Canonical energy backflow in Airy beams

In this work, we are the first, to our knowledge, to demonstrate both analytically and numerically that in the cross-section of a 2D paraxial accelerating Airy beam propagating on a parabolic trajectory, there are areas where a canonical energy backflow occurs. These areas arise in the far sidelobes of the Airy beam characterized by a subwavelength local period (superoscillation areas). At an arbitrary propagation distance from the initial plane, there is a threshold transverse coordinate, such that at all smaller negative values of the transverse coordinate, the longitudinal component of the canonical energy flow is negative. For a nonparaxial Airy beam, we derive an explicit analytical expression for the canonical energy flow near the initial plane. The energy backflow is revealed to occur near the intensity null, where a phase jump of π takes place. Near the intensity null, the phase derivative with respect to the longitudinal coordinate takes large negative values, leading to the longitudinal wavevector projection becoming larger than the wavenumber of light and directed oppositely. The maximum energy backflow is found to be about one-fifth of the direct maximum energy flow. Results of the numerical simulation for the paraxial and non-paraxial Airy beams are shown to agree with the theoretical prediction.