Abstract

The geometrical theory of free-space radiative energy transfer is extended to include the case of partially coherent, radiating sources. For quasi-homogeneous sources an explicit formula for the generalized specific intensity that applies both inside and outside the source is given. Such sources are shown to radiate mainly according to classical theory, and the spectral energy flux density is given by the same expression everywhere. Outside the source only nonevanescent, traveling waves exist, and both the energy density function and the cross-spectral density function of the field are explicitly expressed in terms of the generalized specific intensity. Within a quasi-homogeneous-wave model all the energy expressions then reduce to those of classical theory. Inside the source there are also evanescent, standing waves, but their contributions to the cross-spectral density function and the energy expressions are shown to be negligible.

© 1992 Optical Society of America

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References

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  1. E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary sources,”J. Opt. Soc. Am. 68, 953–964 (1978).
    [CrossRef]
  2. W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional sources. I: General theory,” Opt. Acta 28, 227–244 (1981).
    [CrossRef]
  3. W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional sources. II: Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
    [CrossRef]
  4. W. H. Carter, E. Wolf, “Far-zone behavior of electromagnetic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
    [CrossRef] [PubMed]
  5. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  6. L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
    [CrossRef]
  7. E. Wolf, “Coherence and radiometry,”J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  8. 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.
    [CrossRef]
  9. H. M. Pedersen, “Exact geometrical theory of free-space radiative energy transfer,” J. Opt. Soc. Am. A 8, 176–185 (1991); errata J. Opt. Soc. Am. A 8, 1518 (1991).
    [CrossRef]
  10. H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1982).
    [CrossRef]
  11. L. S. Dolin, “Beam description of weakly-inhomogeneous wavefields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559–563 (1964).
  12. E. Wolf, W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205–209 (1975).
    [CrossRef]
  13. W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,”J. Opt. Soc. Am. 67, 785–796 (1977).
    [CrossRef]
  14. E. Wolf, “Noncosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
    [CrossRef]
  15. E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlation,” Opt. Commun. 62, 12–16 (1987).
    [CrossRef]
  16. A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988).
    [CrossRef]
  17. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

1991 (1)

1988 (1)

A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988).
[CrossRef]

1987 (3)

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

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

W. H. Carter, E. Wolf, “Far-zone behavior of electromagnetic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

1982 (1)

H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1982).
[CrossRef]

1981 (2)

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional sources. I: General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional sources. II: Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

1978 (2)

1977 (1)

1975 (1)

E. Wolf, W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205–209 (1975).
[CrossRef]

1965 (1)

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

1964 (1)

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Carter, W. H.

W. H. Carter, E. Wolf, “Far-zone behavior of electromagnetic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional sources. I: General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional sources. II: Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary sources,”J. Opt. Soc. Am. 68, 953–964 (1978).
[CrossRef]

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

E. Wolf, W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205–209 (1975).
[CrossRef]

Dolin, L. S.

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

Gamliel, A.

A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

Mandel, L.

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Pedersen, H. M.

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

H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1982).
[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.
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

Wolf, E.

A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988).
[CrossRef]

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

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

W. H. Carter, E. Wolf, “Far-zone behavior of electromagnetic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional sources. II: Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional sources. I: General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

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

E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary sources,”J. Opt. Soc. Am. 68, 953–964 (1978).
[CrossRef]

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

E. Wolf, W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205–209 (1975).
[CrossRef]

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

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

J. Opt. Soc. Am. (3)

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

Nature (London) (1)

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

Opt. Acta (3)

H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1982).
[CrossRef]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional sources. I: General theory,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional sources. II: Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

Opt. Commun. (3)

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

A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988).
[CrossRef]

E. Wolf, W. H. Carter, “Angular distribution of radiant intensity from sources of different degrees of spatial coherence,” Opt. Commun. 13, 205–209 (1975).
[CrossRef]

Phys. Rev. A (1)

W. H. Carter, E. Wolf, “Far-zone behavior of electromagnetic fields generated by fluctuating current distributions,” Phys. Rev. A 36, 1258–1269 (1987).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Other (3)

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

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.
[CrossRef]

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Equations (81)

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2 V ( r , t ) - 1 c 2 2 t 2 V ( r , t ) = s ( r , t ) .
V ( r , t ) = - 1 4 π 1 r s ( r - r , t - r / c ) d 3 r ,
Γ v ( r 1 , r 2 , τ ) = V * ( r 1 , t - τ ) V ( r 2 , t ) ,
Γ s ( r 1 , r 2 , τ ) = s * ( r 1 , t - τ ) s ( r 2 , t ) .
Γ v ( r 1 , r 2 , τ ) = ( 1 4 π ) 2 1 r 1 r 2 × Γ s [ r 1 - r 1 , r 2 - r 2 , τ - ( r 2 - r 1 ) / c ] d 3 r 1 d 3 r 2 .
W v ( r 1 , r 2 , ω ) = 1 2 π - Γ v ( r 1 , r 2 , τ ) exp ( i ω τ ) d τ ,
W s ( r 1 , r 2 , ω ) = 1 2 π - Γ s ( r 1 , r 2 , τ ) exp ( i ω τ ) d τ
W v ( r 1 , r 2 , ω ) = ( 1 4 π ) 2 W s ( r 1 - r 1 , r 2 - r 2 , ω ) 1 r 1 r 2 × exp [ i ( ω / c ) ( r 2 - r 1 ) ] d 3 r 1 d 3 r 2 ,
W v ( r 1 , r 2 , ω ) = W s ( r 1 , r 2 , ω ) G * ( r 1 - r 1 ) × G ( r 2 - r 2 ) d 3 r 1 d 3 r 2 ,
G ( r ) = - 1 4 π r exp ( i k r ) ,
2 G ( r ) + k 2 G ( r ) = δ 2 ( r ) ,
1 2 W v ( r 1 , r 2 , ω ) + k 2 W v ( r 1 , r 2 , ω ) = W s ( r 1 , r 2 , ω ) G ( r 2 - r 2 ) d 3 r 2 ,
2 2 W v ( r 1 , r 2 , ω ) + k 2 W v ( r 1 , r 2 , ω ) = W s ( r 1 , r 2 , ω ) G * ( r 1 - r 1 ) d 3 r 1 ,
ξ 2 W ( r , ξ ) + k 2 W ( r , ξ ) + ¼ 2 ( r , ξ ) = 1 2 [ W s ( r - ξ / 2 , ξ - ξ ) G ( ξ ) d 3 ξ + W s ( r + ξ / 2 , ξ - ξ ) G * ( - ξ ) d 3 ξ ] ,
ξ · W ( r , ξ ) = - 1 2 [ W s ( r - ξ / 2 , ξ - ξ ) G ( ξ ) d 3 ξ - W s ( r + ξ / 2 , ξ - ξ ) G * ( - ξ ) d 3 ξ ] ,
ξ 2 W ( r , ξ ) + k 2 W ( r , ξ ) + ¼ 2 W ( r , ξ ) = W s ( r , ξ ) * Re G ( ξ ) ,
ξ · W ( r , ξ ) = - i W s ( r , ξ ) * Im G ( ξ ) ,
E ( r , ω ) = 1 c ( 1 + 1 4 k 2 2 ) W ( r , 0 ) ,
F ( r , ω ) = 1 i k [ ξ W ( r , ξ ) ] ξ = 0 .
I ( r , s ) = ( 1 2 π ) 3 0 W ( r , κ s ) ( κ 3 / k ) d κ ,
W ( r , κ ) = W ( r , ξ ) exp ( - i κ · ξ ) d 3 ξ
F ( r , ω ) = I ( r , s ) s d Ω ( s ) ,
Im G ( ξ ) = - k 4 π sin ( k ξ ) / ( k ξ ) = - k ( 1 4 π ) 2 exp ( i k s · ξ ) d Ω ( s ) ,
κ s · W ( r , κ s ) = W s ( r , κ s ) k ( 1 4 π ) 2 ( 2 π ) 3 δ 3 ( κ s - k s ) d Ω ( s ) ,
W s ( r , κ ) = W s ( r , ξ ) exp ( - i κ · ξ ) d 3 ξ
δ 3 ( κ s - k s ) = ( 1 / k ) 2 δ ( κ - k ) δ 2 ( s - s ) ,
s · I ( r , s ) = W s ( r , k s ) / ( 4 π ) 2 .
I ( r , s ) = ( 1 4 π ) 2 0 W s ( r - σ s , k s ) d σ .
J ( s ) = s 0 · s I ( r , s ) δ ( r · s 0 - p 0 ) d 3 r ,
J ( s ) = I ( r , s ) δ ( r · s - p ) d 3 r ,
J ( s ) = ( 1 4 π ) 2 W s ( r , k s ) d 3 r = ( 1 4 π ) 2 W s ( r 1 , r 2 , ω ) exp [ i k s · ( r 2 - r 1 ) ] d 3 r 1 d 3 r 2 ,
W s ( r , ξ ) = S ( r , ω ) μ ( ξ ) .
W s ( r , κ ) = S ( r , ω ) M ( κ ) ,
M ( κ ) = μ ( ξ ) exp ( - i κ · ξ ) d 3 ξ
I ( r , s ) = ( 1 4 π ) 3 M ( k s ) 0 S ( r - σ s , ω ) d σ ,
J ( s ) = ( 1 4 π ) 2 M ( k s ) S ( r , ω ) d 3 r .
ξ 2 W ( r , ξ ) + k op 2 W ( r , ξ ) = 0 ,
k op = ( k 2 + ¼ 2 ) 1 / 2
W ( r , ξ ) = exp ( i k op s · ξ ) H ( r , s ) d Ω ( s ) ,
W ( r , ξ ) = k k op exp ( i k op s · ξ ) I ( r , s ) d Ω ( s ) .
E ( r , ω ) = k op k 1 c I ( r , s ) d Ω ( s ) ,
F ( r , ω ) = I ( r , s ) s d Ω ( s ) ,
J ( s ) = s · s 0 I ( r , s ) δ ( r · s 0 - p 0 ) d 3 r ,
I ( r , s ) = ( 1 4 π ) 2 0 W s ( r - σ s , k s ) d σ .
W ( r , ξ ) = I ( r , s ) exp ( i k s · ξ ) d Ω ( s ) ,
E ( r , ω ) = 1 c I ( r , s ) d Ω ( s ) .
ξ 2 W ( r , ξ ) + k op 2 W ( r , ξ ) = W s ( r , ξ ) * Re G ( ξ ) .
W ( r , ξ ) = W h ( r , ξ ) + W p ( r , ξ ) ,
W h ( r , ξ ) = k k op exp ( i k op s · ξ ) I h ( r , s ) d Ω ( s ) ,
W p ( r , ξ ) = G op ( ξ ) * W s ( r , ξ ) * Re G ( ξ ) ,
G o p ( ξ ) = 1 4 π ξ exp ( i k op ξ )
W p ( r , ξ ) = - 1 k op 2 - k 2 Re [ G op ( ξ ) - G ( ξ ) ] * W s ( r , ξ ) ,
μ ( ξ ) = exp ( - α ξ ) ,
W p ( r , ξ ) = - 1 k op 2 - k 2 Re [ G op ( ξ ) - G ( ξ ) ] * μ ( ξ ) S ( r , ω ) = - 1 k op 2 - k 2 ( M ( k op s ) { Re G op ( ξ ) + μ ( ξ ) [ 1 / ( 4 π ξ ) + ( k op 2 + α 2 ) / ( 8 π α ) ] } - M ( k s ) { Re G ( ξ ) + μ ( ξ ) [ 1 / ( 4 π ξ ) + ( k 2 + α 2 ) / ( 8 π α ) ] } ) S ( r , ω ) .
W p ( r , ξ ) = - M ( k s ) { ξ Im G ( ξ ) / ( 2 k ) + μ ( ξ ) / ( 8 π α ) + 2 [ Re G ( ξ ) + μ ( ξ ) / ( 4 π ξ ) ] / ( k 2 + α 2 ) } S ( r , ω ) .
W p ( r , 0 ) = M ( k s ) S ( r , ω ) 3 α 2 - k 2 8 π α ( k 2 + α 2 ) .
W p ( r , ξ ) M ( k s ) S ( r , ω ) sin ( k ξ ) / ( 8 π k ) .
I h ( r , s ) = I ( r , s ) - I p ( r , s ) .
W p ( r , κ ) = - 1 k op 2 - k 2 [ g op ( κ ) - g ( κ ) ] M ( κ ) S ( r , ω ) ,
I p ( r , s ) = ( 1 2 π ) 3 0 W p ( r , κ s ) ( κ 3 / k ) d κ = - 1 k op 2 - k 2 ( 1 2 π ) 3 0 [ g op ( κ s ) - g ( κ s ) ] × M ( κ s ) ( κ 3 / k ) d κ S ( r , ω ) .
I p ( r , s ) = 2 α / k k op 2 - k 2 ( 1 2 π ) 2 0 ( 1 κ 2 - k op 2 - 1 κ 2 - k 2 ) × 2 κ 3 d κ ( κ 2 + α 2 ) 2 S ( r , ω ) = 2 α / k k op 2 - k 2 ( 1 2 π ) 2 0 ( 1 x - k op 2 - 1 x - k 2 ) × x d x ( x + α 2 ) 2 S ( r , ω ) .
I p ( r , s ) = 2 α / k k op 2 - k 2 ( 1 2 π ) 2 × [ k op 2 ( α 2 + k op 2 ) 2 log ( α 2 / k op 2 ) + 1 α 2 + k op 2 - k 2 ( α 2 + k 2 ) 2 log ( α 2 / k 2 ) - 1 α 2 + k 2 ] S ( r , ω ) = M ( k s ) ( 1 2 π ) 3 1 k { [ α 2 - k 2 α 2 + k 2 log ( α / k ) - 1 ] S ( r , ω ) - 1 α 2 + k 2 [ α 2 - k 2 α 2 + k 2 log ( α / k ) + α 2 - 5 k 2 8 k 2 ] × 1 2 2 S ( r , ω ) + } ,
I p ( r , s ) = ( 1 2 π ) 4 M ( k s ) S ( r , ω ) λ [ α 2 - k 2 α 2 + k 2 log ( α / k ) - 1 ] .
I p ( r , s ) = ( 1 4 π ) 2 M ( k s ) S ( r , ω ) δ σ ,
δ σ = ( λ / π 2 ) [ α 2 - k 2 α 2 + k 2 log ( α / k ) - 1 ] = ( λ / π ) 2 { λ 2 - ( 2 π l c ) 2 λ 2 + ( 2 π l c ) 2 log [ λ / ( 2 π l c ) ] - 1 } .
I ( r , s ) = I h ( r , s ) + I p ( r , s ) = ( 1 4 π ) 2 M ( k s ) 0 S ( r - σ s , ω ) d σ .
I h ( r , s ) = I ( r , s ) .
I ( r , s ) = I h ( r , s ) = ( 1 4 π ) 2 M ( k s ) S ( r , ω ) r s .
W h ( r , ξ ) = 1 4 π M ( k s ) S ( r , ω ) r s sin ( k ξ ) / ( k ξ ) .
W p ( r , 0 ) / W h ( r , 0 ) = | 3 α 2 - k 2 2 α r s ( k 2 - α 2 ) | = l c 2 r s | 3 λ 2 - ( 2 π l c ) 2 λ 2 + ( 2 π l c ) 2 | 1 ,
μ ( ξ ) * G ( ξ ) = μ ( ξ ) G ( ξ - ξ ) d 3 ξ
M ( κ ) = μ ( ξ ) exp ( - i κ · ξ ) d 3 ξ = 2 π i κ [ 1 ( κ - i α ) 2 - 1 ( κ + i α ) 2 ] = 8 π α ( κ 2 + α 2 ) 2 .
g ( κ ) = G ( ξ ) exp ( - i κ · ξ ) d 3 ξ = - 1 / ( κ 2 - k 2 ) ,
μ ( ξ ) * G ( ξ ) = ( 1 2 π ) 3 M ( κ ) g ( κ ) exp ( i κ · ξ ) d 3 κ = ( 1 2 π ) 2 0 M ( κ ) g ( κ ) [ - 1 1 exp ( i κ ξ x ) d x ] κ 2 d κ = ( 1 2 π ) 2 1 i ξ - M ( κ ) g ( κ ) exp ( i κ ξ ) κ d κ ,
u ( ξ ) * G ( ξ ) = 1 2 π ξ - 1 κ 2 - k 2 [ 1 ( κ - i α ) 2 - 1 ( κ + i α ) 2 ] × exp ( i κ ξ ) d κ .
u ( ξ ) * G ( ξ ) = 8 π α ( k 2 + α 2 ) 2 { - exp ( i k ξ ) / ( 4 π ξ ) + exp ( - α ξ ) [ 1 / ( 4 π ξ ) + ( k 2 + α 2 ) / ( 8 π α ) ] } = M ( k s ) { G ( ξ ) + μ ( ξ ) [ 1 / ( 4 π ξ ) + ( k 2 + α 2 ) / ( 8 π α ) ] } .
μ ( 0 ) * G ( 0 ) = 8 π α ( k 2 + α 2 ) 2 [ - 2 i k α + ( k 2 - α 2 ) ] / ( 8 π α ) = M ( k s ) [ - 2 i k α + ( k 2 - α 2 ) ] / ( 8 π α ) .
J = 0 x d x ( x - k 2 ) ( x + α 2 ) 2 = 0 d x ( x + α 2 ) 2 + k 2 - k 2 d x ( x + α 2 + k 2 ) 2 x = 0 d x ( x + α 2 ) 2 + k 2 - k 2 k 2 d x ( x + α 2 + k 2 ) 2 x + k 2 k 2 d x ( x + α 2 + k 2 ) 2 x = α 2 d x x 2 + 2 k 2 ( k 2 + α 2 ) 0 k 2 d x [ x 2 - ( α 2 + k 2 ) 2 ] 2 + k 2 k 2 d x ( x + α 2 + k 2 ) 2 x = 1 α 2 - 4 k 2 ( k 2 + α 2 ) 2 0 k 2 / ( α 2 + k 2 ) d x ( 1 - x 2 ) 2 + k 2 k 2 d x ( x + α 2 + k 2 ) 2 x .
J = k 2 ( α 2 + k 2 ) 2 log ( α 2 / k 2 ) + 1 α 2 + k 2 .
J ( k 2 ) = M ( k s ) 1 4 π α [ α 2 - k 2 α 2 + k 2 log ( α / k ) - 1 ] ,
J ( k 2 ) = - M ( k s ) 1 π α 1 α 2 + k 2 × [ α 2 - k 2 α 2 + k 2 log ( α / k ) + α 2 - 5 k 2 8 k 2 ] ,

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