1. Introduction

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 [1]. 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 [6] and metal surfaces [7].

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. 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 [11] are also outlined. They include optical manipulation of small particles [12], generation of curved plasma channels in air [13, 14], and generation of surface plasmons with parabolic trajectories [15–18], to mention a few.

2. Formalism

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/z_R. Here w₀ is the beam width and $z_R=k_0{w}_0^2$ denotes the propagation distance. Specifically, an AiB may be written as [9]

Topologic attributes of AiBs may be drawn without difficulty from its transverse spatial frequency

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 $k^2<k_{\mathit{max}}^2=\left(\text{ln}2\right)/a$, which represents the effective bandwidth of the AiB. Moreover, k_max is no more than the far field beam angle in the normalized coordinates, and k_max = 2.6 in Fig. 1. As a consequence, the length of the caustic is finite and it is observed in |ζ| < 2k_max. Along the beam waist ζ = 0 the energy is mostly localized in the region $a^2-k_{\mathit{max}}^2\le s\le a^2$. 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.

 

Fig. 1 (a) |u| and (c) ϕ for an AiB with a = 0.1. The parabolic curves (4) are drawn in the near field (b) and (d) in dashed lines. In the far field, the numbered straight trajectories of light rays satisfy Eq. (3).

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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)u_PW (s, ζ), valid in the limit |ζ| ≫ k_max for points of the contour C ≡ s = kζ + a² − k² taken from Eq. (3). Note that the paraxial wave field u_PW = 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 φ_G(ζ) is commonly estimated as the cumulative phase difference between a given paraxial field and a plane wave also traveling in the +z direction [4]. Considering a centrosymmetric distribution of u around s = 0, the dephase φ_G(ζ) may be derived analytically as the difference ϕ(ζ) − lim_ζ→−∞ 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 u_PW = exp (iψ) with a given tilt k ≠ 0. Following the discussion given above, the phase fronts of u and u_PW become parallel in the far field around the light ray (3). In order to obtain φ_G(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 φ_G(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.

 

Fig. 2 (a) GP for k = {0, ±0.41, ±1.00, ±1.52}. The curves are displaced by π/2 from one to another, with increasing k down upward. The numbers code agrees with that applied in Fig. 1. (b) Geometrical interpretation of the GP in terms of the local wave vector k⃗_PW of a plane wave and that k⃗_i of a paraxial beam.

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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 k⃗_i = k₀ẑ + ∇ϕ of the AiB and that corresponding to the reference plane wave, k⃗_PW = k₀ẑ + ∇ψ. Note that k⃗_PW approaches k₀ẑ + (k/w₀) x̂ to order k, and that k⃗_PW ‖ dr⃗ over the contour C. This is of relevance since many physical processes, like the generation of curved plasma channels [13] and the optical manipulation of microparticles [12], depends openly on k⃗_i, which is in direct proportion to the electromagnetic momentum and the time-averaged flux of energy [20].

Going from r⃗ to r⃗ +dr⃗ over C leads to a nonnegative contribution of the line integral (5) if (a) the wave vectors k⃗_PW and k⃗_i are nonparallel, and if (b) the wavenumber k₀ of the reference plane wave and that k_i = |k⃗_i| of the field u are different. This is illustrated in Fig. 2(b). For a Gaussian beam φ_G = −π/4 − arctan(ζ)/2 at k = 0, where k⃗_i ‖ k⃗_PW but k_i < k₀. In AiBs, however, both angular and modular detuning of k⃗_i are produced.

To examine the angular detuning, it is illustrative to represent k⃗_i graphically by means of the Poynting vector streamlines (PSLs). Commonly employed with vector fields, the PSLs are tangent to the vector k⃗_i and consequently they satisfy the differential equation k⃗_i × dr⃗ = 0, that is dx/dz = (k⃗_i· x̂)/(k⃗_i· 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.

 

Fig. 3 Spatial distribution of κ and several PSLs for the AiB of Fig. 1. (c) Plots of κ over the numbered contours also representing asymptotes of the PSLs.

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Finally we analyze the modular detuning of the local wave vector with respect to k₀. In fact, modular detuning of k⃗_i implies a local deviation of the phase velocity v_i = ω/k_i with respect to the speed of light c = ω/k₀ in vacuum [19]. Taking into account the paraxial regime, the local wavenumber is given by k_i = k₀ +κ/z_R, where

5. Conclusions

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 [12], 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 [13], 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.