TY - JOUR
T1 - Wavefronts, actions and caustics determined by the probability density of an Airy beam
AU - Espindola-Ramos, Ernesto
AU - Silva-Ortigoza, Gilberto
AU - Sosa-Sánchez, Citlalli Teresa
AU - Julián-Macias, Israel
AU - Cabrera-Rosas, Omar De Jesus
AU - Ortega-Vidals, Paula
AU - Juárez-Reyes, Salvador Alejandro
AU - González-Juárez, Adriana
AU - Silva-Ortigoza, Ramón
N1 - Publisher Copyright:
© 2018 IOP Publishing Ltd.
PY - 2018/7
Y1 - 2018/7
N2 - The main contribution of the present work is to use the probability density of an Airy beam to identify its maxima with the family of caustics associated with the wavefronts determined by the level curves of a one-parameter family of solutions to the Hamilton-Jacobi equation with a given potential. To this end, we give a classical mechanics characterization of a solution of the one-dimensional Schrödinger equation in free space determined by a complete integral of the Hamilton-Jacobi and Laplace equations in free space. That is, with this type of solution, we associate a two-parameter family of wavefronts in the spacetime, which are the level curves of a one-parameter family of solutions to the Hamilton-Jacobi equation with a determined potential, and a one-parameter family of caustics. The general results are applied to an Airy beam to show that the maxima of its probability density provide a discrete set of: caustics, wavefronts and potentials. The results presented here are a natural generalization of those obtained by Berry and Balazs in 1979 for an Airy beam. Finally, we remark that, in a natural manner, each maxima of the probability density of an Airy beam determines a Hamiltonian system.
AB - The main contribution of the present work is to use the probability density of an Airy beam to identify its maxima with the family of caustics associated with the wavefronts determined by the level curves of a one-parameter family of solutions to the Hamilton-Jacobi equation with a given potential. To this end, we give a classical mechanics characterization of a solution of the one-dimensional Schrödinger equation in free space determined by a complete integral of the Hamilton-Jacobi and Laplace equations in free space. That is, with this type of solution, we associate a two-parameter family of wavefronts in the spacetime, which are the level curves of a one-parameter family of solutions to the Hamilton-Jacobi equation with a determined potential, and a one-parameter family of caustics. The general results are applied to an Airy beam to show that the maxima of its probability density provide a discrete set of: caustics, wavefronts and potentials. The results presented here are a natural generalization of those obtained by Berry and Balazs in 1979 for an Airy beam. Finally, we remark that, in a natural manner, each maxima of the probability density of an Airy beam determines a Hamiltonian system.
KW - Airy beam
KW - Hamilton-Jacobi equation
KW - classical mechanics
KW - geometrical optics
KW - quantum mechanics
UR - http://www.scopus.com/inward/record.url?scp=85049390682&partnerID=8YFLogxK
U2 - 10.1088/2040-8986/aac5ba
DO - 10.1088/2040-8986/aac5ba
M3 - Artículo
SN - 2040-8978
VL - 20
JO - Journal of Optics (United Kingdom)
JF - Journal of Optics (United Kingdom)
IS - 7
M1 - 075602
ER -