Abstract

The scattering of p- or s-polarized plane waves incident upon flat surfaces characterized by a randomly varying impedance is studied. The impedance is considered to be one dimensional, i.e., to vary along one of the two coordinates on the surface. Particular attention is paid to the role played by multiple scattering of surface waves in the production of enhanced backscattering at this kind of structure and its relation with the backscatter peak observed in metallic surfaces with height variations. The angular distribution of the scattered field is calculated for different realizations of the same statistical ensemble of impedance variations. A peak in the retroreflection direction is observed in p polarization for an inductive impedance and in s polarization for a capacitive impedance. The peak is clearly distinguished in the double-scattering contribution, but it is also present in higher-order multiple-scattering terms. It is shown that the observation of enhanced backscattering at these structures is always associated with the excitation of surface electromagnetic waves along the scatterer. An analogy between randomly varying impedance planes and actual corrugated surfaces is discussed.

© 1992 Optical Society of America

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References

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  1. Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
    [CrossRef]
  2. L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A 1, 836–840 (1984).
    [CrossRef]
  3. E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from gaussian rough surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  4. K. A. O’Donnell, E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  5. A. Sant, J. C. Dainty, M. J. Kim, “Comparison of surface scattering between identical, randomly rough metal and dielectric diffusers,” Opt. Lett. 14, 1183–1184 (1989).
    [CrossRef] [PubMed]
  6. M. J. Kim, J. C. Dainty, A. T. Friberg, A. J. Sant, “Experimental study of enhanced backscattering from one- and two-dimensional random rough surfaces,” J. Opt. Soc. Am. A 7, 569–577 (1990).
    [CrossRef]
  7. J. C. Dainty, M. J. Kim, A. J. Sant, “Measurements of angular scattering by randomly rough metal and dielectric surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 143–156.
  8. Z. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental study of the opposition effect in the scattering of light from a randomly rough metal surface,” Appl. Opt. 28, 537–543 (1989).
    [CrossRef] [PubMed]
  9. A. Ishimaru, “Experimental and theoretical studies on enhanced backscattering from scatterers and rough surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 1–16.
  10. 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,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990) pp. 157–174.
  11. A. R. McGurn, 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]
  12. A. A. Maradudin, T. Michel, A. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [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. V. Celli, A. A. Maradudin, A. M. Marvin, A. R. McGurn, “Some aspects of light scattering from a randomly rough metal surface,” J. Opt. Soc. Am. A 2, 2225–2239 (1985).
    [CrossRef]
  15. A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  16. 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]
  17. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 100–108.
  18. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987).
  19. J. R. Wait, “The scope of impedance boundary condition in radiopropagation,”IEEE Trans. Geosci. Remote Sensing 28, 721–723 (1990).
    [CrossRef]
  20. R. A. Depine, “A simple theoretical approach to light scattering from absorbing microrough surfaces,” Optik (Stuttgart) 82, 5–8 (1989).
  21. R. A. Depine, “Surface impedance boundary conditions used to study light scattering from metallic surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 239–254.
  22. R. A. Depine, V. L. Brudny, “A simple model for a micro-rough diffraction grating that predicts light bands,” J. Mod. Opt. 36, 1257–1271 (1989).
    [CrossRef]
  23. A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  24. J. Pavageau, J. Bousquet, “Diffraction par un rseau conducteur. Nouvelle mthode de resolution,” Opt. Acta 17, 469–478 (1970).
    [CrossRef]
  25. E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary-extensions of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
    [CrossRef]
  26. F. Borgni, C. Papas, “Electromagnetic waveguides and resonators,” in Electric Fields and Waves, Vol. 16 of Encyclopedia of Physics, S. Flügge, ed. (Springer–Verlag, Berlin, 1958), pp. 285–422.
  27. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

1990 (3)

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

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

J. R. Wait, “The scope of impedance boundary condition in radiopropagation,”IEEE Trans. Geosci. Remote Sensing 28, 721–723 (1990).
[CrossRef]

1989 (6)

1987 (3)

1985 (2)

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

1984 (2)

1982 (1)

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

1970 (1)

J. Pavageau, J. Bousquet, “Diffraction par un rseau conducteur. Nouvelle mthode de resolution,” Opt. Acta 17, 469–478 (1970).
[CrossRef]

1965 (1)

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987).

Borgni, F.

F. Borgni, C. Papas, “Electromagnetic waveguides and resonators,” in Electric Fields and Waves, Vol. 16 of Encyclopedia of Physics, S. Flügge, ed. (Springer–Verlag, Berlin, 1958), pp. 285–422.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 100–108.

Bousquet, J.

J. Pavageau, J. Bousquet, “Diffraction par un rseau conducteur. Nouvelle mthode de resolution,” Opt. Acta 17, 469–478 (1970).
[CrossRef]

Brudny, V. L.

R. A. Depine, V. L. Brudny, “A simple model for a micro-rough diffraction grating that predicts light bands,” J. Mod. Opt. 36, 1257–1271 (1989).
[CrossRef]

Celli, V.

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

Dainty, J. C.

Depine, R. A.

R. A. Depine, “A simple theoretical approach to light scattering from absorbing microrough surfaces,” Optik (Stuttgart) 82, 5–8 (1989).

R. A. Depine, V. L. Brudny, “A simple model for a micro-rough diffraction grating that predicts light bands,” J. Mod. Opt. 36, 1257–1271 (1989).
[CrossRef]

R. A. Depine, “Surface impedance boundary conditions used to study light scattering from metallic surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 239–254.

Dummer, R. S.

Friberg, A. T.

Gu, Z.

Hessel, A.

Ishimaru, A.

L. Tsang, A. Ishimaru, “Backscattering enhancement of random discrete scatterers,” J. Opt. Soc. Am. A 1, 836–840 (1984).
[CrossRef]

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
[CrossRef]

A. Ishimaru, “Experimental and theoretical studies on enhanced backscattering from scatterers and rough surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 1–16.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Kim, M. J.

Kuga, Y.

Liszka, E. G.

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

Maradudin, A. A.

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

Z. Gu, R. S. Dummer, A. A. Maradudin, A. R. McGurn, “Experimental study of the opposition effect in the scattering of light from a randomly rough metal surface,” Appl. Opt. 28, 537–543 (1989).
[CrossRef] [PubMed]

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

Marvin, A. M.

McCoy, J. J.

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

McGurn, A.

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

McGurn, A. R.

Mendez, E. R.

A. A. Maradudin, T. Michel, A. McGurn, E. R. Mendez, “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]

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

E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from gaussian rough surfaces,” Opt. Commun. 61, 91–95 (1987).
[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,” 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. McGurn, E. R. Mendez, “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. Mendez, T. Michel, “Backscattering effects in the elastic scattering of p-polarized light from a large-amplitude random metallic 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.

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

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

Oliner, A. A.

Papas, C.

F. Borgni, C. Papas, “Electromagnetic waveguides and resonators,” in Electric Fields and Waves, Vol. 16 of Encyclopedia of Physics, S. Flügge, ed. (Springer–Verlag, Berlin, 1958), pp. 285–422.

Pavageau, J.

J. Pavageau, J. Bousquet, “Diffraction par un rseau conducteur. Nouvelle mthode de resolution,” Opt. Acta 17, 469–478 (1970).
[CrossRef]

Sant, A.

Sant, A. J.

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

J. C. Dainty, M. J. Kim, A. J. Sant, “Measurements of angular scattering by randomly rough metal and dielectric surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 143–156.

Soto-Crespo, J. M.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987).

Tsang, L.

Wait, J. R.

J. R. Wait, “The scope of impedance boundary condition in radiopropagation,”IEEE Trans. Geosci. Remote Sensing 28, 721–723 (1990).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 100–108.

Ann. Phys. (1)

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

Appl. Opt. (2)

IEEE Trans. Geosci. Remote Sensing (1)

J. R. Wait, “The scope of impedance boundary condition in radiopropagation,”IEEE Trans. Geosci. Remote Sensing 28, 721–723 (1990).
[CrossRef]

J. Acoust. Soc. Am. (1)

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

J. Mod. Opt. (1)

R. A. Depine, V. L. Brudny, “A simple model for a micro-rough diffraction grating that predicts light bands,” J. Mod. Opt. 36, 1257–1271 (1989).
[CrossRef]

J. Opt. Soc. Am. A (6)

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

Opt. Acta (1)

J. Pavageau, J. Bousquet, “Diffraction par un rseau conducteur. Nouvelle mthode de resolution,” Opt. Acta 17, 469–478 (1970).
[CrossRef]

Opt. Commun. (1)

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

Opt. Lett. (2)

Optik (Stuttgart) (1)

R. A. Depine, “A simple theoretical approach to light scattering from absorbing microrough surfaces,” Optik (Stuttgart) 82, 5–8 (1989).

Phys. Rev. B (1)

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

Other (8)

R. A. Depine, “Surface impedance boundary conditions used to study light scattering from metallic surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 239–254.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 100–108.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Norwood, Mass., 1987).

J. C. Dainty, M. J. Kim, A. J. Sant, “Measurements of angular scattering by randomly rough metal and dielectric surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 143–156.

A. Ishimaru, “Experimental and theoretical studies on enhanced backscattering from scatterers and rough surfaces,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990), pp. 1–16.

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,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (North-Holland, Amsterdam, 1990) pp. 157–174.

F. Borgni, C. Papas, “Electromagnetic waveguides and resonators,” in Electric Fields and Waves, Vol. 16 of Encyclopedia of Physics, S. Flügge, ed. (Springer–Verlag, Berlin, 1958), pp. 285–422.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

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Figures (11)

Fig. 1
Fig. 1

Surface impedance calculated at y = 0 for 0 ≤ xd. It is zero for dxd + b if the corrugated surface is perfectly conductive.

Fig. 2
Fig. 2

Angular distribution of the scattered power associated with the double-scattering term. Z0 = 0.0236–0.342i (inductive), λ/c = 4.4, L/σ = 120, Nsup = 200, h = 0.05. The angle of incidence is θ0 = 30°. The arrows labeled B and S indicate the backscatter and the specular directions, respectively. Upper curve, p polarization; lower curve, s polarization.

Fig. 3
Fig. 3

Angular distribution of the scattered power for p polarization. Upper curve, total; lower curve, double-scattering term multiplied by 10. Same parameters as in Fig. 2, except h = 0.1 and Nsup = 2000.

Fig. 4
Fig. 4

Double-scattering contribution to the total scattered powder in p polarization. Same parameters as in Fig. 2, but now the angle of incidence is θ0 = 15°.

Fig. 5
Fig. 5

Mean value of the modulus squared of the unknown function Rp(α) as a function of α/k. Same parameters as in Fig. 2. Only the peaks at the positions corresponding to the propagation constant of a surface wave can be observed.

Fig. 6
Fig. 6

Same as Fig. 5 but with the vertical-axis scale amplified. The small peaks near the α/k axis are the double-scattering contribution.

Fig. 7
Fig. 7

Mean value of the absolute value squared of the unknown function Rs(α) as a function of α/k. Same parameters as in Fig. 2.

Fig. 8
Fig. 8

Angular distribution of the scattered power associated with the double-scattering term. Z0 = 0.03 + 2.92i (capacitive), λ/σ = 4.4, L/σ = 120, Nsup = 200, h = 0.05. The angle of incidence is θ0 = 30°. The arrows labeled B and S indicate the backscatter and the specular directions, respectively. Upper curve, s polarization; lower curve, p polarization.

Fig. 9
Fig. 9

Mean value of the modulus squared of the unknown function Rs(α) as a function of α/k. Same parameters as in Fig. 8.

Fig. 10
Fig. 10

Angular distribution of the scattered power. Upper curve, total; lower curve, double-scattering term. Same parameters as in Fig. 2, except h = 0.5.

Fig. 11
Fig. 11

Angular distribution of the scattered power as calculated with the same parameters as in Fig. 2, except h = 0.8. The upper curve was obtained by iterating four times in the integral equation. The bottom curve corresponds to the single-scattering approximation. Contributions from double and triple scattering were plotted together in the middle curve.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

Z ( x ) = Z 0 + Z r ( x ) .
Z r ( x ) = Z 0 h v ( x ) ,
H ( x , y , t ) = f ( x , y ) exp ( - i ω t ) z .
f = f i + f r + f s .
f i ( x , y ) = exp [ i ( α 0 x - β 0 y ) ] ,
f r ( x , y ) = A p ( α 0 ) exp [ i ( α 0 x + β 0 y ) ] ,
f s ( x , y ) = R p ( α ) exp [ i ( α x + β y ) ] d α ,
E = Z ( x ) y × H
i k f y = Z ( x ) f .
Z 0 f s - i k f s y = - Z r ( f i + f r ) - Z r f s .
( Z 0 + β / k ) R p ( α ) = - Z ˜ r ( α - α 0 ) [ 1 + A p ( α 0 ) ] - Z ˜ r ( α - α ) R p ( α ) d α .
Z ˜ r ( α ) = Z r ( x ) exp ( - i α x ) d x ,
E ( x , y , t ) = f ( x , y ) exp ( - i ω t ) z .
k f s - i Z 0 f s y = i Z r ( f i y + f r y ) + i Z r f s y .
( k + Z 0 β ) R s ( α ) = Z ˜ r ( α - α 0 ) β 0 [ 1 - A s ( α 0 ) ] - β Z ˜ r ( α - α ) R s ( α ) d α ,
d P d α = β β 0 R ( α ) 2 ,
R ( α ) = K ( α ) + N ( α , α ) R ( α ) d α ,
K p ( α ) = - Z ˜ r ( α - α 0 ) 1 + A p ( α 0 ) Z 0 + β / k ,
K s ( α ) = Z ˜ r ( α - α 0 ) β 0 1 - A s ( α 0 ) k + Z 0 β ,
R p ( α ) K p ( α ) + 1 + A p ( α 0 ) Z 0 + β / k Z ˜ r ( α - α ) Z ˜ r ( α - α 0 ) Z 0 + β / k d α ,
R s ( α ) K s ( α ) + β 0 1 - A s ( α 0 ) k + Z 0 β × β Z ˜ r ( α - α ) Z ˜ r ( α - α 0 ) k + Z 0 β d α .
H n ( x , y , t ) = D n cos δ x ( cos γ y - tan γ h sin γ y ) × exp ( - i ω t ) z .
E n ( x , 0 , t ) = i D n k ( - γ cos δ x tan γ h x + δ sin δ x y ) × exp ( - i ω t ) .
Z oc p = E x H z = - i γ k tan γ h .
Z av p = - i γ d ( d + b ) k tan γ h .
H sw = exp [ i ( α x + β y - ω t ) ] z ,             β = i β ,
Z av p = Z f b d + b - i γ d ( b + d ) k tan γ h ,
E sw = exp [ i ( α x + β y - ω t ) ] z ,             β = i β ,
E n ( x , y , t ) = C n sin δ x ( sin γ y + tan γ h cos γ y ) × exp ( - i ω t ) z ,
H n ( x , 0 , t ) = - i C n k ( γ sin δ x x - δ cos δ x tan γ h y ) × exp ( - i ω t ) ,
Z oc s = - E z H x = - i k γ tan γ h .
Z av s = - i k d γ ( d + b ) tan γ h
Z av 2 = Z f b d + b - i k d γ ( d + b ) tan γ h
f ( x , 0 ) = R ( α ) exp ( i α x ) d α ,
f ( x , 0 ) 2 = R ( α ) R ( α ) exp [ i ( α - α ) x ] d α d α .
R ( α ) 2 d α .

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