Abstract

The enhanced backscattering of p-polarized light from a small-rms-height, small-rms-slope, one-dimensional, random metal surface was first predicted on the basis of an infinite-order perturbation calculation that used the techniques of many-body theory and the concepts of weak localization theory. We present an elementary calculation of the contribution to the mean differential reflection coefficient from the incoherent component of the scattered light that is based on the first two nonzero terms in the expansion of this function in powers of the surface-profile function. We show that this approach not only accounts for enhanced backscattering but also gives the correct order of magnitude of the effect predicted by infinite-order perturbation theory.

© 1993 Optical Society of America

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  1. A bibliography of these papers can be found in 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]
  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]
  3. See, for example, E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, R. F. Wallis, “Surface polaritons–electromagnetic waves at interfaces,” in Polaritons, E. Burstein, F. de Martini, eds. (Pergamon, London, 1974), pp. 89–108.
  4. A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.
  5. G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
    [CrossRef]
  6. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 5.
  7. Zu-Han Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental study of the opposition effect in the scattering of light from a randomly rough metal surface,” Appl. Opt. 28, 537–543 (1989).
    [CrossRef] [PubMed]
  8. A. A. Maradudin, A. R. McGurn, “Enhanced backscattering of light from the surface of a liquid metal,” in Elementary Excitations in Solids, J. L. Birman, C. Sébenne, R. F. Wallis, eds. (North-Holland, Amsterdam, 1992), pp. 197–210.

1989 (1)

1985 (2)

A bibliography of these papers can be found in 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]

1983 (1)

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

Brown, G. C.

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

Burstein, E.

See, for example, E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, R. F. Wallis, “Surface polaritons–electromagnetic waves at interfaces,” in Polaritons, E. Burstein, F. de Martini, eds. (Pergamon, London, 1974), pp. 89–108.

Celli, V.

A bibliography of these papers can be found in 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]

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

Coopersmith, M.

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

Dummer, R. S.

Gu, Zu-Han

Haller, M.

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

Hartstein, A.

See, for example, E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, R. F. Wallis, “Surface polaritons–electromagnetic waves at interfaces,” in Polaritons, E. Burstein, F. de Martini, eds. (Pergamon, London, 1974), pp. 89–108.

Maradudin, A. A.

Zu-Han Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental study of the opposition effect in the scattering of light from a randomly rough metal surface,” Appl. Opt. 28, 537–543 (1989).
[CrossRef] [PubMed]

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]

A bibliography of these papers can be found in 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]

See, for example, E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, R. F. Wallis, “Surface polaritons–electromagnetic waves at interfaces,” in Polaritons, E. Burstein, F. de Martini, eds. (Pergamon, London, 1974), pp. 89–108.

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

A. A. Maradudin, A. R. McGurn, “Enhanced backscattering of light from the surface of a liquid metal,” in Elementary Excitations in Solids, J. L. Birman, C. Sébenne, R. F. Wallis, eds. (North-Holland, Amsterdam, 1992), pp. 197–210.

Marvin, A. M.

McGurn, A. R.

Zu-Han Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental study of the opposition effect in the scattering of light from a randomly rough metal surface,” Appl. Opt. 28, 537–543 (1989).
[CrossRef] [PubMed]

A bibliography of these papers can be found in 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]

A. A. Maradudin, A. R. McGurn, “Enhanced backscattering of light from the surface of a liquid metal,” in Elementary Excitations in Solids, J. L. Birman, C. Sébenne, R. F. Wallis, eds. (North-Holland, Amsterdam, 1992), pp. 197–210.

Méndez, E. R.

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

Michel, T.

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

Mills, D. L.

See, for example, E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, R. F. Wallis, “Surface polaritons–electromagnetic waves at interfaces,” in Polaritons, E. Burstein, F. de Martini, eds. (Pergamon, London, 1974), pp. 89–108.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 5.

Schoenwald, J.

See, for example, E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, R. F. Wallis, “Surface polaritons–electromagnetic waves at interfaces,” in Polaritons, E. Burstein, F. de Martini, eds. (Pergamon, London, 1974), pp. 89–108.

Wallis, R. F.

See, for example, E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, R. F. Wallis, “Surface polaritons–electromagnetic waves at interfaces,” in Polaritons, E. Burstein, F. de Martini, eds. (Pergamon, London, 1974), pp. 89–108.

Appl. Opt. (1)

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

Phys. Rev. B (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]

Surf. Sci. (1)

G. C. Brown, V. Celli, M. Coopersmith, M. Haller, “Unitary and reciprocal expansions in the theory of light scattering from a grating,” Surf. Sci. 129, 507–515 (1983).
[CrossRef]

Other (4)

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), Chap. 5.

A. A. Maradudin, A. R. McGurn, “Enhanced backscattering of light from the surface of a liquid metal,” in Elementary Excitations in Solids, J. L. Birman, C. Sébenne, R. F. Wallis, eds. (North-Holland, Amsterdam, 1992), pp. 197–210.

See, for example, E. Burstein, A. Hartstein, J. Schoenwald, A. A. Maradudin, D. L. Mills, R. F. Wallis, “Surface polaritons–electromagnetic waves at interfaces,” in Polaritons, E. Burstein, F. de Martini, eds. (Pergamon, London, 1974), pp. 89–108.

A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

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Figures (4)

Fig. 1
Fig. 1

Contribution to the mean differential reflection coefficient from the incoherent component of the scattered light for p-polarized light of wavelength λ = 457.9 nm incident on a one-dimensional, randomly rough, silver surface, characterized by the parameters δ = 5 nm and a = 100 nm. ∊(ω) = −7.5 + i0.24 and θ0 = 0°. The single- and multiple-scattering contributions studied here are plotted together with their sum.

Fig. 2
Fig. 2

Same as Fig. 1 but for an angle of incidence θ0 = 40°.

Fig. 3
Fig. 3

(a) Contribution to the mean differential reflection coefficient from the incoherent component of the scattered light for p-polarized light of wavelength λ = 457.9 nm incident on a one-dimensional, randomly rough, silver surface, characterized by the parameters δ = 0.2 nm and a = 100 nm. ∊(ω) = −7.5 + i0.24 and θ0 = 0°. (b) Single- and multiple-scattering contributions to the results presented in (a). On the scale of this figure the sum of these contributions coincides with the single-scattering contribution.

Fig. 4
Fig. 4

Comparison of the result of the present study (—) and that of Ref. 2 (---) for the contribution to the mean differential reflection coefficient from the incoherent component of the scattered light for p-polarized light of wavelength λ = 457.9 nm incident on a one-dimensional, randomly rough, silver surface, characterized by the parameters δ = 5 nm and a = 100 nm. ∊(ω) = −7.5 + i0.24 and θ0 = 0°. The inset is an enlargement of these results in the immediate vicinity of the retroreflection direction.

Equations (58)

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ζ ( x 1 ) = 0 ,
ζ ( x 1 ) ζ ( x 1 ) = δ 2 W ( | x 1 x 1 | ) .
W ( | x 1 | ) = exp ( x 1 2 / a 2 )
ζ ( x 1 ) = d k 2 π ζ ̂ ( k ) exp ( i k x 1 ) ,
ζ ̂ ( k ) = 0 ,
ζ ̂ ( k ) ζ ̂ ( k ) = 2 π δ ( k + k ) δ 2 g ( | k | ) .
g ( | k | ) = d x 1 W ( | x 1 | ) exp ( i k x 1 ) .
g ( | k | ) = π 1 / 2 a exp ( k 2 a 2 / 4 ) .
H ( x ; t ) = [ 0 , H 2 ( x 1 , x 3 | ω ) , 0 ] exp ( i ω t ) .
H 2 > ( x 1 , x 3 | ω ) = exp [ i k x 1 i α 0 ( k ω ) x 3 ] + d q 2 π R p ( q k ) exp [ i q x 1 + i α 0 ( q ω ) x 3 ] ,
α 0 ( q ω ) = ( ω 2 c 2 q 2 ) 1 / 2 q 2 < ω 2 c 2
= i ( q 2 ω 2 c 2 ) 1 / 2 q 2 > ω 2 c 2 .
R p θ s incoh = 1 L 1 ω 2 π c cos 2 θ s cos θ 0 [ | R p ( q k ) | 2 | R p ( q k ) | 2 ] ,
k = ω c sin θ 0 , q = ω c sin θ s .
d q 2 π I [ α ( p ω ) α 0 ( q ω ) p q ] α ( p ω ) α 0 ( q ω ) × [ α ( p ω ) α 0 ( q ω ) + p q ] R p ( q k ) = I [ α ( p ω ) + α 0 ( k ω ) p k ] α ( p ω ) + α 0 ( k ω ) [ α ( p ω ) α 0 ( k ω ) p k ] ,
I ( γ Q ) = d x 1 exp ( i Q x 1 ) exp [ i γ ζ ( x 1 ) ] ,
α ( q ω ) = [ ( ω ) ( ω 2 / c 2 ) q 2 ] 1 / 2 Re α ( q ω ) > 0 , Im α ( q ω ) > 0 .
R p ( q k ) = 2 π δ ( q k ) R p ( 0 ) ( k ω ) i G 0 ( q ω ) T ( q k ) G 0 ( k ω ) 2 α 0 ( k ω ) ,
R p ( 0 ) ( k ω ) = ( ω ) α 0 ( k ω ) α ( k ω ) ( ω ) α 0 ( k ω ) + α ( k ω ) ,
G 0 ( k ω ) = i ( ω ) ( ω ) α 0 ( k ω ) + α ( k ω ) .
T ( q k ) = V ( q k ) + d p 2 π V ( q p ) G 0 ( p ω ) T ( p k ) ,
d p 2 π [ M ( q p ) N ( q p ) ] V ( p k ) 2 i α 0 ( p ω ) = 1 2 ( ω ) α 0 ( k ω ) { M ( q k ) [ ( ω ) α 0 ( k ω ) α ( k ω ) ] + N ( q k ) [ ( ω ) α 0 ( k ω ) + α ( k ω ) ] } ,
M ( q p ) = I [ α ( q ω ) α 0 ( p ω ) q p ] α ( q ω ) α 0 ( p ω ) × [ q p + α ( q ω ) α 0 ( p ω ) ] ,
N ( q p ) = I [ α ( q ω ) + α 0 ( p ω ) q p ] α ( q ω ) + α 0 ( p ω ) × [ q p + α ( q ω ) α 0 ( p ω ) ] .
R p θ s incoh = 1 L 1 2 π ( ω c ) 3 cos 2 θ s cos θ 0 | G 0 ( q ω ) | 2 × [ | T ( q k ) | 2 | T ( q k ) | 2 ] | G 0 ( k ω ) | 2 ,
| G 0 ( q ω ) | 2 = c 2 ω 2 | ( ω ) | 2 | ( ω ) cos θ s + [ ( ω ) sin 2 θ s ] 1 / 2 | 2 ,
| G 0 ( k ω ) | 2 = c 2 ω 2 | ( ω ) | 2 | ( ω ) cos θ 0 + [ ( ω ) sin 2 θ 0 ] 1 / 2 | 2 .
V ( q k ) = n = 1 ( i ) n n ! V ( n ) ( q k ) ,
T ( q k ) = n = 1 ( i ) n n ! T ( n ) ( q k ) ,
T ( n ) ( q k ) = V ( n ) ( q k ) + m = 1 n 1 ( n m ) × d p 2 π V ( n m ) ( q p ) G 0 ( p ω ) T ( m ) ( p k )
T ( 1 ) ( q k ) = V ( 1 ) ( q k ) ,
T ( 2 ) ( q k ) = V ( 2 ) ( q k ) + 2 d p 2 π V ( 1 ) ( q p ) × G 0 ( p ω ) V ( 1 ) ( p k ) ,
T ( 3 ) ( q k ) = V ( 3 ) ( q k ) + 3 d p 2 π V ( 2 ) ( q p ) × G 0 ( p ω ) V ( 1 ) ( p k ) + 3 d p 2 π V ( 1 ) ( q p ) G 0 ( p ω ) V ( 2 ) ( p k ) + 6 d p 2 π d r 2 π V ( 1 ) ( q p ) G 0 ( p ω ) × V ( 1 ) ( p r ) G 0 ( r ω ) V ( 1 ) ( r k ) ,
I ( γ Q ) = n = 0 ( i ) n n ! γ n ζ ̂ ( n ) ( Q ) ,
ζ ̂ ( n ) ( Q ) = d x 1 ζ n ( x 1 ) exp ( i Q x 1 ) ,
V ( n ) ( q k ) = i ( ω ) 1 2 ( ω ) G ( n 1 ) ( q k ) ζ ̂ ( n ) ( q k ) ( ω ) 1 ( ω ) m = 1 n 1 ( n m ) d p 2 π × F ( n m 1 ) ( q p ) ζ ̂ ( n m ) ( q p ) V ( m ) ( p k ) ,
F ( n ) ( q p ) = { [ α ( q ω ) α 0 ( p ω ) ] n [ q p + α ( q ω ) α 0 ( p ω ) ] [ α ( q ω ) + α 0 ( p ω ) ] n [ q p α ( q ω ) α 0 ( p ω ) ] } × 1 2 α 0 ( p ω ) ,
G ( n ) ( q k ) = { [ α ( q ω ) α 0 ( k ω ) ] n [ q k + α ( q ω ) α 0 ( k ω ) ] × [ ( ω ) α 0 ( k ω ) α ( k ω ) ] + [ α ( q ω ) + α 0 ( k ω ) ] n × [ q k α ( q ω ) α 0 ( k ω ) ] [ ( ω ) α 0 ( k ω ) + α ( k ω ) ] } × 1 2 α 0 ( k ω ) .
V ( 1 ) ( q k ) = i ( ω ) 1 2 ( ω ) [ ( ω ) q k α ( q ω ) α ( k ω ) ] ζ ̂ ( 1 ) ( q k ) ,
V ( 2 ) ( q k ) = i ( ω ) 1 2 ( ω ) [ α ( q ω ) + α ( k ω ) ] × [ q k α ( q ω ) α ( k ω ) ] ζ ̂ ( 2 ) ( q k ) + 2 i [ ( ω ) 1 ] 2 3 ( ω ) α ( q ω ) d p 2 π × ζ ̂ ( 1 ) ( q p ) α ( p ω ) ζ ̂ ( 1 ) ( p k ) α ( k ω ) ,
V ( 3 ) ( q k ) = i ( ω ) 1 3 ( ω ) α ( q ω ) α ( k ω ) { [ ( ω ) 2 ] × [ α 2 ( q ω ) + α 2 ( k ω ) ] 2 ( ω ) α ( q ω ) α ( k ω ) + 3 2 [ ( ω ) 1 ] ( q 2 + k 2 ) } ζ ̂ ( 3 ) ( q k ) + i ( ω ) 1 3 ( ω ) q k { 2 α ( q ω ) α ( k ω ) + 1 2 ( ω ) [ ( ω ) + 1 ] [ α 2 ( q ω ) + α 2 ( k ω ) ] 1 2 ( ω ) [ ( ω ) 1 ] ( q 2 + k 2 ) } ζ ̂ ( 3 ) ( q k ) 3 i [ ( ω ) 1 ] 2 3 ( ω ) d p 2 π ζ ̂ ( 2 ) ( q p ) × α ( p ω ) [ q p α 2 ( q ω ) ] ζ ̂ ( 1 ) ( p k ) α ( k ω ) 3 i [ ( ω ) 1 ] 2 3 ( ω ) α ( q ω ) d p 2 π ζ ̂ ( 1 ) ( q p ) × [ p k α 2 ( k ω ) ] α ( p ω ) ζ ̂ ( 2 ) ( p k ) + 3 2 i × [ ( ω ) 1 ] 2 3 ( ω ) [ ( ω ) q k α ( q ω ) α ( k ω ) ] × { d p 2 π ζ ̂ ( 2 ) ( q p ) p 2 ζ ̂ ( 1 ) ( p k ) + d p 2 π ζ ̂ ( 1 ) ( q p ) p 2 ζ ̂ ( 2 ) ( p k ) } 6 i [ ( ω ) 1 ] 3 4 ( ω ) α ( q ω ) d p 2 π d r 2 π × ζ ̂ ( 1 ) ( q p ) α ( p ω ) ζ ̂ ( 1 ) ( p r ) α ( r ω ) × ζ ̂ ( 1 ) ( r k ) α ( k ω ) .
( q + k ) ζ ̂ ( 2 ) ( q k ) = 2 d p 2 π ζ ̂ ( 1 ) ( q p ) p ζ ̂ ( 1 ) ( p k ) ,
( q + 2 k ) ζ ̂ ( 3 ) ( q k ) = 3 d p 2 π ζ ̂ ( 2 ) ( q p ) p ζ ̂ ( 1 ) ( p k ) ,
( 2 q + k ) ζ ̂ ( 3 ) ( q k ) = 3 d p 2 π ζ ̂ ( 1 ) ( q p ) p ζ ̂ ( 2 ) ( p k ) ,
d p 2 π ζ ̂ ( 1 ) ( q p ) p 2 ζ ̂ ( 2 ) ( p k ) = d p 2 π ζ ̂ ( 2 ) ( q p ) p 2 ζ ̂ ( 1 ) ( p k ) + 1 3 ( q 2 k 2 ) ζ ̂ ( 3 ) ( q k ) ,
R p θ s incoh = 1 L 1 2 π ( ω c ) 3 cos 2 θ s cos θ 0 | G 0 ( q ω ) | 2 × { | T ( 1 ) ( q k ) | 2 + 1 4 [ | T ( 2 ) ( q k ) | 2 | T ( 2 ) ( q k ) | 2 ] 1 3 Re T ( 1 ) ( q k ) * T ( 3 ) ( q k ) } | G 0 ( k ω ) | 2 ,
[ 2 π δ ( q k ) ] 2 = L 1 2 π δ ( q k ) ,
T ( 22 ) ( q k ) = 2 d p 2 π V ( 1 ) ( q p ) G 0 ( p ω ) V ( 1 ) ( p k ) .
R p θ s incoh = 1 L 1 1 2 π ( ω c ) 3 cos 2 θ s cos θ 0 | G 0 ( q ω ) | 2 × [ | T ( 22 ) ( q k ) | 2 | T ( 22 ) ( q k ) | 2 ] × | G 0 ( k ω ) | 2 .
| T ( 22 ) ( q k ) | 2 | T ( 22 ) ( q k ) | 2 = 4 L 1 δ 4 | ( ω ) 1 2 ( ω ) | 4 [ d p 2 π g ( | q p | ) × g ( | k p | ) | G 0 ( p ω ) | 2 | u ( q p ) | 2 | u ( p k ) | 2 + d p 2 π g ( | q p | ) g ( | k p | ) × G 0 ( p ω ) G 0 * ( q + k p , ω ) × u ( q p ) u ( p k ) u * ( q q + k p ) u * ( q + k p k ) ] ,
u ( q k ) = ( ω ) q k α ( q ω ) α ( k ω ) .
G 0 ( k ω ) C ( ω ) [ 1 k k s p ( ω ) i Δ ( ω ) 1 k + k s p ( ω ) + i Δ ( ω ) ] ,
k s p ( ω ) = ω c [ | 1 ( ω ) | | 1 ( ω ) | 1 ] 1 / 2 ,
Δ ( ω ) = 1 2 2 ( ω ) k s p ( ω ) | 1 ( ω ) | [ | 1 ( ω ) | 1 ] ,
C ( ω ) = | 1 ( ω ) | 3 / 2 1 2 ( ω ) 1 .
| G 0 ( k ω ) | 2 C 2 [ 1 ( k k s p ) 2 + Δ 2 + 1 ( k + k s p ) 2 + Δ 2 ] π C 2 Δ [ δ ( k k s p ) + δ ( k + k s p ) ] ,
G 0 ( k ω ) G 0 * ( x k , ω ) 2 π i C 2 2 i Δ x δ ( k k s p ) + 2 π i C 2 2 i Δ + x δ ( k + k s p ) ,
| T ( 22 ) ( q k ) | 2 | T ( 22 ) ( q k ) | 2 = 4 L 1 δ 4 | ( ω ) 1 2 ( ω ) | 4 C 2 2 Δ [ g ( | q k s p | ) × g ( | k k s p | ) | u ( q k s p ) | 2 | u ( k k s p ) | 2 + g ( | q + k s p | ) g ( | k + k s p | ) × | u ( q k s p ) | 2 | u ( k k s p ) | 2 ] + 4 L 1 δ 4 | ( ω ) 1 2 ( ω ) | 4 2 Δ C 2 ( q + k ) 2 + 4 Δ 2 × [ g ( | k k s p | ) g ( | q k s p | ) × u ( q k s p ) u * ( k k s p ) u ( k k s p ) u * ( q k s p ) + g ( | q k s p | ) g ( | k + k s p | ) u ( q k s p ) × u * ( k k s p ) u ( k k s p ) u * ( q k s p ) ] ,

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