Abstract

We study accelerating and decelerating shape-preserving temporal Airy wave-packets propagating in dispersive media. We explore the effects of causality, and find that, whereas decelerating pulses can asymptotically reach zero group velocity, pulses that accelerate towards infinite group velocity inevitably break up, after a specific critical point. The trajectories and the features of causal pulses are analyzed, along with the requirements for the existence of the critical point and experimental schemes for its observation. Finally, we show that causality imposes similar effects on accelerating pulses in the presence of local Kerr-like nonlinearities.

© 2011 OSA

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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] [PubMed]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [CrossRef] [PubMed]
  4. P. Saari, “Laterally accelerating airy pulses,” Opt. Express 16(14), 10303–10308 (2008).
    [CrossRef] [PubMed]
  5. 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]
  6. K.-Y. Kim, C.-Y. Hwang, and B. Lee, “Slow non-dispersing wavepackets,” Opt. Express 19(3), 2286–2293 (2011).
    [CrossRef] [PubMed]
  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. 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]
  9. R. Y. Chiao and P. W. Milonni, “Fast Light, Slow Light,” Opt. Photonics News 13(6), 26–30 (2002).
    [CrossRef]
  10. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106(21), 213903 (2011).
    [CrossRef] [PubMed]
  11. Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express 19(18), 17298–17307 (2011).
    [CrossRef] [PubMed]
  12. G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics (Academic Press, London, 2001).
  13. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
    [CrossRef] [PubMed]

2011 (4)

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

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[CrossRef] [PubMed]

K.-Y. Kim, C.-Y. Hwang, and B. Lee, “Slow non-dispersing wavepackets,” Opt. Express 19(3), 2286–2293 (2011).
[CrossRef] [PubMed]

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

2010 (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]

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]

2008 (2)

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]

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

2007 (2)

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

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

2002 (1)

R. Y. Chiao and P. W. Milonni, “Fast Light, Slow Light,” Opt. Photonics News 13(6), 26–30 (2002).
[CrossRef]

1979 (1)

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

Abdollahpour, D.

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]

Balazs, N. L.

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

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]

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] [PubMed]

Chiao, R. Y.

R. Y. Chiao and P. W. Milonni, “Fast Light, Slow Light,” Opt. Photonics News 13(6), 26–30 (2002).
[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]

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]

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]

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]

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

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] [PubMed]

Fattal, Y.

Greenfield, E.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[CrossRef] [PubMed]

Hwang, C.-Y.

Kaminer, I.

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

Kim, K.-Y.

Lee, B.

Marom, D. M.

Milonni, P. W.

R. Y. Chiao and P. W. Milonni, “Fast Light, Slow Light,” Opt. Photonics News 13(6), 26–30 (2002).
[CrossRef]

Papazoglou, D. G.

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]

Raz, O.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[CrossRef] [PubMed]

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]

Rudnick, A.

Saari, P.

Segev, M.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[CrossRef] [PubMed]

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

Shaarawi, A. 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]

Siviloglou, G. A.

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

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

Suntsov, S.

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]

Tzortzakis, S.

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]

Walasik, W.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[CrossRef] [PubMed]

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]

Am. J. Phys. (1)

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

Nat. Photonics (1)

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]

Opt. Express (3)

Opt. Lett. (1)

Opt. Photonics News (1)

R. Y. Chiao and P. W. Milonni, “Fast Light, Slow Light,” Opt. Photonics News 13(6), 26–30 (2002).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

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]

Phys. Rev. Lett. (4)

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

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]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
[CrossRef] [PubMed]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics (Academic Press, London, 2001).

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Figures (2)

Fig. 1
Fig. 1

Dynamics of an accelerating Airy wavepacket. (a) Without imposing causal initial conditions, the wavepacket comprises of two counter-propagating pulses, with positive and negative group velocities. (b) Physical propagation: frequencies with negative group velocities are eliminated. (c) Unphysical backward propagation: frequencies with positive group velocities are eliminated. (d) Propagation with third-order dispersion included, and without imposing causal initial conditions. The white dashed line marks the position of the critical point.

Fig. 2
Fig. 2

Schematic trajectories of (a) a decelerating and (b) an accelerating Airy wavepacket. The dashed tangent lines mark the point where acceleration stops due to the finite extent of the pulse. (c) The spectrum of an Airy pulse truncated at the critical frequency, and compared to the carrier frequency.

Equations (6)

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i ψ z + i ψ t / v 0 1 2 k'' ψ tt +γψ=0
ψ=A( t t 0 T z T v 0 k' ' 2 z 2 4 T 4 ) e iθ( z,t )
θ=z( t t 0 )( k'' 2 T 3 ) z 2 ( k'' 2 T 3 v 0 ) z 3 ( k' ' 3 12 T 6 )
v= dz dt = 1 1 v 0 + k' ' 2 z 2 T 3 = v 0 1+ k' ' 2 z 2 T 3 v 0
ω 0 v 0 k''>>1
T ω 0 >> 1 a > T k'' v 0

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