Reflection and refraction of an Airy beam at a dielectric interface

Abstract

Reflection and refraction of a finite-power Airy beam at the interface between two dielectric media are investigated analytically and numerically. The formulation takes into account the paraxial nature of the optical beams to derive convenient field evolution equations in coordinate frames moving along Snell's refraction and reflection axes. Through numerical simulations, the self-accelerating dynamics of the Airy-like refracted and reflected beams are observed. Of special interest are the cases of critical incidence at Brewster and total-internal-reflection (TIR) angles. In the former case, we find that the reflected beam achieves self-healing, despite the severe suppression of a part of its spectrum, while, in the latter case, the beam remains nearly unaffected except for the GoosHänchen shift. The self-accelerating quality persists even if the beam is trapped by multiple TIRs inside a dielectric film. The grazing incidence of an Airy beam at the interface between two media with close refractive indices is also investigated, revealing that the interface can act as a filter depending on the beam scale and tilt. We finally consider reverse refraction and perfect imaging of an Airy beam into a left-handed medium.

Accelerating beam propagation in refractive-index potentials

“…By inspection of σ , we see that for a steep enough potential α 2βγ the reflected Airy wave becomes identical to the incident. This limiting case is in agreement with [32] where the interface dynamics of Airy beams were studied. In the other limit case where the index gradient is relatively small 2βγ α, the coefficient of the parabolic term of the trajectory becomes α/(8β) and is independent of γ .…”

“…In particular, we investigate two different generic classes of potentials, which are (a) spatially localized potentials whose index contrast goes to zero outside a bounded region and (b) sigmoid potentials that have different constant indices outside an intermediate region where the index changes smoothly. We show that in contrast to the case of abrupt dielectric interfaces [32], where the reflected and refracted waves have the same structure as the incident Airy wave, for smoothly varying potentials the dynamics are more complicated. In particular, we find that, in general, both the reflected and refracted beams do not follow parabolic trajectories and thus do not have the functional form of an Airy wave.…”

“…(31)-(34), whereas the trajectory of the reflected beam is given by Eqs. (31), (32), and (35). Specifically, the upper part of the left branch of the cusp (which satisfies the condition x < 0) has a parabolic structure and is given by Eq.…”

“…Specifically, the Airy wave can follow trajectories that are different from the parabolic, depending on the particular choice of f (z) [4,31]. Furthermore, and in connection to this study, reflection and refraction of Airy beams at the interface between two dielectric media was investigated in [32]. The nonlinear problem was also studied numerically by solving the nonlinear Helmholtz equation [33].…”

“…The nonlinear problem was also studied numerically by solving the nonlinear Helmholtz equation [33]. However, in these works [32,33] the study is limited to abrupt interfaces between two different media.…”