Abstract

The phase-perturbation method for the computation of fields scattered by rough surfaces is reviewed and then a method for calculating moments of the field scattered from randomly rough surfaces is investigated. In previous work, the authors have shown that the phase-perturbation method can provide an accurate approximation to the fields scattered from deterministic rough surfaces, even in cases when the rms surface-height variation is significant compared to the radiation wavelength. The present work employs cumulants to develop a systematic averaging procedure that retains this advantage, in the case of scalar-wave scattering from surfaces on which Dirichlet boundary conditions hold. The resulting expressions for the moments of the scattered field are shown to reduce to classical small roughness and physical-optics expressions in the appropriate limits. Numerical results for the magnitude of the coherent reflection coefficient in two test cases are also presented.

© 1985 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic, New York, 1978).
  2. F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, Reading, Mass., 1982), Vol. II.
  3. D. Winebrenner, A. Ishimaru, “Investigation of a surface field phase-perturbation technique for scattering from rough surfaces,” Radio Sci. 20, 161–170 (1985).
    [Crossref]
  4. H. Hoinkes, “The physical interaction potential of gas atoms with single-crystal surfaces, determined from gas-surface diffraction experiments.” Rev. Mod. Phys. 52, 933–970 (1980).
    [Crossref]
  5. D. P. Winebrenner, “A surface field phase-perturbation technique for scattering from rough surfaces,” Ph.D. dissertation (University of Washington, Seattle, Wash., 1985).
  6. B. B. Baker, E. T. Copson, The Mathematical Theory of Huygen’s Principle, 2nd ed. (Oxford U. Press, London, 1950).
  7. P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
    [Crossref]
  8. P. A. Martin, “Acoustic scattering and radiation problems, and the null-field method,” Wave Motion 4, 391–408 (1982).
    [Crossref]
  9. M. Nieto-Veserinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [Crossref]
  10. A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, translated by R. A. Silverman (Prentice-Hall, Englewood Cliffs, N.J., 1962).
  11. J. Shen, A. A. Maradudin, “Multiple scattering of waves from random rough surfaces,” Phys. Rev. B 22, 4234–4240 (1980).
    [Crossref]
  12. M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Hafner, New York, 1963).
  13. R. L. Stratonovich, Topics in the Theory of Random Noise, Part 1, Mathematics and Its Applications Series, Vol. 3, translated from Russian by R. A. Silverman (Gordon and Breach, New York, 1963).
  14. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  15. J. G. Zornig, J. F. McDonald, “Direct measurement of surface-scatter channel coherence by impulse probing,” J. Acoust. Soc. Am. 55, 1205–1211 (1974).
    [Crossref]
  16. O. M. Phillips, The Dynamics of the Upper Ocean, 2nd ed. (Cambridge U. Press, Cambridge, 1980).
  17. P. J. Welton, “Backscattering from randomly rough surfaces,” Rev. Cathedec, no. 60 (1979).
  18. S. T. McDaniel, A. D. Gorman, “An examination of the composite-roughness scattering model,” J. Opt. Soc. Am. 73, 1476–1486(1983).
  19. F. M. Labianca, E. Y. Harper, “Connection between various small waveheight solutions of the problem of scattering from the ocean surface,” J. Acoust. Soc. Am. 62, 1144–1157 (1977).
    [Crossref]

1985 (1)

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

1983 (1)

1982 (1)

P. A. Martin, “Acoustic scattering and radiation problems, and the null-field method,” Wave Motion 4, 391–408 (1982).
[Crossref]

1981 (1)

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

1980 (2)

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

H. Hoinkes, “The physical interaction potential of gas atoms with single-crystal surfaces, determined from gas-surface diffraction experiments.” Rev. Mod. Phys. 52, 933–970 (1980).
[Crossref]

1979 (1)

P. J. Welton, “Backscattering from randomly rough surfaces,” Rev. Cathedec, no. 60 (1979).

1977 (1)

F. M. Labianca, E. Y. Harper, “Connection between various small waveheight solutions of the problem of scattering from the ocean surface,” J. Acoust. Soc. Am. 62, 1144–1157 (1977).
[Crossref]

1975 (1)

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[Crossref]

1974 (1)

J. G. Zornig, J. F. McDonald, “Direct measurement of surface-scatter channel coherence by impulse probing,” J. Acoust. Soc. Am. 55, 1205–1211 (1974).
[Crossref]

Baker, B. B.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygen’s Principle, 2nd ed. (Oxford U. Press, London, 1950).

Copson, E. T.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygen’s Principle, 2nd ed. (Oxford U. Press, London, 1950).

Fung, A. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, Reading, Mass., 1982), Vol. II.

Garcia, N.

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

Gorman, A. D.

Harper, E. Y.

F. M. Labianca, E. Y. Harper, “Connection between various small waveheight solutions of the problem of scattering from the ocean surface,” J. Acoust. Soc. Am. 62, 1144–1157 (1977).
[Crossref]

Hoinkes, H.

H. Hoinkes, “The physical interaction potential of gas atoms with single-crystal surfaces, determined from gas-surface diffraction experiments.” Rev. Mod. Phys. 52, 933–970 (1980).
[Crossref]

Ishimaru, A.

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

A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic, New York, 1978).

Kendall, M. G.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Hafner, New York, 1963).

Labianca, F. M.

F. M. Labianca, E. Y. Harper, “Connection between various small waveheight solutions of the problem of scattering from the ocean surface,” J. Acoust. Soc. Am. 62, 1144–1157 (1977).
[Crossref]

Maradudin, A. A.

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

Martin, P. A.

P. A. Martin, “Acoustic scattering and radiation problems, and the null-field method,” Wave Motion 4, 391–408 (1982).
[Crossref]

McDaniel, S. T.

McDonald, J. F.

J. G. Zornig, J. F. McDonald, “Direct measurement of surface-scatter channel coherence by impulse probing,” J. Acoust. Soc. Am. 55, 1205–1211 (1974).
[Crossref]

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Moore, R. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, Reading, Mass., 1982), Vol. II.

Nieto-Veserinas, M.

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

Phillips, O. M.

O. M. Phillips, The Dynamics of the Upper Ocean, 2nd ed. (Cambridge U. Press, Cambridge, 1980).

Shen, J.

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

Stratonovich, R. L.

R. L. Stratonovich, Topics in the Theory of Random Noise, Part 1, Mathematics and Its Applications Series, Vol. 3, translated from Russian by R. A. Silverman (Gordon and Breach, New York, 1963).

Stuart, A.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Hafner, New York, 1963).

Ulaby, F. T.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, Reading, Mass., 1982), Vol. II.

Waterman, P. C.

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[Crossref]

Welton, P. J.

P. J. Welton, “Backscattering from randomly rough surfaces,” Rev. Cathedec, no. 60 (1979).

Winebrenner, D.

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

Winebrenner, D. P.

D. P. Winebrenner, “A surface field phase-perturbation technique for scattering from rough surfaces,” Ph.D. dissertation (University of Washington, Seattle, Wash., 1985).

Yaglom, A. M.

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, translated by R. A. Silverman (Prentice-Hall, Englewood Cliffs, N.J., 1962).

Zornig, J. G.

J. G. Zornig, J. F. McDonald, “Direct measurement of surface-scatter channel coherence by impulse probing,” J. Acoust. Soc. Am. 55, 1205–1211 (1974).
[Crossref]

J. Acoust. Soc. Am. (3)

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[Crossref]

J. G. Zornig, J. F. McDonald, “Direct measurement of surface-scatter channel coherence by impulse probing,” J. Acoust. Soc. Am. 55, 1205–1211 (1974).
[Crossref]

F. M. Labianca, E. Y. Harper, “Connection between various small waveheight solutions of the problem of scattering from the ocean surface,” J. Acoust. Soc. Am. 62, 1144–1157 (1977).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

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

Phys. Rev. B (1)

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

Radio Sci. (1)

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

Rev. Cathedec (1)

P. J. Welton, “Backscattering from randomly rough surfaces,” Rev. Cathedec, no. 60 (1979).

Rev. Mod. Phys. (1)

H. Hoinkes, “The physical interaction potential of gas atoms with single-crystal surfaces, determined from gas-surface diffraction experiments.” Rev. Mod. Phys. 52, 933–970 (1980).
[Crossref]

Wave Motion (1)

P. A. Martin, “Acoustic scattering and radiation problems, and the null-field method,” Wave Motion 4, 391–408 (1982).
[Crossref]

Other (9)

A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic, New York, 1978).

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Addison-Wesley, Reading, Mass., 1982), Vol. II.

D. P. Winebrenner, “A surface field phase-perturbation technique for scattering from rough surfaces,” Ph.D. dissertation (University of Washington, Seattle, Wash., 1985).

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygen’s Principle, 2nd ed. (Oxford U. Press, London, 1950).

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Hafner, New York, 1963).

R. L. Stratonovich, Topics in the Theory of Random Noise, Part 1, Mathematics and Its Applications Series, Vol. 3, translated from Russian by R. A. Silverman (Gordon and Breach, New York, 1963).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions, translated by R. A. Silverman (Prentice-Hall, Englewood Cliffs, N.J., 1962).

O. M. Phillips, The Dynamics of the Upper Ocean, 2nd ed. (Cambridge U. Press, Cambridge, 1980).

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Figures (4)

Fig. 1
Fig. 1

Plane wave incident in the x–z plane upon a rough surface S.

Fig. 2
Fig. 2

Model spectrum and power-law spectrum for reference.

Fig. 3
Fig. 3

Magnitude of coherent plane-wave reflection coefficient obtained from phase-perturbation approximation (circles) and the Kirchhoff approximation (squares) in first test case with rms height 0.5 cm and spectrum as indicated in text.

Fig. 4
Fig. 4

Magnitude of coherent plane-wave reflection coefficient obtained from phase-perturbation approximation (circles) and the Kirchhoff approximation (squares) in second test case with rms height 1 cm and L = 2 cm.

Equations (76)

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r V 1 , ψ ( r ) = ψ i ( r ) d 2 R ψ ( r s ) n G 0 ( r r s ) ,
r V 2 , 0 = ψ i ( r ) d 2 R ψ ( r s ) n G 0 ( r r s ) ,
R = x x ̂ + y ŷ , d 2 R = d x d y , r s = R + h ( R ) , n ( r s ) = h x x ̂ h y ŷ ,
ψ ( r s ) n = lim z h ( x , y ) 0 + ψ ( r ) n ( r s ) .
ψ i ( r ) = exp [ i K i R k z ( K i ) z ] ,
K i = ( k sin θ i ) x ̂
k z ( K ) = { ( k 2 | K | 2 ) 1 / 2 , | K | k i ( | K | 2 k 2 ) 1 / 2 , | K | > k .
G 0 ( r r ) = i 2 ( 2 π ) 2 d 2 K exp [ i K ( R R ) ] × exp [ i k z ( K ) | z z | ] k z ( K ) ,
K = K x x ̂ + K y ŷ .
ψ s ( r ) = d 2 K exp [ i K R + i k z ( K ) z ] T ( K , K i ) ,
T ( K , K i ) = 1 2 i ( 2 π ) 2 k z ( K ) d 2 R × exp [ i K R i k z ( K ) h ( R ) ] ψ ( r s ) n .
2 i ( 2 π ) 2 k z ( K ) δ ( K K i ) = d 2 R exp [ i K R + i k z ( K ) h ( R ) ] ψ ( r s ) n ,
ψ ( r s ) n = 2 i k z ( K i ) exp ( i K i R ) f ( R ) .
h ( R ) = D ( R ) .
f ( R ; k ) = f ( 0 ) ( R ) + ( k ) f ( 1 ) ( R ) + ( k ) 2 2 f ( 2 ) ( R ) + ( k ) 3 6 f ( 3 ) ( R ) + .
f ( 0 ) ( R ) = 1 ,
f ( 1 ) ( R ) = i ( 2 π ) 2 d 2 κ 1 exp ( i κ 1 R ) D ( κ 1 ) β ( K i + κ 1 ) ,
f ( 2 ) ( R ) = ( i ) 2 ( 2 π ) 4 d 2 κ 1 exp ( i κ 1 R ) β ( K i + κ 1 ) d 2 κ 1 D ( κ 2 ) × D ( κ 1 κ 2 ) [ 2 β ( K i + κ 2 ) β ( K i + κ 1 ) ] ,
β ( K ) = k z ( K ) k
D ( K ) = d 2 R exp ( i K R ) D ( R ) .
f ( R ) = exp φ ( R ) .
1 1 + j = 1 ( k ) j j ! f ( j ) = j = 1 exp [ ( k ) j j ! φ ( j ) ] = { 1 + ( k ) φ ( 1 ) + ( k ) 2 2 [ φ ( 1 ) ] 2 + } × { 1 + ( k ) 2 2 φ ( 2 ) + ( k ) 4 4 [ φ ( 2 ) ] 2 + } × { 1 + ( k ) 3 6 φ ( 3 ) + ( k ) 6 12 [ φ ( 3 ) ] 2 + } .
f ( 1 ) = φ ( 1 ) ,
f ( 2 ) = φ ( 2 ) + [ φ ( 1 ) ] 2 ,
f ( 3 ) = φ ( 3 ) + 3 φ ( 2 ) φ ( 1 ) + [ φ ( 1 ) ] 3 ,
T ( K , K i ) = k z ( K i ) ( 2 π ) 2 k z ( K ) d 2 R exp [ i ( K K i ) R ] × exp [ i k 2 ( K ) h ( R ) + φ ( R ) ] .
ψ s ( r ) = d 2 K exp [ i K R + i k z ( K ) z ] T ( K , K i ) ,
T ( K , K i ) = k z ( K i ) ( 2 π ) 2 k z ( K ) d 2 R exp [ i ( K K i ) R ] × exp [ i k z ( K ) h ( R ) + φ ( R ) ] .
i k z ( K ) h ( R ) + φ ( R ) = i ( k ) [ β ( K ) D ( R ) i k φ ( R ) ] .
exp ( i γ Λ ) = exp [ n = 1 r ( i γ ) n n ! ν n ] ,
ν 1 = μ 1 ,
ν 2 = μ 2 μ 1 2 ,
ν 3 = μ 3 3 μ 2 μ 1 + 2 μ 2 3 ,
ν 4 = μ 4 4 μ 3 μ 1 μ 2 2 + 12 μ 2 μ 1 2 6 μ 1 4 ,
Λ = β ( K ) D ( R ) i k φ ( R )
γ = k .
exp { i ( k ) [ β ( K ) D ( R ) i k φ ( R ) ] } = exp [ n ( i k ) n n ! ν n ] .
φ ( R ) ( k ) φ ( 1 ) ( R ) + ( k ) 2 2 φ ( 2 ) ( R ) .
φ ( 1 ) ( R ) = i ( 2 π ) 2 d 2 κ 1 exp ( i κ 1 R ) D ( κ 1 ) β ( K i + κ 1 )
φ ( 2 ) ( R ) = ( i ) 2 ( 2 π ) 4 d 2 κ 1 exp ( i κ 1 R ) d 2 κ 2 D ( κ 2 ) D ( κ 1 κ 2 ) × [ 2 β ( K i + κ 2 ) β ( K i + κ 1 ) β 2 ( K i + κ 1 ) β ( K i + κ 1 κ 2 ) β ( K i + κ 2 ) ] .
ν 1 = Λ β ( K ) D ( R ) i φ ( 1 ) ( R ) i ( k ) 2 φ ( 2 ) ( R ) .
ν ( 1 ) i ( k ) 2 φ ( 2 ) ( R ) .
ν 2 = Λ 2 Λ 2 β 2 ( K ) D 2 ( R ) + 2 i β ( K ) D ( R ) φ ( 1 ) ( R ) [ φ ( 1 ) ( R ) ] 2 .
n ( i k ) n n ! ν n ( k ) 2 2 N 2 ( K ) ,
N 2 ( K ) = [ β 2 ( K ) β 2 ( K i ) ] + 2 [ β ( K ) + β ( K i ) ] ( 2 π ) 2 d 2 κ β ( K i + κ ) W ( κ ) ,
1 = 1 ( 2 π ) 2 d 2 κ W ( κ ) .
T ( K , K i ) δ ( K K i ) exp [ ( k ) 2 2 N 2 ( K i ) ] δ ( K K i ) exp [ 2 ( k ) 2 β ( K i ) ( 2 π ) 2 d 2 κ β ( K i + κ ) W ( κ ) ] .
β ( K ) = { ( 1 | K | 2 k 2 ) 1 / 2 , | K | k i ( | K | 2 k 2 1 ) 1 / 2 , | K | > k ,
β ( K i + κ ) = β ( K i ) + ( κ ) β | K i + 1 2 ( κ ) 2 β | K i + = β ( K i ) + tan θ i κ x k 1 2 cos 3 θ i ( κ x k ) 2 1 2 cos θ i ( κ y k ) 2 + .
N 2 ( K i ) 4 β 2 ( K i ) ( 2 π ) 2 d 2 κ W ( κ ) 4 β 2 ( K i ) .
R ( K i ) exp [ 2 ( k ) 2 β 2 ( K i ) ] = exp [ 2 ( k ) 2 cos 2 θ i ] .
W ( κ ) = j = 1 N a j W j ( κ ) , W j ( κ ) = 2 π L j 2 exp ( | κ | 2 L j 2 2 ) ,
a 1 = 0.6563 , L 1 = 2.381 cm , a 2 = 0.2026 , L 2 = 1.429 cm , a 3 = 0.1048 , L 3 = 0.7692 cm , a 4 = 0.03629 , L 4 = 0.3774 cm .
N 2 ( K ) = j = 1 N a j N 2 j ( K ) .
T I T ( K , K i ) T * ( K , K i ) T ( K , K i ) T * ( K , K i ) .
T I = k z 2 ( K i ) ( 2 π ) 4 k z 2 ( K ) d 2 R c d 2 R d exp [ i ( K K i ) R d ] × exp ( i γ 1 Λ 1 + i γ 2 Λ 2 ) exp ( i γ 1 Λ 1 ) exp ( i γ 2 Λ 2 ) ,
R c = ½ ( R 1 + R 2 ) , R d = R 1 = R 2 , γ 1 = k , Λ 1 = β ( K ) D ( R c + ½ R d ) i k φ ( R c + ½ R d ) ,
γ 2 = k , Λ 2 = β ( K ) D ( R c ½ R d ) + i k φ * ( R c ½ R d ) .
σ ( K , K i ) = k z 2 ( K i ) π d 2 R d exp [ i ( K K i ) R d ] × [ exp ( i γ 1 Λ 1 + i γ 2 Λ 2 ) exp ( i γ 1 Λ 1 ) exp ( i γ 2 Λ 2 ) ] .
exp ( i γ 1 Λ 1 ) = exp [ n ( i γ 1 ) n n ! ν n , 0 ]
exp ( i γ 2 Λ 2 ) = exp [ n ( i γ 2 ) n n ! ν 0 , n ] ,
n ( i k ) n n ! ν n , 0 ( k ) 2 2 N 2 ( K ) ,
exp ( i γ 2 Λ 2 ) exp [ ( k ) 2 2 N 2 * ( K ) ] .
exp ( i γ 1 Λ 1 ) exp ( i γ 2 Λ 2 ) exp { ( k ) 2 Re [ N 2 ( K ) ] } .
exp ( i γ 1 Λ 1 + i γ 2 Λ 2 ) = exp [ n = 1 i n m = 0 n ν n m , n ( n m ) ! m ! γ 1 n m γ 2 m ] ,
ν 1 , 0 = Λ 1 , ν 0 , 1 = Λ 2 , ν 2 , 0 = Λ 1 2 Λ 1 2 , ν 0 , 2 = Λ 2 2 Λ 2 2 ,
ν 1 , 1 = Λ 1 Λ 2 Λ 1 Λ 2 .
exp n = 1 i n m = 0 n ν n m , n ( n m ) ! m ! γ 1 n m γ 2 m exp { ( k ) 2 Re { N 2 ( K ) ] } × exp [ ( k ) 2 N 11 ( K i , K , R d ) ] ,
N 11 ( K i , K , R d ) = [ β ( K ) D ( R c + ½ R d ) i φ ( 1 ) ( R c + ½ R d ) ] × [ β ( K ) D ( R c + ½ R d ) + i φ ( 1 ) * ( R c + ½ R d ) ] = 1 ( 2 π ) 2 d 2 κ exp ( i κ R d ) | β ( K i + κ ) + β ( K ) | 2 W ( κ ) .
σ ( K , K i ) = k 2 2 ( K i ) π exp { ( k ) 2 Re [ N 2 ( K ) ] } × d 2 R d exp [ i ( K K i ) R d ] × { exp [ ( k ) 2 N 11 ( K i , K , R d ) ] 1 } .
σ B ( K i ) = k z 2 ( K i ) π exp { ( k ) 2 Re [ N 2 ( K i ) ] } × d 2 R d × exp [ 2 i K i R d ] { exp [ ( k ) 2 N 11 ( K i , R d ) ] 1 } ,
σ B ( K i ) = k z 2 ( K i ) π d 2 R d exp ( 2 i K i R d ) ( k ) 2 N 11 ( K i , R d ) = ( k ) 2 k z 2 ( K i ) π ( 2 π ) 2 d 2 R exp ( 2 i K i R d ) d 2 κ exp ( i κ R d ) × | β ( K i + κ ) + β ( K i ) | 2 W ( κ ) , σ B ( K i ) = k z 4 ( K i ) π 2 W ( 2 K i ) = k 4 cos 4 θ i π 2 W ( 2 k sin θ i , 0 ) .
β ( K i + κ ) = β ( K i ) + tan θ i κ x k 1 2 cos 3 θ i ( κ x k ) 2 1 2 cos θ i ( κ y k ) 2 + .
N 2 ( K i ) 4 β 2 ( K i ) = 4 cos 2 θ i
N 11 ( K i , R d ) 4 β 2 ( K i ) ( 2 π ) 2 d 2 κ exp ( i κ R d ) W ( κ ) cos 2 θ i C ( R d ) ,
σ B ( K i ) = k z 2 ( K i ) π exp [ 4 ( k ) 2 cos 2 θ i ] d 2 R d exp [ 2 i K i R d ] × { exp [ 4 ( k ) 2 cos 2 θ i C ( R d ) ] 1 } .

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