Abstract

We demonstrate the existence of elliptic vortices of electromagnetic scalar wave fields. The corresponding intensity profiles are formed by propagation-invariant confocal elliptic rings. We have found that copropagation of this kind of vortex occurs without interaction. The results presented here also apply for physical systems described by the 2+1-dimensional Schrödinger equation.

© 2001 Optical Society of America

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  2. Y. S. Kivshar and B. Luther-Davis, Phys. Rep. 298, 81–197 (1998).
    [CrossRef]
  3. D. L. Feder, A. A. Svidzinsky, A. L. Fetter, and C. W. Clark, Phys. Rev. Lett. 86, 564–567 (2001).
    [CrossRef] [PubMed]
  4. I. S. Aranson, A. R. Bishop, I. Daruka, and V. M. Vinokur, Phys. Rev. Lett. 80, 1770–1773 (1998).
    [CrossRef]
  5. C. O. Weis, M. Vaupel, K. Staliunas, G. Slekys, and V. B. Taranenko, Appl. Phys. B 68, 151–168 (1999).
    [CrossRef]
  6. S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225–233 (1996).
    [CrossRef]
  7. K. T. Gahagan and G. A. Swartzlander, J. Opt. Soc. Am. B 16, 533–537 (1999).
    [CrossRef]
  8. G. A. Swartzlander, Opt. Lett. 26, 497–499 (2001).
    [CrossRef]
  9. G. Indebetouw, J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  10. D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Phys. Rev. Lett. 79, 3399–3402 (1997).
    [CrossRef]
  11. D. Rozas and G. A. Swartzlander, Opt. Lett. 25, 126–128 (2000).
    [CrossRef]
  12. G. Molina-Terriza, J. Recolons, and L. Torner, Opt. Lett. 25, 1135–1137 (2000).
    [CrossRef]
  13. G. Molina-Terriza, E. M. Wright, and L. Torner, Opt. Lett. 26, 163–165 (2001).
    [CrossRef]
  14. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  15. N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford U. Press, London, 1951).
  16. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1927), Chaps.  XVIII, XIX.
  17. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 5th ed. (Academic, London, 1994).
  18. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 20, pp. 721–750.
  19. S. Chávez-Cerda, J. Mod. Opt. 46, 923–930 (1999).
  20. A. Vasara, J. Turunen, and A. T. Friberg, J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  21. A. A. Inayat-Hussain, J. Math. Phys. 32, 669–675 (1991).
    [CrossRef]
  22. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493–1495 (2000).
    [CrossRef]
  23. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, Opt. Commun. 195, 35–40 (2001).
    [CrossRef]
  24. N. B. Simpson, K. Dholakia, L. Allen, and M. Padgett, Opt. Lett. 22, 52–54 (1997).
    [CrossRef] [PubMed]
  25. H. C. Manoharan, C. P. Lutz, and D. M. Eigler, Nature 403, 512–515 (2000).
    [CrossRef] [PubMed]

2001 (4)

D. L. Feder, A. A. Svidzinsky, A. L. Fetter, and C. W. Clark, Phys. Rev. Lett. 86, 564–567 (2001).
[CrossRef] [PubMed]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, Opt. Commun. 195, 35–40 (2001).
[CrossRef]

G. Molina-Terriza, E. M. Wright, and L. Torner, Opt. Lett. 26, 163–165 (2001).
[CrossRef]

G. A. Swartzlander, Opt. Lett. 26, 497–499 (2001).
[CrossRef]

2000 (4)

1999 (3)

K. T. Gahagan and G. A. Swartzlander, J. Opt. Soc. Am. B 16, 533–537 (1999).
[CrossRef]

C. O. Weis, M. Vaupel, K. Staliunas, G. Slekys, and V. B. Taranenko, Appl. Phys. B 68, 151–168 (1999).
[CrossRef]

S. Chávez-Cerda, J. Mod. Opt. 46, 923–930 (1999).

1998 (2)

I. S. Aranson, A. R. Bishop, I. Daruka, and V. M. Vinokur, Phys. Rev. Lett. 80, 1770–1773 (1998).
[CrossRef]

Y. S. Kivshar and B. Luther-Davis, Phys. Rep. 298, 81–197 (1998).
[CrossRef]

1997 (2)

N. B. Simpson, K. Dholakia, L. Allen, and M. Padgett, Opt. Lett. 22, 52–54 (1997).
[CrossRef] [PubMed]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Phys. Rev. Lett. 79, 3399–3402 (1997).
[CrossRef]

1996 (1)

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225–233 (1996).
[CrossRef]

1993 (1)

G. Indebetouw, J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

1991 (1)

A. A. Inayat-Hussain, J. Math. Phys. 32, 669–675 (1991).
[CrossRef]

1989 (1)

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 20, pp. 721–750.

Allen, L.

Aranson, I. S.

I. S. Aranson, A. R. Bishop, I. Daruka, and V. M. Vinokur, Phys. Rev. Lett. 80, 1770–1773 (1998).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Bishop, A. R.

I. S. Aranson, A. R. Bishop, I. Daruka, and V. M. Vinokur, Phys. Rev. Lett. 80, 1770–1773 (1998).
[CrossRef]

Chávez-Cerda, S.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, Opt. Commun. 195, 35–40 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

S. Chávez-Cerda, J. Mod. Opt. 46, 923–930 (1999).

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Clark, C. W.

D. L. Feder, A. A. Svidzinsky, A. L. Fetter, and C. W. Clark, Phys. Rev. Lett. 86, 564–567 (2001).
[CrossRef] [PubMed]

Daruka, I.

I. S. Aranson, A. R. Bishop, I. Daruka, and V. M. Vinokur, Phys. Rev. Lett. 80, 1770–1773 (1998).
[CrossRef]

Dholakia, K.

Eigler, D. M.

H. C. Manoharan, C. P. Lutz, and D. M. Eigler, Nature 403, 512–515 (2000).
[CrossRef] [PubMed]

Feder, D. L.

D. L. Feder, A. A. Svidzinsky, A. L. Fetter, and C. W. Clark, Phys. Rev. Lett. 86, 564–567 (2001).
[CrossRef] [PubMed]

Fetter, A. L.

D. L. Feder, A. A. Svidzinsky, A. L. Fetter, and C. W. Clark, Phys. Rev. Lett. 86, 564–567 (2001).
[CrossRef] [PubMed]

Friberg, A. T.

Gahagan, K. T.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 5th ed. (Academic, London, 1994).

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, Opt. Commun. 195, 35–40 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

Inayat-Hussain, A. A.

A. A. Inayat-Hussain, J. Math. Phys. 32, 669–675 (1991).
[CrossRef]

Indebetouw, G.

G. Indebetouw, J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Iturbe-Castillo, M. D.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, Opt. Commun. 195, 35–40 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and B. Luther-Davis, Phys. Rep. 298, 81–197 (1998).
[CrossRef]

Luther-Davis, B.

Y. S. Kivshar and B. Luther-Davis, Phys. Rep. 298, 81–197 (1998).
[CrossRef]

Lutz, C. P.

H. C. Manoharan, C. P. Lutz, and D. M. Eigler, Nature 403, 512–515 (2000).
[CrossRef] [PubMed]

Manoharan, H. C.

H. C. Manoharan, C. P. Lutz, and D. M. Eigler, Nature 403, 512–515 (2000).
[CrossRef] [PubMed]

McDonald, G. S.

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225–233 (1996).
[CrossRef]

McLachlan, N. W.

N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford U. Press, London, 1951).

Molina-Terriza, G.

New, G. H. C.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, Opt. Commun. 195, 35–40 (2001).
[CrossRef]

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Padgett, M.

Recolons, J.

Rodríguez-Dagnino, R. M.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, Opt. Commun. 195, 35–40 (2001).
[CrossRef]

Rozas, D.

D. Rozas and G. A. Swartzlander, Opt. Lett. 25, 126–128 (2000).
[CrossRef]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Phys. Rev. Lett. 79, 3399–3402 (1997).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 5th ed. (Academic, London, 1994).

Sacks, Z. S.

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Phys. Rev. Lett. 79, 3399–3402 (1997).
[CrossRef]

Simpson, N. B.

Slekys, G.

C. O. Weis, M. Vaupel, K. Staliunas, G. Slekys, and V. B. Taranenko, Appl. Phys. B 68, 151–168 (1999).
[CrossRef]

Staliunas, K.

C. O. Weis, M. Vaupel, K. Staliunas, G. Slekys, and V. B. Taranenko, Appl. Phys. B 68, 151–168 (1999).
[CrossRef]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 20, pp. 721–750.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Svidzinsky, A. A.

D. L. Feder, A. A. Svidzinsky, A. L. Fetter, and C. W. Clark, Phys. Rev. Lett. 86, 564–567 (2001).
[CrossRef] [PubMed]

Swartzlander, G. A.

Taranenko, V. B.

C. O. Weis, M. Vaupel, K. Staliunas, G. Slekys, and V. B. Taranenko, Appl. Phys. B 68, 151–168 (1999).
[CrossRef]

Tepichín, E.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, Opt. Commun. 195, 35–40 (2001).
[CrossRef]

Torner, L.

Turunen, J.

Vasara, A.

Vaupel, M.

C. O. Weis, M. Vaupel, K. Staliunas, G. Slekys, and V. B. Taranenko, Appl. Phys. B 68, 151–168 (1999).
[CrossRef]

Vinokur, V. M.

I. S. Aranson, A. R. Bishop, I. Daruka, and V. M. Vinokur, Phys. Rev. Lett. 80, 1770–1773 (1998).
[CrossRef]

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1927), Chaps.  XVIII, XIX.

Weis, C. O.

C. O. Weis, M. Vaupel, K. Staliunas, G. Slekys, and V. B. Taranenko, Appl. Phys. B 68, 151–168 (1999).
[CrossRef]

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1927), Chaps.  XVIII, XIX.

Wright, E. M.

Appl. Phys. B (1)

C. O. Weis, M. Vaupel, K. Staliunas, G. Slekys, and V. B. Taranenko, Appl. Phys. B 68, 151–168 (1999).
[CrossRef]

J. Math. Phys. (1)

A. A. Inayat-Hussain, J. Math. Phys. 32, 669–675 (1991).
[CrossRef]

J. Mod. Opt. (2)

S. Chávez-Cerda, J. Mod. Opt. 46, 923–930 (1999).

G. Indebetouw, J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

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

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

Nature (1)

H. C. Manoharan, C. P. Lutz, and D. M. Eigler, Nature 403, 512–515 (2000).
[CrossRef] [PubMed]

Opt. Commun. (2)

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, Opt. Commun. 195, 35–40 (2001).
[CrossRef]

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Opt. Lett. (6)

Phys. Rep. (1)

Y. S. Kivshar and B. Luther-Davis, Phys. Rep. 298, 81–197 (1998).
[CrossRef]

Phys. Rev. Lett. (3)

D. L. Feder, A. A. Svidzinsky, A. L. Fetter, and C. W. Clark, Phys. Rev. Lett. 86, 564–567 (2001).
[CrossRef] [PubMed]

I. S. Aranson, A. R. Bishop, I. Daruka, and V. M. Vinokur, Phys. Rev. Lett. 80, 1770–1773 (1998).
[CrossRef]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Phys. Rev. Lett. 79, 3399–3402 (1997).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Other (5)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford U. Press, London, 1951).

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge U. Press, Cambridge, 1927), Chaps.  XVIII, XIX.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 5th ed. (Academic, London, 1994).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 20, pp. 721–750.

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Figures (2)

Fig. 1
Fig. 1

Intensities and elliptic phases of higher-order Mathieu beams. The ellipticity of the phase is best observed in the central region of the ringed pattern.

Fig. 2
Fig. 2

Breakup of a fourth-order elliptic vortex as a result of excessive stretching of the elliptic ring pattern by an increase of the value of interfocal distance h of the elliptic coordinates such that now the coefficients Amq and Bmq in Eq.  (4) are not equal, as is the case in Fig.  1. Observe that new vortices are created and that some of them rotate in the opposite direction to the original.

Equations (4)

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

2Rξξ2-a-2qcosh2ξRξ=0,
2Φηη2+a-2qcos2ηΦη=0,
Er=expikzz02πAφ×expiktxcosφ+ysinφdφ,
Eξ,η,z;q=AmqJemξ;qcemη;q+iBmqJomξ;qsemη;qexpikzz,

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