Abstract

Based on a geometric caustic argument and diffraction catastrophe theory, we generate a novel form of accelerating beams using a symmetric 3/2 phase-only pattern. Such beams can be called accelerating quad Airy beams (AQABs) because they look very much like four face-to-face combined Airy beams. Optical characteristics of AQABs are subsequently investigated. The research results show that the beams have axial-symmetrical and centrosymmetrical transverse intensity patterns and quasi-diffraction-free propagation features for their four main lobes while undergoing transverse shift along parabolic trajectories. Moreover, we also demonstrate that AQABs possess self-construction ability when local areas are blocked. The unique optical properties of these beams will make them useful tools for future scientific applications.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
  2. D. N. Christodoulides, “Optical trapping riding along an Airy beam,” Nat. Photonics 2(11), 652–653 (2008).
    [CrossRef]
  3. P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental tests of the new paradigm for laser filamentation in gases,” Phys. Rev. Lett. 106(15), 153902 (2011).
    [CrossRef] [PubMed]
  4. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]

2013 (2)

J. A. Davis, D. M. Cottrell, and J. M. Zinn, “Direct generation of abruptly focusing vortex beams using a 3/2 radial phase-only pattern,” Appl. Opt. 52(9), 1888–1891 (2013).
[CrossRef] [PubMed]

Z. J. Ren, X. D. Li, C. J. Fan, and Z. Q. Xu, “Generation of optical Accelerating quinary-cusp beams and their optical characteristics,” Chin. Phys. Lett. 30(11), 114208 (2013).
[CrossRef]

2012 (7)

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012).
[CrossRef] [PubMed]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108(16), 163901 (2012).
[CrossRef] [PubMed]

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109(20), 203902 (2012).
[CrossRef] [PubMed]

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[CrossRef] [PubMed]

M. A. Alonso and M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett. 37(24), 5175–5177 (2012).
[CrossRef] [PubMed]

Z. J. Ren, L. W. Dong, C. F. Ying, and C. J. Fan, “Generation of optical accelerating regular triple-cusp beams and their topological structures,” Opt. Express 20(28), 29276–29283 (2012).
[CrossRef] [PubMed]

Y. Kaganovsky and E. Heyman, “Nonparaxial wave analysis of three-dimensional Airy beams,” J. Opt. Soc. Am. A 29(5), 671–688 (2012).
[CrossRef] [PubMed]

2011 (4)

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]

L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express 19(17), 16455–16465 (2011).
[CrossRef] [PubMed]

P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental tests of the new paradigm for laser filamentation in gases,” Phys. Rev. Lett. 106(15), 153902 (2011).
[CrossRef] [PubMed]

J. X. Li, X. L. Fan, W. P. Zang, and J. G. Tian, “Vacuum electron acceleration driven by two crossed Airy beams,” Opt. Lett. 36(5), 648–650 (2011).
[CrossRef] [PubMed]

2010 (4)

2009 (3)

D. M. Cottrell, J. A. Davis, and T. M. Hazard, “Direct generation of accelerating Airy beams using a 3/2 phase-only pattern,” Opt. Lett. 34(17), 2634–2636 (2009).
[CrossRef] [PubMed]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[CrossRef] [PubMed]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

2008 (2)

2007 (2)

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]

M. V. Berry, “Looking at coalescing images and poorly resolved caustics,” J. Opt. A, Pure Appl. Opt. 9(7), 649–657 (2007).
[CrossRef]

1998 (1)

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Aleahmad, P.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109(20), 203902 (2012).
[CrossRef] [PubMed]

Alonso, M. A.

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Bandres, M. A.

Barwick, S.

Baumgartl, J.

Bekenstein, R.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108(16), 163901 (2012).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, “Looking at coalescing images and poorly resolved caustics,” J. Opt. A, Pure Appl. Opt. 9(7), 649–657 (2007).
[CrossRef]

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[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]

Cannan, D.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[CrossRef] [PubMed]

Capron, B. A.

Carruthers, A. E.

Chan, V. C.

Chen, Z. G.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[CrossRef] [PubMed]

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

Choi, D.

Christodoulides, D. N.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109(20), 203902 (2012).
[CrossRef] [PubMed]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[CrossRef] [PubMed]

D. N. Christodoulides, “Optical trapping riding along an Airy beam,” Nat. Photonics 2(11), 652–653 (2008).
[CrossRef]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[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]

Cizmár, T.

Cottrell, D. M.

Courvoisier, F.

Davis, J. A.

Dennis, M. R.

Dholakia, K.

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[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]

Dong, L. W.

Dudley, J. M.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Fan, C. J.

Z. J. Ren, X. D. Li, C. J. Fan, and Z. Q. Xu, “Generation of optical Accelerating quinary-cusp beams and their optical characteristics,” Chin. Phys. Lett. 30(11), 114208 (2013).
[CrossRef]

Z. J. Ren, L. W. Dong, C. F. Ying, and C. J. Fan, “Generation of optical accelerating regular triple-cusp beams and their topological structures,” Opt. Express 20(28), 29276–29283 (2012).
[CrossRef] [PubMed]

Fan, X. L.

Froehly, L.

Furfaro, L.

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Giust, R.

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]

Hazard, T. M.

Heyman, E.

Hu, Y.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[CrossRef] [PubMed]

Hwang, C.-Y.

Jacquot, M.

Kaganovsky, Y.

Kaminer, I.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109(20), 203902 (2012).
[CrossRef] [PubMed]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108(16), 163901 (2012).
[CrossRef] [PubMed]

Kim, K.-Y.

Kolesik, M.

P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental tests of the new paradigm for laser filamentation in gases,” Phys. Rev. Lett. 106(15), 153902 (2011).
[CrossRef] [PubMed]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[CrossRef] [PubMed]

Lacourt, P. A.

Lee, B.

Li, J. X.

Li, T. C.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[CrossRef] [PubMed]

Li, X. D.

Z. J. Ren, X. D. Li, C. J. Fan, and Z. Q. Xu, “Generation of optical Accelerating quinary-cusp beams and their optical characteristics,” Chin. Phys. Lett. 30(11), 114208 (2013).
[CrossRef]

Lindberg, J.

Mathis, A.

Mazilu, M.

McNeely, W.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Mills, M. S.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109(20), 203902 (2012).
[CrossRef] [PubMed]

Miri, M. A.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109(20), 203902 (2012).
[CrossRef] [PubMed]

Moloney, J. V.

P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental tests of the new paradigm for laser filamentation in gases,” Phys. Rev. Lett. 106(15), 153902 (2011).
[CrossRef] [PubMed]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[CrossRef] [PubMed]

Morandotti, R.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[CrossRef] [PubMed]

Mourka, A.

Nemirovsky, J.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108(16), 163901 (2012).
[CrossRef] [PubMed]

Polynkin, P.

P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental tests of the new paradigm for laser filamentation in gases,” Phys. Rev. Lett. 106(15), 153902 (2011).
[CrossRef] [PubMed]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[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]

Ren, Z. J.

Z. J. Ren, X. D. Li, C. J. Fan, and Z. Q. Xu, “Generation of optical Accelerating quinary-cusp beams and their optical characteristics,” Chin. Phys. Lett. 30(11), 114208 (2013).
[CrossRef]

Z. J. Ren, L. W. Dong, C. F. Ying, and C. J. Fan, “Generation of optical accelerating regular triple-cusp beams and their topological structures,” Opt. Express 20(28), 29276–29283 (2012).
[CrossRef] [PubMed]

Ring, J. D.

Segev, M.

P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109(20), 203902 (2012).
[CrossRef] [PubMed]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108(16), 163901 (2012).
[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]

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
[CrossRef] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[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]

Tian, J. G.

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[CrossRef]

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]

Wright, E. M.

Xu, Z. Q.

Z. J. Ren, X. D. Li, C. J. Fan, and Z. Q. Xu, “Generation of optical Accelerating quinary-cusp beams and their optical characteristics,” Chin. Phys. Lett. 30(11), 114208 (2013).
[CrossRef]

Yin, X. B.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[CrossRef] [PubMed]

Ying, C. F.

Zang, W. P.

Zhang, P.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[CrossRef] [PubMed]

Zhang, X.

P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[CrossRef] [PubMed]

Zinn, J. M.

Appl. Opt. (1)

Chin. Phys. Lett. (1)

Z. J. Ren, X. D. Li, C. J. Fan, and Z. Q. Xu, “Generation of optical Accelerating quinary-cusp beams and their optical characteristics,” Chin. Phys. Lett. 30(11), 114208 (2013).
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J. Opt. A, Pure Appl. Opt. (1)

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[CrossRef]

J. Opt. Soc. Am. A (1)

Nat. Photonics (2)

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[CrossRef]

D. N. Christodoulides, “Optical trapping riding along an Airy beam,” Nat. Photonics 2(11), 652–653 (2008).
[CrossRef]

Opt. Commun. (1)

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[CrossRef]

Opt. Express (7)

Opt. Lett. (4)

Phys. Rev. Lett. (7)

P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental tests of the new paradigm for laser filamentation in gases,” Phys. Rev. Lett. 106(15), 153902 (2011).
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J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108(16), 163901 (2012).
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P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109(20), 203902 (2012).
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P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
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E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011).
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Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009).
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Figures (6)

Fig. 1
Fig. 1

2D geometrical caustic model for accelerating quad Airy beams. The dashed lines represent the caustic lines (rainbow lines), and the solid lines represent their generating rays. The z–axis is optical axis, and the phase–modulated plane of SLM is overlapping with the x–axis.

Fig. 2
Fig. 2

Schematic of principle for directly generating accelerating beams using phase-only patterns encoded onto SLM. BS, beam splitter; SLM, Spatial light modulation; PMM, Phase modulated mask; CCD, Charge-coupled device.

Fig. 3
Fig. 3

Experimental results for the 2D optical intensity distribution of an AQAB at distances of z = {30, 40, 50, 60, 70, 80} cm (a)–(f), and corresponding numerical simulation results at these same distances (g)–(l).

Fig. 4
Fig. 4

Three-dimensional skeleton frame of optical AQABs. Black lines correspond to propagating trajectory of four main lobes of an AQAB. Blue lines are 2D cross-section outline of AQABs in given position.

Fig. 5
Fig. 5

Self-construction of an AQAB when the main lobe of the top left corner is blocked. Observed intensity profile at Δz = {0, 14, 28, 42} mm from the obstacle.

Fig. 6
Fig. 6

Self-construction of an AQAB when one of its four Airy beams (1/4 of the total area of an AQAB) is blocked. Observed intensity profile at Δz = {0, 6, 12, 18} cm from the obstacle.

Equations (12)

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r= { [x x 0 ] 2 + [y y 0 ] 2 + [zφ( x 0 , y 0 )] 2 } 1/2 .
r( x 0 , y 0 ,x,y,z)=zφ( x 0 , y 0 )+ x 2 + y 2 2z x 0 x+ y 0 y z + x 0 2 + y 0 2 2z .
r / x 0 =0, r / y 0 =0.
φ x 0 = x 0 x z , φ y 0 = y 0 y z .
U(x,y)= 1 jλz exp(jkz)exp[j k 2z ( x 2 + y 2 )] × U( x 0 , y 0 ) exp[j k 2z ( x 0 2 + y 0 2 )]exp[j 2π λz ( x 0 x+ y 0 y)]d x 0 d y 0 ,
U( x 0 , y 0 )=exp[jkφ( x 0 , y 0 )].
φ( x 0 , y 0 )=(4w/3)( x 0 3/2 + y 0 3/2 ) , x 0 , y 0 (0,σ),
φ( x 0 , y 0 )=(4w/3)( | x 0 | 3/2 + | y 0 | 3/2 ) , x 0 , y 0 (σ/2,σ/2).
φ( x 0 , y 0 )=φ( y 0 , x 0 ) , φ(± x 0 ,± y 0 )=φ( x 0 , y 0 ).
U(x,y)=U(y,x) , U(±x,±y)=U(x,y).
Φ=kφ( x 0 , y 0 )=8πw( | MΔ | 3/2 + | NΔ | 3/2 )/3λ,
ΔL=2 w 2 z 2 .

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