We reexamine the Gouy phase in ballistic Airy beams (AiBs). A physical interpretation of our analysis is derived in terms of the local phase velocity and the Poynting vector streamlines. Recent experiments employing AiBs are consistent with our results. We provide an approach which potentially applies to any finite-energy paraxial wave field that lacks a beam axis.
© 2012 Optical Society of America
It is well-known that finite-energy 2D wave fields propagating in free space undergo an overall phase shift of π/2 rads (π rads for 3D waves) if they are compared with the transit of un-truncated plane waves. For aberration-free focused waves, Gouy first realized that the phase is delayed within its focal region . The origin of the Gouy phase (GP) shift is ascribed to the spatial confinement of the optical beam [2, 3], which leads to deviations in the wave front and, therefore, local alterations of the wavenumber in the near field [4,5]. The interest in the analysis of this effect persists nowadays because of its implication in many ultrafast phenomena that are dependent directly on the electric field rather than the pulse envelope such as electron emission from ionized atoms  and metal surfaces .
Recently Pang et al. studied the phase behavior of Airy beams (AiBs) . 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. In this Paper we reexamine the GP in AiBs. A physical interpretation of the GP is given in terms of the local phase velocity and the Poynting vector streamlines of AiBs. Potential applications of our results to recent experiments employing AiBs  are also outlined. They include optical manipulation of small particles , generation of curved plasma channels in air [13, 14], and generation of surface plasmons with parabolic trajectories [15–18], to mention a few.
Let us consider a (1 + 1)D wave field exp(ik0z) u which evolves along the z-axis with a carrier spatial frequency k0 = ω/c. Thus the wave function u satisfies the paraxial wave equation 2i∂ζu + ∂ssu = 0 expressed in terms of the normalized spatial coordinates s = x/w0 and ζ = z/zR. Here w0 is the beam width and denotes the propagation distance. Specifically, an AiB may be written as 
Topologic attributes of AiBs may be drawn without difficulty from its transverse spatial frequencyEquation (1) is directly derived from the Fresnel-Kirchhoff (FK) diffraction integral u = (2π)−1 ∫ũ(k)exp(iψ)dk, being ψ = ks − k2ζ/2 the phase distribution of a plane-wave spectral component with transverse spatial frequency k. The phase ϕ̃ + ψ of the integrand is highly oscillating, where ϕ̃ = arg(ũ). However it reaches a stationary point, that is ∂k[ϕ̃ +ψ] = 0, for two frequencies k± satisfying 19] (PSP) establishes that the FK diffraction integral has a predominant contribution of frequencies in the vicinities of k±. Inside the geometrical shadow s > a2 + ζ2/4 no stationary points may be found. In the light area, constructive interference is attained if the phase ϕ̃ + ψ at frequencies k+ and k− differs by an amount 2πm, where m is an integer. This condition provides the locus of points with peaks in intensity, which leads to
From geometrical grounds, the parameter k represents a particular light “ray” with linear trajectory (3) in the sζ-plane. The envelope (caustic) of this family of rays results from the solution given in Eq. (4) for m = 0, providing the ballistic signature of an AiB.
In Fig. 1 we show the spatial distribution of the magnitude |u| and the phase ϕ = arg(u) of the wave function given in Eq. (1) corresponding to an AiB of a = 0.1. The accelerating behavior of the AiB is limited, and out of the near field the interference-driven parabolic peaks fade away. To establish the boundaries of the near field, we point out that |ũ| falls off less than a half of its maximum in the interval , which represents the effective bandwidth of the AiB. Moreover, kmax is no more than the far field beam angle in the normalized coordinates, and kmax = 2.6 in Fig. 1. As a consequence, the length of the caustic is finite and it is observed in |ζ| < 2kmax. Along the beam waist ζ = 0 the energy is mostly localized in the region . In Fig. 1 the near field is bounded within the region |ζ| < 5.2 and |s| < 6.9, where 4 interference peaks are clearly formed.
We point out that the maximum of intensity is not placed exactly at points that belong to the curves (4) but they are slightly shifted to lower values of the transverse spatial coordinate s. This effect is not caused by the finite energy of the beam since it is observed for a = 0, but it occurs by a non-even symmetry of its spatial spectrum (2).
In the far field, however, the behavior of the AiB is completely different. Applying the PSP, no more than one spatial frequency k is of relevance. The resultant Fraunhofer pattern is u → (2πiζ)−1/2 ũ(k)uPW (s, ζ), valid in the limit |ζ| ≫ kmax for points of the contour C ≡ s = kζ + a2 − k2 taken from Eq. (3). Note that the paraxial wave field uPW = exp (iks − ik2ζ/2) corresponds to a non-truncated plane wave.
3. The Gouy phase
Let us go back with the evaluation of the GP in AiBs. The GP φG(ζ) is commonly estimated as the cumulative phase difference between a given paraxial field and a plane wave also traveling in the +z direction . Considering a centrosymmetric distribution of u around s = 0, the dephase φG(ζ) may be derived analytically as the difference ϕ(ζ) − limζ→−∞ ϕ evaluated along the beam axis. Due to the particular acceleration of the AiB, however, one cannot encounter a beam axis in this case.
In a more general approach, the GP might account for dephasing between the wave field u given in (1) and a plane wave uPW = exp (iψ) with a given tilt k ≠ 0. Following the discussion given above, the phase fronts of u and uPW become parallel in the far field around the light ray (3). In order to obtain φG(ζ) for the normalized angle k we employ ϕ − ks + k2ζ/2 instead of ϕ, which is now evaluated at points of the contour C. In Fig. 2(a) we plot φG(ζ) for different values of the zenith angle k. As expected, the value of the GP varies rapidly in the near field. Well beyond the near field, in the limit ζ → +∞, the GP shift approaches −π/2.
4. Physical interpretation
Let us give a physical interpretation of our approach, which in principle may be applied to any finite-energy wave field that lacks a beam axis. For that purpose, it is illustrative to rewrite the GP as the line integral
where Δk⃗ = ∇ϕ − ∇ψ and C represents a contour of integration (3) with startpoint at ζ → −∞. The form is exactly the same as that encountered when we calculate the work done by a resultant force Δk⃗ that varies along the path C. Here Δk⃗ is understood as the difference of the local wave vector k⃗i = k0ẑ + ∇ϕ of the AiB and that corresponding to the reference plane wave, k⃗PW = k0ẑ + ∇ψ. Note that k⃗PW approaches k0ẑ + (k/w0) x̂ to order k, and that k⃗PW ‖ dr⃗ over the contour C. This is of relevance since many physical processes, like the generation of curved plasma channels  and the optical manipulation of microparticles , depends openly on k⃗i, which is in direct proportion to the electromagnetic momentum and the time-averaged flux of energy .
Going from r⃗ to r⃗ +dr⃗ over C leads to a nonnegative contribution of the line integral (5) if (a) the wave vectors k⃗PW and k⃗i are nonparallel, and if (b) the wavenumber k0 of the reference plane wave and that ki = |k⃗i| of the field u are different. This is illustrated in Fig. 2(b). For a Gaussian beam φG = −π/4 − arctan(ζ)/2 at k = 0, where k⃗i ‖ k⃗PW but ki < k0. In AiBs, however, both angular and modular detuning of k⃗i are produced.
To examine the angular detuning, it is illustrative to represent k⃗i graphically by means of the Poynting vector streamlines (PSLs). Commonly employed with vector fields, the PSLs are tangent to the vector k⃗i and consequently they satisfy the differential equation k⃗i × dr⃗ = 0, that is dx/dz = (k⃗i· x̂)/(k⃗i· ẑ). The PSLs indicate the direction of wave propagation since they are perpendicular to the phase fronts. Under the paraxial approximation the equation for the PSLs reduces to ds/dζ = ∂sϕ in normalized coordinates. For an AiB we finally have
Some PSLs of our AiB are drawn in solid lines in Fig. 3(a) and 3(b). In contrast with the trajectories C of light rays (dashed lines), the PSLs hold ds/dζ = 0 at ζ = 0. Therefore, PSLs approach a parabola s = s0 + s0″ζ2/2 in the neighbourhood of the beam waist, where s″0 = as0 − aAi′(s0)2/Ai(s0)2 + 1/2. If a ≪ 1 and s0 ≥ 0 then s0″ ≈ 1/2 featuring a regular parabolic trajectory. However for sufficiently large values of −s0 ≫ 1 then s0″ < 0 revealing a concavity inversion along the semi-axis. Moreover, in the far field ds/dζ → k leading to exact solutions in the form of Eq. (3) as |ζ| → ∞. These straight lines represent the asymptotes of the PSLs.
Finally we analyze the modular detuning of the local wave vector with respect to k0. In fact, modular detuning of k⃗i implies a local deviation of the phase velocity vi = ω/ki with respect to the speed of light c = ω/k0 in vacuum . Taking into account the paraxial regime, the local wavenumber is given by ki = k0 +κ/zR, whereFig. 3 we plot the parameter κ operating as a trend indicator of the spatial variation of the phase velocity of the AiB depicted in Fig. 1. In the far field κ vanishes leading to a wave field with wavenumber ki = k0 and phase velocity vi = c. However κ presents a more complex behavior in the near field. Out of the geometrical shadow, κ < 0 and it drops near the peaks of intensity. This effect is associated with superluminality, which is well known in Gaussian beams and other kind of focused beams [21, 22]. Over the caustic of the AiB, however, κ ≈ 0 and it strictly vanishes if a = 0. On the contrary, κ grows sharply around the valleys of intensity, and for a = 0 it diverges due to the presence of phase singularities.
In conclusion, we have rewritten the Gouy phase as a line integral, whose form is exactly the same as that encountered when we calculate the work done by a resultant force Δk⃗ that varies along the given path. Here Δk⃗ is understood as the difference of the local wave vector of the AiB, which is in direct proportion to the electromagnetic momentum and the time-averaged flux of energy, and that corresponding to the reference plane wave. The equations describing the Poynting vector streamlines and the spatial variation of the phase velocity for these beams provide a general platform for exploring the flow of electromagnetic energy. These ideas can be used for a variety of applications. For example, our procedure facilitates a means to optically control the movement of particles in air and fluid , enabling precision calibration of the direction of forces exerted over trapped objects. In the case of the curved filaments produced in gases by self-bending AiBs , the forward emissions coming from different points and propagating along angularly resolved trajectories might be derived in a simple manner with extremely-high accuracy. As such, we envision that the work presented here will add a broad understanding to this new emerging field.
This research was funded by the Spanish Ministry of Economy and Competitiveness under the project TEC2009-11635.
References and links
1. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” Compt. Rendue Acad. Sci. (Paris) 110, 1251–1253 (1890).
2. S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26, 485–487 (2001). [CrossRef]
3. T. D. Visser and E. Wolf, “The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields,” Opt. Commun. 283, 3371–3375 (2010). [CrossRef]
4. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. 70, 877–880 (1980). [CrossRef]
5. A. E. Siegman, Lasers (University Science Books, Mill Valley, 1986).
6. G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, “Measurement of the phase of few-cycle laser pulses,” Phys. Rev. Lett. 91, 253004 (2003). [CrossRef]
7. A. Apolonski, P. Dombi, G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. Hänsch, and F. Krausz, “Observation of light-phase-sensitive photoemission from a metal,” Phys. Rev. Lett. 92, 073902 (2004). [CrossRef] [PubMed]
11. Y. Hu, G. Siviloglou, P. Zhang, N. Efremidis, D. Christodoulides, and Z. Chen, “Self-accelerating Airy beams: generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012), vol. 170, pp. 1–46.
12. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Natura Photon. 2, 675–678 (2008). [CrossRef]
13. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009). [CrossRef] [PubMed]
14. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010). [CrossRef]
15. L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy Beam generated by in-plane diffraction,” Phys. Rev. Lett 107, 1–4 (2011). [CrossRef]
16. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett 107, 116802 (2011). [CrossRef] [PubMed]
18. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36, 1164–1166 (2011). [CrossRef] [PubMed]
19. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 1999).
22. C. J. Zapata-Rodríguez and M. A. Porras, “Controlling the carrier-envelope phase of few-cycle focused laser beams with a dispersive beam expander,” Opt. Express 16, 22090–22098 (2008). [CrossRef] [PubMed]