## Negative propagation effect in nonparaxial Airy beams |

Optics Express, Vol. 20, Issue 24, pp. 26913-26921 (2012)

http://dx.doi.org/10.1364/OE.20.026913

Acrobat PDF (1273 KB)

### Abstract

Negative propagation is an unusual effect concerning the local sign change in the Poynting vector components of an optical beam under free propagation. We report this effect for finite-energy Airy beams in a subwavelength nonparaxial regime. This effect is due to a coupling process between propagating and evanescent plane waves forming the beam in the spectral domain and it is demonstrated for a single TE or TM mode. This is contrary to what happens for vector Bessel beams and vector X-waves, for which a complex superposition of TE and TM modes is mandatory. We also show that evanescent waves cannot contribute to the energy flux density by themselves such that a pure evanescent Airy beam is not physically realizable. The break of the shape-preserving and diffraction-free properties of Airy beams in a nonparaxial regime is exclusively caused by the propagating waves. The negative propagation effect in subwavelength nonparaxial Airy beams opens new capabilities in optical traps and tweezers, optical detection of invisibility cloacks and selective on-chip manipulation of nanoparticles.

© 2012 OSA

## 1. Introduction

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

2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**, 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**, 213901 (2007). [CrossRef]

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. **48**, 3170–3176 (2009). [CrossRef] [PubMed]

*Airylike*beams, variants of that original finite-energy Airy beam, were widely analyzed from theoretical and experimental viewpoint [6

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. **48**, 3170–3176 (2009). [CrossRef] [PubMed]

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

11. P. Vaveliuk, G. F. Zebende, M. A. Moret, and B. Ruiz, “Propagating free-space nonparaxial beams,” J. Opt. Soc. Am. A **24**, 3297–3302 (2007). [CrossRef]

12. A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. **34**, 3430–3432 (2009). [CrossRef] [PubMed]

*evanescent Airy beam*, consisting exclusively of its own evanescent waves, was proposed as a fundamental result under strong nonparaxial conditions [12

12. A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. **34**, 3430–3432 (2009). [CrossRef] [PubMed]

13. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. **108**, 163901 (2012). [CrossRef] [PubMed]

13. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. **108**, 163901 (2012). [CrossRef] [PubMed]

14. Z. Chen, “Viewpoint: light bends itself into an arcs,” Phys. **5**, 44 (2012). [CrossRef]

15. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A **24**, 2844–2849 (2007). [CrossRef]

16. M. A. Salem and H. Bağci, “Energy flow characteristics of vector X-waves,” Opt. Express **19**, 8526–8532 (2011). [CrossRef] [PubMed]

*evanescent Airy beam*[12

12. A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. **34**, 3430–3432 (2009). [CrossRef] [PubMed]

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

18. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett. **107**, 116802 (2011). [CrossRef] [PubMed]

20. P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett. **36**, 3191–3193 (2011). [CrossRef] [PubMed]

18. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett. **107**, 116802 (2011). [CrossRef] [PubMed]

21. A. Salandrino and D. N. Christodoulides, “Viewpoint: Airy plasmons defeat diffraction on the surface,” Phys. **4**, 69 (2011). [CrossRef]

22. T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett. **104**, 197203 (2010). [CrossRef] [PubMed]

*μ*m) [23

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

## 2. Theoretical background and paraxial-nonparaxial limit for finite-energy Airy beams

*E⃗*=

*E*

_{0}

*U*(

*x, z*)

*e*where

^{iωt}ŷ*ω*is the frequency,

*E*

_{0}is a constant with electric field dimensions and

*U*is the dimensionless field obeying the HEq in a medium of permittivity

*ε*: with

*λ*being the wavelength. We introduce the spacial dimensionless variables

*x*̃ =

*ε*

^{1/2}

*x*/

*λ*and

*z*̃ =

*ε*

^{1/2}

*x*/

*λ*. This normalization simplifies the mathematics and allows for easier interpretation of the results. The field

*U*can be analyzed by employing the angular spectrum formalism [24].

*z*̃ = 0) and the dimensionless spatial frequency

*p*̃ =

*λp*/

*ε*

^{1/2}is the conjugate variable of

*x*̃. For AiBs,

*𝒰*(

*p*̃;0) is Gaussian and involves a cubic phase [2

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

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

25. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. **32**, 2447–2449 (2007). [CrossRef] [PubMed]

*x*

_{0}(we use the dimensionless size

*x*̃

_{0}=

*ε*

^{1/2}

*x*

_{0}/

*λ*) that accounts for the size of the beam central lobe and the exponential truncation factor,

*a*. This guarantees the square integrability of the beam and controls the spreading properties [2

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

25. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. **32**, 2447–2449 (2007). [CrossRef] [PubMed]

*U*can be physically interpreted as having contributions of both propagating plane waves

*U*and evanescent plane waves

_{pr}*U*separated by the critical spatial frequency |

_{ev}*p*̃

*| = 1: The angular spectrum approach was also employed in Ref. [12*

_{c}**34**, 3430–3432 (2009). [CrossRef] [PubMed]

*z*-component of the time-averaged Poynting vector. This can be expressed, except for a dimensional constant, as [26

26. A. Lencina and P. Vaveliuk, “Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs,” Phys. Rev. E **71**, 056614 (2005). [CrossRef]

27. P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett. **32**, 927–929 (2007). [CrossRef] [PubMed]

*j*=

*pr,ev*. The energy flow density is then constituted by three terms: the propagating and evanescent plane waves and a

*interference-like*term, product of the coupled contribution of both components. Now, we have the full machinery to analyze the propagation dynamics of nonparaxial Airy beams. But before we do that, it is necessary to accurately delimit the paraxial-nonparaxial range in terms of the AiB parameters. Such a range can be quantified by a useful parameter, the

*paraxial estimator*(

*𝒫*) introduced in Ref. [27

27. P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett. **32**, 927–929 (2007). [CrossRef] [PubMed]

*𝒫*lies between the interval (−∞, 1), and the limit

*𝒫*→ 1 guarantees the paraxial approximation validity [27

27. P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett. **32**, 927–929 (2007). [CrossRef] [PubMed]

*𝒫*departures from unit, the beam becomes more and more nonparaxial [27

**32**, 927–929 (2007). [CrossRef] [PubMed]

28. P. Vaveliuk and O. Martinez-Matos, “Physical interpretation of the paraxial estimator,” Opt. Commun. **285**, 4816–4820 (2012). [CrossRef]

*𝒫*can be calculated from the angular spectrum of the beam [29

29. P. Vaveliuk, “Quantifying the paraxiality for laser beams from the *M*^{2}-factor,” Opt. Lett. **34**, 340–342 (2009). [CrossRef] [PubMed]

*𝒫*can be easily obtained from the AiB angular spectrum for

*z*̃ ≥ 0:

*paraxiality scale*of

*𝒫*follows the used for the fundamental Gaussian mode as emphasized in [30

30. P. Vaveliuk and O. Martinez-Matos, “Effect of ABCD transformations on beam paraxiality,” Opt. Express **19**, 25944–25953 (2011). [CrossRef]

*𝒫*-values lower than 0.94. Figure 1 depicts a density plot of

*𝒫*in terms of

*x*̃

_{0}and

*a*. The interval of both parameters covers a wide range of feasible experimental values. The points in Fig. 1 indicate certain configurations of interest that will be analyzed here. The configuration Ai1 represents a typical setup for Aibs generated in free space by mask encoded methods:

*x*̃

_{0}≈ 100 and

*a*≈ 0.08 [3

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

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

**32**, 979–981 (2007). [CrossRef] [PubMed]

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

## 3. Nonparaxial Airy beams and negative propagation effect

*z*-component of the time-averaged Poynting vector Spresents negative values along the propagation coordinates. The spatial range of this phenomenon (up to

*z*̃ ≈ 0.25) is approximately 25% of the total propagation range of the beam as indicates the green region in Fig. 3(a). The amplitude of the major of negative peaks is greater than 10% of the principal peak. These data suggest that this effect would be highly detectable if this configuration would be carried out experimentally. Of course, this effect will be even greater if the beam size decreases further. The coupling term is fully responsible by such a phenomenon as shown by the comparison between Sand S

*[Fig. 3(b)]. The magnitudes are very different in the near field region where the evanescent waves exert influence. As the beam propagates away from the near field region, the dynamics is exclusively governed by the propagating waves and both profiles begin to be practically equivalents after*

_{pr}*z*̃ = 0.25. A great advantage is that this negative propagation takes place for a single TE or TM mode contrary to what happens for Bessel beams and X-waves [15

15. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A **24**, 2844–2849 (2007). [CrossRef]

16. M. A. Salem and H. Bağci, “Energy flow characteristics of vector X-waves,” Opt. Express **19**, 8526–8532 (2011). [CrossRef] [PubMed]

31. Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. **50**, 43–49 (2011). [CrossRef] [PubMed]

32. M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature Phys. **3**, 477–480 (2007). [CrossRef]

33. B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. **103**, 243901 (2009). [CrossRef]

## 4. Concluding remarks

## Acknowledgments

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

5. | D. M. Cottrell, J. A. Davis, and T. M. Hazard, “Direct generation of accelerating Airy beams using a 3/2 phase-only pattern,” Opt. Lett. |

6. | J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. |

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

8. | J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express |

9. | J. E. Morris, M. Mazilu, J. Baumgartl, T. Ciz̃már, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express |

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

11. | P. Vaveliuk, G. F. Zebende, M. A. Moret, and B. Ruiz, “Propagating free-space nonparaxial beams,” J. Opt. Soc. Am. A |

12. | A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. |

13. | I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. |

14. | Z. Chen, “Viewpoint: light bends itself into an arcs,” Phys. |

15. | A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A |

16. | M. A. Salem and H. Bağci, “Energy flow characteristics of vector X-waves,” Opt. Express |

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

18. | A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett. |

19. | L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. |

20. | P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett. |

21. | A. Salandrino and D. N. Christodoulides, “Viewpoint: Airy plasmons defeat diffraction on the surface,” Phys. |

22. | T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett. |

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

24. | J. W. Goodman, |

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

26. | A. Lencina and P. Vaveliuk, “Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs,” Phys. Rev. E |

27. | P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett. |

28. | P. Vaveliuk and O. Martinez-Matos, “Physical interpretation of the paraxial estimator,” Opt. Commun. |

29. | P. Vaveliuk, “Quantifying the paraxiality for laser beams from the |

30. | P. Vaveliuk and O. Martinez-Matos, “Effect of ABCD transformations on beam paraxiality,” Opt. Express |

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

32. | M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature Phys. |

33. | B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(260.2110) Physical optics : Electromagnetic optics

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: August 9, 2012

Revised Manuscript: October 17, 2012

Manuscript Accepted: October 19, 2012

Published: November 14, 2012

**Citation**

Pablo Vaveliuk and Oscar Martinez-Matos, "Negative propagation effect in nonparaxial Airy beams," Opt. Express **20**, 26913-26921 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26913

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### References

- M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264 –267 (1979). [CrossRef]
- 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]
- D. M. Cottrell, J. A. Davis, and T. M. Hazard, “Direct generation of accelerating Airy beams using a 3/2 phase-only pattern,” Opt. Lett.34, 2634–2636 (2009). [CrossRef] [PubMed]
- J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt.48, 3170–3176 (2009). [CrossRef] [PubMed]
- M. A. Bandres and J. C. Gutierrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express15, 16719–16728 (2007). [CrossRef] [PubMed]
- J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express16, 12866–12871 (2008). [CrossRef] [PubMed]
- J. E. Morris, M. Mazilu, J. Baumgartl, T. Ciz̃már, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express17, 13236–13245 (2009). [CrossRef] [PubMed]
- M. I. Carvalho and M. Facão, “Propagation of Airy-related beams,” Opt. Express18, 21938–21949 (2010). [CrossRef] [PubMed]
- P. Vaveliuk, G. F. Zebende, M. A. Moret, and B. Ruiz, “Propagating free-space nonparaxial beams,” J. Opt. Soc. Am. A24, 3297–3302 (2007). [CrossRef]
- A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett.34, 3430–3432 (2009). [CrossRef] [PubMed]
- I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett.108, 163901 (2012). [CrossRef] [PubMed]
- Z. Chen, “Viewpoint: light bends itself into an arcs,” Phys.5, 44 (2012). [CrossRef]
- A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A24, 2844–2849 (2007). [CrossRef]
- M. A. Salem and H. Bağci, “Energy flow characteristics of vector X-waves,” Opt. Express19, 8526–8532 (2011). [CrossRef] [PubMed]
- A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett.35, 2082–2084 (2010). [CrossRef] [PubMed]
- A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011). [CrossRef] [PubMed]
- L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett.107, 126804 (2011). [CrossRef] [PubMed]
- P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett.36, 3191–3193 (2011). [CrossRef] [PubMed]
- A. Salandrino and D. N. Christodoulides, “Viewpoint: Airy plasmons defeat diffraction on the surface,” Phys.4, 69 (2011). [CrossRef]
- T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett.104, 197203 (2010). [CrossRef] [PubMed]
- W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Yu. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett.36, 1164–1166 (2011). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968).
- I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett.32, 2447–2449 (2007). [CrossRef] [PubMed]
- A. Lencina and P. Vaveliuk, “Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs,” Phys. Rev. E71, 056614 (2005). [CrossRef]
- P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett.32, 927–929 (2007). [CrossRef] [PubMed]
- P. Vaveliuk and O. Martinez-Matos, “Physical interpretation of the paraxial estimator,” Opt. Commun.285, 4816–4820 (2012). [CrossRef]
- P. Vaveliuk, “Quantifying the paraxiality for laser beams from the M2-factor,” Opt. Lett.34, 340–342 (2009). [CrossRef] [PubMed]
- P. Vaveliuk and O. Martinez-Matos, “Effect of ABCD transformations on beam paraxiality,” Opt. Express19, 25944–25953 (2011). [CrossRef]
- Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt.50, 43–49 (2011). [CrossRef] [PubMed]
- M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature Phys.3, 477–480 (2007). [CrossRef]
- B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett.103, 243901 (2009). [CrossRef]

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