Abstract

The backscattering cross section of a perfectly conducting, roughened sphere is presented. It is derived from the calculation of the electric-field cross correlation at two temporal frequencies and at two points in the backscatter direction. Careful treatment has been given to the calculation of the ensemble average of the surface height and slope terms, particularly in the backscatter from the off-axis regions of the sphere where the angles of illumination become large. At these large angles significant portions of the surface are in shadow and fail to contribute to the backscatter. The effect of shadowing is treated as a multiplying factor of the percentage of the surface that is illuminated. The cross correlation of the field is reduced to give the backscattering cross section. Both Gaussian and exponential surface correlations are considered, representing extremes in surface structure. The cross sections of both surfaces are shown to increase for certain ranges of rms roughness and correlation length. This increase is due to the backscatter from the annular regions of the sphere, which, when viewed as an image, would appear as a bright ring or halo. These results are supported by experimental measurements taken from roughened disks as a function of the illumination angle.

© 1994 Optical Society of America

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    [CrossRef]
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
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    [CrossRef]
  5. B. R. Levy, J. B. Keller, “Diffraction by a smooth object,” Commun. Pure Appl. Math. 12, 159–209 (1959).
    [CrossRef]
  6. D. L. Sengupta, “The sphere,” in Electromagnetic and Acoustic Scattering by Simple Shapes, J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, eds. (North-Holland, Amsterdam, 1969), Chap. 10, pp. 353–415.
  7. T. Hagfors, “Relations between rough surfaces and their scattering properties as applied to radar astronomy,” in Radar Astronomy, J. V. Evans, T. Hagfors, eds. (McGraw-Hill, New York, 1968), Chap. 4 pp. 187–209.
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    [CrossRef]
  9. R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1967).
    [CrossRef]
  10. B. G. Smith, “Geometric shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
    [CrossRef]
  11. M. J. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
    [CrossRef]
  12. R. Schiffer, K. O. Thielheim, “Light reflection from randomly oriented convex particles with rough surface,” J. Appl. Phys. 53, 2825–2830 (1982).
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  13. E. Bahar, S. Chakrabarti, “Scattering and depolarization by large conducting spheres with rough surfaces,” Appl. Opt. 24, 1820–1825 (1985).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  16. N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
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    [CrossRef]
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  22. R. Schiffer, “The coherent scattering cross-section of a slightly rough sphere,” Opt. Acta 33, 959–980 (1986).
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  23. R. Schiffer, “Light scattering by perfectly conducting statistically irregular particles,” J. Opt. Soc. Am. A 6, 385–402 (1989).
    [CrossRef]
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  28. J. Rheinstein, “Backscatter from spheres: a short pulse view,” IEEE Trans. Antennas Propag. AP-16, 89–97 (1968).
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  29. D. J. Schertler, N. George, “Backscattering cross section of a tilted, roughened disk,” J. Opt. Soc. Am. A 9, 2056–2066 (1992).
    [CrossRef]
  30. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
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    [CrossRef]
  32. L. Shaw, “Comments on ‘Shadowing of random surfaces,’” IEEE Trans. Antennas Propag. AP-14, 253 (1966).
    [CrossRef]
  33. R. A. Brockelman, T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. AP-14, 621–629 (1966).
    [CrossRef]
  34. P. J. Welton, K. E. Hawker, H. G. Frey, “Experimental shadowing measurements on randomly rough surfaces,” J. Acoust. Soc. Am. 54, 446–450 (1973).
    [CrossRef]
  35. G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970).
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  37. J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).
  38. T. Hagfors, “Backscattering from an undulating surface with applications to radar returns from the moon,”J. Geophys. Res. 69, 3779–3784 (1964).
    [CrossRef]
  39. A. Stogryn, “Electromagnetic scattering from rough, finitely conducting surfaces,” Radio Sci. 2, 415–428 (1967).
  40. D. E. Barrick, W. H. Peake, “A review of scattering from surfaces with different roughness scales,” Radio Sci. 3, 865–868 (1968).
  41. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978).
  42. M. K. Abdelazeez, “Wave scattering from a large sphere with rough surface,” IEEE Trans. Antennas Propag. AP-31, 375–377 (1983).
    [CrossRef]

1992 (1)

1991 (2)

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

N. C. Bruce, J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471–1481 (1991).
[CrossRef]

1989 (1)

1986 (1)

R. Schiffer, “The coherent scattering cross-section of a slightly rough sphere,” Opt. Acta 33, 959–980 (1986).
[CrossRef]

1985 (1)

1984 (1)

R. Schiffer, K. O. Thielheim, “Scattering of scalar waves by an impenetrable rough sphere,” Opt. Acta 31, 1085–1100 (1984).
[CrossRef]

1983 (2)

M. K. Abdelazeez, “Wave scattering from a large sphere with rough surface,” IEEE Trans. Antennas Propag. AP-31, 375–377 (1983).
[CrossRef]

E. Bahar, D. E. Barrick, “Scattering cross sections for composite surfaces that cannot be treated as perturbed-physical optics problems,” Radio Sci. 18, 129–137 (1983).
[CrossRef]

1982 (1)

R. Schiffer, K. O. Thielheim, “Light reflection from randomly oriented convex particles with rough surface,” J. Appl. Phys. 53, 2825–2830 (1982).
[CrossRef]

1981 (1)

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

1976 (3)

1974 (1)

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

1973 (1)

P. J. Welton, K. E. Hawker, H. G. Frey, “Experimental shadowing measurements on randomly rough surfaces,” J. Acoust. Soc. Am. 54, 446–450 (1973).
[CrossRef]

1969 (1)

M. J. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
[CrossRef]

1968 (2)

J. Rheinstein, “Backscatter from spheres: a short pulse view,” IEEE Trans. Antennas Propag. AP-16, 89–97 (1968).
[CrossRef]

D. E. Barrick, W. H. Peake, “A review of scattering from surfaces with different roughness scales,” Radio Sci. 3, 865–868 (1968).

1967 (3)

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

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1967).
[CrossRef]

B. G. Smith, “Geometric shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

1966 (2)

L. Shaw, “Comments on ‘Shadowing of random surfaces,’” IEEE Trans. Antennas Propag. AP-14, 253 (1966).
[CrossRef]

R. A. Brockelman, T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. AP-14, 621–629 (1966).
[CrossRef]

1965 (2)

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. AP-13, 384–388 (1965).
[CrossRef]

E. M. Kennaugh, D. L. Moffatt, “Transient and impulse response approximations,” Proc. IEEE 53, 893–901 (1965).
[CrossRef]

1964 (1)

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

1960 (1)

R. E. Hiatt, T. B. A. Senior, V. H. Weston, “A study of surface roughness and its effects on the backscattering cross section of spheres,” Proc. IRE 48, 2008–2016 (1960).
[CrossRef]

1959 (2)

V. H. Weston, “Pulse return from a sphere,” IRE Trans. Antennas Propag. AP-7, S43–S51 (1959).
[CrossRef]

B. R. Levy, J. B. Keller, “Diffraction by a smooth object,” Commun. Pure Appl. Math. 12, 159–209 (1959).
[CrossRef]

1949 (1)

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Physik 25, 377–445 (1908).
[CrossRef]

Abdelazeez, M. K.

M. K. Abdelazeez, “Wave scattering from a large sphere with rough surface,” IEEE Trans. Antennas Propag. AP-31, 375–377 (1983).
[CrossRef]

Bahar, E.

E. Bahar, S. Chakrabarti, “Scattering and depolarization by large conducting spheres with rough surfaces,” Appl. Opt. 24, 1820–1825 (1985).
[CrossRef] [PubMed]

E. Bahar, D. E. Barrick, “Scattering cross sections for composite surfaces that cannot be treated as perturbed-physical optics problems,” Radio Sci. 18, 129–137 (1983).
[CrossRef]

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

Barrick, D. E.

E. Bahar, D. E. Barrick, “Scattering cross sections for composite surfaces that cannot be treated as perturbed-physical optics problems,” Radio Sci. 18, 129–137 (1983).
[CrossRef]

D. E. Barrick, W. H. Peake, “A review of scattering from surfaces with different roughness scales,” Radio Sci. 3, 865–868 (1968).

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

Beal, M. M.

M. M. Beal, “Serrated circular apertures: optical Fourier transforms and fractal analysis,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1990).

Beckmann, P.

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. AP-13, 384–388 (1965).
[CrossRef]

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

Bowman, J. J.

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

Brillouin, L.

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

Brockelman, R. A.

R. A. Brockelman, T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. AP-14, 621–629 (1966).
[CrossRef]

Bruce, N. C.

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

N. C. Bruce, J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471–1481 (1991).
[CrossRef]

Chakrabarti, S.

Dainty, J. C.

N. C. Bruce, J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471–1481 (1991).
[CrossRef]

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

Erdmann, J. C.

Frey, H. G.

P. J. Welton, K. E. Hawker, H. G. Frey, “Experimental shadowing measurements on randomly rough surfaces,” J. Acoust. Soc. Am. 54, 446–450 (1973).
[CrossRef]

Gellert, R. I.

George, N.

D. J. Schertler, N. George, “Backscattering cross section of a tilted, roughened disk,” J. Opt. Soc. Am. A 9, 2056–2066 (1992).
[CrossRef]

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

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

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

Hagfors, T.

R. A. Brockelman, T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. AP-14, 621–629 (1966).
[CrossRef]

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

T. Hagfors, “Relations between rough surfaces and their scattering properties as applied to radar astronomy,” in Radar Astronomy, J. V. Evans, T. Hagfors, eds. (McGraw-Hill, New York, 1968), Chap. 4 pp. 187–209.

Hawker, K. E.

P. J. Welton, K. E. Hawker, H. G. Frey, “Experimental shadowing measurements on randomly rough surfaces,” J. Acoust. Soc. Am. 54, 446–450 (1973).
[CrossRef]

Hiatt, R. E.

R. E. Hiatt, T. B. A. Senior, V. H. Weston, “A study of surface roughness and its effects on the backscattering cross section of spheres,” Proc. IRE 48, 2008–2016 (1960).
[CrossRef]

Ishimaru, A.

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

Jain, A.

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

Keller, J. B.

B. R. Levy, J. B. Keller, “Diffraction by a smooth object,” Commun. Pure Appl. Math. 12, 159–209 (1959).
[CrossRef]

Kennaugh, E. M.

E. M. Kennaugh, D. L. Moffatt, “Transient and impulse response approximations,” Proc. IEEE 53, 893–901 (1965).
[CrossRef]

E. M. Kennaugh, “The impulse response concept in single-body scattering,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1965), pp. 217–236.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Krichbaum, C. K.

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

Levy, B. R.

B. R. Levy, J. B. Keller, “Diffraction by a smooth object,” Commun. Pure Appl. Math. 12, 159–209 (1959).
[CrossRef]

Livanos, A. C.

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

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Physik 25, 377–445 (1908).
[CrossRef]

Moffatt, D. L.

E. M. Kennaugh, D. L. Moffatt, “Transient and impulse response approximations,” Proc. IEEE 53, 893–901 (1965).
[CrossRef]

Papas, C. H.

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

Peake, W. H.

D. E. Barrick, W. H. Peake, “A review of scattering from surfaces with different roughness scales,” Radio Sci. 3, 865–868 (1968).

Rheinstein, J.

J. Rheinstein, “Backscatter from spheres: a short pulse view,” IEEE Trans. Antennas Propag. AP-16, 89–97 (1968).
[CrossRef]

Roth, J. A.

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

Ruck, G. T.

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

Sancer, M. J.

M. J. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
[CrossRef]

Schertler, D. J.

Schiffer, R.

R. Schiffer, “Light scattering by perfectly conducting statistically irregular particles,” J. Opt. Soc. Am. A 6, 385–402 (1989).
[CrossRef]

R. Schiffer, “The coherent scattering cross-section of a slightly rough sphere,” Opt. Acta 33, 959–980 (1986).
[CrossRef]

R. Schiffer, K. O. Thielheim, “Scattering of scalar waves by an impenetrable rough sphere,” Opt. Acta 31, 1085–1100 (1984).
[CrossRef]

R. Schiffer, K. O. Thielheim, “Light reflection from randomly oriented convex particles with rough surface,” J. Appl. Phys. 53, 2825–2830 (1982).
[CrossRef]

Sengupta, D. L.

D. L. Sengupta, “The sphere,” in Electromagnetic and Acoustic Scattering by Simple Shapes, J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, eds. (North-Holland, Amsterdam, 1969), Chap. 10, pp. 353–415.

Senior, T. B. A.

R. E. Hiatt, T. B. A. Senior, V. H. Weston, “A study of surface roughness and its effects on the backscattering cross section of spheres,” Proc. IRE 48, 2008–2016 (1960).
[CrossRef]

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

Shaw, L.

L. Shaw, “Comments on ‘Shadowing of random surfaces,’” IEEE Trans. Antennas Propag. AP-14, 253 (1966).
[CrossRef]

Smith, B. G.

B. G. Smith, “Geometric shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

Spizzichino, A.

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

Stogryn, A.

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

Stuart, W. D.

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

Thielheim, K. O.

R. Schiffer, K. O. Thielheim, “Scattering of scalar waves by an impenetrable rough sphere,” Opt. Acta 31, 1085–1100 (1984).
[CrossRef]

R. Schiffer, K. O. Thielheim, “Light reflection from randomly oriented convex particles with rough surface,” J. Appl. Phys. 53, 2825–2830 (1982).
[CrossRef]

Uslenghi, P. L. E.

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

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Wagner, R. J.

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1967).
[CrossRef]

Welton, P. J.

P. J. Welton, K. E. Hawker, H. G. Frey, “Experimental shadowing measurements on randomly rough surfaces,” J. Acoust. Soc. Am. 54, 446–450 (1973).
[CrossRef]

Weston, V. H.

R. E. Hiatt, T. B. A. Senior, V. H. Weston, “A study of surface roughness and its effects on the backscattering cross section of spheres,” Proc. IRE 48, 2008–2016 (1960).
[CrossRef]

V. H. Weston, “Pulse return from a sphere,” IRE Trans. Antennas Propag. AP-7, S43–S51 (1959).
[CrossRef]

Ann. Physik (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Physik 25, 377–445 (1908).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

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

Commun. Pure Appl. Math. (1)

B. R. Levy, J. B. Keller, “Diffraction by a smooth object,” Commun. Pure Appl. Math. 12, 159–209 (1959).
[CrossRef]

IEEE Trans. Antennas Propag. (7)

B. G. Smith, “Geometric shadowing of a random rough surface,” IEEE Trans. Antennas Propag. AP-15, 668–671 (1967).
[CrossRef]

M. J. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1969).
[CrossRef]

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. AP-13, 384–388 (1965).
[CrossRef]

L. Shaw, “Comments on ‘Shadowing of random surfaces,’” IEEE Trans. Antennas Propag. AP-14, 253 (1966).
[CrossRef]

R. A. Brockelman, T. Hagfors, “Note on the effect of shadowing on the backscattering of waves from a random rough surface,” IEEE Trans. Antennas Propag. AP-14, 621–629 (1966).
[CrossRef]

J. Rheinstein, “Backscatter from spheres: a short pulse view,” IEEE Trans. Antennas Propag. AP-16, 89–97 (1968).
[CrossRef]

M. K. Abdelazeez, “Wave scattering from a large sphere with rough surface,” IEEE Trans. Antennas Propag. AP-31, 375–377 (1983).
[CrossRef]

IRE Trans. Antennas Propag. (1)

V. H. Weston, “Pulse return from a sphere,” IRE Trans. Antennas Propag. AP-7, S43–S51 (1959).
[CrossRef]

J. Acoust. Soc. Am. (2)

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1967).
[CrossRef]

P. J. Welton, K. E. Hawker, H. G. Frey, “Experimental shadowing measurements on randomly rough surfaces,” J. Acoust. Soc. Am. 54, 446–450 (1973).
[CrossRef]

J. Appl. Phys. (2)

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

R. Schiffer, K. O. Thielheim, “Light reflection from randomly oriented convex particles with rough surface,” J. Appl. Phys. 53, 2825–2830 (1982).
[CrossRef]

J. Geophys. Res. (1)

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

J. Mod. Opt. (2)

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

N. C. Bruce, J. C. Dainty, “Multiple scattering from rough dielectric and metal surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 1471–1481 (1991).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Acta (3)

R. Schiffer, K. O. Thielheim, “Scattering of scalar waves by an impenetrable rough sphere,” Opt. Acta 31, 1085–1100 (1984).
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R. Schiffer, “The coherent scattering cross-section of a slightly rough sphere,” Opt. Acta 33, 959–980 (1986).
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N. George, A. C. Livanos, J. A. Roth, C. H. Papas, “Remote sensing of large roughened spheres,” Opt. Acta 23, 367–387 (1976).
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Proc. IEEE (1)

E. M. Kennaugh, D. L. Moffatt, “Transient and impulse response approximations,” Proc. IEEE 53, 893–901 (1965).
[CrossRef]

Proc. IRE (1)

R. E. Hiatt, T. B. A. Senior, V. H. Weston, “A study of surface roughness and its effects on the backscattering cross section of spheres,” Proc. IRE 48, 2008–2016 (1960).
[CrossRef]

Radio Sci. (4)

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

E. Bahar, D. E. Barrick, “Scattering cross sections for composite surfaces that cannot be treated as perturbed-physical optics problems,” Radio Sci. 18, 129–137 (1983).
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A. Stogryn, “Electromagnetic scattering from rough, finitely conducting surfaces,” Radio Sci. 2, 415–428 (1967).

D. E. Barrick, W. H. Peake, “A review of scattering from surfaces with different roughness scales,” Radio Sci. 3, 865–868 (1968).

Other (10)

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

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

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

M. M. Beal, “Serrated circular apertures: optical Fourier transforms and fractal analysis,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1990).

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

D. L. Sengupta, “The sphere,” in Electromagnetic and Acoustic Scattering by Simple Shapes, J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, eds. (North-Holland, Amsterdam, 1969), Chap. 10, pp. 353–415.

T. Hagfors, “Relations between rough surfaces and their scattering properties as applied to radar astronomy,” in Radar Astronomy, J. V. Evans, T. Hagfors, eds. (McGraw-Hill, New York, 1968), Chap. 4 pp. 187–209.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

E. M. Kennaugh, “The impulse response concept in single-body scattering,” in Electromagnetic Scattering, R. L. Rowell, R. S. Stein, eds. (Gordon & Breach, New York, 1965), pp. 217–236.

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Figures (10)

Fig. 1
Fig. 1

Illuminated roughened sphere centered at the origin of the coordinate system. The surface is defined by the vector r and is illuminated by a plane wave traveling in the −z direction.

Fig. 2
Fig. 2

Coordinate transformation in which the point (θ″, ϕ″) on the surface is expressed in relation to the point (θ′, ϕ′) by the polar coordinates ξα. The angle α is referenced to the meridian defined by ϕ′.

Fig. 3
Fig. 3

Comparison of the shadowing functions S(θ′) of Beckmann,8 Wagner, 9 and Smith10 for rms slope values of 2, 0.5, and 0.1 versus incident angle.

Fig. 4
Fig. 4

Simulated results for the shadowing versus incident angle of (a) a Gaussian surface with an rms roughness σ of 2λ and correlation lengths of 50λ and 100λ; (b) an exponential surface with correlation lengths of 50λ, 100λ, and 200λ. Error bars, standard deviation for the 20 surfaces used in the simulation; solid curves, theoretical prediction of Smith.10

Fig. 5
Fig. 5

Normalized backscattering cross section versus normalized rms roughness for the Gaussian surface correlation: (a) lc/λ = 1, 2, 4, and 8; (b) lc/λ = 10, 50, 200, and 1000.

Fig. 6
Fig. 6

Normalized backscattering cross section versus normalized rms roughness for the exponential surface correlation: (a) lc/λ = 1, 2, 4, and 8; (b) lc/λ = 10, 50, 200, and 1000.

Fig. 7
Fig. 7

Normalized backscattering cross section versus normalized rms roughness for (a) the Gaussian and (b) the exponential surfaces, showing the contributions from various annular regions.

Fig. 8
Fig. 8

Simulated spheres based on the integrand of Eq. (35) below for the Gaussian surface. The logarithm of the imaged intensity is shown for a fixed correlation length of 100λ and for several values of rms roughness.

Fig. 9
Fig. 9

Simulated spheres based on the integrand of Eq. (36) for the exponential surface: (a) the logarithm of the imaged intensity is shown for a fixed correlation length of 100λ and for several values of rms roughness; (b) a linear representation of the imaged intensity for a correlation length of 100λ, showing the rapid change in the appearance of the sphere with small changes in the rms roughness.

Fig. 10
Fig. 10

Simulated spheres with use of the backscatter measurements from roughened disks as a function of disk tilt angle, which correspond to annular regions of a sphere. The first simulation comes from a disk with a diffuse white surface, the second from a bead-blasted surface, and the others from ion-beam-etched surfaces.

Equations (39)

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r = [ a + ρ ( θ , ϕ ) ] e r ,
H ( x , y , z ; t ) = 1 4 π × s d s n × H [ x , y , ( a + ρ ) cos θ ; t R / c ] 1 R ,
exp ( ikR ) R = ( 1 R + i k ) exp ( ikR ) R R ̂ ,
H ( x , y , z ; t ) = exp ( i ω t ) 2 π s d s R ̂ × n × e x H inc [ x , y , ( a + ρ ) cos θ ] × ( 1 R + i k ) exp ( ikR ) R
R R 0 x x + y y R 0 ( a + ρ ) cos θ cos θ s ,
n d s = r θ × r ϕ d θ d ϕ = a 2 sin θ d θ d ϕ × ( e r 1 a ρ θ e θ 1 a sin θ ρ ϕ e ϕ ) ,
E ( 0 , 0 , z ; ν ) = i k E 0 2 π z exp ( ikz ) s d s e z × [ e z × ( n × e x ) ] exp [ i k ( a + ρ ) cos θ ] .
E ( 0 , 0 , z ; ν ) = i k a 2 E 0 2 π z exp ( ikz ) sin θ d θ d ϕ × ( cos θ + 1 a ρ θ sin θ ) × exp [ i 2 k ( a + ρ ) cos θ ] .
R 12 ( z 1 , ν 1 ; z 2 , ν 2 ) = E y ( 0 , 0 , z 1 ; ν 1 ) E y * ( 0 , 0 , z 2 ; ν 2 ) = k 1 k 2 a 4 | E 0 | 2 4 π 2 z 1 z 2 exp [ i ( k 1 z 1 k 2 z 2 ) ] × sin θ d θ d ϕ sin θ d θ d ϕ × ( cos θ + 1 a ρ θ sin θ ) × ( cos θ + 1 a ρ θ sin θ ) × exp [ i 2 ( k 1 ρ cos θ k 2 ρ cos θ ) ] × exp [ i 2 a ( k 1 cos θ k 2 cos θ ) ] ,
A = i k E 0 2 π z exp ( ikz ) ,
b = cos θ ,
c = sin θ ,
η = 2 ν c cos θ ,
( b 1 + 1 a ρ θ c 1 ) ( b 2 + 1 a ρ θ c 2 ) × exp [ i 2 π ( η 1 ρ η 2 ρ ) ] = F 2 [ η 1 η 2 ; r ( ξ ) ] [ b 1 b 2 i σ 2 ( b 1 c 2 η 1 b 2 c 1 η 2 ) × cos α d r d ξ c 1 c 2 σ 2 ( cos 2 α d 2 r d ξ 2 + sin 2 α ξ d r d ξ ) + c 1 c 2 η 1 η 2 σ 4 cos 2 α ( d r d ξ ) 2 ]
F 2 [ η 1 , η 2 ; r ( ξ ) ] = exp [ i 2 π ( η 1 ρ η 2 ρ ) ] = exp { 2 π 2 ( η 1 η 2 ) 2 σ 2 4 π 2 η 1 η 2 σ 2 [ 1 r ( ξ ) ] } .
2 a ( k 1 cos θ k 2 cos θ ) 2 a cos θ ( k 1 k 2 ) + 2 k 2 ξ sin θ cos α .
R 12 ( z 1 , ν 1 ; z 2 ν 2 ) = A 1 A 2 * 2 ( π a ) 2 0 π / 2 d θ sin θ × exp [ i 2 a cos θ ( k 1 k 2 ) ] S ( θ ) × 0 d ξ ξ { 2 cos 2 θ J 0 ( ξ γ / l c ) + 4 π σ 2 sin θ cos θ d r d ξ ( η 1 η 2 ) × J 1 ( ξ γ / l c ) σ 2 sin 2 θ × [ d 2 r d ξ 2 β ( d r d ξ ) 2 ] × [ J 0 ( ξ γ / l c ) J 2 ( ξ γ / l c ) ] σ 2 sin 2 θ 1 ξ d r d ξ [ J 0 ( ξ γ / l c ) + J 2 ( ξ γ / l c ) ] } F 2 ( η 1 η 2 ; r ) ,
β = 4 π 2 σ 2 η 1 η 2 ,
γ = 4 π l c λ 2 sin θ
S B ( θ ) = exp [ 1 4 tan θ erfc ( q ) ] ,
S W ( θ ) = 2 erfc ( q ) ( 1 / q π ) exp ( q 2 ) erfc ( q ) × ( 1 exp { 1 2 [ 1 q π exp ( q 2 ) erfc ( q ) ] } ) ,
S S ( θ ) = 1 ½ erfc ( q ) 1 + ½ [ ( 1 / q π ) exp ( q 2 ) erfc ( q ) ] ,
q = cot θ 2 m
m = 1 a [ d ρ d θ d ρ d θ ] θ θ 1 / 2 = σ [ d 2 r ( 0 ) d ξ 2 ] 1 / 2
r G ( ξ ) = exp ( ξ 2 / l c 2 ) ,
r E ( ξ ) = exp ( | ξ | / l c ) ,
m G = 2 σ / l c .
r ( ξ ) = 2 l c δ ( ξ ) + 1 l c 2 exp ( | ξ | / l c ) ,
r ( ξ ) ( 2 / l c ) f ( ξ ) ,
m E 2 = 8 σ 2 λ l c or m E = 2 σ 2 λ l c .
R 12 G ( z 1 , ν 1 ; z 2 , ν 2 ) = A 1 A 2 * 2 ( π a l c ) 2 d θ sin θ exp [ i 2 a cos θ ( k 1 k 2 ) ] S ( θ ) exp [ 2 π 2 σ 2 ( η 1 η 2 ) 2 ] × { cos 2 θ β exp ( γ 2 4 β ) 4 π 2 σ 2 λ 2 ( β + 1 ) 2 sin θ cos 2 θ ( η 1 η 2 ) exp [ γ 2 4 ( β + 1 ) ] + 2 σ 2 l c 2 ( β + 1 ) 2 sin 2 θ [ β + γ 2 4 ( β + 1 ) + 4 π 2 l c 2 λ 2 2 ( β + 1 ) ] exp [ γ 2 4 ( β + 1 ) ] + 2 σ 2 β l c 2 ( β + 2 ) 2 sin 2 θ [ 1 γ 2 4 ( β + 2 ) 4 π 2 l c 2 β λ 2 2 ( β + 2 ) ] exp [ γ 2 4 ( β + 2 ) ] }
R 12 E ( z 1 , ν 1 ; z 2 , ν 2 ) = A 1 A 2 * 2 ( π a l c ) 2 d θ sin θ exp [ i 2 a cos θ ( k 1 k 2 ) ] S ( θ ) exp [ 2 π 2 σ 2 ( η 1 η 2 ) 2 ] × [ 2 cos θ β ( β 2 + γ 2 ) 3 / 2 64 π 3 σ 2 sin θ cos 2 θ ( η 1 η 2 ) λ 2 2 [ ( β + 1 ) 2 + γ 2 ] 3 / 2 + σ 2 l c 2 sin 2 θ × { β ( β + 1 ) + γ 2 [ ( β + 1 ) 2 + γ 2 ] 3 / 2 + β ( β + 2 ) [ ( β + 1 ) 2 + γ 2 ] 3 / 2 } + 16 π 2 σ 2 λ 2 2 γ 2 sin 2 θ cos 2 θ × ( { [ ( β + 1 ) 2 + γ 2 ] 1 / 2 ( β + 1 ) } 2 γ 2 [ ( β + 1 ) 2 + γ 2 ] 3 / 2 { 2 [ ( β + 1 ) 2 + γ 2 ] 1 / 2 + ( β + 1 ) ( β + 2 ) + γ 2 } ) 16 π 2 σ 2 β λ 2 2 γ 2 sin 2 θ cos 2 θ ( { [ ( β + 2 ) 2 + γ 2 ] 1 / 2 ( β + 2 ) } 2 γ 2 [ ( β + 2 ) 2 + γ 2 ] 3 / 2 { 2 [ ( β + 2 ) 2 + γ 2 ] 3 / 2 + ( β + 2 ) } ) ] ,
σ S = 4 π R 0 2 E y ( 0 , 0 , z ; ν ) E y * ( 0 , 0 , z ; ν ) | E 0 | 2 ,
σ S = 4 π R 0 2 R 11 ( z 1 , ν 1 ; z 1 , ν 1 ) | E 0 | 2 .
σ S = 0 π / 2 2 π a 2 sin θ d θ σ ̂ D ( θ i = θ ) ,
σ S G π a 2 = 2 0 π / 2 d θ sin θ S ( θ ) ( ( l c 2 σ ) 2 exp ( γ 2 4 β ) + 2 k 2 σ 2 sin 2 θ { 1 ( β + 1 ) 2 [ β + γ 2 2 ( β + 1 ) ] × exp [ γ 2 4 ( β + 1 ) ] + β ( β + 2 ) 2 [ 1 γ 2 2 ( β + 2 ) ] × exp [ γ 2 4 ( β + 2 ) ] } ) .
σ S E π a 2 = 2 0 π / 2 d θ sin θ S ( θ ) [ 2 k 2 l c 2 cos 2 θ β ( β 2 + γ 2 ) 3 / 2 + k 2 σ 2 sin 2 θ ( β ( β + 1 ) + γ 2 [ ( β + 1 ) 2 + γ 2 ] 3 / 2 + β ( β + 2 ) [ ( β + 2 ) 2 + γ 2 ] 3 / 2 + { [ ( β + 1 ) 2 + γ 2 ] 1 / 2 ( β + 1 ) } 2 γ 2 [ ( β + 1 ) 2 + γ 2 ] 3 / 2 × { 2 [ ( β + 1 ) 2 + γ 2 ] 1 / 2 + ( β + 1 ) ( β + 2 ) + γ 2 } β { [ ( β + 1 ) 2 + γ 2 ] 1 / 2 ( β + 2 ) } 2 γ 2 [ ( β + 1 ) 2 + γ 2 ] 3 / 2 × { 2 [ ( β + 2 ) 2 + γ 2 ] 1 / 2 + ( β + 2 ) } ) ] .
β = ( 4 π σ λ cos θ ) 2 ,
γ = 4 π l c λ sin θ .

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