1. Introduction

In the past several years, Airy beams [1, 2] have attracted great attentions because of their unusual features such as nondiffracting, self-accelerating and self-healing [3]. These features bring about tremendous potential applications of the Airy beams in many areas including optical trapping and manipulation, laser filamentation, and nonlinear optics [4–13]. There are also other families of Airy beams synthesized by manipulating the Airy function, with a typical example being the abruptly autofocusing Airy beams [14–18]. These autofocusing Airy beams are circularly symmetric and can be obtained by wrapping around a one-dimensional (1D) Airy beam. The “circular Airy” beams possess features differing from the two-dimensional (2D) Airy beams formed by merely multiplying two orthogonal 1D Airy functions. In a recent study, a new family of Airy beams, namely, the three-Airy beams, has been proposed by multiplying three Airy functions [19]. This new family of Airy beams exhibits zero net self-acceleration, different from conventional 1D or 2D Airy beams, as their far-field propagation turns into either Laguerre-Gaussian or super-Gaussian-like profiles. Thus far, the three-Airy beams have not been explored in experiment.

In this paper we report on the first experimental observation of the three-Airy beams. Typically, the ideal 1D or 2D infinite-energy Airy beams cannot be obtained experimentally, so in practice only finite-energy Airy beams are created due to an exponential truncation factor. However, the three-Airy beams without the truncation factor exhibit finite energy and they can be realized in experiment without the use of truncation. Our experimental results show that these three-Airy beams no longer have the properties of nondiffracting and self-accelerating in free space. We provide an intuitive explanation by considering the accelerations of Airy beams as vectors. In addition, we study linear and nonlinear propagation of the three-Airy beams in a photorefractive medium. It is shown that a three-Airy beam turns into three intensity spots under a self-defocusing nonlinearity while it evolves into a self-trapped channel under a self-focusing nonlinearity.

2. Theory

First, let us introduce the three-Airy beams theoretically. With properly selected parameters, a three-Airy beam in the real space can be expressed as [19]:

 

Fig. 1 The intensity (central bright pattern) and phase (background pattern) distribution of F[tAi(r; b,c)](K) for (a1) a₁ = −1.108, (a2) a₂ = −3.10, (a3) a₁ = −4.82, (b1) a₁′ = −2.320, (b2) a₂′ = −4.00, (b3) a₃′ = −5.52. (c) The intensity profile of the three-Airy beam (solid line) in the Fourier space for a₁ = −1.108 and its Gaussian approximation (dashed line).

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As shown in Figs. 1(a) and 1(b), the parameter 3^2/3c determines the structure of the three-Airy beam and the number of rings in the intensity patterns. The center of the three-Airy beam is bright for 3^2/3c = a_n and dark for 3^2/3c = a_n′. The number of the rings increases with the parameter 3^2/3c. The three-Airy beam pattern in the Fourier space is similar to that of a Laguerre-Gaussian beam but the former has no angular momentum. In addition, the three-Airy beam has a cubic phase distribution in spectrum and a radially symmetric intensity shape with a super-Gaussian decrease. Figure 1(c) shows the intensity profile of the three-Airy beam in the Fourier space when a₁ = −1.108 and its Gaussian approximation. From Eq. (2), F[tAi(r; b,c)](K) ∝ Ai(K²), as discussed in [19]:

3. Experimental setup

Our experimental setup for generation of the three-Airy beams is depicted in Fig. 2. Similar to the case of our prior work [10–12], the Fourier transformation method is used to generate the three-Airy beams by using a phase mask from the function F[tAi(r; b,c)](K) with a₁ = −1.108. Specifically, an extraordinarily polarized Gaussian beam with a wavelength of λ = 532nm is launched to a spatial light modulator (SLM), where the cubic-phase pattern is imposed in order to generate a desired three-Airy beam. A Fourier transform lens is placed at a distance f behind the SLM to obtain the beam pattern. A biased 1-cm-long photorefractive SBN crystal is placed at a distance f behind the lens. By switching the polarity of the bias field, self-focusing and self-defocusing nonlinearity could be achieved [10]. The three-Airy beam and its Fourier spectrum are monitored by CCD cameras.

 

Fig. 2 Experimental setup for observation of the three-Airy beams. PC: personal computer; Mask: the phase pattern of the three-Airy beam for a₁ = −1.108; SLM: spatial light modulator; BS: beam splitter; L:lens; SBN: strontium-barium-niobate crystal.

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4. Results and discussion

4.1. Linear propagation of three-Airy beams

For the case of linear propagation in free space (without the biased SBN crystal), experimental results are shown in Fig. 3. The transverse intensity profile of the three-Airy beam at the origin (z = 0) is triangularly shaped [Fig. 3(a)] (with short tails hardly visible since the beam intensity is more localized as compared with that of a conventional Airy beam.But we can find the tails and the main lobe of the three-Airy beam are out of phase [Fig. 3(b)]). Figures 3(c) and 3(d) depict the corresponding transverse intensity profiles of the three-Airy beam at z = 2 mm, 4 mm, respectively. Notice the triangular pattern is inverted in Fig. 3(d). The far field pattern is shown in Fig. 3(e). From Figs. 3(a)– 3(e), we see that the three-Airy beam changes its shape during propagation with the location of the beam center unchanged. In addtion, as expected, the shape remains invariant by the rotation of 2π/3 at a given propagation distance. As such,the three-Airy beam does not exhibit the ability to freely accelerate, as compared with the conventional 1D or 2D Airy beams. This can be explained intuitively as follows:

 

Fig. 3 Experimental results (top row) of transverse intensity patterns for the three-Airy beam after linear propagation to different distances (a) z = 0 mm,(b) the phase relationship of the the three-Airy beam at z = 0 mm, (c) z = 2 mm, (d) z = 4 mm, (e) z = 100 mm. (f)–(j) Corresponding numerical simulation for the same distances. Patterns in (e, j) are zoomed out by a factor of 14.

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For a product of three Airy beams, the field can be expressed as [19]:

This suggests that one might be able to analyze the behavior of the family of Airy beams by considering their accelerations as vectors in the 2D plane. The total magnitude and direction of the Airy beams are determined by the components and the angle between them, which can be used to control the behavior of different families of the Airy beams. For instance, we can manipulate the angle between the components of the conventional 2D Airy Beams. The 2D Airy beams could have acceleration with different magnitude. We point out that the propagation dynamics of the three-Airy beam could be better explained through investigating the Poynting vector [20, 21]. Figures 3(f)– 3(j) depict the corresponding simulation results of the three-Airy beam in good agreement with experiments. Apparently, the initial beam structure and subsequent propagation behavior of the three-Airy beam are quite different from those of the simple superposition of three Airy beams [22]. In addition, we also note a similar kind of triangular beam called accelerating regular polygon beams [23]. Accelerating regular polygon beams can allow higher sampling densities in certain application because multiple accelerating intensity peaks appear at the vertices of the beams. These are different from the three-Airy beam whose intensity peak is in the center.

4.2. Nonlinear propagation of three-Airy beams

Next, we consider the nonlinear propagation of the three-Airy beam in a biased SBN crystal. Although the magnitude of nonlinearity cannot be easily measured in experiment, in such a biased crystal, the saturable nonlinearity for an e-polarized beam can be determined by $\mathrm{\Delta }n=-0.5n_e^3{\gamma }_{33}{E}_0/\left(1+I\right)$, in which n_e = 2.3 is the unperturbed refractive index, γ₃₃ = 280pm/V, E₀ is the amplitude of the bias field and I is the intensity of the beam normalized to the background illumination(The incident average intensity of the three-Airy beam 〈I₀〉 ≈ 0.5W/cm² here) [10].The three-Airy beam is launched along the z direction with its input shown Figs. 4(a1–c1). While there is no bias field, the three-Airy beam of undergoes linear propagation just as in free space [Figs. 4(a)].When a positive dc field of E₀ = 50 ×10³V/m is applied, the three-Airy beam experiences a self-focusing nonlinearity, and the beam self-traps into a more circularly symmetric channel[Fig. 4(b2)]. In addition, its Fourier spectrum in k-space is focused more toward the center with three tails in the directions [Fig. 4(b3)] equally angled as in the input [Fig. 4(b1)]. By changing the polarity of the bias field (to E₀ = −35 ×10³V/m), the three-Airy beam experiences a self-defocusing nonlinearity. In this latter case, the shape of the three-Airy beam is less affected but overall the beam spreads much more as compared to the linear case[Fig. 4(c2)]. Furthermore, its k-space spectrum reshapes into three major spots, as shown in Fig. 4(c3). These experimental results are corroborated with our numerical simulation, as shown below.

 

Fig. 4 Experimental results of a three-Airy beam propagating through a 1 cm-long nonlinear crystal. (a1)–(c1) input, (a2)–(c2) output intensity patterns, and (a3)–(c3) k-space spectrum corresponding to the second column. (a) without nonlinearity, (b) with self-focusing nonlinearity, and (c) with self-defocusing nonlinearity.

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In Fig. 5, numerical results of the propagation of the three-Airy beam are shown under similar conditions corresponding to Fig. 4. Moreover, we point out that the triangular shape of the three-Airy beam cannot maintain after long enough propagation. In the linear case, it diffracts into a super-Gaussian-like pattern, but it turns into three intensity spots under the self-defocusing nonlinearity [Figs. 5(a4) and 5(c4)]. In contrast, the beam could preserve its self-trapped pattern for a long distance under the self-focusing nonlinearity, i.e., the three-Airy beam could evolve into a self-trapped channel as seen in Fig. 5(b4).

 

Fig. 5 Numerical simulation corresponding to Fig. 4. (a1)–(c1) input, (a2)–(c2) output intensity patterns, (a3)–(c3) k-space spectrum corresponding to the second column. (a) no nonlinearity, (b) with self-focusing nonlinearity, and (c) with self-defocusing nonlinearity. (a4)–(c4) Side view of 4 cm propagation through the SBN crystal

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5. Conclusion

In conclusion, we have realized the first experimental observation of three-Airy beams and studied their linear and nonlinear propagation dynamics. It is shown that the initial beam structure of the three-Airy beams cannot maintain under linear propagation or under the action of self-defocusing nonlinearity. Under self-focusing nonlinearity, such beams could evolve into self-trapped channels, much similar to a single Gaussian beam but quite different from a single Airy beam. We also considered the accelerations of Airy beams from the vector point of view to understand the overall behavior of the beam propagation. Our results may find applications in beam shaping and related optical design.