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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 2 — Jan. 28, 2013
  • pp: 1615–1622
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Generation of linear and nonlinear propagation of three-Airy beams

Yi Liang, Zhuoyi Ye, Daohong Song, Cibo Lou, Xinzheng Zhang, Jingjun Xu, and Zhigang Chen  »View Author Affiliations


Optics Express, Vol. 21, Issue 2, pp. 1615-1622 (2013)
http://dx.doi.org/10.1364/OE.21.001615


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Abstract

We report the first experimental demonstration of the so-called three-Airy beams. Such beams represent a two-dimensional field that is a product (rather than simple superposition) of three Airy beams. Our experiments show that, in contrast to conventional Airy beams, this new family of Airy beams can be realized even without the use of truncation by finite apertures. Furthermore, we study linear and nonlinear propagation of the three-Airy beams in a photorefractive medium. It is found that a three-Airy beam tends to linearly diffract into a super-Gaussian-like beam, while under nonlinear propagation it either turns into three intensity spots with a self-defocusing nonlinearity or evolves into a self-trapped channel with a self-focusing nonlinearity.

© 2013 OSA

1. Introduction

In the past several years, Airy beams [1

1. G.A. Siviloglou and D.N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007). [CrossRef] [PubMed]

, 2

2. G.A. Siviloglou, J. Broky, A. Dogariu, and D.N. Christodoulides, “Observation of accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]

] have attracted great attentions because of their unusual features such as nondiffracting, self-accelerating and self-healing [3

3. Y. Hu, G.A. Siviloglou, P Zhang, N.K. Efremidis, D.N. 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, New York, 2012),1–46. [CrossRef]

]. 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

4. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008). [CrossRef]

13

13. I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108, 113903 (2012). [CrossRef] [PubMed]

]. 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

14. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35, 4045–4047 (2010). [CrossRef] [PubMed]

18

18. I. Chremmos, Z. Chen, N. K. Efremidis, and D. N. Christodoulides, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A 85, 023828 (2012). [CrossRef]

]. 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

19. E. Abramochkin and E. Razueva, “Product of three Airy beams,” Opt. Lett. 36, 3732–3734 (2011). [CrossRef] [PubMed]

]. 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

19. E. Abramochkin and E. Razueva, “Product of three Airy beams,” Opt. Lett. 36, 3732–3734 (2011). [CrossRef] [PubMed]

]:
tAi(r;b,c)=Ai(bx3y2+c)Ai(bx3y2+c)Ai(by+c),
(1)
where b and c are nonzero real. The Fourier transform of the three-Airy beam can be described as:
F[tAi(r;b,c)](K)=135/6πb2exp(2iK3sin3ϕ27b3)Ai(32/3c+2K234/3b2),
(2)
in which F[f(r)](K)=12π2exp[i(Kr)]f(r)dr is the 2D Fourier transform, K=kx2+ky2 and ϕ = arctan (ky/kx). From Eq. (1) and Eq. (2), it can be seen that this field remains the same under a rotation by an angle of 2π/3. Especially, the field has a local maximum when 32/3c = an = −1.108, −3.10, −4.82...(an is the nth maximum of Ai(x)), and the field vanishes when 32/3c = an′ = −2.320, −4.00, −5.52...(an′ is the nth zeros of Ai(x)). Figures 1[a(1–3)] and 1[b(1–3)] show the intensity and phase distribution of F[tAi(r; b,c)](K) for 32/3c = an and 32/3c = an′, respectively.

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).

As shown in Figs. 1(a) and 1(b), the parameter 32/3c 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 32/3c = an and dark for 32/3c = an′. The number of the rings increases with the parameter 32/3c. 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

19. E. Abramochkin and E. Razueva, “Product of three Airy beams,” Opt. Lett. 36, 3732–3734 (2011). [CrossRef] [PubMed]

]:
Ai(K2)~12πKexp(2K33),(K),
(3)
so the three-Airy beam has a flat top and its intensity decreases more quickly in the edge, features resulting from the negative-argument asymptotic behavior of the Airy function.

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

10. Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett. 35, 3952–3954 (2010). [CrossRef] [PubMed]

12

12. Y. Hu, Z. Sun, D. Bongiovanni, D. Song, C. Lou, J. Xu, Z. Chen, and R. Morandotti, “Reshaping the trajectory and spectrum of nonlinear Airy beams,” Opt. Lett. 37, 3201–3203 (2012). [CrossRef] [PubMed]

], 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

10. Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett. 35, 3952–3954 (2010). [CrossRef] [PubMed]

]. 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.

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.

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

19. E. Abramochkin and E. Razueva, “Product of three Airy beams,” Opt. Lett. 36, 3732–3734 (2011). [CrossRef] [PubMed]

]:
ψ0=1n3exp(αsn)Ai(sn+cn),sn=(anx+bny).
(4)
To produce the three-Airy beam tAi(r; b,c), s1 + s2 + s3 = 0, sn = bIm[exp(2nπi/3)(x+iy)], cn = c, which means a1″ + a2″ + a3″ = 0, b1 + b2 + b3 = 0 [19

19. E. Abramochkin and E. Razueva, “Product of three Airy beams,” Opt. Lett. 36, 3732–3734 (2011). [CrossRef] [PubMed]

]. Then, one could consider the accelerations of constituting Airy beams at three directions as vectors angled 2π/3 apart in a 2D plane. As a result, the vector-sum of the accelerations for three Airy beams is zero. So the whole three-Airy beam would exhibit no self-acceleration, and diffraction takes over instead.

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

20. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16, 9411–9416 (2008). [CrossRef] [PubMed]

, 21

21. J. Broky, G. A. Siviloglou, A. Dogariu, and D.N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008). [CrossRef] [PubMed]

]. 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

22. Z. Zhang, J. Liu, P. Zhang, P. Ni, J. Prakash, Y. Hu, D. Jiang, D. N. Christodoulides, and Z. Chen, “Trapping aerosols with optical bottle arrays generated through a superposition of multiple Airy beams,” to be published in Chinese Opt. Lett. (2013).

]. In addition, we also note a similar kind of triangular beam called accelerating regular polygon beams [23

23. H. Barwick, “Accelerating regular polygon beams,” Opt. Lett. 35, 4118–4120 (2010). [CrossRef] [PubMed]

]. 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 Δn=0.5ne3γ33E0/(1+I), in which ne = 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

10. Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett. 35, 3952–3954 (2010). [CrossRef] [PubMed]

].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.

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

5. Conclusion

Acknowledgments

This work was supported by the National Key basic Research Program of China ( 2010CB934101, 2013CB328702), the National Science Foundation of China (NSFC) ( 60908002, 10904078), International S&T cooperation program of China( 2011DFA52870), International cooperation program of Tianjin ( 11ZGHHZ01000), the 111 Project ( B07013) and the Fundamental Research Funds for the Central Universities.

References and links

1.

G.A. Siviloglou and D.N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007). [CrossRef] [PubMed]

2.

G.A. Siviloglou, J. Broky, A. Dogariu, and D.N. Christodoulides, “Observation of accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]

3.

Y. Hu, G.A. Siviloglou, P Zhang, N.K. Efremidis, D.N. 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, New York, 2012),1–46. [CrossRef]

4.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008). [CrossRef]

5.

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]

6.

A. Chong, W. Renninger, D.N. Christodoulides, and F.W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010). [CrossRef]

7.

T. Ellenbogen, N. Voloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009). [CrossRef]

8.

S. Jia, J. Lee, G. A. Siviloglou, D. N. Christodoulides, and J. W. Fleischer, “Diffusion-trapped Airy Beams in photorefractive Media,” Phys. Rev. Lett. 104, 253904 (2010). [CrossRef] [PubMed]

9.

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011). [CrossRef] [PubMed]

10.

Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett. 35, 3952–3954 (2010). [CrossRef] [PubMed]

11.

Z. Ye, S. Liu, C. Lou, P. Zhang, Y. Hu, D. Song, J. Zhao, and Z. Chen, “Acceleration control of Airy beams with optically induced refractive-index gradient,” Opt. Lett. 36, 3230–3232 (2011). [CrossRef] [PubMed]

12.

Y. Hu, Z. Sun, D. Bongiovanni, D. Song, C. Lou, J. Xu, Z. Chen, and R. Morandotti, “Reshaping the trajectory and spectrum of nonlinear Airy beams,” Opt. Lett. 37, 3201–3203 (2012). [CrossRef] [PubMed]

13.

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108, 113903 (2012). [CrossRef] [PubMed]

14.

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35, 4045–4047 (2010). [CrossRef] [PubMed]

15.

I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36, 1890–1892 (2011). [CrossRef] [PubMed]

16.

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36, 1842–1844 (2011). [CrossRef] [PubMed]

17.

P. Zhang, J. Prakash, Z. Zhang, M. Mills, N. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36, 2883–2885 (2011). [CrossRef] [PubMed]

18.

I. Chremmos, Z. Chen, N. K. Efremidis, and D. N. Christodoulides, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A 85, 023828 (2012). [CrossRef]

19.

E. Abramochkin and E. Razueva, “Product of three Airy beams,” Opt. Lett. 36, 3732–3734 (2011). [CrossRef] [PubMed]

20.

H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16, 9411–9416 (2008). [CrossRef] [PubMed]

21.

J. Broky, G. A. Siviloglou, A. Dogariu, and D.N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008). [CrossRef] [PubMed]

22.

Z. Zhang, J. Liu, P. Zhang, P. Ni, J. Prakash, Y. Hu, D. Jiang, D. N. Christodoulides, and Z. Chen, “Trapping aerosols with optical bottle arrays generated through a superposition of multiple Airy beams,” to be published in Chinese Opt. Lett. (2013).

23.

H. Barwick, “Accelerating regular polygon beams,” Opt. Lett. 35, 4118–4120 (2010). [CrossRef] [PubMed]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(350.5500) Other areas of optics : Propagation

ToC Category:
Physical Optics

History
Original Manuscript: November 5, 2012
Revised Manuscript: December 14, 2012
Manuscript Accepted: December 26, 2012
Published: January 15, 2013

Citation
Yi Liang, Zhuoyi Ye, Daohong Song, Cibo Lou, Xinzheng Zhang, Jingjun Xu, and Zhigang Chen, "Generation of linear and nonlinear propagation of three-Airy beams," Opt. Express 21, 1615-1622 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1615


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References

  1. G.A. Siviloglou and D.N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007). [CrossRef] [PubMed]
  2. G.A. Siviloglou, J. Broky, A. Dogariu, and D.N. Christodoulides, “Observation of accelerating Airy Beams,” Phys. Rev. Lett.99, 213901 (2007). [CrossRef]
  3. Y. Hu, G.A. Siviloglou, P Zhang, N.K. Efremidis, D.N. 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, New York, 2012),1–46. [CrossRef]
  4. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2, 675–678 (2008). [CrossRef]
  5. P. Polynkin, M. Kolesik, J.V. Moloney, G.A. Siviloglou, and D.N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science324, 229–232 (2009). [CrossRef] [PubMed]
  6. A. Chong, W. Renninger, D.N. Christodoulides, and F.W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics4, 103–106 (2010). [CrossRef]
  7. T. Ellenbogen, N. Voloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics3, 395–398 (2009). [CrossRef]
  8. S. Jia, J. Lee, G. A. Siviloglou, D. N. Christodoulides, and J. W. Fleischer, “Diffusion-trapped Airy Beams in photorefractive Media,” Phys. Rev. Lett.104, 253904 (2010). [CrossRef] [PubMed]
  9. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106, 213903 (2011). [CrossRef] [PubMed]
  10. Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett.35, 3952–3954 (2010). [CrossRef] [PubMed]
  11. Z. Ye, S. Liu, C. Lou, P. Zhang, Y. Hu, D. Song, J. Zhao, and Z. Chen, “Acceleration control of Airy beams with optically induced refractive-index gradient,” Opt. Lett.36, 3230–3232 (2011). [CrossRef] [PubMed]
  12. Y. Hu, Z. Sun, D. Bongiovanni, D. Song, C. Lou, J. Xu, Z. Chen, and R. Morandotti, “Reshaping the trajectory and spectrum of nonlinear Airy beams,” Opt. Lett.37, 3201–3203 (2012). [CrossRef] [PubMed]
  13. I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett.108, 113903 (2012). [CrossRef] [PubMed]
  14. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett.35, 4045–4047 (2010). [CrossRef] [PubMed]
  15. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett.36, 1890–1892 (2011). [CrossRef] [PubMed]
  16. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett.36, 1842–1844 (2011). [CrossRef] [PubMed]
  17. P. Zhang, J. Prakash, Z. Zhang, M. Mills, N. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett.36, 2883–2885 (2011). [CrossRef] [PubMed]
  18. I. Chremmos, Z. Chen, N. K. Efremidis, and D. N. Christodoulides, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A85, 023828 (2012). [CrossRef]
  19. E. Abramochkin and E. Razueva, “Product of three Airy beams,” Opt. Lett.36, 3732–3734 (2011). [CrossRef] [PubMed]
  20. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express16, 9411–9416 (2008). [CrossRef] [PubMed]
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