## Dual Airy beam |

Optics Express, Vol. 18, Issue 22, pp. 23504-23516 (2010)

http://dx.doi.org/10.1364/OE.18.023504

Acrobat PDF (1071 KB)

### Abstract

We derive a general form of Airy wave function which satisfies paraxial equation of diffraction. Based on this, we propose a new form of Airy beam, which is composed of two symmetrical Airy beams which accelerate mutually in the opposite directions. This ‘dual’ Airy beam shows several distinguishing features: it has a symmetric transverse intensity pattern and improved self-regeneration property. In addition, we can easily control the propagation direction. We also propose ‘quad’ Airy beam, which forms a rectangular shaped optical array of narrow beams that travel along a straight line. We can control its propagation direction without changing transverse intensity patterns. These kinds of superposed optical beams are expected to be useful for various applications with their unique properties.

© 2010 OSA

## 1. Introduction

1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264–267 (1979). [CrossRef]

3. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express **18**(8), 8440–8452 (2010). [CrossRef] [PubMed]

4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef]

5. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**(3), 207–209 (2008). [CrossRef] [PubMed]

7. Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. **35**(13), 2260–2262 (2010). [CrossRef] [PubMed]

8. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics **4**(2), 103–106 (2010). [CrossRef]

9. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. **35**(12), 2082–2084 (2010). [CrossRef] [PubMed]

10. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics **2**(11), 675–678 (2008). [CrossRef]

11. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**(1), 15–28 (2005). [CrossRef]

12. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. **151**(4-6), 207–211 (1998). [CrossRef]

13. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. **25**(20), 1493–1495 (2000). [CrossRef]

14. H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express **18**(19), 20384–20394 (2010). [CrossRef] [PubMed]

15. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature **419**(6903), 145–147 (2002). [CrossRef] [PubMed]

16. D. G. Grier, “A revolution in optical manipulation,” Nature **424**(6950), 810–816 (2003). [CrossRef] [PubMed]

9. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. **35**(12), 2082–2084 (2010). [CrossRef] [PubMed]

## 2. General form of the Airy wave function

*ϕ*is the Airy wave function,

4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef]

5. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

*k*denotes a wave number which is defined as

*n*is the refractive index of the medium and

*z*is physically meaningful only when we have

*a*,

6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**(3), 207–209 (2008). [CrossRef] [PubMed]

*a*and

7. Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. **35**(13), 2260–2262 (2010). [CrossRef] [PubMed]

## 3. Formulation and numerical results of dual Airy beam

11. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**(1), 15–28 (2005). [CrossRef]

12. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. **151**(4-6), 207–211 (1998). [CrossRef]

3. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express **18**(8), 8440–8452 (2010). [CrossRef] [PubMed]

*α*, and the other a directional factor

*β*. Each Airy beam’s

*β*, independent of

*α*. Figure 3 confirms this property: the added parameter

*β*determines the propagation direction of dual Airy beams. Calculation specifications are shown in Table 3 .

## 4. Numerical results for (2 + 1)D case - quad Airy beam

*z*axis can be calculated explicitly as (see Appendix B)

15. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature **419**(6903), 145–147 (2002). [CrossRef] [PubMed]

## 5. Conclusion

## Appendix A: Proof of Eq. (17)

## Appendix B: Proof of Eq. (19)

*z*axis can be calculated from the following relation:from which we can obtain Eq. (19).

## Acknowledgment

## References and links

1. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

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

3. | Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express |

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

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

6. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. |

7. | Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. |

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

9. | A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. |

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

11. | D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. |

12. | Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. |

13. | J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. |

14. | H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express |

15. | V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature |

16. | D. G. Grier, “A revolution in optical manipulation,” Nature |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(140.3300) Lasers and laser optics : Laser beam shaping

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 29, 2010

Revised Manuscript: October 20, 2010

Manuscript Accepted: October 21, 2010

Published: October 22, 2010

**Citation**

Chi-Young Hwang, Dawoon Choi, Kyoung-Youm Kim, and Byoungho Lee, "Dual Airy beam," Opt. Express **18**, 23504-23516 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23504

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### References

- M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]
- J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef] [PubMed]
- Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]
- G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008). [CrossRef] [PubMed]
- Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35(13), 2260–2262 (2010). [CrossRef] [PubMed]
- A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]
- A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef] [PubMed]
- J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]
- D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005). [CrossRef]
- Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998). [CrossRef]
- J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000). [CrossRef]
- H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express 18(19), 20384–20394 (2010). [CrossRef] [PubMed]
- V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef] [PubMed]
- D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef] [PubMed]

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