Abstract

We present a general theory of three-dimensional non-paraxial spatially-accelerating waves of the Maxwell equations. These waves constitute a two-dimensional structure exhibiting shape-invariant propagation along semicircular trajectories. We provide classification and characterization of possible shapes of such beams, expressed through the angular spectra of parabolic, oblate and prolate spheroidal fields. Our results facilitate the design of accelerating beams with novel structures, broadening scope and potential applications of accelerating beams.

© 2013 OSA

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  1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007).
    [CrossRef] [PubMed]
  2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007).
    [CrossRef]
  3. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9, 1334–1336 (2009).
    [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,” Science324, 229–232 (2009).
    [CrossRef] [PubMed]
  5. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nature Photonics3, 395–398 (2009).
    [CrossRef]
  6. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494, 331–335 (2013).
    [CrossRef] [PubMed]
  7. A. Minovich, A. Klein, N. Janunts, T. Pertsch, D. Neshev, and Y. Kivshar, “Generation and Near-Field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
    [CrossRef] [PubMed]
  8. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett.101, 071110 (2012).
    [CrossRef]
  9. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nature photonics4, 103–106 (2010).
    [CrossRef]
  10. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett.105, 253901 (2010).
    [CrossRef]
  11. I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express19, 23132–23139 (2011).
    [CrossRef] [PubMed]
  12. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106, 213903 (2011).
    [CrossRef] [PubMed]
  13. I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett.108, 113903 (2012).
    [CrossRef] [PubMed]
  14. Y. Hu, Z. Sun, D. Bongiovanni, D. Song, C. Lou, J. Xu, Z. Chen, and R. Morandotti, “Reshaping the trajectory and spectrum of nonlinear Airy beams,” Opt. Lett.37, 3201–3203 (2012).
    [CrossRef] [PubMed]
  15. R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express19, 23706–23715 (2011).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  17. J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express16, 12866–12871 (2008).
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    [CrossRef] [PubMed]
  19. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264–267 (1979).
    [CrossRef]
  20. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett.108, 163901 (2012).
    [CrossRef] [PubMed]
  21. F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett.37, 1736–1738 (2012).
    [CrossRef] [PubMed]
  22. I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
    [CrossRef]
  23. M. A. Bandres and B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys.15, 013054 (2013).
    [CrossRef]
  24. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett.109, 193901 (2012).
    [CrossRef] [PubMed]
  25. 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, 203902 (2012).
    [CrossRef] [PubMed]
  26. M. A. Alonso and M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett.37, 5175–5177 (2012).
    [CrossRef] [PubMed]
  27. I. Kaminer, J. Nemirovsky, and M. Segev, “Self-accelerating self-trapped nonlinear beams of Maxwell’s equations,” Opt. Express20, 18827–18835 (2012).
    [CrossRef] [PubMed]
  28. P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett.37, 2820–2822 (2012).
    [CrossRef] [PubMed]
  29. C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J60, 3580 (1976).
  30. L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E58, 6792–6806 (1998).
    [CrossRef]
  31. J. Stratton, Electromagnetic Theory(Wiley-IEEE, 2007).

2013 (2)

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494, 331–335 (2013).
[CrossRef] [PubMed]

M. A. Bandres and B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys.15, 013054 (2013).
[CrossRef]

2012 (11)

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett.109, 193901 (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, 203902 (2012).
[CrossRef] [PubMed]

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett.108, 113903 (2012).
[CrossRef] [PubMed]

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

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
[CrossRef]

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett.101, 071110 (2012).
[CrossRef]

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett.37, 1736–1738 (2012).
[CrossRef] [PubMed]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett.37, 2820–2822 (2012).
[CrossRef] [PubMed]

Y. Hu, Z. Sun, D. Bongiovanni, D. Song, C. Lou, J. Xu, Z. Chen, and R. Morandotti, “Reshaping the trajectory and spectrum of nonlinear Airy beams,” Opt. Lett.37, 3201–3203 (2012).
[CrossRef] [PubMed]

I. Kaminer, J. Nemirovsky, and M. Segev, “Self-accelerating self-trapped nonlinear beams of Maxwell’s equations,” Opt. Express20, 18827–18835 (2012).
[CrossRef] [PubMed]

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

2011 (4)

I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express19, 23132–23139 (2011).
[CrossRef] [PubMed]

R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express19, 23706–23715 (2011).
[CrossRef] [PubMed]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106, 213903 (2011).
[CrossRef] [PubMed]

A. Minovich, A. Klein, N. Janunts, T. Pertsch, D. Neshev, and Y. Kivshar, “Generation and Near-Field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

2010 (2)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nature photonics4, 103–106 (2010).
[CrossRef]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett.105, 253901 (2010).
[CrossRef]

2009 (4)

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9, 1334–1336 (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,” Science324, 229–232 (2009).
[CrossRef] [PubMed]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nature Photonics3, 395–398 (2009).
[CrossRef]

M. A. Bandres, “Accelerating beams,” Opt. Lett.34, 3791–3793 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007).
[CrossRef]

1998 (1)

L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E58, 6792–6806 (1998).
[CrossRef]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264–267 (1979).
[CrossRef]

1976 (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J60, 3580 (1976).

Abdollahpour, D.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett.105, 253901 (2010).
[CrossRef]

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, 203902 (2012).
[CrossRef] [PubMed]

Alonso, M. A.

Arie, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494, 331–335 (2013).
[CrossRef] [PubMed]

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett.108, 113903 (2012).
[CrossRef] [PubMed]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nature Photonics3, 395–398 (2009).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264–267 (1979).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9, 1334–1336 (2009).
[CrossRef] [PubMed]

Bekenstein, R.

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

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
[CrossRef]

R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express19, 23706–23715 (2011).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264–267 (1979).
[CrossRef]

Bongiovanni, D.

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J60, 3580 (1976).

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007).
[CrossRef]

Cannan, D.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett.37, 2820–2822 (2012).
[CrossRef] [PubMed]

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

Chen, Z.

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nature photonics4, 103–106 (2010).
[CrossRef]

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, 203902 (2012).
[CrossRef] [PubMed]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106, 213903 (2011).
[CrossRef] [PubMed]

I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express19, 23132–23139 (2011).
[CrossRef] [PubMed]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nature photonics4, 103–106 (2010).
[CrossRef]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324, 229–232 (2009).
[CrossRef] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007).
[CrossRef]

Cottrell, D. M.

Courvoisier, F.

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett.37, 1736–1738 (2012).
[CrossRef] [PubMed]

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
[CrossRef]

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett.101, 071110 (2012).
[CrossRef]

Davis, J. A.

Day, D.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9, 1334–1336 (2009).
[CrossRef] [PubMed]

Dholakia, K.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9, 1334–1336 (2009).
[CrossRef] [PubMed]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007).
[CrossRef]

Dolev, I.

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett.108, 113903 (2012).
[CrossRef] [PubMed]

Dudley, J. M.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett.101, 071110 (2012).
[CrossRef]

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
[CrossRef]

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett.37, 1736–1738 (2012).
[CrossRef] [PubMed]

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nature Photonics3, 395–398 (2009).
[CrossRef]

Froehly, L.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett.101, 071110 (2012).
[CrossRef]

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett.37, 1736–1738 (2012).
[CrossRef] [PubMed]

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
[CrossRef]

Furfaro, L.

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett.37, 1736–1738 (2012).
[CrossRef] [PubMed]

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett.101, 071110 (2012).
[CrossRef]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nature Photonics3, 395–398 (2009).
[CrossRef]

Giust, R.

Gover, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494, 331–335 (2013).
[CrossRef] [PubMed]

Greenfield, E.

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
[CrossRef]

Gu, M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9, 1334–1336 (2009).
[CrossRef] [PubMed]

Hannappel, G. M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9, 1334–1336 (2009).
[CrossRef] [PubMed]

Hu, Y.

Jacquot, M.

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett.37, 1736–1738 (2012).
[CrossRef] [PubMed]

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett.101, 071110 (2012).
[CrossRef]

Janunts, N.

A. Minovich, A. Klein, N. Janunts, T. Pertsch, D. Neshev, and Y. Kivshar, “Generation and Near-Field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J60, 3580 (1976).

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, 203902 (2012).
[CrossRef] [PubMed]

I. Kaminer, J. Nemirovsky, and M. Segev, “Self-accelerating self-trapped nonlinear beams of Maxwell’s equations,” Opt. Express20, 18827–18835 (2012).
[CrossRef] [PubMed]

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
[CrossRef]

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

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett.108, 113903 (2012).
[CrossRef] [PubMed]

I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express19, 23132–23139 (2011).
[CrossRef] [PubMed]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106, 213903 (2011).
[CrossRef] [PubMed]

Kivshar, Y.

A. Minovich, A. Klein, N. Janunts, T. Pertsch, D. Neshev, and Y. Kivshar, “Generation and Near-Field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Klein, A.

A. Minovich, A. Klein, N. Janunts, T. Pertsch, D. Neshev, and Y. Kivshar, “Generation and Near-Field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324, 229–232 (2009).
[CrossRef] [PubMed]

Kooi, P.-S.

L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E58, 6792–6806 (1998).
[CrossRef]

Lacourt, P. A.

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett.37, 1736–1738 (2012).
[CrossRef] [PubMed]

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett.101, 071110 (2012).
[CrossRef]

Leong, M.-S.

L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E58, 6792–6806 (1998).
[CrossRef]

Lereah, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494, 331–335 (2013).
[CrossRef] [PubMed]

Li, L.-W.

L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E58, 6792–6806 (1998).
[CrossRef]

Li, T.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett.37, 2820–2822 (2012).
[CrossRef] [PubMed]

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

Lilach, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494, 331–335 (2013).
[CrossRef] [PubMed]

Lou, C.

Lumer, Y.

Mathis, A.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett.101, 071110 (2012).
[CrossRef]

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
[CrossRef]

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett.37, 1736–1738 (2012).
[CrossRef] [PubMed]

Miller, W.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J60, 3580 (1976).

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, 203902 (2012).
[CrossRef] [PubMed]

Minovich, A.

A. Minovich, A. Klein, N. Janunts, T. Pertsch, D. Neshev, and Y. Kivshar, “Generation and Near-Field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Mintry, M. J.

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, 203902 (2012).
[CrossRef] [PubMed]

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324, 229–232 (2009).
[CrossRef] [PubMed]

Morandotti, R.

Nemirovsky, J.

I. Kaminer, J. Nemirovsky, and M. Segev, “Self-accelerating self-trapped nonlinear beams of Maxwell’s equations,” Opt. Express20, 18827–18835 (2012).
[CrossRef] [PubMed]

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

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
[CrossRef]

Neshev, D.

A. Minovich, A. Klein, N. Janunts, T. Pertsch, D. Neshev, and Y. Kivshar, “Generation and Near-Field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Papazoglou, D. G.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett.105, 253901 (2010).
[CrossRef]

Pertsch, T.

A. Minovich, A. Klein, N. Janunts, T. Pertsch, D. Neshev, and Y. Kivshar, “Generation and Near-Field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324, 229–232 (2009).
[CrossRef] [PubMed]

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nature photonics4, 103–106 (2010).
[CrossRef]

Rodríguez-Lara, B. M.

M. A. Bandres and B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys.15, 013054 (2013).
[CrossRef]

Salandrino, A.

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, 203902 (2012).
[CrossRef] [PubMed]

I. Kaminer, J. Nemirovsky, and M. Segev, “Self-accelerating self-trapped nonlinear beams of Maxwell’s equations,” Opt. Express20, 18827–18835 (2012).
[CrossRef] [PubMed]

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
[CrossRef]

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

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett.108, 113903 (2012).
[CrossRef] [PubMed]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106, 213903 (2011).
[CrossRef] [PubMed]

R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express19, 23706–23715 (2011).
[CrossRef] [PubMed]

I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express19, 23132–23139 (2011).
[CrossRef] [PubMed]

Shapira, A.

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett.108, 113903 (2012).
[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,” Science324, 229–232 (2009).
[CrossRef] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007).
[CrossRef]

Song, D.

Stevenson, D. J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9, 1334–1336 (2009).
[CrossRef] [PubMed]

Stratton, J.

J. Stratton, Electromagnetic Theory(Wiley-IEEE, 2007).

Sun, Z.

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett.105, 253901 (2010).
[CrossRef]

Tan, K.-Y.

L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E58, 6792–6806 (1998).
[CrossRef]

Tzortzakis, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett.105, 253901 (2010).
[CrossRef]

Voloch-Bloch, N.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494, 331–335 (2013).
[CrossRef] [PubMed]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nature Photonics3, 395–398 (2009).
[CrossRef]

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nature photonics4, 103–106 (2010).
[CrossRef]

Xu, J.

Yeo, T.-S.

L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E58, 6792–6806 (1998).
[CrossRef]

Yin, X.

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

Zhang, P.

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

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett.37, 2820–2822 (2012).
[CrossRef] [PubMed]

Zhang, X.

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett.37, 2820–2822 (2012).
[CrossRef] [PubMed]

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

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264–267 (1979).
[CrossRef]

Appl. Phys. Lett. (1)

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett.101, 071110 (2012).
[CrossRef]

Lab Chip (1)

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9, 1334–1336 (2009).
[CrossRef] [PubMed]

Nagoya Math. J (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J60, 3580 (1976).

Nature (1)

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494, 331–335 (2013).
[CrossRef] [PubMed]

Nature Photonics (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nature Photonics3, 395–398 (2009).
[CrossRef]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nature photonics4, 103–106 (2010).
[CrossRef]

New J. Phys. (1)

M. A. Bandres and B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New J. Phys.15, 013054 (2013).
[CrossRef]

Opt. Express (4)

Opt. Lett. (7)

Opt. Photon. News (1)

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, M. Segev, A. Mathis, L. Froehly, F. Courvoisier, and J. M. Dudley, “Accelerating beyond the horizon,” Opt. Photon. News23, 26–26 (2012).
[CrossRef]

Phys. Rev. E (1)

L.-W. Li, M.-S. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E58, 6792–6806 (1998).
[CrossRef]

Phys. Rev. Lett. (8)

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett.109, 193901 (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, 203902 (2012).
[CrossRef] [PubMed]

A. Minovich, A. Klein, N. Janunts, T. Pertsch, D. Neshev, and Y. Kivshar, “Generation and Near-Field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011).
[CrossRef] [PubMed]

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

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett.105, 253901 (2010).
[CrossRef]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106, 213903 (2011).
[CrossRef] [PubMed]

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett.108, 113903 (2012).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007).
[CrossRef]

Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science324, 229–232 (2009).
[CrossRef] [PubMed]

Other (1)

J. Stratton, Electromagnetic Theory(Wiley-IEEE, 2007).

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Figures (6)

Fig. 1
Fig. 1

Intensity cross-sections of three-dimensional nonparaxial accelerating beams and their corresponding generating functions g(θ). Top row: Amplitudes of the generating functions as a function of the k-space angle θ. Middle row: Intensity cross-section at z = 0 presenting the shape-invariant profile of each beam. Bottom row: Top-view plot showing the intensity cross-section at plane y = 0 highlighting the circular trajectory. All lengths are in units of k−1.

Fig. 2
Fig. 2

Three-dimensional rotational coordinate systems for which the Helmholtz equation is separable.

Fig. 3
Fig. 3

Parabolic accelerating waves with different “translation” values of β. (a,e) Intensity cross-section at the y = 0 plane, (b–d,f–h) intensity profiles at z = 0 planes. The white line parabolas in (d,h) depict the caustic cross sections. All sections are of size 200 × 200 and all lengths are in units of k−1.

Fig. 4
Fig. 4

Prolate spheroidal accelerating beams of different orders n. (a,e) Intensity cross-section at the y = 0 plane. (b–d,f–h) Intensity profiles at the z = 0 plane. The beam of order n has exactly n+1 stripes. The white line hyperbolas and ellipses in (c,g) depict the caustic cross sections. All subfigures are for m = 120, of size 200×200, and all lengths are in units of k−1.

Fig. 5
Fig. 5

Oblate spheroidal accelerating beams of outer-type and inner-type. (a,e,i) Intensity cross-section at the y = 0 plane. (b–d,f–h,j–l) Intensity profiles at the z = 0 plane. The white line hyperbolas and ellipses in (d,g,h,k,l) depict the caustic cross sections. The white dots correspond to the foci. All subfigures are for m = 100, of size 200 × 200, and all lengths are in units of k−1.

Fig. 6
Fig. 6

Comparison between vector parabolic, prolate, and oblate accelerating beams. Each row shows the electric field intensity of the radial ρ̂ and angular ϕ̂ components of a single electromagnetic accelerating wave at the z = 0 and y = 0 planes, over sections of size 200 × 200. The white arrows in the first row depict the vector component. All lengths are in units of k−1.

Equations (16)

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

ψ ( r ) = A ( θ , ϕ ) exp ( i k r u ) d Ω ,
ψ ( r ) = 0 π π / 2 π / 2 g ( θ ) exp ( i m ϕ ) exp ( i k r u ) sin θ d θ d ϕ ,
g β ( θ ) = 1 2 π [ tan ( θ / 2 ) ] i β sin θ , < β < ,
u C 2 = ( β + β 2 + m 2 ) / k , v C 2 = ( β + β 2 + m 2 ) / k ,
x = u v sin ϕ , y = 1 2 ( u 2 v 2 ) , z = u v cos ϕ ,
g n m ( θ ; γ ) = S m + n m ( cos θ , γ ) , γ k f ,
d d ν [ ( 1 ν 2 ) d d ν S l m ( ν , γ ) ] + ( Λ l m γ 2 ν 2 m 2 1 ν 2 ) S l m ( ν , γ ) = 0 ,
sin 2 η C + = ( Λ γ 2 ) + ( Λ γ 2 ) 2 + 4 γ 2 m 2 2 γ 2 , sinh 2 ξ C = ( Λ γ 2 ) + ( Λ γ 2 ) 2 + 4 γ 2 m 2 2 γ 2 ,
x = f sinh ξ sin η sin ϕ , y = f cosh ξ cos η , z = f sinh ξ sin η cos ϕ ,
g n m ( θ ; i γ ) = S m + n m ( cos θ , i γ ) , γ k f ,
sin 2 η C O = ( Λ + γ 2 ) ( Λ + γ 2 ) 2 4 γ 2 m 2 2 γ 2 , cosh 2 ξ C O = ( Λ + γ 2 ) + ( Λ + γ 2 ) 2 4 γ 2 m 2 2 γ 2 ,
x = f cosh ξ sin η sin ϕ , y = f sinh ξ cos η , z = f cosh ξ sin η cos ϕ ,
sin 2 η C I + = ( Λ + γ 2 ) ( Λ + γ 2 ) 2 4 γ 2 m 2 2 γ 2 , sin 2 η C I = ( Λ + γ 2 ) + ( Λ + γ 2 ) 2 4 γ 2 m 2 2 γ 2 .
H = i ω ε × Π e , E = k 2 Π e + ( Π e ) ,
E = i ω μ × Π m , H = k 2 Π m + ( Π m ) ,
E = i ω μ ( ρ ^ ϕ ρ + ϕ ^ ρ ) ψ ,

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