Abstract

We study periodic inversion and phase transition of normal, displaced, and chirped finite energy Airy beams propagating in a parabolic potential. This propagation leads to an unusual oscillation: for half of the oscillation period the Airy beam accelerates in one transverse direction, with the main Airy beam lobe leading the train of pulses, whereas in the other half of the period it accelerates in the opposite direction, with the main lobe still leading – but now the whole beam is inverted. The inversion happens at a critical point, at which the beam profile changes from an Airy profile to a Gaussian one. Thus, there are two distinct phases in the propagation of an Airy beam in the parabolic potential – the normal Airy and the single-peak Gaussian phase. The length of the single-peak phase is determined by the size of the decay parameter: the smaller the decay, the smaller the length. A linear chirp introduces a transverse displacement of the beam at the phase transition point, but does not change the location of the point. A quadratic chirp moves the phase transition point, but does not affect the beam profile. The two-dimensional case is discussed briefly, being equivalent to a product of two one-dimensional cases.

© 2015 Optical Society of America

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References

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  1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [Crossref] [PubMed]
  2. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [Crossref]
  3. 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]
  4. T. J. Eichelkraut, G. A. Siviloglou, I. M. Besieris, and D. N. Christodoulides, “Oblique Airy wave packets in bidispersive optical media,” Opt. Lett. 35, 3655–3657 (2010).
    [Crossref] [PubMed]
  5. N. K. Efremidis, “Airy trajectory engineering in dynamic linear index potentials,” Opt. Lett. 36, 3006–3008 (2011).
    [Crossref] [PubMed]
  6. Y. Q. Zhang, M. Belić, Z. K. Wu, H. B. Zheng, K. Q. Lu, Y. Y. Li, and Y. P. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38, 4585–4588 (2013).
    [Crossref] [PubMed]
  7. R. Driben, Y. Hu, Z. Chen, B. A. Malomed, and R. Morandotti, “Inversion and tight focusing of Airy pulses under the action of third-order dispersion,” Opt. Lett. 38, 2499–2501 (2013).
    [Crossref] [PubMed]
  8. W.-P. Zhong, M. Belić, Y. Q. Zhang, and T. Huang, “Accelerating Airy-Gauss-Kummer localized wave packets,” Ann. Phys. 340, 171–178 (2014).
    [Crossref]
  9. Y. Q. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, Y. Y. Li, and Y. P. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express 22, 7160–7171 (2014).
    [Crossref] [PubMed]
  10. I. Besieris and A. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E 78, 046605 (2008).
    [Crossref]
  11. R. Driben and T. Meier, “Regeneration of Airy pulses in fiber-optic links with dispersion management of the two leading dispersion terms of opposite signs,” Phys. Rev. A 89, 043817 (2014).
    [Crossref]
  12. S. Wang, D. Fan, X. Bai, and X. Zeng, “Propagation dynamics of Airy pulses in optical fibers with periodic dispersion modulation,” Phys. Rev. A 89, 023802 (2014).
    [Crossref]
  13. J. Rogel-Salazar, H. A. Jiménez-Romero, and S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89, 023807 (2014).
    [Crossref]
  14. G. Zhou, R. Chen, and G. Ru, “Propagation of an Airy beam in a strongly nonlocal nonlinear media,” Laser Phys. Lett. 11, 105001 (2014).
    [Crossref]
  15. 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, 1164–1166 (2011).
    [Crossref] [PubMed]
  16. Z. Ye, S. Liu, C. Lou, P. Zhang, Y. Hu, D. Song, J. Zhao, and Z. Chen, “Acceleration control of Airy beams with optically induced refractive-index gradient,” Opt. Lett. 36, 3230–3232 (2011).
    [Crossref] [PubMed]
  17. F. Xiao, B. Li, M. Wang, W. Zhu, P. Zhang, S. Liu, M. Premaratne, and J. Zhao, “Optical Bloch oscillations of an Airy beam in a photonic lattice with a linear transverse index gradient,” Opt. Express 22, 22763–22770 (2014).
    [Crossref] [PubMed]
  18. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
    [Crossref]
  19. S. Ponomarenko and G. Agrawal, “Do solitonlike self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006).
    [Crossref] [PubMed]
  20. W. Zhong and L. Yi, “Two-dimensional Laguerre-Gaussian soliton family in strongly nonlocal nonlinear media,” Phys. Rev. A 75, 061801 (2007).
    [Crossref]
  21. B. Yang, W.-P. Zhong, and M. R. Belić, “Self-similar Hermite-Gaussian spatial solitons in two-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010).
    [Crossref]
  22. G. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
    [Crossref]
  23. C. Bernardini, F. Gori, and M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free particle states,” Eur. J. Phys 16, 58–62 (1995).
    [Crossref]
  24. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001
  25. O. Vallée and M. Soares, Airy functions and applications to physics (Imperial College Press, Singapore, 2010), 2.
    [Crossref]
  26. Y. Q. Zhang, M. R. Belić, H. B. Zheng, Z. K. Wu, Y. Y. Li, K. Q. Lu, and Y. P. Zhang, “Fresnel diffraction patterns as accelerating beams,” Europhys. Lett. 104, 34007 (2013).
    [Crossref]
  27. L. Zhang, K. Liu, H. Zhong, J. Zhang, Y. Li, and D. Fan, “Effect of initial frequency chirp on Airy pulse propagation in an optical fiber,” Opt. Express 23, 2566–2576 (2015).
    [Crossref]
  28. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
    [Crossref] [PubMed]
  29. Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35, 2260–2262 (2010).
    [Crossref] [PubMed]
  30. R. Driben, V. V. Konotop, and T. Meier, “Coupled Airy breathers,” Opt. Lett. 39, 5523–5526 (2014).
    [Crossref] [PubMed]

2015 (1)

2014 (8)

Y. Q. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, Y. Y. Li, and Y. P. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express 22, 7160–7171 (2014).
[Crossref] [PubMed]

F. Xiao, B. Li, M. Wang, W. Zhu, P. Zhang, S. Liu, M. Premaratne, and J. Zhao, “Optical Bloch oscillations of an Airy beam in a photonic lattice with a linear transverse index gradient,” Opt. Express 22, 22763–22770 (2014).
[Crossref] [PubMed]

R. Driben, V. V. Konotop, and T. Meier, “Coupled Airy breathers,” Opt. Lett. 39, 5523–5526 (2014).
[Crossref] [PubMed]

R. Driben and T. Meier, “Regeneration of Airy pulses in fiber-optic links with dispersion management of the two leading dispersion terms of opposite signs,” Phys. Rev. A 89, 043817 (2014).
[Crossref]

S. Wang, D. Fan, X. Bai, and X. Zeng, “Propagation dynamics of Airy pulses in optical fibers with periodic dispersion modulation,” Phys. Rev. A 89, 023802 (2014).
[Crossref]

J. Rogel-Salazar, H. A. Jiménez-Romero, and S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89, 023807 (2014).
[Crossref]

G. Zhou, R. Chen, and G. Ru, “Propagation of an Airy beam in a strongly nonlocal nonlinear media,” Laser Phys. Lett. 11, 105001 (2014).
[Crossref]

W.-P. Zhong, M. Belić, Y. Q. Zhang, and T. Huang, “Accelerating Airy-Gauss-Kummer localized wave packets,” Ann. Phys. 340, 171–178 (2014).
[Crossref]

2013 (3)

2011 (3)

2010 (3)

2008 (2)

I. Besieris and A. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E 78, 046605 (2008).
[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]

2007 (4)

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

W. Zhong and L. Yi, “Two-dimensional Laguerre-Gaussian soliton family in strongly nonlocal nonlinear media,” Phys. Rev. A 75, 061801 (2007).
[Crossref]

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

2006 (1)

S. Ponomarenko and G. Agrawal, “Do solitonlike self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006).
[Crossref] [PubMed]

1997 (1)

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

1995 (1)

C. Bernardini, F. Gori, and M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free particle states,” Eur. J. Phys 16, 58–62 (1995).
[Crossref]

1994 (1)

G. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[Crossref]

Agarwal, G.

G. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[Crossref]

Agrawal, G.

S. Ponomarenko and G. Agrawal, “Do solitonlike self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006).
[Crossref] [PubMed]

Bai, X.

S. Wang, D. Fan, X. Bai, and X. Zeng, “Propagation dynamics of Airy pulses in optical fibers with periodic dispersion modulation,” Phys. Rev. A 89, 023802 (2014).
[Crossref]

Bandres, M. A.

Belic, M.

Belic, M. R.

Y. Q. Zhang, M. R. Belić, H. B. Zheng, H. X. Chen, C. B. Li, Y. Y. Li, and Y. P. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express 22, 7160–7171 (2014).
[Crossref] [PubMed]

Y. Q. Zhang, M. R. Belić, H. B. Zheng, Z. K. Wu, Y. Y. Li, K. Q. Lu, and Y. P. Zhang, “Fresnel diffraction patterns as accelerating beams,” Europhys. Lett. 104, 34007 (2013).
[Crossref]

B. Yang, W.-P. Zhong, and M. R. Belić, “Self-similar Hermite-Gaussian spatial solitons in two-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010).
[Crossref]

Bernardini, C.

C. Bernardini, F. Gori, and M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free particle states,” Eur. J. Phys 16, 58–62 (1995).
[Crossref]

Besieris, I.

I. Besieris and A. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E 78, 046605 (2008).
[Crossref]

Besieris, I. M.

Broky, J.

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

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

Chávez-Cerda, S.

J. Rogel-Salazar, H. A. Jiménez-Romero, and S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89, 023807 (2014).
[Crossref]

Chen, H. X.

Chen, R.

G. Zhou, R. Chen, and G. Ru, “Propagation of an Airy beam in a strongly nonlocal nonlinear media,” Laser Phys. Lett. 11, 105001 (2014).
[Crossref]

Chen, Z.

Christodoulides, D.

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

Christodoulides, D. N.

Dogariu, A.

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

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

Driben, R.

Efremidis, N. K.

Eichelkraut, T. J.

Fan, D.

L. Zhang, K. Liu, H. Zhong, J. Zhang, Y. Li, and D. Fan, “Effect of initial frequency chirp on Airy pulse propagation in an optical fiber,” Opt. Express 23, 2566–2576 (2015).
[Crossref]

S. Wang, D. Fan, X. Bai, and X. Zeng, “Propagation dynamics of Airy pulses in optical fibers with periodic dispersion modulation,” Phys. Rev. A 89, 023802 (2014).
[Crossref]

Gori, F.

C. Bernardini, F. Gori, and M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free particle states,” Eur. J. Phys 16, 58–62 (1995).
[Crossref]

Gutiérrez-Vega, J. C.

Hu, Y.

Huang, S.

Huang, T.

W.-P. Zhong, M. Belić, Y. Q. Zhang, and T. Huang, “Accelerating Airy-Gauss-Kummer localized wave packets,” Ann. Phys. 340, 171–178 (2014).
[Crossref]

Jiménez-Romero, H. A.

J. Rogel-Salazar, H. A. Jiménez-Romero, and S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89, 023807 (2014).
[Crossref]

Kivshar, Y. S.

Konotop, V. V.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001

Li, B.

Li, C. B.

Li, Y.

Li, Y. Y.

Liu, K.

Liu, S.

Liu, W.

Lou, C.

Lu, K. Q.

Y. Q. Zhang, M. R. Belić, H. B. Zheng, Z. K. Wu, Y. Y. Li, K. Q. Lu, and Y. P. Zhang, “Fresnel diffraction patterns as accelerating beams,” Europhys. Lett. 104, 34007 (2013).
[Crossref]

Y. Q. Zhang, M. Belić, Z. K. Wu, H. B. Zheng, K. Q. Lu, Y. Y. Li, and Y. P. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38, 4585–4588 (2013).
[Crossref] [PubMed]

Malomed, B. A.

Meier, T.

R. Driben, V. V. Konotop, and T. Meier, “Coupled Airy breathers,” Opt. Lett. 39, 5523–5526 (2014).
[Crossref] [PubMed]

R. Driben and T. Meier, “Regeneration of Airy pulses in fiber-optic links with dispersion management of the two leading dispersion terms of opposite signs,” Phys. Rev. A 89, 043817 (2014).
[Crossref]

Miroshnichenko, A. E.

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Morandotti, R.

Neshev, D. N.

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001

Ponomarenko, S.

S. Ponomarenko and G. Agrawal, “Do solitonlike self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006).
[Crossref] [PubMed]

Premaratne, M.

Rogel-Salazar, J.

J. Rogel-Salazar, H. A. Jiménez-Romero, and S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89, 023807 (2014).
[Crossref]

Ru, G.

G. Zhou, R. Chen, and G. Ru, “Propagation of an Airy beam in a strongly nonlocal nonlinear media,” Laser Phys. Lett. 11, 105001 (2014).
[Crossref]

Santarsiero, M.

C. Bernardini, F. Gori, and M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free particle states,” Eur. J. Phys 16, 58–62 (1995).
[Crossref]

Shaarawi, A.

I. Besieris and A. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E 78, 046605 (2008).
[Crossref]

Shadrivov, I. V.

Simon, R.

G. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[Crossref]

Siviloglou, G.

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

Siviloglou, G. A.

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Soares, M.

O. Vallée and M. Soares, Airy functions and applications to physics (Imperial College Press, Singapore, 2010), 2.
[Crossref]

Song, D.

Vallée, O.

O. Vallée and M. Soares, Airy functions and applications to physics (Imperial College Press, Singapore, 2010), 2.
[Crossref]

Wang, M.

Wang, S.

S. Wang, D. Fan, X. Bai, and X. Zeng, “Propagation dynamics of Airy pulses in optical fibers with periodic dispersion modulation,” Phys. Rev. A 89, 023802 (2014).
[Crossref]

Wu, Z. K.

Y. Q. Zhang, M. R. Belić, H. B. Zheng, Z. K. Wu, Y. Y. Li, K. Q. Lu, and Y. P. Zhang, “Fresnel diffraction patterns as accelerating beams,” Europhys. Lett. 104, 34007 (2013).
[Crossref]

Y. Q. Zhang, M. Belić, Z. K. Wu, H. B. Zheng, K. Q. Lu, Y. Y. Li, and Y. P. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38, 4585–4588 (2013).
[Crossref] [PubMed]

Xiao, F.

Xu, J.

Yang, B.

B. Yang, W.-P. Zhong, and M. R. Belić, “Self-similar Hermite-Gaussian spatial solitons in two-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010).
[Crossref]

Ye, Z.

Yi, L.

W. Zhong and L. Yi, “Two-dimensional Laguerre-Gaussian soliton family in strongly nonlocal nonlinear media,” Phys. Rev. A 75, 061801 (2007).
[Crossref]

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001

Zeng, X.

S. Wang, D. Fan, X. Bai, and X. Zeng, “Propagation dynamics of Airy pulses in optical fibers with periodic dispersion modulation,” Phys. Rev. A 89, 023802 (2014).
[Crossref]

Zhang, J.

Zhang, L.

Zhang, P.

Zhang, Y. P.

Zhang, Y. Q.

Zhao, J.

Zheng, H. B.

Zhong, H.

Zhong, W.

W. Zhong and L. Yi, “Two-dimensional Laguerre-Gaussian soliton family in strongly nonlocal nonlinear media,” Phys. Rev. A 75, 061801 (2007).
[Crossref]

Zhong, W.-P.

W.-P. Zhong, M. Belić, Y. Q. Zhang, and T. Huang, “Accelerating Airy-Gauss-Kummer localized wave packets,” Ann. Phys. 340, 171–178 (2014).
[Crossref]

B. Yang, W.-P. Zhong, and M. R. Belić, “Self-similar Hermite-Gaussian spatial solitons in two-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010).
[Crossref]

Zhou, G.

G. Zhou, R. Chen, and G. Ru, “Propagation of an Airy beam in a strongly nonlocal nonlinear media,” Laser Phys. Lett. 11, 105001 (2014).
[Crossref]

Zhu, W.

Ann. Phys. (1)

W.-P. Zhong, M. Belić, Y. Q. Zhang, and T. Huang, “Accelerating Airy-Gauss-Kummer localized wave packets,” Ann. Phys. 340, 171–178 (2014).
[Crossref]

Commun. Theor. Phys. (1)

B. Yang, W.-P. Zhong, and M. R. Belić, “Self-similar Hermite-Gaussian spatial solitons in two-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010).
[Crossref]

Eur. J. Phys (1)

C. Bernardini, F. Gori, and M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free particle states,” Eur. J. Phys 16, 58–62 (1995).
[Crossref]

Europhys. Lett. (1)

Y. Q. Zhang, M. R. Belić, H. B. Zheng, Z. K. Wu, Y. Y. Li, K. Q. Lu, and Y. P. Zhang, “Fresnel diffraction patterns as accelerating beams,” Europhys. Lett. 104, 34007 (2013).
[Crossref]

Laser Phys. Lett. (1)

G. Zhou, R. Chen, and G. Ru, “Propagation of an Airy beam in a strongly nonlocal nonlinear media,” Laser Phys. Lett. 11, 105001 (2014).
[Crossref]

Opt. Commun. (1)

G. Agarwal and R. Simon, “A simple realization of fractional Fourier transform and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[Crossref]

Opt. Express (4)

Opt. Lett. (10)

T. J. Eichelkraut, G. A. Siviloglou, I. M. Besieris, and D. N. Christodoulides, “Oblique Airy wave packets in bidispersive optical media,” Opt. Lett. 35, 3655–3657 (2010).
[Crossref] [PubMed]

N. K. Efremidis, “Airy trajectory engineering in dynamic linear index potentials,” Opt. Lett. 36, 3006–3008 (2011).
[Crossref] [PubMed]

Y. Q. Zhang, M. Belić, Z. K. Wu, H. B. Zheng, K. Q. Lu, Y. Y. Li, and Y. P. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38, 4585–4588 (2013).
[Crossref] [PubMed]

R. Driben, Y. Hu, Z. Chen, B. A. Malomed, and R. Morandotti, “Inversion and tight focusing of Airy pulses under the action of third-order dispersion,” Opt. Lett. 38, 2499–2501 (2013).
[Crossref] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[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, 1164–1166 (2011).
[Crossref] [PubMed]

Z. Ye, S. Liu, C. Lou, P. Zhang, Y. Hu, D. Song, J. Zhao, and Z. Chen, “Acceleration control of Airy beams with optically induced refractive-index gradient,” Opt. Lett. 36, 3230–3232 (2011).
[Crossref] [PubMed]

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

Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35, 2260–2262 (2010).
[Crossref] [PubMed]

R. Driben, V. V. Konotop, and T. Meier, “Coupled Airy breathers,” Opt. Lett. 39, 5523–5526 (2014).
[Crossref] [PubMed]

Phys. Rev. A (4)

R. Driben and T. Meier, “Regeneration of Airy pulses in fiber-optic links with dispersion management of the two leading dispersion terms of opposite signs,” Phys. Rev. A 89, 043817 (2014).
[Crossref]

S. Wang, D. Fan, X. Bai, and X. Zeng, “Propagation dynamics of Airy pulses in optical fibers with periodic dispersion modulation,” Phys. Rev. A 89, 023802 (2014).
[Crossref]

J. Rogel-Salazar, H. A. Jiménez-Romero, and S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89, 023807 (2014).
[Crossref]

W. Zhong and L. Yi, “Two-dimensional Laguerre-Gaussian soliton family in strongly nonlocal nonlinear media,” Phys. Rev. A 75, 061801 (2007).
[Crossref]

Phys. Rev. E (1)

I. Besieris and A. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E 78, 046605 (2008).
[Crossref]

Phys. Rev. Lett. (2)

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

S. Ponomarenko and G. Agrawal, “Do solitonlike self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006).
[Crossref] [PubMed]

Science (1)

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[Crossref]

Other (2)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001

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[Crossref]

Supplementary Material (5)

» Media 1: MP4 (1244 KB)     
» Media 2: MP4 (1435 KB)     
» Media 3: MP4 (1435 KB)     
» Media 4: MP4 (1464 KB)     
» Media 5: MP4 (1413 KB)     

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Figures (7)

Fig. 1
Fig. 1 (a) Propagation of a finite energy Airy beam ψ(x, 0) = Ai(x)exp(ax) with a = 0.1 in a parabolic potential with α = 0.5 ( Media 1). (b) Intensity profiles at z = D / 4 and z = D / 2, respectively. Solid and dashed curves represent the numerical simulation and the analytical solutions, respectively. Gaussian and Airy profiles refer to the left y-axis and the right y-axis, respectively, as indicated by the circle-arrows.
Fig. 2
Fig. 2 Propagation of finite energy Airy beams with different transverse displacements. (a1)–(a3) x0 = −15, −10, and −5, respectively ( Media 2). (b1)–(b3) x0 = 15, 10, and 5, respectively ( Media 3). Other parameters are a = 0.1 and α = 0.5.
Fig. 3
Fig. 3 (a) Numerical trajectory (a1), velocity (a2) and acceleration (a3) of the Airy beam during propagation. Red, black and blue curves correspond to the transversely displaced beams, with displacements x0 = −10, 0, and 10, respectively. (b) Analytical trajectory (b1), velocity (b2) and acceleration (b3) corresponding to (a). Other parameters are a = 0.1 and α = 0.5.
Fig. 4
Fig. 4 (a) The width of the single-peak phase region versus the decay parameter a, corresponding to Fig. 3. (b) and (c) Same as Fig. 1(a), but over a half of the period and corresponding to the green dots in (a). In the left panel, a = 0.01; in the right, a = 0.05.
Fig. 5
Fig. 5 Propagation of finite energy Airy beams with a linear chirp and different transverse displacements. (a)–(c) x0 = −10, 0, and 10, respectively ( Media 4). Other parameters are a = 0.1, α = 0.5, and β = 5.
Fig. 6
Fig. 6 Propagation of finite energy Airy beams with a quadratic chirp, for different transverse displacements. (a)–(c) x0 = −5, 0, and 5, respectively ( Media 5). Other parameters are a = 0.1, α = 0.5, and β = 0.2.
Fig. 7
Fig. 7 Propagation of a two-dimensional finite energy Airy beam ψ(x, y) = exp(ax)Ai(x)exp(ay)Ai(y) in a parabolic potential. (a) Iso-surface plot. (b) Intensity in the cross section xy = 0. The parameters are a = 0.1 and α = 0.5.

Equations (33)

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i ψ z + 1 2 2 ψ x 2 V ( x ) ψ = 0 ,
V ( x ) = 1 2 α 2 x 2
ψ ( x , z ) = + ψ ( ξ , 0 ) H ( x , ξ , z ) d ξ ,
H ( x , ξ , z ) = i 2 π α csc ( α z ) exp { i α cot ( α z ) [ x 2 + ξ 2 2 x ξ sec ( α z ) ] }
ψ ( x , z ) = f ( x , z ) + [ ψ ( ξ , 0 ) exp ( i b ξ 2 ) ] exp ( i K ξ ) d ξ ,
f ( x , z ) = i 2 π K x exp ( i b x 2 ) .
ψ ^ ( k ) = exp ( a k 2 ) exp [ a 3 3 + i 3 ( k 3 3 a 2 k ) ] ,
i π b exp ( i 4 b k 2 ) ,
Ai ( x ) = 1 2 π i i + i exp ( x t t 3 3 ) d t ,
ψ ( x , z ) = f ( x , z ) i π b exp ( a 3 3 ) Ai ( K 2 b 1 16 b 2 + i a 2 b ) × exp [ ( a + i 4 b ) ( K 2 b 1 16 b 2 + i a 2 b ) ] × exp [ i K 2 4 b 1 3 ( a + i 4 b ) 3 ] .
x = 1 4 α 2 sin 2 ( α z ) cos ( α z ) ,
D = 2 π α ,
ψ ( x , z = 2 m + 1 4 D ) = i s α 2 π exp ( a α 2 x 2 ) exp [ a 3 3 + i s 3 ( α 3 x 3 3 a 2 α x ) ] ,
ψ ^ ( k ) = exp ( i x 0 k ) exp ( a k 2 ) exp [ a 3 3 + i 3 ( k 3 3 a 2 k ) ] ,
ψ ( x , z ) = f ( x , z ) i π b exp ( a 3 3 ) Ai ( K 2 b 1 16 b 2 + i a 2 b x 0 ) × exp [ ( a + i 4 b ) ( K 2 b 1 16 b 2 + i a 2 b x 0 ) ] × exp [ i K 2 4 b 1 3 ( a + i 4 b ) 3 ] ,
ψ ( x , z = 2 m + 1 4 D ) = i s α 2 π exp ( i x 0 α x ) exp ( a α 2 x 2 ) exp [ a 3 3 + i s 3 ( α 3 x 3 3 a 2 α x ) ] .
x = 1 4 α 2 sin 2 ( α z ) cos ( α z ) + x 0 cos ( α z ) ,
x ¯ = + x | ψ ( x ) | 2 d x + | ψ ( x ) | 2 d x .
ψ ( x , 0 ) = Ai ( x x 0 ) exp [ a ( x x 0 ) exp ( i β x ) ] ,
ψ ( x , z ) = f ( x , z ) i π b exp ( a 3 3 ) Ai ( K 2 b 1 16 b 2 + i a 2 b x 0 ) × exp [ ( a + i 4 b ) ( K 2 b 1 16 b 2 + i a 2 b x 0 ) ] × exp [ i K 2 4 b 1 3 ( a + i 4 b ) 3 ] ,
ψ ( x , z = 2 m + 1 4 D ) = i s α 2 π exp [ i x 0 ( α x β ) ] exp [ a ( α x β ) 2 ] × exp { a 3 3 + i s 3 [ ( α x β ) 3 3 a 2 ( α x β ) ] } .
x = 1 4 α 2 sin 2 ( α z ) cos ( α z ) x 0 cos ( α z ) + β α sin ( α z ) ,
ψ ( x , 0 ) = Ai ( x x 0 ) exp [ a ( x x 0 ) ] exp ( i β x 2 ) .
ψ ( x , z ) = f ( x , z ) + [ ψ ( ξ , 0 ) exp ( i b ξ 2 ) ] exp ( i K ξ ) d ξ ,
ψ ( x , z ) = f ( x , z ) i π b exp ( a 3 3 ) Ai ( K 2 b 1 16 b 2 + i a 2 b x 0 ) × exp [ ( a + i 4 b ) ( K 2 b 1 16 b 2 + i a 2 b x 0 ) ] × exp [ i K 2 4 b 1 3 ( a + i 4 b ) 3 ] .
z = 1 α arctan ( α 2 β ) + m 2 D .
x = sin 2 ( α z ) 4 α [ α cos ( α z ) + 2 β sin ( α z ) ] + [ α cos ( α z ) + 2 β sin ( α z ) ] x 0 .
i ψ z + 1 2 ( 2 ψ x 2 + 2 ψ y 2 ) V ( x , y ) ψ = 0 ,
V ( x ) = 1 2 α 2 ( x 2 + y 2 ) .
i X z + 1 2 2 X x 2 1 2 α 2 x 2 X μ X = 0 ,
i Y z + 1 2 2 Y y 2 1 2 α 2 y 2 Y μ Y = 0 ,
i f z + 1 2 2 f x 2 1 2 α 2 x 2 f = 0 ,
i g z + 1 2 2 g y 2 1 2 α 2 y 2 g = 0 ,

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