Abstract

We give theoretical support to the splitting rule, which was recently observed numerically for the scattering from heterogeneous rough surfaces. Under certain general conditions, the incoherent intensity of a composite medium with a rough interface is the sum of the incoherent intensity of the rough homogeneous surface with an effective permittivity and the incoherent intensity of the same composite medium below a flat interface. The coherent intensity is merely that of the rough effective homogeneous surface. The effective permittivity is given accurately by the Bruggemann mixing rule, provided that the scale of fluctuations in the volume is small with respect to the electromagnetic wavelength.

© 2007 Optical Society of America

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  1. T. Elfouhaily and C. A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
    [CrossRef]
  2. M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
    [CrossRef]
  3. L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).
    [CrossRef]
  4. L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, 2000).
    [CrossRef]
  5. L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves, Advanced Topics (Wiley, 200l).
  6. J. W. Elson, "Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity," Phys. Rev. B 30, 5460-5480 (1984).
    [CrossRef]
  7. J. M. Elson, J. M. Bennett, and J. C. Stover, "Wavelength and angular dependence of light scattering from beryllium: comparison of theory and experiment," Atmos.-Ocean. 32, 3362-3376 (1993).
  8. S. Mudaliar, "Electromagnetic waves scattering from a random medium layer with random interface," Waves Random Media 4, 167-176 (1994).
    [CrossRef]
  9. S. Dietrich and A. Haase, "Scattering of x-rays and neutrons at interfaces," Phys. Rep. 260, 1-138 (1995).
    [CrossRef]
  10. R. Carminati and J.-J. Greffet, "Influence of dielectric contrast and topography on the near field scattered by an inhomogeneous surface: boundary conditions for diffusion of light," J. Opt. Soc. Am. A 12, 2716-2725 (l995).
    [CrossRef]
  11. K. Sarabandi, O. Yisok, and F. Ulaby, "A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces," IEEE Trans. Geosci. Remote Sens. 34, 425-432 (1996).
    [CrossRef]
  12. J. M. Elson, "Characteristics of far-field scattering by means of surface roughness and variations in subsurface permittivity," Waves Random Media 7, 303-317 (1997).
    [CrossRef]
  13. O. Calvo-Perez, A. Sentenac, and J.-J. Greffet, "Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory," Radio Sci. 34, 311-335 (1999).
    [CrossRef]
  14. C. M. Lam and A. Ishimaru, "Mueller matrix representation for a slab of random medium with discrete particles and random rough surfaces with moderate roughness," J. Phys. A 260, 111-125 (1993).
  15. J. Caron, C. Andraud, and J. Lafait, "Radiative transfer calculations in multilayer systems with smooth or rough interfaces," J. Mod. Opt. 51, 575-595 (2004).
    [CrossRef]
  16. K. Pak, L. Tsang, L. Li, and C. H. Chan, "Combined random rough surfaces and volume scattering based on Monte Carlo simulations and solutions of Maxwell's equations," Radio Sci. 28, 331-338 (1993).
    [CrossRef]
  17. H. Chanal, J. P. Segaud, P. Borderies, and M. Saillard, "Homogenization and scattering from heterogeneous media based on finite-difference-time-domain Monte Carlo computations," J. Opt. Soc. Am. A 23, 370-381 (2006).
    [CrossRef]
  18. A. Madrazo and M. Nieto-Vesperinas, "Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface," J. Opt. Soc. Am. A 14, 1859-1866 (1997).
    [CrossRef]
  19. G. Zhang, L. Tsang, and K. Pak, "Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional surface," J. Opt. Soc. Am. A 15, 2995-3002 (1998).
    [CrossRef]
  20. H. Giovannini, M. Saillard, and A. Sentenac, "Numerical study of scattering from rough inhomogeneous films," J. Opt. Soc. Am. A 15, 1182-1191 (1998).
    [CrossRef]
  21. A. Sentenac, H. Giovannini, and M. Saillard, "Scattering from rough inhomogeneous media: splitting of surface and volume scattering," J. Opt. Soc. Am. A 19, 727-736 (2002).
    [CrossRef]
  22. O. Calvo-Perez, "Diffusion des ondes electromagnetiques par un film rugueux hétérogène," Ph.D. thesis, dissertation (Ecole Centrale Paris, 1999).
  23. D. A. Yaghjian, "Electric dyadic Green's functions in the source region," Proc. IEEE 68, 248-263 (1980).
    [CrossRef]
  24. S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

2006 (1)

2004 (2)

T. Elfouhaily and C. A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

J. Caron, C. Andraud, and J. Lafait, "Radiative transfer calculations in multilayer systems with smooth or rough interfaces," J. Mod. Opt. 51, 575-595 (2004).
[CrossRef]

2002 (1)

2001 (1)

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[CrossRef]

1999 (1)

O. Calvo-Perez, A. Sentenac, and J.-J. Greffet, "Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory," Radio Sci. 34, 311-335 (1999).
[CrossRef]

1998 (2)

1997 (2)

J. M. Elson, "Characteristics of far-field scattering by means of surface roughness and variations in subsurface permittivity," Waves Random Media 7, 303-317 (1997).
[CrossRef]

A. Madrazo and M. Nieto-Vesperinas, "Scattering of light and other electromagnetic waves from a body buried beneath a highly rough random surface," J. Opt. Soc. Am. A 14, 1859-1866 (1997).
[CrossRef]

1996 (1)

K. Sarabandi, O. Yisok, and F. Ulaby, "A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces," IEEE Trans. Geosci. Remote Sens. 34, 425-432 (1996).
[CrossRef]

1995 (2)

1994 (1)

S. Mudaliar, "Electromagnetic waves scattering from a random medium layer with random interface," Waves Random Media 4, 167-176 (1994).
[CrossRef]

1993 (3)

J. M. Elson, J. M. Bennett, and J. C. Stover, "Wavelength and angular dependence of light scattering from beryllium: comparison of theory and experiment," Atmos.-Ocean. 32, 3362-3376 (1993).

C. M. Lam and A. Ishimaru, "Mueller matrix representation for a slab of random medium with discrete particles and random rough surfaces with moderate roughness," J. Phys. A 260, 111-125 (1993).

K. Pak, L. Tsang, L. Li, and C. H. Chan, "Combined random rough surfaces and volume scattering based on Monte Carlo simulations and solutions of Maxwell's equations," Radio Sci. 28, 331-338 (1993).
[CrossRef]

1984 (1)

J. W. Elson, "Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity," Phys. Rev. B 30, 5460-5480 (1984).
[CrossRef]

1980 (1)

D. A. Yaghjian, "Electric dyadic Green's functions in the source region," Proc. IEEE 68, 248-263 (1980).
[CrossRef]

1956 (1)

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Andraud, C.

J. Caron, C. Andraud, and J. Lafait, "Radiative transfer calculations in multilayer systems with smooth or rough interfaces," J. Mod. Opt. 51, 575-595 (2004).
[CrossRef]

Ao, C. O.

L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).
[CrossRef]

Bennett, J. M.

J. M. Elson, J. M. Bennett, and J. C. Stover, "Wavelength and angular dependence of light scattering from beryllium: comparison of theory and experiment," Atmos.-Ocean. 32, 3362-3376 (1993).

Borderies, P.

Calvo-Perez, O.

O. Calvo-Perez, A. Sentenac, and J.-J. Greffet, "Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory," Radio Sci. 34, 311-335 (1999).
[CrossRef]

O. Calvo-Perez, "Diffusion des ondes electromagnetiques par un film rugueux hétérogène," Ph.D. thesis, dissertation (Ecole Centrale Paris, 1999).

Carminati, R.

Caron, J.

J. Caron, C. Andraud, and J. Lafait, "Radiative transfer calculations in multilayer systems with smooth or rough interfaces," J. Mod. Opt. 51, 575-595 (2004).
[CrossRef]

Chan, C. H.

K. Pak, L. Tsang, L. Li, and C. H. Chan, "Combined random rough surfaces and volume scattering based on Monte Carlo simulations and solutions of Maxwell's equations," Radio Sci. 28, 331-338 (1993).
[CrossRef]

Chanal, H.

Dietrich, S.

S. Dietrich and A. Haase, "Scattering of x-rays and neutrons at interfaces," Phys. Rep. 260, 1-138 (1995).
[CrossRef]

Ding, K.

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, 2000).
[CrossRef]

L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).
[CrossRef]

Elfouhaily, T.

T. Elfouhaily and C. A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

Elson, J. M.

J. M. Elson, "Characteristics of far-field scattering by means of surface roughness and variations in subsurface permittivity," Waves Random Media 7, 303-317 (1997).
[CrossRef]

J. M. Elson, J. M. Bennett, and J. C. Stover, "Wavelength and angular dependence of light scattering from beryllium: comparison of theory and experiment," Atmos.-Ocean. 32, 3362-3376 (1993).

Elson, J. W.

J. W. Elson, "Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity," Phys. Rev. B 30, 5460-5480 (1984).
[CrossRef]

Giovannini, H.

Greffet, J.-J.

O. Calvo-Perez, A. Sentenac, and J.-J. Greffet, "Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory," Radio Sci. 34, 311-335 (1999).
[CrossRef]

R. Carminati and J.-J. Greffet, "Influence of dielectric contrast and topography on the near field scattered by an inhomogeneous surface: boundary conditions for diffusion of light," J. Opt. Soc. Am. A 12, 2716-2725 (l995).
[CrossRef]

Guérin, C. A.

T. Elfouhaily and C. A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

Haase, A.

S. Dietrich and A. Haase, "Scattering of x-rays and neutrons at interfaces," Phys. Rep. 260, 1-138 (1995).
[CrossRef]

Ishimaru, A.

C. M. Lam and A. Ishimaru, "Mueller matrix representation for a slab of random medium with discrete particles and random rough surfaces with moderate roughness," J. Phys. A 260, 111-125 (1993).

Kong, J. A.

L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).
[CrossRef]

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, 2000).
[CrossRef]

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves, Advanced Topics (Wiley, 200l).

Lafait, J.

J. Caron, C. Andraud, and J. Lafait, "Radiative transfer calculations in multilayer systems with smooth or rough interfaces," J. Mod. Opt. 51, 575-595 (2004).
[CrossRef]

Lam, C. M.

C. M. Lam and A. Ishimaru, "Mueller matrix representation for a slab of random medium with discrete particles and random rough surfaces with moderate roughness," J. Phys. A 260, 111-125 (1993).

Li, L.

K. Pak, L. Tsang, L. Li, and C. H. Chan, "Combined random rough surfaces and volume scattering based on Monte Carlo simulations and solutions of Maxwell's equations," Radio Sci. 28, 331-338 (1993).
[CrossRef]

Madrazo, A.

Mudaliar, S.

S. Mudaliar, "Electromagnetic waves scattering from a random medium layer with random interface," Waves Random Media 4, 167-176 (1994).
[CrossRef]

Nieto-Vesperinas, M.

Pak, K.

G. Zhang, L. Tsang, and K. Pak, "Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional surface," J. Opt. Soc. Am. A 15, 2995-3002 (1998).
[CrossRef]

K. Pak, L. Tsang, L. Li, and C. H. Chan, "Combined random rough surfaces and volume scattering based on Monte Carlo simulations and solutions of Maxwell's equations," Radio Sci. 28, 331-338 (1993).
[CrossRef]

Rytov, S. M.

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Saillard, M.

Sarabandi, K.

K. Sarabandi, O. Yisok, and F. Ulaby, "A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces," IEEE Trans. Geosci. Remote Sens. 34, 425-432 (1996).
[CrossRef]

Segaud, J. P.

Sentenac, A.

A. Sentenac, H. Giovannini, and M. Saillard, "Scattering from rough inhomogeneous media: splitting of surface and volume scattering," J. Opt. Soc. Am. A 19, 727-736 (2002).
[CrossRef]

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[CrossRef]

O. Calvo-Perez, A. Sentenac, and J.-J. Greffet, "Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory," Radio Sci. 34, 311-335 (1999).
[CrossRef]

H. Giovannini, M. Saillard, and A. Sentenac, "Numerical study of scattering from rough inhomogeneous films," J. Opt. Soc. Am. A 15, 1182-1191 (1998).
[CrossRef]

Stover, J. C.

J. M. Elson, J. M. Bennett, and J. C. Stover, "Wavelength and angular dependence of light scattering from beryllium: comparison of theory and experiment," Atmos.-Ocean. 32, 3362-3376 (1993).

Tsang, L.

G. Zhang, L. Tsang, and K. Pak, "Angular correlation function and scattering coefficient of electromagnetic waves scattered by a buried object under a two-dimensional surface," J. Opt. Soc. Am. A 15, 2995-3002 (1998).
[CrossRef]

K. Pak, L. Tsang, L. Li, and C. H. Chan, "Combined random rough surfaces and volume scattering based on Monte Carlo simulations and solutions of Maxwell's equations," Radio Sci. 28, 331-338 (1993).
[CrossRef]

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves, Advanced Topics (Wiley, 200l).

L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).
[CrossRef]

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, 2000).
[CrossRef]

Ulaby, F.

K. Sarabandi, O. Yisok, and F. Ulaby, "A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces," IEEE Trans. Geosci. Remote Sens. 34, 425-432 (1996).
[CrossRef]

Yaghjian, D. A.

D. A. Yaghjian, "Electric dyadic Green's functions in the source region," Proc. IEEE 68, 248-263 (1980).
[CrossRef]

Yisok, O.

K. Sarabandi, O. Yisok, and F. Ulaby, "A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces," IEEE Trans. Geosci. Remote Sens. 34, 425-432 (1996).
[CrossRef]

Zhang, G.

Atmos.-Ocean. (1)

J. M. Elson, J. M. Bennett, and J. C. Stover, "Wavelength and angular dependence of light scattering from beryllium: comparison of theory and experiment," Atmos.-Ocean. 32, 3362-3376 (1993).

IEEE Trans. Geosci. Remote Sens. (1)

K. Sarabandi, O. Yisok, and F. Ulaby, "A numerical simulation of scattering from one-dimensional inhomogeneous dielectric random surfaces," IEEE Trans. Geosci. Remote Sens. 34, 425-432 (1996).
[CrossRef]

J. Mod. Opt. (1)

J. Caron, C. Andraud, and J. Lafait, "Radiative transfer calculations in multilayer systems with smooth or rough interfaces," J. Mod. Opt. 51, 575-595 (2004).
[CrossRef]

J. Opt. Soc. Am. A (6)

J. Phys. A (1)

C. M. Lam and A. Ishimaru, "Mueller matrix representation for a slab of random medium with discrete particles and random rough surfaces with moderate roughness," J. Phys. A 260, 111-125 (1993).

Phys. Rep. (1)

S. Dietrich and A. Haase, "Scattering of x-rays and neutrons at interfaces," Phys. Rep. 260, 1-138 (1995).
[CrossRef]

Phys. Rev. B (1)

J. W. Elson, "Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity," Phys. Rev. B 30, 5460-5480 (1984).
[CrossRef]

Proc. IEEE (1)

D. A. Yaghjian, "Electric dyadic Green's functions in the source region," Proc. IEEE 68, 248-263 (1980).
[CrossRef]

Radio Sci. (2)

K. Pak, L. Tsang, L. Li, and C. H. Chan, "Combined random rough surfaces and volume scattering based on Monte Carlo simulations and solutions of Maxwell's equations," Radio Sci. 28, 331-338 (1993).
[CrossRef]

O. Calvo-Perez, A. Sentenac, and J.-J. Greffet, "Light scattering by a two-dimensional, rough penetrable medium: a mean-field theory," Radio Sci. 34, 311-335 (1999).
[CrossRef]

Sov. Phys. JETP (1)

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Waves Random Media (4)

J. M. Elson, "Characteristics of far-field scattering by means of surface roughness and variations in subsurface permittivity," Waves Random Media 7, 303-317 (1997).
[CrossRef]

S. Mudaliar, "Electromagnetic waves scattering from a random medium layer with random interface," Waves Random Media 4, 167-176 (1994).
[CrossRef]

T. Elfouhaily and C. A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves Random Media 14, R1-R40 (2004).
[CrossRef]

M. Saillard and A. Sentenac, "Rigorous solutions for electromagnetic scattering from rough surfaces," Waves Random Media 11, R103-R137 (2001).
[CrossRef]

Other (4)

L. Tsang, J. A. Kong, K. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Numerical Simulations (Wiley, 2001).
[CrossRef]

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves, Theories and Applications (Wiley, 2000).
[CrossRef]

L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves, Advanced Topics (Wiley, 200l).

O. Calvo-Perez, "Diffusion des ondes electromagnetiques par un film rugueux hétérogène," Ph.D. thesis, dissertation (Ecole Centrale Paris, 1999).

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Figures (1)

Fig. 1
Fig. 1

Splitting rule.

Equations (42)

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

( K l ) 3 ε v ε B 2 v 1 ,
( K l ) 3 max ε v ε B 2 ε v ε B 2 v 1 ,
ε v h ( r ) = ε v ( r ) H [ h ( x , y ) z ] + ( 1 H [ h ( x , y ) z ] ) ,
curl curl G h ( r , r ) ε h ( r ) K 2 G h ( r , r ) = δ ( r r ) I ,
ε h ( r ) = ε ¯ H [ h ( x , y ) z ] + ( 1 H [ h ( x , y ) z ] ) .
E v h ( r ) = E h ( r ) + K 2 G h ( r , r ) χ v h ( r ) E v h ( r ) d r ,
E v h scat ( r ) = E h scat ( r ) + E v h c ( r ) ,
E v h scat v h = E h scat h + E v h c v h ,
I coh = δ E v h scat v h 2 .
δ E h scat 2 h + δ E v h c 2 v h + 2 R ( δ E h scat ) * δ E v h c v h ,
δ E h scat = E h scat E h scat h ,
δ E v h c = E v h c E v h c v h .
( δ E h scat ) * δ E v h c v h = ( δ E h scat ) * E v h c v h ,
E v h c v = 0 .
G h χ v h E v h G 0 χ v 0 E v 0 ,
G h ( r , r ) = 1 ε ¯ K 2 L δ ( r r ) + G h pv ( r , r ) ,
d r G h pv ( r , r ) Ψ ( r ) = lim V δ 0 r V δ d r G h ( r , r ) Ψ ( r ) ,
F v h ( r ) = E h ( r ) + G h pv ( r , r ) Γ v h ( r ) F v h ( r ) d r ,
F v h ( r ) = 2 ε ¯ + ε v h ( r ) 3 ε ¯ E v h ( r ) ,
Γ v h ( r ) = 3 K 2 ε ¯ χ v h ( r ) 2 ε ¯ + ε v h ( r ) .
Γ v h ( r ) = Γ v ( r ) Γ h ( r ) ,
Γ v ( r ) = 3 K 2 ε ¯ ε v ( r ) ε ¯ ε v ( r ) + 2 ε ¯ 3 ε ¯ ε v ( r ) + 2 ε ¯ K 2 χ v ( r ) .
Γ v h ( r ) F v h ( r ) v = Γ v ( r ) v Γ h ( r ) E h ( r ) + G h pv ( r , r ) Γ v ( r ) Γ v ( r ) v Γ h ( r ) Γ h ( r ) E h ( r ) d r + d r d r G h pv ( r , r ) G h pv ( r , r ) Γ v ( r ) Γ v ( r ) Γ v ( r ) v Γ h ( r ) Γ h ( r ) Γ h ( r ) E h ( r ) + .
p ( ε v ) ε v ε ¯ ε v + 2 ε ¯ d ε v = 0 ,
r ̃ = K r , d r ̃ = K 3 d r , G h ( r , r ) = K G ̃ h ( r ̃ , r ̃ ) ,
Γ v ( r ) = K 2 Γ ̃ v ( r ̃ ) , Γ h ( r ) = Γ ̃ h ( r ̃ ) , E h ( r ) = E ̃ h ( r ̃ ) .
Γ ̃ v ( r ̃ 1 ) Γ ̃ v ( r ̃ 2 ) Γ ̃ v ( r ̃ n ) v ,
κ p 1 ( r i 1 ̃ , , r i p 1 ̃ ) κ p r ( r i 1 ̃ , , r i p r ̃ ) ,
[ ( K l ) 3 ] p 1 Γ ̃ v p v δ ( r i p ̃ r i 1 ̃ ) δ ( r ̃ i 2 r ̃ i 1 ) ,
Γ ̃ v p 1 Γ ̃ v p r v [ ( K l ) 3 ] p 1 1 [ ( K l ) 3 ] p r 1 ,
β = ( K l ) 3 χ max 2 χ v 2 v , η = ( K l ) 3 χ v 2 .
E v h c v η 2 .
G ̃ h ( r ̃ r ̃ ) Γ ̃ h ( r ̃ ) E ̃ h ( r ̃ ) G ̃ h ( r ̃ r ̃ ) Γ ̃ h ( r ̃ ) E ̃ h ( r ̃ ) .
E v h c 2 v δ E v h c 2 v η .
sup σ η σ η σ 2 + η = sup σ η σ η σ 2 + η = η 2 1 + η O ( η 2 ) ,
I incoh δ E h scat 2 h + δ E v h c 2 v h ,
sup σ η σ n σ 2 + η = 1 ,
F v h ( r ) = ( 1 ε v ( r ) ε ¯ ε ¯ L ) E v h ( r ) ,
Γ v ( r ) = K 2 ( ε v ( r ) ε ¯ ) ( 1 ε v ( r ) ε ¯ ε ¯ L ) 1 ,
Γ v ( z ) = K 2 ( ε v ( z ) ε ¯ ) ( 1 + ( ε ¯ ε v ( z ) ε v ( z ) z ̂ z ̂ ) ) ,
ε ¯ = ε ( I z ̂ z ̂ ) + ( 1 ε ) 1 z ̂ z ̂ .
I incoh = PT + Born + cross term ,

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