Abstract

We introduce a class of three-dimensional nonparaxial optical beams found in a parabolic rotational coordinate system. These beams, representing exact solutions of the nonparaxial Helmholtz equation, have inherent parabolic symmetries. Assisted with a computer-generated holography, we experimentally demonstrate the generation of different modes of these beams. The observed transverse beam patterns along the propagation direction agree well with those from our theoretical predication.

© 2013 Optical Society of America

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  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
    [CrossRef]
  2. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
    [CrossRef]
  3. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, Nat. Mater. 2, 229 (2003).
    [CrossRef]
  4. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  5. A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000).
    [CrossRef]
  6. Q. Cao and X. Deng, J. Opt. Soc. Am. A 15, 1144 (1998).
    [CrossRef]
  7. G. A. Deschamps, Electron. Lett. 7, 684 (1971).
    [CrossRef]
  8. G. Rodríguez-Morales and S. Chavez-Cerda, Opt. Lett. 29, 430 (2004).
    [CrossRef]
  9. Z. Ulanowski and I. K. Ludlow, Opt. Lett. 25, 1792 (2000).
    [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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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
    [CrossRef]
  20. P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
    [CrossRef]
  21. L. C. Lew Yan Voon and M. Willatzen, Math. Comput. Simul. 65, 337 (2004).
    [CrossRef]
  22. M. Willatzen and L. C. Lew Yan Voon, Separable Boundary-Value Problems in Physics (Wiley, 2011).
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    [CrossRef]
  25. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 2883 (2011).
    [CrossRef]

2012

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

2011

2010

2008

2007

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

2006

2004

L. C. Lew Yan Voon and M. Willatzen, Math. Comput. Simul. 65, 337 (2004).
[CrossRef]

G. Rodríguez-Morales and S. Chavez-Cerda, Opt. Lett. 29, 430 (2004).
[CrossRef]

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef]

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, Nat. Mater. 2, 229 (2003).
[CrossRef]

2000

Z. Ulanowski and I. K. Ludlow, Opt. Lett. 25, 1792 (2000).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

1998

Q. Cao and X. Deng, J. Opt. Soc. Am. A 15, 1144 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

1975

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

1971

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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]

Alonso, M. A.

April, A.

Atwater, H. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, Nat. Mater. 2, 229 (2003).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef]

Bekenstein, R.

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

Borghi, R.

Boyack, R.

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, 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]

Cao, Q.

Chavez-Cerda, S.

Chen, Z.

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. 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]

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.

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.

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

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

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

Courvoisier, F.

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

Deng, X.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Di Porto, P.

A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

Dogariu, A.

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

Dudley, J. M.

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

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.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

Giust, R.

Harel, E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, Nat. Mater. 2, 229 (2003).
[CrossRef]

Hernandez, D.

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, 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.

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

Kik, P. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, Nat. Mater. 2, 229 (2003).
[CrossRef]

Koel, B. E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, Nat. Mater. 2, 229 (2003).
[CrossRef]

Lacourt, P. A.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Lekner, J.

Lew Yan Voon, L. C.

L. C. Lew Yan Voon and M. Willatzen, Math. Comput. Simul. 65, 337 (2004).
[CrossRef]

M. Willatzen and L. C. Lew Yan Voon, Separable Boundary-Value Problems in Physics (Wiley, 2011).

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

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]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Ludlow, I. K.

Maier, S. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, Nat. Mater. 2, 229 (2003).
[CrossRef]

Mathis, A.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Meltzer, S.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, Nat. Mater. 2, 229 (2003).
[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]

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

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]

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

Nemirovsky, J.

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

Prakash, J.

Requicha, A. A. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, Nat. Mater. 2, 229 (2003).
[CrossRef]

Rodríguez-Morales, G.

Salandrino, A.

Salazar, M.

Santarsiero, M.

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]

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]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

Ulanowski, Z.

Willatzen, M.

L. C. Lew Yan Voon and M. Willatzen, Math. Comput. Simul. 65, 337 (2004).
[CrossRef]

M. Willatzen and L. C. Lew Yan Voon, Separable Boundary-Value Problems in Physics (Wiley, 2011).

Wolff, P.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

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]

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, 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]

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]

Zhang, Z.

Electron. Lett.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

J. Opt. Soc. Am. A

Math. Comput. Simul.

L. C. Lew Yan Voon and M. Willatzen, Math. Comput. Simul. 65, 337 (2004).
[CrossRef]

Nat. Mater.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, Nat. Mater. 2, 229 (2003).
[CrossRef]

Nature

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, Nature 391, 667 (1998).
[CrossRef]

Opt. Commun.

A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Phys. Rev. Lett.

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]

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

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

Other

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.

M. Willatzen and L. C. Lew Yan Voon, Separable Boundary-Value Problems in Physics (Wiley, 2011).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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Figures (4)

Fig. 1.
Fig. 1.

(a)–(c) Exact (00)-mode; (d)–(f) exact (11)-mode; and (g)–(i) exact (23)-mode nonparaxial beams in parabolic rotational coordinates; (a), (d), (g) normalized amplitude distribution at z = 2 cm ; (b), (e), (h) phase distribution at z = 2 cm ; (c), (f), and (i) the normalized intensity evolution on the x = 0 plane; iso-intensity contour of (j) exact (00)-mode, (k) (11)-mode, and (l) (23)-mode of the nonparaxial beams in parabolic rotational coordinates; z 0 = 2 cm and w 0 = 1 mm .

Fig. 2.
Fig. 2.

Numerical and experimental demonstrations of the exact (00)-mode nonparaxial beams in parabolic rotational coordinates. (a) Computer-generated hologram; (b) numerically simulated side-view propagation of the generated beam; (c) phase of the initial generated beam; (d)–(g) snapshots of the normalized transverse intensity patterns taken at the planes marked by the dashed lines in (b); (h) interference phase of the initial generated beam and a plane wave; (i)–(l) experimentally recorded normalized transverse beam patterns at different positions (marked in dashed curve with z = 11.7 , 24.5, 37.5, and 51 cm) corresponding to (b); z 0 = 37.5 cm .

Fig. 3.
Fig. 3.

Description is the same as that in Fig. 2 except the nonparaxial beams in parabolic rotational coordinates are the (11)-mode.

Fig. 4.
Fig. 4.

Description is the same as that in Fig. 2 except the nonparaxial beams in parabolic rotational coordinates are the exact (23)-mode.

Equations (6)

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

2 U x 2 + 2 U y 2 + 2 U z 2 + k 2 U = 0 ,
d 2 M ( ξ ) d ξ 2 + 1 ξ d M ( ξ ) d ξ + ( m 2 ξ 2 + k 2 ξ 2 α k ) M ( ξ ) = 0 ,
d 2 N ( η ) d η 2 + 1 η d N ( η ) d η + ( m 2 η 2 + k 2 η 2 + α k ) N ( η ) = 0 ,
d 2 Θ ( Φ ) d Φ 2 + m 2 Θ ( Φ ) = 0 ,
U ( ξ , η , Φ ) = ( ξ η / w 0 ) m e ± i k ( ξ 2 + η 2 ) / 2 + i m Φ F 1 1 ( i α 4 + m + 1 2 , m + 1 , i k ξ 2 ) F 1 1 ( i α 4 + m + 1 2 , m + 1 , i k η 2 ) .
U ( x , y , z ) = ( 1 ) m [ ( x ± i y ) / w 0 ] m e ± i k r F 1 1 ( i α 4 + m + 1 2 , m + 1 , i k ( z 0 z + r ) ) F 1 1 ( i α 4 + m + 1 2 , m + 1 , i k ( z z 0 + r ) ) ,

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