Abstract

We carry out computer simulations of the scattering of a beam of light with p or s polarization from a transparent dielectric film (BaSO4) on a perfectly conducting substrate. The dielectric–vacuum interface is assumed to be a random grating; the dielectric–conductor interface is assumed to be planar. It is found that for incident light of either polarization the angular dependence of the incoherent contribution to the mean intensity of the scattered light has a well-defined peak when the scattering angle corresponds to the retroreflection direction (enhanced backscattering). The existence of enhanced backscattering of p-polarized light in this geometry contrasts with its absence in the scattering of p-polarized light from a random grating on a semi-infinite dielectric medium that is characterized by the same roughness parameters and index of refraction. The enhanced backscattering that is observed in the scattering of s-polarized light from a random grating on a semi-infinite dielectric medium is stronger when the scattering takes place from the structure studied. For both polarizations the enhancement is most pronounced when the distance of the perfectly conducting substrate from the mean dielectric–vacuum interface is several (2–10) times the distance of the focusing plane of the rough interface from the mean interface.

© 1991 Optical Society of America

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References

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  1. A. A. Maradudin, E. R. Méndez, and 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]
  2. A. A. Maradudin, E. R. Méndez, and 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 and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.
  3. A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 155–207 (1990).
    [CrossRef]
  4. E. Jakeman, “Enhanced backscattering through a deep random phase screen,” J. Opt. Soc. Am. A 5, 1638–1648 (1988).
    [CrossRef]
  5. P. R. Tapster, A. R. Weeks, and E. Jakeman, “Observation of backscattering enhancement through atmospheric phase screens,” J. Opt. Soc. Am. A 6, 517–522.
  6. E. Jakeman, “The physical optics of enhanced backscattering,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 111–123.
  7. See, for example, J. D. Jackson, Classical Electrodynamics, (Wiley, New York, 1962), pp. 14–15.
  8. 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]
  9. Y. Kuga and A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
    [CrossRef]
  10. M. P. van Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
    [CrossRef] [PubMed]
  11. P. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
    [CrossRef] [PubMed]
  12. A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  13. V. Celli, A. A. Maradudin, A. M. Marvin, and A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
    [CrossRef]
  14. A. R. McGurn and A. A. Maradudin, “Localization effects in the elastic scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
    [CrossRef]
  15. See, for example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 4.4.

1990 (1)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 155–207 (1990).
[CrossRef]

1989 (1)

1988 (2)

E. Jakeman, “Enhanced backscattering through a deep random phase screen,” J. Opt. Soc. Am. A 5, 1638–1648 (1988).
[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]

1987 (1)

1985 (4)

M. P. van Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

P. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

V. Celli, A. A. Maradudin, A. M. Marvin, and A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[CrossRef]

1984 (1)

Born, M.

See, for example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 4.4.

Celli, V.

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

V. Celli, A. A. Maradudin, A. M. Marvin, and A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[CrossRef]

Ishimaru, A.

Jackson, J. D.

See, for example, J. D. Jackson, Classical Electrodynamics, (Wiley, New York, 1962), pp. 14–15.

Jakeman, E.

E. Jakeman, “Enhanced backscattering through a deep random phase screen,” J. Opt. Soc. Am. A 5, 1638–1648 (1988).
[CrossRef]

P. R. Tapster, A. R. Weeks, and E. Jakeman, “Observation of backscattering enhancement through atmospheric phase screens,” J. Opt. Soc. Am. A 6, 517–522.

E. Jakeman, “The physical optics of enhanced backscattering,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 111–123.

Kuga, Y.

Lagendijk, A.

M. P. van Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 155–207 (1990).
[CrossRef]

A. A. Maradudin, E. R. Méndez, and 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. R. McGurn and A. A. Maradudin, “Localization effects in the elastic scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
[CrossRef]

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

V. Celli, A. A. Maradudin, A. M. Marvin, and A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[CrossRef]

A. A. Maradudin, E. R. Méndez, and 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 and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

Maret, G.

P. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Marvin, A. M.

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 155–207 (1990).
[CrossRef]

A. R. McGurn and A. A. Maradudin, “Localization effects in the elastic scattering of light from a randomly rough surface,” J. Opt. Soc. Am. B 4, 910–926 (1987).
[CrossRef]

V. Celli, A. A. Maradudin, A. M. Marvin, and A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
[CrossRef]

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 155–207 (1990).
[CrossRef]

A. A. Maradudin, E. R. Méndez, and 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, and 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 and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 155–207 (1990).
[CrossRef]

A. A. Maradudin, E. R. Méndez, and 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, and 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 and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

Tapster, P. R.

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 Albada, M. P.

M. P. van Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

Weeks, A. R.

Wolf, E.

See, for example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 4.4.

Wolf, P.

P. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Ann. Phys. (N.Y.) (1)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 155–207 (1990).
[CrossRef]

J. Acoust. Soc. Am. (1)

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

J. Opt. Soc. Am. A (4)

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

Opt. Lett. (1)

Phys. Rev. B (1)

A. R. McGurn, A. A. Maradudin, and V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Phys. Rev. Lett. (2)

M. P. van Albada and A. Lagendijk, “Observation of weak localization of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

P. Wolf and G. Maret, “Weak localization and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Other (4)

E. Jakeman, “The physical optics of enhanced backscattering,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 111–123.

See, for example, J. D. Jackson, Classical Electrodynamics, (Wiley, New York, 1962), pp. 14–15.

See, for example, M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Sec. 4.4.

A. A. Maradudin, E. R. Méndez, and 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 and J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 157–174.

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Figures (11)

Fig. 1
Fig. 1

Structure studied in this paper.

Fig. 2
Fig. 2

(a) Differentional reflection coefficient 〈∂Rp/∂θsincoh for the scattering of a p-polarized beam of light from a random grating on the surface of a film of BaSO4 deposited on a perfectly conducting substrate: δ = 1.2 μm, a = 2 μm, d = 4.8 μm, λ = 6328 Å, nd = 1.628 + 0.0003i, L = 25.6 μm, g = 6.4 μm, N = 300, Np = 1000, θ0 = 5°. (b) Same as in (a) except that the random grating is on a semi-infinite BaSO4 substrate.

Fig. 3
Fig. 3

Same as in Fig. 2 except that θ0 = 20°.

Fig. 4
Fig. 4

(a) Differential reflection coefficient 〈∂Rs/∂θsincoh for the scattering of an s-polarized beam of light from a random grating on the surface of a film of BaSO4 deposited on a perfectly conducting substrate: δ = 1.2 μm, a = 2 μm, d = 4.8 μm, λ = 6328 Å, nd = 1.628 + 0.0003i, L = 25.6 μm, g = 6.4 μm, N = 300, Np = 1000, θ0 = 5°. (b) Same as in (a) except that the random grating is on a semi-infinite BaSO4 substrate.

Fig. 5
Fig. 5

Same as in Fig. 4 except that θ0 = 20°.

Fig. 6
Fig. 6

Enhancement mechanisms: (a) coherent enhancement, (b) incoherent, geometrical enhancement.

Fig. 7
Fig. 7

Differential reflection coefficient 〈∂Rp/∂θsincoh for the scattering of a p-polarized beam of light from a random grating on the surface of a film of BaSO4 deposited on a perfectly conducting substrate: δ = 0.6117 μm, a = 2 μm, λ = 6328 Å, nd = 1.628 + 0.0003i, so that lf = 4.8936 μm, L = 25.6 μm, g =6.4 μm, N = 300, Np = 1000, θ0 = 5°. (a) d = ½lf, (b) d = lf, (c) d = 2lf, (d) d = 8lf.

Fig. 8
Fig. 8

Same as in Fig. 7 but for θ0 = 20°.

Fig. 9
Fig. 9

Same as in Fig. 7 but for an incident beam of s polarization.

Fig. 10
Fig. 10

Same as in Fig. 8 but for an incident beam of s polarization.

Fig. 11
Fig. 11

Solid lines, light path that contributes to backscattering when the perfect conductor is placed at x3 = −d; dashed lines, the light path that corresponds to the same incident beam when the perfect conductor is placed at x3 = −2d.

Equations (49)

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H 2 ( 1 ) ( 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 ] ,
R p / θ s incoh = 1 2 ( 2 π ) 3 / 2 c ω w r p ( θ s ) 2 - r p ( θ s ) 2 1 - c 2 ( 1 + 2 tan 2 θ 0 ) / ( 2 ω 2 w 2 ) ,
r p ( θ s ) = - d x 1 exp { - i ω c [ x 1 sin θ s + ζ ( x 1 ) cos θ s ] } × { i ω c [ ζ ( x 1 ) sin θ s - cos θ s ] H ( 1 ) ( x 1 ω ) - L ( 1 ) ( x 1 ω ) } .
{ 1 + [ ζ ( x 1 ) ] 2 } 1 / 2 H 2 ( 1 ) ( x 1 , x 3 ω ) / n x 3 = ζ ( x 1 ) ,
H ( 1 ) ( x 1 ω ) = H ( 1 ) ( x 1 ω ) inc + - d x 1 [ H 0 ( x 1 x 1 ) H ( 1 ) ( x 1 ω ) - L 0 ( x 1 x 1 ) L ( 1 ) ( x 1 ω ) ] ,
0 = - d x 1 [ H 11 ( x 1 x 1 ) H ( 1 ) ( x 1 ω ) - d L 11 ( x 1 x 1 ) L ( 1 ) ( x 1 ω ) - H 12 ( x 1 x 1 ) H ( 2 ) ( x 1 ω ) ] ,
H ( 2 ) ( x 1 ω ) = - d x 1 [ - H 21 ( x 1 x 1 ) H ( 1 ) ( x 1 ω ) + d L 21 ( x 1 x 1 ) L ( 1 ) ( x 1 ω ) + H 22 ( x 1 x 1 ) H ( 2 ) ( x 1 ω ) ] ,
H 0 ( x 1 x 1 ) = i 4 [ - ζ ( x 1 ) x 1 + x 3 ] H 0 ( 1 ) ( ω c R ) | x 3 = ζ ( x 1 ) x 3 = ζ ( x 1 ) + ,
L 0 ( x 1 x 1 ) = i 4 H 0 ( 1 ) ( ω c R ) | x 3 = ζ ( x 1 ) x 3 = ζ ( x 1 ) + ,
H 11 ( x 1 x 1 ) = i 4 [ - ζ ( x 1 ) x 1 + x 3 ] H 0 ( 1 ) × ( n d ω c R ) | x 3 = ζ ( x 1 ) x 3 = ζ ( x 1 ) + ,
L 11 ( x 1 x 1 ) = i 4 H 0 ( 1 ) ( n d ω c R ) | x 3 = ζ ( x 1 ) x 3 = ζ ( x 1 ) + ,
H 12 ( x 1 x 1 ) = i 4 x 3 H 0 ( 1 ) ( n d ω c R ) | x 3 = - d x 3 = ζ ( x 1 ) + ,
H 21 ( x 1 x 1 ) = i 4 [ - ζ ( x 1 ) x 1 + x 3 ] H 0 ( 1 ) ( n d ω c R ) | x 3 = ζ ( x 1 ) x 3 = - d + ,
L 21 ( x 1 x 1 ) = i 4 H 0 ( 1 ) ( n d ω c R ) | x 3 = ζ ( x 1 ) x 3 = - d + ,
H 22 ( x 1 x 1 ) = i 4 x 3 H 0 ( 1 ) ( n d ω c R ) | x 3 = - d x 3 = - d + .
E 2 ( 1 ) ( x 1 , x 3 ω ) inc = exp { i ( ω / c ) ( x 1 sin θ 0 - x 3 cos θ 0 ) × [ 1 + ω ( x 1 , x 3 ) ] } exp [ - ( x 1 cos θ 0 + x 3 sin θ 0 ) 2 / w 2 ]
R s / θ s inch = 1 2 ( 2 π ) 3 / 2 c ω w r s ( θ s ) 2 - r s ( θ s ) 2 1 - c 2 ( 1 + 2 tan 2 θ 0 ) / ( 2 ω 2 w 2 ) ,
r s ( θ s ) = - d x 1 exp { - i ω c [ x 1 sin θ s + ζ ( x 1 ) cos θ s ] } × { i ω c [ ζ ( x 1 ) sin θ s - cos θ s ] e ( 1 ) ( x 1 ω ) - F ( 1 ) ( x 1 ω ) } .
{ 1 + [ ζ ( x 1 ) ] 2 } 1 / 2 E 2 ( 1 ) ( x 1 , x 3 ω ) / n x 3 = ζ ( x 1 )
E ( 1 ) ( x 1 ω ) = E ( 1 ) ( x 1 ω ) inc + - d x 1 [ H 0 ( x 1 x 1 ) E ( 1 ) ( x 1 ω ) - L 0 ( x 1 x 1 ) F ( 1 ) ( x 1 ω ) ] ,
0 = - d x 1 { [ H 11 ( x 1 x 1 ) E ( 1 ) ( x 1 ω ) - L 11 ( x 1 x 1 ) F ( 1 ) ( x 1 ω ) ] + L 12 ( x 1 x 1 ) F ( 2 ) ( x 1 ω ) } ,
0 = - d x 1 { [ H 21 ( x 1 x 1 ) E ( 1 ) ( x 1 ω ) - L 21 ( x 1 x 1 ) × F ( 1 ) ( x 1 ω ) ] + L 22 ( x 1 x 1 ) F ( 2 ) ( x 1 ω ) } ,
L 12 ( x 1 x 1 ) = i 4 H 0 ( 1 ) ( n d ω c R ) | x 3 = - d x 3 = ζ ( x 1 ) + ,
L 22 ( x 1 x 1 ) = i 4 H 0 ( 1 ) ( n d ω c R ) | x 3 = - d x 3 = - d + .
H ( 1 ) ( x m ω ) = 2 H ( 1 ) ( x m ω ) inc + n = 1 N [ H m n ( 0 ) H 1 ( x n ω ) - L m n ( 0 ) L ( 1 ) ( x n ω ) ] ,
0 = H ( 1 ) ( x m ω ) + n = 1 N [ H m n ( 11 ) H ( 1 ) ( x n ω ) - d L m n ( 11 ) L ( 1 ) ( x n ω ) - H m n ( 12 ) H ( 2 ) ( x n ω ) ] ,
H ( 2 ) ( x m ω ) = - n = 1 N [ H m n ( 21 ) H ( 1 ) ( x n ω ) - d L m n ( 21 ) L ( 1 ) ( x n ω ) ]
E ( 1 ) ( x m ω ) = 2 E ( 1 ) ( x m ω ) inc + n = 1 N [ H m n ( 0 ) E ( 1 ) ( x n ω ) - L m n ( 0 ) F ( 1 ) ( x n ω ) ] ,
0 = E ( 1 ) ( x m ω ) + n = 1 N [ H m n ( 11 ) E ( 1 ) ( x n ω ) - L m n ( 11 ) F ( 1 ) ( x n ω ) + L m n ( 12 ) F ( 2 ) ( x n ω ) ] ,
0 = n = 1 N [ H m n ( 21 ) E ( 1 ) ( x n ω ) - L m n ( 21 ) F ( 1 ) ( x n ω ) + H m n ( 22 ) F 2 ( x n ω ) ]
r p ( θ s ) Δ x n = 1 N exp { - i ω c [ x n sin θ s + ζ ( x n ) cos θ s ] } × { i ω c [ ζ ( x n ) sin θ s - cos θ s ] H ( 1 ) ( x n ω ) - L ( 1 ) ( x n ω ) }
r s ( θ s ) Δ x n = 1 N exp { - i ω c [ x n sin θ s + ζ ( x n ) cos θ s ] } × { i ω c [ ζ ( x n ) sin θ s - cos θ s ] E ( 1 ) ( x n ω ) - F ( 1 ) ( x n ω ) } .
l f = n d n d - 1 ρ ,
ρ = [ ζ ( x 1 ) ] 2 - 1 / 2 = 1 δ [ W i v ( 0 ) ] - 1 / 2 .
H m n ( 0 ) = Δ x ( - i 2 ) ω 2 c 2 × H 1 ( 1 ) { ( ω / c ) [ ( x m - x n ) 2 + [ ζ ( x m ) - ζ ( x n ) ] 2 ] 1 / 2 } ( ω / c ) { ( x m - x n ) 2 + [ ζ ( x m ) - ζ ( x n ) ] 2 } 1 / 2 × { ( x m - x n ) ζ ( x n ) - [ ζ ( x m ) - ζ ( x n ) ] } , m n ,
H m n ( 0 ) = Δ x ζ ( x m ) 2 π γ m 2 ,             m = n ,
L m n ( 0 ) = Δ x ( + i 2 ) H 0 ( 1 ) ( ω c { ( x m - x n ) 2 + [ ζ ( x m ) - ζ ( x n ) ] 2 } 1 / 2 ) ,             m n ,
L m n ( 0 ) = Δ x ( + i 2 ) H 0 ( 1 ) ( ω c γ m Δ x 2 e ) ,             m = n ,
H m n ( 11 ) = Δ x ( - i 2 ) n d 2 ω 2 c 2 × H 1 ( 1 ) ( n d ω c { ( x m - x n ) 2 + [ ζ ( x m ) - ζ ( x n ) ] 2 } 1 / 2 ) n d ω c { ( x m - x n ) 2 + [ ζ ( x m ) - ζ ( x n ) ] 2 } 1 / 2 × { ( x m - x n ) ζ ( x n ) - [ ζ ( x m ) - ζ ( x n ) ] } , m n ,
H m n ( 1 ) = Δ x ζ ( x m ) 2 π γ m 2 ,             m = n ,
L m n ( 11 ) = Δ x ( + i 2 ) H 0 ( 1 ) ( n d ω c { ( x m - x n ) 2 + [ ζ ( x m ) - ζ ( x n ) ] 2 } 1 / 2 ) ,             m n ,
L m n ( 11 ) = Δ x ( + i 2 ) H 0 ( 1 ) ( n d ω c γ m Δ x 2 e ) ,             m = n ,
H m n ( 2 ) = Δ x ( - i 2 ) n d 2 ω 2 c 2 × H 1 ( 1 ) ( n d ω c { ( x m - x n ) 2 + [ ζ ( x m ) + d ] 2 } 1 / 2 ) n d ω c { ( x m - x n ) 2 + [ ζ ( x m ) + d ] 2 } 1 / 2 × [ - ζ ( x m ) - d ] ,
L m n ( 12 ) = Δ x ( i 2 ) H 0 ( 1 ) ( n d ω c { ( x m - x n ) 2 + [ ζ ( x m ) + d ] 2 } 1 / 2 ) ,
H m n ( 21 ) = Δ x ( - i 2 ) n d 2 ω 2 c 2 × H 1 ( 1 ) ( n d ω c { ( x m - x n ) 2 + [ d + ζ ( x n ) ] 2 } 1 / 2 ) n d ω c { ( x m - x n ) 2 + [ d + ζ ( x n ) ] 2 } 1 / 2 × { ( x m - x n ) ζ ( x n ) + [ d + ζ ( x n ) ] } ,
L m n ( 21 ) = Δ x ( + i 2 ) H 0 ( 1 ) ( n d ω c { ( x m - x n ) 2 + [ d + ζ ( x n ) ] 2 } 1 / 2 ) ,
L m n ( 22 ) = Δ x ( i 2 ) H 0 ( 1 ) ( n d ω c x m - x n ) ,             m n ,
L m n ( 22 ) = Δ x ( i 2 ) H 0 ( 1 ) ( n d ω c Δ x 2 e ) ,             m = n ,
γ m = { 1 + [ ζ ( x m ) ] 2 } 1 / 2 .

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