Abstract

The total integrated reflectivity of a slightly rough dielectric surface is studied by the Rayleigh–Rice method. Special emphasis is laid on the correct second-order terms of the perturbation expansion; this is important for correlation lengths of the surface structure comparable to the wavelength. The main effects of a surface roughness of this kind are reflection loss, shift of the Brewster angle to smaller values, and slight reddening of the scattered light.

© 1987 Optical Society of America

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  1. G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.
  2. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).
  3. D. E. Barrick, “Relationship between Slope Probability Density Function and the Physical Optics Integral in Rough Surface Scattering,” Proc. IEEE 56, 1728 (1968).
    [Crossref]
  4. R. D. Kodis, “A Note on the Theory of Scattering from an Irregular Surface,” IEEE Trans. Antennas Propag. AP-14, 77 (1966).
    [Crossref]
  5. D. E. Barrick, “Rough Surface Scattering Based on the Specular Point Theory,” IEEE Trans. Antennas Propag. AP-16, 449 (1968).
    [Crossref]
  6. M. I. Sancer, “Shadow-Corrected Electromagnetic Scattering from a Randomly Rough Surface,” IEEE Trans. Antennas Propag. AP-17, 577 (1969).
    [Crossref]
  7. D. O. Muhleman, “Radar Scattering from Venus and the Moon,” Astron. J. 69, 34 (1964).
    [Crossref]
  8. C. C. Sung, J. A. Holzer, “Scattering of Electromagnetic Waves from a Rough Surface,” Appl. Phys. Lett. 28, 429 (1976).
    [Crossref]
  9. J. A. Holzer, C. C. Sung, “Evaluation of the Irradiance of Reflected Light from Rough Surfaces,” J. Appl. Phys. 47, 3363 (1976).
    [Crossref]
  10. J. A. Holzer, C. C. Sung, “Scattering of Electromagnetic Waves from a Rough Surface II,” J. Appl. Phys. 49, 1002 (1978).
    [Crossref]
  11. Lord Rayleigh, Theory of Sound, Vol. 2 (Macmillan, London, 1929), pp. 89–96.
  12. S. O. Rice, “Reflection of Electromagnetic Waves from Slightly Rough Surfaces,” Commun. Pure Appl. Math. 4, 351 (1951).
    [Crossref]
  13. G. R. Valenzuela, “Depolarization of EM Waves by Slightly Rough Surfaces,” IEEE Trans. Antennas Propag. AP-15, 552 (1967).
    [Crossref]
  14. E. Kroger, E. Kretschmann, “Scattering of Light by Slightly Rough Surfaces or Thin Films Including Plasma Resonance Emission,” Z. Phys. 237, 1 (1970).
    [Crossref]
  15. J. M. Elson, R. H. Ritchie, “Photon Interaction at a Rough Metal Surface,” Phys. Rev. B 4, 4129 (1971).
    [Crossref]
  16. A. A. Maradudin, D. L. Mills, “Scattering and Absorption of Electromagnetic Radiation by a Semi-Infinite Medium in the Presence of Surface Roughness,” Phys. Rev. B 11, 1392 (1975).
    [Crossref]
  17. V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
    [Crossref]
  18. A. Marvin, F. Toigo, V. Celli, “Light Scattering from Rough Surfaces: General Incidence Angle and Polarization,” Phys. Rev. B 11, 2777 (1975).
    [Crossref]
  19. R. F. Millar, “The Rayleigh Hypothesis and a Related Least Squares Solution to Scattering Problems for Periodic Surfaces and Other Scatterers,” Radio Sci. 8, 785 (1973).
    [Crossref]
  20. N. R. Hill, V. Celli, “Limits of Convergence of the Rayleigh Method for Surface Scattering,” Phys. Rev. B 17, 2478 (1978).
    [Crossref]
  21. P. M. van den Berg, J. T. Fokkema, “The Rayleigh Hypothesis in the Theory of Reflection by a Grating,” J. Opt. Soc. Am. 69, 27 (1979).
    [Crossref]
  22. W. A. Schlup, “On the Convergence of the Rayleigh Ansatz for Hard-Wall Scattering on Arbitrary Periodic Surface Profiles,” J. Phys. 17, 2607 (1984).
  23. D. Maystre, M. Cadilhac, “Singularities of the Continuation of Fields and Validity of Rayleigh’s Hypothesis,” J. Math. Phys. 26, 2201 (1985).
    [Crossref]
  24. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. 15, 5618 (1977).
    [Crossref]
  25. G. S. Agarwal, “Interaction of Electromagnetic Waves at Rough Dielectric Surfaces,” Phys. Rev. 15, 2371 (1977).
    [Crossref]
  26. N. Garcia, V. Celli, M. Nieto-Vesperinas, “Exact Multiple Scattering of Waves from Random Rough Surfaces,” Opt. Commun. 30, 279 (1979).
    [Crossref]
  27. M. Nieto-Vesperinas, N. Garcia, “A Detailed Study of the Scattering of Scalar Waves from Random Rough Surfaces,” Opt. Acta 28, 1651 (1981).
    [Crossref]
  28. J. Shen, A. A. Maradudin, “Multiple Scattering of Waves from Random Rough Surfaces,” Phys. Rev. B 22, 4234 (1980).
    [Crossref]
  29. N. E. Glass, R. Loudon, A. A. Maradudin, “Propagation of Rayleigh Surface Waves Across a Large-Amplitude Grating,” Phys. Rev. 24, 6843 (1981).
    [Crossref]
  30. D. Maystre, O. Mata Mendez, A. Roger, “A New Electromagnetic Theory for Scattering from Shallow Rough Surfaces,” Opt. Acta 30, 1707 (1983).
    [Crossref]
  31. D. Maystre, “Rigorous Theory of Light Scattering from Rough Surfaces,” J. Opt. 15, 43 (1984).
    [Crossref]
  32. D. Winebrenner, A. Ishimaru, “Investigation of a Surface Field Phase Perturbation Technique for Scattering from Rough Surfaces,” Radio Sci. 20, 161 (1985).
    [Crossref]
  33. J. M. Elson, “Light Scattering from Surfaces with a Single Dielectric Overlayer,” J. Opt. Soc. Am. 66, 682 (1976).
    [Crossref]
  34. J. M. Elson, “Diffraction and Diffuse Scattering from Dielectric Multilayers,” J. Opt. Soc. Am. 69, 48 (1979).
    [Crossref]
  35. J. M. Elson, J. P. Rahn, J. M. Bennett, “Light Scattering from Multilayer Optics: Comparison of Theory and Experiment,” Appl. Opt. 19, 669 (1980).
    [Crossref] [PubMed]
  36. J. M. Elson, J. P. Rahn, J. M. Bennett, “Relationship of the Total Integrated Scattering from Multilayer-Coated Optics to Angle of Incidence, Polarization, Correlation Length, and Roughness Cross-Correlation Properties,” Appl. Opt. 22, 3207 (1983).
    [Crossref] [PubMed]
  37. J. W. Wright, “A New Model for Sea Clutter,” IEEE Trans. Antennas Propag. AP-16, 217 (1968).
    [Crossref]
  38. A. K. Fung, “Backscattering of Waves by Composite Rough Surfaces,” IEEE Trans. Antennas Propag. AP-17, 590 (1969).
    [Crossref]
  39. S. T. Wu, A. K. Fung, “A Noncoherent Model for Microwave Emissions and Backscattering from the Sea Surface,” J. Geophys. Res. 77, 5917 (1972).
    [Crossref]
  40. J. C. Leader, “An Analysis of the Spatial Coherence of Laser Light Scattered from a Surface with Two Scales of Roughness,” J. Opt. Soc. Am. 66, 536 (1976).
    [Crossref]
  41. J. C. Leader, “Incoherent Backscatter from Rough Surfaces: The Two-Scale Model Reexamined,” Radio Sci. 13, 441 (1978).
    [Crossref]
  42. G. S. Brown, “Backscattering from a Gaussian-Distributed Perfectly Conducting Rough Surface,” IEEE Trans. Antennas Propag. AP-28, 943 (1980).
    [Crossref]
  43. F. T. Ulaby, F. Kouyate, A. K. Fung, A. J. Sieber, “A Backscatter Model for a Randomly Perturbed Periodic Surface,” IEEE Trans. Geosci. Remote Sensing GE-20, 518 (1982).
    [Crossref]
  44. E. Bahar, C. L. Rufenach, D. E. Barrick, M. A. Fitzwater, “Scattering Cross Section Modulation for Arbitrarily Oriented Composite Rough Surfaces: Full Wave Approach,” Radio Sci. 18, 675 (1983).
    [Crossref]
  45. G. R. Valenzuela, “The Effective Reflection Coefficients in Forward Scattering from a Dielectric Slightly Rough Surface,” IEEE Proc. 58, 1279 (1970).
    [Crossref]
  46. J. C. Leader, A. K. Fung, “Comments on the Evaluation of the Irradiance of Reflected Light from Rough Surfaces,” J. Appl. Phys. 48, 1736 (1977).
    [Crossref]
  47. J. M. Elson, J. M. Bennett, “Relation Between the Angular Dependence of Scattering and the Statistical Properties of Optical Surfaces,” J. Opt. Soc. Am. 69, 31 (1979).
    [Crossref]
  48. R. Schiffer, K. O. Thielheim, “The Effect of Slight Surface Roughness on the Scattering Properties of ConvexParticles,” J. Appl. Phys. 57, 2437 (1985).
    [Crossref]
  49. R. Schiffer, “The Effect of Surface Roughness on the Spectral Reflectance of Dielectric Particles. Application to the Zodiacal Light,” Astron. Astrophys. 148, 347 (1985).
  50. H. Schade, Z. E. Smith, “Mie Scattering and Rough Surfaces,” Appl. Opt. 24, 3221 (1985).
    [Crossref] [PubMed]

1985 (5)

D. Maystre, M. Cadilhac, “Singularities of the Continuation of Fields and Validity of Rayleigh’s Hypothesis,” J. Math. Phys. 26, 2201 (1985).
[Crossref]

D. Winebrenner, A. Ishimaru, “Investigation of a Surface Field Phase Perturbation Technique for Scattering from Rough Surfaces,” Radio Sci. 20, 161 (1985).
[Crossref]

R. Schiffer, K. O. Thielheim, “The Effect of Slight Surface Roughness on the Scattering Properties of ConvexParticles,” J. Appl. Phys. 57, 2437 (1985).
[Crossref]

R. Schiffer, “The Effect of Surface Roughness on the Spectral Reflectance of Dielectric Particles. Application to the Zodiacal Light,” Astron. Astrophys. 148, 347 (1985).

H. Schade, Z. E. Smith, “Mie Scattering and Rough Surfaces,” Appl. Opt. 24, 3221 (1985).
[Crossref] [PubMed]

1984 (2)

D. Maystre, “Rigorous Theory of Light Scattering from Rough Surfaces,” J. Opt. 15, 43 (1984).
[Crossref]

W. A. Schlup, “On the Convergence of the Rayleigh Ansatz for Hard-Wall Scattering on Arbitrary Periodic Surface Profiles,” J. Phys. 17, 2607 (1984).

1983 (3)

J. M. Elson, J. P. Rahn, J. M. Bennett, “Relationship of the Total Integrated Scattering from Multilayer-Coated Optics to Angle of Incidence, Polarization, Correlation Length, and Roughness Cross-Correlation Properties,” Appl. Opt. 22, 3207 (1983).
[Crossref] [PubMed]

D. Maystre, O. Mata Mendez, A. Roger, “A New Electromagnetic Theory for Scattering from Shallow Rough Surfaces,” Opt. Acta 30, 1707 (1983).
[Crossref]

E. Bahar, C. L. Rufenach, D. E. Barrick, M. A. Fitzwater, “Scattering Cross Section Modulation for Arbitrarily Oriented Composite Rough Surfaces: Full Wave Approach,” Radio Sci. 18, 675 (1983).
[Crossref]

1982 (1)

F. T. Ulaby, F. Kouyate, A. K. Fung, A. J. Sieber, “A Backscatter Model for a Randomly Perturbed Periodic Surface,” IEEE Trans. Geosci. Remote Sensing GE-20, 518 (1982).
[Crossref]

1981 (2)

N. E. Glass, R. Loudon, A. A. Maradudin, “Propagation of Rayleigh Surface Waves Across a Large-Amplitude Grating,” Phys. Rev. 24, 6843 (1981).
[Crossref]

M. Nieto-Vesperinas, N. Garcia, “A Detailed Study of the Scattering of Scalar Waves from Random Rough Surfaces,” Opt. Acta 28, 1651 (1981).
[Crossref]

1980 (3)

J. Shen, A. A. Maradudin, “Multiple Scattering of Waves from Random Rough Surfaces,” Phys. Rev. B 22, 4234 (1980).
[Crossref]

G. S. Brown, “Backscattering from a Gaussian-Distributed Perfectly Conducting Rough Surface,” IEEE Trans. Antennas Propag. AP-28, 943 (1980).
[Crossref]

J. M. Elson, J. P. Rahn, J. M. Bennett, “Light Scattering from Multilayer Optics: Comparison of Theory and Experiment,” Appl. Opt. 19, 669 (1980).
[Crossref] [PubMed]

1979 (4)

1978 (3)

N. R. Hill, V. Celli, “Limits of Convergence of the Rayleigh Method for Surface Scattering,” Phys. Rev. B 17, 2478 (1978).
[Crossref]

J. A. Holzer, C. C. Sung, “Scattering of Electromagnetic Waves from a Rough Surface II,” J. Appl. Phys. 49, 1002 (1978).
[Crossref]

J. C. Leader, “Incoherent Backscatter from Rough Surfaces: The Two-Scale Model Reexamined,” Radio Sci. 13, 441 (1978).
[Crossref]

1977 (3)

J. C. Leader, A. K. Fung, “Comments on the Evaluation of the Irradiance of Reflected Light from Rough Surfaces,” J. Appl. Phys. 48, 1736 (1977).
[Crossref]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. 15, 5618 (1977).
[Crossref]

G. S. Agarwal, “Interaction of Electromagnetic Waves at Rough Dielectric Surfaces,” Phys. Rev. 15, 2371 (1977).
[Crossref]

1976 (4)

J. M. Elson, “Light Scattering from Surfaces with a Single Dielectric Overlayer,” J. Opt. Soc. Am. 66, 682 (1976).
[Crossref]

C. C. Sung, J. A. Holzer, “Scattering of Electromagnetic Waves from a Rough Surface,” Appl. Phys. Lett. 28, 429 (1976).
[Crossref]

J. A. Holzer, C. C. Sung, “Evaluation of the Irradiance of Reflected Light from Rough Surfaces,” J. Appl. Phys. 47, 3363 (1976).
[Crossref]

J. C. Leader, “An Analysis of the Spatial Coherence of Laser Light Scattered from a Surface with Two Scales of Roughness,” J. Opt. Soc. Am. 66, 536 (1976).
[Crossref]

1975 (3)

A. A. Maradudin, D. L. Mills, “Scattering and Absorption of Electromagnetic Radiation by a Semi-Infinite Medium in the Presence of Surface Roughness,” Phys. Rev. B 11, 1392 (1975).
[Crossref]

V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
[Crossref]

A. Marvin, F. Toigo, V. Celli, “Light Scattering from Rough Surfaces: General Incidence Angle and Polarization,” Phys. Rev. B 11, 2777 (1975).
[Crossref]

1973 (1)

R. F. Millar, “The Rayleigh Hypothesis and a Related Least Squares Solution to Scattering Problems for Periodic Surfaces and Other Scatterers,” Radio Sci. 8, 785 (1973).
[Crossref]

1972 (1)

S. T. Wu, A. K. Fung, “A Noncoherent Model for Microwave Emissions and Backscattering from the Sea Surface,” J. Geophys. Res. 77, 5917 (1972).
[Crossref]

1971 (1)

J. M. Elson, R. H. Ritchie, “Photon Interaction at a Rough Metal Surface,” Phys. Rev. B 4, 4129 (1971).
[Crossref]

1970 (2)

E. Kroger, E. Kretschmann, “Scattering of Light by Slightly Rough Surfaces or Thin Films Including Plasma Resonance Emission,” Z. Phys. 237, 1 (1970).
[Crossref]

G. R. Valenzuela, “The Effective Reflection Coefficients in Forward Scattering from a Dielectric Slightly Rough Surface,” IEEE Proc. 58, 1279 (1970).
[Crossref]

1969 (2)

A. K. Fung, “Backscattering of Waves by Composite Rough Surfaces,” IEEE Trans. Antennas Propag. AP-17, 590 (1969).
[Crossref]

M. I. Sancer, “Shadow-Corrected Electromagnetic Scattering from a Randomly Rough Surface,” IEEE Trans. Antennas Propag. AP-17, 577 (1969).
[Crossref]

1968 (3)

D. E. Barrick, “Relationship between Slope Probability Density Function and the Physical Optics Integral in Rough Surface Scattering,” Proc. IEEE 56, 1728 (1968).
[Crossref]

D. E. Barrick, “Rough Surface Scattering Based on the Specular Point Theory,” IEEE Trans. Antennas Propag. AP-16, 449 (1968).
[Crossref]

J. W. Wright, “A New Model for Sea Clutter,” IEEE Trans. Antennas Propag. AP-16, 217 (1968).
[Crossref]

1967 (1)

G. R. Valenzuela, “Depolarization of EM Waves by Slightly Rough Surfaces,” IEEE Trans. Antennas Propag. AP-15, 552 (1967).
[Crossref]

1966 (1)

R. D. Kodis, “A Note on the Theory of Scattering from an Irregular Surface,” IEEE Trans. Antennas Propag. AP-14, 77 (1966).
[Crossref]

1964 (1)

D. O. Muhleman, “Radar Scattering from Venus and the Moon,” Astron. J. 69, 34 (1964).
[Crossref]

1951 (1)

S. O. Rice, “Reflection of Electromagnetic Waves from Slightly Rough Surfaces,” Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, “Interaction of Electromagnetic Waves at Rough Dielectric Surfaces,” Phys. Rev. 15, 2371 (1977).
[Crossref]

Bahar, E.

E. Bahar, C. L. Rufenach, D. E. Barrick, M. A. Fitzwater, “Scattering Cross Section Modulation for Arbitrarily Oriented Composite Rough Surfaces: Full Wave Approach,” Radio Sci. 18, 675 (1983).
[Crossref]

Barrick, D. E.

E. Bahar, C. L. Rufenach, D. E. Barrick, M. A. Fitzwater, “Scattering Cross Section Modulation for Arbitrarily Oriented Composite Rough Surfaces: Full Wave Approach,” Radio Sci. 18, 675 (1983).
[Crossref]

D. E. Barrick, “Relationship between Slope Probability Density Function and the Physical Optics Integral in Rough Surface Scattering,” Proc. IEEE 56, 1728 (1968).
[Crossref]

D. E. Barrick, “Rough Surface Scattering Based on the Specular Point Theory,” IEEE Trans. Antennas Propag. AP-16, 449 (1968).
[Crossref]

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.

Beckmann, P.

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

Bennett, J. M.

Brown, G. S.

G. S. Brown, “Backscattering from a Gaussian-Distributed Perfectly Conducting Rough Surface,” IEEE Trans. Antennas Propag. AP-28, 943 (1980).
[Crossref]

Cadilhac, M.

D. Maystre, M. Cadilhac, “Singularities of the Continuation of Fields and Validity of Rayleigh’s Hypothesis,” J. Math. Phys. 26, 2201 (1985).
[Crossref]

Celli, V.

N. Garcia, V. Celli, M. Nieto-Vesperinas, “Exact Multiple Scattering of Waves from Random Rough Surfaces,” Opt. Commun. 30, 279 (1979).
[Crossref]

N. R. Hill, V. Celli, “Limits of Convergence of the Rayleigh Method for Surface Scattering,” Phys. Rev. B 17, 2478 (1978).
[Crossref]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. 15, 5618 (1977).
[Crossref]

A. Marvin, F. Toigo, V. Celli, “Light Scattering from Rough Surfaces: General Incidence Angle and Polarization,” Phys. Rev. B 11, 2777 (1975).
[Crossref]

V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
[Crossref]

Elson, J. M.

Fitzwater, M. A.

E. Bahar, C. L. Rufenach, D. E. Barrick, M. A. Fitzwater, “Scattering Cross Section Modulation for Arbitrarily Oriented Composite Rough Surfaces: Full Wave Approach,” Radio Sci. 18, 675 (1983).
[Crossref]

Fokkema, J. T.

Fung, A. K.

F. T. Ulaby, F. Kouyate, A. K. Fung, A. J. Sieber, “A Backscatter Model for a Randomly Perturbed Periodic Surface,” IEEE Trans. Geosci. Remote Sensing GE-20, 518 (1982).
[Crossref]

J. C. Leader, A. K. Fung, “Comments on the Evaluation of the Irradiance of Reflected Light from Rough Surfaces,” J. Appl. Phys. 48, 1736 (1977).
[Crossref]

S. T. Wu, A. K. Fung, “A Noncoherent Model for Microwave Emissions and Backscattering from the Sea Surface,” J. Geophys. Res. 77, 5917 (1972).
[Crossref]

A. K. Fung, “Backscattering of Waves by Composite Rough Surfaces,” IEEE Trans. Antennas Propag. AP-17, 590 (1969).
[Crossref]

Garcia, N.

M. Nieto-Vesperinas, N. Garcia, “A Detailed Study of the Scattering of Scalar Waves from Random Rough Surfaces,” Opt. Acta 28, 1651 (1981).
[Crossref]

N. Garcia, V. Celli, M. Nieto-Vesperinas, “Exact Multiple Scattering of Waves from Random Rough Surfaces,” Opt. Commun. 30, 279 (1979).
[Crossref]

Glass, N. E.

N. E. Glass, R. Loudon, A. A. Maradudin, “Propagation of Rayleigh Surface Waves Across a Large-Amplitude Grating,” Phys. Rev. 24, 6843 (1981).
[Crossref]

Hill, N. R.

N. R. Hill, V. Celli, “Limits of Convergence of the Rayleigh Method for Surface Scattering,” Phys. Rev. B 17, 2478 (1978).
[Crossref]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. 15, 5618 (1977).
[Crossref]

Holzer, J. A.

J. A. Holzer, C. C. Sung, “Scattering of Electromagnetic Waves from a Rough Surface II,” J. Appl. Phys. 49, 1002 (1978).
[Crossref]

C. C. Sung, J. A. Holzer, “Scattering of Electromagnetic Waves from a Rough Surface,” Appl. Phys. Lett. 28, 429 (1976).
[Crossref]

J. A. Holzer, C. C. Sung, “Evaluation of the Irradiance of Reflected Light from Rough Surfaces,” J. Appl. Phys. 47, 3363 (1976).
[Crossref]

Ishimaru, A.

D. Winebrenner, A. Ishimaru, “Investigation of a Surface Field Phase Perturbation Technique for Scattering from Rough Surfaces,” Radio Sci. 20, 161 (1985).
[Crossref]

Kodis, R. D.

R. D. Kodis, “A Note on the Theory of Scattering from an Irregular Surface,” IEEE Trans. Antennas Propag. AP-14, 77 (1966).
[Crossref]

Kouyate, F.

F. T. Ulaby, F. Kouyate, A. K. Fung, A. J. Sieber, “A Backscatter Model for a Randomly Perturbed Periodic Surface,” IEEE Trans. Geosci. Remote Sensing GE-20, 518 (1982).
[Crossref]

Kretschmann, E.

E. Kroger, E. Kretschmann, “Scattering of Light by Slightly Rough Surfaces or Thin Films Including Plasma Resonance Emission,” Z. Phys. 237, 1 (1970).
[Crossref]

Krichbaum, C. K.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.

Kroger, E.

E. Kroger, E. Kretschmann, “Scattering of Light by Slightly Rough Surfaces or Thin Films Including Plasma Resonance Emission,” Z. Phys. 237, 1 (1970).
[Crossref]

Leader, J. C.

J. C. Leader, “Incoherent Backscatter from Rough Surfaces: The Two-Scale Model Reexamined,” Radio Sci. 13, 441 (1978).
[Crossref]

J. C. Leader, A. K. Fung, “Comments on the Evaluation of the Irradiance of Reflected Light from Rough Surfaces,” J. Appl. Phys. 48, 1736 (1977).
[Crossref]

J. C. Leader, “An Analysis of the Spatial Coherence of Laser Light Scattered from a Surface with Two Scales of Roughness,” J. Opt. Soc. Am. 66, 536 (1976).
[Crossref]

Loudon, R.

N. E. Glass, R. Loudon, A. A. Maradudin, “Propagation of Rayleigh Surface Waves Across a Large-Amplitude Grating,” Phys. Rev. 24, 6843 (1981).
[Crossref]

Maradudin, A. A.

N. E. Glass, R. Loudon, A. A. Maradudin, “Propagation of Rayleigh Surface Waves Across a Large-Amplitude Grating,” Phys. Rev. 24, 6843 (1981).
[Crossref]

J. Shen, A. A. Maradudin, “Multiple Scattering of Waves from Random Rough Surfaces,” Phys. Rev. B 22, 4234 (1980).
[Crossref]

A. A. Maradudin, D. L. Mills, “Scattering and Absorption of Electromagnetic Radiation by a Semi-Infinite Medium in the Presence of Surface Roughness,” Phys. Rev. B 11, 1392 (1975).
[Crossref]

Marvin, A.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. 15, 5618 (1977).
[Crossref]

V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
[Crossref]

A. Marvin, F. Toigo, V. Celli, “Light Scattering from Rough Surfaces: General Incidence Angle and Polarization,” Phys. Rev. B 11, 2777 (1975).
[Crossref]

Mata Mendez, O.

D. Maystre, O. Mata Mendez, A. Roger, “A New Electromagnetic Theory for Scattering from Shallow Rough Surfaces,” Opt. Acta 30, 1707 (1983).
[Crossref]

Maystre, D.

D. Maystre, M. Cadilhac, “Singularities of the Continuation of Fields and Validity of Rayleigh’s Hypothesis,” J. Math. Phys. 26, 2201 (1985).
[Crossref]

D. Maystre, “Rigorous Theory of Light Scattering from Rough Surfaces,” J. Opt. 15, 43 (1984).
[Crossref]

D. Maystre, O. Mata Mendez, A. Roger, “A New Electromagnetic Theory for Scattering from Shallow Rough Surfaces,” Opt. Acta 30, 1707 (1983).
[Crossref]

Millar, R. F.

R. F. Millar, “The Rayleigh Hypothesis and a Related Least Squares Solution to Scattering Problems for Periodic Surfaces and Other Scatterers,” Radio Sci. 8, 785 (1973).
[Crossref]

Mills, D. L.

A. A. Maradudin, D. L. Mills, “Scattering and Absorption of Electromagnetic Radiation by a Semi-Infinite Medium in the Presence of Surface Roughness,” Phys. Rev. B 11, 1392 (1975).
[Crossref]

Muhleman, D. O.

D. O. Muhleman, “Radar Scattering from Venus and the Moon,” Astron. J. 69, 34 (1964).
[Crossref]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, N. Garcia, “A Detailed Study of the Scattering of Scalar Waves from Random Rough Surfaces,” Opt. Acta 28, 1651 (1981).
[Crossref]

N. Garcia, V. Celli, M. Nieto-Vesperinas, “Exact Multiple Scattering of Waves from Random Rough Surfaces,” Opt. Commun. 30, 279 (1979).
[Crossref]

Rahn, J. P.

Rayleigh, Lord

Lord Rayleigh, Theory of Sound, Vol. 2 (Macmillan, London, 1929), pp. 89–96.

Rice, S. O.

S. O. Rice, “Reflection of Electromagnetic Waves from Slightly Rough Surfaces,” Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

Ritchie, R. H.

J. M. Elson, R. H. Ritchie, “Photon Interaction at a Rough Metal Surface,” Phys. Rev. B 4, 4129 (1971).
[Crossref]

Roger, A.

D. Maystre, O. Mata Mendez, A. Roger, “A New Electromagnetic Theory for Scattering from Shallow Rough Surfaces,” Opt. Acta 30, 1707 (1983).
[Crossref]

Ruck, G. T.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.

Rufenach, C. L.

E. Bahar, C. L. Rufenach, D. E. Barrick, M. A. Fitzwater, “Scattering Cross Section Modulation for Arbitrarily Oriented Composite Rough Surfaces: Full Wave Approach,” Radio Sci. 18, 675 (1983).
[Crossref]

Sancer, M. I.

M. I. Sancer, “Shadow-Corrected Electromagnetic Scattering from a Randomly Rough Surface,” IEEE Trans. Antennas Propag. AP-17, 577 (1969).
[Crossref]

Schade, H.

Schiffer, R.

R. Schiffer, “The Effect of Surface Roughness on the Spectral Reflectance of Dielectric Particles. Application to the Zodiacal Light,” Astron. Astrophys. 148, 347 (1985).

R. Schiffer, K. O. Thielheim, “The Effect of Slight Surface Roughness on the Scattering Properties of ConvexParticles,” J. Appl. Phys. 57, 2437 (1985).
[Crossref]

Schlup, W. A.

W. A. Schlup, “On the Convergence of the Rayleigh Ansatz for Hard-Wall Scattering on Arbitrary Periodic Surface Profiles,” J. Phys. 17, 2607 (1984).

Shen, J.

J. Shen, A. A. Maradudin, “Multiple Scattering of Waves from Random Rough Surfaces,” Phys. Rev. B 22, 4234 (1980).
[Crossref]

Sieber, A. J.

F. T. Ulaby, F. Kouyate, A. K. Fung, A. J. Sieber, “A Backscatter Model for a Randomly Perturbed Periodic Surface,” IEEE Trans. Geosci. Remote Sensing GE-20, 518 (1982).
[Crossref]

Smith, Z. E.

Spizzichino, A.

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

Stuart, W. D.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.

Sung, C. C.

J. A. Holzer, C. C. Sung, “Scattering of Electromagnetic Waves from a Rough Surface II,” J. Appl. Phys. 49, 1002 (1978).
[Crossref]

J. A. Holzer, C. C. Sung, “Evaluation of the Irradiance of Reflected Light from Rough Surfaces,” J. Appl. Phys. 47, 3363 (1976).
[Crossref]

C. C. Sung, J. A. Holzer, “Scattering of Electromagnetic Waves from a Rough Surface,” Appl. Phys. Lett. 28, 429 (1976).
[Crossref]

Thielheim, K. O.

R. Schiffer, K. O. Thielheim, “The Effect of Slight Surface Roughness on the Scattering Properties of ConvexParticles,” J. Appl. Phys. 57, 2437 (1985).
[Crossref]

Toigo, F.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. 15, 5618 (1977).
[Crossref]

A. Marvin, F. Toigo, V. Celli, “Light Scattering from Rough Surfaces: General Incidence Angle and Polarization,” Phys. Rev. B 11, 2777 (1975).
[Crossref]

V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
[Crossref]

Ulaby, F. T.

F. T. Ulaby, F. Kouyate, A. K. Fung, A. J. Sieber, “A Backscatter Model for a Randomly Perturbed Periodic Surface,” IEEE Trans. Geosci. Remote Sensing GE-20, 518 (1982).
[Crossref]

Valenzuela, G. R.

G. R. Valenzuela, “The Effective Reflection Coefficients in Forward Scattering from a Dielectric Slightly Rough Surface,” IEEE Proc. 58, 1279 (1970).
[Crossref]

G. R. Valenzuela, “Depolarization of EM Waves by Slightly Rough Surfaces,” IEEE Trans. Antennas Propag. AP-15, 552 (1967).
[Crossref]

van den Berg, P. M.

Winebrenner, D.

D. Winebrenner, A. Ishimaru, “Investigation of a Surface Field Phase Perturbation Technique for Scattering from Rough Surfaces,” Radio Sci. 20, 161 (1985).
[Crossref]

Wright, J. W.

J. W. Wright, “A New Model for Sea Clutter,” IEEE Trans. Antennas Propag. AP-16, 217 (1968).
[Crossref]

Wu, S. T.

S. T. Wu, A. K. Fung, “A Noncoherent Model for Microwave Emissions and Backscattering from the Sea Surface,” J. Geophys. Res. 77, 5917 (1972).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. C. Sung, J. A. Holzer, “Scattering of Electromagnetic Waves from a Rough Surface,” Appl. Phys. Lett. 28, 429 (1976).
[Crossref]

Astron. Astrophys. (1)

R. Schiffer, “The Effect of Surface Roughness on the Spectral Reflectance of Dielectric Particles. Application to the Zodiacal Light,” Astron. Astrophys. 148, 347 (1985).

Astron. J. (1)

D. O. Muhleman, “Radar Scattering from Venus and the Moon,” Astron. J. 69, 34 (1964).
[Crossref]

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of Electromagnetic Waves from Slightly Rough Surfaces,” Commun. Pure Appl. Math. 4, 351 (1951).
[Crossref]

IEEE Proc (1)

G. R. Valenzuela, “The Effective Reflection Coefficients in Forward Scattering from a Dielectric Slightly Rough Surface,” IEEE Proc. 58, 1279 (1970).
[Crossref]

IEEE Trans. Antennas Propag. (7)

G. R. Valenzuela, “Depolarization of EM Waves by Slightly Rough Surfaces,” IEEE Trans. Antennas Propag. AP-15, 552 (1967).
[Crossref]

R. D. Kodis, “A Note on the Theory of Scattering from an Irregular Surface,” IEEE Trans. Antennas Propag. AP-14, 77 (1966).
[Crossref]

D. E. Barrick, “Rough Surface Scattering Based on the Specular Point Theory,” IEEE Trans. Antennas Propag. AP-16, 449 (1968).
[Crossref]

M. I. Sancer, “Shadow-Corrected Electromagnetic Scattering from a Randomly Rough Surface,” IEEE Trans. Antennas Propag. AP-17, 577 (1969).
[Crossref]

J. W. Wright, “A New Model for Sea Clutter,” IEEE Trans. Antennas Propag. AP-16, 217 (1968).
[Crossref]

A. K. Fung, “Backscattering of Waves by Composite Rough Surfaces,” IEEE Trans. Antennas Propag. AP-17, 590 (1969).
[Crossref]

G. S. Brown, “Backscattering from a Gaussian-Distributed Perfectly Conducting Rough Surface,” IEEE Trans. Antennas Propag. AP-28, 943 (1980).
[Crossref]

IEEE Trans. Geosci. Remote Sensing (1)

F. T. Ulaby, F. Kouyate, A. K. Fung, A. J. Sieber, “A Backscatter Model for a Randomly Perturbed Periodic Surface,” IEEE Trans. Geosci. Remote Sensing GE-20, 518 (1982).
[Crossref]

J. Appl. Phys. (4)

J. A. Holzer, C. C. Sung, “Evaluation of the Irradiance of Reflected Light from Rough Surfaces,” J. Appl. Phys. 47, 3363 (1976).
[Crossref]

J. A. Holzer, C. C. Sung, “Scattering of Electromagnetic Waves from a Rough Surface II,” J. Appl. Phys. 49, 1002 (1978).
[Crossref]

J. C. Leader, A. K. Fung, “Comments on the Evaluation of the Irradiance of Reflected Light from Rough Surfaces,” J. Appl. Phys. 48, 1736 (1977).
[Crossref]

R. Schiffer, K. O. Thielheim, “The Effect of Slight Surface Roughness on the Scattering Properties of ConvexParticles,” J. Appl. Phys. 57, 2437 (1985).
[Crossref]

J. Geophys. Res. (1)

S. T. Wu, A. K. Fung, “A Noncoherent Model for Microwave Emissions and Backscattering from the Sea Surface,” J. Geophys. Res. 77, 5917 (1972).
[Crossref]

J. Math. Phys. (1)

D. Maystre, M. Cadilhac, “Singularities of the Continuation of Fields and Validity of Rayleigh’s Hypothesis,” J. Math. Phys. 26, 2201 (1985).
[Crossref]

J. Opt. (1)

D. Maystre, “Rigorous Theory of Light Scattering from Rough Surfaces,” J. Opt. 15, 43 (1984).
[Crossref]

J. Opt. Soc. Am. (5)

J. Phys. (1)

W. A. Schlup, “On the Convergence of the Rayleigh Ansatz for Hard-Wall Scattering on Arbitrary Periodic Surface Profiles,” J. Phys. 17, 2607 (1984).

Opt. Acta (2)

M. Nieto-Vesperinas, N. Garcia, “A Detailed Study of the Scattering of Scalar Waves from Random Rough Surfaces,” Opt. Acta 28, 1651 (1981).
[Crossref]

D. Maystre, O. Mata Mendez, A. Roger, “A New Electromagnetic Theory for Scattering from Shallow Rough Surfaces,” Opt. Acta 30, 1707 (1983).
[Crossref]

Opt. Commun. (1)

N. Garcia, V. Celli, M. Nieto-Vesperinas, “Exact Multiple Scattering of Waves from Random Rough Surfaces,” Opt. Commun. 30, 279 (1979).
[Crossref]

Phys. Rev. (3)

N. E. Glass, R. Loudon, A. A. Maradudin, “Propagation of Rayleigh Surface Waves Across a Large-Amplitude Grating,” Phys. Rev. 24, 6843 (1981).
[Crossref]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. 15, 5618 (1977).
[Crossref]

G. S. Agarwal, “Interaction of Electromagnetic Waves at Rough Dielectric Surfaces,” Phys. Rev. 15, 2371 (1977).
[Crossref]

Phys. Rev. B (6)

N. R. Hill, V. Celli, “Limits of Convergence of the Rayleigh Method for Surface Scattering,” Phys. Rev. B 17, 2478 (1978).
[Crossref]

J. Shen, A. A. Maradudin, “Multiple Scattering of Waves from Random Rough Surfaces,” Phys. Rev. B 22, 4234 (1980).
[Crossref]

J. M. Elson, R. H. Ritchie, “Photon Interaction at a Rough Metal Surface,” Phys. Rev. B 4, 4129 (1971).
[Crossref]

A. A. Maradudin, D. L. Mills, “Scattering and Absorption of Electromagnetic Radiation by a Semi-Infinite Medium in the Presence of Surface Roughness,” Phys. Rev. B 11, 1392 (1975).
[Crossref]

V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
[Crossref]

A. Marvin, F. Toigo, V. Celli, “Light Scattering from Rough Surfaces: General Incidence Angle and Polarization,” Phys. Rev. B 11, 2777 (1975).
[Crossref]

Proc. IEEE (1)

D. E. Barrick, “Relationship between Slope Probability Density Function and the Physical Optics Integral in Rough Surface Scattering,” Proc. IEEE 56, 1728 (1968).
[Crossref]

Radio Sci. (4)

R. F. Millar, “The Rayleigh Hypothesis and a Related Least Squares Solution to Scattering Problems for Periodic Surfaces and Other Scatterers,” Radio Sci. 8, 785 (1973).
[Crossref]

D. Winebrenner, A. Ishimaru, “Investigation of a Surface Field Phase Perturbation Technique for Scattering from Rough Surfaces,” Radio Sci. 20, 161 (1985).
[Crossref]

E. Bahar, C. L. Rufenach, D. E. Barrick, M. A. Fitzwater, “Scattering Cross Section Modulation for Arbitrarily Oriented Composite Rough Surfaces: Full Wave Approach,” Radio Sci. 18, 675 (1983).
[Crossref]

J. C. Leader, “Incoherent Backscatter from Rough Surfaces: The Two-Scale Model Reexamined,” Radio Sci. 13, 441 (1978).
[Crossref]

Z. Phys. (1)

E. Kroger, E. Kretschmann, “Scattering of Light by Slightly Rough Surfaces or Thin Films Including Plasma Resonance Emission,” Z. Phys. 237, 1 (1970).
[Crossref]

Other (3)

Lord Rayleigh, Theory of Sound, Vol. 2 (Macmillan, London, 1929), pp. 89–96.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.

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

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Figures (19)

Fig. 1
Fig. 1

Geometry defining incident and reflected rays.

Fig. 2
Fig. 2

Second-order incoherent reflectivity for horizontal polarization and a Gaussian correlation function.

Fig. 3
Fig. 3

Second-order incoherent reflectivity for horizontal polarization and an exponential correlation function.

Fig. 4
Fig. 4

Second-order incoherent reflectivity for vertical polarization and a Gaussian correlation function.

Fig. 5
Fig. 5

Second-order incoherent reflectivity for vertical polarization and an exponential correlation function.

Fig. 6
Fig. 6

Second-order coherent reflectivity for horizontal polarization and a Gaussian correlation function.

Fig. 7
Fig. 7

Second-order coherent reflectivity for horizontal polarization and an exponential correlation function.

Fig. 8
Fig. 8

Second-order coherent reflectivity for vertical polarization and a Gaussian correlation function.

Fig. 9
Fig. 9

Second-order coherent reflectivity for vertical polarization and an exponential correlation function.

Fig. 10
Fig. 10

Total reflectivity for horizontal polarization and a Gaussian correlation function.

Fig. 11
Fig. 11

Total reflectivity for vertical polarization and a Gaussian correlation function.

Fig. 12
Fig. 12

Relative total reflectivity as a function of wavelength normalized to 1 for λ = 0.4 μm, for horizontal polarization, a Gaussian correlation function, and a correlation length l = 100 nm.

Fig. 13
Fig. 13

Relative total reflectivity for horizontal polarization, an exponential correlation function, and a correlation length l = 100 nm.

Fig. 14
Fig. 14

Relative total reflectivity for vertical polarization, a Gaussian correlation function, and a correlation length l = 100 nm.

Fig. 15
Fig. 15

Relative total reflectivity for vertical polarization, an exponential correlation function, and a correlation length l = 100 nm.

Fig. 16
Fig. 16

Relative total reflectivity for vertical polarization, a Gaussian correlation function, and a correlation length l = 500 nm.

Fig. 17
Fig. 17

Relative total reflectivity for vertical polarization, an exponential correlation function, and a correlation length l = 500 nm.

Fig. 18
Fig. 18

Ratio of reflectivities at λ = 0.8 and 0.4 μm, respectively, for unpolarized light as a function of the angle of incidence for a Gaussian correlation function.

Fig. 19
Fig. 19

Ratio of reflectivities at λ = 0.8 and 0.4 μm, respectively, for unpolarized light as a function of the angle of incidence for an exponential correlation function.

Equations (26)

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

E s = E s ( 0 ) + E s ( 1 ) + E s ( 2 ) + ,
E s 2 = E s ( 0 ) 2 + E s ( 1 ) 2 + 2 Re [ E s ( 0 ) · E s ( 2 ) * ] + ,
exp ( - 4 h 2 k 2 cos 2 ψ i ) h k 1 1 - 4 h 2 k 2 cos 2 ψ i ,
k l 1.
h k 1
h k 1 ;
z = f ( x , y ) ,
f ( x , y ) = 0 ,
f ( x , y ) f ( x + Δ x , y + Δ y ) = h 2 ρ [ ( Δ x ) 2 + ( Δ y ) 2 ] ,
ρ ( 0 ) = 1 ,
ρ G ( r ) = exp ( - r 2 l 2 ) ,
ρ E ( r ) = exp ( - r l ) ,
ρ ˜ E ( r ) = 0 T t J 0 ( r l · t ) ( 1 + t 2 ) 3 / 2 d t ,
R H ( 0 ) = ( cos ψ i - - sin 2 ψ i cos ψ i + - sin 2 ψ i ) 2 ,
R V ( 0 ) = ( cos ψ i - - sin 2 ψ i cos ψ i + - sin 2 ψ i ) 2 ,
R H inc = k 4 cos ψ i 0 π / 2 d ψ r sin ψ r cos 2 ψ r 0 2 π d φ ( α H H 2 + α V H 2 ) · W ( k sin ψ i - k sin ψ r cos φ , - k sin ψ r sin φ ) ,
R V inc = k 4 cos ψ i 0 π / 2 d ψ r sin ψ r cos 2 ψ r 0 2 π d φ ( α V V 2 + α H V 2 ) · W ( k sin ψ i - k sin ψ r cos φ , - k sin ψ r sin φ ) ,
W ( p , q ) = h 2 π 2 - d x d y ρ ( x 2 + y 2 ) exp [ i ( p x + q y ) ] = 2 h 2 π 0 d r r ρ ( r ) J 0 ( r p 2 + q 2 )
α H H = - ( - 1 ) cos φ ( cos ψ i + - sin 2 ψ i ) ( cos ψ r + - sin 2 ψ r ) ,
α V V = ( - 1 ) ( sin ψ i sin ψ r - cos φ - sin 2 ψ i - sin 2 ψ r ) ( cos ψ i + - sin 2 ψ i ) ( cos ψ r + - sin 2 ψ r ) ,
α V H = - ( - 1 ) - sin 2 ψ r sin φ ( cos ψ i + - sin 2 ψ i ) ( cos ψ r + - sin 2 ψ r ) ,
α H V = ( - 1 ) - sin 2 ψ i sin φ ( cos ψ i + - sin 2 ψ i ) ( cos ψ r + - sin 2 ψ r ) ,
R H coh = - ( cos ψ i - - sin 2 ψ i cos ψ i + - sin 2 ψ i ) 2 H ,
H = cos ψ i Re [ 4 h 2 k 2 - sin 2 ψ i + k ( - 1 ) 0 d s s 0 2 π d φ W ( s cos φ - k sin ψ i , s sin φ ) · ( s 2 sin 2 φ k 2 - s 2 + k 2 - s 2 - k 2 k 2 - s 2 + k 2 - s 2 ) ]
R V coh = - ( - 1 ) ( cos ψ i - - sin 2 ψ i ) ( cos ψ i + - sin 2 ψ i ) 3 H ,
H = cos ψ i Re ( 4 h 2 k 2 - sin 2 ψ i + k 2 0 d s s 0 2 π d φ W ( s cos φ - k sin ψ i , s sin φ ) · { - ( - 1 ) k ( - sin 2 ψ i ) 1 k 2 - s 2 + k 2 - s 2 + - 1 k [ + ( - 1 ) sin 2 ψ i ] s 2 k 2 - s 2 + k 2 - s 2 - - 1 k ( - sin 2 ψ i ) s 2 sin 2 φ k 2 - s 2 + k 2 - s 2 - 2 k sin ψ i - sin 2 ψ i s cos φ k 2 - s 2 k 2 - s 2 + s 2 } ) .

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