们给出了实验结果，展示了在阻挡平面波、(m = 5)阶贝塞尔光束以及(m = 5)阶且(q = 50)的马蒂厄光束后得到的图案，该马蒂厄光束的复横向振幅由(Ce(ξ,50)ce(η,50)+iSe(ξ,50)se(η,50))给出，其中((ξ, η))是平面上的椭圆坐标。
