Abstract

The time-dependent inverse source problem with far-field data is investigated within a limited-view Radon inversion framework, analogous to that of a limited-view computed tomography reconstruction problem. We investigate the domains in the Radon and Fourier spaces within which data are available for the reconstruction of the space–time structure of the source. Using a linear inversion formalism we derive a filtered backprojectionlike procedure to reconstruct the minimum-energy source consistent with prescribed far-field data. The source inversion technique developed in the paper is illustrated with a numerical example. The paper also contains a new description of nonradiating sources in the time domain.

© 1999 Optical Society of America

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References

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  1. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  2. N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
    [CrossRef]
  3. A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
    [CrossRef]
  4. A. J. Devaney, R. P. Porter, “Holography and the inverse source problem. Part II: inhomogeneous media,” J. Opt. Soc. Am. A 2, 2006–2011 (1985).
    [CrossRef]
  5. E. Heyman, A. J. Devaney, “Time-dependent multipoles and their application for radiation from volume source distributions,” J. Math. Phys. 37, 682–692 (1996).
    [CrossRef]
  6. P. R. Smith, T. M. Peters, R. H. T. Bates, “Image reconstruction from finite number of projections,” J. Phys. A Math. Nucl. Gen. 6, 361–382 (1973).
    [CrossRef]
  7. A. K. Louis, “Incomplete data problems in x-ray computerized tomography. I. Singular value decomposition of the limited angle transform,” Numer. Math. 48, 251–262 (1986).
    [CrossRef]
  8. A. D. Yaghjian, T. B. Hansen, “Time-domain far fields,” J. Appl. Phys. 79, 2822–2830 (1996).
    [CrossRef]
  9. F. G. Friedlander, “An inverse problem for radiation fields,” Proc. London Math. Soc. 3, 551–576 (1973).
    [CrossRef]
  10. M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. I: general formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
    [CrossRef]
  11. I. J. LaHaie, “The inverse source problem for three-dimensional partially coherent sources and fields,” J. Opt. Soc. Am. A 2, 35–45 (1985).
    [CrossRef]
  12. H. E. Moses, “Solution of Maxwell’s equations in terms of a spinor notation: the direct and inverse problem,” Phys. Rev. 113, 1670–1679 (1959).
    [CrossRef]
  13. H. E. Moses, “The time-dependent inverse source problem for the acoustic and electromagnetic equations in the one- and three-dimensional cases,” J. Math. Phys. 25, 1905–1923 (1984).
    [CrossRef]
  14. R. W. Deming, A. J. Devaney, “A filtered backpropagation algorithm for GPR,” J. Env. Eng. Geo. 0, 113–123 (1996).
    [CrossRef]
  15. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).
  16. D. N. Ghosh Roy, Methods of Inverse Problems in Physics, 2nd ed. (CRC Press, Boca Raton, Fla., 1991).
  17. M. Bertero, “Linear inverse and ill-posed problems,” in P. W. Hawkes, ed., Advances in Electronics and Electron Physics (Academic, New York, 1989), Vol. 75, pp. 1–120.
  18. R. P. Porter, A. J. Devaney, “Generalized holography and computational solutions to inverse source problems,” J. Opt. Soc. Am. 72, 1707–1713 (1982).
    [CrossRef]
  19. E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate,” IEEE Trans. Antennas Propag. 45, 1098–1107 (1997).
    [CrossRef]
  20. E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate: Part II—square pulse excitation,” IEEE Trans. Antennas Propag. 46, 243–250 (1998).
    [CrossRef]
  21. K. T. Ladas, A. J. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
    [CrossRef]
  22. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985).
  23. A. K. Louis, “Ghosts in tomography—The null space of the Radon transform,” Math. Methods Appl. Sci. 3, 1–10 (1981).
    [CrossRef]
  24. A. K. Louis, “Orthogonal function series expansions and the null space of the Radon transform,” SIAM J. Math. Anal. 15, 621–633 (1984).
    [CrossRef]
  25. K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
    [CrossRef]
  26. A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1531 (1978).
    [CrossRef]
  27. K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
    [CrossRef]

1998

E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate: Part II—square pulse excitation,” IEEE Trans. Antennas Propag. 46, 243–250 (1998).
[CrossRef]

1997

E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate,” IEEE Trans. Antennas Propag. 45, 1098–1107 (1997).
[CrossRef]

1996

R. W. Deming, A. J. Devaney, “A filtered backpropagation algorithm for GPR,” J. Env. Eng. Geo. 0, 113–123 (1996).
[CrossRef]

E. Heyman, A. J. Devaney, “Time-dependent multipoles and their application for radiation from volume source distributions,” J. Math. Phys. 37, 682–692 (1996).
[CrossRef]

A. D. Yaghjian, T. B. Hansen, “Time-domain far fields,” J. Appl. Phys. 79, 2822–2830 (1996).
[CrossRef]

1991

K. T. Ladas, A. J. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
[CrossRef]

1986

A. K. Louis, “Incomplete data problems in x-ray computerized tomography. I. Singular value decomposition of the limited angle transform,” Numer. Math. 48, 251–262 (1986).
[CrossRef]

K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
[CrossRef]

1985

1984

H. E. Moses, “The time-dependent inverse source problem for the acoustic and electromagnetic equations in the one- and three-dimensional cases,” J. Math. Phys. 25, 1905–1923 (1984).
[CrossRef]

A. K. Louis, “Orthogonal function series expansions and the null space of the Radon transform,” SIAM J. Math. Anal. 15, 621–633 (1984).
[CrossRef]

1982

1981

A. K. Louis, “Ghosts in tomography—The null space of the Radon transform,” Math. Methods Appl. Sci. 3, 1–10 (1981).
[CrossRef]

1978

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1531 (1978).
[CrossRef]

1977

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[CrossRef]

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

1973

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

P. R. Smith, T. M. Peters, R. H. T. Bates, “Image reconstruction from finite number of projections,” J. Phys. A Math. Nucl. Gen. 6, 361–382 (1973).
[CrossRef]

F. G. Friedlander, “An inverse problem for radiation fields,” Proc. London Math. Soc. 3, 551–576 (1973).
[CrossRef]

1959

H. E. Moses, “Solution of Maxwell’s equations in terms of a spinor notation: the direct and inverse problem,” Phys. Rev. 113, 1670–1679 (1959).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985).

Bates, R. H. T.

P. R. Smith, T. M. Peters, R. H. T. Bates, “Image reconstruction from finite number of projections,” J. Phys. A Math. Nucl. Gen. 6, 361–382 (1973).
[CrossRef]

Bertero, M.

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. I: general formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

M. Bertero, “Linear inverse and ill-posed problems,” in P. W. Hawkes, ed., Advances in Electronics and Electron Physics (Academic, New York, 1989), Vol. 75, pp. 1–120.

Bleistein, N.

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

Cohen, J. K.

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

De Mol, C.

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. I: general formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

Deming, R. W.

R. W. Deming, A. J. Devaney, “A filtered backpropagation algorithm for GPR,” J. Env. Eng. Geo. 0, 113–123 (1996).
[CrossRef]

Devaney, A. J.

E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate: Part II—square pulse excitation,” IEEE Trans. Antennas Propag. 46, 243–250 (1998).
[CrossRef]

E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate,” IEEE Trans. Antennas Propag. 45, 1098–1107 (1997).
[CrossRef]

R. W. Deming, A. J. Devaney, “A filtered backpropagation algorithm for GPR,” J. Env. Eng. Geo. 0, 113–123 (1996).
[CrossRef]

E. Heyman, A. J. Devaney, “Time-dependent multipoles and their application for radiation from volume source distributions,” J. Math. Phys. 37, 682–692 (1996).
[CrossRef]

K. T. Ladas, A. J. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
[CrossRef]

A. J. Devaney, R. P. Porter, “Holography and the inverse source problem. Part II: inhomogeneous media,” J. Opt. Soc. Am. A 2, 2006–2011 (1985).
[CrossRef]

R. P. Porter, A. J. Devaney, “Generalized holography and computational solutions to inverse source problems,” J. Opt. Soc. Am. 72, 1707–1713 (1982).
[CrossRef]

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1531 (1978).
[CrossRef]

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

Friedlander, F. G.

F. G. Friedlander, “An inverse problem for radiation fields,” Proc. London Math. Soc. 3, 551–576 (1973).
[CrossRef]

Ghosh Roy, D. N.

D. N. Ghosh Roy, Methods of Inverse Problems in Physics, 2nd ed. (CRC Press, Boca Raton, Fla., 1991).

Hansen, T. B.

A. D. Yaghjian, T. B. Hansen, “Time-domain far fields,” J. Appl. Phys. 79, 2822–2830 (1996).
[CrossRef]

Heyman, E.

E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate: Part II—square pulse excitation,” IEEE Trans. Antennas Propag. 46, 243–250 (1998).
[CrossRef]

E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate,” IEEE Trans. Antennas Propag. 45, 1098–1107 (1997).
[CrossRef]

E. Heyman, A. J. Devaney, “Time-dependent multipoles and their application for radiation from volume source distributions,” J. Math. Phys. 37, 682–692 (1996).
[CrossRef]

Jackson, J. D.

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

Kim, K.

K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
[CrossRef]

Ladas, K. T.

K. T. Ladas, A. J. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
[CrossRef]

LaHaie, I. J.

Louis, A. K.

A. K. Louis, “Incomplete data problems in x-ray computerized tomography. I. Singular value decomposition of the limited angle transform,” Numer. Math. 48, 251–262 (1986).
[CrossRef]

A. K. Louis, “Orthogonal function series expansions and the null space of the Radon transform,” SIAM J. Math. Anal. 15, 621–633 (1984).
[CrossRef]

A. K. Louis, “Ghosts in tomography—The null space of the Radon transform,” Math. Methods Appl. Sci. 3, 1–10 (1981).
[CrossRef]

Marengo, E. A.

E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate: Part II—square pulse excitation,” IEEE Trans. Antennas Propag. 46, 243–250 (1998).
[CrossRef]

E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate,” IEEE Trans. Antennas Propag. 45, 1098–1107 (1997).
[CrossRef]

Moses, H. E.

H. E. Moses, “The time-dependent inverse source problem for the acoustic and electromagnetic equations in the one- and three-dimensional cases,” J. Math. Phys. 25, 1905–1923 (1984).
[CrossRef]

H. E. Moses, “Solution of Maxwell’s equations in terms of a spinor notation: the direct and inverse problem,” Phys. Rev. 113, 1670–1679 (1959).
[CrossRef]

Peters, T. M.

P. R. Smith, T. M. Peters, R. H. T. Bates, “Image reconstruction from finite number of projections,” J. Phys. A Math. Nucl. Gen. 6, 361–382 (1973).
[CrossRef]

Pike, E. R.

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. I: general formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

Porter, R. P.

Smith, K. T.

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[CrossRef]

Smith, P. R.

P. R. Smith, T. M. Peters, R. H. T. Bates, “Image reconstruction from finite number of projections,” J. Phys. A Math. Nucl. Gen. 6, 361–382 (1973).
[CrossRef]

Solmon, D. C.

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[CrossRef]

Wagner, S. L.

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[CrossRef]

Wolf, E.

K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
[CrossRef]

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

Yaghjian, A. D.

A. D. Yaghjian, T. B. Hansen, “Time-domain far fields,” J. Appl. Phys. 79, 2822–2830 (1996).
[CrossRef]

Bull. Am. Math. Soc.

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[CrossRef]

IEEE Trans. Antennas Propag.

E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate,” IEEE Trans. Antennas Propag. 45, 1098–1107 (1997).
[CrossRef]

E. A. Marengo, A. J. Devaney, E. Heyman, “Analysis and characterization of ultrawideband, scalar volume sources and the fields they radiate: Part II—square pulse excitation,” IEEE Trans. Antennas Propag. 46, 243–250 (1998).
[CrossRef]

Inverse Probl.

K. T. Ladas, A. J. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
[CrossRef]

M. Bertero, C. De Mol, E. R. Pike, “Linear inverse problems with discrete data. I: general formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

J. Appl. Phys.

A. D. Yaghjian, T. B. Hansen, “Time-domain far fields,” J. Appl. Phys. 79, 2822–2830 (1996).
[CrossRef]

J. Env. Eng. Geo.

R. W. Deming, A. J. Devaney, “A filtered backpropagation algorithm for GPR,” J. Env. Eng. Geo. 0, 113–123 (1996).
[CrossRef]

J. Math. Phys.

H. E. Moses, “The time-dependent inverse source problem for the acoustic and electromagnetic equations in the one- and three-dimensional cases,” J. Math. Phys. 25, 1905–1923 (1984).
[CrossRef]

E. Heyman, A. J. Devaney, “Time-dependent multipoles and their application for radiation from volume source distributions,” J. Math. Phys. 37, 682–692 (1996).
[CrossRef]

N. Bleistein, J. K. Cohen, “Nonuniqueness in the inverse source problem in acoustics and electromagnetics,” J. Math. Phys. 18, 194–201 (1977).
[CrossRef]

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1531 (1978).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A Math. Nucl. Gen.

P. R. Smith, T. M. Peters, R. H. T. Bates, “Image reconstruction from finite number of projections,” J. Phys. A Math. Nucl. Gen. 6, 361–382 (1973).
[CrossRef]

Math. Methods Appl. Sci.

A. K. Louis, “Ghosts in tomography—The null space of the Radon transform,” Math. Methods Appl. Sci. 3, 1–10 (1981).
[CrossRef]

Numer. Math.

A. K. Louis, “Incomplete data problems in x-ray computerized tomography. I. Singular value decomposition of the limited angle transform,” Numer. Math. 48, 251–262 (1986).
[CrossRef]

Opt. Commun.

K. Kim, E. Wolf, “Non-radiating monochromatic sources and their fields,” Opt. Commun. 59, 1–6 (1986).
[CrossRef]

Phys. Rev.

H. E. Moses, “Solution of Maxwell’s equations in terms of a spinor notation: the direct and inverse problem,” Phys. Rev. 113, 1670–1679 (1959).
[CrossRef]

Phys. Rev. D

A. J. Devaney, E. Wolf, “Radiating and nonradiating classical current distributions and the fields they generate,” Phys. Rev. D 8, 1044–1047 (1973).
[CrossRef]

Proc. London Math. Soc.

F. G. Friedlander, “An inverse problem for radiation fields,” Proc. London Math. Soc. 3, 551–576 (1973).
[CrossRef]

SIAM J. Math. Anal.

A. K. Louis, “Orthogonal function series expansions and the null space of the Radon transform,” SIAM J. Math. Anal. 15, 621–633 (1984).
[CrossRef]

Other

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985).

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

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

D. N. Ghosh Roy, Methods of Inverse Problems in Physics, 2nd ed. (CRC Press, Boca Raton, Fla., 1991).

M. Bertero, “Linear inverse and ill-posed problems,” in P. W. Hawkes, ed., Advances in Electronics and Electron Physics (Academic, New York, 1989), Vol. 75, pp. 1–120.

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Figures (5)

Fig. 1
Fig. 1

(a) Schematization of ρ(X) and its Radon projection (Rρ)(Vs, ξ) in a viewing direction Vs, showing the relationship between Vs, s, and the rotated coordinate axis ξ used in the definition of the Radon transform in 3D space–time. (b) The light cone in 3D space–time.

Fig. 2
Fig. 2

(a) Plots of the normalized singular values σ¯l2 versus l and parameterized by ωa/c for a source in the spherical volume V:ra. (b)–(d) Plots of σl2(ω)/αl2(ω) versus l and parameterized by ωa/c for sources in the spherical shells defined by bra with b=0.5a, b=0.9a and b=0.97a, respectively.

Fig. 3
Fig. 3

Source spatial profile at different times.

Fig. 4
Fig. 4

Results for the time in Fig. 3(a): (a) original source spatial profile, (b) reconstructed source spatial profile obtained from the filtered backprojection algorithm.

Fig. 5
Fig. 5

Results for the time in Fig. 3(c): (a) original source spatial profile, (b) reconstructed source spatial profile obtained from the filtered backprojection algorithm.

Equations (72)

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

2-1c2 2t2U(r, t)=-4πQ(r, t),
U(r, t)=-dtd3r Q(r, t)δt+|r-r|c-t|r-r|
F(s, τ)=-dtd3rQ(r, t)δ(t-rs/c-τ).
F(s, τ)=12 d4XM(X)ρ(X)δ(cτ/2-XVs),
Vs=Vs0=12,Vs1=-12sx,Vs2=-12sy,
Vs3=-12sz,
M(X)=1ifXD0otherwise,
M(X)=1ifbra0otherwise.
(Rρ)(V, ξ)=d4Xρ(X)δ(ξ-XV).
f(s, ω)=1c d4XM(X)ρ(X)expi2ωc VsX.
F=Pρ,
ρ1, ρ2X=d4Xρ1*(X)ρ2(X),
F, PρY=PF, ρX,
(PF)(X)=M(X)1c S2dsF(s,2VsX/c).
(PF)(X)=M(X)1c S2dsF(s, t-sr/c),
2-1c2 2t2(PF)(X)=0,
ρME(X)=(PF¯)(X),
(PPF¯)(s, τ)=F(s, τ).
ρME=P[PP+βI]-1F,
2-1c2 2t2ρME(X)=0
(PPF¯)(s, τ)=1c dtd3rM(r)δ(τ+rs/c-t)×S2dsF¯(s, t-rs/c)=1c S2dsdtF¯(s, t)d3rM(r)×δ[t-τ-r(s-s)/c]=1c S2dsF¯(s, τ)h(s-s, τ),
h(s-s, τ)=d3rM(r)δ[τ+(s-s)r/c].
1c S2dsF¯(s, τ)h(s-s, τ)=F(s, τ).
Q(r, t)=M(r)δ(t-rs/c).
1c S2dsf¯(s, ω)h˜(s-s, ω)=f(s, ω),
h˜(s-s, ω)=d3rM(r)exp-iωcr(s-s).
expiωcrr^s
=4πl=0m=-lliljlωcrYl,m(r^)Yl,m*(s),
h˜(s-s, ω)=l=0m=-llσl2(ω)Yl,m(s)Yl,m*(s),
σl2(ω)=(4π)2badrr2jl2ωcr=αl2(ω)-βl2(ω),
αl2(ω)=8π2a3jl2ωca-jl-1ωcajl+1ωca,
βl2(ω)=8π2b3jl2ωcb-jl-1ωcbjl+1ωcb.
f(s, ω)=l=0m=-ll(-i)la˜l,m(ω)Yl,m(s),
f¯(s, ω)=l=0m=-ll(-i)la˜¯l,m(ω)Yl,m(s),
a˜l,m(ω)=ilS2dsf(s, ω)Yl,m*(s),
a˜¯l,m(ω)=ilS2dsf¯(s, ω)Yl,m*(s).
a˜l,m(ω)=1cσl2(ω)a˜¯l,m(ω),
QME(r, t)=M(r)12π -dω exp(-iωt)×l=0m=-ll4πa˜l,m(ω)σl-2(ω)jlωcrYl,m(rˆ)=12π -dω exp(-iωt)S2dsf(s, ω)H˜(r,s, ω)=S2dsF(s, t)H(r, s, t),
H˜(r,s, ω)
=4πM(r)l=0m=-llilσl-2(ω)jlωcrYl,m(rˆ)Yl,m*(s),
H(r, s, t)
=2M(r)-dω exp(-iωt)×l=0m=-llilσl-2(ω)jlωcrYl,m(rˆ)Yl,m*(s).
-dtDd3r|QME(r, t)|2
=12π -dωl=0m=-ll|a˜l,m(ω)|2σl-2(ω).
Q(r, t)=M(r)G(t-zˆr/c)
h˜(s-s, ω)=h˜(ϕ, ϕ, ω)=axaz sincωax2c(sx-sx)×sincωaz2c(sz-sz)=axaz sincωax2c(sin ϕ-sin ϕ)×sincωaz2c(cos ϕ-cos ϕ),
(1/c)Γ(1, 1)Γ(1,2)Γ(1, n)Γ(2, 1)Γ(2, 2)Γ(2, n)Γ(n, 1)Γ(n, 2)Γ(n, n) f¯(1, ω)f¯(2, ω)f¯(n, ω)
=f(1, ω)f(2, ω)f(n, ω)
Γ(i, j)=axay sincωax2c(sin ϕi-sin ϕj)×sincωaz2c(cos ϕi-cos ϕj).
U(r, t)=1r l=0m=0lj=1,2mSl,m(j)(rˆ)Ll(r, t)ql,m(j)(t),
Ll(r, t)ql,m(j)(t)=n=0l (l+n)!n!(l-n)!2rc-nt-nql,m(j)(τ),
t-1ql,m(j)(t)=-tdtql,m(j)(t).
U(rs, t)1r l=0m=0lj=1,2mSl,m(j)(s)ql,m(j)(τ)asr
F(s, τ)=l=0m=0lj=1,2mSl,m(j)(s)ql,m(j)(τ).
S2dsSl,m(j)(s)Sl,m(j)(s)=δl,lδm,mδj,j1m,
ql,m(j)(t)=S2dsF(s, t)Sl,m(j)(s).
E-dτS2ds|F(s, τ)|2=-dtl=0m=0lj=1,2m[ql,m(j)(t)]2=0.
Q(r, t)=Πn=0Nsn+1c tA(r, t),
-dt exp(iωt)Dd3r exp(-iωsr/c)Q(r, t)
=Πn=0Ni ωc(sns-1)-dt exp(iωt)×Dd3r exp(-iωsr/c)A(r, t),
2-1c2 2t2UR(r, t)=-4πQR(r, t),
2-1c2 2t2UNR(r, t)=-4πQNR(r, t).
UNR(r, t)2-1c2 2t2UR(r, t)
=-4πUNR(r, t)QR(r, t),
UR(r, t)2-1c2 2t2UNR(r, t)
=-4πUR(r, t)QNR(r, t).
Ddtd3rUR(r, t)QNR(r, t)=0,
2-1c2 2t2UR(r, t)=0if(r, t)D,
G(r-r, t-t)=δ(t+|r-r|/c-t)/|r-r|
2-1c2 2t2G(r-r, t-t)=0
Ddtd3rG(r-r, t-t)QNR(r, t)=0
if(r, t)D.

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