Abstract

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.

© 2021 Optical Society of America

Paraxial optical fields whose intensity pattern skeletons are stable caustics

Ernesto Espíndola-Ramos, Gilberto Silva-Ortigoza, Citlalli Teresa Sosa-Sánchez, Israel Julián-Macías, Omar de Jesús Cabrera-Rosas, Paula Ortega-Vidals, Adriana González-Juárez, Ramón Silva-Ortigoza, Mercedes Paulina Velázquez-Quesada, and G. F. Torres del Castillo
J. Opt. Soc. Am. A 36(11) 1820-1828 (2019)

Exact and geometrical optics energy trajectories in Bessel beams via the quantum potential

Gilberto Silva-Ortigoza, Israel Julián-Macías, Ernesto Espíndola-Ramos, and Ramón Silva-Ortigoza
J. Opt. Soc. Am. B 40(3) 620-630 (2023)

Pearcey beam tuning and caustic evolution

Yihao Wang
J. Opt. Soc. Am. A 38(11) 1726-1731 (2021)

Properties of Hermite–Gaussian beams via the quantum potential

Gilberto Silva-Ortigoza, Ernesto Espíndola-Ramos, Edna Gabriela Gochicoa-Fuentes, and Ramón Silva-Ortigoza
J. Opt. Soc. Am. B 40(10) 2706-2715 (2023)

Wavefronts and caustics associated with Mathieu beams

Israel Julián-Macías, Carolina Rickenstorff-Parrao, Omar de Jesús Cabrera-Rosas, Ernesto Espíndola-Ramos, Salvador Alejandro Juárez-Reyes, Paula Ortega-Vidals, Gilberto Silva-Ortigoza, and Citlalli Teresa Sosa-Sánchez
J. Opt. Soc. Am. A 35(2) 267-274 (2018)