Abstract

Light diffracting to different diffraction orders of a periodic grating generally propagates along a set of straight trajectories. Here we show that certain quasi-periodic gratings can produce curved diffraction orders. These curved lobes are created by the caustic interference of the originally straight diffraction orders and manifest themselves as accelerating beams. Both numerical simulations and experimental results demonstrate the validity of multiple accelerating beam generation with a single binary grating. Our work makes a quantitative link between the quasi-periodicity of a grating and the resulting caustic diffraction orders. Furthermore, the use of binary devices has important applications in acoustics, x-ray optics, and electron beam engineering and is also useful when high optical power is needed.

© 2013 Optical Society of America

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N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[CrossRef]

Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
[CrossRef]

L. Li, T. Li, S. M. Wang, and S. N. Zhu, Phys. Rev. Lett. 110, 046807 (2013).
[CrossRef]

2012 (4)

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (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.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

2011 (4)

2009 (2)

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

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

2008 (1)

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

2007 (2)

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

W. D. Furlan, G. Saavedra, and J. A. Monsoriu, Opt. Lett. 32, 2109 (2007).
[CrossRef]

2003 (1)

1992 (1)

W. R. McKinney, Rev. Sci. Instrum. 63, 1410 (1992).
[CrossRef]

1987 (1)

1985 (1)

1979 (2)

M. V. Berry, J. Phys. A 12, 781 (1979).
[CrossRef]

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

1977 (1)

H. Dammann and E. Klotz, Opt. Acta 24, 505 (1977).
[CrossRef]

1974 (1)

Agrawal, A.

Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
[CrossRef]

Aleahmad, P.

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

Arie, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[CrossRef]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Baumgartl, J.

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

Bekenstein, R.

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

Berry, M. V.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

M. V. Berry, J. Phys. A 12, 781 (1979).
[CrossRef]

Bowyer, S.

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]

Capasso, F.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Chen, Z.

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

Christodoulides, D. N.

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

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

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

Cluzel, B.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Courvoisier, F.

Dammann, H.

H. Dammann and E. Klotz, Opt. Acta 24, 505 (1977).
[CrossRef]

de Fornel, F.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Dellinger, J.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Dholakia, K.

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

Dogariu, A.

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

Dudley, J. M.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Froehl, L.

Froehly, L.

Furfaro, L.

Furlan, W. D.

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Gao, N.

Genevet, P.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Giust, R.

Gover, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[CrossRef]

Greenfield, E.

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

Hettrick, M. C.

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]

Jacquot, M.

Kaminer, I.

P. Aleahmad, M.-A. 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]

Klotz, E.

H. Dammann and E. Klotz, Opt. Acta 24, 505 (1977).
[CrossRef]

Koleskik, M.

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

Lacourt, P. A.

Lee, W. H.

Lereah, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[CrossRef]

Li, L.

L. Li, T. Li, S. M. Wang, and S. N. Zhu, Phys. Rev. Lett. 110, 046807 (2013).
[CrossRef]

Li, T.

L. Li, T. Li, S. M. Wang, and S. N. Zhu, Phys. Rev. Lett. 110, 046807 (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]

Lilach, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[CrossRef]

Lin, J.

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

Malina, R. F.

Martin, C.

Mathis, A.

Mazilu, M.

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

McKinney, W. R.

W. R. McKinney, Rev. Sci. Instrum. 63, 1410 (1992).
[CrossRef]

Mills, M. S.

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

Miri, M.-A.

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

Moloney, J. V.

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

Monsoriu, J. A.

Morandotti, R.

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

Mrowka, S.

Nahata, A.

Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
[CrossRef]

Nemirovsky, J.

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

Ouelette, F.

Polynkin, P.

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

Raz, O.

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

Saavedra, G.

Segev, M.

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

P. Aleahmad, M.-A. 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]

Siviloglou, G. A.

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

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

Vardeny, Z. V.

Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
[CrossRef]

Voloch-Bloch, N.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[CrossRef]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Walasik, W.

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

Wang, S. M.

L. Li, T. Li, S. M. Wang, and S. N. Zhu, Phys. Rev. Lett. 110, 046807 (2013).
[CrossRef]

Xie, C.

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.

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, 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, Y.

Zhu, S. N.

L. Li, T. Li, S. M. Wang, and S. N. Zhu, Phys. Rev. Lett. 110, 046807 (2013).
[CrossRef]

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Appl. Opt. (3)

J. Phys. A (1)

M. V. Berry, J. Phys. A 12, 781 (1979).
[CrossRef]

Nat. Photonics (3)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Z. V. Vardeny, A. Nahata, and A. Agrawal, Nat. Photonics 7, 177 (2013).
[CrossRef]

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

Nature (1)

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, Nature 494, 331 (2013).
[CrossRef]

Opt. Acta (1)

H. Dammann and E. Klotz, Opt. Acta 24, 505 (1977).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. Lett. (7)

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett. 109, 093904 (2012).
[CrossRef]

L. Li, T. Li, S. M. Wang, and S. N. Zhu, Phys. Rev. Lett. 110, 046807 (2013).
[CrossRef]

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]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (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.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

Rev. Sci. Instrum. (1)

W. R. McKinney, Rev. Sci. Instrum. 63, 1410 (1992).
[CrossRef]

Science (1)

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

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Figures (3)

Fig. 1.
Fig. 1.

(a) Structure of a quasi-periodic grating. (b) Generation of curved caustics. (c) Relation between different diffraction orders. (d) Multiple accelerating beam generation with a single grating.

Fig. 2.
Fig. 2.

(a) The quasi-periodic grating with a duty cycle of 0.1. (b) Its normalized diffraction pattern in the x and z plane. (c) The experimental setup. (d) Microscopic image of part of the grating. (e) The recorded diffraction profile in the xz plane. The x and z coordinates are given in units of mm. The inset plots the cross section along the yellow solid line together with the fitted Airy(x)2 curve.

Fig. 3.
Fig. 3.

(a) 2D quasi-periodic grating with a 0.14 duty ratio. Its normalized diffraction pattern in the xy plane at (b) z=66mm, (c) z=100mm, and (d) z=200mm. The x and y coordinates are given in units of mm.

Equations (10)

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

αm(x)=sin1[mλΛ(x)],
tan[αm(x)]=fm(z)=fm(z)xz,
mλΛ(x)=fm[zm(x)]=fm[zm(x)]xzm(x).
zm(x)=z1(x)m,
fm(z)=f1(mz),
[(n+1)1/sD,n1/sD](n0,1<s<2),
Λ(x)=Dss(x)s1.
fm(z)=(s12s)s12sDss2[(2s)smλz]12s.
fm(z)=916D3(mλz)2,
K(x,y)=2πΛ2D(x,y)n(x,y)=2πΛ(x)ex+2πΛ(y)ey,

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