Abstract

We investigate the propagation properties of finite-energy Airy beams in nonlocal nonlinear atomic vapor. The nonlocality is caused by the transportation of optically excited atoms and described with a Green function. By conducting numerical simulation on Airy beams in the case of one-dimension, two-dimensions, and ring geometry, respectively, we find that the beams have a more stable shape and a longer propagation distance in the presence of nonlocality. The results show that the nonlocality can slow down the beams’ distortion induced by the self-focusing effect and contributes to a more stable propagation of Airy beams in the nonlinear media.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  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]
  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]
  4. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optical mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
    [Crossref]
  5. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
    [Crossref]
  6. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
    [Crossref]
  7. 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]
  8. I. M. Allayarov and E. N. Tsoy, “Dynamics of Airy beams in nonlinear media,” Phys. Rev. A 90(2), 023852 (2014).
    [Crossref]
  9. C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy–Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
    [Crossref]
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    [Crossref]
  13. D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48(6), 4583–4587 (1993).
    [Crossref]
  14. F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13(8), 284–286 (1968).
    [Crossref]
  15. C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
    [Crossref]
  16. R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express 19(24), 23706–23715 (2011).
    [Crossref]
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    [Crossref]
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  21. R. W. Boyd, “Nonlinear Optics (3rd),” 278–280 (2007).
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    [Crossref]
  23. S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98(26), 263902 (2007).
    [Crossref]
  24. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045 (2010).
    [Crossref]
  25. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842 (2011).
    [Crossref]
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    [Crossref]

2017 (2)

Q. Kong, N. Wei, C. Z. Fan, J. L. Shi, and M. Shen, “Suppression of collapse for two-dimensional Airy beam in nonlocal nonlinear media,” Sci. Rep. 7(1), 4198 (2017).
[Crossref]

Z. K. Wu, P. Li, and Y. Z. Gu, “Propagation dynamics of finite-energy Airy beams in nonlocal nonlinear media,” Front. Phys. 12(5), 124203 (2017).
[Crossref]

2016 (1)

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy–Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

2015 (1)

2014 (2)

2013 (1)

2011 (4)

2010 (2)

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045 (2010).
[Crossref]

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]

2009 (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

2008 (1)

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

2007 (3)

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

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

S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98(26), 263902 (2007).
[Crossref]

2006 (1)

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2(11), 769–774 (2006).
[Crossref]

2003 (1)

C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
[Crossref]

1997 (1)

A. W. Snyder and D. J. Mitchell, “Accessible Solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

1993 (1)

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48(6), 4583–4587 (1993).
[Crossref]

1979 (1)

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

1968 (1)

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13(8), 284–286 (1968).
[Crossref]

1967 (1)

D. H. Close, “Strong-field saturation effects in laser media,” Phys. Rev. 153(2), 360–371 (1967).
[Crossref]

Alfassi, B.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2(11), 769–774 (2006).
[Crossref]

Allayarov, I. M.

I. M. Allayarov and E. N. Tsoy, “Dynamics of Airy beams in nonlinear media,” Phys. Rev. A 90(2), 023852 (2014).
[Crossref]

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
[Crossref]

Balazs, N. L.

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

Baumgartl, J.

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

Bekenstein, R.

Belic, M. R.

Berry, M. V.

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

Blasberg, T.

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48(6), 4583–4587 (1993).
[Crossref]

Boyd, R. W.

R. W. Boyd, “Nonlinear Optics (3rd),” 278–280 (2007).

Broky, J.

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

Cao, M.

Chen, B.

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy–Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

Chen, C.

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy–Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

Chong, A.

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]

Chremmos, I.

Christodoulides, D. N.

I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36(10), 1890 (2011).
[Crossref]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
[Crossref]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842 (2011).
[Crossref]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045 (2010).
[Crossref]

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]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

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

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

Close, D. H.

D. H. Close, “Strong-field saturation effects in laser media,” Phys. Rev. 153(2), 360–371 (1967).
[Crossref]

Cohen, O.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2(11), 769–774 (2006).
[Crossref]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
[Crossref]

Dabby, F. W.

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13(8), 284–286 (1968).
[Crossref]

Deng, D.

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy–Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

Dholakia, K.

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

Dogariu, A.

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

Efremidis, N. K.

Fan, C. Z.

Q. Kong, N. Wei, C. Z. Fan, J. L. Shi, and M. Shen, “Suppression of collapse for two-dimensional Airy beam in nonlocal nonlinear media,” Sci. Rep. 7(1), 4198 (2017).
[Crossref]

Gao, H.

Gu, Y. Z.

Z. K. Wu, P. Li, and Y. Z. Gu, “Propagation dynamics of finite-energy Airy beams in nonlocal nonlinear media,” Front. Phys. 12(5), 124203 (2017).
[Crossref]

Guo, W.

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

Kong, Q.

Q. Kong, N. Wei, C. Z. Fan, J. L. Shi, and M. Shen, “Suppression of collapse for two-dimensional Airy beam in nonlocal nonlinear media,” Sci. Rep. 7(1), 4198 (2017).
[Crossref]

Krolikowski, W.

S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98(26), 263902 (2007).
[Crossref]

Li, F.

Li, P.

Z. K. Wu, P. Li, and Y. Z. Gu, “Propagation dynamics of finite-energy Airy beams in nonlocal nonlinear media,” Front. Phys. 12(5), 124203 (2017).
[Crossref]

Li, Y. Y.

Lu, K. Q.

Mazilu, M.

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

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, “Accessible Solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

Papazoglou, D. G.

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
[Crossref]

Peng, X.

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy–Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

Peng, Y.

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy–Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

Renninger, W. H.

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]

Rotschild, C.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2(11), 769–774 (2006).
[Crossref]

Saffman, M.

S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98(26), 263902 (2007).
[Crossref]

Segev, M.

R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express 19(24), 23706–23715 (2011).
[Crossref]

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2(11), 769–774 (2006).
[Crossref]

Shen, M.

Q. Kong, N. Wei, C. Z. Fan, J. L. Shi, and M. Shen, “Suppression of collapse for two-dimensional Airy beam in nonlocal nonlinear media,” Sci. Rep. 7(1), 4198 (2017).
[Crossref]

Shi, J. L.

Q. Kong, N. Wei, C. Z. Fan, J. L. Shi, and M. Shen, “Suppression of collapse for two-dimensional Airy beam in nonlocal nonlinear media,” Sci. Rep. 7(1), 4198 (2017).
[Crossref]

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

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

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

Skupin, S.

S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98(26), 263902 (2007).
[Crossref]

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, “Accessible Solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

Suter, D.

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48(6), 4583–4587 (1993).
[Crossref]

Tsoy, E. N.

I. M. Allayarov and E. N. Tsoy, “Dynamics of Airy beams in nonlinear media,” Phys. Rev. A 90(2), 023852 (2014).
[Crossref]

Tzortzakis, S.

Wei, D.

Wei, N.

Q. Kong, N. Wei, C. Z. Fan, J. L. Shi, and M. Shen, “Suppression of collapse for two-dimensional Airy beam in nonlocal nonlinear media,” Sci. Rep. 7(1), 4198 (2017).
[Crossref]

Whinnery, J. R.

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13(8), 284–286 (1968).
[Crossref]

Wise, F. W.

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]

Wu, Z. K.

Yang, X.

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy–Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

Ye, F.

Yu, Y.

Zhang, L.

Zhang, P.

Zhang, S.

Zhang, Y. P.

Zhang, Y. Q.

Zheng, H. B.

Zhou, M.

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy–Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

Am. J. Phys. (1)

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

Appl. Phys. Lett. (1)

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13(8), 284–286 (1968).
[Crossref]

Front. Phys. (1)

Z. K. Wu, P. Li, and Y. Z. Gu, “Propagation dynamics of finite-energy Airy beams in nonlocal nonlinear media,” Front. Phys. 12(5), 124203 (2017).
[Crossref]

J. Opt. (1)

C. Chen, X. Peng, B. Chen, Y. Peng, M. Zhou, X. Yang, and D. Deng, “Propagation of an Airy–Gaussian vortex beam in linear and nonlinear media,” J. Opt. 18(5), 055505 (2016).
[Crossref]

Nat. Photonics (2)

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]

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

Nat. Phys. (1)

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2(11), 769–774 (2006).
[Crossref]

Opt. Express (1)

Opt. Lett. (8)

Phys. Rev. (1)

D. H. Close, “Strong-field saturation effects in laser media,” Phys. Rev. 153(2), 360–371 (1967).
[Crossref]

Phys. Rev. A (2)

I. M. Allayarov and E. N. Tsoy, “Dynamics of Airy beams in nonlinear media,” Phys. Rev. A 90(2), 023852 (2014).
[Crossref]

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48(6), 4583–4587 (1993).
[Crossref]

Phys. Rev. Lett. (3)

C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91(7), 073901 (2003).
[Crossref]

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

S. Skupin, M. Saffman, and W. Krolikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98(26), 263902 (2007).
[Crossref]

Sci. Rep. (1)

Q. Kong, N. Wei, C. Z. Fan, J. L. Shi, and M. Shen, “Suppression of collapse for two-dimensional Airy beam in nonlocal nonlinear media,” Sci. Rep. 7(1), 4198 (2017).
[Crossref]

Science (2)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

A. W. Snyder and D. J. Mitchell, “Accessible Solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

Other (1)

R. W. Boyd, “Nonlinear Optics (3rd),” 278–280 (2007).

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Figures (8)

Fig. 1.
Fig. 1. The relationship between the susceptibilities and the intensity with parameters $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$.
Fig. 2.
Fig. 2. The local and nonlocal susceptibilities on the spot diameter for Gaussian field, with parameters ${A_0}=3{\times }10^{5}V/m$, ${w_0}=10{\mu }m$, $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$.
Fig. 3.
Fig. 3. Nonlocal and local propagation of a one-dimensional Airy beams. The first column shows the propagation in local case and the second column shows the propagation in nonlocal case. The propagation distances are (a0)(b0)z=0; (a1)(b1)z=0.5mm; (a2)(b2)z=1mm; (a3)(b3)z=2mm. The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=3.2{\times }10^{4}V/m$, ${w_0}=15{\mu }m$, ${\alpha }=0.1$.
Fig. 4.
Fig. 4. The maximum intensity of the one-dimensional Airy beam at different propagation distances in the local case (the blue line) and nonlocal case (the red dashed line). The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=3.2{\times }10^{4}V/m$, ${w_0}=15{\mu }m$, ${\alpha }=0.1$.
Fig. 5.
Fig. 5. Nonlocal and local propagation of a 2D Airy beam with the input power ${P_0}=0.51mW$. The first column shows the propagation in local case and the second column shows the propagation in nonlocal case. The propagation distances are (a0)(b0)z=0; (a1)(b1)z=2mm; (a2)(b2)z=4mm; (a3)(b3)z=6mm. The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=3.43{\times }10^{4}V/m$, ${w_0}=30{\mu }m$, ${\alpha }=0.1$.
Fig. 6.
Fig. 6. The maximum intensity of the 2D Airy beam at different propagation distances in the local case (the blue line) and nonlocal case (the red dashed line). The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=3.43{\times }10^{4}V/m$, ${w_0}=30{\mu }m$, ${\alpha }=0.1$.
Fig. 7.
Fig. 7. Propagation of ring-Airy beams with the input power ${P_0}=0.58mW$ and width ${w_0}=15{\mu }m$. The three column pictures show the propagation in free space, local and nonlocal nonlinear conditions, respectively. The distances are (a0)(b0)(c0)z=0; (a1)(b1)(c1)z=3mm; (a2)(b2)(a2)z=3.8mm; (a3)(b3)(a3)z=6.3mm. The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=1.2{\times }10^{4}V/m$, ${r_0}=15{\mu }m$, ${\alpha }=0.1$.
Fig. 8.
Fig. 8. The maximum intensity of the ring-Airy beam at different propagation distances in the case of the free space (the black dashed line), local conditions (the blue line) and nonlocal conditions (the red line marked with dots), respectively. The parameters are $p=4.0$, $q=0.0083$, ${\chi _0}=0.03$, ${A_0}=1.2{\times }10^{4}V/m$, ${w_0}=15{\mu }m$, ${\alpha }=0.1$.

Equations (12)

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ρ ˙ b a = i ( ω b a + 1 T 2 ) ρ b a + i V b a W ,
W ˙ = W W e q T 1 2 i ( V b a ρ a b V a b ρ b a ) ,
χ = α 0 ( 0 ) ω b a / c Δ T 2 i 1 + Δ 2 T 2 2 + I / I S ,
ψ z = i 2 k 2 ψ + i k 2 χ ( I ) ψ ,
χ l = χ 0 I m [ ξ ( p + i q ) ] ,
χ n l l o c ( I ) = χ 0 { R e [ ξ ( p + i q I ) ] R e [ ξ ( p + i q ) ] } ,
χ n l l o c ( I ) = χ 0 { I m [ ξ ( p + i q I ) ] 1 + I / I S I m [ ξ ( p + i q ) ] } ,
χ n l n o n l o c ( r ) = γ 2 g ( r , r 0 ; γ / 2 ) χ n l l o c ( r 0 ) d r 0 ,
χ n l n o n l o c ( r ) = γ g ( r , r 0 ; γ ) χ n l l o c ( r 0 ) d r 0 ,
ψ ( x , 0 ) = A 0 A i ( x ) e x p ( α x / w 0 ) ,
ψ ( x , y , 0 ) = A 0 A i ( x ) A i ( y ) e x p ( α x / w 0 ) e x p ( α y / w 0 ) ,
ψ ( x , y , 0 ) = A 0 A i [ ( r 0 r ) / w 0 ] e x p [ α ( r 0 r ) / w 0 ] ,

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