Abstract

We study linear and nonlinear self-accelerating beams propagating along circular trajectories beyond the paraxial approximation. Such nonparaxial accelerating beams are exact solutions of the Helmholtz equation, preserving their shapes during propagation even under nonlinearity. We generate experimentally and observe directly these large-angle bending beams in colloidal suspensions of polystyrene nanoparticles.

© 2012 Optical Society of America

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  1. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [CrossRef]
  2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  3. J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef]
  4. Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012).
  5. Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, Opt. Lett. 35, 3952 (2010).
    [CrossRef]
  6. I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213903 (2011).
    [CrossRef]
  7. A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 84, 021807 (2011).
    [CrossRef]
  8. A. V. Novitsky, and D. V. Novitsky, Opt. Lett. 34, 3430 (2009).
    [CrossRef]
  9. A. Torre, Opt. Commun. 283, 4146 (2010).
    [CrossRef]
  10. L. Froehl, L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, Opt. Express 19, 16455 (2011).
    [CrossRef]
  11. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
    [CrossRef]
  12. I. Kaminer and M. Segev, “Sub-wavelength nonlinear accelerating beams,” submitted to OSA Topical Meeting, Nonlinear Photonics (2012).
  13. Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, Opt. Lett. 35, 2260 (2010).
    [CrossRef]
  14. P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 1491 (2011).
    [CrossRef]
  15. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 2883 (2011).
    [CrossRef]
  16. A. Ashkin, J. M. Dziedzic, and P. W. Smith, Opt. Lett. 7, 276 (1982).
    [CrossRef]
  17. P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, Opt. Lett. 36, 3191 (2011).
    [CrossRef]
  18. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
    [CrossRef]
  19. L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
    [CrossRef]

2012 (1)

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

2011 (8)

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[CrossRef]

I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213903 (2011).
[CrossRef]

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 84, 021807 (2011).
[CrossRef]

P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 1491 (2011).
[CrossRef]

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

P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, Opt. Lett. 36, 3191 (2011).
[CrossRef]

L. Froehl, L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, Opt. Express 19, 16455 (2011).
[CrossRef]

2010 (3)

2009 (1)

2007 (2)

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

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

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

1982 (1)

Abdollahpour, D.

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 84, 021807 (2011).
[CrossRef]

Ashkin, A.

Bekenstein, R.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Broky, J.

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

Chen, Z.

Christodoulides, D. N.

P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 1491 (2011).
[CrossRef]

I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213903 (2011).
[CrossRef]

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

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]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012).

Couairon, A.

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 84, 021807 (2011).
[CrossRef]

Courvoisier, F.

Dogariu, A.

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

Dudley, J. M.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Dziedzic, J. M.

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Efremidis, N. K.

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

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012).

Faccio, D.

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 84, 021807 (2011).
[CrossRef]

Froehl, L.

Froehly, L.

Furfaro, L.

Giust, R.

Hernandez, D.

Hu, Y.

Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, Opt. Lett. 35, 3952 (2010).
[CrossRef]

Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, Opt. Lett. 35, 2260 (2010).
[CrossRef]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012).

Huang, S.

Jacquot, M.

Janunts, N.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

Kaminer, I.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213903 (2011).
[CrossRef]

I. Kaminer and M. Segev, “Sub-wavelength nonlinear accelerating beams,” submitted to OSA Topical Meeting, Nonlinear Photonics (2012).

Kivshar, Y. S.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

Klein, A. E.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

Lacourt, P. A.

Li, L.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[CrossRef]

Li, T.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[CrossRef]

Liu, Y.

Lotti, A.

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 84, 021807 (2011).
[CrossRef]

Lou, C.

Lu, C.

Mathis, A.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Mills, M. S.

Minovich, A.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

Nemirovsky, J.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Neshev, D. N.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

Novitsky, A. V.

Novitsky, D. V.

Panagiotopoulos, P.

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 84, 021807 (2011).
[CrossRef]

Papazoglou, D. G.

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 84, 021807 (2011).
[CrossRef]

Pertsch, T.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

Prakash, J.

Salazar, M.

Segev, M.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213903 (2011).
[CrossRef]

I. Kaminer and M. Segev, “Sub-wavelength nonlinear accelerating beams,” submitted to OSA Topical Meeting, Nonlinear Photonics (2012).

Siviloglou, G. A.

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

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

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012).

Smith, P. W.

Torre, A.

A. Torre, Opt. Commun. 283, 4146 (2010).
[CrossRef]

Tzortzakis, S.

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 84, 021807 (2011).
[CrossRef]

Wang, S.

Wang, S. M.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[CrossRef]

Xu, J.

Yin, X.

Zhang, C.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[CrossRef]

Zhang, P.

Zhang, X.

Zhang, Z.

Zhu, S. N.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[CrossRef]

Opt. Commun. (1)

A. Torre, Opt. Commun. 283, 4146 (2010).
[CrossRef]

Opt. Express (1)

Opt. Lett. (8)

Phys. Rev. A (1)

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 84, 021807 (2011).
[CrossRef]

Phys. Rev. Lett. (6)

I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213903 (2011).
[CrossRef]

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

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[CrossRef]

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett. 107, 126804 (2011).
[CrossRef]

Other (2)

I. Kaminer and M. Segev, “Sub-wavelength nonlinear accelerating beams,” submitted to OSA Topical Meeting, Nonlinear Photonics (2012).

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012).

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Figures (5)

Fig. 1.
Fig. 1.

(a) Amplitude of a linear NAB (solid, blue) and an Airy beam (dashed, red) with the same main lobe size; (b) intensity profiles of the NAB and Airy beam in (a) after linear propagation of z=R/2; (c) phase profile of Fourier spectra of the NAB and the Airy beam in (a); (d), (e) Sideview propagation of the Airy beam (d) and the NAB (e) in (a).

Fig. 2.
Fig. 2.

Numerical solution (a) and its propagation (b) for a nonlinear NAB; (c)–(e) depicts nonlinear evolution of the NAB with initial condition shown in Fig. 1(a), where (c) and (d) are intensity profiles after linear (dashed) and nonlinear (solid) propagation to z=0 and R/2, as marked by the dashed lines in the side view (e).

Fig. 3.
Fig. 3.

Experimental setup for the generation of NABs. SLM, spatial light modulator; H, computer generated hologram; L, lens; CL, cylindrical lens; O, microscope objective.

Fig. 4.
Fig. 4.

Experimental side-view photography of the linear propagation of (a) an Airy beam and (b) a NAB taken from scattered light; (c), (d) transverse patterns taken at z=0 and z=0.1mm positions as marked by dashed lines in (a), (b) where (c) and (d) correspond to the Airy beam and the NAB, and top and bottom panels correspond to z=0 (z=0.1mm) propagation distance; (e), (f) Intensity profiles taken along the dashed line marked in (c), (d) from the Airy beam (dashed) and the NAB (solid). (e) is taken from z=0 and (f) from z=0.1mm.

Fig. 5.
Fig. 5.

Experimental side-view photography of (a) linear and (b) nonlinear NAB. (c) Transverse output patterns taken at about z=1mm as marked by dashed line in (a), (b), where top and bottom panels correspond to patterns from (a) and (b), respectively. (d) Linear (dashed) and nonlinear (solid) output intensity profiles taken along dashed line in (c).

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

2Ux2+2Uz2+k02(n0+Δn)2U=0,
d2udr2+1rdudrβ2r2u+k02n02u+2k02n0γu3=0.
U(x,z)=Jβ(kx2+z2)exp(iβarctan(z/x)).
U(x,0)=exp(αx)H(x+β/k)Jβ(kx+β).
ψ(fx)=(k2fx2)1/2exp(iβ(fx/karcsin(fx/k))),

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