Abstract

The standard dyadic Green function description of the electromagnetic field generated by an electric point dipole is modified (and corrected) so that a rigorous classical theory for the attached and radiated parts of the near field appears. The present propagator formalism follows from analysis of the transverse and longitudinal dipole electrodynamics. Elimination of both the transverse and the longitudinal self-fields leads to a description of the radiated dipole field that enables one to obtain the associated energy flux in the near- and mid-field zones also and that is correctly retarded (with the vacuum speed of light) everywhere in space. The related retarded transverse propagator exists in the time (space) domain, whereas the standard propagator exists only in the frequency (space) domain. As a forerunner to an analysis of the Weyl expansions for the standard, longitudinal self-field and retarded transverse propagators, the plane-wave mode expansions of these propagators are investigated, and contour integrations are specified in such a manner that the rigorous Green function description is regained. It is found that, in order for the retarded transverse propagator description to be consistent in the near-field zone, the Weyl expansion for this propagator has to contain evanescent components not only for wave numbers larger than the vacuum wave number but in the entire angular spectrum. The present theory may influence our view of optical near-field phenomena and (classical) photon tunneling because in both of these fields a proper identification of attached and radiated fields seems needed.

© 1999 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  52. J. R. Ackerhalt, P. Knight, and J. H. Eberly, Phys. Rev. Lett. 30, 456 (1973).
    [CrossRef]
  53. K. Wodkiewicz and J. H. Eberly, Ann. Phys. (Leipzig) 101, 574 (1976).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  57. R. G. Woolley, Mol. Phys. 22, 1013 (1971).
    [CrossRef]

1998 (4)

O. Keller, Phys. Rev. A 58, 3407 (1998).
[CrossRef]

T. Andersen and O. Keller, Phys. Rev. B 57, 14793 (1998).
[CrossRef]

T. Andersen and O. Keller, Phys. Scr. 58, 132 (1998).
[CrossRef]

E. Wolf and J. T. Foley, Opt. Lett. 23, 16 (1998).
[CrossRef]

1997 (2)

M. Xiao, J. Mod. Opt. 44, 327 (1997).
[CrossRef]

M. Xiao, J. Mod. Opt. 44, 1609 (1997).
[CrossRef]

1996 (5)

M. Xiao, Opt. Commun. 132, 403 (1996).
[CrossRef]

M. Xiao, Chem. Phys. Lett. 258, 363 (1996).
[CrossRef]

O. Keller, J. Nonlinear Opt. Phys. Mater. 5, 109 (1996).
[CrossRef]

O. Keller, Phys. Rep. 268, 85 (1996).
[CrossRef]

C. Girard and A. Dereux, Progr. Phys. 59, 657 (1996), and references herein.
[CrossRef]

1995 (3)

H. F. Arnoldus and T. F. George, Phys. Rev. A 51, 4250 (1995).
[CrossRef] [PubMed]

G. S. Agarwal and S. D. Gupta, Opt. Commun. 119, 591 (1995).
[CrossRef]

E. Goldstein, K. Plättner, and P. Meystre, Quantum Semiclassic. Opt. 7, 743 (1995).
[CrossRef]

1994 (1)

1993 (1)

O. Keller, M. Xiao, and S. I. Bozhevolnyi, Surf. Sci. 280, 217 (1993).
[CrossRef]

1991 (2)

C. Girard and X. Boujou, J. Chem. Phys. 95, 2056 (1991).
[CrossRef]

J. Van Bladel, IEEE Trans. Antennas Propag. 33, 69 (1991).
[CrossRef]

1990 (2)

1988 (1)

O. Keller and P. Sønderkær, in Optical Testing and Metrology II, C. P. Grover, ed., Proc. SPIE 954, 344 (1988).
[CrossRef]

1986 (1)

O. Keller, Phys. Rev. B 34, 3883 (1986).
[CrossRef]

1985 (1)

1983 (2)

1982 (4)

1980 (1)

1977 (1)

H. J. Kimble and L. Mandel, Phys. Rev. A 15, 689 (1977).
[CrossRef]

1976 (1)

K. Wodkiewicz and J. H. Eberly, Ann. Phys. (Leipzig) 101, 574 (1976).
[CrossRef]

1973 (1)

J. R. Ackerhalt, P. Knight, and J. H. Eberly, Phys. Rev. Lett. 30, 456 (1973).
[CrossRef]

1971 (1)

R. G. Woolley, Mol. Phys. 22, 1013 (1971).
[CrossRef]

1967 (1)

O. L. Brill and B. Goodman, Am. J. Phys. 35, 382 (1967).
[CrossRef]

1959 (1)

E. A. Power and S. Zinau, Philos. Trans. R. Soc. London, Ser. A 251, 427 (1959).
[CrossRef]

1919 (1)

H. Weyl, Ann. Phys. (Leipzig) 60, 481 (1919).
[CrossRef]

1867 (1)

L. V. Lorenz, Philos. Mag. 34, 287 (1867).

Ackerhalt, J. R.

J. R. Ackerhalt, P. Knight, and J. H. Eberly, Phys. Rev. Lett. 30, 456 (1973).
[CrossRef]

Agarwal, G. S.

Andersen, T.

T. Andersen and O. Keller, Phys. Rev. B 57, 14793 (1998).
[CrossRef]

T. Andersen and O. Keller, Phys. Scr. 58, 132 (1998).
[CrossRef]

Arnoldus, H. F.

H. F. Arnoldus and T. F. George, Phys. Rev. A 51, 4250 (1995).
[CrossRef] [PubMed]

Boujou, X.

C. Girard and X. Boujou, J. Chem. Phys. 95, 2056 (1991).
[CrossRef]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, O. Keller, and I. I. Smolyaninov, Opt. Lett. 19, 1 (1994).
[CrossRef]

O. Keller, M. Xiao, and S. I. Bozhevolnyi, Surf. Sci. 280, 217 (1993).
[CrossRef]

Brill, O. L.

O. L. Brill and B. Goodman, Am. J. Phys. 35, 382 (1967).
[CrossRef]

Courjon, D.

Dereux, A.

C. Girard and A. Dereux, Progr. Phys. 59, 657 (1996), and references herein.
[CrossRef]

Drummond, P. D.

Eberly, J. H.

K. Wodkiewicz and J. H. Eberly, Ann. Phys. (Leipzig) 101, 574 (1976).
[CrossRef]

J. R. Ackerhalt, P. Knight, and J. H. Eberly, Phys. Rev. Lett. 30, 456 (1973).
[CrossRef]

Foley, J. T.

Friberg, A. T.

George, T. F.

H. F. Arnoldus and T. F. George, Phys. Rev. A 51, 4250 (1995).
[CrossRef] [PubMed]

Girard, C.

C. Girard and A. Dereux, Progr. Phys. 59, 657 (1996), and references herein.
[CrossRef]

C. Girard and X. Boujou, J. Chem. Phys. 95, 2056 (1991).
[CrossRef]

C. Girard and D. Courjon, Phys. Rev. B 42, 9340 (1990).
[CrossRef]

B. Labani, C. Girard, D. Courjon, and D. van Labeke, J. Opt. Soc. Am. B 7, 936 (1990).
[CrossRef]

Goldstein, E.

E. Goldstein, K. Plättner, and P. Meystre, Quantum Semiclassic. Opt. 7, 743 (1995).
[CrossRef]

Goodman, B.

O. L. Brill and B. Goodman, Am. J. Phys. 35, 382 (1967).
[CrossRef]

Gupta, S. D.

G. S. Agarwal and S. D. Gupta, Opt. Commun. 119, 591 (1995).
[CrossRef]

Keller, O.

T. Andersen and O. Keller, Phys. Scr. 58, 132 (1998).
[CrossRef]

O. Keller, Phys. Rev. A 58, 3407 (1998).
[CrossRef]

T. Andersen and O. Keller, Phys. Rev. B 57, 14793 (1998).
[CrossRef]

O. Keller, Phys. Rep. 268, 85 (1996).
[CrossRef]

O. Keller, J. Nonlinear Opt. Phys. Mater. 5, 109 (1996).
[CrossRef]

S. I. Bozhevolnyi, O. Keller, and I. I. Smolyaninov, Opt. Lett. 19, 1 (1994).
[CrossRef]

O. Keller, M. Xiao, and S. I. Bozhevolnyi, Surf. Sci. 280, 217 (1993).
[CrossRef]

O. Keller and P. Sønderkær, in Optical Testing and Metrology II, C. P. Grover, ed., Proc. SPIE 954, 344 (1988).
[CrossRef]

O. Keller, Phys. Rev. B 34, 3883 (1986).
[CrossRef]

Kimble, H. J.

H. J. Kimble and L. Mandel, Phys. Rev. A 15, 689 (1977).
[CrossRef]

Knight, P.

J. R. Ackerhalt, P. Knight, and J. H. Eberly, Phys. Rev. Lett. 30, 456 (1973).
[CrossRef]

Labani, B.

Lorenz, L. V.

L. V. Lorenz, Philos. Mag. 34, 287 (1867).

Mandel, L.

H. J. Kimble and L. Mandel, Phys. Rev. A 15, 689 (1977).
[CrossRef]

Meystre, P.

E. Goldstein, K. Plättner, and P. Meystre, Quantum Semiclassic. Opt. 7, 743 (1995).
[CrossRef]

Nieto-Vesperinas, M.

Plättner, K.

E. Goldstein, K. Plättner, and P. Meystre, Quantum Semiclassic. Opt. 7, 743 (1995).
[CrossRef]

Power, E. A.

E. A. Power and S. Zinau, Philos. Trans. R. Soc. London, Ser. A 251, 427 (1959).
[CrossRef]

Smolyaninov, I. I.

Sønderkær, P.

O. Keller and P. Sønderkær, in Optical Testing and Metrology II, C. P. Grover, ed., Proc. SPIE 954, 344 (1988).
[CrossRef]

Van Bladel, J.

J. Van Bladel, IEEE Trans. Antennas Propag. 33, 69 (1991).
[CrossRef]

van Labeke, D.

Weyl, H.

H. Weyl, Ann. Phys. (Leipzig) 60, 481 (1919).
[CrossRef]

Wodkiewicz, K.

K. Wodkiewicz and J. H. Eberly, Ann. Phys. (Leipzig) 101, 574 (1976).
[CrossRef]

Wolf, E.

Woolley, R. G.

R. G. Woolley, Mol. Phys. 22, 1013 (1971).
[CrossRef]

Xiao, M.

M. Xiao, J. Mod. Opt. 44, 1609 (1997).
[CrossRef]

M. Xiao, J. Mod. Opt. 44, 327 (1997).
[CrossRef]

M. Xiao, Chem. Phys. Lett. 258, 363 (1996).
[CrossRef]

M. Xiao, Opt. Commun. 132, 403 (1996).
[CrossRef]

O. Keller, M. Xiao, and S. I. Bozhevolnyi, Surf. Sci. 280, 217 (1993).
[CrossRef]

Zinau, S.

E. A. Power and S. Zinau, Philos. Trans. R. Soc. London, Ser. A 251, 427 (1959).
[CrossRef]

Am. J. Phys. (1)

O. L. Brill and B. Goodman, Am. J. Phys. 35, 382 (1967).
[CrossRef]

Ann. Phys. (Leipzig) (2)

K. Wodkiewicz and J. H. Eberly, Ann. Phys. (Leipzig) 101, 574 (1976).
[CrossRef]

H. Weyl, Ann. Phys. (Leipzig) 60, 481 (1919).
[CrossRef]

Chem. Phys. Lett. (1)

M. Xiao, Chem. Phys. Lett. 258, 363 (1996).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. Van Bladel, IEEE Trans. Antennas Propag. 33, 69 (1991).
[CrossRef]

J. Chem. Phys. (1)

C. Girard and X. Boujou, J. Chem. Phys. 95, 2056 (1991).
[CrossRef]

J. Mod. Opt. (2)

M. Xiao, J. Mod. Opt. 44, 327 (1997).
[CrossRef]

M. Xiao, J. Mod. Opt. 44, 1609 (1997).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

O. Keller, J. Nonlinear Opt. Phys. Mater. 5, 109 (1996).
[CrossRef]

J. Opt. Soc. Am. (5)

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

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

Mol. Phys. (1)

R. G. Woolley, Mol. Phys. 22, 1013 (1971).
[CrossRef]

Opt. Commun. (4)

G. S. Agarwal, A. T. Friberg, and E. Wolf, Opt. Commun. 43, 446 (1982).
[CrossRef]

G. S. Agarwal, Opt. Commun. 42, 205 (1982).
[CrossRef]

G. S. Agarwal and S. D. Gupta, Opt. Commun. 119, 591 (1995).
[CrossRef]

M. Xiao, Opt. Commun. 132, 403 (1996).
[CrossRef]

Opt. Lett. (2)

Philos. Mag. (1)

L. V. Lorenz, Philos. Mag. 34, 287 (1867).

Philos. Trans. R. Soc. London, Ser. A (1)

E. A. Power and S. Zinau, Philos. Trans. R. Soc. London, Ser. A 251, 427 (1959).
[CrossRef]

Phys. Rep. (1)

O. Keller, Phys. Rep. 268, 85 (1996).
[CrossRef]

Phys. Rev. A (3)

H. F. Arnoldus and T. F. George, Phys. Rev. A 51, 4250 (1995).
[CrossRef] [PubMed]

O. Keller, Phys. Rev. A 58, 3407 (1998).
[CrossRef]

H. J. Kimble and L. Mandel, Phys. Rev. A 15, 689 (1977).
[CrossRef]

Phys. Rev. B (3)

O. Keller, Phys. Rev. B 34, 3883 (1986).
[CrossRef]

T. Andersen and O. Keller, Phys. Rev. B 57, 14793 (1998).
[CrossRef]

C. Girard and D. Courjon, Phys. Rev. B 42, 9340 (1990).
[CrossRef]

Phys. Rev. Lett. (1)

J. R. Ackerhalt, P. Knight, and J. H. Eberly, Phys. Rev. Lett. 30, 456 (1973).
[CrossRef]

Phys. Scr. (1)

T. Andersen and O. Keller, Phys. Scr. 58, 132 (1998).
[CrossRef]

Proc. SPIE (1)

O. Keller and P. Sønderkær, in Optical Testing and Metrology II, C. P. Grover, ed., Proc. SPIE 954, 344 (1988).
[CrossRef]

Progr. Phys. (1)

C. Girard and A. Dereux, Progr. Phys. 59, 657 (1996), and references herein.
[CrossRef]

Quantum Semiclassic. Opt. (1)

E. Goldstein, K. Plättner, and P. Meystre, Quantum Semiclassic. Opt. 7, 743 (1995).
[CrossRef]

Surf. Sci. (1)

O. Keller, M. Xiao, and S. I. Bozhevolnyi, Surf. Sci. 280, 217 (1993).
[CrossRef]

Other (19)

M. Nieto-Vesperinas and N. Garcia, eds., Optics at the Nanometer Scale (Kluwer, Dordrecht, The Netherlands, 1996), and references herein.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970).

E. Wolf, in Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, 1973), p. 339.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, New York, 1994).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

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

O. Keller, in Notions and Perspectives of Nonlinear Optics, O. Keller, ed. (World Scientific, London, 1996), p. 140.

A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, Oxford, 1966).

R. Penrose and W. Rindler, Spinors and Space-Time (Cambridge, 1984), Vol. I.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles, Vol. 1 of Advanced Series of Applied Physics (World Scientific, Singapore, 1988), and references herein.

O. Keller, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1997), Vol. XXXVII, p. 257, and references herein.

O. Keller, presented at the Seminar on Problems of Quantum Optics, Dubna, USSR, September 30–October 4, 1991.

S. R. de Groot and L. G. Suttorp, Foundations of Electrodynamics (North-Holland, Amsterdam, 1972).

J. Van Kranendonk and J. E. Sipe, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. XV, p. 245.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

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

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Figures (2)

Fig. 1
Fig. 1

Contour integrations in a complex q plane along the specified contours are used to determine a, the transverse propagator D0T; b, the longitudinal self-field propagator gL; and c, the standard propagator D0 from their respective integral representations. Filled circles, positions of poles, all of first order. The contour chosen for the D0T integration consists as shown of two straight-line sections running parallel to the real axis at distances infinitesimally above and below this axis and a semicircle in the upper half plane. When the radius of the semicircle tends toward infinity the contribution to the integral from the semicircle vanishes. With the pole at q=0 placed inside the contour one ensures that only retarded (with c0) interactions are included. The pole at q=q0 (=ω/c0) relates to the outward propagating dipole field. The noncausal ingoing contribution is eliminated when the pole at q=-q0 is located outside the contour. The integral representation for gL has a pole only at q=0, and the specified contour runs around this pole in the manner shown. The D0 propagator has no pole at q=0 in its integral representation, and this leads to a dipole-attached contribution in the interaction described by the standard propagator. The retarded outgoing (pole at q=q0) and ingoing (pole at q=-q0) fields are included and excluded, respectively, in D0 when the specified contour for the integration is employed.

Fig. 2
Fig. 2

Dyadic mode amplitudes belonging to the Weyl expansions of the longitudinal self-field (gL) and standard (D0) propagators determined from their respective integral representations by specified contour integrations in the complex q plane. a, Contours employed for gL; b, c, those for D0. In all cases the contours consist of expanding semicircles plus straight-line sections located just above and below the real q axis; cf. the caption to Fig. 1. The gL mode amplitude has first-order poles in its integral representation on the imaginary axis at q=±iq, and the specified contours (solid and dashed curves) enclose either the upper (for sgn Z=+1) or lower (for sgn Z=-1) pole. The D0-contour integrations specified for an evanescent [q>q0(=ω/c0)] mode are shown in b. The (first-order) poles are located on the imaginary axis at q=±iα0 [=i(q2-q02)1/2]; solid and dashed curves, contours specified for sgn Z=+1 and sgn Z=-1, respectively. For a propagating (qq0) mode the poles are located on the real axis at q=±q0 [=(q02-q2)1/2], and the D0 integrations run along the solid (for sgn Z=+1) and dashed (for sgn Z=-1) contours. Including only one of the poles inside the contour (as shown) retains outgoing (causal) fields from the source in the propagator description but eliminates ingoing (noncausal) fields.

Tables (1)

Tables Icon

Table 1 Schematic Exposition Indicating Whether Propagating and Evanescent Modes Are Present in the Weyl Expansions of the Standard (D0), Longitudinal Self-Field (gL), and Retarded Transverse (D0T) Propagators in the Two Angular Spectral Regions 0<qq0 (=ω/c0) and q0<q<a

Equations (79)

Equations on this page are rendered with MathJax. Learn more.

J(r, t)=J(r, t)d3rδ(r-r0)-J(r, t)(r-r0)d3r·δ(r-r0)+ ,
J(r, t)JED(r, t)=J0(t)δ(r-r0),
J0(t)=J(r, t)d3r
JED(r, t)=PED(r, t)t,
JED(r; ω)=J0(ω)δ(r-r0),
PED(r; ω)=P0(ω)δ(r-r0),
×E(r, t)=-B(r, t)t,
×B(r, t)=μ0J(r, t)+1c02Et
2-1c022t2E(r, t)-·E(r, t)=μ0J(r, t)t
(2+q02)E(r; ω)-·E(r; ω)=-iμ0ωJ(r; ω),
E(r; ω)=-iμ0ωG0(r-r; ω)·J(r; ω)d3r,
E(r; ω)=-iμ0ωD0(r-r0; ω)·J0(ω),rr0
D0(R; ω)=q04πi1iq0R(U-eReR)-1(iq0R)2-1(iq0R)3(U-3eReR)×exp(iq0R),R0,
D0(R; ω)=-(U+q0-2)d0(R; ω),
d0(R; ω)=exp(iq0R)4πR
D0F(R; ω)=exp(iq0R)4πR(eReR-U),
D0F(R; τ)=12π-D0F(R; ω)exp(-iωτ)dω=14πRδRc0-τ(eReR-U).
·E(r, t)=10ρ(r, t)
ρ(r; ω)=(iω)-1J0(ω)·δ(r-r0)
E(r; ω)=ET(r; ω)+EL(r; ω),
J(r; ω)=JT(r; ω)+JL(r; ω),
(2+q02)ET(r; ω)=-iμ0ωJT(r; ω),
i0ωEL(r; ω)=JL(r; ω).
Uδ(r-r0)=δT(r-r0)+δL(r-r0),
δT(R)=23δ(R)U-14πR3(U-3eReR),
δL(R)=13δ(R)U+14πR3(U-3eReR)
JLED(r; ω)=δL(r-r0)·J0(ω)
EL(r; ω)=-iμ0ωgL(r-r0; ω)·J0(ω),
gL(r-r0; ω)=1q02δL(r-r0)
gL(R; ω)=14πq02R3(U-3eReR),R0.
ET(r; ω)=-iμ0ωD0T(r-r0; ω)·J0(ω),rr0,
D0T(R; ω)=D0(R; ω)-gL(R; ω),R0
D0T(R; ω)=q04πiexp(iq0R)iq0R(U-eReR)-exp(iq0R)(iq0R)2-exp(iq0R)-1(iq0R)3×(U-3eReR)
D0T(R, τ)=-14πR(U-eReR)δRc0-τ+c02τ4πR3(U-3eReR)Θ(τ)ΘRc0-τ,
ET(r; ω)=-iμ0ωd0(r-r; ω)·JT(r; ω)d3r,
d0(r-r; ω)=-d0(r-r; ω)U,
d0(R, τ)=-14πRδRc0-τU
JTED(r; ω)=δT(r-r0)·J0(ω),
D0T(r-r0)=d0(r-r; ω)·δT(r-r0)d3r,
rr0.
JT(r; ω)=δT(r-r)·J(r; ω)d3r
ET(r; ω)=-iμ0ωG0T(r-r; ω)·J(r; ω)d3r,
G0T(R; ω)=gT(R; ω)+D0T(R; ω).
gT(R; ω)=13q02δT(R),
ETSF(r; ω)=-iμ0ωgT(r-r0; ω)·J0(ω).
ESF(r; ω)=-iμ0ω[gL(r-r0; ω)+gT(r-r0; ω)]·J0(ω).
ESF(r; ω)=1i0ω59δ(r-r0)U+16πR3(U-3eReR)·J0(ω).
T(R; ω)=1(2π)3-T(q, ω)exp(iq·R)d3q,R0.
D0T(q, ω)=U-eqeqq02-q2,
gL(q, ω)=eqeqq02,
D0(q, ω)=U-eqeqq02-q2+eqeqq02,
D0(q, ω)=U-(q/q0)2eqeqq02-q2,
1R=12π2-exp(iq·R)q2d3q,
gL(R; ω)=-1q0214πR,
D0T(R; ω)=q0-2d0(R; 0)-(U+q0-2)d0(R; ω).
T(R; ω)=1(2π)2-T(Z; q, ω)exp(iq·R)d2q,
T(Z; q, ω)=12π-T(q, ω)exp(iqZ)dq
gL(Z; q, ω)=12πq02-1q2(q+qez)×(q+qez)exp(iqZ)dq,
gL(Z; q, ω)=q2q02exp(-q|Z|)[eqeq-ezez+i(eqez+ezeq)sgn Z],z0,
D0(Z; q, ω)=12πq02-q02U-qqq02-q2exp(iqZ)dq,
D0(Z; q, ω)=exp(iκ0|Z|)2iκ0q02[(κ0)2eqeq+q2ezez-q02ez×eqeq×ez-κ0q(eqez+ezeq)sgn Z],Z0.
D0T(Z; q, ω)=12π-U-eqeqq02-q2exp(iqZ)dq,
D0T(Z; q, ω)=D0(Z; q, ω)-gL(Z; q, ω),
gT(R; ω)=-13gL(R; ω),R0,
gT(Z; q, ω)=-gL(Z; g, ω),Z0,
gL(R, Z; ω)=18π2q02002πq2 exp(-q|Z|)exp(iqR cos α)×cos2 α0i cos α sgn Z0sin2 α0i cos α sgn Z0-1dαdq,
gL(R, Z; ω)=14πq020q2 exp(-q|Z|)×½[J0(qR)-J2(qR)]0-J1(qR)sgn Z0½[J0(qR)+J2(qR)]0-J1(qR)sgn Z0-J0(qR)dq
D0(R, Z; ω)=18π2q02002πqiκ0exp(iκ0|Z|)exp(iqR cos α)×(κ0)2 cos2 α+q02 sin2 α0-κ0q cos α sgn Z0(κ0)2 sin2 α+q02 cos2 α0-κ0q cos α sgn Z0q2dαdq,
D0(R, Z; ω)=14πq020qiκ0exp(iκ0|Z|)×q02J0(qR)-q22[J0(qR)-J2(qR)]0-iκ0qJ1(qR)sgn Z0q02J0(qR)-q22[J0(qR)+J2(qR)]0-iκ0qJ1(qR)sgn Z0q2J0(qR)dq.
D0(R, Z; ω)=0q0D0(q, κ0=q0)dq+q0D0(q, κ0=iα0)dq.
gL(R, Z; ω)=0GL(q)dq
D0T(R, Z; ω)=0q0D0(q, κ0=q0)dq-0q0 GL(q)dq+q0[D0(q, κ0=iα0)-GL(q)]dq.
GL(q)=limqD0(q, κ0).
limc0D0T(R, Z; ω)=0.
EˆT(r; ω)=-iμ0ωd0(r-r; ω)·JˆT(r; ω)d3r,
EˆT(r; ω)=13i0ωJˆT(r; ω)-iμ0ωD0T(r-r; ω)·Jˆ(r; ω)d3r
EˆT(z, q, ω)=-iμ0ωd0(z-z; q, ω)·JˆT(z; q, ω)dz,
d0(z-z; q, ω)=12iκ0exp(iκ0|z-z|)U
EˆT(z; q, ω)=-iμ0ωD0T(z-z; q, ω)·Jˆ(z; q, ω)dz,

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