Abstract

We propose and demonstrate the generation of periodic self-accelerating beams through both phase and amplitude modulation in the Fourier space. For small amplitude variations, an accelerating beam still follows a smooth convex trajectory, which can be traced by acting on the spectral phase only. However, large modulations such as those generated from the Heaviside function with a zero amplitude distribution partially modify the convex trajectory due to the appearance of straight-line paths. Furthermore, periodic self-accelerating beams along convex trajectories are realized by employing an array of “spectral wells” in both the paraxial and nonparaxial regimes.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. Y. Hu, G. Siviloglou, P. Zhang, N. Efremidis, D. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012), Vol. 170, pp. 1–46.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2013 (3)

2012 (6)

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

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

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, Opt. Lett. 37, 1736 (2012).
[CrossRef]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Aleahmad, M. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

2011 (3)

2010 (2)

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

J. Morris, T. Cizmar, H. Dalgarno, R. Marchington, F. Gunn-Moore, and K. Dholakia, J. Opt. 12, 124002 (2010).
[CrossRef]

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]

1995 (1)

Aleahmad, P.

P. Aleahmad, M. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

Barwick, S.

Bekenstein, R.

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

Bongiovanni, D.

Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, “Multi-path multi-component self-accelerating beams through spectrum-engineered position mapping,” arXiv:1307.0891 (2013).

Broky, J.

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

Cannan, D.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

Chen, Z.

J. Zhao, P. Zhang, D. Deng, J. Liu, Y. Gao, I. D. Chremmos, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 38, 498 (2013).
[CrossRef]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

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

Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, “Multi-path multi-component self-accelerating beams through spectrum-engineered position mapping,” arXiv:1307.0891 (2013).

Y. Hu, G. Siviloglou, P. Zhang, N. Efremidis, D. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012), Vol. 170, pp. 1–46.

Chremmos, I. D.

Christodoulides, D.

Y. Hu, G. Siviloglou, P. Zhang, N. Efremidis, D. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012), Vol. 170, pp. 1–46.

Christodoulides, D. N.

J. Zhao, P. Zhang, D. Deng, J. Liu, Y. Gao, I. D. Chremmos, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 38, 498 (2013).
[CrossRef]

P. Aleahmad, M. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

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]

Cizmar, T.

J. Morris, T. Cizmar, H. Dalgarno, R. Marchington, F. Gunn-Moore, and K. Dholakia, J. Opt. 12, 124002 (2010).
[CrossRef]

Courvoisier, F.

Dalgarno, H.

J. Morris, T. Cizmar, H. Dalgarno, R. Marchington, F. Gunn-Moore, and K. Dholakia, J. Opt. 12, 124002 (2010).
[CrossRef]

Deng, D.

Dennis, M. R.

Dholakia, K.

J. Morris, T. Cizmar, H. Dalgarno, R. Marchington, F. Gunn-Moore, and K. Dholakia, J. Opt. 12, 124002 (2010).
[CrossRef]

Dogariu, A.

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

Dudley, J. M.

Efremidis, N.

Y. Hu, G. Siviloglou, P. Zhang, N. Efremidis, D. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012), Vol. 170, pp. 1–46.

Efremidis, N. K.

Froehly, L.

Furfaro, L.

Gao, Y.

Giust, R.

Greenfield, E.

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

E. Greenfield, I. Kaminer, and M. Segev, in Frontiers in Optics Conference, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW1A.2.

Gunn-Moore, F.

J. Morris, T. Cizmar, H. Dalgarno, R. Marchington, F. Gunn-Moore, and K. Dholakia, J. Opt. 12, 124002 (2010).
[CrossRef]

Howls, C. J.

Hu, Y.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

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

Y. Hu, G. Siviloglou, P. Zhang, N. Efremidis, D. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012), Vol. 170, pp. 1–46.

Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, “Multi-path multi-component self-accelerating beams through spectrum-engineered position mapping,” arXiv:1307.0891 (2013).

Huang, S.

Jacquot, M.

Kaminer, I.

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

P. Aleahmad, M. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

E. Greenfield, I. Kaminer, and M. Segev, in Frontiers in Optics Conference, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW1A.2.

Lacourt, P. A.

Li, T.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Liu, J.

Lou, C.

Marchington, R.

J. Morris, T. Cizmar, H. Dalgarno, R. Marchington, F. Gunn-Moore, and K. Dholakia, J. Opt. 12, 124002 (2010).
[CrossRef]

Mathis, A.

Mills, M. S.

P. Aleahmad, M. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

Miri, M.

P. Aleahmad, M. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

Morandotti, R.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, “Multi-path multi-component self-accelerating beams through spectrum-engineered position mapping,” arXiv:1307.0891 (2013).

Morris, J.

J. Morris, T. Cizmar, H. Dalgarno, R. Marchington, F. Gunn-Moore, and K. Dholakia, J. Opt. 12, 124002 (2010).
[CrossRef]

Nemirovsky, J.

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

Raz, O.

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

Ring, J. D.

Rosen, J.

Salandrino, A.

Segev, M.

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

P. Aleahmad, M. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

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

E. Greenfield, I. Kaminer, and M. Segev, in Frontiers in Optics Conference, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW1A.2.

Siviloglou, G.

Y. Hu, G. Siviloglou, P. Zhang, N. Efremidis, D. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012), Vol. 170, pp. 1–46.

Siviloglou, G. A.

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]

Walasik, W.

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

Xu, J.

Yariv, A.

Yin, X.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Zhang, P.

J. Zhao, P. Zhang, D. Deng, J. Liu, Y. Gao, I. D. Chremmos, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 38, 498 (2013).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

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

Y. Hu, G. Siviloglou, P. Zhang, N. Efremidis, D. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012), Vol. 170, pp. 1–46.

Zhang, X.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, Opt. Lett. 37, 2820 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Zhao, J.

J. Opt. (1)

J. Morris, T. Cizmar, H. Dalgarno, R. Marchington, F. Gunn-Moore, and K. Dholakia, J. Opt. 12, 124002 (2010).
[CrossRef]

Opt. Express (1)

Opt. Lett. (9)

Phys. Rev. A (1)

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

Phys. Rev. Lett. (5)

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

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Aleahmad, M. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

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

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

Other (3)

E. Greenfield, I. Kaminer, and M. Segev, in Frontiers in Optics Conference, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW1A.2.

Y. Hu, G. Siviloglou, P. Zhang, N. Efremidis, D. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, 2012), Vol. 170, pp. 1–46.

Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, “Multi-path multi-component self-accelerating beams through spectrum-engineered position mapping,” arXiv:1307.0891 (2013).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Schematics of the beam path composed of straight and convex lines and (b) |μ| related to a Heaviside-spectral distribution.

Fig. 2.
Fig. 2.

Numerical simulations (first row) and experimental observations (second row) of Airy beams generated (a), (d) without and (b), (e) with a Heaviside-shaped amplitude modulation. (c), (f) Spectral components related to the main lobes in (b) and (e), respectively. The dashed white lines/curves in (a)–(c) are calculated from theory.

Fig. 3.
Fig. 3.

Generation of periodic accelerating beams along a parabolic curve by employing (a), (b) one spectral well and (c), (d) an array of spectral wells. (a), (c) k-space amplitude modulation relative to the results in (b) and (d), where the upper and bottom panels correspond to numerical and experimental results, respectively.

Fig. 4.
Fig. 4.

Nonparaxial periodic self-accelerating beams. (a) Imposed phases and (b)–(d) amplitude modulations employed for generating periodic accelerating beams along (e)–(g) circular, elliptic, and parabolic trajectories, respectively.

Equations (5)

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

E˜(kx,z)=A(kx)exp[ikx2z/(2k)+iρ(kx)],
x=kxz/kρ(kx),
μ(kx)=z/k+ρ(kx)=0,
A(kx)={1kxkx00kx>kx0.
x=kx0z/kρ(kx0).

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