Abstract

The energy flow lines (field lines of the Poynting vector) of electric dipole radiation exhibit a vortex structure in the near field when the dipole moment of the source is in circular rotation. The spatial extend of this vortex is smaller than a wavelength and may not be observable by a measurement in the near field. We show that the rotation of the field lines close to the source affects the image of the dipole in the far field, and this opens the possibility for observation of this vortex by a measurement in the far field.

© 2008 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 3.
  2. H. F. Schouten, T. D. Visser, G. Gbur, and D. Lenstra, Phys. Rev. Lett. 93, 173901 (2004).
    [CrossRef] [PubMed]
  3. I. V. Lindell, Methods for Electromagnetic Field Analysis (Oxford U. Press, 1992), Section 1.4.
  4. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), p. 411.
  5. H. F. Arnoldus, and J. T. Foley, Opt. Commun. 231, 115 (2004).
    [CrossRef]
  6. H. F. Arnoldus, X. Li, and J. Shu, Opt. Lett. 33, 1446 (2008).
    [CrossRef] [PubMed]
  7. J. Shu, X. Li, and H. F. Arnoldus, J. Mod. Opt. 55, 2457 (2008).
    [CrossRef]

2008 (2)

J. Shu, X. Li, and H. F. Arnoldus, J. Mod. Opt. 55, 2457 (2008).
[CrossRef]

H. F. Arnoldus, X. Li, and J. Shu, Opt. Lett. 33, 1446 (2008).
[CrossRef] [PubMed]

2004 (2)

H. F. Schouten, T. D. Visser, G. Gbur, and D. Lenstra, Phys. Rev. Lett. 93, 173901 (2004).
[CrossRef] [PubMed]

H. F. Arnoldus, and J. T. Foley, Opt. Commun. 231, 115 (2004).
[CrossRef]

Arnoldus, H. F.

H. F. Arnoldus, X. Li, and J. Shu, Opt. Lett. 33, 1446 (2008).
[CrossRef] [PubMed]

J. Shu, X. Li, and H. F. Arnoldus, J. Mod. Opt. 55, 2457 (2008).
[CrossRef]

H. F. Arnoldus, and J. T. Foley, Opt. Commun. 231, 115 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 3.

Foley, J. T.

H. F. Arnoldus, and J. T. Foley, Opt. Commun. 231, 115 (2004).
[CrossRef]

Gbur, G.

H. F. Schouten, T. D. Visser, G. Gbur, and D. Lenstra, Phys. Rev. Lett. 93, 173901 (2004).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), p. 411.

Lenstra, D.

H. F. Schouten, T. D. Visser, G. Gbur, and D. Lenstra, Phys. Rev. Lett. 93, 173901 (2004).
[CrossRef] [PubMed]

Li, X.

J. Shu, X. Li, and H. F. Arnoldus, J. Mod. Opt. 55, 2457 (2008).
[CrossRef]

H. F. Arnoldus, X. Li, and J. Shu, Opt. Lett. 33, 1446 (2008).
[CrossRef] [PubMed]

Lindell, I. V.

I. V. Lindell, Methods for Electromagnetic Field Analysis (Oxford U. Press, 1992), Section 1.4.

Schouten, H. F.

H. F. Schouten, T. D. Visser, G. Gbur, and D. Lenstra, Phys. Rev. Lett. 93, 173901 (2004).
[CrossRef] [PubMed]

Shu, J.

H. F. Arnoldus, X. Li, and J. Shu, Opt. Lett. 33, 1446 (2008).
[CrossRef] [PubMed]

J. Shu, X. Li, and H. F. Arnoldus, J. Mod. Opt. 55, 2457 (2008).
[CrossRef]

Visser, T. D.

H. F. Schouten, T. D. Visser, G. Gbur, and D. Lenstra, Phys. Rev. Lett. 93, 173901 (2004).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 3.

J. Mod. Opt. (1)

J. Shu, X. Li, and H. F. Arnoldus, J. Mod. Opt. 55, 2457 (2008).
[CrossRef]

Opt. Commun. (1)

H. F. Arnoldus, and J. T. Foley, Opt. Commun. 231, 115 (2004).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

H. F. Schouten, T. D. Visser, G. Gbur, and D. Lenstra, Phys. Rev. Lett. 93, 173901 (2004).
[CrossRef] [PubMed]

Other (3)

I. V. Lindell, Methods for Electromagnetic Field Analysis (Oxford U. Press, 1992), Section 1.4.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), p. 411.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 3.

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Figures (5)

Fig. 1
Fig. 1

Shown is the field line of the Poynting vector for θ o = π 4 , ϕ o = π , and τ = 1 . The field line lies on the cone θ o = π 4 . Near the origin, the field line swirls around the z axis numerous times, and far away from the origin it approaches the line l . The line m is parallel to the line l and intersects the x y plane at the origin. When viewed from the far field, the field line appears to be displaced over vector q d as compared to a ray that would come from the location of the dipole. This displacement gives a shift of the image of the dipole.

Fig. 2
Fig. 2

Image plane in the observation direction ( θ o , ϕ o ) is perpendicular to the radial unit vector r ̂ o . The Cartesian coordinates ( λ , μ ) in this plane are defined as shown. The field line of the Poynting vector that runs into this direction intersects the image plane at the point indicated by the displacement vector q d , which is along the μ axis (and is in the direction shown for τ = 1 ).

Fig. 3
Fig. 3

Shown is the intensity distribution I ( r ) in the λ ¯ μ ¯ plane for θ o = π 2 and τ = 1 . The peak of the distribution is at the negative side of the μ ¯ axis, and this is a result of the spiraling of the field lines in the counterclockwise direction near the position of the dipole.

Fig. 4
Fig. 4

Shown is the displacement μ ¯ d of the field line in the observation direction ( θ o , ϕ o ) and the location of the peak μ ¯ p of the intensity distribution in the same direction, both as a function of the polar angle of observation θ o and for τ = 1 . For τ = 1 , both curves change sign but are otherwise the same.

Fig. 5
Fig. 5

Graph shows the intensity I ( r ) as a function of μ ¯ for λ ¯ = 0 . The value of the observation angle is θ o = π 2 . For τ = 1 and τ = 1 the maxima are at μ ¯ p = 2 3 and μ ¯ p = 2 3 , respectively.

Equations (5)

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d ( t ) = d o Re ( ε e i ω t )
ε = 1 2 ( τ e x + i e y ) ,
S ( r ) = 3 P o 8 π r 2 [ ( 1 1 2 sin 2 θ ) r ̂ + τ q ( 1 + 1 q 2 ) e ϕ sin θ ] ,
μ ¯ d = 2 τ sin θ o 1 + cos 2 θ o .
μ ¯ p = 2 τ sin θ o 3 + 5 cos 2 θ o .

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