The Gouy phase of Airy beams

Abstract

The phase behavior of Airy beams is studied, and their Gouy phase is defined. Analytic expressions for the idealized, infinite-energy type beam are derived. They are shown to be excellent approximations for finite-energy beams generated under typical experimental conditions. © 2011 Optical Society of America OCIS codes: 050.1960, 350.5030. Beams that do not spread on propagation, so-called nondiffracting beams, have attracted considerable attention since they were discovered by Durnin et al. [1][2][3]. A special type of such beams are the Airy beams described by Berry and Balazs in the context of quantum mechanics [4]. These beams have the remarkable property that they "accelerate" away from the original direction of propagation. Airy beams are idealizations, because they carry an infinite amount of energy. Siviloglou and Christodoulides discussed how an exponentially modulated Airy function source would produce a finite-energy beam, which would retain its nondiffracting and accelerating behavior over an appreciable propagation distance [5]. After the experimental realization of such a beam [6], several studies have been devoted to their properties [7][8][9][10], and a number of applications are being pursued. For instance, the "self-healing" capacity of Airy beams [11] makes them excellent candidates for optical communication through turbulent media [12]. Other intriguing applications are the generation of curved plasma channels [13], and the manipulation of particles along bends in labs-on-achip [14]. Traditionally, the term Gouy phase describes how the phase of a monochromatic, focused field differs from that of a plane wave with the same frequency (see [15] and the references therein). Recently, however, it has also been used to describe the phase of a nondiffracting Bessel beam [16]. In this Letter we study the phase behavior of both finite-energy and infinite-energy Airy beams. By comparing their phase to that of a suitable reference field, their Gouy phase can be defined. A good understanding of the phase properties of Airy beams is of importance in interferometric or remote-sensing applications employing them.Consider a monochromatic, one-dimensional beamlike wave field Uðx; z; ωÞ that propagates in the positive zdirection, and can be written as Uðx; z; ωÞ ¼ ϕðx; zÞe iðkz−ωtÞ ;with the envelope ϕðx; zÞ a solution of the paraxial wave equationHere k ¼ ω=c is the wavenumber associated with frequency ω, c denotes the speed of light, and t the time.A possible solution of Eq. (2) is the so-called Airy beam, given by the expression [4] ϕðs; ξÞwith Ai the Airy function, s ¼ x=x 0 a dimensionless transverse coordinate, and ξ ¼ z=kx 2 0 a normalized propagation distance. In the remainder the constant x 0 is taken to be positive, and the time-dependent part of the wave field is suppressed. An example of the intensity distribution of an Airy beam is shown in Fig. 1, from which both the diffraction-free propagation and the transverse acceleration can be seen.Because of its curved trajectory, we define the Gouy p...

Phase anomalies in Talbot light carpets of self-images

“…In optical waves, various classes of beams exhibit the Gouy phase. Examples are general higher Gaussian modes like Hermite-Gaussian and Laguerre-Gaussian beams [44,[57][58][59][60], more specifically a vortex beam [61,62], a radially polarized beam [63], the Airy beam [64], and the Bessel beam [65,66]. In addition to such optical beams, surface plasmon-polaritons [67], matter waves [41], scattered hotspots (i.e., a photonic nanojet) [32], and diffracted hotspots (i.e., the spot of Arago) [68,69] also show axial phase shifts.…”

Considerations on the electromagnetic flow in Airy beams based on the Gouy phase

“…Recently Pang et al studied the phase behavior of Airy beams (AiBs) [8]. Because of its curved trajectory [9,10], they defined the GP of an AiB as the difference between its phase and that of a diverging cylindrical wave, the latter considered a suitable reference field.…”

Wavefront spacing and Gouy phase in presence of primary spherical aberration

“…Its physical origin has been discussed in [9]. Recently, it has been theoretically investigated in a variety of applications, such as highnumerical aperture systems [10,11], nondiffracting beams [12,13], and partially coherent focused fields [14]. Experimental observations were reported in, e.g., [15][16][17][18][19][20][21].…”