Abstract

The analogs of the generalized radiances introduced in two previous manuscripts [J. Opt. Soc. Am. A 18, 902 (2001) and J. Opt. Soc. Am. A 18, 910 (2001)] for fully coherent fields in two- and three-dimensional free space are given here for the case of partial coherence. These functions are exactly conserved along rays and are suitable for the description of fields with components propagating in any direction. Also defined here is a global measure of coherence, which can be expressed in terms of the new functions. The cases of radiation in a blackbody cavity and partially coherent focused waves are considered as examples.

© 2001 Optical Society of America

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References

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  1. L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).
  2. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  3. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973).
    [CrossRef]
  4. Some of the most relevant papers in the subject are compiled in A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, Vol. MS69 of Milestone Series (SPIE Optical Engineering Press, Bellingham, Wash., 1993).
  5. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. X, pp. 554–632.
  6. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, UK, 1995) Chaps. 4 and 5, pp. 147–374.
  7. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 13–27.
  8. A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).
    [CrossRef]
  9. In the following paper, as well as in Refs. 10-12, it was shown that, for quasi-homogeneous planar sources and in the limit of small wavelength, Walther’s two definitions of generalized radiance are nonnegative: A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).
    [CrossRef]
  10. 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]
  11. K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. A 4, 1233–1236 (1987).
    [CrossRef]
  12. 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 (1987).
    [CrossRef]
  13. Walther’s two generalized radiances, as well as the more general family to which they belong to, are asymptotically conserved along rays in the limit of small wavelength and for planar quasi-homogeneous sources, as shown in Refs. 9-12. Definitions of generalized radiance that are explicitly conserved along rays are given in Refs. 26-30below.
  14. M. A. Alonso, “Radiometry and wide-angle wave fields. I. Coherent fields in two dimensions,” J. Opt. Soc. Am. A 18, 902–909 (2001).
    [CrossRef]
  15. M. A. Alonso, “Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions,” J. Opt. Soc. Am. A 18, 910–918 (2001).
    [CrossRef]
  16. M. A. Alonso, “Measurement of Helmholtz wave fields,” J. Opt. Soc. Am. A 17, 1256–1264 (2000).
    [CrossRef]
  17. See Ref. 5, p. 610, and Ref. 6, pp. 162–163.
  18. See Ref. 5, p. 567, and Ref. 6, p. 170. The cross-spectral density is often denoted by the letter W. However, in order to avoid confusion with the Wigner function used here, we denote the cross-spectral density by G, as in Ref. 5.
  19. See Ref. 6, p. 170.
  20. These properties are found in Ref. 6, pp. 183 and 215.
  21. See Ref. 6, pp. 214–216. Notice that the orthogonality relation there differs from the one to be given here: Whereas Eq. (4.7–12) of Ref. 6 involves integration over all space, Eq. (2.17) of this paper involves integration over all directions.
  22. See Ref. 6, p. 290.
  23. This condition for the region of interest in explained in Ref. 14.
  24. This function was previously proposed in Ref. 16. A similar function was defined in E. Marchand, E. Wolf, “Angular correlation in the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972). However, that function corresponds to the double Fourier transform in the transverse spatial variables of the cross-spectral density at a plane of fixed z. It therefore differs from the form used here by an obliquity factor and a mapping, and it assumes forward propagation.
    [CrossRef]
  25. S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981). An interpretation of this inner product in the two-dimensional case is given in Appendix A of Ref. 14.
    [CrossRef]
  26. K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
    [CrossRef]
  27. G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972).
    [CrossRef]
  28. H. M. Pedersen, “Exact geometrical description of free space radiative energy transfer for scalar wavefields,” in Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990), pp. 883–887.
  29. H. M. Pedersen, “Exact theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991); errata 8, 1518 (1991).
    [CrossRef]
  30. H. M. Pedersen, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A 25, 5263–5278 (1992).
    [CrossRef]
  31. R. G. Littlejohn, R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993).
    [CrossRef]
  32. See, for example, R. Eisberg, R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (Wiley, New York, 1985), pp. 13–19.
  33. W. H. Carter, A. Gamliel, E. Wolf, “Coherence properties of the field produced by an infinitely large, uniform, planar, secondary Lambertian source,” J. Mod. Opt. 41, 1973–1981 (1994).
    [CrossRef]
  34. W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
    [CrossRef]
  35. Remember that the denominator is given by the square of the total power defined in Eq. (4.2). The fact that this quantity diverges, however, does not mean that the radiation inside a blackbody cavity is infinite, because the field does not extend over all space but only inside the cavity.
  36. A distribution of this form was used in C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998). An alternative periodic Gaussian analog is given in Ref. 26.
    [CrossRef]

2001

2000

1999

1998

A distribution of this form was used in C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998). An alternative periodic Gaussian analog is given in Ref. 26.
[CrossRef]

1994

W. H. Carter, A. Gamliel, E. Wolf, “Coherence properties of the field produced by an infinitely large, uniform, planar, secondary Lambertian source,” J. Mod. Opt. 41, 1973–1981 (1994).
[CrossRef]

1993

1992

H. M. Pedersen, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A 25, 5263–5278 (1992).
[CrossRef]

1991

1987

K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. A 4, 1233–1236 (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 (1987).
[CrossRef]

1985

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]

1981

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981). An interpretation of this inner product in the two-dimensional case is given in Appendix A of Ref. 14.
[CrossRef]

In the following paper, as well as in Refs. 10-12, it was shown that, for quasi-homogeneous planar sources and in the limit of small wavelength, Walther’s two definitions of generalized radiance are nonnegative: A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).
[CrossRef]

1979

1975

1973

1972

1968

1964

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).

Agarwal, G. S.

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 (1987).
[CrossRef]

Alonso, M. A.

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. X, pp. 554–632.

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 13–27.

Carter, W. H.

W. H. Carter, A. Gamliel, E. Wolf, “Coherence properties of the field produced by an infinitely large, uniform, planar, secondary Lambertian source,” J. Mod. Opt. 41, 1973–1981 (1994).
[CrossRef]

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
[CrossRef]

Dolin, L. S.

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).

Eisberg, R.

See, for example, R. Eisberg, R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (Wiley, New York, 1985), pp. 13–19.

Foley, J. T.

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 (1987).
[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]

Forbes, G. W.

Friberg, A. T.

In the following paper, as well as in Refs. 10-12, it was shown that, for quasi-homogeneous planar sources and in the limit of small wavelength, Walther’s two definitions of generalized radiance are nonnegative: A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).
[CrossRef]

A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).
[CrossRef]

Gamliel, A.

W. H. Carter, A. Gamliel, E. Wolf, “Coherence properties of the field produced by an infinitely large, uniform, planar, secondary Lambertian source,” J. Mod. Opt. 41, 1973–1981 (1994).
[CrossRef]

Kim, K.

Littlejohn, R. G.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, UK, 1995) Chaps. 4 and 5, pp. 147–374.

Marchand, E.

Ovchinnikov, G. I.

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972).
[CrossRef]

Pedersen, H. M.

H. M. Pedersen, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A 25, 5263–5278 (1992).
[CrossRef]

H. M. Pedersen, “Exact theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991); errata 8, 1518 (1991).
[CrossRef]

H. M. Pedersen, “Exact geometrical description of free space radiative energy transfer for scalar wavefields,” in Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990), pp. 883–887.

Resnick, R.

See, for example, R. Eisberg, R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (Wiley, New York, 1985), pp. 13–19.

Saghafi, S.

A distribution of this form was used in C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998). An alternative periodic Gaussian analog is given in Ref. 26.
[CrossRef]

Sheppard, C. J. R.

A distribution of this form was used in C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998). An alternative periodic Gaussian analog is given in Ref. 26.
[CrossRef]

Steinberg, S.

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981). An interpretation of this inner product in the two-dimensional case is given in Appendix A of Ref. 14.
[CrossRef]

Tatarskii, V. I.

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972).
[CrossRef]

Walther, A.

Winston, R.

Wolf, E.

W. H. Carter, A. Gamliel, E. Wolf, “Coherence properties of the field produced by an infinitely large, uniform, planar, secondary Lambertian source,” J. Mod. Opt. 41, 1973–1981 (1994).
[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 (1987).
[CrossRef]

K. Kim, E. Wolf, “Propagation law for Walther’s first generalized radiance function and its short-wavelength limit with quasi-homogeneous sources,” J. Opt. Soc. Am. A 4, 1233–1236 (1987).
[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]

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
[CrossRef]

This function was previously proposed in Ref. 16. A similar function was defined in E. Marchand, E. Wolf, “Angular correlation in the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972). However, that function corresponds to the double Fourier transform in the transverse spatial variables of the cross-spectral density at a plane of fixed z. It therefore differs from the form used here by an obliquity factor and a mapping, and it assumes forward propagation.
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, UK, 1995) Chaps. 4 and 5, pp. 147–374.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. X, pp. 554–632.

Wolf, K. B.

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981). An interpretation of this inner product in the two-dimensional case is given in Appendix A of Ref. 14.
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

L. S. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).

J. Math. Phys.

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981). An interpretation of this inner product in the two-dimensional case is given in Appendix A of Ref. 14.
[CrossRef]

J. Mod. Opt.

W. H. Carter, A. Gamliel, E. Wolf, “Coherence properties of the field produced by an infinitely large, uniform, planar, secondary Lambertian source,” J. Mod. Opt. 41, 1973–1981 (1994).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

H. M. Pedersen, “Coherence and radiative energy transfer for linear surface gravity waves in water of constant depth,” J. Phys. A 25, 5263–5278 (1992).
[CrossRef]

Opt. Acta

In the following paper, as well as in Refs. 10-12, it was shown that, for quasi-homogeneous planar sources and in the limit of small wavelength, Walther’s two definitions of generalized radiance are nonnegative: A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).
[CrossRef]

Opt. Commun.

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]

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 (1987).
[CrossRef]

Phys. Rev. A

A distribution of this form was used in C. J. R. Sheppard, S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998). An alternative periodic Gaussian analog is given in Ref. 26.
[CrossRef]

Radiophys. Quantum Electron.

G. I. Ovchinnikov, V. I. Tatarskii, “On the problem of the relationship between coherence theory and the radiation-transfer equation,” Radiophys. Quantum Electron. 15, 1087–1089 (1972).
[CrossRef]

Other

H. M. Pedersen, “Exact geometrical description of free space radiative energy transfer for scalar wavefields,” in Coherence and Quantum Optics VI, J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990), pp. 883–887.

Walther’s two generalized radiances, as well as the more general family to which they belong to, are asymptotically conserved along rays in the limit of small wavelength and for planar quasi-homogeneous sources, as shown in Refs. 9-12. Definitions of generalized radiance that are explicitly conserved along rays are given in Refs. 26-30below.

Some of the most relevant papers in the subject are compiled in A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, Vol. MS69 of Milestone Series (SPIE Optical Engineering Press, Bellingham, Wash., 1993).

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. X, pp. 554–632.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, UK, 1995) Chaps. 4 and 5, pp. 147–374.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983), pp. 13–27.

See Ref. 5, p. 610, and Ref. 6, pp. 162–163.

See Ref. 5, p. 567, and Ref. 6, p. 170. The cross-spectral density is often denoted by the letter W. However, in order to avoid confusion with the Wigner function used here, we denote the cross-spectral density by G, as in Ref. 5.

See Ref. 6, p. 170.

These properties are found in Ref. 6, pp. 183 and 215.

See Ref. 6, pp. 214–216. Notice that the orthogonality relation there differs from the one to be given here: Whereas Eq. (4.7–12) of Ref. 6 involves integration over all space, Eq. (2.17) of this paper involves integration over all directions.

See Ref. 6, p. 290.

This condition for the region of interest in explained in Ref. 14.

See, for example, R. Eisberg, R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. (Wiley, New York, 1985), pp. 13–19.

Remember that the denominator is given by the square of the total power defined in Eq. (4.2). The fact that this quantity diverges, however, does not mean that the radiation inside a blackbody cavity is infinite, because the field does not extend over all space but only inside the cavity.

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Figures (5)

Fig. 1
Fig. 1

Global degree of coherence for the focused field with angular correlation function given in Eq. (6.1), as a function of the angular correlation width w, in both two (solid curve) and three (dashed curve) dimensions. Notice that these curves differ noticeably only for small w, where they grow as wN-1.

Fig. 2
Fig. 2

Plots of (a) (2π/k)M(0)/F as a function of k|l| and (b) (2π/k)2M(0)/F as a function of k|L|, for the field with angular correlation function given in Eq. (6.1) in two and three dimensions, respectively. Both are plotted for w between π/16 and 4π at steps of π/16. Curves with lower intersection with the vertical axis correspond to smaller w.

Fig. 3
Fig. 3

Plots of (a) (2π/k)M(n)/F as a function of k|l| (two dimensions) and (b) (2π/k)2M(2)/F as a function of k|L| (three dimensions), for the same fields as in Fig. 2. Both are plotted for w=π/16, π/4, 4π and n=0,1,2.

Fig. 4
Fig. 4

Plots of (a) S/kF (two dimensions) and (b) πS/k2F (three dimensions), as functions of k|r| for the same fields as in Fig. 2. Both are plotted for w between π/16 and 4π at steps of π/16. Curves with lower intersection with the vertical axis correspond to smaller w.

Fig. 5
Fig. 5

Plots of (S-2k-2H)/S(0;k) as a function of k|r| for the (a) two- and (b) three-dimensional fields with angular correlation function given in Eq. (6.1), plotted for w between π/16 and 4π at steps of π/16. Curves with lower intersection with the vertical axis correspond to smaller w.

Equations (74)

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

Γ(r1, r2; τ)  V*(r1; t)V(r2; t+τ),
G(r1, r2; k)=Γ(r1, r2; τ)exp(iωτ)dτ,
(12+k2)G(r1, r2; k)=(22+k2)G(r1, r2; k)=0,
G(r1, r2; k)=G*(r2, r1; k).
(2+k2)Um(r; k)=0,
G(r1, r2; k)=mβm(k)Um*(r1; k)Um(r2; k),
S(r; k)=G(r, r; k)=mβm(k)Um*(r; k)Um(r; k)=mβm(k)Sm(r; k),
F(r; k)=kc2iGr-r2, r+r2; kr=0=mβm(k)kc4i[Um*(r; k)Um(r; k)-Um*(r; k)Um(r; k)]=mβm(k)Fm(r; k),
H(r;k)=-122Gr-r2, r+r2; kr=0=mβm(k)14[k2Um*(r; k)Um(r; k)+Um*(r; k)·Um(r; k)]=mβm(k)Hm(r; k),
I(r)=0S(r; k)dk,
J(r)=0F(r; k)dk,
E(r)=0H(r; k)dk.
G(r1, r2; k)=k2πN-12(N-1)πΞ(u1, u2; k)×exp[-ik(r1·u1-r2·u2)]×dN-1α1dN-1α2,
Ξ(u1, u2; k)=Ξ*(u2, u1; k).
Ξ(u1, u2; k)=mβm(k)φm*(u1; k)φm(u2; k),
Um(r; k)=k2π(N-1)/22(N-1)πφm(u; k)×exp(ikr·u)dN-1α.
2(N-1)πφm*(u; k)φn(u; k)dN-1α=δm,n,
W(r, p; k)=k2πNNU*r-r2; kUr+r2; k×exp(-ikr·p)dNr,
B(n)(r, u; k)0W(r, pu; k)pN+n-1dp,
u·B(n)(r, u; k)=0.
S(r; k)=U*(r; k)U(r; k)=2(N-1)πB(0)(r, u; k)dN-1α,
F(r; k)=kc4i[U*(r; k)U(r; k)-U*(r; k)U(r; k)]=k2c22(N-1)πB(1)(r, u; k)udN-1α,
H(r; k)=14[k2U*(r; k)U(r; k)+U*(r; k)·U(r; k)]=k222(N-1)πB(2)(r, u; k)dN-1α.
U(r; k)=k2π(N-1)/22(N-1)πφ(u; k)×exp(ikr·u)dN-1α,
AnB(n)(r, u; k)dN-1r=|φ(u; k)|2u·n.
W(r, p; k)mβm(k)Wm(r, p; k)=k2πNNGr-r2, r+r2; k×exp(-ikr·p)dNr,
u·B(n)(r, u; k)=0,
S(r; k)=mβm(k)Sm(r; k)=2(N-1)πB(0)(r, u; k)dN-1α,
F(r; k)=mβm(k)Fm(r; k)=k2c22(N-1)πB(1)(r, u; k)udN-1α,
H(r; k)=mβm(k)Hm(r; k)=k222(N-1)πB(2)(r, u; k)dN-1α,
AnB(n)(r, u; k)dN-1r=1u·nmβm(k)|φm(u; k)|2=Ξ(u, u; k)u·n.
B(n)[r, u(θ); k]=M(n)[r·u(θ),θ; k],
M(n)(l, θ; k)k2π-ππΞ¯θ-α2, θ+α2; k×exp2ikl sinα2cosnα2dα,
B(n)(r, u; k)=M(n)(r×u, u; k),
M(n)(L, u; k)k2π22π0πΞcosα2u-sinα2w(u, θ), cosα2u+sinα2w(u, θ); k×exp2ikL·wu, θ-π2sinα2×sin α cosnα2dαdθ,
Ua|Ub2(N-1)πφa*(u; k)φb(u; k)dN-1α.
Φ2(N-1)πΞ(u, u; k)dN-1α=mβm(k)×2(N-1)π|φm(u; k)|2dN-1α=mβm(k),
Ga|Gb2(N-1)πΞa(u1, u2; k)×Ξb(u2, u1; k)dN-1α1dN-1α2.
G|G=2(N-1)π|Ξ(u1, u2; k)|2dN-1α1dN-1α2=mβm2(k),
C(k)G|GΦ2=2(N-1)π|Ξ(u1, u2; k)|2dN-1α1dN-1α22(N-1)πΞ(u, u; k)dN-1α2.
0C(k)=mβm2(k)mβm(k)21.
Φ=2πM(n)(l, θ; k)dθdl,
Ga|Gb=2πk2πMa(n)(l, θ; k)Mb(1-n)(l, θ; k)dθdl.
C(k)=2πk2πM(0)(l, θ; k)M(1)(l, θ; k)dθdl2πM(n)(l, θ; k)dθdl2,
Φ=4πAuM(n)(L, u; k)d2LdΩ,
Ga|Gb=2πk24πAuMa(0)(L, u; k)×Mb(0)(L, u; k)d2LdΩ.
C(k)=2πk24πAu[M(0)(L, u; k)]2d2LdΩ4πAuM(0)(L, u; k)d2LdΩ2.
M(n)(l, θ; k)=F(k),
mM(n)lm(l, θ; k)0(2ik)mk2π-ππΞ¯θ-α2, θ+α2; k×exp2ikl sinα2sinmα2cosnα2dα
Ξ¯θ-α2, θ+α2; k=f(θ, k)δ(α).
Ξ¯(θ1, θ2; k)=2πkF(k)comb(θ1-θ2),
G(r1, r2; k)=2πF(k)J0(k|r2-r1|),
Ξ(u1, u2; k)=2πk2F(k)δ(1-u1·u2)π.
G(r1, r2; k)=4πF(k)sin(k|r2-r1|)k|r2-r1|.
Ξ(u1, u2; k)=F(k)2(N-1)πexp-1-u1·u2w2.
Ξ¯(θ1, θ2;k)=F(k)2πexp-1-cos(θ2-θ1)w2=F(k)2πexp-2w2sin2θ2-θ12.
C(k)=12π2πexp-2w2(1-cos θ)dθ=exp-2w2I02w2,
M(n)(l, θ; k)=Fk4π2-ππexp-2w2sin2α2×cos2kl sinα2cosnα2dα.
C(k)=1(4π)24πexp-2w2(1-u1·u2)dΩ1dΩ2=1(4π)24π2π0π×sin ϕ exp-2w2(1-cos ϕ)dϕdθdΩ=w241-exp-4w2.
M(n)(L, u; k)=Fk2(2π)32π0πexp-2w2sin2α2×exp2ikL·wu, θ-π2sinα2×sinα2cosn+1α2dαdθ=Fk2(2π)20πexp-2w2sin2α2×J02k|L|sinα2sinα2cosn+1α2dα.
S(r;k)=2πM(0)[r·u(θ),θ; k]dθ.
I(r)=2πM(0)[r·u(θ),θ; k]dθdk.
f(0)[kl, ku(θ)]  k-1M(0)(l, θ; k).
I(r)=2f(0)(kzx-kx z, k)d2k.
f(1)[kl, ku(θ)]c2M(1)(l, θ; k),
f(2)[kl, ku(θ)]k2M(2)(l, θ; k).
J(r)=2f(1)(kzx-kxz, k)kd2k,
E(r)=2f(2)(kzx-kxz, k)d2k.
I4πg(u1, u2)dΩ1dΩ2,
I=4π2π0πgcosα2u-sinα2w(u, θ),cosα2u+sinα2w(u, θ)h(α)dαdθdΩ,
g(u1, u2)=δ(α-α)=δ[arccos(u1·u2)-α]=δ(u1·u2-cos α)sin α.
4πg(u1, u2)dΩ1dΩ2=4π2π0πgcosα2u-sinα2w(u, θ),cosα2u+sinα2w(u, θ)sin αdαdθdΩ.
2πk24πAuMa(0)(L, u; k)Mb(0)(L, u; k)d2LdΩ=4π2π0πΞacosα2u-sinα2w(u, θ),cosα2u+sinα2w(u, θ); k×Ξbcosα2u+sinα2w(u, θ), cosα2u-sinα2w(u, θ); ksin αdαdθdΩ
=4Ξa(u1, u2; k)×Ξb(u2, u1; k)dΩ1dΩ2=Ga|Gb,

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