## Self-healing properties of optical Airy beams

Optics Express, Vol. 16, Issue 17, pp. 12880-12891 (2008)

http://dx.doi.org/10.1364/OE.16.012880

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### Abstract

We investigate both theoretically and experimentally the self-healing properties of accelerating Airy beams. We show that this class of waves tends to reform during propagation in spite of the severity of the imposed perturbations. In all occasions the reconstruction of these beams is interpreted through their internal transverse power flow. The robustness of these optical beams in scattering and turbulent environments is also studied experimentally. Our observations are in excellent agreement with numerical simulations.

© 2008 Optical Society of America

## 1. Introduction

6. K. Dholakia, “Optics: Against the spread of the light,” Nature **451**, 413 (2008). [CrossRef] [PubMed]

4. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. **32**, 2447–2249 (2007). [CrossRef] [PubMed]

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

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

9. M. Asorey, P. Facchi, V. I. Man’ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Generalized tomographic maps,” Phys. Rev. A. **77**, 042115 (2008). [CrossRef]

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]

10. A. V. Gorbach and D. V. Skryabin, “Soliton self-frequency shift, non-solitonic radiation and self-induced transparency in air-core fibers,” Opt. Express **16**, 4858–4865 (2008). [CrossRef] [PubMed]

12. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987). [CrossRef] [PubMed]

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

14. R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. **122**, 169–177 (1996). [CrossRef]

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

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

## 2. Dynamics of optical Airy beams

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

*φ*is the electric field envelope and

*k*=2

*πn*/

*λ*

_{0}is the wavenumber of the optical wave. Following the approach of Ref. [1

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

*θ*, and the scales

_{m}*x*

_{0},

*y*

_{0}. Clearly, for zero launch angles

*θ*and if

_{m}*x*

_{0}=

*y*

_{0}, the main lobe of the Airy beam will move on a parabola (projected along the 45° axis in the

*x*-

*y*plane). On the other hand, a “boomerang-like” curve may result if for example the “launch” angles are chosen to have opposite signs, say

*θ*=-2

_{x}*mrad*and

*θ*=2

_{y}*mrad*(while

*x*

_{0}=

*y*

_{0}=77

*µm*), as shown in Fig. 2. What is also very interesting is the fact that these displacements vary quadratically with the wavelength

*λ*

_{0}.

*z*=0, can be explained from Babinet’s principle [15

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

*ε*(

*x*,

*y*), i.e.

*ϕ*(

*x*,

*y*,

*z*=0)=

*U*(

_{ND}*x*,

*y*,

*z*=0)-

*ε*(

*x*,

*y*,

*z*=0), then from Eq. (1) one finds that

*iε*+(1/2

_{z}*k*)Δ

^{2}

_{⊥}

*ε*=0. As a result the perturbation

*ε*is expected to rapidly diffract as opposed to the non-diffracting beam that remains invariant during propagation. As a consequence, at large distances |

*ϕ*(

*x*,

*y*,

*z*)|

^{2}≅

*U*(

_{ND}*x*,

*y*,

*z*,)|

^{2}and hence the ND beam reforms during propagation. This argument holds for all ND fields including the accelerating Airy beam.

*S⃗*associated with Airy optical beams. In the paraxial regime,

*S⃗*is given by [18

18. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A. **45**, 8185–8189 (1992). [CrossRef] [PubMed]

*S⃗*denotes the longitudinal component of the Poynting vector whereas

_{z}*S⃗*

_{⊥}the transverse. From Eqs. (3, 4) one can directly obtain the direction of the Poynting vector associated with an ideal 2D Airy (

*a*=0) beam, as schematically indicated in Fig. 3.

_{m}*ψ*the projection of

*S⃗*makes with respect to

*x*axis is given by:

*S⃗*relative to the

*z*axis is given by:

*S⃗*is at every point parallel to the unit tangent vector

*l̂*of the trajectory curve of Eq. (3) (see Fig. 2). This statement is also valid for finite energy Airy beams during the quasi-diffraction free stage of propagation. At larger distances however small deviations are expected to occur as shown in Fig. 4 for the 1D case. In addition one can show that the polarization of the beam can evolve in a similar manner.

*S⃗*

_{⊥}.

## 3. Experimental Set-up

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

*Ai*(

*x*/

*x*

_{0})exp(

*ax*/

*x*

_{0})

*Ai*(

*y*/

*y*

_{0})exp(

*ay*/

*y*

_{0}) is produced by Fourier transforming a broad Gaussian beam when a 2D cubic phase modulation is imposed. A linearly polarized Gaussian beam from an Argon-ion laser at 488

*nm*is collimated using a beam expander to a beam width of 6.7

*mm*. The cubic phase (Fig. 5(b)) was introduced using a computer controlled spatial light modulator (SLM). A spherical lens

*L*with a focal length

*f*=1

*m*was placed at a distance

*f*in front of the SLM phase array in order to generate the finite energy two-dimensional Airy wavepacket. The exponentially truncated Airy function (

*x*

_{0}=

*y*

_{0}≅77

*µm*,

*a*=0.08) is Fourier generated at a distance

*f*after the lens (Fig. 5(c)). The propagation dynamics of these beams can then be recorded, as a function of propagation distance, by translating the imaging apparatus. In order to block the Airy pattern in a controlled manner a rectangular opaque obstacle was inserted at this point using a micro-positioner.

## 4. Self-healing properties of Airy beams

*µm*corresponding to

*x*

_{0}=

*y*

_{0}≈77

*µm*and

*a*=0.08. Figure 6(b) depicts the reformation of this Airy beam after a distance of

*z*=11

*cm*. The self-healing of this beam is apparent. The main lobe is reborn at the corner and persists undistorted up to a distance of 30

*cm*(Fig. 6(c)). In our set up, this latter distance (30

*cm*) corresponds approximately to four diffraction lengths of the corner lobe. Our experimental observations are in excellent agreement with numerical results presented in Figs. 6(d)–6(f) for the same propagation distances.

*z*=0

*cm*and after

*z*=30

*cm*(Fig 7(b)). The corresponding numerical simulations are shown in Figs. 7(c) and 7(d). This is another manifestation of the non-diffracting nature of Airy beams.

*S⃗*

_{⊥}within the perturbed Airy beam. To do so we use the result of Eq. (4). Figure 8(a) depicts the transverse flow within the Airy beam at

*z*=1

*cm*when the main lobe has been removed. Evidently the power flows from the side lobes towards the corner in order to facilitate self-healing. On the other hand, once reconstruction has been reached (at

*z*=11

*cm*), then the internal power density around the newly-formed main lobe flows along the 45° axis in the

*x*-

*y*plane (for

*x*

_{0}=

*y*

_{0}) in order to enable the acceleration dynamics of the Airy beam (Fig. 8(b)).

*z*=16

*cm*of propagation the beam self-heals and reconstructs in detail its fine intensity structure as depicted in Fig. 9(b). Figures 9(c) and 9(d) show the corresponding calculated intensity profiles for these same distances.

*z*=1

*cm*, the Poynting vector provides energy towards the blocked region for rebirth to occur while on the main lobe is directed along 45° in the

*x*-

*y*plane in order to enable the self-bending of the Airy beam.

*z*=24

*cm*(Fig. 11(b)). Compared to the other two cases, this self-healing distance is somewhat longer since the perturbation is now more severe (given the ratio of the power blocked over the incident power). Figures 11(c) and 11(d) show the corresponding simulated patterns while Fig. 11(e) depicts the transverse power flow at

*z*=1

*cm*.

*y*axis as shown in Fig. 12(a). Interestingly, in this physical setting, the beam not only self-heals itself after

*z*=18

*cm*but also the initially blocked part is reborn even brighter when compared to its surroundings (Fig. 12(b)). This is a clear manifestation of the non-diffracting character of the Airy beam. The corresponding numerical results (Figs. 12(c) and 12(d)) are in excellent agreement with the experiments.

## 5. Propagation of Airy beams in adverse environments

### 5.1 Airy beams in scattering media

*n*=1.45) suspended in pure water(

*n*=1.33). The size of the dielectric micro-particles was 0.5

*µm*and 1.5

*µm*in diameter and thus light scattering was predominantly of the Mie type [19]. Both suspensions were 0.2% in weight concentration while the volume filling factor was 0.1% (the specific gravity of the silica particles is

*µm*

^{2}and 3.76

*µm*

^{2}[20

20. S. Prahl, “Mie Scattering Calculator,” (2008). http://omlc.ogi.edu/calc/mie_calc.html

*cm*in the water-silica mixture (diameter of 0.5

*µm*) (Fig. 13(b)). A longer (10

*cm*) cell was used to observe the complete reformation of the Airy pattern in the same scattering media. Figure 13(c) depicts the self-healing of an Airy beam after propagating 10

*cm*in the same environment. Besides the anticipated drop in the beam intensity due to Mie scattering, the beam still exhibits in every respect its characteristic pattern.

*µm*microspheres having a much larger scattering cross-section and hence more pronounced scattering effects. The self-reconstructed beam at the end of a 10

*cm*glass cell is depicted in Fig. 14.

### 5.2 Airy beams in turbulent media

*cm*. In all our experiments the resilience of the Airy beam (without any initial amplitude distortions) against turbulence was remarkable (Fig. 15(a) and the associated video file). To some extent this robustness can be qualitatively understood if one considers the phase structure of the beam: alternations in phase between 0 ’s and

*π*’s result in zero-intensity regions and these singularities can be in turn extremely stable [21

21. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A **336**, 165–190 (1974). [CrossRef]

22. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. **28**, 10–12 (2003). [CrossRef] [PubMed]

## 6. Conclusions

## Acknowledgment

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

4. | I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. |

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

6. | K. Dholakia, “Optics: Against the spread of the light,” Nature |

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

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

9. | M. Asorey, P. Facchi, V. I. Man’ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, “Generalized tomographic maps,” Phys. Rev. A. |

10. | A. V. Gorbach and D. V. Skryabin, “Soliton self-frequency shift, non-solitonic radiation and self-induced transparency in air-core fibers,” Opt. Express |

11. | P. Saari, “Laterally accelerating Airy pulses,” Opt. Express |

12. | J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

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

14. | R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun. |

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

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

17. | O. Vallée and M. Soares, |

18. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A. |

19. | H.C. van de Hulst, |

20. | S. Prahl, “Mie Scattering Calculator,” (2008). http://omlc.ogi.edu/calc/mie_calc.html |

21. | J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A |

22. | A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(290.5850) Scattering : Scattering, particles

(290.7050) Scattering : Turbid media

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 2, 2008

Revised Manuscript: August 1, 2008

Manuscript Accepted: August 5, 2008

Published: August 8, 2008

**Virtual Issues**

Vol. 3, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

John Broky, Georgios A. Siviloglou, Aristide Dogariu, and Demetrios N. Christodoulides, "Self-healing properties of optical Airy beams," Opt. Express **16**, 12880-12891 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-17-12880

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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]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of Airy beams," Opt. Lett. 33, 207-209 (2008). [CrossRef] [PubMed]
- I. M. Besieris and A. M. Shaarawi, "A note on an accelerating finite energy Airy beam," Opt. Lett. 32, 2447-2249 (2007). [CrossRef] [PubMed]
- M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979). [CrossRef]
- K. Dholakia, "Optics: Against the spread of the light," Nature 451, 413 (2008). [CrossRef] [PubMed]
- M. A. Bandres and J. C. Gutiérrez-Vega, "Airy-Gauss beams and their transformation by paraxial optical systems," Opt. Express 15, 16719-16728 (2007). [CrossRef] [PubMed]
- H. I. Sztul and R. R. Alfano, "The Poynting vector and angular momentum of Airy beams," Opt. Express 16, 9411-9416 (2008). [CrossRef] [PubMed]
- M. Asorey, P. Facchi, V. I. Man'ko, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, "Generalized tomographic maps," Phys. Rev. A. 77, 042115 (2008). [CrossRef]
- A. V. Gorbach and D. V. Skryabin, "Soliton self-frequency shift, non-solitonic radiation and self-induced transparency in air-core fibers," Opt. Express 16, 4858-4865 (2008). [CrossRef] [PubMed]
- P. Saari, "Laterally accelerating airy pulses," Opt. Express 16, 10303-10308 (2008). [CrossRef] [PubMed]
- J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005). [CrossRef]
- R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, "Interboard optical data distribution by Bessel beam shadowing," Opt. Commun. 122, 169-177 (1996). [CrossRef]
- Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207-211 (1998). [CrossRef]
- V. Garces-Chavez, D. McGloin, H. Melville,W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002). [CrossRef] [PubMed]
- O. Vallée and M. Soares, Airy Functions and Applications to Physics, (Imperial College Press, London, 2004).
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A. 45, 8185-8189 (1992). [CrossRef] [PubMed]
- H. C. van de Hulst, Light Scattering by Small Particles, (Dover Publication Inc., New York, 1981).
- S. Prahl, "Mie Scattering Calculator," (2008). http://omlc.ogi.edu/calc/mie_calc.html
- J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974). [CrossRef]
- A. Dogariu and S. Amarande, "Propagation of partially coherent beams: turbulence-induced degradation," Opt. Lett. 28, 10-12 (2003). [CrossRef] [PubMed]

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