Abstract

We study theoretically the Stokes matrix of a perfectly conducting, one-dimensional rough surface that is illu-by a polarized light beam of finite width whose plane of incidence is perpendicular to the grooves of the minated surface. An exact expression for the scattered field derived from Green’s second integral theorem is used to the angular distribution of the Stokes matrix that has eight nonzero elements, four of which are compute unique. Results are presented for the numerical calculation of each matrix element averaged over an ensemble of surface profiles that are realizations of a stationary, Gaussian stochastic process. All four unique matrix elements are significant, with the diagonal elements displaying enhanced backscattering and the off-diagonal elements having complicated angular dependences including structures in the retroreflection direction. With the use of a single source function evaluated through the iteration of the surface integral equation obtained from the extinction theorem for the p-polarized field, we derive an approximate expression for the Stokes matrix that indicates that multiple scattering plays an important role in the polarized scattering from a perfectly conducting rough surface that displays enhanced backscattering. The numerical calculation of each of the con-to the Stokes matrix, taking into account single-, double-, and triple-scattering processes, enables us tributions to assign the main features of the Stokes matrix to particular multiple-scattering processes. Experimental measurements of the matrix elements are presented for a one-dimensional Gaussian surface fabricated in gold-photoresist. The results are found to be reasonably consistent with the theory, although we suggest that coated differences in one matrix element may be due to the finite conductivity of the experimental surface.

© 1992 Optical Society of America

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  1. E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  2. K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  3. M.-J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
    [CrossRef]
  4. A. J. Sant, J. C. Dainty, M.-J. Kim, “Comparison of surface scattering between identical, randomly rough metal and dielectric diffusers,” Opt. Lett. 14, 1183–1185 (1989).
    [CrossRef] [PubMed]
  5. M. E. Knotts, K. A. O’Donnell, “Anomalous scattering from a perturbed grating,” Opt. Lett. 15, 1485–1487 (1990).
    [CrossRef] [PubMed]
  6. K. A. O’Donnell, M. E. Knotts, “The polarization dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
    [CrossRef]
  7. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [CrossRef]
  8. Y.-Q. Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
    [CrossRef]
  9. A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Media 1, 21–34 (1991).
    [CrossRef]
  10. J. S. Chen, A. Ishimaru, “Numerical simulation of the second-order Kirchhoff approximation from very rough surfaces and a study of backscattering enhancement,”J. Acoust. Soc. Am. 88, 1846–1850 (1990).
    [CrossRef]
  11. M. Nieto-Vesperinas, J. M. Soto-Crespo, “Monte-Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces,” Opt. Lett. 12, 979–981 (1987).
    [CrossRef] [PubMed]
  12. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep reflection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
    [CrossRef]
  13. A. A. Maradudin, E. R. Mendez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
    [CrossRef] [PubMed]
  14. A. A. Maradudin, E. R. Méndez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random grating,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.
  15. W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).
  16. E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
    [CrossRef]
  17. E. Hecht, “Note on an operational definition of the Stokes parameters,” Am. J. Phys. 38, 1156–1158 (1970).
    [CrossRef]
  18. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 269–278.
  19. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 554.
  20. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  21. 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]
  22. P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
    [CrossRef]
  23. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).
  24. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 24.

1991

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

K. A. O’Donnell, M. E. Knotts, “The polarization dependence of scattering from one-dimensional rough surfaces,” J. Opt. Soc. Am. A 8, 1126–1131 (1991).
[CrossRef]

1990

M.-J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
[CrossRef]

M. E. Knotts, K. A. O’Donnell, “Anomalous scattering from a perturbed grating,” Opt. Lett. 15, 1485–1487 (1990).
[CrossRef] [PubMed]

J. S. Chen, A. Ishimaru, “Numerical simulation of the second-order Kirchhoff approximation from very rough surfaces and a study of backscattering enhancement,”J. Acoust. Soc. Am. 88, 1846–1850 (1990).
[CrossRef]

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

Y.-Q. Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[CrossRef]

1989

1988

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]

1987

1982

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[CrossRef]

1978

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

1970

E. Hecht, “Note on an operational definition of the Stokes parameters,” Am. J. Phys. 38, 1156–1158 (1970).
[CrossRef]

1956

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 24.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 554.

Chen, J. S.

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

J. S. Chen, A. Ishimaru, “Numerical simulation of the second-order Kirchhoff approximation from very rough surfaces and a study of backscattering enhancement,”J. Acoust. Soc. Am. 88, 1846–1850 (1990).
[CrossRef]

Dainty, J. C.

Friberg, A. T.

Gray, P. F.

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

Hecht, E.

E. Hecht, “Note on an operational definition of the Stokes parameters,” Am. J. Phys. 38, 1156–1158 (1970).
[CrossRef]

Ishimaru, A.

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

J. S. Chen, A. Ishimaru, “Numerical simulation of the second-order Kirchhoff approximation from very rough surfaces and a study of backscattering enhancement,”J. Acoust. Soc. Am. 88, 1846–1850 (1990).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 269–278.

Jin, Y.-Q.

Y.-Q. Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[CrossRef]

Kim, M.-J.

Knotts, M. E.

Lax, M.

Y.-Q. Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[CrossRef]

Liszka, E. G.

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[CrossRef]

Maradudin, A. A.

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

A. A. Maradudin, E. R. Mendez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef] [PubMed]

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

McCoy, J. J.

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[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. 203, 255–307 (1990).
[CrossRef]

Meecham, W. C.

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).

Mendez, E. R.

Méndez, E. R.

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

K. A. O’Donnell, E. R. Méndez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

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

Michel, T.

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

A. A. Maradudin, E. R. Mendez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic grating,” Opt. Lett. 14, 151–153 (1989).
[CrossRef] [PubMed]

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

Nieto-Vesperinas, M.

O’Donnell, K. A.

Sant, A. J.

Soto-Crespo, J. M.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 24.

Thorsos, E. I.

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]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 554.

Am. J. Phys.

E. Hecht, “Note on an operational definition of the Stokes parameters,” Am. J. Phys. 38, 1156–1158 (1970).
[CrossRef]

Ann. Phys.

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

J. Acoust. Soc. Am.

J. S. Chen, A. Ishimaru, “Numerical simulation of the second-order Kirchhoff approximation from very rough surfaces and a study of backscattering enhancement,”J. Acoust. Soc. Am. 88, 1846–1850 (1990).
[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]

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[CrossRef]

J. Opt. Soc. Am. A

J. Rat. Mech. Anal.

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).

Opt. Acta

P. F. Gray, “A method of forming optical diffusers of simple known statistical properties,” Opt. Acta 25, 765–775 (1978).
[CrossRef]

Opt. Commun.

E. R. Méndez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Opt. Lett.

Phys. Rev. B

Y.-Q. Jin, M. Lax, “Backscattering enhancement from a randomly rough surface,” Phys. Rev. B 42, 9819–9829 (1990).
[CrossRef]

Waves Random Media

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

Other

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

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), pp. 269–278.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 554.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), p. 24.

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Figures (11)

Fig. 1
Fig. 1

Geometry of scattering by a one-dimensional rough surface. k ^ inc = sin θ 0 x ^ 1 - cos θ 0 x ^ 3, and k ^ sc = sin θ s x ^ 1 - cos θ s x ^ 3.

Fig. 2
Fig. 2

The four unique elements of the Stokes matrix, S11(θs), S12(θs), S33(θs), and S34(θs), for a one-dimensional, perfectly conducting rough surface at normal incidence. δ/a = 0.6, L/g = 4, a/λ = 3, g/λ = 20; Nx = 600, Np = 4000.

Fig. 3
Fig. 3

The four unique elements of the Stokes matrix S11(θs), S12 (θs), S33(θs), and S34(θs) for the same parameters as in Fig. 2 but with θ0 = 10°.

Fig. 4
Fig. 4

The four unique elements of the Stokes matrix, S11(θs), S12(θs), S33(θs), and S34(θs), for a one-dimensional, perfectly conducting rough surface at normal incidence. δ/a = 0.6, L/g = 4, a = λ, g/λ = 20; Nx = 600, Np = 4000.

Fig. 5
Fig. 5

The four unique elements of the Stokes matrix, S11(θs), S12(θs), S33(θs), and S34(θs), for the same parameters as in Fig. 4 but with θ0 = 10°.

Fig. 6
Fig. 6

Scattered powers I+(θs) and I(θs) for a +45° linearly polarized field incident at normal incidence upon a one-dimensional, perfectly conducting rough surface. δ/a = 0.6, L/g = 4, a/λ = 3, g/λ = 20; Nx = 600, Np = 4000.

Fig. 7
Fig. 7

(a)–(c) The eight contributions to the Stokes matrix elements computed with the single-, the double-, and the triple-scattering amplitudes for a one-dimensional, perfectly conducting rough surface at normal incidence. δ/a = 0.6, L/g = 4, a/λ = 3 g/λ = 20; Nx = 600, Np = 4000. A, 2C〈|Δrp(1)(θs)|2〉2; B, 2C〈|Δrp(2)(θs)|2ψ;C, 2C〈|Δrp(3)(θs)|2〉; D, 4Cre〈Δrp(2)(θsrp(3)*(θs)〉; E, 4Ce〈Δrp(1)*(θsrp(2)(θs)〉; F, 4Ce〈Δrp(2)(θsrp(3)(θs)〉; G, 4C Fmrp(1)*(θsrp(2)(θs)〉; H, 4C Fm〈Δrp(2)(θs) Δrp(3)*(θs)〉.

Fig. 8
Fig. 8

Scattering instrument used in the measurements of rough surface scattering. The incident beam is brought over the detector by a series of mirrors and, after orientation of the polarization direction, is incident upon the rough surface. The detector arm may be moved out of the plane of the figure to measure the intensity at all the scattering angles. M denotes a mirror, P denotes a linear, polarizer, and W denotes a half-wave plate.

Fig. 9
Fig. 9

Measured values of the four unique elements of the Stokes matrix, S11(θs), S12(θs), S33(θs) and S34(θs) for the one- dimensional surface at normal incidence.

Fig. 10
Fig. 10

Measured values of the four unique elements of the Stokes matrix, S11(θs), S12(θs), S33(θs), and S34(θs), for the one-dimensional surface; θ0 = 10°.

Fig. 11
Fig. 11

Measured values of the four unique element of the Stokes matrix, S11(θs), S12(θs), S33(θs), and S34(θs), for the one-dimensional surface; θ0 = 30°.

Equations (74)

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E ( x 1 , x 3 ω ) inc = ω w 2 π c - π / 2 π / 2 d θ exp [ - ( ω 2 w 2 / 4 c 2 ) ( θ - θ 0 ) 2 ] × [ E p e ^ p ( θ ) inc + E s e ^ p ( θ ) inc ] exp [ i ( ω / c ) ( x 1 sin θ - x 3 cos θ ) ] , θ 0 < π 2 ,
e ^ s ( θ ) inc = x ^ 2 ,
e ^ p ( θ ) inc = - x ^ 1 cos θ - x ^ 3 sin θ ,
E ( x 1 , x 3 ω ) inc = i c ω × [ E p ϕ ( x 1 , x 3 ω ) inc x ^ 2 ] + E s ϕ ( x 1 , x 3 ω ) inc x ^ 2 ,
ϕ ( x 1 , x 3 ω ) inc = ω w 2 π c - π / 2 π / 2 d θ exp [ - ( ω 2 w 2 / 4 c 2 ) ( θ - θ 0 ) 2 ] × exp [ i ( ω / c ) ( x 1 sin θ - x 3 cos θ ) ] ,
E ( x 1 , x 3 ω ) sc = i 4 π - π / 2 π / 2 d θ s × [ E p r p ( θ s ) e ^ p ( θ s ) sc + E s r s ( θ s ) e ^ s ( θ s ) sc ] × exp [ i ( ω / c ) ( x 1 sin θ s + x 3 cos θ s ) ] ,
e ^ s ( θ s ) sc = x ^ 2 ,
e ^ p ( θ s ) sc = x ^ 1 cos θ s - x ^ 3 sin θ s .
e ^ + = 1 2 ( e ^ p + e ^ s ) ,
e ^ - = 1 2 ( e ^ p - e ^ s ) ,
e ^ R = 1 2 ( e ^ p - i e ^ s ) ,
e ^ L = 1 2 ( e ^ p + i e ^ s ) ,
I = P p + P s = P + + P - = P R + P L ,
Q = P p - P s ,
U = P + - P - ,
V = P R - P L ,
V inc = ( I inc Q inc U inc V inc ) = P inc [ E p 2 + E s 2 E p 2 - E s 2 2 R e ( E p E s * ) 2 F m ( E p E s * ) ] ,
v ( θ s ) sc = L 2 c 2 64 π 2 ω [ r p ( θ s ) E p 2 + r s ( θ s ) E s 2 r p ( θ s ) E p 2 - r s ( θ s ) E s 2 2 R e [ r p ( θ s ) E p r s * ( θ s ) E s * ] 2 F m [ r p ( θ s ) E p r s * ( θ s ) E s * ] ] .
[ R p θ s + R s θ s R p θ s - R s θ s R + θ s - R - θ s R R θ s - R L θ s ] = v ( θ s ) sc P inc ( E p 2 + E s 2 ) .
v ( θ s ) sc = s ( θ s ) V inc
s ( θ s ) = [ s 11 ( θ s ) s 12 ( θ s ) 0 0 s 12 ( θ s ) s 11 ( θ s ) 0 0 0 0 s 33 ( θ s ) s 34 ( θ s ) 0 0 - s 34 ( θ s ) s 33 ( θ s ) ] ,
s 11 ( θ s ) = C [ r p ( θ s ) 2 + r s ( θ s ) 2 ] ,
s 12 ( θ s ) = C [ r p ( θ s ) 2 - r s ( θ s ) 2 ] ,
s 33 ( θ s ) = C { 2 R e [ r p ( θ s ) r s * ( θ s ) ] } ,
s 34 ( θ s ) = C { - 2 F m [ r p ( θ s ) r s * ( θ s ) ] } ,
C = L 2 c 2 32 ( 2 π ) 2 ω P inc .
V ( θ s ) sc = v ( θ s ) sc ,
S ( θ s ) = s ( θ s )
V ( θ s ) sc = S ( θ s ) V inc ;
S ( θ s ) = S ( θ s ) coh + S ( θ s ) incoh ,
ϕ ( x 1 , x 3 ω ) inc = exp { i ( ω / c ) ( x 1 sin θ 0 - x 3 cos θ 0 ) × [ 1 + W ( x 1 , x 3 ) ] } × exp [ - ( x 1 cos θ 0 + x 3 sin θ 0 ) 2 / w 2 ] ,
W ( x 1 , x 3 ) = c 2 w 2 ω 2 { [ 2 w 2 ( x 1 cos θ 0 + x 3 sin θ 0 ) 2 ] - 1 } .
P inc = L 2 c w 8 ( 2 π ) 1 / 2 [ 1 - c 2 2 ω 2 w 2 ( 1 + 2 tan 2 θ 0 ) ] .
V ( θ s ) sc = P inc [ S 11 ( θ s ) S 12 ( θ s ) S 33 ( θ s ) - S 34 ( θ s ) ] .
I ± ( θ s ) = ( 1 / 2 ) [ S 11 ( θ s ) ± S 33 ( θ s ) ] .
S 12 ( θ s ) = ½ [ I p ( θ s ) - I s ( θ s ) ] ,
S 34 ( θ s ) = ½ [ I L ( θ s ) - I R ( θ s ) ] .
r α ( θ s ) = r α ( 1 ) ( θ s ) + r α ( 2 ) ( θ s ) + r α ( 3 ) ( θ s ) + ,             α = p , s ,
r s ( n ) ( θ s ) = ( - 1 ) n r p ( n ) ( θ s ) ,             n > 0 ,
s 11 ( θ s ) 2 C { r p ( 1 ) ( θ s ) 2 + r p ( 2 ) ( θ s ) 2 + r p ( 3 ) ( θ s ) 2 + 2 R e [ r p ( 1 ) ( θ s ) r p ( 3 ) * ( θ s ) ] } ,
s 12 ( θ s ) 4 C { R e [ r p ( 1 ) * ( θ s ) r p ( 2 ) ( θ s ) ] + R e [ r p ( 2 ) ( θ s ) r p ( 3 ) * ( θ s ) ] } ,
s 33 ( θ s ) 2 C { - r p ( 1 ) ( θ s ) 2 + r p ( 2 ) ( θ s ) 2 - r p ( 3 ) ( θ s ) 2 - 2 R e [ r p ( 1 ) ( θ s ) r p ( 3 ) * ( θ s ) ] } s ,
s 34 ( θ s ) 4 C { F m [ r p ( 1 ) * ( θ s ) r p ( 2 ) ( θ s ) ] + F m [ r p ( 2 ) ( θ s ) r p ( 3 ) * ( θ s ) ] } ,
I - ( θ s ) 2 C { r p ( 1 ) ( θ s ) 2 + r p ( 3 ) ( θ s ) 2 + 2 R e [ r p ( 1 ) ( θ s ) r p ( 3 ) * ( θ s ) ] } ,
I + ( θ s ) 2 C r p ( 2 ) ( θ s ) 2 .
( 2 x 1 2 + 2 x 2 2 + 2 x 3 2 + ω 2 c 2 ) ψ ( x 1 , x 2 , x 3 ω ) = 0 ,
ψ D ( x 1 , x 2 , x 3 ω ) x 3 = ζ ( x 1 , x 2 ) = 0
n ψ N ( x 1 , x 2 , x 3 ω ) x 3 = ζ ( x 1 , x 2 ) = 0 ,
n = { 1 + [ ζ ( x 1 + x 2 ) x 1 ] 2 + [ ζ ( x 1 + x 2 ) x 2 ] 2 } - 1 / 2 × [ - ζ ( x 1 + x 2 ) x 1 x 1 - ζ ( x 1 + x 2 ) x 2 x 2 + x 3 ]
( 2 x 1 2 + 2 x 2 2 + 2 x 3 2 + ω 2 c 2 ) G ( x 1 , x 2 , x 3 x 1 , x 2 , x 3 ) = - 4 π δ ( x 1 - x 1 ) δ ( x 2 - x 2 ) δ ( x 3 - x 3 )
Θ [ x 3 - ζ ( X ) ] ψ D ( x ω ) = ψ ( x ω ) inc - 1 4 π S d s G ( x x ) ψ D ( x ω ) n ,
Θ [ x 3 - ζ ( X ) ψ N ( x ω ) = ψ ( x ω ) inc + 1 4 π S d s G ( x x ) n ψ N ( x ω ) ,
ψ N ( x ω ) = 2 ψ ( x ω ) inc + 2 P 4 π S d s G ( x x ) n ψ N ( x ω ) ,
ψ D ( x ω ) n = 2 ψ ( x ω ) inc n - 2 P 4 π S d s G ( x x ) n ψ D ( x ω ) n
ψ D ( x ω ) n = D ( 1 ) ( x ω ) + D ( 2 ) ( x ω ) + D ( 3 ) ( x ω ) + ,
D ( 1 ) ( x ω ) = 2 ψ ( x ω ) inc n ,
D ( m ) ( x ω ) = - 2 P 4 π S d s G ( x x ) n D ( m - 1 ) ( x ω ) ,             m > 1 ;
ψ N ( x ω ) = N ( 1 ) ( x ω ) + N ( 2 ) ( x ω ) + N ( 3 ) ( x ω ) + ,
N ( 1 ) ( x ω ) = 2 ψ ( x ω ) inc ,
N ( m ) ( x ω ) = 2 P 4 π S d s G ( x x ) n N ( m - 1 ) ( x ω ) ,             m > 1.
ψ D ( x ω ) sc = ψ D ( 1 ) ( x ω ) sc + ψ D ( 2 ) ( x ω ) sc + ψ D ( 3 ) ( x ω ) sc + ,
ψ D ( m ) ( x ω ) sc = - 1 4 π S d s G ( x x ) D ( m ) ( x ω ) ,             m > 0 ;
ψ N ( x ω ) sc = ψ N ( 1 ) ( x ω ) sc + ψ N ( 2 ) ( x ω ) sc + ψ N ( 3 ) ( x ω ) sc + ,
ψ N ( m ) ( x ω ) sc = 1 4 π S d s G ( x x ) n N ( m ) ( x ω ) ,             m > 0.
D ( m ) ( x ω ) = 2 ψ D ( m - 1 ) ( x ω ) sc n - D ( m - 1 ) ( x ω ) ,
N ( m ) ( x ω ) = 2 ψ D ( m - 1 ) ( x ω ) sc - N ( m - 1 ) ( x ω ) ,
ψ D ( m ) ( x ω ) sc = ( - 1 ) m ψ N ( m ) ( x ω ) sc .
ψ D ( 1 ) ( x ω ) sc = - 1 4 π S d s G ( x x ) 2 ψ ( x ω ) inc n ,
ψ N ( 1 ) ( x ω ) sc = 1 4 π S d s G ( x x ) n 2 ψ ( x ω ) inc ,
ψ D ( 1 ) ( x ω ) sc + ψ N ( 1 ) ( x ω ) sc = - 2 4 π S d s [ G ( x x ) ψ ( x ω ) inc n - G ( x x ) n ψ ( x ω ) inc ] .
ψ D ( m ) ( x ω ) sc = - 2 4 π S d s G ( x x ) ψ D ( m - 1 ) ( x ω ) sc n - ψ D ( m - 1 ) ( x ω ) sc ,
ψ N ( m ) ( x ω ) sc = 2 4 π S d s G ( x x ) n ψ N ( m - 1 ) ( x ω ) sc - ψ N ( m - 1 ) ( x ω ) sc ,
ψ D ( m ) ( x ω ) sc - ( - 1 ) m ψ N ( m ) ( x ω ) sc = - 2 4 π S d s × [ G ( x x ) ψ D ( m - 1 ) ( x ω ) sc n - G ( x x ) n ψ D ( m - 1 ) ( x ω ) sc ] - 2 ψ D ( m - 1 ) ( x ω ) sc .
ψ D ( m ) ( x ω ) sc = ( - 1 ) m ψ N ( m ) ( x ω ) sc

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