Abstract

Established mathematical techniques to model the energy density of high-frequency waves in random media by radiative transfer equations and to model the small mean-free-path limit of radiative transfer solutions by diffusion equations are reviewed. These techniques are then applied to the derivation of radiative transfer and diffusion equations for the radiance, also known as specific intensity, of electromagnetic waves in situations where the refractive index of the underlying structure varies smoothly in space.

© 2006 Optical Society of America

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  1. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  3. L. Erdös and H. T. Yau, 'Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation,' Commun. Pure Appl. Math. 53, 667-735 (2000).
    [CrossRef]
  4. H. Spohn, 'Derivation of the transport equation for electrons moving through random impurities,' J. Stat. Phys. 17, 385-412 (1977).
    [CrossRef]
  5. G. Bal, 'Kinetic models for scalar wave fields in random media,' Wave Motion 43, 132-157 (2005).
  6. 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]
  7. A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, 'Boundary layers and homogenization of transport processes,' Res. Inst. Math. Sci. Kyoto Univ. 15, 53-157 (1979).
    [CrossRef]
  8. R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology (Springer-Verlag, 1993), Vol. 6.
    [CrossRef]
  9. E. W. Larsen and J. B. Keller, 'Asymptotic solution of neutron transport problems for small mean free paths,' J. Math. Phys. 15, 75-81 (1974).
    [CrossRef]
  10. H. A. Ferwerda, 'The radiative transfer equation for scattering media with a spatially varying refractive index,' J. Opt. A, Pure Appl. Opt. 1, L1-L2 (1999).
    [CrossRef]
  11. T. Khan and H. Jiang, 'A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,' J. Opt. A, Pure Appl. Opt. 5, 137-141 (2003).
    [CrossRef]
  12. T. Khan and A. Thomas, 'Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,' Opt. Commun. 255, 130-166 (2005).
    [CrossRef]
  13. L. Martí-López, J. Bouza-Domínguez, J. C. Hebden, A. S. R. Arridge, and R. A. Martinéz-Celerio, 'Validity conditions for the radiative transfer equation,' J. Opt. Soc. Am. A 20, 2046-2056 (2003).
    [CrossRef]
  14. M. Premaratne, E. Premaratne, and A. J. Lowery, 'The photon transport equation for turbid biological media with spatially varying isotropic refractive index,' Opt. Express 13, 389-399 (2005).
    [CrossRef] [PubMed]
  15. J. Tualle and E. Tenet, 'Derivation of the radiative transfer equation for scattering media with spatially varying refractive index,' Opt. Commun. 228, 33-38 (2003).
    [CrossRef]
  16. M. L. Shendeleva, 'Radiative transfer in a turbid medium with a varying refractive index: comment,' J. Opt. Soc. Am. A 21, 2464-2467 (2004).
    [CrossRef]
  17. K. Kline and W. I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley-Interscience, 1965).
  18. G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, 1974).
  19. P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, 'Homogenization limits and Wigner transforms,' Commun. Pure Appl. Math. 50, 323-380 (1997).
    [CrossRef]
  20. P.-L. Lions and T. Paul, 'Sur les mesures de Wigner,' Rev. Mat. Iberoam. 9, 553-618 (1993).
    [CrossRef]
  21. G. Bal, J. B. Keller, G. Papanicolaou, and L. Ryzhik, 'Transport theory for waves with reflection and transmission at interfaces,' Wave Motion 30, 303-327 (1999).
    [CrossRef]
  22. G. Bal, G. Papanicolaou, and L. Ryzhik, 'Probabilistic theory of transport processes with polarization,' SIAM J. Appl. Math. 60, 1639-1666 (2000).
    [CrossRef]
  23. J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, 1969).
  24. G. Bal and M. Moscoso, 'Polarization effects of seismic waves on the basis of radiative transport theory,' Geophys. J. Int. 142, 571-585 (2000).
    [CrossRef]

2005 (3)

G. Bal, 'Kinetic models for scalar wave fields in random media,' Wave Motion 43, 132-157 (2005).

T. Khan and A. Thomas, 'Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,' Opt. Commun. 255, 130-166 (2005).
[CrossRef]

M. Premaratne, E. Premaratne, and A. J. Lowery, 'The photon transport equation for turbid biological media with spatially varying isotropic refractive index,' Opt. Express 13, 389-399 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (3)

J. Tualle and E. Tenet, 'Derivation of the radiative transfer equation for scattering media with spatially varying refractive index,' Opt. Commun. 228, 33-38 (2003).
[CrossRef]

L. Martí-López, J. Bouza-Domínguez, J. C. Hebden, A. S. R. Arridge, and R. A. Martinéz-Celerio, 'Validity conditions for the radiative transfer equation,' J. Opt. Soc. Am. A 20, 2046-2056 (2003).
[CrossRef]

T. Khan and H. Jiang, 'A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,' J. Opt. A, Pure Appl. Opt. 5, 137-141 (2003).
[CrossRef]

2000 (3)

L. Erdös and H. T. Yau, 'Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation,' Commun. Pure Appl. Math. 53, 667-735 (2000).
[CrossRef]

G. Bal, G. Papanicolaou, and L. Ryzhik, 'Probabilistic theory of transport processes with polarization,' SIAM J. Appl. Math. 60, 1639-1666 (2000).
[CrossRef]

G. Bal and M. Moscoso, 'Polarization effects of seismic waves on the basis of radiative transport theory,' Geophys. J. Int. 142, 571-585 (2000).
[CrossRef]

1999 (2)

G. Bal, J. B. Keller, G. Papanicolaou, and L. Ryzhik, 'Transport theory for waves with reflection and transmission at interfaces,' Wave Motion 30, 303-327 (1999).
[CrossRef]

H. A. Ferwerda, 'The radiative transfer equation for scattering media with a spatially varying refractive index,' J. Opt. A, Pure Appl. Opt. 1, L1-L2 (1999).
[CrossRef]

1997 (1)

P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, 'Homogenization limits and Wigner transforms,' Commun. Pure Appl. Math. 50, 323-380 (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]

1993 (1)

P.-L. Lions and T. Paul, 'Sur les mesures de Wigner,' Rev. Mat. Iberoam. 9, 553-618 (1993).
[CrossRef]

1979 (1)

A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, 'Boundary layers and homogenization of transport processes,' Res. Inst. Math. Sci. Kyoto Univ. 15, 53-157 (1979).
[CrossRef]

1977 (1)

H. Spohn, 'Derivation of the transport equation for electrons moving through random impurities,' J. Stat. Phys. 17, 385-412 (1977).
[CrossRef]

1974 (1)

E. W. Larsen and J. B. Keller, 'Asymptotic solution of neutron transport problems for small mean free paths,' J. Math. Phys. 15, 75-81 (1974).
[CrossRef]

Arridge, A. S. R.

Bal, G.

G. Bal, 'Kinetic models for scalar wave fields in random media,' Wave Motion 43, 132-157 (2005).

G. Bal, G. Papanicolaou, and L. Ryzhik, 'Probabilistic theory of transport processes with polarization,' SIAM J. Appl. Math. 60, 1639-1666 (2000).
[CrossRef]

G. Bal and M. Moscoso, 'Polarization effects of seismic waves on the basis of radiative transport theory,' Geophys. J. Int. 142, 571-585 (2000).
[CrossRef]

G. Bal, J. B. Keller, G. Papanicolaou, and L. Ryzhik, 'Transport theory for waves with reflection and transmission at interfaces,' Wave Motion 30, 303-327 (1999).
[CrossRef]

Bensoussan, A.

A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, 'Boundary layers and homogenization of transport processes,' Res. Inst. Math. Sci. Kyoto Univ. 15, 53-157 (1979).
[CrossRef]

Bouza-Domínguez, J.

Chandrasekhar, S.

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

Dautray, R.

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology (Springer-Verlag, 1993), Vol. 6.
[CrossRef]

Erdös, L.

L. Erdös and H. T. Yau, 'Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation,' Commun. Pure Appl. Math. 53, 667-735 (2000).
[CrossRef]

Ferwerda, H. A.

H. A. Ferwerda, 'The radiative transfer equation for scattering media with a spatially varying refractive index,' J. Opt. A, Pure Appl. Opt. 1, L1-L2 (1999).
[CrossRef]

Gelbard, E. M.

J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, 1969).

Gérard, P.

P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, 'Homogenization limits and Wigner transforms,' Commun. Pure Appl. Math. 50, 323-380 (1997).
[CrossRef]

Hebden, J. C.

Ishimaru, A.

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

Jiang, H.

T. Khan and H. Jiang, 'A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,' J. Opt. A, Pure Appl. Opt. 5, 137-141 (2003).
[CrossRef]

Kay, W. I.

K. Kline and W. I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley-Interscience, 1965).

Keller, J. B.

G. Bal, J. B. Keller, G. Papanicolaou, and L. Ryzhik, 'Transport theory for waves with reflection and transmission at interfaces,' Wave Motion 30, 303-327 (1999).
[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]

E. W. Larsen and J. B. Keller, 'Asymptotic solution of neutron transport problems for small mean free paths,' J. Math. Phys. 15, 75-81 (1974).
[CrossRef]

Khan, T.

T. Khan and A. Thomas, 'Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,' Opt. Commun. 255, 130-166 (2005).
[CrossRef]

T. Khan and H. Jiang, 'A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,' J. Opt. A, Pure Appl. Opt. 5, 137-141 (2003).
[CrossRef]

Kline, K.

K. Kline and W. I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley-Interscience, 1965).

Larsen, E. W.

E. W. Larsen and J. B. Keller, 'Asymptotic solution of neutron transport problems for small mean free paths,' J. Math. Phys. 15, 75-81 (1974).
[CrossRef]

Lions, J.-L.

A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, 'Boundary layers and homogenization of transport processes,' Res. Inst. Math. Sci. Kyoto Univ. 15, 53-157 (1979).
[CrossRef]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology (Springer-Verlag, 1993), Vol. 6.
[CrossRef]

Lions, P.-L.

P.-L. Lions and T. Paul, 'Sur les mesures de Wigner,' Rev. Mat. Iberoam. 9, 553-618 (1993).
[CrossRef]

Lowery, A. J.

Markowich, P. A.

P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, 'Homogenization limits and Wigner transforms,' Commun. Pure Appl. Math. 50, 323-380 (1997).
[CrossRef]

Martí-López, L.

Martinéz-Celerio, R. A.

Mauser, N. J.

P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, 'Homogenization limits and Wigner transforms,' Commun. Pure Appl. Math. 50, 323-380 (1997).
[CrossRef]

Moscoso, M.

G. Bal and M. Moscoso, 'Polarization effects of seismic waves on the basis of radiative transport theory,' Geophys. J. Int. 142, 571-585 (2000).
[CrossRef]

Papanicolaou, G.

G. Bal, G. Papanicolaou, and L. Ryzhik, 'Probabilistic theory of transport processes with polarization,' SIAM J. Appl. Math. 60, 1639-1666 (2000).
[CrossRef]

G. Bal, J. B. Keller, G. Papanicolaou, and L. Ryzhik, 'Transport theory for waves with reflection and transmission at interfaces,' Wave Motion 30, 303-327 (1999).
[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]

Papanicolaou, G. C.

A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, 'Boundary layers and homogenization of transport processes,' Res. Inst. Math. Sci. Kyoto Univ. 15, 53-157 (1979).
[CrossRef]

Paul, T.

P.-L. Lions and T. Paul, 'Sur les mesures de Wigner,' Rev. Mat. Iberoam. 9, 553-618 (1993).
[CrossRef]

Poupaud, F.

P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, 'Homogenization limits and Wigner transforms,' Commun. Pure Appl. Math. 50, 323-380 (1997).
[CrossRef]

Premaratne, E.

Premaratne, M.

Ryzhik, L.

G. Bal, G. Papanicolaou, and L. Ryzhik, 'Probabilistic theory of transport processes with polarization,' SIAM J. Appl. Math. 60, 1639-1666 (2000).
[CrossRef]

G. Bal, J. B. Keller, G. Papanicolaou, and L. Ryzhik, 'Transport theory for waves with reflection and transmission at interfaces,' Wave Motion 30, 303-327 (1999).
[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]

Shendeleva, M. L.

Spanier, J.

J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, 1969).

Spohn, H.

H. Spohn, 'Derivation of the transport equation for electrons moving through random impurities,' J. Stat. Phys. 17, 385-412 (1977).
[CrossRef]

Tenet, E.

J. Tualle and E. Tenet, 'Derivation of the radiative transfer equation for scattering media with spatially varying refractive index,' Opt. Commun. 228, 33-38 (2003).
[CrossRef]

Thomas, A.

T. Khan and A. Thomas, 'Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,' Opt. Commun. 255, 130-166 (2005).
[CrossRef]

Tualle, J.

J. Tualle and E. Tenet, 'Derivation of the radiative transfer equation for scattering media with spatially varying refractive index,' Opt. Commun. 228, 33-38 (2003).
[CrossRef]

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, 1974).

Yau, H. T.

L. Erdös and H. T. Yau, 'Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation,' Commun. Pure Appl. Math. 53, 667-735 (2000).
[CrossRef]

Commun. Pure Appl. Math. (2)

L. Erdös and H. T. Yau, 'Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation,' Commun. Pure Appl. Math. 53, 667-735 (2000).
[CrossRef]

P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, 'Homogenization limits and Wigner transforms,' Commun. Pure Appl. Math. 50, 323-380 (1997).
[CrossRef]

Geophys. J. Int. (1)

G. Bal and M. Moscoso, 'Polarization effects of seismic waves on the basis of radiative transport theory,' Geophys. J. Int. 142, 571-585 (2000).
[CrossRef]

J. Math. Phys. (1)

E. W. Larsen and J. B. Keller, 'Asymptotic solution of neutron transport problems for small mean free paths,' J. Math. Phys. 15, 75-81 (1974).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

H. A. Ferwerda, 'The radiative transfer equation for scattering media with a spatially varying refractive index,' J. Opt. A, Pure Appl. Opt. 1, L1-L2 (1999).
[CrossRef]

T. Khan and H. Jiang, 'A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,' J. Opt. A, Pure Appl. Opt. 5, 137-141 (2003).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Stat. Phys. (1)

H. Spohn, 'Derivation of the transport equation for electrons moving through random impurities,' J. Stat. Phys. 17, 385-412 (1977).
[CrossRef]

Opt. Commun. (2)

T. Khan and A. Thomas, 'Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices,' Opt. Commun. 255, 130-166 (2005).
[CrossRef]

J. Tualle and E. Tenet, 'Derivation of the radiative transfer equation for scattering media with spatially varying refractive index,' Opt. Commun. 228, 33-38 (2003).
[CrossRef]

Opt. Express (1)

Res. Inst. Math. Sci. Kyoto Univ. (1)

A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, 'Boundary layers and homogenization of transport processes,' Res. Inst. Math. Sci. Kyoto Univ. 15, 53-157 (1979).
[CrossRef]

Rev. Mat. Iberoam. (1)

P.-L. Lions and T. Paul, 'Sur les mesures de Wigner,' Rev. Mat. Iberoam. 9, 553-618 (1993).
[CrossRef]

SIAM J. Appl. Math. (1)

G. Bal, G. Papanicolaou, and L. Ryzhik, 'Probabilistic theory of transport processes with polarization,' SIAM J. Appl. Math. 60, 1639-1666 (2000).
[CrossRef]

Wave Motion (3)

G. Bal, J. B. Keller, G. Papanicolaou, and L. Ryzhik, 'Transport theory for waves with reflection and transmission at interfaces,' Wave Motion 30, 303-327 (1999).
[CrossRef]

G. Bal, 'Kinetic models for scalar wave fields in random media,' Wave Motion 43, 132-157 (2005).

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 (6)

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology (Springer-Verlag, 1993), Vol. 6.
[CrossRef]

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

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

J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, 1969).

K. Kline and W. I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley-Interscience, 1965).

G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, 1974).

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Equations (44)

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ϵ ε E ε t = × H ε , ϵ ε E ε = 0 ,
μ ε H ε t = × E ε , μ ε H ε = 0 ,
E ε ( t ) = 1 2 R 3 [ ϵ ε ( x ) E ε 2 ( t , x ) + μ ε ( x ) H ε 2 ( t , x ) ] d x = E ε ( 0 ) .
E ε ( t , x ) = 1 2 [ ϵ ε ( x ) E ε 2 ( t , x ) + μ ε ( x ) H ε 2 ( t , x ) ] .
ϵ ε ( x ) = ϵ 0 , μ ε ( x ) = μ 0 ( x ) + ε μ 1 ( x ε ) .
c ( x ) = 1 ϵ 0 μ 0 ( x ) , n ( x ) = c c ( x ) ,
α ( t , x , k ) = 1 2 [ I + Q U + i V U i V I Q ] ( t , x , k ) ,
lim ε 0 E ε ( t , x ) = R 3 Tr α ( t , x , k ) d k = R 3 I ( t , x , k ) d k .
α t + { ω , α } + N α α N + π ω 2 ( x , k ) 2 ( 2 π ) 3 R 3 R ̂ ( k q ) T ( k , q ) ( α ( k ) α ( q ) ) T ( q , k ) δ ( ω ( x , k ) ω ( x , q ) ) d q = 0 ,
ω ( x , k ) = c ( x ) k .
{ ω , α } ( x , k ) = k ω ( x , k ) x α ( x , k ) x ω ( x , k ) k α ( x , k ) = c ( x ) k ̂ x α ( x , k ) k c ( x ) k α ( x , k ) ,
k ̂ = k k .
I t + { ω , I } + Σ ( x , k ) I = R 3 σ ( x , k , q ) I ( t , x , q ) × δ ( ω ( x , k ) ω ( x , q ) ) d q ,
Σ s ( x , k ) = R 3 σ ( x , k , q ) δ ( ω ( x , k ) ω ( x , q ) ) d q .
S 2 0 1 c ( x ) L ( t , x , Ω , ω ) d ω d Ω = lim ε 0 E ε ( t , x ) = R 3 I ( t , x , k ) d k .
R 3 I ( t , x , k ) d k = R 3 I ( t , x , k ) k 2 d k d k ̂ = S 2 0 I ( t , x , ω c ( x ) Ω ) ω 2 c 3 ( x ) d ω d Ω .
ω 2 c 2 ( x ) I ( t , x , ω c ( x ) Ω ) = L ( t , x , Ω , ω ) .
I ( t , x , Ω , ω ) = I ( t , x , ω c ( x ) Ω ) .
x x + c c ω ω ,
k c ( x ) Ω ω + c ( x ) ω ( I 3 Ω Ω ) Ω .
{ ω , I } ( t , x , k ) ( c ( x ) Ω x I c ( x ) ( I 3 Ω Ω ) Ω I ) ( t , x , Ω , ω ) .
I t + ( c ( x ) Ω x I c ( x ) ( I 3 Ω Ω ) Ω I ) + Σ ̃ ( x , Ω , ω ) I = S 2 σ ̃ ( x , Ω , Ω , ω ) I ( t , x , Ω , ω ) d Ω .
Σ ̃ ( x , Ω , ω ) = Σ ( x , ω Ω c ( x ) ) ,
σ ̃ ( x , Ω , Ω , ω ) = σ ( x , ω c ( x ) Ω , ω c ( x ) Ω ) .
L ( t , x , Ω , ω ) = ω 2 c 2 ( x ) I ( t , x , Ω , ω ) .
1 c 3 ( x ) c ( x ) Ω c 2 ( x ) = 2 Ω c ( x ) c ( x ) ,
1 c ( x ) L t + ( Ω x L c ( x ) c ( x ) ( I 3 Ω Ω ) Ω L ) + 2 Ω c ( x ) c ( x ) L = Σ ̃ ( x , ω ) c ( x ) L + S 2 σ ̃ ( x , Ω , Ω , ω ) c ( x ) L ( t , x , Ω , ω ) d Ω .
μ a ( x ) = Σ a ( x ) c ( x ) , μ s ( x , ω ) θ ( x , Ω Ω ) = σ ̃ ( x , Ω , Ω , ω ) c ( x ) ,
n ( x ) c L t + Ω x L + n ( x ) n ( x ) ( I 3 Ω Ω ) Ω L 2 Ω n ( x ) n ( x ) L = ( μ s ( x , ω ) + μ a ( x ) ) L + μ s ( x , ω ) × S 2 θ ( x , Ω Ω ) L ( t , x , Ω , ω ) d Ω .
l ( x , ω ) = η μ t ( x , ω ) L , μ t ( x , ω ) = μ s ( x , ω ) + η 2 μ a ( x ) ,
η c 0 ( x ) I η t + ( Ω x I η c ( x ) c ( x ) ( I 3 Ω Ω ) Ω I η ) + η μ a ( x ) I η = μ s ( x , ω ) η S 2 θ ( x , Ω Ω ) ( I η ( t , x , Ω , ω ) I η ( t , x , Ω , ω ) ) d Ω ,
Ω x I 0 = μ s ( x , ω ) S 2 θ ( x , Ω Ω ) ( I 1 ( t , x , Ω , ω ) I 1 ( t , x , Ω , ω ) ) d Ω
λ 1 ( x ) Ω = S 2 θ ( x , Ω Ω ) Ω d Ω .
I 1 ( t , x , Ω , ω ) = 1 μ s ( x , ω ) ( 1 λ 1 ( x ) ) Ω I 0 ( t , x , ω ) .
l * ( x , ω ) = 1 μ t ( x , ω ) ( 1 λ 1 ( x ) ) = l ( x , ω ) 1 λ 1 ( x ) .
1 c ( x ) I 0 t + S 2 ( Ω x I 1 c ( x ) c ( x ) ( I 3 Ω Ω ) Ω I 1 ) d Ω 4 π + μ a ( x ) I 0 = 0 .
S 2 Ω l * ( x , ω ) Ω d Ω 4 π = l * ( x , ω ) 3 = D ( x , ω ) ,
D ( x , ω ) = l * ( x , ω ) 3 = 1 3 μ t ( x , ω ) ( 1 λ 1 ( x ) ) .
S 2 ( I 3 Ω Ω ) Ω Ω d Ω 4 π = S 2 ( I 3 Ω Ω ) d Ω 4 π = I 3 1 3 I 3 = 2 3 I 3 .
1 c ( x ) I 0 t D ( x , ω ) I 0 2 D ( x ) c ( x ) c ( x ) I 0 + μ a ( x ) I 0 = 0 ,
1 c 3 ( x ) I 0 t ( D ( x ) c 2 ( x ) I 0 ) + μ a ( x ) c 2 ( x ) I 0 = 0 .
I η = I 0 η l * ( x , ω ) Ω I 0 + O ( η 2 ) ,
L η = L 0 η l * ( x , ω ) c 2 ( x ) Ω [ c 2 ( x ) L 0 ] + O ( η 2 ) ,
1 c 0 ( x ) L 0 t ( D ( x , ω ) c 2 ( x ) [ c 2 ( x ) L 0 ] ) + μ a ( x ) L 0 = 0 ,

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