Abstract

The propagation of decelerating Airy pulses in non-instantaneous cubic medium is investigated both theoretically and numerically. In a Debye model, at variance with the case of accelerating Airy and Gaussian pulses, a decelerating Airy pulse evolves into a single soliton for weak and general non-instantaneous response. Airy pulses can hence be used to control soliton generation by temporal shaping. The effect is critically dependent on the response time, and could be used as a way to measure the Debye type response function. For highly non-instantaneous response, we theoretically find a decelerating Airy pulse is still transformed into Airy wave packet with deceleration. The theoretical predictions are confirmed by numerical simulations.

© 2017 Optical Society of America

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References

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2016 (2)

L. Zhang, K. Liu, H. Zhong, J. Zhang, Y. Hu, J. Deng, D. Lei, Y. Li, and D. Fan, “Discriminating the role of Raman effects in the propagation of decelerating and accelerating Airy pulses by time–frequency analysis,” J. Opt. 18(1), 015505 (2016).
[Crossref]

F. Deng, W. Hong, and D. Deng, “Airy-type solitary wave in highly noninstantaneous Kerr media,” Opt. Express 24(14), 15997–16002 (2016).
[Crossref] [PubMed]

2015 (3)

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(3), 2566–2576 (2015).
[Crossref] [PubMed]

L. Zhang, K. Liu, H. Zhong, J. Zhang, J. Deng, Y. Li, and D. Fan, “Engineering deceleration and acceleration of soliton emitted from Airy pulse with quadratic phase modulation in optical fibers without high-order effects,” Sci. Rep. 5, 11843 (2015).
[Crossref] [PubMed]

Y. Hu, A. Tehranchi, S. Wabnitz, R. Kashyap, Z. Chen, and R. Morandotti, “Improved intrapulse raman scattering control via asymmetric airy pulses,” Phys. Rev. Lett. 114(7), 073901 (2015).
[Crossref] [PubMed]

2014 (6)

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(2), 023802 (2014).
[Crossref]

T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref] [PubMed]

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref] [PubMed]

L. Zhang, J. Zhang, Y. Chen, A. Liu, and G. Liu, “Dynamic propagation of finite-energy Airy pulses in the presence of higher-order effects,” J. Opt. Soc. Am. B 31(4), 889–897 (2014).
[Crossref]

L. Zhang and H. Zhong, “Modulation instability of finite energy Airy pulse in optical fiber,” Opt. Express 22(14), 17107–17115 (2014).
[Crossref] [PubMed]

L. Zhang, H. Zhong, Y. Li, and D. Fan, “Manipulation of Raman-induced frequency shift by use of asymmetric self-accelerating Airy pulse,” Opt. Express 22(19), 22598–22607 (2014).
[Crossref] [PubMed]

2013 (5)

2012 (4)

C. Ament, M. Kolesik, J. Moloney, and P. Polynkin, “Self-focusing dynamics of ultraintense Airy waveforms in water,” Phys. Rev. A 86(4), 043842 (2012).
[Crossref]

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108(11), 113903 (2012).
[Crossref] [PubMed]

B. Kibler, C. Michel, J. Garnier, and A. Picozzi, “Temporal dynamics of incoherent waves in noninstantaneous response nonlinear Kerr media,” Opt. Lett. 37(13), 2472–2474 (2012).
[Crossref] [PubMed]

I. Kaminer, J. Nemirovsky, and M. Segev, “Self-accelerating self-trapped nonlinear beams of Maxwell’s equations,” Opt. Express 20(17), 18827–18835 (2012).
[Crossref] [PubMed]

2011 (8)

2010 (8)

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[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. 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]

L. Zhang, S. Wen, X. Fu, J. Deng, J. Zhang, and D. Fan, “Spatiotemporal instability in dispersive nonlinear Kerr medium with a finite response time,” Opt. Commun. 283(10), 2251–2257 (2010).
[Crossref]

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105(26), 263902 (2010).
[Crossref] [PubMed]

C. Conti, S. Stark, P. St. J. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by the Raman effect in photonic crystal fibers,” Phys. Rev. A 82(1), 013838 (2010).
[Crossref]

R. V. J. Raja, A. Husakou, J. Hermann, and K. Porsezian, “Supercontinuum generation in liquid-filled photonic crystal fiber with slow nonlinear response,” J. Opt. Soc. Am. B 27(9), 1763–1768 (2010).
[Crossref]

M. Vieweg, T. Gissibl, S. Pricking, B. T. Kuhlmey, D. C. Wu, B. J. Eggleton, and H. Giessen, “Ultrafast nonlinear optofluidics in selectively liquid-filled photonic crystal fibers,” Opt. Express 18(24), 25232–25240 (2010).
[Crossref] [PubMed]

2009 (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] [PubMed]

P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009).
[Crossref] [PubMed]

2008 (3)

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

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]

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]

2007 (4)

2006 (1)

1989 (1)

J. A. Giannini and R. I. Joseph, “The role of the second Painleve transcendent in nonlinear optics,” Phys. Lett. A 141(8–9), 417–419 (1989).
[Crossref]

1979 (1)

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

1972 (1)

J. A. Fleck and R. L. Carman, “Effect of relaxation on small-scale filament formation by ultrashort light pulses,” Appl. Phys. Lett. 20(8), 290–293 (1972).
[Crossref]

1969 (1)

J. A. Fleck and P. L. Kelley, “Temporal aspects of the self-focusing of light beams,” Appl. Phys. Lett. 15(10), 313–315 (1969).
[Crossref]

Abdollahpour, D.

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A 84(2), 021807 (2011).
[Crossref]

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]

Ament, C.

C. Ament, M. Kolesik, J. Moloney, and P. Polynkin, “Self-focusing dynamics of ultraintense Airy waveforms in water,” Phys. Rev. A 86(4), 043842 (2012).
[Crossref]

C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with femtosecond self-healing Airy pulses,” Phys. Rev. Lett. 107(24), 243901 (2011).
[Crossref] [PubMed]

Arie, A.

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108(11), 113903 (2012).
[Crossref] [PubMed]

Azaña, J.

Bache, M.

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(2), 023802 (2014).
[Crossref]

Balazs, N. L.

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

Bang, O.

Baumgartl, J.

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

Bekenstein, R.

Belic, M.

Berry, M. V.

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

Besieris, I. M.

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]

Biancalana, F.

C. Conti, S. Stark, P. St. J. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by the Raman effect in photonic crystal fibers,” Phys. Rev. A 82(1), 013838 (2010).
[Crossref]

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105(26), 263902 (2010).
[Crossref] [PubMed]

Boguslawski, M.

P. Rose, F. Diebel, M. Boguslawski, and C. Denz, “Airy beam induced optical routing,” Appl. Phys. Lett. 102(10), 101101 (2013).
[Crossref]

Bongiovanni, D.

Broky, J.

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]

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]

Carman, R. L.

J. A. Fleck and R. L. Carman, “Effect of relaxation on small-scale filament formation by ultrashort light pulses,” Appl. Phys. Lett. 20(8), 290–293 (1972).
[Crossref]

Chen, Y.

Chen, Z.

Chong, A.

A. Chong, W. 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]

Christodoulides, D. N.

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106(21), 213903 (2011).
[Crossref] [PubMed]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011).
[Crossref] [PubMed]

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]

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[Crossref] [PubMed]

A. Chong, W. 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] [PubMed]

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]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
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L. Zhang, K. Liu, H. Zhong, J. Zhang, Y. Hu, J. Deng, D. Lei, Y. Li, and D. Fan, “Discriminating the role of Raman effects in the propagation of decelerating and accelerating Airy pulses by time–frequency analysis,” J. Opt. 18(1), 015505 (2016).
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L. Zhang, K. Liu, H. Zhong, J. Zhang, J. Deng, Y. Li, and D. Fan, “Engineering deceleration and acceleration of soliton emitted from Airy pulse with quadratic phase modulation in optical fibers without high-order effects,” Sci. Rep. 5, 11843 (2015).
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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(3), 2566–2576 (2015).
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L. Zhang, K. Liu, H. Zhong, J. Zhang, Y. Hu, J. Deng, D. Lei, Y. Li, and D. Fan, “Discriminating the role of Raman effects in the propagation of decelerating and accelerating Airy pulses by time–frequency analysis,” J. Opt. 18(1), 015505 (2016).
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L. Zhang, K. Liu, H. Zhong, J. Zhang, J. Deng, Y. Li, and D. Fan, “Engineering deceleration and acceleration of soliton emitted from Airy pulse with quadratic phase modulation in optical fibers without high-order effects,” Sci. Rep. 5, 11843 (2015).
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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(3), 2566–2576 (2015).
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A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A 84(2), 021807 (2011).
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C. Ament, M. Kolesik, J. Moloney, and P. Polynkin, “Self-focusing dynamics of ultraintense Airy waveforms in water,” Phys. Rev. A 86(4), 043842 (2012).
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P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
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P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
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A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A 84(2), 021807 (2011).
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Polynkin, P.

C. Ament, M. Kolesik, J. Moloney, and P. Polynkin, “Self-focusing dynamics of ultraintense Airy waveforms in water,” Phys. Rev. A 86(4), 043842 (2012).
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C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with femtosecond self-healing Airy pulses,” Phys. Rev. Lett. 107(24), 243901 (2011).
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P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009).
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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).
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C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105(26), 263902 (2010).
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C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, “Highly noninstantaneous solitons in liquid-core photonic crystal fibers,” Phys. Rev. Lett. 105(26), 263902 (2010).
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S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
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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).
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C. Conti, S. Stark, P. St. J. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by the Raman effect in photonic crystal fibers,” Phys. Rev. A 82(1), 013838 (2010).
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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).
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Y. Hu, A. Tehranchi, S. Wabnitz, R. Kashyap, Z. Chen, and R. Morandotti, “Improved intrapulse raman scattering control via asymmetric airy pulses,” Phys. Rev. Lett. 114(7), 073901 (2015).
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Tzortzakis, S.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, “Sharply autofocused ring-Airy beams transforming into non-linear intense light bullets,” Nat. Commun. 4, 2622 (2013).
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A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A 84(2), 021807 (2011).
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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).
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T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
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Wabnitz, S.

Y. Hu, A. Tehranchi, S. Wabnitz, R. Kashyap, Z. Chen, and R. Morandotti, “Improved intrapulse raman scattering control via asymmetric airy pulses,” Phys. Rev. Lett. 114(7), 073901 (2015).
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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(2), 023802 (2014).
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L. Zhang, S. Wen, X. Fu, J. Deng, J. Zhang, and D. Fan, “Spatiotemporal instability in dispersive nonlinear Kerr medium with a finite response time,” Opt. Commun. 283(10), 2251–2257 (2010).
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A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
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L. Zhang, K. Liu, H. Zhong, J. Zhang, J. Deng, Y. Li, and D. Fan, “Engineering deceleration and acceleration of soliton emitted from Airy pulse with quadratic phase modulation in optical fibers without high-order effects,” Sci. Rep. 5, 11843 (2015).
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Figures (8)

Fig. 1
Fig. 1 Temporal evolution of decelerating Airy pulse with a = 0.03 in non-instantaneous Kerr nonlinear medium with N = 2 and different response time T r .
Fig. 2
Fig. 2 Stable temporal evolution of decelerating Airy pulse with a = 0.03 in non-instantaneous Kerr nonlinear medium with different response time T r and nonlinearity N .
Fig. 3
Fig. 3 The critical values of the response time T r c is plotted as a function of (a) nonlinearity N and (b) the truncation coefficient N .
Fig. 4
Fig. 4 Temporal evolutions of decelerating Airy pulse with different values of truncation coefficient in non-instantaneous Kerr nonlinear medium with the parameters T r = 4 and N = 2 .
Fig. 5
Fig. 5 Temporal evolution of decelerating Airy pulse with a = 0.03 and N = 2 in (a-c) highly non-instantaneous cubic media and (d) linear media. The corresponding (e) Peak intensity (PI) and (f) position of PI as a function of propagation distance.
Fig. 6
Fig. 6 Temporal evolution of accelerating Airy pulse with a = 0.03 in non-instantaneous Kerr nonlinear medium with different response time T r and nonlinearity N .
Fig. 7
Fig. 7 The change of nonlinear refractive δ n ( T , Z ) as a function of propagation distance for (a) accelerating and (b) decelerating Airy pulses propagation in non-instantaneous Kerr nonlinear medium.
Fig. 8
Fig. 8 Temporal evolution of (a) Gaussian pulse and (b) decelerating Airy pulse (DAiP) with a = 0.03 in non-instantaneous nonlinear medium. (c) Peak intensity (PI) and the position of PI along propagation. (d) Output pulse shapes at Z = 15 .

Equations (11)

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i U Z + 1 2 s 2 U T 2 + σ N 2 U + R ( T T ) | U ( T ) | 2 d T = 0.
T = t z / v g t 0 , Z = z L D , N = γ P 0 L D , s = sgn ( β 2 ) , σ = sgn ( γ ) .
i U Z + s 2 2 U T 2 + σ R ( T ) U = 0.
i U Z + s 2 2 U T 2 = 0.
U ( Z , T ) = A i ( T Z 2 4 + i s a 1 Z ) exp [ a 1 ( T Z 2 2 ) i s 3 Z 3 12 + i s a 1 2 Z 2 + i s T Z 2 ] .
i U Z + s 2 2 U T 2 + σ Ε T r U = 0.
U ( Z , T ) = A i ( T Z 2 4 + i s a 2 Z ) exp [ a 2 ( T Z 2 2 ) i s 3 Z 3 12 + i s a 2 2 Z 2 + i s T Z 2 + i σ Ε Z T r ] .
s a 1 2 2 = s a 2 2 2 + σ Ε T r + 2 n π .
a 2 2 2 = a 1 2 2 s σ Ε T r > 0 ,
a 1 2 2 Ε T r .
( Ε T r ) c r = a 1 2 2 .

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