## Quantitative description of the self-healing ability of a beam |

Optics Express, Vol. 22, Issue 6, pp. 6899-6904 (2014)

http://dx.doi.org/10.1364/OE.22.006899

Acrobat PDF (1444 KB)

### Abstract

Quantitative description of the self-healing ability of a beam is very important for studying or comparing the self-healing ability of different beams. As describing the similarity by using the angle of two infinite-dimensional complex vectors in Hilbert space, the angle of two intensity profiles is proposed to quantitatively describe the self-healing ability of different beams. As a special case, quantitative description of the self-healing ability of a Bessel-Gaussian beam is studied. Results show that the angle of two intensity profiles can be used to describe the self-healing ability of arbitrary beams as the reconstruction distance for quantitatively describing the self-healing ability of Bessel beam. It offers a new method for studying or comparing the self-healing ability of different beams.

© 2014 Optical Society of America

## 1. Introduction

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

7. S. Vyas, Y. Kozawa, and S. Sato, “Self-healing of tightly focused scalar and vector Bessel-Gauss beams at the focal plane,” J. Opt. Soc. Am. A **28**(5), 837–843 (2011). [CrossRef] [PubMed]

12. 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]

7. S. Vyas, Y. Kozawa, and S. Sato, “Self-healing of tightly focused scalar and vector Bessel-Gauss beams at the focal plane,” J. Opt. Soc. Am. A **28**(5), 837–843 (2011). [CrossRef] [PubMed]

8. J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express **20**(17), 18955–18966 (2012). [CrossRef] [PubMed]

9. R. Cao, Y. Hua, C. Min, S. Zhu, and X. C. Yuan, “Self-healing optical pillar array,” Opt. Lett. **37**(17), 3540–3542 (2012). [CrossRef] [PubMed]

10. M. Anguiano-Morales, A. Martínez, M. D. Iturbe-Castillo, S. Chávez-Cerda, and N. Alcalá-Ochoa, “Self-healing property of a caustic optical beam,” Appl. Opt. **46**(34), 8284–8290 (2007). [CrossRef] [PubMed]

11. P. Vaity and R. P. Singh, “Self-healing property of optical ring lattice,” Opt. Lett. **36**(15), 2994–2996 (2011). [CrossRef] [PubMed]

12. 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]

13. I. A. Litvin, M. G. Mclaren, and A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. **282**(6), 1078–1082 (2009). [CrossRef]

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

13. I. A. Litvin, M. G. Mclaren, and A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. **282**(6), 1078–1082 (2009). [CrossRef]

14. M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel
beams,” Opt. Eng. **46**(7), 078001 (2007). [CrossRef]

15. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. **106**(21), 213902 (2011). [CrossRef] [PubMed]

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

17. R. Martínez-Herrero, I. Juvells, and A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. **38**(12), 2065–2067 (2013). [CrossRef] [PubMed]

## 2. Quantitative description of the self-healing ability

*S*is named as similarity of two functions in general.

## 3. Special case: self-healing ability of a Bessel-Gaussian beam

13. I. A. Litvin, M. G. Mclaren, and A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. **282**(6), 1078–1082 (2009). [CrossRef]

14. M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel
beams,” Opt. Eng. **46**(7), 078001 (2007). [CrossRef]

*R*before reconstruction occurs [13

**282**(6), 1078–1082 (2009). [CrossRef]

14. M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel
beams,” Opt. Eng. **46**(7), 078001 (2007). [CrossRef]

*α*is the axicon angle (see Fig. 1). For comparison with the existing results in a special case, as an example, quantitative description of the self-healing ability of a Bessel-Gaussian beam is studied in the following. As shown in Fig. 1, Gaussian beam passing through an axicon is used to generate the Bessel-Gaussian beam [13

**282**(6), 1078–1082 (2009). [CrossRef]

*δ*() is the DiracDelta function and

*u*and

*v*being the component of the angular spectrum along

*x*and

*y*-axis),

*k*= 2

*π/λ*is the wavenumber.

*I*

_{0}is the zeroth-order modified Bessel function of the first kind,

*w*

_{0}is the waist width of the Gaussian beam. By using the transfer function of the angular spectrum in free spacewe can obtain the angular at

*z*-plane asFrom Eq. (16) and using the inverse Fourier transform we can get the optical field of Gaussian- Bessel beam in free space aswhere

*J*

_{0}is the zeroth-order Bessel function of the first kind, and

*x*,

*y*) being the transverse coordinates]. When we set

*z*= 0 or

*R*is the radius of the obstacle. With the same method as in Eq. (17), the optical field of a Bessel-Gaussian beam at z-plane with an obstacle can be given as

*S*is investigated. From Eqs. (11), (17) and (21) the similarity can be calculated. Figure 4 shows the variation of the similarity

*S*during propagation with different parameters.

*S*with different

*w*

_{0}where

*R*= 0.8

*μ*m. It can be seen that the similarity is large with large

*w*

_{0}when the propagation distance is short. With the increase of propagation distance the difference of the similarity corresponding different

*w*

_{0}become small. For comparison with the reconstruction distance in Eq. (12),

*S*with different

*R*where

*w*

_{0}= 10

*μ*m. We can see that the speed of the self-healing is different with different

*R*. When R is small, the distance is small to reconstruct its shape. We also can see from Fig. 4(b) that can also be used to describe the distance where the reconstruction has completed.

**282**(6), 1078–1082 (2009). [CrossRef]

## 4. Conclusion

## Acknowledgment

## References and links

1. | H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., |

2. | V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, and K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. |

3. | X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. |

4. | M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett. |

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

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

7. | S. Vyas, Y. Kozawa, and S. Sato, “Self-healing of tightly focused scalar and vector Bessel-Gauss beams at the focal plane,” J. Opt. Soc. Am. A |

8. | J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express |

9. | R. Cao, Y. Hua, C. Min, S. Zhu, and X. C. Yuan, “Self-healing optical pillar array,” Opt. Lett. |

10. | M. Anguiano-Morales, A. Martínez, M. D. Iturbe-Castillo, S. Chávez-Cerda, and N. Alcalá-Ochoa, “Self-healing property of a caustic optical beam,” Appl. Opt. |

11. | P. Vaity and R. P. Singh, “Self-healing property of optical ring lattice,” Opt. Lett. |

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

13. | I. A. Litvin, M. G. Mclaren, and A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. |

14. | M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel
beams,” Opt. Eng. |

15. | E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. |

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

17. | R. Martínez-Herrero, I. Juvells, and A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. |

18. | J. James, |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 26, 2014

Revised Manuscript: March 11, 2014

Manuscript Accepted: March 11, 2014

Published: March 17, 2014

**Citation**

Xiuxiang Chu and Wei Wen, "Quantitative description of the self-healing ability of a beam," Opt. Express **22**, 6899-6904 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6899

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### References

- H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves: Theory and. Applications (John Wiley, 2008).
- V. Garcés-Chávez, D. Roskey, M. D. Summers, H. Melville, D. McGloin, E. M. Wright, K. Dholakia, “Optical levitation in a Bessel light beam,” Appl. Phys. Lett. 85(18), 4001–4003 (2004). [CrossRef]
- X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007). [CrossRef]
- M. Boguslawski, P. Rose, C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett. 98(6), 061111 (2011). [CrossRef]
- A. Chong, W. H. Renninger, D. N. Christodoulides, F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]
- M. V. Berry, N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]
- S. Vyas, Y. Kozawa, S. Sato, “Self-healing of tightly focused scalar and vector Bessel-Gauss beams at the focal plane,” J. Opt. Soc. Am. A 28(5), 837–843 (2011). [CrossRef] [PubMed]
- J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012). [CrossRef] [PubMed]
- R. Cao, Y. Hua, C. Min, S. Zhu, X. C. Yuan, “Self-healing optical pillar array,” Opt. Lett. 37(17), 3540–3542 (2012). [CrossRef] [PubMed]
- M. Anguiano-Morales, A. Martínez, M. D. Iturbe-Castillo, S. Chávez-Cerda, N. Alcalá-Ochoa, “Self-healing property of a caustic optical beam,” Appl. Opt. 46(34), 8284–8290 (2007). [CrossRef] [PubMed]
- P. Vaity, R. P. Singh, “Self-healing property of optical ring lattice,” Opt. Lett. 36(15), 2994–2996 (2011). [CrossRef] [PubMed]
- J. Broky, G. A. Siviloglou, A. Dogariu, D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef] [PubMed]
- I. A. Litvin, M. G. Mclaren, A. Forbes, “A conical wave approach to calculating Bessel–Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009). [CrossRef]
- M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46(7), 078001 (2007). [CrossRef]
- E. Greenfield, M. Segev, W. Walasik, O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef] [PubMed]
- Y. Kaganovsky, E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010). [CrossRef] [PubMed]
- R. Martínez-Herrero, I. Juvells, A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38(12), 2065–2067 (2013). [CrossRef] [PubMed]
- J. James, Mathematics Dictionary Mathematics Dictionary, 5th ed. (Springer, 1992).

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