Stable and unstable Airy-related caustics and beams

Abstract

Optical beams with an underlying caustic structure are stable under perturbation if the caustics belong to the catastrophe-theory classification; otherwise they are unstable. The original Airy beam in two spatial dimensions, with its curved caustic, is stable in this sense. But the separable Airy-product beam in three-dimensions is unstable: under separability-breaking perturbation, it unfolds into the hyperbolic umbilic diffraction catastrophe, which is stable. By including initial phase factors, a variety of new exact solutions of the paraxial wave equation can be generated, corresponding to Pearcey and higher-catastrophe beams with stable caustics, and with the associated diffraction catastrophes appearing in their canonical forms or as deformations of these.

Minimal analytical model for undular tidal bore profile; quantum and Hawking effect analogies

“…In optics, an Airy function that everyone can see describes 'supernumerary rainbows', which are wave oscillations decorating the directional caustic [30,31] resulting from focusing of sunlight by raindrops. Currently much studied [32,33], also in optics, are 'Airy beams': propagating along curved paths without spreading. And on a cosmic scale, gravitational lensing of light is dominated by caustics [34,35], but the associated Airy functions are either too small to see, or obscured by decoherence [36].…”

Airy wavepackets are Perelomov coherent states

“…one surprisingly discovers that, these Perelomov coherent states are actually the Airy It is worth mentioning that unlike harmonic oscillator coherent states, the peak of the probability density of these coherent states does not traverse a classical trajectory, which in this system is the one without any acceleration, but rather traverses along a caustic of classical trajectories, as elucidated by Berry and Balazs, 1 and Berry. 21 It is worth emphasizing that, from this treatment it becomes apparent that the Galilean invariance of free Schrödinger equation has played a vital role in realisation of these coherent states in the system. The behaviour of Schrödinger equation under Galilean transformation and its generalisation, in this context, from the view point of gravitation has been studied in detail by Greenberger.…”

Shaping light beams in nonlinear processes using structured light and patterned crystals

“…  which will create an accelerating Airy beam in 2D. We note that this is a specific separable solution of the equation, which is unstable to separability-breaking perturbations, where it unfolds into the hyperbolic umbilic diffraction catastrophe [78]. Luckily, separating the different dimensions in optical setups that rely on free-space propagation is very simple, so the issue of stability of the solution is usually of no direct consequence.…”