## Nonparaxial diffraction analysis of Airy and SAiry beams

Optics Express, Vol. 17, Issue 25, pp. 22432-22441 (2009)

http://dx.doi.org/10.1364/OE.17.022432

Acrobat PDF (552 KB)

### Abstract

We theoretically analyze Airy beams by solving the exact vectorial Helmholtz equation using boundary conditions at a diffraction aperture. As result, the diffracted beams are obtained in the whole space; thus, we demonstrate that the parabolic trajectories are larger than those previously reported, showing that the Airy beams start to form before the Fourier plane. We also demonstrate the possibility of using a new type of Airy beams (SAiry beams) with finite energy that can be generated at the focal plane of the lens due to diffraction by a circular aperture of a spherical wave modified by a cubic phase. The finite energy ensured by the principle of conservation of energy of a diffracted beam.

© 2009 Optical Society of America

## 1. Introduction

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

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

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

4. J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip **9**, 1334–1336 (
2009). [CrossRef] [PubMed]

5. P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science **324**, 229–232 (
2009). [CrossRef] [PubMed]

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

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

8. 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]

9. M. A. Bandres and J. Gutierrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express **15**, 16719–16728 (
2007). [CrossRef] [PubMed]

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

## 2. Nonparaxial vectorial diffraction integrals

*E*⃗

*on the plane z=0:*

_{o}*R*=[(

*x*-

*xo*)

^{2}+(

*y*-

*yo*)

^{2}+

*z*

^{2}]

^{1/2}. The propagation properties of vectorial beams can be studied using the Eqs. (2) and (3). Assuming that after the lens (for analysis of experimental set-up see for example references [1

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

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

4. J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip **9**, 1334–1336 (
2009). [CrossRef] [PubMed]

*a*is the aperture radius and

*D*={(

*x*,

_{o}*y*)\(

_{o}*x*

^{2}

*+*

_{o}*y*

^{2}

*)≤*

_{o}*a*

^{2}}.

*α*is the linearly polarized angle. In this work, we are going to analyze two types of diffraction patterns, the first one corresponds to those previously reported (see for example [1

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

*β*≠0 and

*wo*=

*finite*(Airy beam) and the second one which is characterized by

*β*≠0 and

*wo*→∞ (SAiry beam). In both cases we assumed diffraction by a finite aperture as can be seen in the transmittance function.

## 3. Numerical results and discussion

*E*⃗

*(*

_{o}*x*,

_{o}*y*,0) which is

_{o}*x*̂ polarized, so

*α*=0, and Eq. (3) can be reduced to:

**99**, 213901 (
2007). [CrossRef]

## 3.1. Airy beams

**99**, 213901 (
2007). [CrossRef]

*cm*longer than predicted by the paraxial solution. These theoretical results justify the experimental measurements described in Ref. [5

5. P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channels generation using ultraintense Airy beams,” Science **324**, 229–232 (
2009). [CrossRef] [PubMed]

*cm*the intensity patterns are identical in both methods. However (since the condition of paraxial approximation

*cm*differences may be seen between the two method, such as the intensity distribution and the maximum intensity reached. However, these differences may greater when the

*a*/

*f*ratio tends to one, as was recently noted for spherical waves [1

**99**, 213901 (
2007). [CrossRef]

*x*,

_{max}*y*) are the Cartesian coordinates of the maximum intensity at each plane z. As can be seen the parabolic trajectory is observed in the -0.2 m 0.2 m interval, and there is no acceleration outside this region. It is important to remark that outside the -0.15 m, 0.15 m z interval the maximum intensity is lower than 80% of the maximum obtained at the focal plane, so the effects of the peak intensity for several applications could may be negligible although the trajectory is still parabolic in the -0.2 m 0.2 m interval as mentioned above. For values of |

_{max}*Z*|>0.2 m the parabolic trajectory disappears.

## 3.2. SAiry beams

*t*(

*x*,

_{o}*y*)|=1

_{o}*onD*than if

*E*(

_{o}*x*,

_{o}*y*,

_{o}*o*) in the integral 7. As can be seen the most significant difference between the transmittance that generates the Airy and that which generates the SAiry beams is that the high and low spatial frequencies have the same weight in SAiry beams, whereas the high spatial frequencies are nearly null for the Airy beams, and the contrast is lower for SAiry beams in the other of spatial frequencies.

*cm*and in all cases the intensity of the main peak is higher than that the obtained in the same plane with an Airy beam as can be clearly observed in Fig. 5. Moreover it is important to point out that the behavior of the intensity is quite different between the two kinds of beams, for example, the maximum intensity is not obtained at the focal plane, but at a plane situated at -14

*cm*from the focus. Also the growth and decreasing slopes are very different in PRF and PF regions. Figure 6 shows the parabolic trajectory of the SAiry beam, which as can be seen is greater than that the observed for the Airy beam and approximately coincides in the -20, 20

*cm*Z region. Moreover, an observed drawback with SAiry beams is that the spread of the diffraction pattern is always wider than for Airy beams. In any case, from Figs. 5 and 6 it may be deduced that for some applications where the spread of the diffraction pattern is not important and the peak maximum and parabolic trajectory are the main parameters, SAiry beams are better than Airy beams.

## 4. Conclusions

## Acknowledgments

## References and links

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

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

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

4. | J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microcells,” Lab on a Chip |

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

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

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

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

9. | M. A. Bandres and J. Gutierrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express |

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

11. | R. K. Luneburg, |

12. | J. Guo, X. Zhao, and Y. Min, “The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture,” Opt. Comm. |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(260.2110) Physical optics : Electromagnetic optics

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 27, 2009

Manuscript Accepted: October 28, 2009

Published: November 23, 2009

**Citation**

Luis Carretero, Pablo Acebal, Salvador Blaya, Celia García, Antonio Fimia, Roque Madrigal, and Angel Murciano, "Nonparaxial diffraction analysis of Airy and SAiry beams," Opt. Express **17**, 22432-22441 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-22432

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### References

- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]
- G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979-981 (2007). [CrossRef] [PubMed]
- J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nat. Photonics 2, 675-678 (2008). [CrossRef]
- J. Baumgartl, G. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, "Optical redistribution of microparticles and cells between microcells," Lab Chip 9, 1334-1336 (2009). [CrossRef] [PubMed]
- P. Polynkin, M. Koleskik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, "Curved plasma channels generation using ultraintense Airy beams," Science 324, 229-232 (2009). [CrossRef] [PubMed]
- T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, "Nonlinear generation and manipulation of Airy beams," Nat. Photonics 3, 395-398 (2009). [CrossRef]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of Airy beams," Opt. Lett. 33, 207-209 (2008). [CrossRef] [PubMed]
- 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]
- M. A. Bandres and J. Gutierrez-Vega, "Airy-Gauss beams and their transformation by paraxial optical systems," Opt. Express 15, 16719-16728 (2007). [CrossRef] [PubMed]
- H. Sztul and R. Alfano, "The Poynting vector and angular momentum of Airy beams," Opt. Express 16, 9411-9416 (2008). [CrossRef] [PubMed]
- R. K. Luneburg, Mathematical theory of optics (University California Press, Berkeley, California, 1964).
- J. Guo, X. Zhao, and Y. Min, "The general integral expressions for on-axis nonparaxial vectorial spherical waves diffracted at a circular aperture," Opt. Comm. 282, 1511-1515 (2009). [CrossRef]

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