Abstract

We consider the theory of radiative transport in quasi-homogeneous random media. We derive the radiative transport equation that governs the propagation of light in such media. This result provides conditions under which it is justified to apply radiative transport theory to spatially inhomogeneous media.

© 2018 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  3. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  4. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).
  5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  6. L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer (Gordon and Breach Publishers, 1996).
  7. M. I. Mischenko and L. D. Travis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).
  8. Y. N. Barabanenkov and V. M. Finkelberg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587–591 (1968).
  9. D. Vollhardt and P. Wolfle, “Diagrammatic self-consistent treatment of the Anderson localization problem in d ≤ 2 dimensions,” Phys. Rev. B 22, 4666–4679 (1980).
    [Crossref]
  10. U. Frisch, Probabilistic Methods in Applied Mathematics, A. T. Barucha-Reid, ed. (Academic, 1968).
  11. F. C. MacKintosh and S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
    [Crossref]
  12. L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
    [Crossref]
  13. G. Bal, T. Komorowski, and L. Ryzhik, “Kinetic limits for waves in a random medium,” Kinet. Relat. Models 3, 529–644 (2010).
    [Crossref]
  14. A. Caze and J. C. Schotland, “Diagrammatic and asymptotic approaches to the origins of radiative transport theory: tutorial,” J. Opt. Soc. Am. A 32, 1475–1484 (2015).
    [Crossref]
  15. S. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Prob. 25, 123010 (2009).
    [Crossref]
  16. R. A. Silverman, “Scattering of plane waves by locally homogeneous dielectric noise,” Proc. Cambridge Philos. Soc. 54, 530–537 (1958).
    [Crossref]
  17. W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
    [Crossref]
  18. D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11, 1128–1135 (1994).
    [Crossref]
  19. D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
    [Crossref]
  20. T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23, 1631–1638 (2006).
    [Crossref]
  21. Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett. 35, 4000–4003 (2010).
    [Crossref]
  22. D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294, 43–48 (2013).
    [Crossref]
  23. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1997).
  24. E. Wolf, “New theory of radiative transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
    [Crossref]

2015 (1)

2013 (1)

D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294, 43–48 (2013).
[Crossref]

2010 (2)

Y. Xin, Y. He, Y. Chen, and J. Li, “Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media,” Opt. Lett. 35, 4000–4003 (2010).
[Crossref]

G. Bal, T. Komorowski, and L. Ryzhik, “Kinetic limits for waves in a random medium,” Kinet. Relat. Models 3, 529–644 (2010).
[Crossref]

2009 (1)

S. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Prob. 25, 123010 (2009).
[Crossref]

2006 (1)

1997 (1)

D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[Crossref]

1996 (1)

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[Crossref]

1994 (1)

1989 (1)

F. C. MacKintosh and S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[Crossref]

1988 (1)

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[Crossref]

1980 (1)

D. Vollhardt and P. Wolfle, “Diagrammatic self-consistent treatment of the Anderson localization problem in d ≤ 2 dimensions,” Phys. Rev. B 22, 4666–4679 (1980).
[Crossref]

1976 (1)

E. Wolf, “New theory of radiative transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
[Crossref]

1968 (1)

Y. N. Barabanenkov and V. M. Finkelberg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587–591 (1968).

1958 (1)

R. A. Silverman, “Scattering of plane waves by locally homogeneous dielectric noise,” Proc. Cambridge Philos. Soc. 54, 530–537 (1958).
[Crossref]

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

Apresyan, L. A.

L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer (Gordon and Breach Publishers, 1996).

Arridge, S.

S. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Prob. 25, 123010 (2009).
[Crossref]

Bal, G.

G. Bal, T. Komorowski, and L. Ryzhik, “Kinetic limits for waves in a random medium,” Kinet. Relat. Models 3, 529–644 (2010).
[Crossref]

Barabanenkov, Y. N.

Y. N. Barabanenkov and V. M. Finkelberg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587–591 (1968).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1997).

Carter, W. H.

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[Crossref]

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Caze, A.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Chen, Y.

Finkelberg, V. M.

Y. N. Barabanenkov and V. M. Finkelberg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587–591 (1968).

Fischer, D. G.

Frisch, U.

U. Frisch, Probabilistic Methods in Applied Mathematics, A. T. Barucha-Reid, ed. (Academic, 1968).

He, Y.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

John, S.

F. C. MacKintosh and S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[Crossref]

Keller, J. B.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[Crossref]

Komorowski, T.

G. Bal, T. Komorowski, and L. Ryzhik, “Kinetic limits for waves in a random medium,” Kinet. Relat. Models 3, 529–644 (2010).
[Crossref]

Kravtsov, Y. A.

L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer (Gordon and Breach Publishers, 1996).

Kuebel, D.

D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294, 43–48 (2013).
[Crossref]

Li, J.

MacKintosh, F. C.

F. C. MacKintosh and S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Mischenko, M. I.

M. I. Mischenko and L. D. Travis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Montambaux, G.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

Papanicolaou, G.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[Crossref]

Ryzhik, L.

G. Bal, T. Komorowski, and L. Ryzhik, “Kinetic limits for waves in a random medium,” Kinet. Relat. Models 3, 529–644 (2010).
[Crossref]

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[Crossref]

Schotland, J. C.

Silverman, R. A.

R. A. Silverman, “Scattering of plane waves by locally homogeneous dielectric noise,” Proc. Cambridge Philos. Soc. 54, 530–537 (1958).
[Crossref]

Travis, L. D.

M. I. Mischenko and L. D. Travis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Visser, T. D.

D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294, 43–48 (2013).
[Crossref]

T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23, 1631–1638 (2006).
[Crossref]

Vollhardt, D.

D. Vollhardt and P. Wolfle, “Diagrammatic self-consistent treatment of the Anderson localization problem in d ≤ 2 dimensions,” Phys. Rev. B 22, 4666–4679 (1980).
[Crossref]

Wolf, E.

D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294, 43–48 (2013).
[Crossref]

T. D. Visser, D. G. Fischer, and E. Wolf, “Scattering of light from quasi-homogeneous sources by quasi-homogeneous media,” J. Opt. Soc. Am. A 23, 1631–1638 (2006).
[Crossref]

D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[Crossref]

D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11, 1128–1135 (1994).
[Crossref]

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[Crossref]

E. Wolf, “New theory of radiative transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1997).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wolfle, P.

D. Vollhardt and P. Wolfle, “Diagrammatic self-consistent treatment of the Anderson localization problem in d ≤ 2 dimensions,” Phys. Rev. B 22, 4666–4679 (1980).
[Crossref]

Xin, Y.

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Inverse Prob. (1)

S. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Prob. 25, 123010 (2009).
[Crossref]

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

Kinet. Relat. Models (1)

G. Bal, T. Komorowski, and L. Ryzhik, “Kinetic limits for waves in a random medium,” Kinet. Relat. Models 3, 529–644 (2010).
[Crossref]

Opt. Commun. (3)

D. G. Fischer and E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[Crossref]

W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[Crossref]

D. Kuebel, T. D. Visser, and E. Wolf, “Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media,” Opt. Commun. 294, 43–48 (2013).
[Crossref]

Opt. Lett. (1)

Phys. Rev. B (2)

D. Vollhardt and P. Wolfle, “Diagrammatic self-consistent treatment of the Anderson localization problem in d ≤ 2 dimensions,” Phys. Rev. B 22, 4666–4679 (1980).
[Crossref]

F. C. MacKintosh and S. John, “Diffusing-wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[Crossref]

Phys. Rev. D (1)

E. Wolf, “New theory of radiative transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
[Crossref]

Proc. Cambridge Philos. Soc. (1)

R. A. Silverman, “Scattering of plane waves by locally homogeneous dielectric noise,” Proc. Cambridge Philos. Soc. 54, 530–537 (1958).
[Crossref]

Sov. Phys. JETP (1)

Y. N. Barabanenkov and V. M. Finkelberg, “Radiation transport equation for correlated scatterers,” Sov. Phys. JETP 26, 587–591 (1968).

Wave Motion (1)

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[Crossref]

Other (9)

U. Frisch, Probabilistic Methods in Applied Mathematics, A. T. Barucha-Reid, ed. (Academic, 1968).

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

L. A. Apresyan and Y. A. Kravtsov, Radiation Transfer (Gordon and Breach Publishers, 1996).

M. I. Mischenko and L. D. Travis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1997).

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Figures (3)

Fig. 1.
Fig. 1. Lowest-order self-energy diagram.
Fig. 2.
Fig. 2. Lowest-order irreducible-vertex diagram.
Fig. 3.
Fig. 3. Correlation function G G * in terms of ladder diagrams. A double line with a left-pointing arrow denotes a factor of G and a double line with a right-pointing arrow denotes a factor of G * .

Equations (62)

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s ^ · I + μ e I = μ s d s ^ A ( s ^ , s ^ ) I ( r , s ^ ) ,
2 U ( r ) + k 0 2 ( 1 + 4 π η ( r ) ) U ( r ) = 4 π S ( r ) ,
η ( r ) = 0 ,
η ( r ) η ( r ) = C ( r , r ) ,
C ( r , r ) = C f ( | r r | ) C s ( ( r + r ) / 2 ) ,
C f ( r ) = C 0 e r 2 / l f 2 , C s ( r ) = e r 2 / l s 2 ,
U ( r ) = d 3 r G ( r , r ) S ( r ) .
G ( r , r ) = G 0 ( r , r ) + k 0 2 d 3 r G 0 ( r , r ) η ( r ) G ( r , r ) ,
G 0 ( r , r ) = e i k 0 | r r | | r r | .
G 0 ( r , r ) = 4 π d 3 k ( 2 π ) 3 e i k · ( r r ) k 2 k 0 2 i ε ,
G ( r , r ) = G 0 ( r , r ) + d 3 r 1 d 3 r 2 G 0 ( r , r 1 ) Σ ( r 1 , r 2 ) G ( r 2 , r ) ,
Σ ( r 1 , r 2 ) = k 0 4 G 0 ( r 1 , r 2 ) C ( r 1 , r 2 ) .
R = r + r 2 , R = r r .
G ( R , R ) = G 0 ( R ) + d 3 r 1 d 3 r 2 G 0 ( R + R / 2 r 1 ) Σ ( r 1 , r 2 ) × G ( ( r 2 + R R / 2 ) / 2 , r 2 R + R / 2 ) ,
G ( R , R ) = G ( R + R / 2 , R R / 2 ) , G 0 ( r r ) = G 0 ( r , r ) .
G ˜ ( R , k ) = d 3 R e i k · R G ( R , R ) ,
G ˜ ( R , k ) = G ˜ 0 ( k ) + C s ( R ) G ˜ 0 ( k ) Σ 0 ( k ) G ˜ ( R , k ) ,
G ˜ 0 ( k ) = 4 π k 2 k 0 2 i ε ,
Σ 0 ( k ) = k 0 4 d 3 k ( 2 π ) 3 C ˜ f ( k k ) k 2 k 0 2 i ε ,
C s ( R + R ) C s ( R ) , G ˜ ( R + R , k ) G ˜ ( R , k ) ,
G ˜ ( R , k ) = 4 π k 2 k 0 2 4 π C s ( R ) Σ 0 ( k ) i ε .
1 k 2 k 0 2 i ε = P 1 k 2 k 0 2 + i π δ ( k 2 k 0 2 ) ,
δ ( k 2 k 0 2 ) = 1 2 k 0 ( δ ( k k 0 ) + δ ( k + k 0 ) ) ,
Re Σ 0 ( k ) = k 0 4 P d 3 k ( 2 π ) 3 C ˜ f ( k k ) k 2 k 0 2 ,
Im Σ 0 ( k ) = k 0 5 d s ^ 4 π C ˜ f ( k k 0 s ^ ) ,
1 s = k 0 4 d s ^ C ˜ f ( k 0 | s ^ s ^ | ) ,
G ˜ ( R , k ) = 4 π k 2 κ 2 ( R ) i ε ,
κ ( R ) = k 0 ( 1 + i C s ( R ) 2 k 0 s ) .
G ( r , r ) = e i k 0 | r r | | r r | exp [ C s ( ( r + r ) / 2 ) | r r | / 2 l s ] .
U ( r ) U * ( r ) = d 3 r 1 d 3 r 2 G ( r , r 1 ) G * ( r , r 2 ) S ( r 1 ) S * ( r 2 ) ,
G ( r 1 , r 2 ) G * ( r 1 , r 2 ) = G ( r 1 , r 2 ) G * ( r 1 , r 2 ) + d 3 r d 3 r G ( r 1 , r ) G * ( r 1 , r ) Γ ( r , r ) × G ( r , r 2 ) G * ( r , r 2 ) ,
Γ ( r , r ) = k 0 4 C ( r , r ) .
W ( r , k ) = d 3 r ( 2 π ) 3 e i k · r U ( r r / 2 ) U * ( r + r / 2 ) .
I ( r ) = d 3 k W ( r , k ) .
U ( r ) U * ( r ) = d 3 k e k · ( r r ) W ( ( r + r ) / 2 , k ) .
W ˜ ( q , k ) = e i q · r U ˜ ( k + q / 2 ) U ˜ * ( k + q / 2 ) / ( 2 π ) 3 + k 0 4 e i q · r G ˜ ( r , k / 2 + q ) G ˜ * ( r , k / 2 + q ) C s ( r ) × d 3 k ( 2 π ) 3 C ˜ f ( k k ) W ˜ ( q , k ) ,
a b = a b 1 / b 1 / a ,
G ˜ ( r , k / 2 + q ) G ˜ * ( r , k / 2 + q ) = Δ G ( r , q , k ) 2 k · q + Δ Σ ( r , q , k ) ,
Δ G ( r , q , k ) = 4 π ( G ˜ ( r , k + q / 2 ) G ˜ * ( r , k + q / 2 ) ) ,
Δ Σ ( r , q , k ) = 4 π C s ( r ) ( Δ Σ 0 ( k + q / 2 ) Δ Σ 0 * ( k + q / 2 ) ) .
( 2 k · q + Δ Σ ( r , q , k ) ) W ˜ ( q , k ) = k 0 4 Δ G ( r , q , k ) C s ( r ) × d 3 k ( 2 π ) 3 C ˜ f ( k k ) W ˜ ( q , k ) + 1 ( 2 π ) 3 Δ G ( r , q , k ) S ˜ ( k + q / 2 ) S ˜ * ( k + q / 2 ) .
k · r W ( r , k ) + 1 2 i d 3 q ( 2 π ) 3 e i q · r Δ Σ ( r , q , k ) W ˜ ( q , k ) = k 0 4 2 i d 3 q ( 2 π ) 3 e i q · r d 3 k ( 2 π ) 3 C s ( r ) Δ G ( r , q , k ) C ˜ f ( k k ) × W ˜ ( q , k ) + 1 2 i d 3 q ( 2 π ) 6 e i q · r Δ G ( r , q , k ) S ˜ ( k + q / 2 ) S * ˜ ( k + q / 2 ) .
Δ G ( r , 0 , k ) = 2 i ( 2 π ) 3 k 0 δ ( k k 0 ) .
Δ Σ ( r , 0 , k ) = 2 i k C s ( r ) / l s .
δ ( k k 0 ) I ( r , s ^ ) = k 0 W ( r , k s ^ ) ,
A ( s ^ , s ^ ) = k 0 4 l s C ˜ f ( k 0 ( s ^ s ^ ) ) ,
μ s ( r ) = C s ( r ) / l s ,
I 0 ( r , s ^ ) = 1 k 0 2 d 3 q ( 2 π ) 3 e i q · r S ˜ ( k 0 s ^ + q / 2 ) S ˜ * ( k 0 s ^ + q / 2 ) ,
s ^ · r I ( r , s ^ ) + μ s ( r ) I ( r , s ^ ) = μ s ( r ) d s ^ A ( s ^ , s ^ ) I ( r , s ^ ) + I 0 ( r , s ^ ) .
· D ( r ) u ( r ) = Q ( r ) .
u ( r ) = 1 c d s ^ I ( r , s ^ ) , Q ( r ) = d s ^ I 0 ( r , s ^ ) ,
D ( r ) = c 3 ( 1 g ) μ s ( r ) , g = s ^ · s ^ A ( s ^ , s ^ ) d s ^ ,
U ( r ) U ( r ) = U ( r ) U ( r ) + k 0 4 d 3 R d 3 R G ( r , R ) G * ( r , R ) C f ( R R ) C s ( ( R + R ) / 2 ) U ( R ) U ( R ) .
W ˜ ( q , k ) = W ˜ 1 ( q , k ) + W ˜ 2 ( q , k ) ,
W ˜ 1 ( q , k ) = 1 ( 2 π ) 3 d 3 Δ r d 3 r e i q · Δ r e i k · r U ( r + Δ r r / 2 ) U * ( r + Δ r + r / 2 )
W ˜ 2 ( q , k ) = k 0 4 ( 2 π ) 3 d 3 Δ r d 3 r d 3 R d 3 R e i q · Δ r e i k · r G ( r + Δ r r / 2 , R ) G * ( r + Δ r + r / 2 , R ) × C f ( R R ) C s ( ( R + R ) / 2 ) U ( R ) U ( R ) .
W ˜ 1 ( q , k ) = e i q · r ( 2 π ) 3 U ˜ ( k + q / 2 ) U ˜ ( k + q / 2 ) ,
W ˜ 2 ( q , k ) = k 0 4 ( 2 π ) 3 d 3 Δ r d 3 r d 3 R d 3 R e i q · Δ r e i k · r G ( ( r + Δ r r / 2 + R ) / 2 , r + Δ r r / 2 R ) × G * ( ( r + Δ r + r / 2 + R ) / 2 , r + Δ r + r / 2 R ) C f ( R R ) C s ( ( R + R ) / 2 ) U ( R ) U ( R ) .
G ( r + Δ r , r ) G ( r , r ) ,
C s ( r + Δ r ) C s ( r ) ,
W ˜ 2 ( q , k ) = k 0 4 ( 2 π ) 3 d 3 Δ r d 3 r d 3 R d 3 R e i q · ( Δ r + R ) e i k · ( r R ) G ( r , r r / 2 + R ) × G * ( r , r + r / 2 R ) C s ( r ) K ( r + R , R ) .
W ˜ 2 ( q , k ) = k 0 4 e i q · r G ˜ ( r , k / 2 + q ) G * ˜ ( r , k / 2 + q ) C s ( r ) d 3 k ( 2 π ) 3 C ˜ f ( k k ) W ˜ ( q , k ) .