Abstract

The energy flow lines (field lines of the Poynting vector) for radiation emitted by a dipole are in general curves, rather than straight lines. For a linear dipole the field lines are straight, but when the dipole moment of a source rotates, the field lines wind numerous times around an axis, which is perpendicular to the plane of rotation, before asymptotically approaching a straight line. We consider an elliptical dipole moment, representing the most general state of oscillation, and this includes the linear dipole as a special case. Due to the spiraling near the source, for the case of a rotating dipole moment, the field lines in the far field are displaced with respect to the outward radial direction, and this leads to a shift of the intensity distribution of the radiation in the far field. This shift is shown to be independent of the distance to the source and, although of nanoscale dimension, should be experimentally observable.

© 2009 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. Arnoldus, X. Li, and J. Shu, “Subwavelength displacement of the far-field image of a radiating dipole,” Opt. Lett. 33, 1446-1448 (2008).
    [CrossRef] [PubMed]
  3. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1991), p. 411.
  4. C. T. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977), Vol. 1, p. 838.
  5. H. F. Arnoldus and J. T. Foley, “The dipole vortex,” Opt. Commun. 231, 115-128 (2004).
    [CrossRef]
  6. I. V. Lindell, Methods for Electromagnetic Field Analysis (Oxford U. Press, 1992), Sec. 1.4.
  7. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 469.
  8. J. Shu, X. Li, and H. F. Arnoldus, “Energy flow lines for the radiation emitted by a dipole,” J. Mod. Opt. 55, 2457-2471 (2008).
    [CrossRef]
  9. K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
    [CrossRef]
  10. Y. Ohdaira, T. Inoue, H. Hori, and K. Kitahara, “Local circular polarization observed in surface vortices of optical near-fields,” Opt. Express 16, 2915-2921 (2008).
    [CrossRef] [PubMed]

2008 (3)

2007 (1)

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

2004 (1)

H. F. Arnoldus and J. T. Foley, “The dipole vortex,” Opt. Commun. 231, 115-128 (2004).
[CrossRef]

Arnoldus, H. F.

H. F. Arnoldus, X. Li, and J. Shu, “Subwavelength displacement of the far-field image of a radiating dipole,” Opt. Lett. 33, 1446-1448 (2008).
[CrossRef] [PubMed]

J. Shu, X. Li, and H. F. Arnoldus, “Energy flow lines for the radiation emitted by a dipole,” J. Mod. Opt. 55, 2457-2471 (2008).
[CrossRef]

H. F. Arnoldus and J. T. Foley, “The dipole vortex,” Opt. Commun. 231, 115-128 (2004).
[CrossRef]

Born, M.

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

Choi, S. B.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Choi, W. J.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Cohen-Tannoudji, C. T.

C. T. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977), Vol. 1, p. 838.

Diu, B.

C. T. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977), Vol. 1, p. 838.

Foley, J. T.

H. F. Arnoldus and J. T. Foley, “The dipole vortex,” Opt. Commun. 231, 115-128 (2004).
[CrossRef]

Hori, H.

Inoue, T.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1991), p. 411.

Kihm, H. W.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Kihm, J. E.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Kim, D. S.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Kim, H.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Kim, J.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Kitahara, K.

Laloë, F.

C. T. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977), Vol. 1, p. 838.

Lee, B.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Lee, K. G.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Li, X.

H. F. Arnoldus, X. Li, and J. Shu, “Subwavelength displacement of the far-field image of a radiating dipole,” Opt. Lett. 33, 1446-1448 (2008).
[CrossRef] [PubMed]

J. Shu, X. Li, and H. F. Arnoldus, “Energy flow lines for the radiation emitted by a dipole,” J. Mod. Opt. 55, 2457-2471 (2008).
[CrossRef]

Lienau, C.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Lindell, I. V.

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

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 469.

Ohdaira, Y.

Park, D. J.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Park, Q. H.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Ropers, C.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Shu, J.

H. F. Arnoldus, X. Li, and J. Shu, “Subwavelength displacement of the far-field image of a radiating dipole,” Opt. Lett. 33, 1446-1448 (2008).
[CrossRef] [PubMed]

J. Shu, X. Li, and H. F. Arnoldus, “Energy flow lines for the radiation emitted by a dipole,” J. Mod. Opt. 55, 2457-2471 (2008).
[CrossRef]

Wolf, E.

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 469.

Woo, D. H.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Yoon, Y. C.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

J. Mod. Opt. (1)

J. Shu, X. Li, and H. F. Arnoldus, “Energy flow lines for the radiation emitted by a dipole,” J. Mod. Opt. 55, 2457-2471 (2008).
[CrossRef]

Nat. Photonics (1)

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1, 53-56 (2007).
[CrossRef]

Opt. Commun. (1)

H. F. Arnoldus and J. T. Foley, “The dipole vortex,” Opt. Commun. 231, 115-128 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Other (5)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1991), p. 411.

C. T. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977), Vol. 1, p. 838.

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 469.

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

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Figures (10)

Fig. 1
Fig. 1

Field line of the Poynting vector for a rotating dipole moment in the x y plane. At a large distance the field line approaches the line l. Line m is parallel to line l, but it starts at the origin of coordinates. The rotation of the field line near the source leads to an effective displacement of the field line in the far field, as compared to an optical ray that would emanate from the location of the dipole. We use dimensionless coordinates x ¯ = k o x , y ¯ = k o y , and z ¯ = k o z .

Fig. 2
Fig. 2

Several field lines in the x y plane of the Poynting vector for the case of a counterclockwise rotating circular dipole in the x y plane. A bundle of field lines, as in the figure, determines the intensity distribution on an image plane (line y ¯ = 4 ). The bold field line is approximately perpendicular to the image plane, and runs asymptotically into the observation direction ( θ o , ϕ o ) , which is ( π 2 , π 2 ) in this illustration.

Fig. 3
Fig. 3

Image plane is spanned by the unit vectors e θ o and e ϕ o , and λ and μ are the corresponding Cartesian coordinates in the plane. Field lines of the Poynting vector that cross this plane determine the intensity profile, formed on the plane. The bold field line runs asymptotically in the r ̂ o direction, and crosses the plane at the location given by the displacement vector q d with respect to the origin of the plane. This is the same q d as in Fig. 1. Angle γ is the angle between the observation direction ( θ o , ϕ o ) , represented by r ̂ o , and the angular location of the field point r in the observation plane, as seen from the site of the source.

Fig. 4
Fig. 4

Graph of the intensity distribution in an image plane perpendicular to the y axis, for a dipole moment which oscillates linearly along the y axis. The dimensionless distance between the plane and the dipole is q o = 2 , and the dimensionless radius of the ring is 1.63.

Fig. 5
Fig. 5

Solid curve shows α, the solution of Eq. (28), as a function of θ o , and the dashed curve is the approximation given by Eq. (29).

Fig. 6
Fig. 6

Graph shows the far-field intensity distribution for a rotating dipole with β = 1 for observation along the x y plane. The maximum is located on the μ ¯ axis at μ ¯ p = 2 3 .

Fig. 7
Fig. 7

Shift μ ¯ p of the peak in the intensity distribution and the displacement μ ¯ d of the central field line as a function of the observation angle θ o , both for a circular dipole with β = 1 . For β = 1 both functions change sign.

Fig. 8
Fig. 8

Location of the maxima along the coordinate axes in the λ ¯ μ ¯ plane as a function of β . For β < 2 5 there are two maxima along both axes, and there is a hole in the middle. For β > 2 3 there is a single peak near the origin of coordinates. In the region indicated by the double-headed arrow there is a minimum along the λ ¯ direction and a maximum along the μ ¯ direction near the origin.

Fig. 9
Fig. 9

The shift μ ¯ p for β 0 of either the hole or the peak with respect to the origin of coordinates. In the transition region 2 5 < β < 2 3 , there is neither a hole nor a peak in the intensity distribution.

Fig. 10
Fig. 10

Near-field intensity distribution for a dipole with β = 1 , observed in an image plane perpendicular to the y axis ( θ o = ϕ o = π 2 ) , shows a positive and a negative extremum. This is due to the fact that the field lines of the Poynting vector cross the plane in the outward direction at the negative μ ¯ side, and re-enter the image plane at the positive μ ¯ side. This profile is a result of the numerous rotations of the field lines around the z axis close to the source, as shown in Fig. 2.

Equations (36)

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d ( t ) = d o Re ( ε e i ω t ) ,
E ( r , t ) = Re [ E ( r ) e i ω t ] ,
E ( r ) = d o k o 3 4 π ε o { ε ( ε r ̂ ) r ̂ + [ ε 3 ( ε r ̂ ) r ̂ ] i q ( 1 + i q ) } e i q q ,
B ( r ) = d o k o 3 4 π ε o c ε × r ̂ ( 1 + i q ) e i q q ,
S ( r ) = 1 2 μ o Re [ E ( r ) × B ( r ) * ] ,
S ( r ) = 3 P o 8 π r 2 { [ 1 ( r ̂ ε ) ( r ̂ ε * ) ] r ̂ 2 q ( 1 + 1 q 2 ) Im [ ( r ̂ ε ) ε * ] } ,
P o = c k o 4 d o 2 12 π ε o ,
S ( r ) = 3 P o 8 π r 2 [ 1 ( r ̂ ε ) 2 ] r ̂ .
d P d Ω = r o 2 S ( r ) r ̂ ,
d P d Ω = 3 P o 8 π [ 1 ( r ̂ ε ) ( r ̂ ε * ) ] .
r ̂ = ( e x cos ϕ + e y sin ϕ ) sin θ + e z cos θ .
r = r o + λ e θ o + μ e ϕ o .
I ( r o ; λ , μ ) = S ( r ) r ̂ o ,
q = q o 2 + λ ¯ 2 + μ ¯ 2 ,
r ̂ = 1 q ( q o r ̂ o + λ ¯ e θ o + μ ¯ e ϕ o ) ,
I ( r o ; λ , μ ) = I o ( q o q ) 3 { 1 ( r ̂ ε ) ( r ̂ ε * ) 2 q o ( 1 + 1 q 2 ) Im [ ( r ̂ ε ) ( r ̂ o ε * ) ] } ,
I o = 3 P o 8 π r o 2 .
I ( r o ; λ , μ ) = I o cos 3 γ sin 2 γ ,
ε = 1 β 2 + 1 ( β e x + i e y ) , β real .
e θ o = ( e x cos ϕ o + e y sin ϕ o ) cos θ o e z sin θ o ,
e ϕ o = e x sin ϕ o + e y cos ϕ o ,
I ( r o ; λ , μ ) = I o ( q o q ) 3 [ 1 ( r ̂ ε ) ( r ̂ ε * ) 1 q o q ( 1 + 1 q 2 ) 2 β β 2 + 1 μ ¯ sin θ o ] .
( r ̂ ε ) ( r ̂ ε * ) = 1 q 2 1 β 2 + 1 [ β 2 ( ρ ¯ cos ϕ o μ ¯ sin ϕ o ) 2 + ( ρ ¯ sin ϕ o + μ ¯ cos ϕ o ) 2 ] .
ρ ¯ = q o sin θ o + λ ¯ cos θ o .
I ( r o ; λ , μ ) = I o ( q o q ) 3 [ 1 1 2 q 2 ( ρ ¯ 2 + μ ¯ 2 ) β q o q ( 1 + 1 q 2 ) μ ¯ sin θ o ] .
μ ¯ [ 4 5 2 q 2 ( μ ¯ 2 + ρ ¯ 2 ) ] = β sin θ o q o q ( q o 2 + λ ¯ 2 3 μ ¯ 2 ) ,
3 λ ¯ + 5 λ ¯ 2 q 2 ( μ ¯ 2 + ρ ¯ 2 ) ρ ¯ cos θ o = β 4 λ ¯ μ ¯ q o q sin θ o ,
5 2 α ( sin θ o + α cos θ o ) 2 = ( 1 + α 2 ) [ sin θ o cos θ o + α ( 3 + cos 2 θ o ) ] ,
α sin ( 2 θ o ) 7 sin 2 θ o 8 ,
μ ¯ p = β ( 1 + α 2 ) 3 2 2 sin θ o 8 ( 1 + α 2 ) 5 ( sin θ o + α cos θ o ) 2 ,
μ ¯ d = β 2 sin θ o 2 sin 2 θ o .
I ( r o ; λ , μ ) = I o ( q o q ) 3 { 1 1 q 2 ( β 2 + 1 ) [ β 2 μ ¯ 2 + q o 2 + 2 β μ ¯ q q o ( 1 + 1 q 2 ) ] } .
λ ¯ q o = ± 2 3 β 2 3 ( β 2 + 1 ) , β < 2 3 .
μ ¯ [ 5 β 2 + 3 5 q 2 ( q o 2 + β 2 μ ¯ 2 ) ] = 2 β q o q ( q o 2 3 μ ¯ 2 ) .
μ ¯ q o = ± 2 5 β 2 3 , β < 2 5 .
λ ¯ p = 0 , μ ¯ p = 2 β 2 5 β 2 .

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