## Time-spatial drift of decelerating electromagnetic pulses |

Optics Express, Vol. 21, Issue 14, pp. 17366-17371 (2013)

http://dx.doi.org/10.1364/OE.21.017366

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### Abstract

A time dependent electromagnetic pulse generated by a current running laterally to the direction of the pulse propagation is considered in paraxial approximation. It is shown that the pulse envelope moves in the time-spatial coordinates on the surface of a parabolic cylinder for the Airy pulse and a hyperbolic cylinder for the Gaussian. These pulses propagate in time with deceleration along the dominant propagation direction and drift uniformly in the lateral direction. The Airy pulse stops at infinity while the asymptotic velocity of the Gaussian is nonzero.

© 2013 OSA

27*Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables*, M. Abramowitz and I. Stegun eds. (Nat. Bureau of Standards, 1964).

## 1. Introduction

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

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

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

4. P. Saari, “Laterally accelerating airy pulses,” Opt. Express **16**(14), 10303–10308 (2008). [CrossRef] [PubMed]

7. M. I. Carvalho and M. Facão, “Propagation of Airy-related beams,” Opt. Express **18**(21), 21938–21949 (2010). [CrossRef] [PubMed]

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

14. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. **36**(7), 1164–1166 (2011). [CrossRef] [PubMed]

*x*axis. In this case a parabolic dependence exists between the lateral

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

4. P. Saari, “Laterally accelerating airy pulses,” Opt. Express **16**(14), 10303–10308 (2008). [CrossRef] [PubMed]

15. I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(4), 046605 (2008). [CrossRef] [PubMed]

21. I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express **19**(23), 23132–23139 (2011). [CrossRef] [PubMed]

16. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. **105**(25), 253901 (2010). [CrossRef] [PubMed]

17. 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]

20. Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express **19**(18), 17298–17307 (2011). [CrossRef] [PubMed]

*x*axis,

## 2. The master equation

## 3. Decelerating Airy pulse

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

*x.*The retarded time

*x,*x ˙ = 0 , the velocity in the transverse direction is constant

*y*and

*z*are constants. This gives the envelope velocity along the propagation direction

## 4. Decelerating Gaussian

*x*) the envelope movement, as in the case of the Airy pulse, is uniform and its velocity is constant:

*y*and

*z*the envelope movement is decelerating along the longitudinal axis

## 5. Conclusions

## References and links

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

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

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

4. | P. Saari, “Laterally accelerating airy pulses,” Opt. Express |

5. | M. A. Bandres, “Accelerating beams,” Opt. Lett. |

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

7. | M. I. Carvalho and M. Facão, “Propagation of Airy-related beams,” Opt. Express |

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

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

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

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

12. | J.-X. Li, W.-P. Zang, and J.-G. Tian, “Vacuum laser-driven acceleration by Airy beams,” Opt. Express |

13. | Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. |

14. | W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. |

15. | I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

16. | D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. |

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

18. | T. J. Eichelkraut, G. A. Siviloglou, I. M. Besieris, and D. N. Christodoulides, “Oblique Airy wave packets in bidispersive optical media,” Opt. Lett. |

19. | Y. Kaganovsky and E. Heyman, “Airy pulsed beams,” J. Opt. Soc. Am. A |

20. | Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express |

21. | I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express |

22. | A. Yariv, |

23. | L. Novotny and B. Hecht, |

24. | Y. B. Band, |

25. | V. S. Vladimirov, |

26. | A. N. Tychonov and A. A. Samarski, |

**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: May 9, 2013

Revised Manuscript: July 2, 2013

Manuscript Accepted: July 3, 2013

Published: July 12, 2013

**Citation**

Alexander G. Nerukh and Dmitry A. Nerukh, "Time-spatial drift of decelerating electromagnetic pulses," Opt. Express **21**, 17366-17371 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-17366

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### References

- M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264–267 (1979). [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, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007). [CrossRef] [PubMed]
- P. Saari, “Laterally accelerating airy pulses,” Opt. Express16(14), 10303–10308 (2008). [CrossRef] [PubMed]
- M. A. Bandres, “Accelerating beams,” Opt. Lett.34(24), 3791–3793 (2009). [CrossRef] [PubMed]
- Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express18(8), 8440–8452 (2010). [CrossRef] [PubMed]
- M. I. Carvalho and M. Facão, “Propagation of Airy-related beams,” Opt. Express18(21), 21938–21949 (2010). [CrossRef] [PubMed]
- J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008). [CrossRef]
- T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics3(7), 395–398 (2009). [CrossRef]
- A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett.35(12), 2082–2084 (2010). [CrossRef] [PubMed]
- H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express18(19), 20384–20394 (2010). [CrossRef] [PubMed]
- J.-X. Li, W.-P. Zang, and J.-G. Tian, “Vacuum laser-driven acceleration by Airy beams,” Opt. Express18(7), 7300–7306 (2010). [CrossRef] [PubMed]
- Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt.50(1), 43–49 (2011). [CrossRef] [PubMed]
- W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett.36(7), 1164–1166 (2011). [CrossRef] [PubMed]
- I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.78(4), 046605 (2008). [CrossRef] [PubMed]
- D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett.105(25), 253901 (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. Photonics4(2), 103–106 (2010). [CrossRef]
- T. J. Eichelkraut, G. A. Siviloglou, I. M. Besieris, and D. N. Christodoulides, “Oblique Airy wave packets in bidispersive optical media,” Opt. Lett.35(21), 3655–3657 (2010). [CrossRef] [PubMed]
- Y. Kaganovsky and E. Heyman, “Airy pulsed beams,” J. Opt. Soc. Am. A28(6), 1243–1255 (2011). [CrossRef] [PubMed]
- Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express19(18), 17298–17307 (2011). [CrossRef] [PubMed]
- I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express19(23), 23132–23139 (2011). [CrossRef] [PubMed]
- A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, 1997).
- L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).
- Y. B. Band, Light and Matter (John Wiley & Sons, 2007).
- V. S. Vladimirov, Equations of Mathematical Physics (Marcel Dekker Inc., 1971).
- A. N. Tychonov and A. A. Samarski, Partial Differential Equations in Mathematical Physics (Holden-Day, 1964).

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