Flux trajectory analysis of Airy-type beams

Airy beams are solutions to the paraxial Helmholtz equation known for exhibiting shape invariance along their self-accelerated propagation in free space. These two properties are associated with the fact that they are not square integrable, that is, they carry infinite energy. To circumvent this drawback, families of so-called finite-energy Airy-type beams have been proposed in the literature and, in some cases, also implemented in the laboratory. Here an analysis of the propagation of this type of structured light beam is presented from a flux trajectory perspective with the purpose of better understanding the mechanisms that make infinite and finite energy beams exhibit different behaviors. As is shown, while the foremost part of the beam can be clearly and unambiguously associated with the well-known accelerating term, the rear part of the beam corresponds to a nearly homogeneous distribution of flow trajectories, particularly for long propagation distances. This is shown to be related to an effective transfer of trajectories between adjacent lobes (gradually, from the fore part of the beam to its rear part), which leads to smearing out the transverse flow along the rear part of the beam. This is in sharp contrast to the situation found in ideal Airy beams, where trajectories belonging to a given lobe of the intensity distribution remain the same all along the propagation. The analysis is supplemented with an also trajectory-based description of Young's experiment performed with finite-energy Airy beams to provide a dynamical understanding of the autofocusing phenomenon observed with circular Airy beams.