## Generation of linear and nonlinear propagation of three-Airy beams |

Optics Express, Vol. 21, Issue 2, pp. 1615-1622 (2013)

http://dx.doi.org/10.1364/OE.21.001615

Acrobat PDF (1815 KB)

### 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

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]

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]

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]

## 2. Theory

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

*b*and

*c*are nonzero real. The Fourier transform of the three-Airy beam can be described as: in which

*ϕ*= arctan (

*k*/

_{y}*k*). From Eq. (1) and Eq. (2), it can be seen that this field remains the same under a rotation by an angle of 2

_{x}*π*/3. Especially, the field has a local maximum when 3

^{2/3}

*c*=

*a*= −1.108, −3.10, −4.82...(

_{n}*a*is the

_{n}*n*th maximum of Ai(x)), and the field vanishes when 3

^{2/3}

*c*=

*a*′ = −2.320, −4.00, −5.52...(

_{n}*a*′ is the

_{n}*n*th 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 3

^{2/3}

*c*=

*a*and 3

_{n}^{2/3}

*c*=

*a*′, respectively.

_{n}^{2/3}

*c*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

^{2/3}

*c*=

*a*and dark for 3

_{n}^{2/3}

*c*=

*a*′. The number of the rings increases with the parameter 3

_{n}^{2/3}

*c*. 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

*a*

_{1}= −1.108 and its Gaussian approximation. From Eq. (2),

**F**[

*tAi*(

**r**;

*b*,

*c*)](

**K**) ∝

*Ai*(

*K*

^{2}), as discussed in [19

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

## 3. Experimental setup

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. 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]

**F**[

*tAi*(

**r**;

*b*,

*c*)](

**K**) with

*a*

_{1}= −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]

## 4. Results and discussion

### 4.1. Linear propagation of three-Airy beams

*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 2

*π*/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:

**36**, 3732–3734 (2011). [CrossRef] [PubMed]

*tAi*(

**r**;

*b*,

*c*),

*s*

_{1}+

*s*

_{2}+

*s*

_{3}= 0,

*s*=

_{n}*b*Im[exp(2

*nπi*/3)(

*x*+

*iy*)],

*c*=

_{n}*c*, which means

*a*

_{1}″ +

*a*

_{2}″ +

*a*

_{3}″ = 0,

*b*

_{1}+

*b*

_{2}+

*b*

_{3}= 0 [19

**36**, 3732–3734 (2011). [CrossRef] [PubMed]

*π*/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.

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]

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

### 4.2. Nonlinear propagation of three-Airy beams

*n*= 2.3 is the unperturbed refractive index,

_{e}*γ*

_{33}= 280

*pm*/

*V*,

*E*

_{0}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 〈

*I*

_{0}〉 ≈ 0.5

*W*/

*cm*

^{2}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]

*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

*E*

_{0}= 50 ×10

^{3}V/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

*E*

_{0}= −35 ×10

^{3}V/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.

## 5. Conclusion

## Acknowledgments

## References and links

1. | G.A. Siviloglou and D.N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

2. | G.A. Siviloglou, J. Broky, A. Dogariu, and D.N. Christodoulides, “Observation of accelerating Airy Beams,” Phys. Rev. Lett. |

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 |

4. | J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics |

5. | P. Polynkin, M. Kolesik, J.V. Moloney, G.A. Siviloglou, and D.N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science |

6. | A. Chong, W. Renninger, D.N. Christodoulides, and F.W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics |

7. | T. Ellenbogen, N. Voloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics |

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. |

9. | I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. |

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. |

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. |

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. |

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. |

14. | N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. |

15. | I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. |

16. | D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. |

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. |

18. | I. Chremmos, Z. Chen, N. K. Efremidis, and D. N. Christodoulides, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A |

19. | E. Abramochkin and E. Razueva, “Product of three Airy beams,” Opt. Lett. |

20. | H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express |

21. | J. Broky, G. A. Siviloglou, A. Dogariu, and D.N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express |

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. |

**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

- G.A. Siviloglou and D.N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007). [CrossRef] [PubMed]
- G.A. Siviloglou, J. Broky, A. Dogariu, and D.N. Christodoulides, “Observation of accelerating Airy Beams,” Phys. Rev. Lett.99, 213901 (2007). [CrossRef]
- 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]
- J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2, 675–678 (2008). [CrossRef]
- 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]
- 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]
- T. Ellenbogen, N. Voloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics3, 395–398 (2009). [CrossRef]
- 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]
- I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106, 213903 (2011). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett.35, 4045–4047 (2010). [CrossRef] [PubMed]
- I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett.36, 1890–1892 (2011). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- E. Abramochkin and E. Razueva, “Product of three Airy beams,” Opt. Lett.36, 3732–3734 (2011). [CrossRef] [PubMed]
- H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express16, 9411–9416 (2008). [CrossRef] [PubMed]
- J. Broky, G. A. Siviloglou, A. Dogariu, and D.N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16, 12880–12891 (2008). [CrossRef] [PubMed]
- 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).
- H. Barwick, “Accelerating regular polygon beams,” Opt. Lett.35, 4118–4120 (2010). [CrossRef] [PubMed]

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