Abstract

The Hopf-Ranãda linked and knotted light beam solution, which has been interpreted physically and extended analytically by Irvine and Bouwmeester recently, is viewed in this Letter as a null electromagnetic field. It is shown, in particular, that the Hopf-Ranãda solution is a variant of a luminal null electromagnetic wave due originally to Robinson and Troutman and reported by Bialynicki-Birula recently. This analogy is motivated by means of a method due to Whittaker and Bateman, and a relationship to well-known scalar luminal localized waves is examined.

© 2009 Optical Society of America

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  1. W. T. M. Irvine and D. Bouwmeester, Nat. Phys. 4, 716 (2008).
    [CrossRef]
  2. A. F. Ranãda, J. Phys. A 23, L815 (1990).
    [CrossRef]
  3. A. F. Ranãda, J. Phys. A 25, 1621 (1992).
    [CrossRef]
  4. A. F. Ranãda and J. L. Trueba, Mod. Nonlin. Opt. 119, 197 (2001).
    [CrossRef]
  5. H. Hopf, Math. Ann. 104, 637 (1931).
    [CrossRef]
  6. H. Bateman, Proc. London Math. Soc. 7, 70 (1909).
    [CrossRef]
  7. L. Silberstein, Ann. Phys. 22, 579 (1907).
    [CrossRef]
  8. I. M. Besieris and A. M. Shaarawi, PIER B 8, 1 (2008).
    [CrossRef]
  9. H. Bateman, The Mathematical Analysis of Electrical and Optical Wave-Motion on the Basis of Maxwell's Equations (Dover, 1955).
  10. I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, PIER 48, 201 (2004).
    [CrossRef]
  11. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, J. Opt. Soc. Am. A 10, 75 (1993).
    [CrossRef]
  12. K. Reivelt and P. Saari, Phys. Rev. E 65, 046622 (2002).
    [CrossRef]
  13. I. Bialynicki-Birula, J. Opt. A 6, S181 (2004).
    [CrossRef]

2008

W. T. M. Irvine and D. Bouwmeester, Nat. Phys. 4, 716 (2008).
[CrossRef]

I. M. Besieris and A. M. Shaarawi, PIER B 8, 1 (2008).
[CrossRef]

2004

I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, PIER 48, 201 (2004).
[CrossRef]

I. Bialynicki-Birula, J. Opt. A 6, S181 (2004).
[CrossRef]

2002

K. Reivelt and P. Saari, Phys. Rev. E 65, 046622 (2002).
[CrossRef]

2001

A. F. Ranãda and J. L. Trueba, Mod. Nonlin. Opt. 119, 197 (2001).
[CrossRef]

1993

1992

A. F. Ranãda, J. Phys. A 25, 1621 (1992).
[CrossRef]

1990

A. F. Ranãda, J. Phys. A 23, L815 (1990).
[CrossRef]

1931

H. Hopf, Math. Ann. 104, 637 (1931).
[CrossRef]

1909

H. Bateman, Proc. London Math. Soc. 7, 70 (1909).
[CrossRef]

1907

L. Silberstein, Ann. Phys. 22, 579 (1907).
[CrossRef]

Attiya, A. M.

I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, PIER 48, 201 (2004).
[CrossRef]

Bateman, H.

H. Bateman, Proc. London Math. Soc. 7, 70 (1909).
[CrossRef]

H. Bateman, The Mathematical Analysis of Electrical and Optical Wave-Motion on the Basis of Maxwell's Equations (Dover, 1955).

Besieris, I. M.

I. M. Besieris and A. M. Shaarawi, PIER B 8, 1 (2008).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, PIER 48, 201 (2004).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, J. Opt. Soc. Am. A 10, 75 (1993).
[CrossRef]

Bialynicki-Birula, I.

I. Bialynicki-Birula, J. Opt. A 6, S181 (2004).
[CrossRef]

Bouwmeester, D.

W. T. M. Irvine and D. Bouwmeester, Nat. Phys. 4, 716 (2008).
[CrossRef]

Hopf, H.

H. Hopf, Math. Ann. 104, 637 (1931).
[CrossRef]

Irvine, W. T. M.

W. T. M. Irvine and D. Bouwmeester, Nat. Phys. 4, 716 (2008).
[CrossRef]

Ranãda, A. F.

A. F. Ranãda and J. L. Trueba, Mod. Nonlin. Opt. 119, 197 (2001).
[CrossRef]

A. F. Ranãda, J. Phys. A 25, 1621 (1992).
[CrossRef]

A. F. Ranãda, J. Phys. A 23, L815 (1990).
[CrossRef]

Reivelt, K.

K. Reivelt and P. Saari, Phys. Rev. E 65, 046622 (2002).
[CrossRef]

Saari, P.

K. Reivelt and P. Saari, Phys. Rev. E 65, 046622 (2002).
[CrossRef]

Shaarawi, A. M.

I. M. Besieris and A. M. Shaarawi, PIER B 8, 1 (2008).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, PIER 48, 201 (2004).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, J. Opt. Soc. Am. A 10, 75 (1993).
[CrossRef]

Silberstein, L.

L. Silberstein, Ann. Phys. 22, 579 (1907).
[CrossRef]

Trueba, J. L.

A. F. Ranãda and J. L. Trueba, Mod. Nonlin. Opt. 119, 197 (2001).
[CrossRef]

Ziolkowski, R. W.

Ann. Phys.

L. Silberstein, Ann. Phys. 22, 579 (1907).
[CrossRef]

J. Opt. A

I. Bialynicki-Birula, J. Opt. A 6, S181 (2004).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

A. F. Ranãda, J. Phys. A 23, L815 (1990).
[CrossRef]

A. F. Ranãda, J. Phys. A 25, 1621 (1992).
[CrossRef]

Math. Ann.

H. Hopf, Math. Ann. 104, 637 (1931).
[CrossRef]

Mod. Nonlin. Opt.

A. F. Ranãda and J. L. Trueba, Mod. Nonlin. Opt. 119, 197 (2001).
[CrossRef]

Nat. Phys.

W. T. M. Irvine and D. Bouwmeester, Nat. Phys. 4, 716 (2008).
[CrossRef]

Phys. Rev. E

K. Reivelt and P. Saari, Phys. Rev. E 65, 046622 (2002).
[CrossRef]

PIER

I. M. Besieris, A. M. Shaarawi, and A. M. Attiya, PIER 48, 201 (2004).
[CrossRef]

PIER B

I. M. Besieris and A. M. Shaarawi, PIER B 8, 1 (2008).
[CrossRef]

Proc. London Math. Soc.

H. Bateman, Proc. London Math. Soc. 7, 70 (1909).
[CrossRef]

Other

H. Bateman, The Mathematical Analysis of Electrical and Optical Wave-Motion on the Basis of Maxwell's Equations (Dover, 1955).

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