Abstract

Expressions for the s and p mean reflected amplitudes of light incident onto a two-dimensional random dielectric surface are derived. From these expressions the influence of roughness on the specular component of the field was studied. The effect of the transverse structure of the surface and the role of surface plasmon in the decay of the reflectivity are investigated. The shift of the Brewster angle is studied and applications to the determination of surface parameters are discussed. Finally, it is shown that roughness induces nonspecular effects on dielectric surfaces near the Brewster angle.

© 1993 Optical Society of America

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References

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  1. C. Eckart, “The scattering of sound from the sea surface,”J. Acoust. Soc. Am. 25, 556–570 (1953).
    [CrossRef]
  2. D. Winebrenner, A. Ishimaru, “Application of the phase perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
    [CrossRef]
  3. J. Sheng, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
    [CrossRef]
  4. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).
  5. J. A. Sánchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
    [CrossRef]
  6. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1988).
    [CrossRef]
  7. M. Saillard, D. Maystre, “Scattering from metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  8. M. Saillard, “A characterization tool for dielectric rough surfaces: Brewster’s phenomenon,” Waves Random Media 2, 67–79 (1992).
    [CrossRef]
  9. J. J. Greffet, “Theoretical model of the shift of the Brewster angle on a rough surface,” Opt. Lett. 17, 238–240 (1992).
    [CrossRef] [PubMed]
  10. F. Falco, T. Tamir, “Improved analysis of nonspecular phenomena in beams reflected from stratified media,” J. Opt. Soc. Am. A 7, 185–190 (1990).
    [CrossRef]
  11. G. C. Brown, V. Celli, M. Haller, A. Marvin, “Vector theory of light scattering from a rough surface: unitary and reciprocal expansions,” Surface Sci. 136, 381–397 (1984).
    [CrossRef]
  12. G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
    [CrossRef]
  13. S. L. Broschat, E. I. Thorsos, A. Ishimaru, “The phase perturbation technique vs. an exact numerical method for random rough surface scattering,”J. Electromag. Waves Appl. 3, 237–256 (1989).
    [CrossRef]
  14. J. M. Soto-Crespo, M. Nieto-Vesperinas, A. T. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small pertubation method,” J. Opt. Soc. Am. A 7, 1185–1201 (1990).
    [CrossRef]
  15. M. Nieto-Vesperinas, N. García, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [CrossRef]
  16. J. M. Bennett, “Measurement of the rms roughness, autocovariance function and other statistical properties of optical surfaces using a feco scanning interferometer,” Appl. Opt. 15, 2705–2721 (1976).
    [CrossRef] [PubMed]
  17. J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
    [CrossRef]
  18. J. J. Greffet, C. Baylard, “Nonspecular reflection from a lossy dielectric,” Opt. Lett. 18, 1129–1131 (1993).
    [CrossRef] [PubMed]
  19. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961).

1993 (1)

1992 (3)

J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

M. Saillard, “A characterization tool for dielectric rough surfaces: Brewster’s phenomenon,” Waves Random Media 2, 67–79 (1992).
[CrossRef]

J. J. Greffet, “Theoretical model of the shift of the Brewster angle on a rough surface,” Opt. Lett. 17, 238–240 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (3)

1989 (1)

S. L. Broschat, E. I. Thorsos, A. Ishimaru, “The phase perturbation technique vs. an exact numerical method for random rough surface scattering,”J. Electromag. Waves Appl. 3, 237–256 (1989).
[CrossRef]

1988 (1)

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

1985 (2)

D. Winebrenner, A. Ishimaru, “Application of the phase perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
[CrossRef]

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (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,” Surface Sci. 136, 381–397 (1984).
[CrossRef]

1981 (1)

M. Nieto-Vesperinas, N. García, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

1980 (1)

J. Sheng, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[CrossRef]

1976 (1)

1953 (1)

C. Eckart, “The scattering of sound from the sea surface,”J. Acoust. Soc. Am. 25, 556–570 (1953).
[CrossRef]

Baylard, C.

J. J. Greffet, C. Baylard, “Nonspecular reflection from a lossy dielectric,” Opt. Lett. 18, 1129–1131 (1993).
[CrossRef] [PubMed]

J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

Bennett, J. M.

Broschat, S. L.

S. L. Broschat, E. I. Thorsos, A. Ishimaru, “The phase perturbation technique vs. an exact numerical method for random rough surface scattering,”J. Electromag. Waves Appl. 3, 237–256 (1989).
[CrossRef]

Brown, G.

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

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,” Surface Sci. 136, 381–397 (1984).
[CrossRef]

Celli, V.

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

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

Eckart, C.

C. Eckart, “The scattering of sound from the sea surface,”J. Acoust. Soc. Am. 25, 556–570 (1953).
[CrossRef]

Falco, F.

Friberg, A. T.

García, N.

M. Nieto-Vesperinas, N. García, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Greffet, J. J.

Haller, M.

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

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

Ishimaru, A.

S. L. Broschat, E. I. Thorsos, A. Ishimaru, “The phase perturbation technique vs. an exact numerical method for random rough surface scattering,”J. Electromag. Waves Appl. 3, 237–256 (1989).
[CrossRef]

D. Winebrenner, A. Ishimaru, “Application of the phase perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
[CrossRef]

Maradudin, A. A.

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

J. Sheng, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[CrossRef]

Marvin, A.

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

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

Maystre, D.

Merzbacher, E.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961).

Nieto-Vesperinas, M.

Ogilvy, J. A.

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

Saillard, M.

M. Saillard, “A characterization tool for dielectric rough surfaces: Brewster’s phenomenon,” Waves Random Media 2, 67–79 (1992).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

Sánchez-Gil, J. A.

Sheng, J.

J. Sheng, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[CrossRef]

Soto-Crespo, J. M.

Tamir, T.

Thorsos, E. I.

S. L. Broschat, E. I. Thorsos, A. Ishimaru, “The phase perturbation technique vs. an exact numerical method for random rough surface scattering,”J. Electromag. Waves Appl. 3, 237–256 (1989).
[CrossRef]

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

Winebrenner, D.

Appl. Opt. (1)

J. Acoust. Soc. Am. (2)

C. Eckart, “The scattering of sound from the sea surface,”J. Acoust. Soc. Am. 25, 556–570 (1953).
[CrossRef]

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

J. Electromag. Waves Appl. (1)

S. L. Broschat, E. I. Thorsos, A. Ishimaru, “The phase perturbation technique vs. an exact numerical method for random rough surface scattering,”J. Electromag. Waves Appl. 3, 237–256 (1989).
[CrossRef]

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

Opt. Acta (1)

M. Nieto-Vesperinas, N. García, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Opt. Commun. (1)

J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (2)

J. Sheng, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
[CrossRef]

G. Brown, V. Celli, M. Haller, A. A. Maradudin, A. Marvin, “Resonant light scattering from a randomly rough surface,” Phys. Rev. B 31, 4993–5005 (1985).
[CrossRef]

Surface Sci. (1)

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

Waves Random Media (1)

M. Saillard, “A characterization tool for dielectric rough surfaces: Brewster’s phenomenon,” Waves Random Media 2, 67–79 (1992).
[CrossRef]

Other (2)

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

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961).

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Figures (12)

Fig. 1
Fig. 1

Modulus of the coherent reflection factor for a flat surface (dotted curves) and a rough surface (solid curves) versus the angle of incidence. = 2.25 + i0.0, δ = 0.05λ, a = 2λ. (a) p polarization, (b) s polarization.

Fig. 2
Fig. 2

Modulus of the coherent reflection factor versus the angle of incidence. Dotted curves, flat surface; solid curves, rough surface ( = −18 + i0.5, δ = 0.05λ); dashed curves, Fresnel reflection factor in p polarization multiplied by exp(−2k02δ2 cos2θ0). (a) p polarization, (b) s polarization.

Fig. 3
Fig. 3

Modulus of the ratio Rs/Rp at normal incidence. a = λ, θ0 = 0.

Fig. 4
Fig. 4

Solid curves, modulus of the coherent intensity reflection factor at normal incidence for a rough silver surface divided by the modulus of the coherent intensity reflection factor for a flat surface versus 4k02δ2 and for various Z = k02a2/4; dashed curve, the decay factor exp(−4k03δ2). = −18 + i0.5. (a) p polarization, (b) s polarization.

Fig. 5
Fig. 5

Same as Fig. 4 but for a rough dielectric surface ( = 2.25 + i0.0).

Fig. 6
Fig. 6

Modulus of the coherent reflection factor for p polarization for a flat surface (dotted curve) and a rough surface (solid curve) versus the angle of incidence near the Brewster angle. = 2.25 + i0.0, δ = 0.05λ, a = 2λ.

Fig. 7
Fig. 7

Angular shift of the Brewster angle (a) versus δ and (b) versus a. ∊ = 2.25 + i0.0.

Fig. 8
Fig. 8

Modulus of the coherent reflection factor at its minimum for the scattering of p-polarized light from a rough dielectric surface (a) versus δ and (b) versus a. ∊ = 2.25 + i0.0.

Fig. 9
Fig. 9

Definition of the nonspecular effects. δx, lateral shift; δz, focal shift; δθ, angular shift.

Fig. 10
Fig. 10

(a) Lateral shift and (b) focal shift divided by the wavelength λ versus the angle of incidence for a rough surface (dashed curves, δ = 0.05λ, a = 2λ) and a flat surface (solid horizontal lines). = 2.25 + i0.0, w = 10λ.

Fig. 11
Fig. 11

(a) Lateral shift and (b) focal shift divided by the wavelength λ versus the angle of incidence, for a rough surface (crosses, δ = 0.05λ, a = 2λ) and a flat surface (solid curves). = −18 + i0.5, w = 10λ, p polarization.

Fig. 12
Fig. 12

Angular shift versus the angle of incidence, for a rough surface (crosses and dashed curve, δ = 0.05λ, a = 2λ) and a flat surface (solid curves). (a) = −18 + i0.5, (b) = 2.25 + i0.0, w = 10λ.

Equations (66)

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ζ ( x ) = 0 ,
ζ ( x ) ζ ( x ) = δ 2 W ( x - x ) ,
δ = ζ 2 ( x ) 1 / 2 .
W ( x ) = exp ( - x 2 / a 2 ) ,
ζ ( x ) = d 2 Q ( 2 π ) 2 ζ ^ ( Q ) exp ( i Q · x ) ,
ζ ^ ( Q ) = 0
ζ ^ ( Q ) ζ ^ ( Q ) = ( 2 π ) 2 δ ( Q + Q ) δ 2 g ( Q ) .
g ( Q ) = d 2 x W ( x ) exp ( - i Q · x ) .
g ( Q ) = π a 2 exp ( - Q 2 a 2 / 4 ) .
E ( x ; t ) = E ( i ) ( x ; t ) + E ( s ) ( x ; t ) ,
E ( i ) ( x ; t ) = { - c ω [ k ^ α 0 ( k ω ) + x ^ 3 k ] × B ( k ω ) + ( x ^ 3 × k ^ ) 3 B ( k ω ) } × exp { i [ k - x ^ 3 α 0 ( k ω ) ] · x - i ω t } ,
E ( s ) ( x ; t ) = d 2 q ( 2 π ) 2 { c ω [ q ^ α 0 ( q ω ) - x ^ 3 q ] × A ( q ω ) + ( x ^ 3 × q ^ ) 3 A ( q ω ) } × exp { i [ q + x ^ 3 α 0 ( q ω ) ] · x - i ω t } .
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 ,
A α ( q ω ) = β R α β ( q k ) B β ( k ω ) ,
d 2 q ( 2 π ) 2 I [ α ( p ω ) - α 0 ( q ω ) p - q ] α ( p ω ) - α 0 ( q ω ) M ( p q ) R ( q k ) = - I [ α ( p ω ) + α 0 ( k ω ) p - k ] α ( p ω ) + α 0 ( k ω ) N ( p k ) ,
I ( γ Q ) = d 2 x exp ( - i Q · x ) exp [ - i γ ζ ( x ) ] ,
M ( p q ) = [ [ p q + α ( p ω ) p ^ · q ^ α 0 ( k ω ) ] - ω c α ( p ω ) ( p ^ × q ^ ) 3 ω c ( p ^ × q ^ ) 3 α 0 ( q ω ) ω 2 c 2 p ^ · q ^ ] ,
N ( p k ) = [ [ p k - α ( p ω ) p ^ · k ^ α 0 ( k ω ) ] - ω c α ( p ω ) ( p ^ × k ^ ) 3 - ω c ( p ^ × k ^ ) 3 α 0 ( k ω ) ω 2 c 2 p ^ · k ^ ] .
α ( 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 α β ( q k ) = ( 2 π ) 2 δ ( q - k ) δ α β R α ( 0 ) ( q ω ) - i G α ( 0 ) ( q ω ) T α β ( q k ) G β ( 0 ) ( k ω ) 2 α 0 ( k ω ) ,
R p ( 0 ) ( q ω ) = α 0 ( q ω ) - α ( q ω ) α 0 ( q ω ) + α ( q ω ) ,
R s ( 0 ) ( q ω ) = α 0 ( q ω ) - α ( q ω ) α 0 ( q ω ) + α ( q ω )
G p ( 0 ) ( q ω ) = i α 0 ( q ω ) + α ( q ω ) ,
G s ( 0 ) ( q ω ) = i α 0 ( q ω ) + α ( q ω )
T α β ( q k ) = V α β ( q k ) + γ d 2 p ( 2 π ) 2 V α β ( q p ) G γ ( 0 ) ( p ω ) T γ β ( p k )
= V α β ( q k ) + γ d 2 p ( 2 π ) 2 T α β ( q p ) G γ ( 0 ) ( p ω ) V γ β ( p k ) .
d 2 p ( 2 π ) 2 { I [ α ( q ω ) - α 0 ( p ω ) q - p ] α ( q ω ) - α 0 ( p ω ) M ( q p ) - I [ α ( q ω ) + α 0 ( p ω ) q - p ] α ( q ω ) + α 0 ( p ω ) N ( q p ) } V ( p k ) 2 i α 0 ( p ω ) = { I [ α ( q ω ) - α 0 ( k ω ) q - k ] α ( q ω ) - α 0 ( k ω ) P ( q k ) + I [ α ( q ω ) + α 0 ( k ω ) q - k ] α ( q ω ) + α 0 ( k ω ) Q ( q k ) } 1 2 α 0 ( k ω ) ,
P ( q k ) = [ 1 [ q k + α ( q ω ) q ^ · k ^ α 0 ( k ω ) ] [ α 0 ( k ω ) - α ( k ω ) ] - ω c α ( q ω ) ( q ^ × k ^ ) 3 [ α 0 ( k ω ) - α ( k ω ) ] 1 ω c ( q ^ × k ^ ) 3 α 0 ( k ω ) [ α 0 ( k ω ) - α ( k ω ) ] ω 2 c 2 q ^ · k ^ [ α 0 ( k ω ) - α ( k ω ) ] ] ,
Q ( q k ) = [ 1 [ q k - α ( q ω ) q ^ · k ^ α 0 ( k ω ) ] [ α 0 ( k ω ) + α ( k ω ) ] - ω c α ( q ω ) ( q ^ × k ^ ) 3 [ α 0 ( k ω ) + α ( k ω ) ] - 1 ω c ( q ^ × k ^ ) 3 α 0 ( k ω ) [ α 0 ( k ω ) + α ( k ω ) ] ω 2 c 2 q ^ · k ^ [ α 0 ( k ω ) + α ( k ω ) ] ] .
V ( q k ) = V ( 1 ) ( q k ) + V ( 2 ) ( q k ) + ,
I ( γ Q ) = n = 0 ( - i ) n n ! γ n ζ ^ ( n ) ( Q ) ,
ζ ^ ( n ) ( Q ) = d 2 x exp ( - i Q · x ) ζ n ( x ) ,
V ( 1 ) ( q k ) = ζ ^ ( q - k ) - 1 × [ 1 [ q k - α ( q ω ) q ^ · k ^ α ( k ω ) ] - ω c α ( q ω ) ( q ^ × k ^ ) 3 - ω c ( q ^ × k ^ ) 3 α ( k ω ) ω 2 c 2 q ^ · k ^ ] ,
V ( 2 ) ( q k ) = - i 2 - 1 ζ ^ ( 2 ) ( q - k ) [ α ( q ω ) + α ( k ω ) ] [ 1 [ q k - α ( q ω ) q ^ · k ^ α ( k ω ) ] - ω c α ( q ω ) ( q ^ × k ^ ) 3 - ω c ( q ^ × k ^ ) 3 α ( k ω ) ω 2 c 2 q ^ · k ^ ] - i ( - 1 ) 2 d 2 p ( 2 π ) 2 ζ ^ ( q - p ) ζ ^ ( p - k ) [ 1 α ( q ω ) q ^ · p ^ α ( p ω ) p ^ · k ^ α ( k ω ) ω c α ( q ω ) q ^ · p ^ α ( p ω ) ( p ^ × k ^ ) 3 ω c ( q ^ × p ^ ) 3 α ( p ω ) p ^ · k ^ α ( k ω ) ( q ^ × p ^ ) 3 α ( p ω ) ( p ^ × k ^ ) 3 ω 2 c 2 ] .
G α β ( q k ) = ( 2 π ) 2 δ ( q - k ) δ α β G α ( 0 ) ( q ω ) + G α ( 0 ) ( q ω ) γ d 2 p ( 2 π ) 2 V α β ( q p ) G γ β ( p k ) .
G α β ( q k ) = ( 2 π ) 2 δ ( q - k ) δ α β G α ( 0 ) ( q ω ) + G α ( 0 ) ( q ω ) T α β ( q k ) G β ( 0 ) ( k ω ) ,
G α β ( q k ) = ( 2 π ) 2 δ ( q - k ) δ α β G α ( 0 ) ( q ω ) + G α ( 0 ) ( q ω ) γ d 2 p ( 2 π ) 2 M α γ ( q p ) G γ β ( p k ) ,
M α β ( q k ) = V α β ( q k ) + γ d 2 p ( 2 π ) 2 V α γ ( q p ) G γ ( 0 ) ( p ω ) Q M γ β ( p k ) .
G α β ( 0 ) ( q p ) = ( 2 π ) 2 δ ( q - k ) δ α β G α ( 0 ) ( q ω )
G α β ( q k ) = ( 2 π ) 2 δ ( q - k ) δ α β G α ( q ω ) .
M α β ( q k ) = ( 2 π ) 2 δ ( q - k ) δ α β M α ( q ω ) ,
T α β ( q k ) = ( 2 π ) 2 δ ( q - k ) δ α β T α ( q ω ) .
G α ( q ω ) = G α ( 0 ) ( q ω ) + G α ( 0 ) ( q ω ) M α ( q ω ) G α ( q ω ) .
G α ( q ω ) = G α ( 0 ) ( q ω ) + G α ( 0 ) ( q ω ) T α ( q ω ) G α ( 0 ) ( q ω ) .
G α ( q ω ) = 1 G α ( 0 ) ( q ω ) - 1 - M α ( q ω ) ,
T α ( q ω ) G α ( 0 ) ( q ω ) = M α ( q ω ) G α ( q ω ) .
R α β ( q k ) = ( 2 π ) 2 δ ( q + k ) δ α β R α ( k ω ) ,
R α ( k ω ) = R α ( 0 ) ( k ω ) - i G α ( 0 ) ( k ω ) T α ( k ω ) G α ( 0 ) ( k ω ) 2 α 0 ( k ω ) .
R α ( k ω ) = R α ( 0 ) ( k ω ) - i G α ( 0 ) ( k ω ) M α ( k ω ) 2 α 0 ( k ω ) G α ( 0 ) ( k ω ) - 1 - M α ( k ω ) .
R p ( k ω ) = α 0 ( k ω ) - α ( k ω ) + i M p ( k ω ) α 0 ( k ω ) + α ( k ω ) - i M p ( k ω ) ,
R s ( k ω ) = α 0 ( k ω ) - α ( k ω ) + i M s ( k ω ) α 0 ( k ω ) + α ( k ω ) - i M s ( k ω ) .
M p ( k ω ) = - i δ 2 ( - 1 2 ) α ( k ω ) ( 2 k 2 - ω 2 c 2 ) - i δ 2 ( - 1 ) 2 3 α 2 ( k ω ) d 2 p ( 2 π ) 2 g ( k - p ) × ( k ^ · p ^ ) 2 α ( p ω ) + δ 2 ( - 1 ) 2 × d 2 p ( 2 π ) 2 g ( k - p ) × { G p ( 0 ) ( p ω ) 1 2 [ k p - α ( p ω ) p ^ · k ^ α ( k ω ) ] 2 - G s ( 0 ) ( p ω ) ω 2 c 2 α 2 ( k ω ) ( k ^ × p ^ ) 3 2 } ,
M s ( k ω ) = - i δ 2 ( - 1 ) ω 2 c 2 α ( k ω ) + i δ 2 ( - 1 ) 2 ω 2 c 2 d 2 p ( 2 π ) 2 g ( k - p ) α ( p ω ) × ( k ^ × p ^ ) 3 2 + δ 2 ( - 1 ) 2 d 2 p ( 2 π ) 2 g ( k - p ) × [ - ω 2 c 2 G p ( 0 ) ( p ω ) ( k ^ × p ^ ) 3 2 α 2 ( p ω ) + 2 ω 4 c 4 G s ( 0 ) ( p ω ) ( k ^ · p ^ ) 2 ] .
k = k ( cos θ k , sin θ k ) ,             p = p ( cos θ p , sin θ p ) ,
g ( k - p ) = l = - g l ( k p ) exp [ i l ( θ k - θ p ) ] .
g l ( k p ) = π a 2 exp [ - a 2 4 ( k 2 + p 2 ) ] I l ( a 2 2 k p ) ,
k = ω c sin θ 0 ,
M p ( k ω ) = - i ω c δ 2 a 2 ( ω a c ) 2 - 1 2 ( - sin 2 θ 0 ) 1 / 2 × ( 2 sin 2 θ 0 - ) - i 4 π δ 2 a 2 ( a ω c ) 2 ( - 1 ) 2 3 × ( - sin 2 θ 0 ) 0 d p p α ( p ω ) × [ g 0 ( k p ) + g 2 ( k p ) ] + 1 2 π δ 2 ( - 1 ) 2 0 d p p ( g 0 ( k p ) × { 1 2 G p ( 0 ) ( p ω ) [ 2 k 2 p 2 + 1 2 α 2 ( p ω ) α 2 ( k ω ) ] - 1 2 ω 4 c 4 ( - sin 2 θ 0 ) G s ( 0 ) ( p ω ) } - g 1 ( k p ) 2 k α ( k ω ) G p ( 0 ) ( p ω ) p α ( p ω ) + g 2 ( k p ) [ 1 2 2 α 2 ( k ω ) G p ( 0 ) ( p ω ) α 2 ( p ω ) + 1 2 ω 4 c 4 ( - sin 2 θ 0 ) G s ( 0 ) ( p ω ) ] ) .
M s ( k ω ) = - i ω c δ 2 a 2 ( ω a c ) 2 ( - 1 ) ( - sin 2 θ 0 ) 1 / 2 + i 4 π δ 2 a 2 ( ω a c ) 2 ( - 1 ) 2 0 d p p α ( p ω ) × [ g 0 ( k p ) - g 2 ( k p ) ] + 1 4 π δ 2 a 2 ( ω a c ) 2 × ( - 1 ) 2 0 d p p { g 0 ( k p ) × [ - G p ( 0 ) ( p ω ) α 2 ( p ω ) + 2 ω 2 c 2 G s ( 0 ) ( p ω ) ] + g 2 ( k p ) [ G p ( 0 ) ( p ω ) α 2 ( p ω ) + 2 ω 2 c 2 G s ( 0 ) ( p ω ) ] } .
| R s R p | = 1 + C ( a , ) ( 2 π δ λ ) 4 ,
C ( a , ) = | 1 - M p 2 k 0 2 | .
k sp = k 0 sin θ i + q ,
( k sp - k 0 sin θ 0 ) a < 2             or             sin θ 0 > k sp k 0 - λ π a .
R α ( k ω ) = - 1 - 2 i α 0 ( k ω ) G α ( k ω )

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