Abstract

The electromagnetic field, generated by a source, has four typical components: the far field, the middle field, the near field, and the self-field. This decomposition is studied with the help of the dyadic Green’s function for the electric field and its representation in reciprocal (k) space. The representations in k space involve three universal functions, which we call the T(q) functions. Various representations of these functions are presented, and an interesting sum rule is derived. It is shown that the magnetic field can be split in a similar way, leading to a middle field and a far field only.

© 2001 Optical Society of America

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References

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  1. D. W. Pohl and D. Courjon, eds., Near Field Optics, Vol. 242 of Proceedings of the NATO Advanced Research Workshop on Near Field Optics, Series E, Applied Sciences (Kluwer, Dordrecht, 1993).
  2. M. A. Paesler and P. J. Moyer, Near Field Optics, Theory, Instrumentation, and Applications (Wiley, New York, 1996).
  3. M. Ohtsu, ed., Near-Field Nano/Atom Optics and Technology (Springer, Berlin, 1998).
  4. J. van Kranendonk and J. E. Sipe, “Foundations of the macroscopic electromagnetic theory of dielectric media,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. 15, pp. 246–350.
  5. M. Born and E. Wolf, Principles of Optics, 6th ed., (Pergamon, Oxford, 1980), Appendix 5.
  6. O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
    [CrossRef]
  7. O. Keller, “Electromagnetic propagators in micro- and mesoscopic optics,” in Computational Studies of New Materials, D. A. Jelski and T. F. George, eds. (World Scientific, Singapore, 1999), pp. 375–439.
  8. C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, 1971).
  9. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 141.
  10. O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
    [CrossRef]
  11. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7.
  12. Appendix 3 of Ref. 5.
  13. D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics (Academic, London, 1984).
  14. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons & Atoms (Wiley, New York, 1989).

1999 (1)

1996 (1)

O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
[CrossRef]

Keller, O.

O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
[CrossRef]

O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
[CrossRef]

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

Phys. Rep. (1)

O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
[CrossRef]

Other (12)

O. Keller, “Electromagnetic propagators in micro- and mesoscopic optics,” in Computational Studies of New Materials, D. A. Jelski and T. F. George, eds. (World Scientific, Singapore, 1999), pp. 375–439.

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, 1971).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 141.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7.

Appendix 3 of Ref. 5.

D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics (Academic, London, 1984).

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons & Atoms (Wiley, New York, 1989).

D. W. Pohl and D. Courjon, eds., Near Field Optics, Vol. 242 of Proceedings of the NATO Advanced Research Workshop on Near Field Optics, Series E, Applied Sciences (Kluwer, Dordrecht, 1993).

M. A. Paesler and P. J. Moyer, Near Field Optics, Theory, Instrumentation, and Applications (Wiley, New York, 1996).

M. Ohtsu, ed., Near-Field Nano/Atom Optics and Technology (Springer, Berlin, 1998).

J. van Kranendonk and J. E. Sipe, “Foundations of the macroscopic electromagnetic theory of dielectric media,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. 15, pp. 246–350.

M. Born and E. Wolf, Principles of Optics, 6th ed., (Pergamon, Oxford, 1980), Appendix 5.

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Figures (3)

Fig. 1
Fig. 1

Real and imaginary parts of the function T(q)FF.

Fig. 2
Fig. 2

Real and imaginary parts of the function T(q)MF.

Fig. 3
Fig. 3

Real and imaginary parts of the function T(q)NF.

Equations (111)

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Eˆ(r, ω)=-dtE(r, t)exp(iωt).
E(r, t)=1π Re 0dωEˆ(r, ω)exp(-iωt).
·Eˆ=ρˆ/0,
×Eˆ=iωBˆ,
·Bˆ=0,
×Bˆ=-iωc2 Eˆ+μ0jˆ.
Eˆ(r)=iωμ04π d3rg(r-r)jˆ(r)+iωμ04πk02·d3rg(r-r)jˆ(r),
Bˆ(r)=μ04π×d3rg(r-r)jˆ(r),
g(r)=exp(ik0r)/r,
(2+k02)d3rg(r-r)jˆ(r)=-4πjˆ(r).
·d3rg(r-r)jˆ(r)
=-4π3 jˆ(r)+d3r{·[g(r-r)jˆ(r)]}.
{·[g(r-r)jˆ(r)]}=[g(r-r)]·jˆ(r),
Eˆ(r)=-i30ω jˆ(r)+iωμ04π d3 rd(r-r)·jˆ(r).
d(r)=I+1k02g(r),
d(r)=1+ik0r-1k02r2Ig(r)+-1-3ik0r+3k02r2rˆrˆg(r),
g(r)=-4π3k02δ(r)I+d(r).
Eˆ(r)=iωμ04π d3rg(r-r)·jˆ(r).
k02Eˆ-×(×Eˆ)=-iωμ0jˆ
k02g(r)-×[×g(r)]=-4πδ(r)I,
F(k)=d3rf(r)exp(-ik·r),
f(r)=1(2π)3 d3kF(k)exp(ik·r).
G(k)=d3r exp(-ik·r) exp(ik0r)r.
G(k)=2πik 0dr{exp[i(k0-k)r]-exp[i(k0+k)r]},
G(k)=4πk2-k02-i,
g(r)=-iπr 0dk kk2-k02-i[exp(ikr)-exp(-ikr)].
g(r)=-iπr -dk kk2-k02-i exp(ikr).
Eˆ(k)=iωμ04π Jˆ(k)-1k02 k[k·Jˆ(k)]G(k),
Eˆ(k)=iωμ04πG(k)·Jˆ(k),
G(k)=G(k)I-1k02 kk.
g(r)SF=-4π3k02δ(r)I,
g(r)NF=-1k02r3(I-3rˆrˆ)exp(ik0r),
g(r)MF=ik0r2(I-3rˆrˆ)exp(ik0r),
g(r)FF=1r(I-rˆrˆ)exp(ik0r),
Eˆ(r)=αEˆ(r)αF,
Eˆ(r)αF=iωμ04π d3rg(r-r)αF·jˆ(r), α=S,N,M,or F,
g(r)αFG(k)αF,α=S,N,M,or F,
Eˆ(k)αF=iωμ04πG(k)αF·Jˆ(k),α=S,N,M,or F.
G(k)SF=-4π3k02I,
G(k)NF=-1k02 d3r exp(-ik·r) exp(ik0r)r3(I-3rˆrˆ).
G(k)NF=-4πk02k(I-3kˆkˆ)0dr exp(ik0r)r3 
×3k cos(kr)+r-3k2rsin(kr),
G(k)NF=4πk02(I-3kˆkˆ)T(k0/k)NF,
T(q)NF=-0 dtt3 3 cos t+t-3tsin texp(iqt),
G(k)MF=4πk02(I-3kˆkˆ)T(k0/k)MF,
G(k)FF=(I-kˆkˆ)G(k)+4πk02(I-3kˆkˆ)T(k0/k)FF,
T(q)MF=iq0 dtt2 3 cos t+t-3tsin texp(iqt),
T(q)FF=q20 dtt cos t-sin ttexp(iqt).
T(q)NF=13-iq0 dtt3(t cos t-sin t)exp(iqt),
T(q)NF=13-12q2-iq2(q2-1)0 dtt exp(iqt)sin t.
T(q)NF=13-12q2+14q(q2-1)ln1+q1-q-iπ4 q(q2-1),0q<10,q>1.
T(q)MF=-q ddqT(q)NF,
T(q)MF=32q2-14q(3q2-1)ln1+q1-q+iπ4 q(3q2-1),0q<10,q>1.
T(q)MF=iq0 dtt3(t cos t-sin t)exp(iqt)-q20 dtt2(t cos t-sin t)exp(iqt).
T(q)NF+T(q)MF+T(q)FF=13,
T(q)FF=-q2+12q3 ln1+q1-q-iπ4 2q3,0q<10,q>1.
G(k)NF+G(k)MF+G(k)FF=(I-kˆkˆ)G(k)+4π3k02(I-3kˆkˆ).
αG(k)αF=(I-kˆkˆ)G(k)-4πk02 kˆkˆ,
T(q)NF=13-12q2-14q(q2-1)ln q-1q+1,
T(q)MF=32q2+14q(3q2-1)ln q-1q+1,
T(q)FF=-q2-12q3 ln q-1q+1.
1k02r3(I-3rˆrˆ)[(1-ik0r)exp(ik0r)-1]
4πk02(I-3kˆkˆ)T(k0/k)FF.
Bˆ(r)=μ04π d3rc(r-r)×jˆ(r),
c(r)=g(r).
c(r)=ik0-1rrˆg(r),
Bˆ(r)αF=μ04π d3rc(r-r)αF×jˆ(r),α=M,F,
c(r)MF=-1r rˆg(r),
c(r)FF=ik0rˆg(r).
C(k)=ikG(k)
Bˆ(k)=μ04π C(k)×Jˆ(k),
c(r)αFC(k)αF,α=M,F,
Bˆ(k)αF=μ04π C(k)αF×Jˆ(k),α=M,F.
C(k)αF=d3rc(r)αF exp(-ik·r).
C(k)MF=-4πk02ikT(k0/k)FF,
C(k)FF=ikG(k)+4πk02T(k0/k)FF.
·Bˆ(r)αF=0,
Eˆ(k)FF=iωμ04πG(k){Jˆ(k)-kˆ[kˆ·Jˆ(k)]}+iωμ0k02T(k0/k)FF{Jˆ(k)-3kˆ[kˆ·Jˆ(k)]},
1ω k×Eˆ(k)FF=iμ04π G(k)+4πk02T(k0/k)FFk×Jˆ(k).
×Eˆ(r)FF=iωBˆ(r)FF,
1ω k×Eˆ(k)MF=iμ0k02T(k0/k)MFk×Jˆ(k),
Bˆ(k)MF=-iμ0k02T(k0/k)FFk×Jˆ(k),
ik·Eˆ(k)FF=20ω[k·Jˆ(k)]T(k0/k)FF.
ik·Eˆ(k)FF=20Rˆ(k)T(k0/k)FF.
ik·Eˆ(k)SF=130Rˆ(k),
αik·Eˆ(k)αF=10Rˆ(k),
·Eˆ(r)SF=130ρˆ(r),
12π 0dk k2k2-k02-i×(I+rˆrˆ) 1ikr[exp(ikr)-exp(-ikr)]+(I-3rˆrˆ)1ikr+2(ikr)3[exp(ikr)-exp(-ikr)]-2(ikr)2[exp(ikr)+exp(-ikr)].
12π 0dk k2 exp(ikr)k2-k02-i (I+rˆrˆ) 1ikr
+(I-3rˆrˆ)1ikr-2(ikr)2+2(ikr)3.
(I-rˆrˆ) exp(ik0r)r-1k02r3(I-3rˆrˆ)
×[(1-ik0r)exp(ik0r)-1](I-kˆkˆ)G(k).
2πk02r3(I-3rˆrˆ)I(k0r)FF4πk02(I-3kˆkˆ)T(k0/k)FF,
I(b)FF=0duT(b/u)FFu-3usin u+3 cos u.
I(b)FF=b20 dvv exp(iv)0 duu2 u-3usin u+3 cos ucos(pu)-1pu sin(pu).
0 duu2 u-3usin u+3 cos u
×cos(pu)-1pu sin(pu)
=12πp2,0p<10,p>1.
I(b)FF=π2[(1-ib)exp(ib)-1],
I(b)MF=ib20 dvv2 exp(iv)0 duu2 u-3usin u+3 cos u×pu-3pusin(pu)+3 cos(pu).
0 duu2 u-3usin u+3 cos upu-3pusin(pu)+3 cos(pu)=p0umaxdu sin u sin(pu).
I(b)MF=-b4 0umaxdu sin u lnu+bu-b2+ibπ2 0umaxdu sin u.
0du sin u lnu+bu-b2=40du cos u 1b2-u2,
0du sin u lnu+bu-b2=2π sin b.
I(b)MF=12ibπ[exp(ib)-cos(kmaxr)].
g(r)NF4π3k02(I-3kˆkˆ)
+4πk02(I-3kˆkˆ)[T(k0/k)NF-1/3].
δ(r)l=13δ(r)+14πr3(I-3rˆrˆ),
-1k02r3(I-3rˆrˆ)4π3k02(I-3kˆkˆ),
1k02r3(I-3rˆrˆ)[1-exp(ik0r)]
4πk02(I-3kˆkˆ)[T(k0/k)NF-1/3],

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