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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References

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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).
  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).
    [PubMed]
  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 (2)

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 (2)

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 (1)

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

2001 (1)

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

1993 (1)

1992 (1)

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

1990 (1)

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

1931 (1)

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

1909 (1)

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

1907 (1)

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).
[PubMed]

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).

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).

Ziolkowski, R. W.

Ann. Phys. (1)

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

J. Opt. A (1)

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

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

J. Phys. A (2)

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

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

Math. Ann. (1)

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

Mod. Nonlin. Opt. (1)

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

Nat. Phys. (1)

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

Phys. Rev. E (1)

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

PIER (1)

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

PIER B (1)

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

Proc. London Math. Soc. (1)

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

Other (1)

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

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Figures (2)

Fig. 1
Fig. 1

The electric (uppermost; red online) and magnetic (leftmost; blue online) field lines corresponding to a null Riemann–Silberstein vector arising from the functions ϕ 1 = β , ϕ 2 = α 2 for the parameter values b = 10 2 , b + = 10 .

Fig. 2
Fig. 2

The electric (lowermost; red online) and magnetic (other; blue online) field lines corresponding to a null Riemann–Silberstein vector arising from the functions ϕ 1 = β 3 , ϕ 2 = α 4 for the parameter values b = 1 , b + = 10 .

Equations (14)

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E ( r , t ) = 1 4 π i ς × ς * ( 1 + ς ς * ) 2 , B ( r , t ) = 1 4 π i η × η * ( 1 + η η * ) 2 ;
ς ( r , t ) = ( A x + t y ) + i ( A z + t ( A 1 ) ) ( t x A y ) + i ( A ( A 1 ) t z ) ;
η ( r , t ) = ( A z + t ( A 1 ) + i ( t x A y ) ) ( A x + t y ) + i ( A ( A 1 ) t z ) ;
A ( r , t ) = 1 2 ( x 2 + y 2 + z 2 t 2 + 1 ) ,
× F = i t F , F = 0 ,
P E × B = i F * × F ,
w e m 1 2 ( E E + B B ) = F * F ,
J r × P = i r × ( F * × F ) .
F = 2 π 1 ( x 2 + y 2 + z 2 ( t i ) 2 ) 3 ( i ( 1 + t 2 x 2 2 i x y + y 2 2 i z + z 2 + 2 t ( i + z ) ) ( 1 + t 2 + x 2 + 2 i x y y 2 2 i z + z 2 + 2 t ( i + z ) ) 2 ( i x + y ) ( i + t + z ) )
α ± ( r , t ) = b ± i ( z t ) + x 2 + y 2 b ± i ( z ± t ) ,
β ± ( r , t ) = x + i y b ± i ( z ± t ) .
( ϕ x ) 2 + ( ϕ y ) 2 + ( ϕ z ) 2 ( ϕ t ) 2 = 0 .
ϕ 1 + = 1 2 π | β + | ( b ± = 1 , z z , t t ) = 1 2 π x + i y i + t + z ,
ϕ 2 + = ( i | α + | ( b ± = 1 , z z , t t ) ) 2 = ( i t + z + x 2 + y 2 i + t + z ) 2 .

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