Accelerating parabolic beams

Abstract

We demonstrate the existence of accelerating parabolic beams that constitute, together with the Airy beams, the only orthogonal and complete families of solutions of the two-dimensional paraxial wave equation that exhibit the unusual ability to remain diffraction-free and freely accelerate during propagation. Since the accelerating parabolic beams, like the Airy beams, carry infinite energy, we present exact finite-energy accelerating parabolic beams that still retain their unusual features over several diffraction lengths. © 2008 Optical Society of America OCIS codes: 050.1940, 260.1960.5500.Recently Siviloglou et al. [1][2][3] called the attention of the optics community by demostrating the existence of the accelerating Airy beams. These beams are finite-energy solutions of the paraxial wave equation that have the unusual ability to remain diffraction free while having a quadratic transverse shift during propagation over long distances. The pure Airy beams are perfectly diffractionless and have a quadratic transverse shift during propagation but cannot be physically realizable, because they have infinite energy. However, the Airy beams have finite energy and retain these unusual properties over long distances.In this Letter, we introduce the accelerating parabolic beams and prove that they are, together with the two-dimensional Airy beams, the only orthogonal and complete family of explicit solutions of the twodimensional paraxial wave equation that remain diffraction free and freely accelerate during propagation. We also introduce a finite-energy version of the accelerating parabolic beams that can be physically realizable and exhibit the unusual properties of the ideal accelerating parabolic beams over a finite propagation distance.We begin our analysis by considering the normalized paraxial wave equation ٌ 2 + i‫ץ‬ s = 0, where ٌ 2 = ‫ץ‬ uu + ‫ץ‬ vv , ͑u , v͒ = ͑x , y͒ / represents the dimensionless transverse coordinates, is an arbitrary transverse scale, s = z /2k 2 is the normalized propagation distance, and k is the wave number.We know that the desired properties of the beam, that is, to remain diffraction free and to exhibit a transverse translation during propagation, can be expressed by ͉͑u,v,s͉͒ = ͉͑u + ⌬͑s͒,v,0͉͒, ͑1͒

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“…Despite their not being explicitly Gaussian, other types of propagation-invariant beams, such as Bessel [6,7], Mathieu [8], Airy [4,5], and parabolic [56] beams, correspond to the limits of the structured Gaussian beams described here. These other beams are idealized solutions that involve infinite power, corresponding to the limit N → ∞ in particular regions of the physical disk.…”

Bessel-like optical beams with arbitrary trajectories

“…These waves emerged in 2007 with the introduction of Airy beams [6], a concept stimulated by quantum mechanics [7]. Parabolic accelerating beams [8] are another characteristic example. These beams (mostly Airy) have also found several applications for light trajectory control and navigation around objects, micromanipulation, surface plasmon routing and curved plasma filaments and autofocusing (see [9] for a recent review).…”

Engineering light-matter interaction for emerging optical manipulation applications

“…Examples of accelerating beam include Airy beams [24] and other parabolic beams [86], which are quasi-invariant beams: their intensity patterns appear invariant in an accelerating frame (i.e., the overall pattern is preserved but exhibits transverse shifts during propagation). The parabolic trajectory of an Airy beam in the axial section is shown in Figure 5A from [24], which also shows a finite quasi-invariant range due to truncations from limited apertures of the optical system.…”