Abstract

A general connection between interferometry and radiometry is studied on the basis of a new relationship between Walther’s first generalized radiance function and the mutual coherence function for polychromatic fields. New results include a definition of the generalized radiance function that is valid even for nonstationary fields and a principle of novel radiometry that retrieves three-dimensional radiance in the direction-frequency domain from the three-dimensional mutual coherence function. Radiometric interpretations of interferometric multispectral imaging are given as illustrative examples of this new principle.

© 1997 Optical Society of America

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References

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  1. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  2. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  3. L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
    [CrossRef]
  4. A. T. Friberg, B. J. Thompson, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series, MS69 (SPIE, Bellingham, Wash., 1993).
  5. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Sec. 5.
  6. See, for example, Ref. 5, p. 170.
  7. See, for example, Ref. 5, p. 160.
  8. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [CrossRef]
  9. A. Walther, “Propagation of the generalized radiance through lenses,” J. Opt. Soc. Am. 68, 1606–1610 (1978).
    [CrossRef]
  10. T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).
  11. T. Jannson, “Radiance transfer function,” J. Opt. Soc. Am. 70, 1544–1549 (1980).
    [CrossRef]
  12. R. G. Littlejohn, R. Winston, “Correction to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).
    [CrossRef]
  13. K. Itoh, Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A 3, 94–100 (1986).
    [CrossRef]
  14. J.-M. Mariotti, S. T. Ridgeway, “Double Fourier spatio–spectral interferometry: combining high spectral and spatial resolution in the near infrared,” Astron. Astrophys. 195, 350–363 (1988).
  15. B. Clark, “Synthesis imaging,” course notes, National Radio Astronomy Observatory Summer School, R. A. Perley, F. F. Schwab, A. H. Bridle, eds. (National Radio Astronomy Observatory, Socorro, N. M., 1985), pp. 1–7.
  16. P. Zhao, J.-M. Mariotti, P. Lena, V. Coude du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
    [CrossRef]
  17. A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
    [CrossRef]
  18. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  19. G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1986).
    [CrossRef]
  20. K. Itoh, T. Inoue, T. Yoshida, Y. Ichioka, “Interferometric supermultispectral imaging,” Appl. Opt. 29, 1625–1630 (1990).
    [CrossRef] [PubMed]
  21. K. Itoh, “Interferometric Multispectral Imaging,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1996), Vol. XXXV, pp. 145–196.
  22. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), Sec. 10.4.2.
  23. F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 92–166 (1988).
    [CrossRef]
  24. A. A. Michelson, “On the application of interference-methods to spectroscopic measurements,” Philos. Mag. 31, 338–346 (1891).
    [CrossRef]
  25. H. W. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  26. F. Gori, “Quasi-homogeneous source and geometrical optics,” Opt. Lett. 4, 354–356 (1979).
    [CrossRef] [PubMed]
  27. A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1980).
    [CrossRef]
  28. J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
    [CrossRef]
  29. M. N. Vesperinas, “Classical radiometry and radiative transfer theory: a short-wavelength limit of a general mapping of cross-spectral densities in second-order co-herence theory,” J. Opt. Soc. Am. A 3, 1354–1359 (1986).
    [CrossRef]
  30. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
    [CrossRef]
  31. E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
    [CrossRef]

1994 (1)

P. Zhao, J.-M. Mariotti, P. Lena, V. Coude du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

1993 (1)

1992 (1)

1990 (1)

1988 (2)

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 92–166 (1988).
[CrossRef]

J.-M. Mariotti, S. T. Ridgeway, “Double Fourier spatio–spectral interferometry: combining high spectral and spatial resolution in the near infrared,” Astron. Astrophys. 195, 350–363 (1988).

1987 (2)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[CrossRef]

1986 (3)

1985 (1)

J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
[CrossRef]

1984 (1)

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[CrossRef]

1980 (3)

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

T. Jannson, “Radiance transfer function,” J. Opt. Soc. Am. 70, 1544–1549 (1980).
[CrossRef]

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1980).
[CrossRef]

1979 (1)

1978 (3)

1977 (1)

1973 (1)

1968 (1)

1891 (1)

A. A. Michelson, “On the application of interference-methods to spectroscopic measurements,” Philos. Mag. 31, 338–346 (1891).
[CrossRef]

Agarwal, G. S.

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1986).
[CrossRef]

Apresyan, L. A.

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), Sec. 10.4.2.

Carter, H. W.

Clark, B.

B. Clark, “Synthesis imaging,” course notes, National Radio Astronomy Observatory Summer School, R. A. Perley, F. F. Schwab, A. H. Bridle, eds. (National Radio Astronomy Observatory, Socorro, N. M., 1985), pp. 1–7.

Coude du Foresto, V.

P. Zhao, J.-M. Mariotti, P. Lena, V. Coude du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

Foley, J. T.

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1986).
[CrossRef]

J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
[CrossRef]

Friberg, A. T.

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1980).
[CrossRef]

Gori, F.

Ichioka, Y.

Inoue, T.

Itoh, K.

Janicki, R.

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

Jannson, T.

T. Jannson, “Radiance transfer function,” J. Opt. Soc. Am. 70, 1544–1549 (1980).
[CrossRef]

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

Kravtsov, Yu. A.

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[CrossRef]

Lena, P.

P. Zhao, J.-M. Mariotti, P. Lena, V. Coude du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

Littlejohn, R. G.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Sec. 5.

Mariotti, J.-M.

P. Zhao, J.-M. Mariotti, P. Lena, V. Coude du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

J.-M. Mariotti, S. T. Ridgeway, “Double Fourier spatio–spectral interferometry: combining high spectral and spatial resolution in the near infrared,” Astron. Astrophys. 195, 350–363 (1988).

Michelson, A. A.

A. A. Michelson, “On the application of interference-methods to spectroscopic measurements,” Philos. Mag. 31, 338–346 (1891).
[CrossRef]

Ohtsuka, Y.

Ridgeway, S. T.

J.-M. Mariotti, S. T. Ridgeway, “Double Fourier spatio–spectral interferometry: combining high spectral and spatial resolution in the near infrared,” Astron. Astrophys. 195, 350–363 (1988).

Roddier, F.

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 92–166 (1988).
[CrossRef]

Vesperinas, M. N.

Walther, A.

Winston, R.

Wolf, E.

A. T. Friberg, G. S. Agarwal, J. T. Foley, E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386–1393 (1992).
[CrossRef]

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1986).
[CrossRef]

J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
[CrossRef]

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
[CrossRef]

H. W. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), Sec. 10.4.2.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Sec. 5.

Yoshida, T.

Zhao, P.

P. Zhao, J.-M. Mariotti, P. Lena, V. Coude du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

Zhou, B.

P. Zhao, J.-M. Mariotti, P. Lena, V. Coude du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (1)

J.-M. Mariotti, S. T. Ridgeway, “Double Fourier spatio–spectral interferometry: combining high spectral and spatial resolution in the near infrared,” Astron. Astrophys. 195, 350–363 (1988).

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (3)

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

Nature (London) (1)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[CrossRef]

Opt. Acta (1)

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1980).
[CrossRef]

Opt. Commun. (5)

J. T. Foley, E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236–241 (1985).
[CrossRef]

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

G. S. Agarwal, J. T. Foley, E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67–72 (1986).
[CrossRef]

P. Zhao, J.-M. Mariotti, P. Lena, V. Coude du Foresto, B. Zhou, “Double Fourier interferometry with IR single-mode fiber optics,” Opt. Commun. 110, 497–502 (1994).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

T. Jannson, R. Janicki, “An eigenvalue formulation of inverse theory of scalar diffraction,” Optik (Stuttgart) 56, 429–441 (1980).

Philos. Mag. (1)

A. A. Michelson, “On the application of interference-methods to spectroscopic measurements,” Philos. Mag. 31, 338–346 (1891).
[CrossRef]

Phys. Rep. (1)

F. Roddier, “Interferometric imaging in optical astronomy,” Phys. Rep. 170, 92–166 (1988).
[CrossRef]

Usp. Fiz. Nauk (1)

L. A. Apresyan, Yu. A. Kravtsov, “Photometry and coherence: wave aspects of the theory of radiation transport,” Usp. Fiz. Nauk 142, 689–711 (1984) [Sov. Phys. Usp. 27, 301–313 (1984)].
[CrossRef]

Other (7)

A. T. Friberg, B. J. Thompson, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series, MS69 (SPIE, Bellingham, Wash., 1993).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Sec. 5.

See, for example, Ref. 5, p. 170.

See, for example, Ref. 5, p. 160.

K. Itoh, “Interferometric Multispectral Imaging,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1996), Vol. XXXV, pp. 145–196.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), Sec. 10.4.2.

B. Clark, “Synthesis imaging,” course notes, National Radio Astronomy Observatory Summer School, R. A. Perley, F. F. Schwab, A. H. Bridle, eds. (National Radio Astronomy Observatory, Socorro, N. M., 1985), pp. 1–7.

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Figures (2)

Fig. 1
Fig. 1

Schematic of a rotational-shear volume interferometer with the coordinate system. Rotational shear is introduced by twisting the two right-angle prisms around the optical axis. Longitudinal shear is created by shifting the prisms along the optical axis. The light source is assumed to be a planar, secondary, polychromatic source in the source plane z=0. The z axis is taken to be the optical axis of the interferometer. The overall optical distance from the observation plane to the origin is taken to be Rz. The Rx axis on the observation plane is taken as a reflected geometry.

Fig. 2
Fig. 2

Principle of coherence spectroradiometry. The three-dimensional Fourier transform of the mutual intensity with respect to the separation is proportional to the generalized (spectral) radiance at the mean position R0=(0, 0, Rz), which coincides with the generalized (spectral) radiance of the light source at the point r=-Rzs/sz.

Equations (76)

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Γ(r, r; τ)=0+dω exp(-iωτ)W(r, r; ω),
(2+k2)Uω(r)=0,
W(z)(r, r; ω)Uω*(r, z)Uω(r, z),
B(r, s; ω)=szk2π2d2ρ exp(-ik·ρ)×W(z)(r-ρ/2, r+ρ/2; ω)=szk2π2d2q exp(iq·r)×W˜(z)(k-q/2, k+q/2; ω),
W˜(z)(k, k; ω)= d2rd2r(2π)2×exp[-i(k·r-k·r)]×W(z)(r, r; ω).
Uω(r)= d2k2πexp(ik·r)U˜ω(k, z).
2z2+kz2U˜ω(k, z)=0,
kz=[k2-k2]1/2ifkki[k2-k2]1/2ifkk ,
U˜ω(k, z)=exp[ikz(z-z)]U˜ω(k, z).
W˜(z)(k, k; ω)=U˜ω*(k, z)U˜ω(k, z).
W˜(z)(k, k; ω)=exp[i(kz-kz)(z-z)]×W˜(z)(k, k; ω),
W(r-ρ/2, r+ρ/2; ω)Uω*(r-ρ/2)Uω(r+ρ/2).
W(r-ρ/2, r+ρ/2; ω)
=dΩ(s)exp(ik·ρ)B(r, s; ω),
W(r-ρ/2, r+ρ/2; ω)
= d2kd2k(2π)2exp[i(k-k)·r+i(k+k)·ρ/2]U˜ω*(k, z-ρz/2)×U˜ω(k, z+ρz/2),
U˜ω*(k, z-ρz/2)U˜ω(k, z+ρz/2)
=exp[i(kz+kz)ρz/2]W˜(z)(k, k; ω),
W(r-ρ/2, r+ρ/2; ω)
= d2kd2k(2π)2exp[i(k-k)·r+i(k+k)·ρ/2]W˜(z)(k, k; ω).
W(r-ρ/2, r+ρ/2; ω)
= d2kd2q(2π)2exp[ik·ρ+iq·r]×W˜(z)(k-q/2, k+q/2; ω).
W(r-ρ/2, r+ρ/2; ω)
= d2kszk2exp(ik·ρ)B(r, s; ω).
d2kszk2=d2ssz=dΩ(s)
B(r, s; ω)=szk2π2d2ρ exp(-ik·ρ)×W(r-ρ/2, r+ρ/2; ω).
Γ(r-ρ/2, r+ρ/2; τ)
=0+dωdΩ(s)exp[i(k·ρ-ωτ)]B(r, s; ω).
Γ(r-ρ/2, r+ρ/2; τ)
=c d3kk2exp[i(k·ρ-ωτ)]B(r, s; ω).
B(r, s; ω)=k2(2π)3cd3ρ exp[-i(k·ρ-ωτ)]×Γ(r-ρ/2, r+ρ/2; τ).
Γ(r-ρ/2, r+ρ/2)=c d3kk2exp(ik·ρ)×B(r, s; ω),
B(r, s; ω)=k2(2π)3cd3ρ exp(-ik·ρ)×Γ(r-ρ/2, r+ρ/2).
K(r, k)=d3ρ exp(-ik·ρ)Γ(r-ρ/2, r+ρ/2)=d3q exp(iq·r)Γ˜(k-q/2, k+q/2).
Γ˜(k, k)= d3rd3r(2π)3exp[-i(k·r-k·r)]×Γ(r, r).
B(r, s; ω)=k2(2π)3cK(r, k).
 d3k(2π)3K(r, k)=Γ(r, r)I(r),
 d3r(2π)3K(r, k)=Γ˜(k, k)I˜(k),
0+dωdΩ(s)B(r, s; ω)=I(r),
d3rB(r, s; ω)=k2cI˜(k).
I(X)=14[Γ(R-ρ/2, R+ρ/2)+Γ*(R-ρ/2, R+ρ/2)+Γ(R-ρ/2, R-ρ/2)+Γ(R+ρ/2, R+ρ/2)],
X=(X, Z),
R=(X cos θ, Rz),
ρ=(2Y sin θ,-2X sin θ, 2Z)
I(ρ)=14[Γ(R0-ρ/2, R0+ρ/2)+Γ*(R0-ρ/2, R0+ρ/2)+2I(R0)],
 d3ρ(2π)3exp(-ik·ρ)I(ρ)
=c4k2[B(R0, s; ω)+B(R0,-s; ω)]+12I(R0)δ3(k),
W(0)(r, r; ω)=S(0)r+r2; ωμ(0)(r-r; ω),
λlL.
B(0)(r, s; ω)=(2π)-1szk2S(0)(r; ω)μ˜(0)(k; ω),
μ˜(0)(k; ω)= d2r2πexp(-ik·r)μ(0)(r; ω).
B(R, s; ω)=B(0)R-Rzszs, s; ω,
s·RB(R, s; ω)=0.
r=-Rzszs.
Γ(R0-ρ/2, R0+ρ/2; τ)
=ΓR0-ρ/2, R0+ρ/2; τ-ρzc,
B(R0, s;ω)
=k2(2π)3cd2ρdρz exp-ik·ρ+iωτ-ρzc×Γ(R0-ρ/2, R0+ρ/2; τ),
kzk=ωc.
B(R0, s; ω)=k2(2π)3d2ρdτ×exp[-i(k·ρ-ωτ)]×Γ(R0-ρ/2, R0+ρ/2; τ).
Bc(r, s; ω)=(2π)-1szk2 exp(ik·r)×Uω*(r)U˜ω(k, z)=szk2π2d2ρ exp(-ik·ρ)×W(z)(r, r+ρ; ω)=szk2π2d2q exp(iq·r)×W˜(z)(k-q, k; ω).
W(r, r+ρ; ω)=Uω*(r)Uω(r+ρ).
Uω(r+ρ)= d2k2πexp[i(k·ρ+k·r)]U˜ω(k, z),
W(r, r+ρ; ω)=dΩ(s)exp(ik·ρ)Bc(r, s; ω),
Γ(r, r+ρ; τ)=c d3kk2exp[i(k·ρ-ωτ)]Bc(r, s; ω),
Bc(r, s; ω)=k2(2π)3cd3ρ exp[-i(k·ρ-ωτ)]×Γ(r, r+ρ; τ),
Γ(r, r+ρ)=c d3kk2exp(ik·ρ)Bc(r, s; ω),
Bc(r, s; ω)=k2(2π)3cd3ρ exp(-ik·ρ)×Γ(r, r+ρ).
B(r, s; ω)Re[Bc(r, s; ω)],
W˜(0)(k, k; ω)=S˜(0)(k-k; ω)μ˜(0)k+k2; ω.
S˜(0)(k; ω)= d2r2πexp(-ik·r)S(0)(r; ω)
W˜(Rz)(k, k; ω)=exp[i(kz-kz)Rz]W˜(0)(k, k; ω).
B(R, s; ω)=szk2π2μ˜(0)(k; ω)d2q exp[iq·R+i(κ(+)-κ(-))Rz]S˜(0)(q; ω),
κ(±)=[k2-(k±q/2)2]1/2.
κ(±)kz±k·q2kz.
B(R, s; ω)=szk2π2μ˜(0)(k; ω)d2q×expiq·R-RzkzkS˜(0)(q; ω)=(2π)-1szk2S(0)R-Rzszs; ω×μ˜(0)(k; ω).

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