Abstract

The reflectivity of p- and s-polarized light incident upon a one-dimensional, randomly rough, dielectric surface is calculated for several angles of incidence, when the plane of incidence is normal to the generators of the surface, by a second-order small-amplitude perturbation theory, by a second-order self-energy perturbation theory, and by a second-order phase perturbation theory. The wavelength of the incident light is λ = 0.6328 μm, and the dielectric constant of the scattering medium is = 2.25. The surface roughness is characterized by a rms height δ and a transverse correlation length a. From a comparison of the results of these approximate calculations with those obtained by a numerical simulation approach for each polarization of the incident light, and for several angles of incidence, curves of δ/λ as a function of a/λ are constructed, below which each perturbative method is valid with an error that is smaller than 2.5%. It is found that, for a given value of a/λ, the reflectivity of s- and p-polarized light is given most accurately by the phase perturbation theory and least accurately by small-amplitude perturbation theory.

© 1995 Optical Society of America

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References

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  1. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  2. Rayleigh, The Theory of Sound, 2nd ed. (Macmillan, London, 1896), Vol. II, pp. 89, 96.
  3. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
    [CrossRef]
  4. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991), p. 39.
  5. 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 perturbation method,” J. Opt. Soc. Am. A 7, 1185–1201 (1990).
    [CrossRef]
  6. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (NY) 203, 255–307 (1990).
    [CrossRef]
  7. J. A. Sánchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
    [CrossRef]
  8. A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
    [CrossRef]
  9. 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]
  10. J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
    [CrossRef]
  11. D. P. Winebrenner, A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20, 161–170 (1985).
    [CrossRef]
  12. D. P. Winebrenner, A. Ishimaru, “Application of the phase-perturbation technique to randomly rough surfaces,” J. Opt. Soc. Am. A 2, 2285–2294 (1985).
    [CrossRef]
  13. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [CrossRef]
  14. A. A. Maradudin, “Electromagnetic surface excitations on rough surfaces,” in Electromagnetic Surface Excitations, R. F. Wallis, G. I. Stegeman, eds. (Springer-Verlag, New York, 1986), pp. 57–131.
    [CrossRef]
  15. N. García, A. A. Maradudin, V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. 45, 287–298 (1982).
    [CrossRef]
  16. R. Kubo, “Generalized cumulant expansion method,” J. Phys. Soc. Jpn. 17, 1100–1120 (1962).
    [CrossRef]
  17. N. García, M. Nieto-Vesperinas, “Rough surface retrieval from the specular intensity of multiply-scattered waves,” Phys. Rev. Lett. 71, 3645–3648 (1993).
    [CrossRef] [PubMed]

1993

A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

N. García, M. Nieto-Vesperinas, “Rough surface retrieval from the specular intensity of multiply-scattered waves,” Phys. Rev. Lett. 71, 3645–3648 (1993).
[CrossRef] [PubMed]

1991

1990

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 perturbation method,” J. Opt. Soc. Am. A 7, 1185–1201 (1990).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (NY) 203, 255–307 (1990).
[CrossRef]

1985

D. P. Winebrenner, A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20, 161–170 (1985).
[CrossRef]

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

1983

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]

1982

N. García, A. A. Maradudin, V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. 45, 287–298 (1982).
[CrossRef]

1980

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

1977

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

1962

R. Kubo, “Generalized cumulant expansion method,” J. Phys. Soc. Jpn. 17, 1100–1120 (1962).
[CrossRef]

1951

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

1907

Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[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]

Celli, V.

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]

N. García, A. A. Maradudin, V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. 45, 287–298 (1982).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[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]

Friberg, A. T.

García, N.

N. García, M. Nieto-Vesperinas, “Rough surface retrieval from the specular intensity of multiply-scattered waves,” Phys. Rev. Lett. 71, 3645–3648 (1993).
[CrossRef] [PubMed]

N. García, A. A. Maradudin, V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. 45, 287–298 (1982).
[CrossRef]

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]

Hill, N. R.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Ishimaru, A.

D. P. Winebrenner, A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20, 161–170 (1985).
[CrossRef]

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

Kubo, R.

R. Kubo, “Generalized cumulant expansion method,” J. Phys. Soc. Jpn. 17, 1100–1120 (1962).
[CrossRef]

Luna, R. E.

A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (NY) 203, 255–307 (1990).
[CrossRef]

N. García, A. A. Maradudin, V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. 45, 287–298 (1982).
[CrossRef]

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

A. A. Maradudin, “Electromagnetic surface excitations on rough surfaces,” in Electromagnetic Surface Excitations, R. F. Wallis, G. I. Stegeman, eds. (Springer-Verlag, New York, 1986), pp. 57–131.
[CrossRef]

Marvin, A.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (NY) 203, 255–307 (1990).
[CrossRef]

Méndez, E. R.

A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (NY) 203, 255–307 (1990).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (NY) 203, 255–307 (1990).
[CrossRef]

Nieto-Vesperinas, M.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991), p. 39.

Rayleigh,

Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Rayleigh, The Theory of Sound, 2nd ed. (Macmillan, London, 1896), Vol. II, pp. 89, 96.

Rice, S. O.

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

Sánchez-Gil, J. A.

Shen, J.

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

Soto-Crespo, J. M.

Toigo, F.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

Winebrenner, D. P.

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

D. P. Winebrenner, A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20, 161–170 (1985).
[CrossRef]

Ann. Phys. (NY)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (NY) 203, 255–307 (1990).
[CrossRef]

Commun. Pure Appl. Math.

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

J. Opt. Soc. Am. A

J. Phys. Soc. Jpn.

R. Kubo, “Generalized cumulant expansion method,” J. Phys. Soc. Jpn. 17, 1100–1120 (1962).
[CrossRef]

Philos. Mag.

N. García, A. A. Maradudin, V. Celli, “On the Debye–Waller factor in atom-surface scattering,” Philos. Mag. 45, 287–298 (1982).
[CrossRef]

Phys. Rev. B

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[CrossRef]

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

Phys. Rev. Lett.

N. García, M. Nieto-Vesperinas, “Rough surface retrieval from the specular intensity of multiply-scattered waves,” Phys. Rev. Lett. 71, 3645–3648 (1993).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A

Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Radio Sci.

D. P. Winebrenner, A. Ishimaru, “Investigation of a surface field phase perturbation technique for scattering from rough surfaces,” Radio Sci. 20, 161–170 (1985).
[CrossRef]

Surf. Sci.

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]

Waves Random Media

A. A. Maradudin, R. E. Luna, E. R. Méndez, “The Brewster effect for a one-dimensional random surface,” Waves Random Media 3, 51–60 (1993).
[CrossRef]

Other

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991), p. 39.

Rayleigh, The Theory of Sound, 2nd ed. (Macmillan, London, 1896), Vol. II, pp. 89, 96.

A. A. Maradudin, “Electromagnetic surface excitations on rough surfaces,” in Electromagnetic Surface Excitations, R. F. Wallis, G. I. Stegeman, eds. (Springer-Verlag, New York, 1986), pp. 57–131.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Reflectivity in p polarization as a function of δ/λ for a/λ = 0.3 and θ0 = 20° calculated by the use of the self-energy perturbation theory (SEPT), the small-amplitude perturbation theory (SAPT), the phase perturbation theory (PTT), the exponential factor (EXP), and the numerical simulation method (center solid curve). The numerical simulation result reduced and augmented by 2.5% is also shown (lower and upper solid curves, respectively), and its intersections with the perturbation-theoretic results are marked by open circles.

Fig. 2
Fig. 2

Curves of δ/λ as a function of a/λ below which each perturbative method is valid with an error smaller than 2.5%: p polarization, θ0 = 0°.

Fig. 3
Fig. 3

Same as Fig. 2 but for θ0 = 20°.

Fig. 4
Fig. 4

Same as Fig. 2 but for θ0 = 40°.

Fig. 5
Fig. 5

Same as Fig. 2 but in s polarization.

Fig. 6
Fig. 6

Same as Fig. 2 but in s polarization for θ0 = 20°.

Fig. 7
Fig. 7

Same as Fig. 2 but in s polarization for θ0 = 40°.

Fig. 8
Fig. 8

Reflectivity as a function of δ/λ in s polarization for θ0 = 0°. The result of numerical simulation method is represented by the solid curve. (a) a/λ = 1, (b) a/λ = 2.

Equations (105)

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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 Q 2 π ζ ^ ( Q ) exp ( i Q x 1 ) .
ζ ^ ( Q ) = 0 ,
ζ ^ ( Q ) ζ ^ ( Q ) = 2 π δ ( Q + Q ) δ 2 g ( Q ) ,
g ( Q ) = - d x 1 W ( x 1 ) exp ( - i Q x 1 ) .
g ( Q ) = π a exp ( - Q 2 a 2 / 4 ) .
k ζ ( x 1 ) < < 1 ,
ζ ( x 1 ) < < 1 ,
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 .
E 2 > ( x 1 , x 3 ω ) = exp [ i k x 1 - i α 0 ( k , ω ) x 3 ] + - d q 2 π R s ( q k ) exp [ i q x 1 + i α 0 ( q , ω ) x 3 ] .
R p , s θ s spe = 1 L 1 ω 2 π c cos 2 θ s cos θ 0 R p , s ( 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 ] ,
α ( q , ω ) = [ ( ω 2 / c 2 ) - q 2 ] 1 / 2 ,             Re [ α ( q , ω ) ] > 0 ,             Im [ α ( q , ω ) ] > 0 ,
I ( γ Q ) = - d x 1 exp [ - i γ ζ ( x 1 ) ] exp ( i Q x 1 ) .
- d q 2 π I [ α ( p , ω ) - α 0 ( q , ω ) p - q ] α ( p , ω ) - α 0 ( q , ω ) R s ( q k ) = - I [ α ( p , ω ) + α 0 ( k , ω ) p - k ] α ( q , ω ) + α 0 ( k , ω ) .
I ( γ Q ) = n = 0 ( - i γ ) n n ! ζ ^ ( n ) ( Q ) ,
ζ ^ ( n ) ( Q ) = - d x 1 ζ n ( x 1 ) exp ( - i Q x 1 ) .
R p , s ( q k ) = n = 0 ( - i ) n n ! R p , s ( n ) ( q k ) ,
R p ( 0 ) ( q k ) = 2 π δ ( q - k ) α 0 ( k , ω ) - α ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) ,
R p ( n ) ( q k ) = - 1 α 0 ( q , ω ) + α ( q , ω ) × ( { [ α ( q , ω ) α 0 ( k , ω ) - q k ] [ α 0 ( k , ω ) + α ( k , ω ) ] × [ α ( q , ω ) + α 0 ( k , ω ) ] n - 1 - [ α ( q , ω ) α 0 ( k , ω ) + q k ] × [ α 0 ( k , ω ) - α ( k , ω ) ] [ α ( q , ω ) - α 0 ( k , ω ) ] n - 1 } × ζ ^ ( n ) ( q - k ) α 0 ( k , ω ) + α ( k , ω ) - m = 1 n - 1 ( n m ) - d r 2 π [ α ( q , ω ) α 0 ( r , ω ) + q r ] × [ α ( q , ω ) - α 0 ( r , ω ) ] n - m - 1 ζ ^ ( n - m ) ( q - r ) R p ( m ) ( r k ) ) ,             n 1 ;
R ( 0 ) ( q k ) = 2 π δ ( q - k ) α 0 ( k , ω ) - α ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) ,
R s ( n ) ( q k ) = [ α 0 ( q , ω ) - α ( q , ω ) ] × { [ α ( q , ω ) + α 0 ( k , ω ) ] n - 1 ζ ^ ( n ) ( q - k ) + m = 0 n - 1 ( n m ) - d r 2 π [ α ( q , ω ) - α 0 ( r , ω ) ] n - m - 1 ζ ^ ( n - m ) ( q - r ) R s ( m ) ( r k ) } ,             n 1.
R p ( 0 ) ( q k ) = 2 π δ ( q - k ) α 0 ( k , ω ) - α ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) ,
R p ( 1 ) ( q k ) = - 1 α 0 ( q , ω ) + α ( q , ω ) ζ ^ ( 1 ) ( q - k ) × [ α ( q , ω ) α ( k , ω ) - q k ] 2 α 0 ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) ,
R p ( 2 ) ( q k ) = - 1 α 0 ( q , ω ) + α ( q , ω ) ζ ^ ( 2 ) ( q - k ) × [ α ( q , ω ) α ( k , ω ) ] - q k ] [ α ( q , ω ) + α ( k , ω ) ] × 2 α 0 ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) - 2 ( - 1 ) 2 × α ( q , ω ) α 0 ( q , ω ) + α ( q , ω ) - d r 2 π × ζ ^ ( 1 ) ( q - r ) α ( r , ω ) ζ ^ ( 1 ) ( r - k ) × α ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) 2 α 0 ( k , ω ) + 2 ( - 1 ) 2 × 1 α 0 ( q , ω ) + α ( q , ω ) - d r 2 π ζ ^ ( 1 ) ( q - r ) × [ α ( q , ω ) α ( r , ω ) - q r ] [ α ( r , ω ) α ( k , ω ) - r k ] α 0 ( r , ω ) + α ( r , ω ) × ζ ^ ( 1 ) ( r - k ) 2 α 0 ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) ;
R s ( 0 ) ( q k ) = 2 π δ ( q - k ) α 0 ( k , ω ) - α ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) ,
R p ( 1 ) ( q k ) = 1 α 0 ( q , ω ) + α ( q , ω ) ζ ^ ( 1 ) ( q - k ) × ω 2 c 2 ( 1 - ) 2 α 0 ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) ,
R s ( 2 ) ( q k ) = 1 α 0 ( q , ω ) + α ( q , ω ) × { ζ ^ ( 2 ) ( q - k ) ω 2 c 2 ( 1 - ) [ α ( q , ω ) + α ( k , ω ) ] + 2 [ ω 2 c 2 ( 1 - ) ] 2 - d r 2 π × ζ ^ ( 1 ) ( q - r ) ζ ^ ( 1 ) ( r - k ) α 0 ( r , ω ) + α ( r , ω ) } × 2 α 0 ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) .
R p , s ( q k ) 2 = [ R p , s ( 0 ) ( q k ) ] 2 - R p , s ( 0 ) ( q k ) Re R p . s ( 2 ) ( q k ) ,
R p ( 2 ) ( q k ) = 2 π δ ( q - k ) δ 2 a 2 4 cos θ 0 [ cos θ 0 + ( - sin 2 θ 0 ) 1 / 2 ] 2 × { ( - 1 ) ( ω a c ) 2 ( - sin 2 θ 0 ) 1 / 2 × ( - 2 sin 2 θ 0 ) - ( - 1 ) 2 ( ω a c ) 2 ( - sin 2 θ 0 ) × - d x ω 2 π c g ( ω c sin θ 0 - x ) ( - x 2 ) 1 / 2 + ( - 1 ) 2 ( ω a c ) 2 - d x ω 2 π c g ( ω c sin θ 0 - x ) × [ ( - x 2 ) 1 / 2 ( - sin 2 θ 0 ) 1 / 2 - x sin θ 0 ] 2 ( 1 - x 2 ) 1 / 2 + ( - x 2 ) 1 / 2 } 2 π δ ( q - k ) δ 2 a 2 4 cos θ 0 [ cos θ 0 + ( + sin 2 θ 0 ) 1 / 2 ] 2 × μ p [ θ 0 , ( ω a / c ) ] ,
R s ( 2 ) ( q k ) = 2 π δ ( q - k ) δ 2 a 2 4 cos θ 0 [ cos θ 0 + ( - sin 2 θ 0 ) 1 / 2 ] 2 × { - ( - 1 ) ( ω a c ) 2 ( - sin 2 θ 0 ) 1 / 2 + ( - 1 ) 2 ( ω a c ) 2 - d x ω 2 π c × g [ ( ω / c ) sin θ 0 - x ] ( 1 - x 2 ) 1 / 2 + ( - x 2 ) 1 / 2 } 2 π δ ( q - k ) δ 2 a 2 4 cos θ 0 [ cos θ 0 + ( - sin 2 θ 0 ) 1 / 2 ] 2 × μ s [ θ 0 , ( ω a / c ) ] .
R p θ s spe = δ ( θ s - θ 0 ) R p [ θ 0 , ( ω a / c ) , ( δ 2 / a 2 ) ]
R p [ θ 0 , ( ω a / c ) , ( δ 2 / a 2 ) ] = [ cos θ 0 - ( - sin 2 θ 0 ) 1 / 2 cos θ 0 + ( - sin 2 θ 0 ) 1 / 2 ] 2 - 4 δ 2 a 2 ( cos θ 0 ) × cos θ 0 - ( - sin 2 θ 0 ) 1 / 2 [ cos θ 0 - ( - sin 2 θ 0 ) 1 / 2 ] 3 Re { μ p [ θ 0 , ( ω a / c ) ] } .
R s θ s spe = δ ( θ s - θ 0 ) R s [ θ 0 , ( ω a / c ) , ( δ 2 / a 2 ) ]
R s [ θ 0 , ( ω a / c ) , ( δ 2 / a 2 ) ] = [ cos θ 0 - ( - sin 2 θ 0 ) 1 / 2 cos θ 0 + ( - sin 2 θ 0 ) 1 / 2 ] 2 - 4 δ 2 a 2 ( cos θ 0 ) × cos θ 0 - ( - sin 2 θ 0 ) 1 / 2 [ cos θ 0 + ( - sin 2 θ 0 ) 1 / 2 ] 3 Re { μ s [ θ 0 , ( ω a / c ) ] } .
[ 2 π δ ( q - k ) ] 2 = 2 π δ ( 0 ) 2 π δ ( q - k ) = L 1 2 π δ ( q - k ) ,
δ ( q - k ) = c ω δ ( θ s - θ 0 ) cos θ 0
μ p [ θ 0 , ( ω a / c ) ] = ( - 1 ) ( ω a c ) 2 ( - sin 2 θ 0 ) 1 / 2 ( - 2 sin 2 θ 0 ) - 1 2 π ( - 1 ) 2 ( ω a c ) 3 ( - sin 2 θ 0 ) × - d x ( - x 2 ) 1 / 2 exp [ - ( ω a / 2 c ) 2 ( sin θ 0 - x ) 2 ] + 1 2 π ( - 1 ) 2 ( ω a c ) 3 - d x × [ x sin θ 0 - ( - x 2 ) 1 / 2 ( - sin 2 θ 0 ) 1 / 2 ] 2 ( 1 - x 2 ) 1 / 2 + ( - x 2 ) 1 / 2 × exp [ - ( ω a / 2 c ) 2 ( sin θ 0 - x 2 ) ] ,
μ s [ θ 0 , ( ω a / c ) ] = - ( - 1 ) ( ω a c ) 2 ( - sin 2 θ 0 ) 1 / 2 + 1 2 π ( - 1 ) 2 ( ω a c ) 3 × - d x exp [ - ( ω a / 2 c ) 2 ( sin θ 0 - x ) 2 ] ( 1 - x 2 ) 1 / 2 + ( - x 2 ) 1 / 2 .
R p , s ( q k ) = 2 π δ ( q - k ) R p , s ( 0 ) ( k , ω ) - 2 i G p , s ( 0 ) ( q , ω ) T p , s ( q k ) G p , s ( 0 ) ( k , ω ) α 0 ( k , ω ) ,
R p ( 0 ) ( k , ω ) = α 0 ( k , ω ) - α ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) ,
R s ( 0 ) ( k , ω ) = α 0 ( k , ω ) - α ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) ,
G p ( 0 ) ( k , ω ) = i α 0 ( k , ω ) + α ( k , ω ) ,
G s ( 0 ) ( k , ω ) = i α 0 ( k , ω ) + α ( k , ω ) .
T p , s ( q k ) = V p , s ( q k ) + - d r 2 π V p , s ( q r ) G p , s ( 0 ) ( r , ω ) T p , s ( r k )
= V p , s ( q k ) + - d r 2 π T p , s ( q r ) G p , s ( 0 ) ( r , ω ) V p , s ( r k ) .
- d r 2 π [ M ( q r ) - N ( q r ) ] V p ( r k ) 2 i α 0 ( r , ω ) = 1 2 α 0 ( k , ω ) { M ( q k ) [ α 0 ( k , ω ) - α ( k , ω ) ] + N ( q k ) [ α 0 ( k , ω ) + α ( k , ω ) ] } ,
M ( q r ) = I [ α ( q , ω ) - α 0 ( r , ω ) q - r ] α ( q , ω ) - α 0 ( r , ω ) × [ q r + α ( q , ω ) α 0 ( r , ω ) ] ,
N ( q r ) = I [ α ( q , ω ) + α 0 ( r , ω ) q - r ] α ( q , ω ) - α 0 ( r , ω ) × [ q r - α ( q , ω ) α 0 ( r , ω ) ] ,
- d r 2 π { I [ α ( q , ω ) - α 0 ( r , ω ) q - r ] α ( q , ω ) - α 0 ( r , ω ) - I [ α ( q , ω ) + α 0 ( r , ω ) q - r ] α ( q , ω ) + α 0 ( r , ω ) } V s ( r k ) 2 i α 0 ( r , ω ) = 1 2 α 0 ( k , ω ) { I [ α ( q , ω ) - α 0 ( k , ω ) q - k ] α ( q , ω ) - α 0 ( k , ω ) × [ α 0 ( k , ω ) - α ( k , ω ) ] + I [ α ( q , ω ) + α 0 ( k , ω ) q - k ] α ( q , ω ) + α 0 ( k , ω ) × [ α 0 ( k , ω ) + α ( k , ω ) ] } ,
G p , s ( q k ) = 2 π δ ( q - k ) G p , s ( 0 ) ( k , ω ) + G p , s ( 0 ) ( q , ω ) - d r 2 π V p , s ( q r ) G p , s ( r k )
= 2 π δ ( q - k ) G p , s ( 0 ) ( k , ω ) + G p , s ( 0 ) ( q , ω ) T p , s ( q k ) G p , s ( 0 ) ( k , ω ) .
R p , s ( q k ) = - 2 π δ ( q - k ) - 2 i G p , s ( q k ) α 0 ( k , ω ) .
G p , s ( q k ) = 2 π δ ( q - k ) G p , s ( 0 ) ( k , ω ) + G p , s ( 0 ) ( q , ω ) - d r 2 π M p , s ( q r ) G p , s ( r k ) ,
M p , s ( q k ) = V p , s ( q k ) + - d r 2 π V p , s ( q r ) G p , s ( 0 ) ( r , ω ) Q M p , s ( r k ) .
M p , s ( q k = 2 π δ ( q - k ) M p , s ( k , ω ) .
G p , s ( q k ) = 2 π δ ( q - k ) G p , s ( k , ω ) .
G p ( k , ω ) = i α 0 ( k , ω ) + α ( k , ω ) - i M p ( k , ω ) ,
G s ( k , ω ) = i α 0 ( k , ω ) + α ( k , ω ) - i M s ( k , ω ) .
R p ( q k ) = 2 π δ ( q - k ) × 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 ( q k ) = 2 π δ ( q - k ) α 0 ( k , ω ) - α ( k , ω ) + i M s ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) - i M s ( k , ω ) .
R p , s θ s spe = δ ( θ s - θ 0 ) R p , s [ θ 0 , ( ω a / c ) , ( δ 2 / a 2 ) ] ,
R p [ θ 0 , ( ω a / c ) , ( δ 2 / a 2 ) ] = | α 0 ( k , ω ) - α ( k , ω ) + i M p ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) - i M p ( k , ω ) | 2 ,
R s [ θ 0 , ( ω a / c ) , ( δ 2 / a 2 ) ] = | α 0 ( k , ω ) - α ( k , ω ) + i M s ( k , ω ) α 0 ( k , ω ) + α ( k , ω ) - i M s ( k , ω ) | 2 ,
V p , s ( q k ) = n = 1 V p , s ( n ) ( q k ) ,
V p ( 1 ) ( q k ) = - 1 2 [ q k - α ( q , ω ) α ( k , ω ) ] ζ ^ ( 1 ) ( q - k ) ,
V p ( 2 ) ( q k ) = - i 2 - 1 2 [ α ( q , ω ) + α ( k , ω ) ] × [ q k - α ( q , ω ) α ( k , ω ) ] ζ ^ ( 2 ) ( q - k ) - i ( - 1 ) 2 3 α ( q , ω ) - d r 2 π ζ ^ ( 1 ) ( q - r ) × α ( r , ω ) ζ ^ ( 1 ) ( r - k ) α ( k , ω ) ;
V s ( 1 ) ( q k ) = ω 2 c 2 ( - 1 ) ζ ^ ( 1 ) ( q - k ) ,
V s ( 2 ) ( q k ) = - i 2 ω 2 c 2 ( - 1 ) [ α ( q , ω ) + α ( k , ω ) ] ζ ^ ( 2 ) ( q - k ) .
M p , s ( q k ) = V p , s ( 2 ) ( q k ) + - d r 2 π V p , s ( 1 ) ( q r ) V p , s ( 1 ) ( r k ) G p , s ( 0 ) ( r , ω ) .
M p ( k , ω ) = i ω c δ 2 a 2 μ p [ θ 0 , ( ω a / c ) ] ,
M s ( k , ω ) = i ω c δ 2 a 2 μ s [ θ 0 , ( ω a / c ) ] ,
R p [ θ 0 , ( ω a / c ) , ( δ 2 / a 2 ) ] = | cos θ 0 - ( - sin 2 θ 0 ) 1 / 2 - ( δ 2 / a 2 ) μ p [ θ 0 , ( ω a / c ) ] cos θ 0 + ( - sin 2 θ 0 ) 1 / 2 + ( δ 2 / a 2 ) μ p [ θ 0 , ( ω a / c ) ] | 2 ,
R s [ θ 0 , ( ω a / c ) , ( δ 2 / a 2 ) ] = | cos θ 0 - ( - sin 2 θ 0 ) 1 / 2 - ( δ 2 / a 2 ) μ s [ θ 0 , ( ω a / c ) ] cos θ 0 + ( - sin 2 θ 0 ) 1 / 2 + ( δ 2 / a 2 ) μ s [ θ 0 , ( ω a / c ) ] | 2 .
S p , s ( q k ) = [ α 0 ( q , ω ) α 0 ( k , ω ) ] 1 / 2 R p , s ( q k ) = n = 0 ( - i ) n n ! S p , s ( n ) ( q k ) ,
S p , s ( n ) ( q k ) = [ α 0 ( q , ω ) α 0 ( k , ω ) ] 1 / 2 R p , s ( n ) ( q k ) .
R p , s θ s spe = 1 L 1 ω 2 π c ( cos θ s ) S p , s ( q k ) 2 .
S p , s ( q k ) = S p , s ( - k - q ) ,
S p , s ( q k ) = [ R p , s ( 0 ) ( q , ω ) ] 1 / 2 × n = 0 ( - i ) n n ! H p , s ( n ) ( q k ) [ R p , s ( 0 ) ( k , ω ) ] 1 / 2 ,
H p , s ( n ) ( q k ) = S p , s ( n ) ( q k ) [ R p , s ( 0 ) ( q , ω ) ] 1 / 2 [ R p , s ( 0 ) ( k , ω ) ] 1 / 2
H p , s ( 0 ) ( q k ) = 2 π δ ( q - k ) .
S p , s ( q k ) = [ R p , s ( 0 ) ( q , ω ) ] 1 / 2 - d x 1 exp [ - i ( q - k ) x 1 ] × n = 0 ( - i ) n n ! H p , s ( n ) ( q x 1 k ) [ R p , s ( 0 ) ( k , ω ) ] 1 / 2 ,
H p , s ( 0 ) ( q x 1 k ) = 1
H p ( 1 ) ( q x 1 k ) = 2 + 1 α 0 1 / 2 ( q , ω ) d ( q , ω ) ζ ( x 1 ) [ α ( q , ω ) α ( k , ω ) ] - q k ] α 0 1 / 2 ( k , ω ) d ( k , ω ) ,
H p ( 2 ) ( q x 1 k ) = 2 + 1 α 0 1 / 2 ( q , ω ) d ( q , ω ) { ζ 2 ( x 1 ) [ α ( q , ω ) α ( k , ω ) - q k ] × [ α ( q , ω ) + α ( k , ω ) ] - - 1 α ( q , ω ) × - d r 2 π α ( r , ω ) - d u exp ( i r u ) ζ ( x 1 + u ) ζ ( x 1 ) × [ exp ( - i q u ) + exp ( i k u ) ] α ( k , ω ) + - 1 - d r 2 π × [ α ( q , ω ) α ( r , ω ) - q r ] [ α ( r , ω ) α ( k , ω ) - r k ] α 0 ( r , w ) + α ( r , ω ) × - d u ζ ( x 1 + u ) ζ ( x 1 ) [ exp ( - i q u ) exp ( i r u ) + exp ( i k u ) exp ( - i r u ) ] } α 0 1 / 2 ( k , ω ) d ( k , ω ) ,
d ( k , ω ) = ( + 1 ω 2 c 2 - k 2 ) 1 / 2
H s ( 1 ) ( q x 1 k ) = 2 α 0 1 / 2 ( q , ω ) ζ ( x 1 ) α 0 1 / 2 ( k , ω ) ,
H s ( 2 ) ( q x 1 k ) = 2 α 0 1 / 2 ( q , ω ) { ζ 2 ( x 1 ) [ α ( q , ω ) + α ( k , ω ) ] + ω 2 c 2 ( 1 - ) × - d r 2 π 1 α 0 ( r , w ) + α ( r , ω ) × - d u exp ( i r u ) ζ ( x 1 + u ) ζ ( x 1 ) × [ exp ( - i q u ) + exp ( i k u ) ] } α 0 1 / 2 ( k , ω ) ,
H p , s ( n ) ( q x 1 k ) = H p , s ( n ) ( - k x 1 - q ) .
S p , s ( q k ) = [ R p , s ( 0 ) ( q , ω ) ] 1 / 2 - d x 1 exp [ - i ( q - k ) x 1 ] × exp [ n = 1 ( - i ) n n ! G p , s ( n ) ( q x 1 k ) ] [ R p , s ( 0 ) ( k , ω ) ] 1 / 2 .
G ( n ) = H ( n ) - s = 1 n - 1 ( n - 1 s - 1 ) H ( n - s ) G ( s ) ,
G p , s ( 1 ) ( q x 1 k ) = H p , s ( 1 ) ( q x 1 k ) ,
G p , s ( 2 ) ( q x 1 k ) = H p , s ( 2 ) ( q x 1 k ) - [ H p , s ( 1 ) ( q x 1 k ) ] 2 ,
S p , s ( q k ) = 2 π δ ( q - k ) R p , s ( 0 ) ( k , ω ) × exp [ - 1 2 H p , s ( 2 ) ( k x 1 k ) ] .
H p ( 2 ) ( k x 1 k ) = 4 δ 2 a 2 - 1 × cos θ 0 cos 2 θ 0 - sin 2 θ 0 μ p [ θ 0 , ( ω a / c ) ] ,
H s ( 2 ) ( k x 1 k ) = 4 δ 2 a 2 cos θ 0 1 - μ s [ θ 0 , ( ω a / c ) ] ,
R p [ θ 0 , ( ω a / c ) , ( δ 2 / a 2 ) ] = [ cos θ 0 - ( - sin 2 θ 0 ) 1 / 2 cos θ 0 + ( - sin 2 θ 0 ) 1 / 2 ] 2 × exp ( - 4 δ 2 a 2 - 1 cos θ 0 cos 2 θ 0 - sin 2 θ 0 × Re { μ p [ θ 0 , ( ω a / c ) ] } ) ,
R s [ θ 0 , ( ω a / c ) , ( δ 2 / a 2 ) ] = [ cos θ 0 - ( - sin 2 θ 0 ) 1 / 2 cos θ 0 + ( - sin 2 θ 0 ) 1 / 2 ] 2 × exp ( - 4 δ 2 a 2 cos θ 0 1 - Re { μ s [ θ 0 , ( ω a / c ) ] } ) .
R p , s [ θ 0 , ( δ / λ ) ] = R p , s ( θ 0 ) 2 exp [ - ( 4 π δ / λ ) 2 cos 2 θ 0 ] ,
reflectivity p , s = - π / 2 π / 2 R p , s θ s spe d θ s .

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