Abstract

An expression is derived for the cross correlation of the electric field scattered from a roughened metallic disk to two points in the far zone. Both exponential and Gaussian surface correlations are considered. In the calculation of the cross correlation, careful treatment is given to the terms involving the surface slope, and our new results are applicable to a broader range of surface roughness and observation angle than were earlier approximate methods. The cross correlation is then reduced to give the backscattering cross section of the disk as a function of the tilt or incident angle for both types of surface correlation. It is found that the cross section for the Gaussian surface falls off rapidly with increasing tilt angle. A particularly interesting new finding is that the cross section for the exponential surface actually increases under certain roughness conditions as the tilt angle is increased. While the effect of shadowing is not included in the analysis, we derive an expression showing the limiting value of the tilt angle below which shadowing effects are negligible. In illustration of these basic results, experiments are presented for the backscattering from roughened surfaces prepared by bead blasting and by ion-beam etching.

© 1992 Optical Society of America

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References

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  1. N. George, “Speckle from rough, moving objects,”J. Opt. Soc. Am. 66, 1182–1194 (1976).
    [CrossRef]
  2. N. George, A. C. Livanos, J. A. Roth, C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
    [CrossRef]
  3. R. J. Papa, J. F. Lennon, R. L. Taylor, “The variation of bistatic rough surface scattering cross section for a physical optics model,”IEEE Trans. Antennas Propag. AP-34, 1229–1237 (1986).
    [CrossRef]
  4. E. Bahar, “Scattering cross sections for random rough surfaces: full wave analysis,” Radio Sci. 16, 331–341 (1981).
    [CrossRef]
  5. E. Bahar, “Scattering cross sections for composite random surfaces: full wave analysis,” Radio Sci. 16, 1327–1335 (1981).
    [CrossRef]
  6. E. Bahar, M. A. Fitzwater, “Like- and cross-polarized scattering cross sections for random rough surfaces: theory and experiment,” J. Opt. Soc. Am. A 2, 2295–2303 (1985).
    [CrossRef]
  7. S. L. Broschat, E. I. Thorsos, A. Ishimaru, “A heuristic algorithm for the bistatic radar cross section of random rough surface scattering,”IEEE Trans. Geosci. Remote Sensing 28, 202–206 (1990).
    [CrossRef]
  8. L. G. Shirley, N. George, “Diffuser radiation patterns over a large dynamic range. 1: Strong diffusers,” Appl. Opt. 27, 1850–1861 (1988).
    [CrossRef] [PubMed]
  9. V. A. Hughes, “Diffraction theory applied to radio wave scattering from the lunar surface,” Proc. Phys. Soc. 80, 1117–1127(1962).
    [CrossRef]
  10. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  11. A. K. Fung, “Theory of radar scatter from rough surfaces, bistatic and monostatic, with applications to lunar radar return,”J. Geophys. Res. 69, 1063–1073 (1964).
    [CrossRef]
  12. T. Hagfors, “Backscattering from an undulating surface with applications to radar returns from the moon,”J. Geophys. Res. 69, 3779–3784 (1964).
    [CrossRef]
  13. J. C. Leader, “The relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-19, 786–788 (1971).
    [CrossRef]
  14. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vol. 2.
  15. G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).
  16. J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).
  17. T. Hagfors, “Relations between rough surfaces and their scattering properties as applied to radar astronomy,” in Radar Astronomy, J. V. Evans, T. Hagfors, eds. (McGraw-Hill, New York, 1968), Chap. 2.
  18. A. Stogryn, “Electromagnetic scattering from rough, finitely conducting surfaces,” Radio Sci. 2, 415–428 (1967).
  19. See, for example, W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).
  20. N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
    [CrossRef]
  21. K. J. Allardyce, N. George, “Diffraction analysis of rough reflective surfaces,” Appl. Opt. 26, 2364–2375 (1987).
    [CrossRef] [PubMed]

1990 (1)

S. L. Broschat, E. I. Thorsos, A. Ishimaru, “A heuristic algorithm for the bistatic radar cross section of random rough surface scattering,”IEEE Trans. Geosci. Remote Sensing 28, 202–206 (1990).
[CrossRef]

1988 (1)

1987 (1)

1986 (1)

R. J. Papa, J. F. Lennon, R. L. Taylor, “The variation of bistatic rough surface scattering cross section for a physical optics model,”IEEE Trans. Antennas Propag. AP-34, 1229–1237 (1986).
[CrossRef]

1985 (1)

1981 (2)

E. Bahar, “Scattering cross sections for random rough surfaces: full wave analysis,” Radio Sci. 16, 331–341 (1981).
[CrossRef]

E. Bahar, “Scattering cross sections for composite random surfaces: full wave analysis,” Radio Sci. 16, 1327–1335 (1981).
[CrossRef]

1976 (2)

N. George, A. C. Livanos, J. A. Roth, C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
[CrossRef]

N. George, “Speckle from rough, moving objects,”J. Opt. Soc. Am. 66, 1182–1194 (1976).
[CrossRef]

1974 (1)

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

1971 (1)

J. C. Leader, “The relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-19, 786–788 (1971).
[CrossRef]

1967 (1)

A. Stogryn, “Electromagnetic scattering from rough, finitely conducting surfaces,” Radio Sci. 2, 415–428 (1967).

1964 (2)

A. K. Fung, “Theory of radar scatter from rough surfaces, bistatic and monostatic, with applications to lunar radar return,”J. Geophys. Res. 69, 1063–1073 (1964).
[CrossRef]

T. Hagfors, “Backscattering from an undulating surface with applications to radar returns from the moon,”J. Geophys. Res. 69, 3779–3784 (1964).
[CrossRef]

1962 (1)

V. A. Hughes, “Diffraction theory applied to radio wave scattering from the lunar surface,” Proc. Phys. Soc. 80, 1117–1127(1962).
[CrossRef]

Allardyce, K. J.

Bahar, E.

E. Bahar, M. A. Fitzwater, “Like- and cross-polarized scattering cross sections for random rough surfaces: theory and experiment,” J. Opt. Soc. Am. A 2, 2295–2303 (1985).
[CrossRef]

E. Bahar, “Scattering cross sections for random rough surfaces: full wave analysis,” Radio Sci. 16, 331–341 (1981).
[CrossRef]

E. Bahar, “Scattering cross sections for composite random surfaces: full wave analysis,” Radio Sci. 16, 1327–1335 (1981).
[CrossRef]

Barrick, D. E.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Bowman, J. J.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Broschat, S. L.

S. L. Broschat, E. I. Thorsos, A. Ishimaru, “A heuristic algorithm for the bistatic radar cross section of random rough surface scattering,”IEEE Trans. Geosci. Remote Sensing 28, 202–206 (1990).
[CrossRef]

Davenport, W. B.

See, for example, W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Fitzwater, M. A.

Fung, A. K.

A. K. Fung, “Theory of radar scatter from rough surfaces, bistatic and monostatic, with applications to lunar radar return,”J. Geophys. Res. 69, 1063–1073 (1964).
[CrossRef]

George, N.

Hagfors, T.

T. Hagfors, “Backscattering from an undulating surface with applications to radar returns from the moon,”J. Geophys. Res. 69, 3779–3784 (1964).
[CrossRef]

T. Hagfors, “Relations between rough surfaces and their scattering properties as applied to radar astronomy,” in Radar Astronomy, J. V. Evans, T. Hagfors, eds. (McGraw-Hill, New York, 1968), Chap. 2.

Hughes, V. A.

V. A. Hughes, “Diffraction theory applied to radio wave scattering from the lunar surface,” Proc. Phys. Soc. 80, 1117–1127(1962).
[CrossRef]

Ishimaru, A.

S. L. Broschat, E. I. Thorsos, A. Ishimaru, “A heuristic algorithm for the bistatic radar cross section of random rough surface scattering,”IEEE Trans. Geosci. Remote Sensing 28, 202–206 (1990).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vol. 2.

Jain, A.

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

Krichbaum, C. K.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Leader, J. C.

J. C. Leader, “The relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-19, 786–788 (1971).
[CrossRef]

Lennon, J. F.

R. J. Papa, J. F. Lennon, R. L. Taylor, “The variation of bistatic rough surface scattering cross section for a physical optics model,”IEEE Trans. Antennas Propag. AP-34, 1229–1237 (1986).
[CrossRef]

Livanos, A. C.

N. George, A. C. Livanos, J. A. Roth, C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
[CrossRef]

Papa, R. J.

R. J. Papa, J. F. Lennon, R. L. Taylor, “The variation of bistatic rough surface scattering cross section for a physical optics model,”IEEE Trans. Antennas Propag. AP-34, 1229–1237 (1986).
[CrossRef]

Papas, C. H.

N. George, A. C. Livanos, J. A. Roth, C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
[CrossRef]

Root, W. L.

See, for example, W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Roth, J. A.

N. George, A. C. Livanos, J. A. Roth, C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
[CrossRef]

Ruck, G. T.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Senior, T. B. A.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Shirley, L. G.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Stogryn, A.

A. Stogryn, “Electromagnetic scattering from rough, finitely conducting surfaces,” Radio Sci. 2, 415–428 (1967).

Stuart, W. D.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

Taylor, R. L.

R. J. Papa, J. F. Lennon, R. L. Taylor, “The variation of bistatic rough surface scattering cross section for a physical optics model,”IEEE Trans. Antennas Propag. AP-34, 1229–1237 (1986).
[CrossRef]

Thorsos, E. I.

S. L. Broschat, E. I. Thorsos, A. Ishimaru, “A heuristic algorithm for the bistatic radar cross section of random rough surface scattering,”IEEE Trans. Geosci. Remote Sensing 28, 202–206 (1990).
[CrossRef]

Uslenghi, P. L. E.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Appl. Opt. (2)

Appl. Phys. (1)

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

R. J. Papa, J. F. Lennon, R. L. Taylor, “The variation of bistatic rough surface scattering cross section for a physical optics model,”IEEE Trans. Antennas Propag. AP-34, 1229–1237 (1986).
[CrossRef]

J. C. Leader, “The relationship between the Kirchhoff approach and small perturbation analysis in rough surface scattering theory,”IEEE Trans. Antennas Propag. AP-19, 786–788 (1971).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (1)

S. L. Broschat, E. I. Thorsos, A. Ishimaru, “A heuristic algorithm for the bistatic radar cross section of random rough surface scattering,”IEEE Trans. Geosci. Remote Sensing 28, 202–206 (1990).
[CrossRef]

J. Geophys. Res. (2)

A. K. Fung, “Theory of radar scatter from rough surfaces, bistatic and monostatic, with applications to lunar radar return,”J. Geophys. Res. 69, 1063–1073 (1964).
[CrossRef]

T. Hagfors, “Backscattering from an undulating surface with applications to radar returns from the moon,”J. Geophys. Res. 69, 3779–3784 (1964).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

N. George, A. C. Livanos, J. A. Roth, C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
[CrossRef]

Proc. Phys. Soc. (1)

V. A. Hughes, “Diffraction theory applied to radio wave scattering from the lunar surface,” Proc. Phys. Soc. 80, 1117–1127(1962).
[CrossRef]

Radio Sci. (3)

E. Bahar, “Scattering cross sections for random rough surfaces: full wave analysis,” Radio Sci. 16, 331–341 (1981).
[CrossRef]

E. Bahar, “Scattering cross sections for composite random surfaces: full wave analysis,” Radio Sci. 16, 1327–1335 (1981).
[CrossRef]

A. Stogryn, “Electromagnetic scattering from rough, finitely conducting surfaces,” Radio Sci. 2, 415–428 (1967).

Other (6)

See, for example, W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vol. 2.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

T. Hagfors, “Relations between rough surfaces and their scattering properties as applied to radar astronomy,” in Radar Astronomy, J. V. Evans, T. Hagfors, eds. (McGraw-Hill, New York, 1968), Chap. 2.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

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Figures (7)

Fig. 1
Fig. 1

Three disks tilted at various angles with respect to the illumination: a, zero; b, 20°; c, 40°; and d, 70°. The large disk has a diffuse white surface. The smaller two are 1-in.-diameter silver disks with rms roughness of 2 and 8 μm for the upper and lower disks, respectively.

Fig. 2
Fig. 2

Roughened disk in the plane z = 0 illuminated by a plane wave incident at an angle θi, as shown, with the scattering to be calculated at the observation point P.

Fig. 3
Fig. 3

Coordinate transformation for points D′(x′, y′) and D″(x″, y″) shown in Fig. 2.

Fig. 4
Fig. 4

Backscattering cross section of a roughened disk versus incident angle. The curves are for lc/λ = 100 and for a Gaussian surface correlation.

Fig. 5
Fig. 5

Backscattering cross section of a roughened disk for an exponential surface correlation. Again, lc/λ = 100.

Fig. 6
Fig. 6

Backscattering cross-section measurements versus incident angle for the three disks shown in Fig. 1.

Fig. 7
Fig. 7

Graphs of Eqs. (B6) and (B8) for the backscattering cross-section results for the integration-by-parts method: (a) Gaussian surface correlation; (b) exponential correlation.

Equations (86)

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H inc = H 0 ( cos θ i e x - sin θ i e z ) × exp [ i ( ω t + k sin θ i x + k cos θ i z ) ] ,
E inc = e y E 0 exp [ i ( ω t + k sin θ i x + k cos θ i z ) ] .
× A = μ H .
2 A - ɛ μ 2 A t 2 = - μ J ,
A ( x , y , z ; t ) = μ 4 π v d v [ J ] R ,
A = μ 0 4 π s d s [ K ] R ,
R 2 = ( x - x ) 2 + ( y - y ) 2 + ( z - h ) 2 .
K = n × H = 2 n × H inc ,
H ( x , y , z ; t ) = 1 2 π × s n × H inc ( x , y , h ; t - R / c ) 1 R d s .
exp ( - i k R ) R = - ( 1 R + i k ) exp ( - i k R ) R R ^ ,
n d s = ( - h x x ^ - h y y ^ + z ^ ) d x d y .
R R 0 - x x + y y R 0 - h cos θ ,
E y ( x , y , z ; λ ) = - i k 2 π R 0 μ 0 / ɛ 0 H 0 exp ( - i k R 0 ) × d x d y [ x 2 + z 2 R 0 2 ( cos θ i - h x sin θ i ) - ( x y R 0 2 sin θ i + y z R 0 2 cos θ i ) h y ] × exp { i k [ x ( sin θ cos ϕ + sin θ i ) + y sin θ sin ϕ + h ( cos θ + cos θ i ) ] } .
R 12 ( x 1 , y 1 , z 1 ; λ 1 : x 2 , y 2 , z 2 ; λ 2 ) = E y ( x 1 , y 1 , z 1 ; λ 1 ) E y * ( x 2 , y 2 , z 2 ; λ 2 ) .
g = - d u f ( u ) g ( u ) ,
A = - i k E 0 2 π R 0 exp ( - i k R 0 ) ,
b = x 2 + z 2 R 0 2 cos θ i = ( sin 2 θ cos 2 ϕ + cos 2 θ ) cos θ i ,
c = x 2 + z 2 R 0 2 sin θ i = ( sin 2 θ cos 2 ϕ + cos 2 θ ) sin θ i ,
d = x y R 0 2 sin θ i + y z R 0 2 cos θ i = sin 2 θ cos ϕ sin ϕ sin θ i + sin θ sin ϕ cos θ cos θ i ,
ξ = k ( sin θ cos ϕ + sin θ i ) ,
ζ = k sin θ sin ϕ ,
η = k ( cos θ + cos θ i )
R 12 = A 1 A 2 * d x d x d y d y × ( b 1 - c 1 h x - d 1 h y ) ( b 2 - c 2 h x - d 2 h y ) × exp { i [ η 1 h - η 2 h ) ] } × exp [ i ( ξ 1 x - ξ 2 x + ζ 1 y - ζ 2 y ) ] ,
σ BS = 4 π R 0 2 E y ( x , y , z ; λ ) E y * ( x , y , z ; λ ) E 0 2 ,
σ BS = 4 π R 0 2 R 11 ( x , y , z ; λ ) E 0 2 ,
= ( b 1 b 2 - b 1 c 2 h x - b 1 d 2 h y - b 2 c 2 h x - b 2 d 1 h y + c 1 c 2 h x h x + c 1 d 2 h x h y + c 2 d 1 h x h y + d 1 d 2 h y h y ) exp [ i ( η 1 h - η 2 h ) ] .
E 2 ( η 1 , - η 2 ; r ) = exp [ i ( η 1 h - η 2 h ) ] = exp [ - 1 2 ( η 1 h - η 2 h ) 2 ] .
r = h h σ 2 ,
f ( u ) = 1 ( 2 π ) n / 2 C 1 / 2 exp [ - 1 2 ( u - u ) t C - 1 ( u - u ) ] ,
C i k = ( u i - u i ) ( u k - u k ) .
h x exp ( i Δ h ˜ ) = i h x Δ h ˜ F 2 ( η 1 , - η 2 ; r ) .
h x h x exp ( i Δ h ˜ ) = ( h x h x - h x Δ h ˜ h x Δ h ˜ ) F 2 ( η 1 , - η 2 ; r ) .
( Δ h ˜ ) 2 = ( η 1 h - η 2 h ) 2 = σ 2 [ η 1 2 + η 2 2 - 2 η 1 η 2 r ( ρ ) ] ,
h x h x = - σ 2 [ ( x - x ρ ) 2 d 2 r d ρ 2 + ( y - y ) 2 ρ 3 d r d ρ ] ,
h x Δ h ˜ = - σ 2 η 2 d r d ρ ( x - x ρ ) ,
h x Δ h ˜ = σ 2 η 1 d r d ρ ( x - x ρ ) ,
h x h y = h x h y = σ 2 ( x - x ρ ) ( y - y ρ ) ( 1 ρ d r d ρ - d 2 r d ρ 2 ) .
= { b 1 b 2 - i σ 2 ( b 1 c 2 η 1 - b 2 c 1 η 2 ) d r d ρ ( x - x ρ ) - i σ 2 ( b 1 d 2 η 1 - b 2 d 1 η 2 ) d r d ρ ( y - y ρ ) - c 1 c 2 σ 2 [ ( x - x ρ ) 2 d 2 r d ρ 2 + ( y - y ) 2 ρ 3 d r d ρ ] + c 1 c 2 η 1 η 2 σ 4 ( d r d ρ ) 2 ( x - x ρ ) 2 - d 1 d 2 σ 2 [ ( y - y ρ ) 2 d 2 r d ρ 2 + ( x - x ) 2 ρ 3 d r d ρ ] + d 1 d 2 η 1 η 2 σ 4 ( d r d ρ ) 2 ( y - y ρ ) 2 + ( c 1 d 2 + c 2 d 1 ) σ 2 [ 1 ρ d r d ρ - d 2 r d ρ 2 + η 1 η 2 σ 2 ( d r d ρ ) 2 ] × ( x - x ρ ) ( y - y ρ ) } F 2 ( η 1 , - η 2 ; r ) .
x - x = - ρ cos α ,
y - y = - ρ sin α ,
R 12 = A 1 A 2 * d x d y exp { i [ x ( ξ 1 - ξ 2 ) + y ( ζ 1 - ζ 2 ) ] } × d ρ d α ρ { b 1 b 2 + i σ 2 d r d ρ [ ( b 1 c 2 η 1 - b 2 c 1 η 2 ) × cos α + ( b 1 d 2 η 1 - b 2 d 1 η 2 ) sin α ] - c 1 c 2 σ 2 ( cos 2 α d 2 r d ρ 2 + sin 2 α 1 ρ d r d ρ ) + c 1 c 2 η 1 η 2 σ 4 ( d r d ρ ) 2 cos 2 α - d 1 d 2 σ 2 ( sin 2 α d 2 r d ρ 2 + cos 2 α 1 ρ d r d ρ ) + d 1 d 2 η 1 η 2 σ 4 ( d r d ρ ) 2 sin 2 α + ( c 1 d 2 + c 2 d 1 ) σ 2 × [ 1 ρ d r d ρ - d 2 r d ρ 2 + η 1 η 2 σ 2 ( d r d ρ ) 2 ] cos α sin α } × F 2 ( η 1 , - η 2 ; r ) exp [ - i ( ρ ξ 2 cos α + ρ ζ 2 sin α ) ] .
β = σ 2 η 1 η 2 ,
γ = l c ( ξ 2 2 + ζ 2 2 ) 1 / 2 ,
R 12 = A 1 A 2 * ( π a ) 2 Jinc { a [ ( ξ 1 - ξ 2 ) 2 + ( ζ 1 - ζ 2 ) 2 ] 1 / 2 } × 0 d ρ ρ { 2 b 1 b 2 J 0 ( ρ γ / l c ) - 2 σ 2 ( ξ 2 2 + ζ 2 2 ) 1 / 2 d r d ρ × [ ( b 1 c 2 η 1 - b 2 c 1 η 2 ) ξ 2 + ( b 1 d 2 η 1 - b 2 d 1 η 2 ) ζ 2 ] J 1 ( ρ γ / l c ) - σ 2 [ c 1 c 2 d 2 r d ρ 2 - c 1 c 2 β ( d r d ρ ) 2 + d 1 d 2 1 ρ d r d ρ ] × [ J 0 ( ρ γ / l c ) - ξ 2 2 - ζ 2 2 ξ 2 2 + ζ 2 2 J 2 ( ρ γ / l c ) ] - σ 2 [ d 1 d 2 d 2 r d ρ 2 - d 1 d 2 β ( d r d ρ ) 2 + c 1 c 2 1 ρ d r d ρ ] × [ J 0 ( ρ γ / l c ) + ξ 2 2 - ζ 2 2 ξ 2 2 + ζ 2 2 J 2 ( ρ γ / l c ) ] - σ 2 ( c 1 d 2 + c 2 d 1 ) [ 1 ρ d r d ρ - d 2 r d ρ 2 + β ( d r d ρ ) 2 ] × 2 ξ 2 ζ 2 ξ 2 2 + ζ 2 2 J 2 ( ρ γ / l c ) } F 2 ( η 1 , - η 2 ; r ) .
r G ( ρ ) = exp ( - ρ 2 / l c 2 ) ,
r E ( ρ ) = exp ( - ρ / l c ) .
R 12 G = A 1 A 2 * ( π a ) 2 l c 2 Jinc { a [ ( ξ 1 - ξ 2 ) 2 + ( ζ 1 - ζ 2 ) 2 ] 1 / 2 } × exp [ - 1 2 σ 2 ( η 1 - η 2 ) 2 ] × { b 1 b 2 β exp ( - γ 2 4 β ) + σ 2 ( β + 1 ) 2 [ ( b 1 c 2 η 1 - b 2 c 1 η 2 ) ξ 2 + ( b 1 d 2 η 1 - b 2 d 1 η 2 ) ζ 2 ] exp [ - γ 2 4 ( β + 1 ) ] + 2 σ 2 l c 2 ( c 1 c 2 + d 1 d 2 ) [ β ( β + 1 ) 2 + γ 2 4 ( β + 1 ) 3 ] × exp [ - γ 2 4 ( β + 1 ) ] + σ 2 2 ( β + 1 ) 3 × [ c 1 c 2 - d 1 d 2 ) ( ξ 2 2 - ζ 2 2 ) + 2 ( c 1 d 2 + c 2 d 1 ) ξ 2 ζ 2 ] × exp [ - γ 2 4 ( β + 1 ) ] + 2 σ 2 l c 2 ( c 1 c 2 + d 1 d 2 ) β × [ 1 ( β + 2 ) 2 - γ 2 4 ( β + 2 ) 3 ] exp [ - γ 2 4 ( β + 2 ) ] - σ 2 β 2 ( β + 2 ) 3 [ ( c 1 c 2 - d 1 d 2 ) ( ξ 2 2 - ζ 2 2 ) + 2 ( c 1 d 2 + c 2 d 1 ) ξ 2 ζ 2 ] exp [ - γ 2 4 ( β + 2 ) ] } .
σ BSG π a 2 = ( l c 2 σ ) 2 exp ( - γ 2 4 β ) + 2 k 2 σ 2 × sin 2 θ i { 1 ( β + 1 ) 2 [ β + γ 2 2 ( β + 1 ) ] × exp [ - γ 2 4 ( β + 1 ) ] + β ( β + 2 ) 2 [ 1 - γ 2 2 ( β + 2 ) ] × exp [ - γ 2 4 ( β + 2 ) ] } .
β = ( 4 π σ λ cos θ i ) 2 ,
γ = 4 π l c λ sin θ i .
σ BSG π a 2 = ( l c 2 σ ) 2 ,
R 12 E = A 1 A 2 * ( π a ) 2 l c 2 Jinc { a [ ξ 1 - ξ 2 ) 2 + ( ζ 1 - ζ 2 ) 2 ] 1 / 2 } × exp [ - 1 2 σ 2 ( η 1 - η 2 ) 2 ] [ 2 b 1 b 2 β ( β 2 + γ 2 ) 3 / 2 + 2 σ 2 [ ( b 1 c 2 η 1 - b 2 c 1 η 2 ) ξ 2 + ( b 1 d 2 η 1 - b 2 d 1 η 2 ) ζ 2 ] × ( ξ 2 2 + ζ 2 2 ) 1 / 2 [ ( β + 1 ) 2 + γ 2 ] 3 / 2 + σ 2 l c 2 ( c 1 c 2 + d 1 d 2 ) × { β ( β + 1 ) + γ 2 [ ( β + 1 ) 2 + γ 2 ] 3 / 2 + β ( β + 2 ) [ ( β + 2 ) 2 + γ 2 ] 3 / 2 } + σ 2 γ 2 [ ( c 1 c 2 - d 1 d 2 ) ( ξ 2 2 - ζ 2 2 ) + 2 ( c 1 d 2 + c 2 d 1 ) ξ 2 ζ 2 ] × ( { [ ( β + 1 ) 2 + γ 2 ] 1 / 2 - ( β + 1 ) } 2 γ 2 [ ( β + 1 ) 2 + γ 2 ] 3 / 2 × { 2 [ ( β + 1 ) 2 + γ 2 ] 1 / 2 + ( β + 1 ) ( β + 2 ) + γ 2 } ) - σ 2 β γ 2 [ ( c 1 c 2 - d 1 d 2 ) ( ξ 2 2 - ζ 2 2 ) + 2 ( c 1 d 2 + c 2 d 1 ) ξ 2 ζ 2 ] × ( { [ ( β + 2 ) 2 + γ 2 ] 1 / 2 - ( β + 2 ) } 2 γ 2 [ ( β + 2 ) 2 + γ 2 ] 3 / 2 × { 2 [ ( β + 2 ) 2 + γ 2 ] 1 / 2 + ( β + 2 ) } ) ] ,
σ BSG π a 2 = 2 k 2 l c 2 cos 2 θ i β ( β 2 + γ 2 ) 3 / 2 + k 2 σ 2 × sin 2 θ i ( β ( β + 1 ) + γ 2 [ ( β + 1 ) 2 + γ 2 ] 3 / 2 + β ( β + 2 ) [ ( β + 2 ) 2 + γ 2 ] 3 / 2 + { [ ( β + 1 ) 2 + γ 2 ] 1 / 2 - ( β + 1 ) 2 } γ 2 [ ( β + 1 ) 2 + γ 2 ] 3 / 2 × { 2 [ ( β + 1 ) 2 + γ 2 ] 1 / 2 + ( β + 1 ) ( β + 2 ) + γ 2 } - β { [ ( β + 2 ) 2 + γ 2 ] 1 / 2 - ( β + 2 ) } 2 γ 2 [ ( β + 2 ) 2 + γ 2 ] 3 / 2 × { 2 [ ( β + 2 ) 2 + γ 2 ] 1 / 2 + ( β + 2 ) } ) ,
σ BSE π a 2 = l c 2 8 k 2 σ 4 ,
h x h x exp ( i Δ h ˜ ) = d h x d h x d Δ h ˜ h x h x exp ( i Δ h ˜ ) f ( h x , h x , Δ h ˜ ) .
u = [ h x h x Δ h ˜ ]
C = [ h x 2 h x h x h x Δ h ˜ h x h x h x 2 h x Δ h ˜ h x Δ h ˜ h x Δ h ˜ Δ h ˜ 2 ] .
C 11 - 1 = 1 C ( h x 2 Δ h ˜ 2 - h x Δ h ˜ 2 2 ) ,
C 12 - 1 = C 21 - 1 = 1 C ( h x Δ h ˜ h x Δ h ˜ - h x h x Δ h ˜ 2 ) ,
C 13 - 1 = C 31 - 1 = 1 C ( h x h x h x Δ h ˜ - h x 2 h x Δ h ˜ ) ,
C 22 - 1 = 1 C ( h x 2 Δ h ˜ 2 - h x Δ h ˜ 2 ) ,
C 23 - 1 = C 32 - 1 = 1 C ( h x h x h x Δ h ˜ - h x 2 h x Δ h ˜ ) ,
C 33 - 1 = 1 C ( h x 2 2 - h x h x 2 ) .
f ( h x , h x , Δ h ˜ ) = 1 ( 2 π ) 3 / 2 C 1 / 2 × exp [ - ½ ( C 11 - 1 h x 2 + C 22 - 1 h x 2 + C 33 - 1 Δ h ˜ 2 + 2 C 12 - 1 h x h x + 2 C 13 - 1 h x Δ h ˜ + 2 C 23 - 1 h x Δ h ˜ ) ] .
h x h x exp ( i Δ h ˜ ) = 1 ( 2 π ) 3 / 2 C 1 / 2 - d h x d h x d Δ h ˜ h x h x exp ( i Δ h ˜ ) × exp [ - ½ ( C 11 - 1 h x 2 + C 22 - 1 h x 2 + C 33 - 1 Δ h ˜ 2 + 2 C 12 - 1 h x h x + 2 C 13 - 1 h x Δ h ˜ + 2 C 23 - 1 h x Δ h ˜ ) ] .
L = - d h x h x × exp { - ½ [ C 22 - 1 h x 2 + 2 h x ( C 12 - 1 h x + C 23 - 1 Δ h ˜ ) ] } ,
L = - 2 π C 22 - 1 ( C 12 - 1 h x + C 23 - 1 Δ h ˜ C 22 - 1 ) × exp [ 1 2 C 22 - 1 ( C 12 - 1 h x + C 23 - 1 Δ h ˜ ) 2 ] .
M = - 2 π C 22 - 1 exp [ ( C 23 - 1 Δ h ˜ ) 2 / 2 C 22 - 1 ] × - d h x h x ( C 12 - 1 h x + C 23 - 1 Δ h ˜ C 22 - 1 ) × exp [ ( C 12 - 1 h x ) 2 / 2 C 22 - 1 ] × exp ( C 12 - 1 C 23 - 1 h x Δ h ˜ / C 22 - 1 ) × exp [ - ½ ( C 11 - 1 h x 2 + 2 C 13 - 1 h x Δ h ˜ ) ] .
M = - 2 π ( ζ C 22 - 1 ) 1 / 2 C 12 - 1 C 22 - 1 [ 1 ζ + ( X 3 - X C 23 - 1 C 12 - 1 ) Δ h ˜ 2 ] × exp [ 1 2 ( ( C 23 - 1 ) 2 C 22 - 1 + ζ X 2 ) Δ h ˜ 2 ] ,
ζ = C 11 - 1 - ( C 12 - 1 ) 2 C 22 - 1
X = C 13 - 1 - C 12 - 1 C 23 - 1 / C 22 - 1 ζ .
N = - 2 π ( ζ C 22 - 1 ) 1 / 2 C 12 - 1 C 22 - 1 × - d Δ h ˜ [ 1 ζ + ( X 2 - X C 23 - 1 C 12 - 1 ) Δ h ˜ 2 ] × exp { - 1 2 [ C 33 - 1 - ( C 23 - 1 ) 2 C 22 - 1 - ζ X 2 ] Δ h ˜ 2 } ,
N = - ( 2 π ) 3 / 2 ( ζ Ψ C 22 - 1 ) C 12 - 1 C 22 - 1 × [ 1 ζ + 1 Ψ ( X 2 - X C 23 - 1 C 12 - 1 ) ( 1 - 1 Ψ ) ] exp ( - 1 2 Ψ ) ,
Ψ = C 33 - 1 - ( C 23 - 1 ) 2 C 22 - 1 - ζ X 2 .
ζ = Δ h ˜ 2 h x 2 Δ h ˜ 2 - h x Δ h ˜ 2 ,
X = - h x Δ h ˜ Δ h ˜ 2 ,
Ψ = 1 Δ h ˜ 2 .
h x h x exp ( i Δ h ˜ ) = ( h x h x - h x Δ h ˜ h x Δ h ˜ ) exp ( - ½ Δ h ˜ 2 ) .
E y ( x , y , z ; λ ) = A d x d y ( b - c h x - d h y ) × exp [ i ( ξ x + ζ y + η h ) ] .
E y ( x , y , z ; λ ) = A ( ( b - c ξ + d ζ η ) d x d y × exp [ i ( ξ x + ζ y + η h ) ] - c i η d y { exp [ i ( ξ x + ζ y + η h ) ] } x limits - d i η d x { exp [ i ( ξ x + ζ y + η h ) ] } y limits ) ,
R 12 = A 1 A 2 * ( b 1 - c 1 ξ 1 + d 1 ζ 1 η 1 ) ( b 2 - c 2 ξ 2 + d 2 ζ 2 η 2 ) × d x d y d x d y × exp [ i ( ξ 1 x - ξ 2 x + ζ 1 y - ζ 2 y ) ] F 2 ( η 1 , - η 2 ; r ) .
R 12 = 2 A 1 A 2 * ( π a ) 2 Jinc { a [ ( ξ 1 - ξ 2 ) 2 + ( ζ 1 - ζ 2 ) 2 ] 1 / 2 } × ( b 1 - c 1 ξ 1 + d 1 ζ 1 η 1 ) ( b 2 - c 2 ξ 2 + d 2 ζ 2 η 2 ) × - d ρ ρ J 0 ( ρ γ / l c ) F 2 ( η 1 , - η 2 ; r ) .
R 12 G = A 1 A 2 * ( π a ) 2 l c 2 Jinc { a [ ( ξ 1 - ξ 2 ) 2 + ( ζ 1 - ζ 2 ) 2 ] 1 / 2 } × ( b 1 - c 1 ξ 1 + d 1 ζ 1 η 1 ) ( b 2 - c 2 ξ 2 + d 2 ζ 2 η 2 ) × exp [ - 1 2 σ 2 ( η 1 - η 2 ) 2 ] 1 β exp ( - γ 2 4 β ) ,
σ BSG π a 2 = ( l c 2 σ cos 2 θ i ) 2 exp ( - γ 2 4 β ) .
R 12 E = 2 A 1 A 2 * ( π a ) 2 l c 2 Jinc { a [ ( ξ 1 - ξ 2 ) 2 + ( ζ 1 - ζ 2 ) 2 ] 1 / 2 } × ( b 1 - c 1 ξ 1 + d 1 ζ 1 η 1 ) ( b 2 - c 2 ξ 2 + d 2 ζ 2 η 2 ) × exp [ - 1 2 σ 2 ( η 1 - η 2 ) 2 ] β ( β 2 + γ 2 ) 3 / 2 ,
σ BSE π a 2 = 2 k 2 l c 2 1 cos 2 θ i β ( β 2 + γ 2 ) 3 / 2 .

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