Espíndola-Ramos Ernesto, Silva-Ortigoza Gilberto, Sosa-Sánchez Citlalli Teresa, Julián-Macías Israel, González-Juárez Adriana, Cabrera-Rosas Omar de Jesús, Ortega-Vidals Paula, Rickenstorff-Parrao Carolina, Silva-Ortigoza Ramón
J Opt Soc Am A Opt Image Sci Vis. 2021 Mar 1;38(3):303-312. doi: 10.1364/JOSAA.411094.
From a geometric perspective, the caustic is the most classical description of a wave function since its evolution is governed by the Hamilton-Jacobi equation. On the other hand, according to the Madelung-de Broglie-Bohm equations, the most classical description of a solution to the Schrödinger equation is given by the zeros of the Madelung-Bohm potential. In this work, we compare these descriptions, and, by analyzing how the rays are organized over the caustic, we find that the wave functions with fold caustic are the most classical beams because the zeros of the Madelung-Bohm potential coincide with the caustic. For another type of beam, the Madelung-Bohm potential is in general distinct to zero over the caustic. We have verified these results for the one-dimensional Airy and Pearcey beams, which, according to the catastrophe theory, have stable caustics. Similarly, we introduce the optical Madelung-Bohm potential, and we show that if the optical beam has a caustic of the fold type, then its zeros coincide with the caustic. We have verified this fact for the Bessel beams of nonzero order. Finally, we remark that for certain cases, the zeros of the Madelung-Bohm potential are linked with the superoscillation phenomenon.
从几何角度来看，焦散是波函数最经典的描述，因为其演化由哈密顿 - 雅可比方程支配。另一方面，根据马德隆 - 德布罗意 - 玻姆方程，薛定谔方程解的最经典描述由马德隆 - 玻姆势的零点给出。在这项工作中，我们比较了这些描述，并通过分析光线在焦散上的组织方式，发现具有折叠焦散的波函数是最经典的光束，因为马德隆 - 玻姆势的零点与焦散重合。对于另一种类型的光束，马德隆 - 玻姆势在焦散上通常不为零。我们已经针对一维艾里光束和皮尔西光束验证了这些结果，根据突变理论，它们具有稳定的焦散。类似地，我们引入了光学马德隆 - 玻姆势，并表明如果光束具有折叠型焦散，那么其零点与焦散重合。我们已经针对非零阶贝塞尔光束验证了这一事实。最后，我们指出在某些情况下，马德隆 - 玻姆势的零点与超振荡现象有关。
