Abstract

The radiative transport equation (RTE) is used widely to describe the propagation of multiply scattered light in disordered media. In this tutorial, we present two derivations of the RTE for scalar wave fields. The first derivation is based on diagrammatic perturbation theory, while the second stems from an asymptotic multiscale expansion. Although the two approaches are quite distinct mathematically, some common ground can be found and is discussed.

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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. S. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
    [Crossref]
  4. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  5. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University, 2007).
  6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  7. Y. N. Barabanenkov and V. M. Finkelberg, “Radiation transport equation for correlated scatterers,” Sov. Phys. J. Exp. Theor. Phys. 26, 587 (1968).
  8. 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]
  9. U. Frisch, Probabilistic Methods in Applied Mathematics, A. T. Barucha-Reid, ed. (Academic, 1968).
  10. 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]
  11. 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]
  12. G. Bal, T. Komorowski, and L. Ryzhik, “Kinetic limits for waves in a random medium,” Kinet. Relat. Models 3, 529–644 (2010).
    [Crossref]
  13. E. Wolf, “New theory of radiative transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976).
    [Crossref]
  14. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1997).
  15. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [Crossref]
  16. A. Nayfeh, Perturbation Methods (Wiley, 1973).
  17. L. Erdos and H. T. Yau, “Linear Boltzmann equation as the weak coupling limit of a random Schrodinger equation,” Comm. Pure Appl. Math. 53, 667–735 (2000).
    [Crossref]

2010 (1)

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 Probl. 25, 123010 (2009).
[Crossref]

2000 (1)

L. Erdos and H. T. Yau, “Linear Boltzmann equation as the weak coupling limit of a random Schrodinger equation,” Comm. Pure Appl. Math. 53, 667–735 (2000).
[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]

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]

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. J. Exp. Theor. Phys. 26, 587 (1968).

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Akkermans, E.

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

Arridge, S.

S. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 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. J. Exp. Theor. Phys. 26, 587 (1968).

Born, M.

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

Case, K. M.

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

Chandrasekhar, S.

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

Erdos, L.

L. Erdos and H. T. Yau, “Linear Boltzmann equation as the weak coupling limit of a random Schrodinger equation,” Comm. Pure Appl. Math. 53, 667–735 (2000).
[Crossref]

Finkelberg, V. M.

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

Frisch, U.

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

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]

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).

Montambaux, G.

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

Nayfeh, A.

A. Nayfeh, Perturbation Methods (Wiley, 1973).

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.

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

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Wolf, E.

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]

Yau, H. T.

L. Erdos and H. T. Yau, “Linear Boltzmann equation as the weak coupling limit of a random Schrodinger equation,” Comm. Pure Appl. Math. 53, 667–735 (2000).
[Crossref]

Zweifel, P. F.

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

Comm. Pure Appl. Math. (1)

L. Erdos and H. T. Yau, “Linear Boltzmann equation as the weak coupling limit of a random Schrodinger equation,” Comm. Pure Appl. Math. 53, 667–735 (2000).
[Crossref]

Inverse Probl. (1)

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

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]

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Phys. Rev. B (2)

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]

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]

Phys. Rev. D (1)

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

Sov. Phys. J. Exp. Theor. Phys. (1)

Y. N. Barabanenkov and V. M. Finkelberg, “Radiation transport equation for correlated scatterers,” Sov. Phys. J. Exp. Theor. Phys. 26, 587 (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 (8)

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

A. Nayfeh, Perturbation Methods (Wiley, 1973).

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).

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Tables (1)

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Table 1. Diagrammatic Rules

Equations (109)

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s ^ · I ( r , s ^ ) + μ s I ( r , s ^ ) = 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 ) ,
· J = 0 ,
J = 1 2 i ( U * U U U * ) .
W ( r , k ) = d r ( 2 π ) 3 e i k · r U ( r r / 2 ) U * ( r + r / 2 ) .
I ( r ) = W ( r , k ) d k .
J ( r ) = k W ( r , k ) d k .
U ( r ) = A e i k 0 s ^ · r ,
W ( r , k ) = | A | 2 δ ( k k 0 s ^ ) ,
I = | A | 2 , J = | A | 2 s ^ .
η = 0 ,
η ( r ) η ( r ) = C ( r r ) ,
U ( r ) = G ( r , r ) S ( r ) d r .
2 G ( r , r ) + k 0 2 ( 1 + 4 π η ( r ) ) G ( r , r ) = 4 π δ ( r r ) .
G ( r , r ) = G 0 ( r , r ) + k 0 2 d r 1 G 0 ( r , r 1 ) η ( r 1 ) G ( r 1 , r ) ,
G 0 ( r , r ) = exp ( i k 0 | r r | ) | r r | .
G ( r , r ) = G 0 ( r , r ) + k 0 2 d r 1 G 0 ( r , r 1 ) η ( r 1 ) G 0 ( r 1 , r ) + k 0 4 d r 1 d r 2 G 0 ( r , r 1 ) η ( r 1 ) G 0 ( r 1 , r 2 ) η ( r 2 ) G 0 ( r 2 , r ) + k 0 6 d r 1 d r 2 d r 3 G 0 ( r , r 1 ) η ( r 1 ) G 0 ( r 1 , r 2 ) × η ( r 2 ) G 0 ( r 2 , r 3 ) η ( r 3 ) G 0 ( r 3 , r ) + .
η ( r 1 ) η ( r p ) = π η ( r π ( 1 ) ) η ( r π ( 2 ) ) η ( r π ( p 1 ) η ( r π ( p ) ) ,
η ( r 1 ) η ( r 2 ) η ( r 3 ) η ( r 4 ) = η ( r 1 ) η ( r 2 ) η ( r 3 ) η ( r 4 ) + η ( r 1 ) η ( r 3 ) η ( r 2 ) η ( r 4 ) + η ( r 1 ) η ( r 4 ) η ( r 2 ) η ( r 3 ) .
Σ = Σ 1 + Σ 2 a + Σ 2 b + ,
Σ 1 ( r 1 , r 2 ) = k 0 4 C ( r 1 r 2 ) G 0 ( r 1 , r 2 ) ,
Σ 2 a ( r 1 , r 2 ) = k 0 8 d r 1 d r 2 C ( r 1 r 2 ) C ( r 1 r 2 ) × G 0 ( r 1 , r 1 ) G 0 ( r 1 , r 2 ) G 0 ( r 2 , r 2 ) ,
Σ 2 b ( r 1 , r 2 ) = k 0 8 C ( r 1 r 2 ) d r 1 d r 2 C ( r 1 r 2 ) × G 0 ( r 1 , r 1 ) G 0 ( r 1 , r 2 ) G 0 ( r 2 , r 2 ) .
G ( r , r ) = G 0 ( r , r ) + d r 1 d r 2 G 0 ( r , r 1 ) Σ ( r 1 , r 2 ) G ( r 2 , r ) .
G ( r , r ) = G 0 ( r , r ) + d r 1 d r 2 G 0 ( r , r 1 ) Σ ( r 1 , r 2 ) G 0 ( r 2 , r ) + d r 1 d r 2 d r 3 d r 4 G 0 ( r , r 1 ) Σ ( r 1 , r 2 ) × G 0 ( r 2 , r 3 ) Σ ( r 3 , r 4 ) G 0 ( r 4 , r ) + .
G ˜ ( k , k ) = 1 ( 2 π ) 3 d r d r e i k · r i k · r G ( r , r )
= G ( k ) δ ( k k ) ,
Σ ˜ ( k , k ) = Σ ( k ) δ ( k k ) , G ˜ 0 ( k , k ) = G 0 ( k ) δ ( k k ) .
G 0 ( k ) = 4 π k 2 k 0 2 i θ ,
G ( k ) = G 0 ( k ) + G 0 ( k ) Σ ( k ) G ( k ) .
G ( k ) = 4 π k 2 k 0 2 4 π Σ ( k ) i θ .
Σ ( k ) = k 0 4 d k ( 2 π ) 3 C ˜ ( k k ) G 0 ( k ) .
1 k 2 k 0 2 ± i θ = 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 ( k ) = 4 π k 0 4 P d k ( 2 π ) 3 C ˜ ( k k ) k 2 k 0 2 ,
Im Σ ( k ) = k 0 5 d s ^ 4 π C ˜ ( k k 0 s ^ ) ,
1 s = k 0 4 d s ^ C ˜ ( k 0 ( s ^ s ^ ) ) .
G ( k ) = 4 π k 2 κ 2 i θ ,
κ = k 0 ( 1 + i 2 k 0 s ) .
G ( r r ) = exp ( i k 0 | r r | ) | r r | exp ( | r r | / 2 s ) .
U ( r ) U * ( r ) = d r 1 d r 2 G ( r , r 1 ) G * ( r , r 2 ) × S ( r 1 ) S * ( r 2 ) ,
Γ = Γ 1 + Γ 2 a + Γ 2 b + Γ 2 c + ,
Γ 1 ( R 1 , R 2 ; R 1 , R 2 ) = k 0 4 C ( R 1 R 1 ) × δ ( R 1 R 2 ) δ ( R 1 R 2 ) ,
Γ 2 a ( R 1 , R 2 ; R 1 , R 2 ) = k 0 8 G 0 ( R 1 , R 2 ) G 0 * ( R 1 , R 2 ) × C ( R 1 R 2 ) C ( R 2 R 1 ) ,
Γ 2 b ( R 1 , R 2 ; R 1 , R 2 ) = k 0 8 δ ( R 1 R 2 ) C ( R 1 R 2 ) × d R C ( R 1 R ) G 0 * ( R 1 , R ) G 0 * ( R , R 2 ) .
G ( r 1 , r 2 ) G * ( r 1 , r 2 ) = G ( r 1 , r 2 ) G * ( r 1 , r 2 ) + d R 1 d R 1 d R 2 d R 2 G ( r 1 , R 1 ) G * ( r 1 , R 1 ) × Γ ( R 1 , R 2 ; R 1 , R 2 ) G ( R 2 , r 2 ) G * ( R 2 , r 2 ) .
G ( r 1 , r 2 ) G * ( r 1 , r 2 ) = G ( r 1 , r 2 ) G * ( r 1 , r 2 ) + k 0 4 d R 1 d R 1 G ( r 1 , R 1 ) G * ( r 1 , R 1 ) × C ( R 1 R 1 ) G ( R 1 , r 2 ) G * ( R 1 , r 2 ) .
W ( r , k ) = d r ( 2 π ) 3 e i k · r U ( r r / 2 ) U * ( r + r / 2 ) ,
W ˜ ( Q , k ) = U ( k + Q / 2 ) U * ( k + Q / 2 ) / ( 2 π ) 3 + k 0 4 G ( k + Q / 2 ) G * ( k + Q / 2 ) × d k ( 2 π ) 3 C ˜ ( k k ) W ˜ ( Q , k ) ,
G ( k + Q / 2 ) G * ( k + Q / 2 ) = G ( k + Q / 2 ) G * ( k + Q / 2 ) G * ( k + Q / 2 ) 1 G ( k + Q / 2 ) 1 ,
( Q / 2 + k ) 2 ( Q / 2 k ) 2 = 2 k · Q
G ( k + Q / 2 ) G * ( k + Q / 2 ) = Δ G ( k , Q ) 2 k · Q + Δ Σ ( k , Q ) .
Δ Σ ( Q , k ) = 4 π [ Σ ( k + Q / 2 ) Σ * ( k + Q / 2 ) ] ,
Δ G ( Q , k ) = 4 π [ G ( k + Q / 2 ) G * ( k + Q / 2 ) ] .
( 2 k · Q + Δ Σ ( Q , k ) ) W ˜ ( Q , k ) = k 0 4 Δ G ( Q , k ) d k ( 2 π ) 3 C ˜ ( k k ) W ˜ ( Q , k ) + 1 ( 2 π ) 3 Δ G ( Q , k ) S ˜ ( k + Q / 2 ) S ˜ * ( k + Q / 2 ) .
k · W ( r , k ) + 1 2 i d Q ( 2 π ) 3 e i Q · r Δ Σ ( Q , k ) W ˜ ( Q , k ) = k 0 4 2 i d Q ( 2 π ) 3 e i Q · r d k ( 2 π ) 3 Δ G ( Q , k ) C ˜ ( k k ) W ˜ ( Q , k ) + 1 2 i d Q ( 2 π ) 6 e i Q · r Δ G ( Q , k ) S ˜ ( k + Q / 2 ) S ˜ * ( k + Q / 2 ) .
Δ G ( 0 , k ) = 2 i ( 2 π ) 3 k 0 δ ( k k 0 ) .
Δ Σ ( 0 , k ) = 2 i k μ s ,
k · W ( r , k ) + k μ s W ( r , k ) = k 0 3 δ ( k k 0 ) d k C ˜ ( k k ) W ( r , k ) + 1 k 0 δ ( k k 0 ) d Q ( 2 π ) 3 e i Q · r S ˜ ( k + Q / 2 ) S ˜ * ( k + Q / 2 ) .
δ ( k k 0 ) I ( r , s ^ ) = k 0 W ( r , k s ^ ) , A ( s ^ , s ^ ) = k 0 4 μ s C ˜ ( k 0 ( s ^ s ^ ) ) .
s ^ · I ( r , s ^ ) + μ s I ( r , s ^ ) = μ s d s ^ A ( s ^ , s ^ ) I ( r , s ^ ) + I 0 ( r , s ^ ) ,
I 0 ( r , s ^ ) = 1 k 0 d Q ( 2 π ) 3 e i Q · r S ˜ ( k 0 s ^ + Q / 2 ) S ˜ * ( k 0 s ^ + Q / 2 ) .
μ s = 4 π k 0 4 C 0 , A = 1 / ( 4 π ) ,
η ( r ) = i v ( r r i ) ,
C ( r r ) = ρ d R v ( r R ) v ( r R )
μ s = ρ σ s , A = d σ s d Ω / σ s ,
2 U ( r ) + k 0 2 U ( r ) = 0 .
ϵ 2 2 U ϵ ( r ) + k 0 2 U ϵ ( r ) = 0 ,
U ϵ ( r ) = A exp ( i k 0 ϵ s ^ · r ) ,
W ( r , k ) = | A | 2 δ ( k ( k 0 / ϵ ) s ^ ) ,
W ϵ ( r , k ) = d r ( 2 π ) 3 e i k · r U ϵ ( r ϵ r / 2 ) U ε * ( r + ϵ r / 2 ) ,
W ϵ ( r , k ) = | A | 2 δ ( k k 0 s ^ ) ,
ϵ 2 2 U ϵ ( r ) + k 0 2 ( 1 + 4 π ϵ η ( r / ϵ ) ) U ϵ ( r ) = 0 .
ϵ 2 r 1 2 Φ ϵ + k 0 2 Φ ϵ = 4 π k 0 2 ϵ η ( r 1 / ϵ ) Φ ϵ ,
ϵ 2 r 2 2 Φ ϵ + k 0 2 Φ ϵ = 4 π k 0 2 ϵ η ( r 2 / ϵ ) Φ ϵ .
ϵ 2 ( r 1 2 r 2 2 ) Φ ϵ ( r 1 , r 2 ) = 4 π k 0 2 ϵ ( η ( r 1 / ϵ ) η ( r 2 / ϵ ) ) Φ ϵ ( r 1 , r 2 ) .
r 1 = r ϵ r / 2 , r 2 = r + ϵ r / 2
r 2 2 r 1 2 = 2 ϵ r · r .
ϵ r · r Φ ϵ + 2 π k 0 2 ϵ ( η ( r / ϵ + r / 2 ) η ( r / ϵ r / 2 ) ) Φ ϵ = 0 .
k · r W ϵ ( r , k ) + 1 ϵ L W ϵ ( r , k ) = 0 ,
L W ϵ ( r , k ) = 2 i π k 0 2 d k ( 2 π ) 3 e i k · r / ϵ × η ˜ ( k ) [ W ϵ ( r , k + k / 2 ) W ϵ ( r , k k / 2 ) ] .
W ϵ ( r , k ) = W 0 ( r , k ) + ϵ W 1 ( r , R , k ) + ,
r r + 1 ϵ R .
ϵ k · r W ϵ ( r , R , k ) + k · R W ϵ ( r , R , k ) + ϵ L W ϵ ( r , R , k ) = 0 .
k · R W 1 + L W 0 = 0 .
W ˜ 1 ( r , q , k ) = 2 π k 0 2 η ˜ ( q ) [ W 0 ( r , k + q / 2 ) W 0 ( r , k q / 2 ) ] q · k + i θ ,
W ˜ 1 ( r , q , k ) = d R e i q · R W 1 ( r , R , k ) ,
k · r W 0 + k · R W 2 + L W 1 = 0.
k · r W n 2 + k · R W n + L W n 1 = 0
k · r W 0 ( r , k ) + 2 π i k 0 2 d p ( 2 π ) 3 e i p · R × η ˜ ( p ) [ W 1 ( r , R , k + p / 2 ) W 1 ( r , R , k p / 2 ) ] = 0 .
η ˜ ( p ) η ˜ ( q ) = ( 2 π ) 3 C ˜ ( p ) δ ( p + q ) ,
k · r W 0 ( r , k ) k 0 4 d p 2 π C ˜ ( p k ) × ( W 0 ( r , p ) W 0 ( r , k ) ) 2 θ 1 4 ( p 2 k 2 ) 2 + θ 2 = 0 .
lim θ 0 + θ x 2 + θ 2 = π δ ( x ) ,
δ ( p 2 2 k 2 2 ) = 1 p δ ( p k ) ,
δ ( k k 0 ) I ( r , s ^ ) = W 0 ( r , k s ^ ) ,
s ^ · W 0 ( r , k s ^ ) + k 0 4 d s ^ C ˜ ( k ( s ^ s ^ ) ) W 0 ( r , k s ^ ) = k 0 4 d s ^ C ˜ ( k ( s ^ s ^ ) ) W 0 ( r , k s ^ ) .
s ^ · I ( r , s ^ ) + μ s I ( r , s ^ ) = μ s d s ^ A ( s ^ , s ^ ) I ( r , s ^ ) .
U ( r ) U * ( r ) = U ( r ) U * ( r ) + k 0 4 d R 1 d R 1 G ( r , R 1 ) × G * ( r , R 1 ) C ( R 1 R 1 ) × U ( R 1 ) U * ( R 1 ) .
W ˜ ( Q , k ) = 1 ( 2 π ) 3 d r d r e i Q · r e i k · r U ( r r / 2 ) U * ( r + r / 2 ) := W 1 ( Q , k ) + W 2 ( Q , k ) .
W 1 ( Q , k ) = 1 ( 2 π ) 3 d r d r e i Q · r e i k · r U ( r r / 2 ) U * ( r + r / 2 ) = 1 ( 2 π ) 9 d r d r d k 1 d k 2 e i Q · r e i k · r e i k 1 · ( r r / 2 ) e i k 2 · ( r + r / 2 ) U ( k 1 ) U * ( k 2 ) = 1 ( 2 π ) 3 d k 1 d k 2 δ ( Q k 1 k 2 ) δ ( k + k 1 / 2 k 2 / 2 ) U ( k 1 ) U * ( k 2 ) = 1 ( 2 π ) 3 U ( k + Q / 2 ) U * ( k + Q / 2 ) .
W 2 ( Q , k ) = k 0 4 ( 2 π ) 3 d r d r d R 1 d R 1 e i Q · r e i k · r G ( r r / 2 , R 1 ) G * ( r + r / 2 , R 1 ) C ( R 1 R 1 ) × U ( R 1 ) U * ( R 1 ) = k 0 4 ( 2 π ) 18 d r d r d R 1 d R 1 d k 1 d k 5 e i Q · r e i k · r e i k 1 · ( r r / 2 R 1 ) e i k 2 · ( r + r / 2 R 1 ) e i k 3 · ( R 1 R 1 ) e i k 4 · R 1 e i k 5 · R 1 × G ( k 1 ) G * ( k 2 ) C ˜ ( k 3 ) U ( k 4 ) U * ( k 5 ) = k 0 4 ( 2 π ) 6 d k 1 d k 5 δ ( Q k 1 k 2 ) δ ( k + k 1 / 2 k 2 / 2 ) δ ( k 1 k 3 k 4 ) δ ( k 2 + k 3 k 5 ) × G ( k 1 ) G * ( k 2 ) C ˜ ( k 3 ) U ( k 4 ) U * ( k 5 ) = k 0 4 ( 2 π ) 6 G ( k + Q / 2 ) G * ( k + Q / 2 ) d k 3 d k 4 d k 5 δ ( k + Q / 2 k 3 k 4 ) × δ ( k + Q / 2 + k 3 k 5 ) C ˜ ( k 3 ) U ( k 4 ) U * ( k 5 ) .
W 2 ( Q , k ) = k 0 4 ( 2 π ) 6 G ( k + Q / 2 ) G * ( k + Q / 2 ) d k 3 d k 4 d k 5 δ ( Q / 2 k 3 k 4 ) δ ( Q / 2 + k 3 k 5 ) × C ˜ ( k k 3 ) U ( k 4 ) U * ( k 5 ) = k 0 4 G ( k + Q / 2 ) G * ( k + Q / 2 ) d k 3 ( 2 π ) 6 C ˜ ( k k 3 ) U ( k 3 + Q / 2 ) U * ( k 3 + Q / 2 ) .
W ˜ ( Q , k 3 ) = 1 ( 2 π ) 3 d r d r e i Q · r e i k 3 · r U ( r r 2 ) U * ( r + r 2 ) = 1 ( 2 π ) 9 d r d r d k 1 d k 2 e i Q · r e i k 3 · r e i k 1 · ( r r / 2 ) e i k 2 · ( r + r / 2 ) U ( k 1 ) U * ( k 2 ) = 1 ( 2 π ) 3 d k 1 d k 2 δ ( Q k 1 k 2 ) δ ( k 3 + k 1 / 2 k 2 / 2 ) U ( k 1 ) U * ( k 2 ) = 1 ( 2 π ) 3 U ( k 3 + Q / 2 ) U * ( k 3 + Q / 2 ) ,
W ( Q , k ) = 1 ( 2 π ) 3 U ( k + Q / 2 ) U * ( k + Q / 2 ) + k 0 4 G ( k + Q / 2 ) G * ( k + Q / 2 ) d k ( 2 π ) 3 C ˜ ( k k ) W ( Q , k ) ,
ϵ r · r Φ ϵ ( r ϵ r / 2 , r + ϵ r / 2 ) = 2 π k 0 2 ϵ [ η ( r / ϵ r / 2 ) η ( r / ϵ + r / 2 ) ] Φ ϵ .
ϵ d r ( 2 π ) 3 e i k · r r · r Φ ϵ ( r ϵ r / 2 , r + ϵ r / 2 ) = ϵ r · d r ( 2 π ) 3 e i k · r r d k e i k · r W ϵ ( r , k ) = ϵ r · d r d k ( 2 π ) 3 e i k · r ( i k ) e i k · r W ϵ ( r , k ) = ϵ r · d k δ ( k k ) ( i k ) W ϵ ( r , k ) = i ϵ k · r W ϵ ( r , k ) .
2 π k 0 2 ϵ d r ( 2 π ) 3 e i k · r η ( r / ϵ r / 2 ) Φ ϵ ( r / ϵ r / 2 , r ϵ + r / 2 ) = 2 π k 0 2 ϵ d r ( 2 π ) 3 e i k · r d k ( 2 π ) 3 e i k · [ r / ϵ r / 2 ] η ˜ ( k ) × Φ ϵ ( r ϵ r / 2 , r + ϵ r / 2 ) = 2 π k 0 2 ϵ d k ( 2 π ) 3 e i k · r / ϵ η ˜ ( k ) W ϵ ( r , k + k / 2 ) .
2 π k 0 2 ϵ d r ( 2 π ) 3 e i k · r η ( r + ϵ r / 2 ) Φ ϵ ( r / ϵ r / 2 , r ϵ + r / 2 ) = 2 π k 0 2 ϵ d k ( 2 π ) 3 e i k · r / ϵ η ˜ ( k ) W ϵ ( r , k k / 2 ) .
2 π i k 0 2 d p ( 2 π ) 3 e i p · R / ϵ η ( p ) [ W 1 ( r , R , k + p / 2 ) W 1 ( r , R , k p / 2 ) ] = i ( 2 π ) 2 k 0 4 d p ( 2 π ) 3 C ˜ ( p ) [ W 0 ( r , k ) W 0 ( r , k + p ) p · ( k + p / 2 ) + i θ W 0 ( r , k ) W 0 ( r , k p ) p · ( k p / 2 ) i θ ] = i ( 2 π ) 2 k 0 4 d p ( 2 π ) 3 C ˜ ( p ) ( W 0 ( r , k + p ) W 0 ( r , k ) ) [ 1 p · ( k + p / 2 ) i θ 1 p · ( k + p / 2 ) + i θ ] = k 0 4 d p 2 π C ˜ ( p k ) ( W 0 ( r , p ) W 0 ( r , k ) ) 2 θ 1 4 ( p 2 k 2 ) 2 + θ 2 .

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