Abstract

We study the propagation of Airy beams in transversely linear index potentials with a gradient that is dynamically changing along the propagation direction. We find exact solutions in the case of Airy and apodized (Gaussian and exponentially) Airy beams in 1+1 and 2+1 dimensions. More important, we find that the Airy beam can follow any predefined path, in which case the potential gradient is determined as a function of this path.

© 2011 Optical Society of America

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  1. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [CrossRef] [PubMed]
  2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  3. J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
    [CrossRef]
  4. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
    [CrossRef] [PubMed]
  5. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
    [CrossRef]
  6. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
    [CrossRef]
  7. N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
    [CrossRef] [PubMed]
  8. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
    [CrossRef] [PubMed]
  9. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett 36, 2883(2011).
    [CrossRef] [PubMed]
  10. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, Opt. Lett. 36, 1890 (2011).
    [CrossRef] [PubMed]
  11. E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
    [CrossRef] [PubMed]
  12. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, Opt. Lett. 36, 1164 (2011).
    [CrossRef] [PubMed]
  13. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Express 15, 16719 (2007).
    [CrossRef] [PubMed]

2011 (5)

2010 (3)

N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
[CrossRef] [PubMed]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

2009 (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

2008 (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[CrossRef]

2007 (3)

Abdollahpour, D.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[CrossRef]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Chen, Z.

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett 36, 2883(2011).
[CrossRef] [PubMed]

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Chremmos, I.

Christodoulides, D. N.

I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, Opt. Lett. 36, 1890 (2011).
[CrossRef] [PubMed]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett 36, 2883(2011).
[CrossRef] [PubMed]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[CrossRef] [PubMed]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
[CrossRef] [PubMed]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef] [PubMed]

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[CrossRef]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Efremidis, N. K.

Greenfield, E.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef] [PubMed]

Gutiérrez-Vega, J. C.

Kivshar, Y. S.

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

Liu, W.

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[CrossRef]

Mills, M. S.

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett 36, 2883(2011).
[CrossRef] [PubMed]

Miroshnichenko, A. E.

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

Neshev, D. N.

Papazoglou, D. G.

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[CrossRef] [PubMed]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

Prakash, J.

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett 36, 2883(2011).
[CrossRef] [PubMed]

Raz, O.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef] [PubMed]

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Segev, M.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef] [PubMed]

Shadrivov, I. V.

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef] [PubMed]

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Tzortzakis, S.

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[CrossRef] [PubMed]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Walasik, W.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef] [PubMed]

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Zhang, P.

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett 36, 2883(2011).
[CrossRef] [PubMed]

Zhang, Z.

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett 36, 2883(2011).
[CrossRef] [PubMed]

Nat. Photon. (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[CrossRef]

Nat. Photonics (1)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Opt. Express (1)

Opt. Lett (1)

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett 36, 2883(2011).
[CrossRef] [PubMed]

Opt. Lett. (5)

Phys. Rev. Lett. (3)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef] [PubMed]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[CrossRef]

Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1
Fig. 1

Exponentially apodized Airy beam propagation ( a = 0.05 ) along predefined paths. (a) Power law trajectory x = ( z 1 ) 3 , (b) sinusoidal trajectory given by Eq. (22), (c) logarithmic trajectory x = log ( 1 + 10 z ) , (d) hyperbolic trajectory given by Eq. (23).

Fig. 2
Fig. 2

Schematic illustration of a 2D Airy beam following a spiral trajectory in space.

Equations (28)

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i u z + 1 2 u x x d ( z ) x 2 u = 0 ,
u ˜ = u e i k x d x , u = 1 2 π - u ˜ e i k x d k ,
u ˜ z d ( z ) 2 u ˜ k = i k 2 2 u ˜ .
d z 1 = d k d ( z ) / 2 = d u ˜ i k 2 u ˜ / 2 ,
κ ( z ) = k D ( z ) 2 ,
D ( z ) = 0 z d ( s ) d s .
u ( x , z ) = 1 2 π R u ˜ ( k , 0 ) e i 2 0 z κ ( s ) 2 d s e i κ ( z ) x d k .
u ( x , z = 0 ) = Ai ( γ 1 / 3 x ) ,
u ˜ ( k , z = 0 ) = 1 γ 1 / 3 exp [ i k 3 3 γ ] ,
u ( x , z ) = e ϕ ( x , z ) Ai ( γ 1 / 3 μ ( x , z ) ) ,
μ ( x , z ) = F 1 ( z ) 2 + x γ z 2 4 ,
ϕ = i γ F 1 ( z ) z 4 i D ( z ) x 2 i F 2 ( z ) 8 + i γ z x 2 i γ 2 z 3 12 ,
F 1 ( z ) = 0 z D ( s ) d s , F 2 ( z ) = 0 z D 2 ( s ) d s .
x = F 1 ( z ) 2 + γ z 2 4 ,
x = x 0 + ψ z F 1 ( z ) 2 + γ z 2 4 .
x 0 = g ( 0 ) , ψ = g ( 0 ) , d ( z ) = γ 2 g ( z ) .
u ( x , z ) = Ai [ γ 1 / 3 ( i a z + F 1 2 + x γ z 2 4 ) ] e ϕ ( x , z ) ,
ϕ ( x , z ) = i a 2 z 2 + a F 1 2 + a x a γ z 2 2 i x D 2 + i z γ F 1 4 i F 2 8 + i γ x z 2 i γ 2 z 3 12 ,
u ( x , z ) = e ϕ σ 1 / 2 Ai [ γ 1 / 3 ( 1 σ ( x + F 1 2 ) c 0 - γ z 2 4 σ 2 ) ] ,
ϕ ( x , z ) = σ w 2 ( x + F 1 2 ) 2 i F 2 8 i x D 2 + i z 2 σ [ γ σ ( x + F 1 2 ) γ c 0 γ 2 z 2 6 σ 2 ] .
x = g ( z ) = c ( z z 0 ) b .
x = g ( z ) = 2 ( cos ( π z ) 1 ) .
x = g ( z ) = 2 ( ( z 2 ) 2 + 1 1 ) H ( z 2 ) ,
d ( z ) = 1 4 H ( z 2 ) ( ( z 4 ) z + 5 ) 3 / 2 .
i u z + 1 2 u x x + u y y 1 2 [ d 1 ( z ) x + d 2 ( z ) y ] u = 0 ,
u ( x , y , z ) = v ( x , z ) w ( y , z ) ,
2 i v z + v x x d 1 ( z ) x v = 0 , 2 i w z + w y y d 2 ( z ) y w = 0.
u ( x , y , 0 ) = Ai [ γ 1 / 3 ( x x 0 ) ] Ai [ γ 1 / 3 ( y y 0 ) ] e i ψ 1 x + i ψ 2 y ,

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