Abstract

The angular distribution of the mean diffuse intensity scattered from a metal surface with one-dimensional roughness is studied with perturbation theory. From an approach based on the reduced Rayleigh equations in p polarization, exact perturbation terms up to eighth order in the height parameter are developed for surface roughness consistent with a stationary Gaussian process. The theory is evaluated for a number of cases in which surface plasmon polariton excitation is significant and produces effects such as backscattering enhancement. For surface roughness having a wide Gaussian power spectrum, it is found that the high-order terms lead to roughness-induced broadening of the backscattering peak. For rectangular spectra, two cases are studied in which backscattering effects are due to sixth- and eighth-order terms; both cases provide good comparisons with previously unexplained experimental results. Further, because of an eighth-order term, the diffuse intensity is shown to contain a specular peak that also relies on polariton excitation. This new effect is studied in detail and is found to arise from the constructive interference of contributions produced by multiple-scattering processes, although the time-reversed paths that produce backscattering enhancement are not essential to the specular effect.

© 2001 Optical Society of America

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References

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  1. 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]
  2. 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]
  3. 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]
  4. C. S. West, K. A. O’Donnell, “Scattering by plasmon polaritons on a metal surface with a detuned roughness spectrum,” Opt. Lett. 21, 1–3 (1996).
    [CrossRef] [PubMed]
  5. A. Arsenieva, S. Feng, “Correspondence between correlation functions and enhanced backscattering peak for scattering from smooth random surfaces,” Phys. Rev. B 47, 13047–13050 (1993).
    [CrossRef]
  6. V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
    [CrossRef]
  7. C. S. West, K. A. O’Donnell, “Angular correlation functions of light scattered from weakly rough metal surfaces,” Phys. Rev. B 59, 2393–2406 (1999).
    [CrossRef]
  8. P. Tran, V. Celli, “Monte Carlo calculation of backscattering enhancement for a randomly rough grating,” J. Opt. Soc. Am. A 5, 1635–1637 (1988).
    [CrossRef]
  9. T. R. Michel, “Resonant light scattering from weakly rough random surfaces and imperfect gratings,” J. Opt. Soc. Am. A 11, 1874–1885 (1994).
    [CrossRef]
  10. A. A. Maradudin, E. R. Méndez, “Enhanced backscattering of light from weakly rough, random metal surfaces,” Appl. Opt. 32, 3335–3343 (1993).
    [CrossRef] [PubMed]
  11. V. Freilikher, I. Yurkevich, “Backscattering enhancement from surfaces with random impedance,” Phys. Lett. A 183, 247–252 (1993).
    [CrossRef]
  12. 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]
  13. H. Hanato, H. Ogura, Z. L. Wang, “Scattering from a slightly random, one-dimensional metal surface: 45° linearly polarized incidence, backscattering enhancement and degree of polarization,” Waves Random Media 7, 11–34 (1997).
    [CrossRef]
  14. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 82.
  15. M. Arnold, M. Otto, “Notes on localization of surface plasmon polaritons,” Opt. Commun. 125, 122–136 (1996).
    [CrossRef]
  16. D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
    [CrossRef] [PubMed]
  17. K. A. O’Donnell, C. S. West, E. R. Méndez, “Backscattering enhancement from polariton–polariton coupling on a rough metal surface,” Phys. Rev. B 57, 13209–13219 (1998).
    [CrossRef]
  18. C. S. West, “Backscattering enhancement from plasmon polaritons on rough metal surfaces,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Georgia, 1997), p. 152.
  19. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Connection between blazes from gratings and enhancements from random rough surfaces,” Phys. Rev. B 39, 8193–8197 (1989).
    [CrossRef]
  20. E. R. Méndez, M. A. Ponce, V. Ruiz-Cortés, Z. Gu, “Coherent effects in the scattering of light from random surfaces with symmetry,” Opt. Lett. 16, 123–125 (1991).
    [CrossRef]

1999 (1)

C. S. West, K. A. O’Donnell, “Angular correlation functions of light scattered from weakly rough metal surfaces,” Phys. Rev. B 59, 2393–2406 (1999).
[CrossRef]

1998 (1)

K. A. O’Donnell, C. S. West, E. R. Méndez, “Backscattering enhancement from polariton–polariton coupling on a rough metal surface,” Phys. Rev. B 57, 13209–13219 (1998).
[CrossRef]

1997 (2)

H. Hanato, H. Ogura, Z. L. Wang, “Scattering from a slightly random, one-dimensional metal surface: 45° linearly polarized incidence, backscattering enhancement and degree of polarization,” Waves Random Media 7, 11–34 (1997).
[CrossRef]

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

1996 (3)

C. S. West, K. A. O’Donnell, “Scattering by plasmon polaritons on a metal surface with a detuned roughness spectrum,” Opt. Lett. 21, 1–3 (1996).
[CrossRef] [PubMed]

M. Arnold, M. Otto, “Notes on localization of surface plasmon polaritons,” Opt. Commun. 125, 122–136 (1996).
[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]

1995 (2)

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef] [PubMed]

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]

1994 (1)

1993 (3)

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

V. Freilikher, I. Yurkevich, “Backscattering enhancement from surfaces with random impedance,” Phys. Lett. A 183, 247–252 (1993).
[CrossRef]

A. Arsenieva, S. Feng, “Correspondence between correlation functions and enhanced backscattering peak for scattering from smooth random surfaces,” Phys. Rev. B 47, 13047–13050 (1993).
[CrossRef]

1991 (1)

1989 (1)

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Connection between blazes from gratings and enhancements from random rough surfaces,” Phys. Rev. B 39, 8193–8197 (1989).
[CrossRef]

1988 (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]

Arnold, M.

M. Arnold, M. Otto, “Notes on localization of surface plasmon polaritons,” Opt. Commun. 125, 122–136 (1996).
[CrossRef]

Arsenieva, A.

A. Arsenieva, S. Feng, “Correspondence between correlation functions and enhanced backscattering peak for scattering from smooth random surfaces,” Phys. Rev. B 47, 13047–13050 (1993).
[CrossRef]

Celli, V.

Feng, S.

A. Arsenieva, S. Feng, “Correspondence between correlation functions and enhanced backscattering peak for scattering from smooth random surfaces,” Phys. Rev. B 47, 13047–13050 (1993).
[CrossRef]

Freilikher, V.

V. Freilikher, I. Yurkevich, “Backscattering enhancement from surfaces with random impedance,” Phys. Lett. A 183, 247–252 (1993).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 82.

Gu, Z.

Hanato, H.

H. Hanato, H. Ogura, Z. L. Wang, “Scattering from a slightly random, one-dimensional metal surface: 45° linearly polarized incidence, backscattering enhancement and degree of polarization,” Waves Random Media 7, 11–34 (1997).
[CrossRef]

Lagendijk, A.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef] [PubMed]

Leskova, T. A.

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

Malyshkin, V.

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

Maradudin, A. A.

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (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]

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]

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]

Marvin, A. M.

McGurn, A. R.

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (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]

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]

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]

Méndez, E. R.

Michel, T. R.

Nieto-Vesperinas, M.

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Connection between blazes from gratings and enhancements from random rough surfaces,” Phys. Rev. B 39, 8193–8197 (1989).
[CrossRef]

O’Donnell, K. A.

C. S. West, K. A. O’Donnell, “Angular correlation functions of light scattered from weakly rough metal surfaces,” Phys. Rev. B 59, 2393–2406 (1999).
[CrossRef]

K. A. O’Donnell, C. S. West, E. R. Méndez, “Backscattering enhancement from polariton–polariton coupling on a rough metal surface,” Phys. Rev. B 57, 13209–13219 (1998).
[CrossRef]

C. S. West, K. A. O’Donnell, “Scattering by plasmon polaritons on a metal surface with a detuned roughness spectrum,” Opt. Lett. 21, 1–3 (1996).
[CrossRef] [PubMed]

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]

Ogura, H.

H. Hanato, H. Ogura, Z. L. Wang, “Scattering from a slightly random, one-dimensional metal surface: 45° linearly polarized incidence, backscattering enhancement and degree of polarization,” Waves Random Media 7, 11–34 (1997).
[CrossRef]

Otto, M.

M. Arnold, M. Otto, “Notes on localization of surface plasmon polaritons,” Opt. Commun. 125, 122–136 (1996).
[CrossRef]

Ponce, M. A.

Ruiz-Cortés, V.

Soto-Crespo, J. M.

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Connection between blazes from gratings and enhancements from random rough surfaces,” Phys. Rev. B 39, 8193–8197 (1989).
[CrossRef]

Tran, P.

van Albada, M. P.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef] [PubMed]

van Tiggelen, B. A.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef] [PubMed]

Wang, Z. L.

H. Hanato, H. Ogura, Z. L. Wang, “Scattering from a slightly random, one-dimensional metal surface: 45° linearly polarized incidence, backscattering enhancement and degree of polarization,” Waves Random Media 7, 11–34 (1997).
[CrossRef]

West, C. S.

C. S. West, K. A. O’Donnell, “Angular correlation functions of light scattered from weakly rough metal surfaces,” Phys. Rev. B 59, 2393–2406 (1999).
[CrossRef]

K. A. O’Donnell, C. S. West, E. R. Méndez, “Backscattering enhancement from polariton–polariton coupling on a rough metal surface,” Phys. Rev. B 57, 13209–13219 (1998).
[CrossRef]

C. S. West, K. A. O’Donnell, “Scattering by plasmon polaritons on a metal surface with a detuned roughness spectrum,” Opt. Lett. 21, 1–3 (1996).
[CrossRef] [PubMed]

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]

C. S. West, “Backscattering enhancement from plasmon polaritons on rough metal surfaces,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Georgia, 1997), p. 152.

Wiersma, D. S.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef] [PubMed]

Yurkevich, I.

V. Freilikher, I. Yurkevich, “Backscattering enhancement from surfaces with random impedance,” Phys. Lett. A 183, 247–252 (1993).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

M. Arnold, M. Otto, “Notes on localization of surface plasmon polaritons,” Opt. Commun. 125, 122–136 (1996).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

V. Freilikher, I. Yurkevich, “Backscattering enhancement from surfaces with random impedance,” Phys. Lett. A 183, 247–252 (1993).
[CrossRef]

Phys. Rev. B (5)

K. A. O’Donnell, C. S. West, E. R. Méndez, “Backscattering enhancement from polariton–polariton coupling on a rough metal surface,” Phys. Rev. B 57, 13209–13219 (1998).
[CrossRef]

A. Arsenieva, S. Feng, “Correspondence between correlation functions and enhanced backscattering peak for scattering from smooth random surfaces,” Phys. Rev. B 47, 13047–13050 (1993).
[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]

C. S. West, K. A. O’Donnell, “Angular correlation functions of light scattered from weakly rough metal surfaces,” Phys. Rev. B 59, 2393–2406 (1999).
[CrossRef]

M. Nieto-Vesperinas, J. M. Soto-Crespo, “Connection between blazes from gratings and enhancements from random rough surfaces,” Phys. Rev. B 39, 8193–8197 (1989).
[CrossRef]

Phys. Rev. Lett. (1)

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, “Experimental evidence for recurrent multiple scattering events of light in disordered media,” Phys. Rev. Lett. 74, 4193–4196 (1995).
[CrossRef] [PubMed]

Waves Random Media (3)

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]

H. Hanato, H. Ogura, Z. L. Wang, “Scattering from a slightly random, one-dimensional metal surface: 45° linearly polarized incidence, backscattering enhancement and degree of polarization,” Waves Random Media 7, 11–34 (1997).
[CrossRef]

V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, M. Nieto-Vesperinas, “Speckle correlations in the light scattered from weakly rough random metal surfaces,” Waves Random Media 7, 479–520 (1997).
[CrossRef]

Other (2)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 82.

C. S. West, “Backscattering enhancement from plasmon polaritons on rough metal surfaces,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Georgia, 1997), p. 152.

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Figures (10)

Fig. 1
Fig. 1

Light scattering from a rough metal surface. The angles θi and θs are positive as shown.

Fig. 2
Fig. 2

(A) For order n as indicated, the mean intensities I(n)(θs|θi) for the Gaussian spectrum with σ=5 nm and θi=20°. (B) I(8)(θs|θi) compared with the results of Refs. 1 and 9 (noisy curve).

Fig. 3
Fig. 3

(A) Backscattering peak in I(n)(θs|θi) for the Gaussian spectrum with σ = 5 nm. (B) Peak width (full width at half-maximum) as a function of σ; arrows denote where peaks become unphysical.

Fig. 4
Fig. 4

(A) 2–2 effect in I(n)(θs|θi) for (kmin, kmax)=(0.83 ω/c, 1.29 ω/c), =-9.0+1.29i, σ=10 nm, λ=612 nm, and θi=10°; the three curves overlap closely for θs-41°. (B) Scattering processes responsible for the backscattering peak.

Fig. 5
Fig. 5

(A) The solid curve is the 3–3 effect in I(8)(θs|θi) for (kmin, kmax)=(0.60 ω/c, 0.93 ω/c), =-5.7+0.75i, σ=10.2 nm, λ=442 nm, and θi=14°. Circular points are experimental data from Ref. 18. (B) Scattering processes responsible for the backscattering peak; the straight arrows denote the nonresonant wave.

Fig. 6
Fig. 6

For the θi indicated, intensity contributions from the 3–3 term’s two-dimensional integral in Eq. (A4). Parameters are as in Fig. 5 except that =-12.0+1.0i.

Fig. 7
Fig. 7

(A) 4–4 effect in I(8)(θs|θi) for (kmin, kmax)=(0.91 ω/c, 1.42 ω/c), =-12.6+1.16i, σ=15.5 nm, λ=674 nm, and θi=19°. (B) Peak in I(8)(θs|θi) as compared with experimental data of Ref. 4. (C) Scattering processes responsible for the backscattering peak; the straight arrows denote the nonresonant wave.

Fig. 8
Fig. 8

(A) Mean intensity I(8)(θs|θi) for the Gaussian spectrum with θi=20° and σ = 8 nm. (B) Intensity contribution from the 4–4 term’s one-dimensional integral in Eq. (A7) compared with the approximation of Eq. (11).

Fig. 9
Fig. 9

(A) Specular effect in I(8)(θs|θi) for (kmin, kmax)=(1.70 ω/c, 2.25 ω/c), =-9.5+0.24i, θi=52°, (σ/λ)=2×10-3 (dashed curve), and (σ/λ)=3×10-3 (solid curve). (B) Intensity contribution from the 4–4 term’s one-dimensional integral in Eq. (A7) (solid curve) compared with the simplified term of Eq. (11) (dashed curve).

Fig. 10
Fig. 10

For the specular peak, the three types of scattering processes that contribute.

Equations (67)

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

ζˆ(k)=-dx ζ(x)exp(-ikx).
ζ^*(k)ζˆ(k)=2πσ2δ(k-k)g(k),
T(q|k)=n=1(-i)nn! T(n)(q|k),
T(1)(q|k)=A(1)(q|k)ζˆ(q-k),
T(2)(q|k)=12π-dp A(2)(q|p|k)ζˆ(q-p)ζˆ(p-k),
T(3)(q|k)=1(2π)2-dpdr A(3)(q|p|r|k)×ζˆ(q-p)ζˆ(p-r)ζˆ(r-k),
I(θs|θi)=2πωc3cos2(θs)cos(θi)|G0(q)|2×|ΔT(q|k)|2|G0(k)|2,
|ΔT(q|k)|2=|T(1)(q|k)|2+12!2 |ΔT(2)(q|k)|2-21!3!ReT(1)*(q|k)T(3)(q|k)+13!2 |T(3)(q|k)|2-22!4!ReΔT(2)*(q|k)ΔT(4)(q|k)+21!5!ReT(1)*(q|k)T(5)(q|k)+14!2 |ΔT(4)(q|k)|2-23!5!ReT(3)*(q|k)T(5)(q|k)+22!6!ReΔT(2)*(q|k)ΔT(6)(q|k)-21!7!ReT(1)*(q|k)T(7)(q|k),
G0(k)=iα0(k)+α(k),
g(k)=πa exp(-a2k2/4),
σ8(2π)3-dp[|F(q, p, k)|2+|G(q, p, k)|2]
×g(q-p)g(p-k),
F(q, p, k)=-dr A(4)(q|q+r|p+r|k+r|k)g(r),
G(q, p, k)=-dr A(4)(q|p|p+r|p|k)g(r).
ϕA=kx1-qx4+β14,
ϕB=kx1-qx4+β14.
ΔϕBA=(q-k)(x1-x1).
σ8(2π)3-dp[F(q, p, k)+G(q, p, k)]×g(q-p)g(p-k),
F(q, p, k)=[F*(q, p, k)+F*(q, q+k-p, k)]×F(q, p, k),
G(q, p, k)=[G*(q, p, k)+G*(q, q+k-p, k)]×G(q, p, k).
ΔϕCA=(q-k)(x1-x1)+(q+k)Δx,
|T(1)(q|k)|2
=σ2|A(1)(q|k)|2g(q-k), 
|ΔT(2)(q|k)|2
=σ42π-dp A(2,0)*(q, p, k)A(2)(q|p|k)×g(q-p)g(p-k),
T(1)*(q|k)T(3)(q|k)
=σ42πA(1)*(q|k)A(3,1,1)(q, k)g(q-k),
|T(3)(q|k)|2
=σ6(2π)2|A(3,1,1)(q, k)|2g(q-k)+-dpdrA(3,0)*(q, p, r, k)×A(3)(q|p|r|k)g(q-p)g(p-r)g(r-k),
ΔT(2)*(q|k)ΔT(4)(q|k)
 =σ6(2π)2-dp A(2,0)*(q, p, k)×A(4,1,1)(q, p, k)g(q-p)g(p-k),
T(1)*(q|k)T(5)(q|k)
=σ6(2π)2A(1)*(q|k)A(5,2,2)(q, k)g(q-k),
|ΔT(4)(q|k)|2
=σ8(2π)3-dp[A(4,1,1)*(q, p, k)+A(4,1,1)*(q, q+k-p, k)]A(4,1,1)(q, p, k)g(q-p)g(p-k)+-dpdrds A(4,0)*(q, p, r, s, k)×A(4)(q|p|r|s|k)g(q-p)g(p-r)g(r-s)×g(s-k),
T(3)*(q|k)T(5)(q|k)
=σ8(2π)3A(3,1,1)*(q, k)A(5,2,2)(q, k)g(q-k)+-dpdr A(3,0)*(q, p, r, k)A(5,1,1)(q, p, r, k)×g(q-p)g(p-r)g(r-k),
ΔT(2)*(q|k)ΔT(6)(q|k)
=σ8(2π)3-dp A(2,0)*(q, p, k)×A(6,2,2)(q, p, k)g(q-p)g(p-k),
T(1)*(q|k)T(7)(q|k)
=σ8(2π)3A(1)*(q|k)A(7,3,3)(q, k)g(q-k).
A(3)(q|p|r|k)=A(3)(q|{q-p, p-r, r-k}|k),
S=A(2)(q|p|k)+A(2)(q|q+k-p|k),
S=A(2)(q|{q-p, p-k}|k)+A(2)(q|{p-k, q-p}|k).
S=πA(2)(q|{q-p, p-k}|k),
A(2,0)(q, p, k)=πA(2)(q|{q-p, p-k}|k),
A(3,0)(q, p, r, k)
 =πA(3)(q|{q-p, p-r, r-k}|k),
A(4,0)(q, p, r, s, k)
=πA(4)(q|{q-p, p-r, r-s, s-k}|k),
A(3,1)(q, p, k)=πA(3)(q|{-p, p, q-k}|k),
A(3,1)(q, p, k)=A(3)(q|{-p, p, q-k}|k)+A(3)(q|{-p, q-k, p}|k)+A(3)(q|{q-k,-p, p}|k)=A(3)(q|p+q|q|k)+A(3)(q|p+q|p+k|k)+A(3)(q|k|p+k|k),
A(4,1)(q, p, r, k)=πA(4)(q|{-p, p, q-r, r-k}|k),
A(5,1)(q, p, r, s, k)
=πA(5)(q|{-p, p, q-r, r-s, s-k}|k),
A(5,2)(q, p, r, k)
=πA(5)(q|{-p, -r, r, p, q-k}|k),
A(6,2)(q, p, r, s, k)
=πA(6)(q|{-p, -r, r, p, q-s, s-k}|k),
A(7,3)(q, p, r, s, k)
=πA(7)(q|{-p, -r, -s, s, r, p, q-k}|k),
A(3,1,1)(q, k)=-dp g(p)A(3,1)(q, p, k),
A(4,1,1)(q, r, k)=-dp g(p)A(4,1)(q, p, r, k),
A(5,1,1)(q, r, s, k)=-dp g(p)A(5,1)(q, p, r, s, k),
A(5,2,2)(q, k)=-dpdr g(p)g(r)×A(5,2)(q, p, r, k),
A(6,2,2)(q, s, k)=-dpdr g(p)g(r)×A(6,2)(q, p, r, s, k),
A(7,3,3)(q, k)=-dpdrds g(p)g(r)g(s)×A(7,3)(q, p, r, s, k),

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