Wave analysis of airy beams

Abstract

The Airy beams are analyzed in order to provide a cogent physical explanation to their intriguing features which include weak diffraction, curved propagation trajectories in freespace, and self healing. The asymptotically exact analysis utilizes the method of uniform geometrical optics (UGO), and it is also verified via a uniform asymptotic evaluation of the KirchhoffHuygens integral. Both formulations are shown to fully agree with the exact Airy beam solution in the paraxial zone where the latter is valid, but they are also valid outside this zone. Specifically it is shown that the beam along the curved propagation trajectory is not generated by contributions from the main lobe in the aperture, i.e., it is not described by a local wave-dynamics along this trajectory. Actually, this beam is identified as a caustic of rays that emerge sideways from points in the initial aperture that are located far away from the main lobe. The field of these focusing rays, described here by the UGO, fully agrees with the Airy beam solution. These observations explain that the "weakdiffraction" and the "self healing" properties are generated, in fact, by a continuum of sideways contributions to the field. The uniform ray representation provides a systematic framework to synthesize aperture sources for other beam solutions with similar properties in uniform or in non-uniform media.

Reflection and refraction of an Airy beam at a dielectric interface

“…As first predicted theoretically and later observed experimentally [3], the self-accelerating property is manifested by a parabolic, ballistic-like trajectory that the beam's local intensity features follow in space, giving the impression of a projectile moving under the influence of gravity [4]. This peculiar behavior of light bending in the absence of refractive index gradients is due to the inherent chirped phase modulation of the Airy function, which causes the constituent beam rays to form a parabolic caustic in space [5].…”

“…3(b), the features of the Airy profile have been recovered by depth z r 3 cm. For an Airy beam whose lobes develop to negative x, it is the positive spatial frequencies (ξ > 0) that correspond to the rays forming the parabolic caustic [5]. Hence, if the angle of incidence θ 1 is slightly increased beyond the Brewster angle so that the zero of the reflection coefficient Rξ β 0 occurs at some negative ξ, then the reflected caustic will be essentially unaffected.…”

Accelerating and Abruptly-Autofocusing Beam Waves in the Fresnel Zone of Antenna Arrays

“…An accelerating beam results from the formation of a curved caustic which in turn follows from a chirped-phase modulation of the input condition u(x, 0) [13]. To see this, assume that the latter can be described by a slowly-varying envelope A(x) modulated by the phase q(x), i.e.…”

“…A ray-optics analysis of Airy beams [13] reveals that their spatial acceleration results from the curved caustic that envelopes the rays emitted from the input aperture (Fig. 2).…”

Properties of Airy-Gauss Beams in the Fractional Fourier Transform Plane

“…The Wigner distribution function [2] and wave analysis [3] have been used to explain the intriguing features of an Airy beam. The self-healing [4], Poynting vector and angular momentum [5], ballistic dynamics [6], beam propagation factor [7], fractional Fourier transform (FRFT) [8], and far-field divergent properties [9] of an Airy beam have been investigated, respectively.…”