Abstract

The problem of an electromagnetic wave scattered by a slab with two rough boundaries is solved by the small-perturbation method under the Rayleigh hypothesis. To obtain the perturbative development, we use a systematic procedure that involves integral equations called the reduced Rayleigh equations. Then we show for a dielectric slab deposited on a silver film that the backscattering enhancement can be produced by guided waves that interact with the two rough surfaces.

© 2001 Optical Society of America

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References

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  1. I. Ohlı́dal, K. Navrátil, M. Ohlı́dal, “Scattering of light from multilayer systems with rough boundaries,” Prog. Opt. 34, 251–334 (1995).
  2. I. M. Fuks, A. G. Voronovich, “Wave diffraction by rough interfaces in an arbitrary plane-layered medium,” Waves Random Media 10, 253–272 (2000).
    [Crossref]
  3. R. Garcı́a-Llamas, L. E. Regalado, C. Amra, “Sacttering of light from a two-layer system with a rough surface,” J. Opt. Soc. Am. A 16, 2713–2719 (1999).
    [Crossref]
  4. P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [Crossref]
  5. J. M. Elson, “Multilayer-coated optics: guided-wave coupling and scattering of interface random roughness,” J. Opt. Soc. Am. A 12, 729–742 (1995).
    [Crossref]
  6. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [Crossref]
  7. A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer, Berlin, 1994).
  8. G. C. Brown, V. Celli, M. Haller, A. Marvin, “Vector theory of light scattering from a rough surface: unitary and reciprocal expansions,” Surf. Sci. 136, 381–397 (1984).
    [Crossref]
  9. A. Soubret, G. Berginc, C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scatter-ing by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411-1-20 (2001).
    [Crossref]
  10. A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [Crossref]
  11. V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
    [Crossref]
  12. A. R. McGurn, A. A. Maradudin, “Localization effects in the elastic scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
    [Crossref]
  13. A. A. Maradudin, E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335–3343 (1993).
    [Crossref] [PubMed]
  14. C. S. West, K. A. O’Donnell, “Observations of backscattering enhancement from polaritons on a rough metal surface,” J. Opt. Soc. Am. A 12, 390–397 (1995).
    [Crossref]
  15. V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
    [Crossref]
  16. A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Progress in Electromagnetic Research, M. Tateiba, L. Tsang, eds. (Elsevier, New York, 1996), Vol. 14, pp. 1–36.
  17. L. Tsang, G. T. J. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).
  18. A. R. McGurn, A. A. Maradudin, “Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces,” Waves Random Media 6, 251–267 (1996).
    [Crossref]
  19. J. T. Johnson, “Third-order small-perturbation method for scattering from dielectric rough surfaces,” J. Opt. Soc. Am. A 16, 2720–2736 (1999).
    [Crossref]
  20. J. A. Ogilvy, Theory of Wave Scattering From Random Rough Surfaces (Hilger, Bristol, UK, 1991).

2001 (1)

A. Soubret, G. Berginc, C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scatter-ing by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411-1-20 (2001).
[Crossref]

2000 (1)

I. M. Fuks, A. G. Voronovich, “Wave diffraction by rough interfaces in an arbitrary plane-layered medium,” Waves Random Media 10, 253–272 (2000).
[Crossref]

1999 (2)

1997 (1)

V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[Crossref]

1996 (1)

A. R. McGurn, A. A. Maradudin, “Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces,” Waves Random Media 6, 251–267 (1996).
[Crossref]

1995 (3)

1993 (1)

1987 (1)

1985 (2)

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[Crossref]

V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[Crossref]

1984 (1)

G. C. Brown, V. Celli, M. Haller, A. Marvin, “Vector theory of light scattering from a rough surface: unitary and reciprocal expansions,” Surf. Sci. 136, 381–397 (1984).
[Crossref]

1981 (1)

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[Crossref]

Amra, C.

Berginc, G.

A. Soubret, G. Berginc, C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scatter-ing by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411-1-20 (2001).
[Crossref]

Bourrely, C.

A. Soubret, G. Berginc, C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scatter-ing by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411-1-20 (2001).
[Crossref]

Bousquet, P.

Brown, G. C.

G. C. Brown, V. Celli, M. Haller, A. Marvin, “Vector theory of light scattering from a rough surface: unitary and reciprocal expansions,” Surf. Sci. 136, 381–397 (1984).
[Crossref]

Celli, V.

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[Crossref]

V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[Crossref]

G. C. Brown, V. Celli, M. Haller, A. Marvin, “Vector theory of light scattering from a rough surface: unitary and reciprocal expansions,” Surf. Sci. 136, 381–397 (1984).
[Crossref]

Chan, T. K.

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Progress in Electromagnetic Research, M. Tateiba, L. Tsang, eds. (Elsevier, New York, 1996), Vol. 14, pp. 1–36.

Elson, J. M.

Flory, F.

Freilikher, V.

V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[Crossref]

Fuks, I. M.

I. M. Fuks, A. G. Voronovich, “Wave diffraction by rough interfaces in an arbitrary plane-layered medium,” Waves Random Media 10, 253–272 (2000).
[Crossref]

Garci´a-Llamas, R.

Haller, M.

G. C. Brown, V. Celli, M. Haller, A. Marvin, “Vector theory of light scattering from a rough surface: unitary and reciprocal expansions,” Surf. Sci. 136, 381–397 (1984).
[Crossref]

Ishimaru, A.

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Progress in Electromagnetic Research, M. Tateiba, L. Tsang, eds. (Elsevier, New York, 1996), Vol. 14, pp. 1–36.

Johnson, J. T.

Kanzieper, E.

V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[Crossref]

Kong, G. T. J.

L. Tsang, G. T. J. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).

Kuga, Y.

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Progress in Electromagnetic Research, M. Tateiba, L. Tsang, eds. (Elsevier, New York, 1996), Vol. 14, pp. 1–36.

Le, C.

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Progress in Electromagnetic Research, M. Tateiba, L. Tsang, eds. (Elsevier, New York, 1996), Vol. 14, pp. 1–36.

Maradudin, A. A.

V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[Crossref]

A. R. McGurn, A. A. Maradudin, “Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces,” Waves Random Media 6, 251–267 (1996).
[Crossref]

A. A. Maradudin, E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335–3343 (1993).
[Crossref] [PubMed]

A. R. McGurn, A. A. Maradudin, “Localization effects in the elastic scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
[Crossref]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[Crossref]

V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[Crossref]

Marvin, A.

G. C. Brown, V. Celli, M. Haller, A. Marvin, “Vector theory of light scattering from a rough surface: unitary and reciprocal expansions,” Surf. Sci. 136, 381–397 (1984).
[Crossref]

Marvin, A. M.

McGurn, A. R.

A. R. McGurn, A. A. Maradudin, “Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces,” Waves Random Media 6, 251–267 (1996).
[Crossref]

A. R. McGurn, A. A. Maradudin, “Localization effects in the elastic scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
[Crossref]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[Crossref]

V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[Crossref]

Méndez, E. R.

Navrátil, K.

I. Ohlı́dal, K. Navrátil, M. Ohlı́dal, “Scattering of light from multilayer systems with rough boundaries,” Prog. Opt. 34, 251–334 (1995).

O’Donnell, K. A.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering From Random Rough Surfaces (Hilger, Bristol, UK, 1991).

Ohli´dal, I.

I. Ohlı́dal, K. Navrátil, M. Ohlı́dal, “Scattering of light from multilayer systems with rough boundaries,” Prog. Opt. 34, 251–334 (1995).

Ohli´dal, M.

I. Ohlı́dal, K. Navrátil, M. Ohlı́dal, “Scattering of light from multilayer systems with rough boundaries,” Prog. Opt. 34, 251–334 (1995).

Regalado, L. E.

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[Crossref]

Roche, P.

Sengers, L. A.

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Progress in Electromagnetic Research, M. Tateiba, L. Tsang, eds. (Elsevier, New York, 1996), Vol. 14, pp. 1–36.

Shin, R.

L. Tsang, G. T. J. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).

Soubret, A.

A. Soubret, G. Berginc, C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scatter-ing by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411-1-20 (2001).
[Crossref]

Tsang, L.

L. Tsang, G. T. J. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).

Voronovich, A. G.

I. M. Fuks, A. G. Voronovich, “Wave diffraction by rough interfaces in an arbitrary plane-layered medium,” Waves Random Media 10, 253–272 (2000).
[Crossref]

A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer, Berlin, 1994).

West, C. S.

Appl. Opt. (1)

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

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

Phys. Rep. (1)

V. Freilikher, E. Kanzieper, A. A. Maradudin, “Coherent scattering enhancement in systems bounded by rough surfaces,” Phys. Rep. 288, 127–204 (1997).
[Crossref]

Phys. Rev. B (2)

A. Soubret, G. Berginc, C. Bourrely, “Application of reduced Rayleigh equations to electromagnetic wave scatter-ing by two-dimensional randomly rough surfaces,” Phys. Rev. B 63, 245411-1-20 (2001).
[Crossref]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[Crossref]

Prog. Opt. (1)

I. Ohlı́dal, K. Navrátil, M. Ohlı́dal, “Scattering of light from multilayer systems with rough boundaries,” Prog. Opt. 34, 251–334 (1995).

Surf. Sci. (1)

G. C. Brown, V. Celli, M. Haller, A. Marvin, “Vector theory of light scattering from a rough surface: unitary and reciprocal expansions,” Surf. Sci. 136, 381–397 (1984).
[Crossref]

Waves Random Media (2)

I. M. Fuks, A. G. Voronovich, “Wave diffraction by rough interfaces in an arbitrary plane-layered medium,” Waves Random Media 10, 253–272 (2000).
[Crossref]

A. R. McGurn, A. A. Maradudin, “Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces,” Waves Random Media 6, 251–267 (1996).
[Crossref]

Other (4)

J. A. Ogilvy, Theory of Wave Scattering From Random Rough Surfaces (Hilger, Bristol, UK, 1991).

A. G. Voronovich, Wave Scattering from Rough Surfaces (Springer, Berlin, 1994).

A. Ishimaru, C. Le, Y. Kuga, L. A. Sengers, T. K. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Progress in Electromagnetic Research, M. Tateiba, L. Tsang, eds. (Elsevier, New York, 1996), Vol. 14, pp. 1–36.

L. Tsang, G. T. J. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley-Interscience, New York, 1985).

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Figures (7)

Fig. 1
Fig. 1

New mechanisms responsible for enhanced backscattering. The incident wave is transmitted at A as a perfect interface between the media 0 and 1; then the wave is scattered by the second rough surface at B and by the first one at C. But the wave can now follow this path the other way round. The phase difference between these two waves is zero in the antispecular direction, which produces the peak.

Fig. 2
Fig. 2

Rough surface with an incident wave coming from the medium 0 and scattered by a slab with two rough surfaces.

Fig. 3
Fig. 3

Definition of the scattering vector.

Fig. 4
Fig. 4

Bistatic coefficients for horizontal (TE) and a vertical (TM) polarized normally incident light of wavelength λ=632.8 nm, on a slab with an upper two-dimensional randomly rough surface, characterized by the parameters σ1=15 nm, l1=100 nm, and a bottom rough surface characterized by σ2=5 nm, l2=100 nm. The dielectric constants are 1=2.6896+i0.0075 and 2=-18.3+i0.55. The thickness of the film is H=500 nm. The scattered field is observed in the incident plane. Solid curves, total incoherent scattering γ¯incoh; dotted curves, first order given by I¯(1010)+I¯(0101); dashed curves second order I¯(2020)+I¯(0202)+I¯(1111); dash-dotted curves, third order I¯(3010)+I¯(0301)+I¯(1210)+I¯(2101).

Fig. 5
Fig. 5

Only the second-order contributions of Fig. 4 are depicted. Solid curves, I¯(1111); dashed curves I¯(2020); dotted curves, I¯(0202).

Fig. 6
Fig. 6

Same parameters as in Fig. 4, but with θ0=-20°.

Fig. 7
Fig. 7

Second-order contributions to Fig. 6.

Equations (85)

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

hi(x)=0,
hi(x)hi(x)=Wi(x-x),
h1(x)h2(x)=0,
Wi(x)=σi2 exp(-x2/li2),
hi(p)=0,
hi(p)hi(p)=(2π)2δ(p+p)Wi(p),
h1(p)h2(p)=0,
Wi(p)d2xWi(x)exp(-ip·x)
=πσi2li2 exp(-p2li2/4).
E0(x, z)=Ei(p0)exp[ip0·x-iα0(p0)z]+ d2p(2π)2 Es(p)exp[ip·x+iα0(p)z],
α0(p)(0K02-p2)1/2,
K0=ω/c,
Ei(p0)=EVi(p0)eˆV0-(p0)+EHi(p0)eˆH(p0),
Es(p)=EVs(p)eˆV0+(p)+EHs(p)eˆH(p).
eˆH(p)=eˆz×pˆ,
eˆV0±(p)=±α0(p)0K0 pˆ-p0K0 eˆz,
E1(r)= d2p(2π)2 E1-(p)exp[ip·x-iα1(p)z]+ d2p(2π)2 E1+(p)exp[ip·x+iα1(p)z],
α1(p)(1K02-p2)1/2.
eˆH(p)=eˆz×pˆ,
eˆV1±(p)=±α1(p)1K0 pˆ-p1K0 eˆz.
Es(p)R¯(p|p0)·Ei(p0),
R¯(p|p0)=RVV(p|p0)RVH(p|p0)RHV(p|p0)RHH(p|p0).
f¯g¯=fVVfVHfHVfHHgVVgVHgHVgHHfVVgVV*fVHgVH*Re(fVVgVH*)-Im(fVVgVH*)fHVgHV*fHHgHH*Re(fHVgHH*)-Im(fHVgVH*)2 Re(fVVgHV*)2 Re(fVHgHH*)Re(fVVgVV*+fHVgVH*)-Im(fVVgHH*-fVHgHV*)2 Im(fVVgHV*)2 Im(fVHgHH*)Im(fVVgVV*+fHVgVH*)Re(fVVgHH*-fVHgHV*),
M¯incoh(p|p0)=K02 cos2 θ(2π)2[R¯(p|p0)R¯(p|p0)-R¯(p|p0)R¯(p|p0)].
γ¯incoh(p|p0)1A cos θ0 M¯incoh(p|p0),
R¯=R¯(00)+R¯(10)+R¯(01)+R¯(11)+R¯(20)+R¯(21)+R¯(12)+R¯(22)+R¯(30)+R¯(03)+.
h12p+1(x)=0,
h22p+1(x)=0,papositiveinteger,
γ¯incoh(p|p0)=γ¯uincoh(p|p0)+γ¯dincoh(p|p0)+γ¯udincoh(p|p0),
γ¯uincoh(p|p0)=K02 cos2 θA(2π)2 cos θ0[R¯(10)  R¯(10)+R¯(20)  R¯(20)+R¯(30)  R¯(10)]
γ¯dincoh(p|p0)=K02 cos2 θA(2π)2 cos θ0[R¯(01)  R¯(01)+R¯(02)  R¯(02)+R¯(03)  R¯(01)]
γ¯udincoh(p|p0)=K02 cos2 θA(2π)2 cos θ0[R¯(10)  R¯(12)+R¯(12)  R¯(10)+R¯(01)  R¯(21)+R¯(21)  R¯(01)+R¯(11)  R¯(11)+].
R¯(10)(p|p0)=α0(p0)X¯(10)(p|p0)h1(p-p0),
R¯(01)(p|p0)=α0(p0)X¯(01)(p|p0)h2(p-p0),
R¯(11)(p|p0)=α0(p0) d2p1(2π)2[X¯(11)12(p|p1|p0)h1(p-p1)h2(p1-p0)+X¯(11)21(p|p1|p0)h2(p-p1)h1(p1-p0)],
R¯(20)(p|p0)=α0(p0) d2p1(2π)2 X¯(20)(p|p1|p0)h1(p-p1)h1(p1-p0),
R¯(02)(p|p0)=α0(p0) d2p1(2π)2 X¯(02)(p|p1|p0)h2(p-p1)h2(p1-p0),
R¯(21)(p|p0)=α0(p0) d2p1(2π)2 d2p2(2π)2[X¯(21)112(p|p1|p2|p0)h1(p-p1)h1(p1-p2)h2(p2-p0)+X¯(21)121(p|p1|p2|p0)h1(p-p1)h2(p1-p2)h1(p2-p0)+X¯(21)211(p|p1|p2|p0)h2(p-p1)h1(p1-p2)h1(p2-p0)],
R¯(12)(p|p0)=α0(p0) d2p1(2π)2 d2p2(2π)2[X¯(12)221(p|p1|p2|p0)h2(p-p1)h2(p1-p2)h1(p2-p0)+X¯(12)212(p|p1|p2|p0)h2(p-p1)h1(p1-p2)h2(p2-p0)+X¯(12)122(p|p1|p2|p0)h1(p-p1)h2(p1-p2)h2(p2-p0)],
R¯(30)(p|p0)=α0(p0) d2p1(2π)2 d2p2(2π)2 X¯(30)(p|p1|p2|p0)h1(p-p1)h1(p1-p2)h1(p2-p0),
R¯(03)(p|p0)=α0(p0) d2p1(2π)2 d2p2(2π)2 X¯(03)(p|p1|p2|p0)h2(p-p1)h2(p1-p2)h2(p2-p0).
γ¯uincoh(p|p0)=K04 cos2 θ cos θ0(2π)2[I¯(10-10)(p|p0)+I¯(20-20)(p|p0)+I¯(30-10)(p|p0)],
γ¯dincoh(p|p0)=K04 cos2 θ cos θ0(2π)2[I¯(01-01)(p|p0)+I¯(02-02)(p|p0)+I¯(03-01)(p|p0)],
γ¯udincoh(p|p0)=K04 cos2 θ cos θ0(2π)2[I¯(12-10)(p|p0)+I¯(11-11)(p|p0)+I¯(21-01)(p|p0)],
I¯(10-10)(p|p0)=W1(p-p0)X¯(10)(p|p0)X¯(10)(p|p0)
I¯(20-20)(p|p0)= d2p1(2π)2W1(p-p1)W1(p1-p0)X¯(20)(p|p1|p0) [X¯(20)(p|p1|p0)+X¯(20)(p|p+p0-p1|p0)]
I¯(30-10)(p|p0)=W1(p-p0)[X¯(10)(p|p0)X¯(30)(p|p0)+X¯(30)(p|p0)X¯(10)(p|p0)],
I¯(01-01)(p|p0)=W2(p-p0)X¯(01)(p|p0)X¯(01)(p|p0)
I¯(02-02)(p|p0)= d2p1(2π)2W2(p-p1)W2(p1-p0)X¯(02)(p|p1|p0) [X¯(02)(p|p1|p0)+X¯(02)(p|p+p0-p1|p0)]
I¯(03-01)(p|p0)=W2(p-p0)[X¯(01)(p|p0)X¯(03)(p|p0)+X¯(03)(p|p0)X¯(01)(p|p0)],
I¯(12-10)(p|p0)=W1(p-p0)[X¯(12)(p|p0)X¯(10)(p|p0)+X¯(10)(p|p0)X¯(12)(p|p0)],
I¯(21-01)(p|p0)=W2(p-p0)[X¯(21)(p|p0)X¯(01)(p|p0)+X¯(01)(p|p0)X¯(21)(p|p0)],
I¯(11-11)(p|p0)= d2p1(2π)2{W1(p-p1)W2(p1-p0)X¯(11)12(p|p1|p0)[X¯(11)12(p|p1|p0)+X¯(11)21(p|p+p0-p1|p0)]+W2(p-p1)W1(p1-p0)X¯(11)21(p|p1|p0)[X¯(11)21(p|p1|p0)+X¯(11)12(p|p+p0-p1|p0)]}
X¯(30)(p|p0)= d2p1(2π)2{W1(p1-p0)X¯(30)(p|p0|p1|p0)+W1(p-p1)[X¯(30)(p|p1|p0-p+p1|p0)+X¯(30)(p|p1|p|p0)]},
X¯(03)(p|p0)= dp1(2π)2{W2(p1-p0)X¯(03)(p|p0|p1|p0)+W2(p-p1)[X¯(03)(p|p1|p0-p+p1|p0)+X¯(03)(p|p1|p|p0)]},
X¯(12)(p|p0)= d2p1(2π)2{W2(p-p1)[X¯(12)221(p|p1|p|p0)+X¯(12)212(p|p1|p0-p+p1|p0)]+W2(p1-p0)X¯(12)122(p|p0|p1|p0)},
X¯(21)(p|p0)= d2p1(2π)2{W1(p-p1)[X¯(21)112(p|p1|p|p0)+X¯(21)121(p|p1|p0-p+p1|p0)]+W1(p1-p0)X¯(21)211(p|p0|p1|p0)}.
 d2p(2π)2 M¯h1+,0+(u|p)·R¯(p|p0)·Ei(p0)+M¯h1+,0-(u|p0)·Ei(p0)=2(01)1/2α1(u)(1-0) E1+(u),
 d2p(2π)2 M¯h1-,0+(u|p)·R¯(p|p0)·Ei(p0)+M¯h1-,0-(u|p0)·Ei(p0)=-2(01)1/2α1(u)(1-0) E1-(u),
M¯h1b,0a(u|p)I(bα1(u)-aα0(p)|u-p)bα1(u)-aα0(p) M¯1b,0a(u|p)
M¯1b,0a(u|p)=u p+abα1(u)α0(p)uˆ·pˆ-b01/2K0α1(u)(uˆ×pˆ)za11/2K0α0(p)(uˆ×pˆ)z(01)1/2K02uˆ·pˆ,
I(α|p)d2x exp[-ip·x-iαh1(x)];
E1+(u)= d2u1(2π)2 R¯s1,2H(u|u1)·E1-(u1).
R¯s1,2H(p|p0)=exp{i[α1(p)+α1(p0)]H}R¯s1,2(p|p0).
 d2p(2π)2 M¯h1+,0+(u|p)+ d2u1(2π)2 α1(u)α1(u1)×R¯s1,2H(u|u1)·M¯h1-,0+(u1|p)·R¯(p|p0)=-M¯h1+,0-(u|p0)+ d2u1(2π)2 α1(u)α1(u1)×R¯s1,2H(u|u1)·M¯h1-,0-(u|p0).
I(α|p)=(2π)2δ(p)-iαh1(1)(p)-α22h1(2)(p)-iα33!h1(3)(p)+ ,
h1(n)(p)d2x exp(-ip·x)h1n(x),
R¯s1,2H(p|p0)=(2π)2δ(p-p0)X¯s1,2H(0)(p0)+α0(p0)X¯s1,2H(1)(p|p0)×h2(p-p0)+α0(p0) d2p1(2π)2 X¯s1,2H(2)(p|p1|p0)×h2(p-p1)h2(p1-p0)+α0(p0) d2p1(2π)2 d2p2(2π)2 X¯s1,2H(3)(p|p1|p2|p0)×h2(p-p1)h2(p1-p2)h2(p2-p0).
R¯(n0)(p|p0)=R¯u(n)(p|p0),
R¯(0n)(p|p0)=R¯d(n)(p|p0),
X¯(11)21(p|p1|p0)=T¯10(p)·U¯(0)(p)·X¯s1,2H(1)(p|p0)·[-¯·D¯10-(p1)·X¯(10)(p1|p0)+iS¯+(p1|p0)],
X¯(11)12(p|p1|p0)=iP¯+(p|p0)·X¯(01)(p1|p0),
X¯(21)112(p|p1|p2|p0)=iP¯+(p|p1)·X¯(11)12(p1|p2|p0)+12[α1(p)P¯-(p|p2)-α0(p2)P¯+(p|p2)]·X¯(01)(p2|p0),
X¯(21)121(p|p1|p2|p0)=iP¯+(p|p1)·X¯(11)21(p1|p2|p0),
X¯(21)211(p|p1|p2|p0)=T¯10(p)·U¯(0)(p)·X¯s1,2H(1)(p|p1)·-·¯D¯10-(p1)·X¯(20)(p1|p2|p0)+i(1-0)2(01)1/2 M¯1-,0+(p1|p2)·X¯(10)(p2|p0)-12[α1(p1)S¯+(p1|p0)+α0(p0)S¯-(p1|p0)],
X¯(12)122(p|p1|p2|p0)=iP¯+(p|p1)·X¯(02)(p1|p2|p0),
X¯(12)212(p|p1|p2|p0)=T¯10(p)·U¯(0)(p)·X¯s1,2H(1)(p|p1)·-¯·D¯10-(p1)·X¯(11)12(p1|p2|p0)+i(1-0)2(01)1/2 M¯1-,0+(p1|p2)·X¯(01)(p2|p0),
X¯(12)221(p|p1|p2|p0)=T¯10(p)·U¯(0)(p)·{-X¯s1,2H(1)(p|p1)·¯·D¯10-(p1)·X¯(11)21(p1|p2|p0)+X¯s1,2H(2)(p|p1|p2)·[-¯·D¯10-(p2)·X¯(10)(p2|p0)+iS¯+(p2|p0)]},
2π101/2 σ1λ1,2π211/2 σ2λ1,
σ1l11,σ2l21.
¯12 (01)-1/2001,
S¯±(p|p0)1-02α0(p0)(01)1/2[M¯1-,0+(p|p0)·X¯(00)(p|p0)±M¯1-,0-(p|p0)].
S¯+(pp0)=(1-0)(01)1/2×1p p0FV+(p0)+0α1(p)α1(p0)FV-(p0)pˆ·pˆ001/2K0α1(p)FH+(p0)(pˆ×pˆ0)z-011/2K0α1(p0)FV-(p0)(pˆ×pˆ0)z(01)1/2K02FH+(p0)pˆ·pˆ0·[D¯10+(p0)]-1,
S¯-(p|p0)=(1-0)α0(p0)(01)1/2-0α1(p0)p p0FV-(p0)-01/2K0α1(p)α1(p0)FH-(p0)(pˆ×pˆ0)z-1α1(p)α02(p0)FV+(p0)pˆ·pˆ013/2K0α02(p0)FV+(p0)(pˆ×pˆ0)z-(01)1/2K02α1(p0)FH-(p0)pˆ·pˆ0·[D¯10+(p0)]-1.

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