Abstract

Retroreflection by common surfaces occurs much more frequently and in a much wider variety of curve shapes than existing retroreflection mechanisms (shadowing among opaque surface irregularities) can explain. This paper proposes several new mechanisms. Surfaces may retroreflect from (i) right angle corners and troughs, (ii) optically smooth, convex sections of interface below which scatterers occur, and (iii) opaque inclusions below the air-material interface (below-surface shadowing). Derivations of special cases give retroreflections with observable magnitudes and with curve shapes covering the variety of those observed. The paper also demonstrates a vast variety of ways in which these mechanisms can occur and establishes the existance of the necessary combinations of structures on many of the most commonly exposed natural surfaces. This comprises considerable evidence that the proposed mechansims (perhaps combined with some surface shadowing) can explain the ubiquitous occurrence of retroreflection from common surfaces.

© 1978 Optical Society of America

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References

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  1. B. W. Hapke, “Optical Properties of the Lunar Surface,” in Physics and Astronomy of the Moon, 2nd ed. edited by Z. Kopal (Academic, New York, 1971), Chap. 5.
  2. P. Oetking, “Photometric Studies of Diffusely Reflecting Surfaces with Applications to the Brightness of the Moon,” J. Geophys. Res. 71, 2505–2515 (1966).
    [Crossref]
  3. B. O’Leary and F. Briggs, “Optical Properties of Apollo 11 Moon Samples,” J. Geophys. Res. 75, 6532–6538 (1970).
    [Crossref]
  4. B. Mason and W. G. Melson, The Lunar Rocks (Wiley-Interscience, New York, 1970).
  5. B. W. Hapke and H. Van Horn, “Photometric Studies of Complex Surfaces, with Applications to the Moon,” J. Geophys. Res. 68, 4545–4570 (1963).
    [Crossref]
  6. M. Minnaert, “Photometry of the Moon,” in Planets and Satellites, The Solar System, edited by G. P. Kuiper and B. M. Middlehurst (University of Chicago, Chicago, 1961), Vol. 3, p. 229.
  7. B. W. Hapke, “A Theoretical Photometric Function for the Lunar Surface,” J. Geophys. Res. 68, 4571–4588 (1963).
    [Crossref]
  8. L. Wilson, “A Photometric Investigation of the Packing State of Apollo 11 Lunar Regolith Samples,” Planet. Space Sci. 21, 113–118 (1973).
    [Crossref]
  9. D. S. McKay, R. M. Fruland, and G. H. Heiken, “Grain Size Distribution as an Indicator of the Maturity of Lunar Soils,” in Fifth Lunar Science Conference (Mar. 18–22, 1974) (compiled by the Lunar Science Institute, Houston, 1974), Volume 2, pages 480–482.
  10. T. S. Trowbridge and K. P. Reitz, “Average irregularity representation of a rough surface of ray reflection,” J. Opt. Soc. Am. 65, 531–536 (1975).
    [Crossref]
  11. D. H. Krinsley and J. C. Doornkamp, Atlas of Quartz Sand Surface Textures (Cambridge University, Cambridge, 1973).
  12. E. C. T. Chao, J. A. Boreman, J. A. Minkin, O. B. James, and G. A. Desborough, “Lunar Glasses of Impact Origin: Physical and Chemical Characteristics and Geologic Implications,” J. Geophys. Res. 75, 7445–7479 (1970).
    [Crossref]
  13. J. A. O’Keefe, “Lunar Rays,” Astrophys. J. 126, 466–466 (1957).
    [Crossref]
  14. W. H. Pickering, The Moon (Doubleday, New York, 1903), Pls. 1A–1E, 2A–2E, 15A–15E, and 16A–16E.
  15. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” U. S. Department of Commerce, National Bureau of Standards Monograph 160, Oct. 1977 (U. S. GPO, Washington, 1977) SD Cat. No. C13. 44:160, Price $2.10.
  16. D. B. Judd, “Terms, definitions, and symbols in reflectometry,” J. Opt. Soc. Am. 57, 445–452 (1967).
    [Crossref] [PubMed]
  17. F. E. Nicodemus, “Reflectance Nomenclature and Directional Reflectance and Emissivity,” Appl. Opt. 9, 1474–1475 (1970).
    [Crossref] [PubMed]
  18. H. D. Eckhardt, “Simple Model of Corner Reflector Phenomena,” Appl. Opt. 10, 1559–1566 (1971).
    [Crossref] [PubMed]
  19. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. StegunU. S. Dept. of Commerce, National Bureau of Standards Applied Mathematics Series 55 (U. S. GPO, Washington, 1964), Price $6.50.
  20. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 396.
  21. Reference 20, p. 395.
  22. W. M. Irvine, “The Shadowing Effect in Diffuse Reflection,” J. Geophys. Res. 71, 2931–2937 (1966).
    [Crossref]
  23. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  24. B. F. Armaly and T. T. Lam, “Influence of Refractive Index on Reflectance from a Semi-Infinite Absorbing-Scattering Medium with Collimated Incident Radiation,” Int. J. Heat Mass Transfer 18, 893–899 (1975). [Captions of Figs. 3 and 4 are interchanged, and d0 in Eq. (19) should be b.]
    [Crossref]
  25. Crystallographic properties and forms of occurrence of minerals were mostly obtained from W. A. Deer, R. A. Howie, and J. Zussman, Rock Forming Minerals (Wiley, New York, 1962).
  26. H. Beutelspacher and H. W. Van Der Marel, Atlas of Electron Microscopy of Clay Minerals and Their Admixtures (Elsevier, Amsterdam, London, New York, 1968).
  27. Åke Hillefors, Deep Weathered Rock Material and Sand Grains Under the Scanning Electron Microscope, Lund Studies in Geography, Series A, Physical Geography, No.  49 (1971).
  28. R. D. Harvey, Electron Microscope Study of Microtexture and Grain Surface In Limestones, Illinois State Geological Survey (Urbana, Ill.), Circular 404 (1966).
  29. R. Shoji and R. L. Folk, “Surface Morphology of Some Limestone Types as Revealed by Electron Microscope,” J. Sedimentary Petrology 34, 144–155 (1964).
  30. E. S. Dana, A Textbook of Mineralogy, 4th ed. (Wiley, New York, 1949).

1975 (2)

T. S. Trowbridge and K. P. Reitz, “Average irregularity representation of a rough surface of ray reflection,” J. Opt. Soc. Am. 65, 531–536 (1975).
[Crossref]

B. F. Armaly and T. T. Lam, “Influence of Refractive Index on Reflectance from a Semi-Infinite Absorbing-Scattering Medium with Collimated Incident Radiation,” Int. J. Heat Mass Transfer 18, 893–899 (1975). [Captions of Figs. 3 and 4 are interchanged, and d0 in Eq. (19) should be b.]
[Crossref]

1973 (1)

L. Wilson, “A Photometric Investigation of the Packing State of Apollo 11 Lunar Regolith Samples,” Planet. Space Sci. 21, 113–118 (1973).
[Crossref]

1971 (2)

Åke Hillefors, Deep Weathered Rock Material and Sand Grains Under the Scanning Electron Microscope, Lund Studies in Geography, Series A, Physical Geography, No.  49 (1971).

H. D. Eckhardt, “Simple Model of Corner Reflector Phenomena,” Appl. Opt. 10, 1559–1566 (1971).
[Crossref] [PubMed]

1970 (3)

B. O’Leary and F. Briggs, “Optical Properties of Apollo 11 Moon Samples,” J. Geophys. Res. 75, 6532–6538 (1970).
[Crossref]

E. C. T. Chao, J. A. Boreman, J. A. Minkin, O. B. James, and G. A. Desborough, “Lunar Glasses of Impact Origin: Physical and Chemical Characteristics and Geologic Implications,” J. Geophys. Res. 75, 7445–7479 (1970).
[Crossref]

F. E. Nicodemus, “Reflectance Nomenclature and Directional Reflectance and Emissivity,” Appl. Opt. 9, 1474–1475 (1970).
[Crossref] [PubMed]

1967 (1)

1966 (3)

P. Oetking, “Photometric Studies of Diffusely Reflecting Surfaces with Applications to the Brightness of the Moon,” J. Geophys. Res. 71, 2505–2515 (1966).
[Crossref]

W. M. Irvine, “The Shadowing Effect in Diffuse Reflection,” J. Geophys. Res. 71, 2931–2937 (1966).
[Crossref]

R. D. Harvey, Electron Microscope Study of Microtexture and Grain Surface In Limestones, Illinois State Geological Survey (Urbana, Ill.), Circular 404 (1966).

1964 (1)

R. Shoji and R. L. Folk, “Surface Morphology of Some Limestone Types as Revealed by Electron Microscope,” J. Sedimentary Petrology 34, 144–155 (1964).

1963 (2)

B. W. Hapke and H. Van Horn, “Photometric Studies of Complex Surfaces, with Applications to the Moon,” J. Geophys. Res. 68, 4545–4570 (1963).
[Crossref]

B. W. Hapke, “A Theoretical Photometric Function for the Lunar Surface,” J. Geophys. Res. 68, 4571–4588 (1963).
[Crossref]

1957 (1)

J. A. O’Keefe, “Lunar Rays,” Astrophys. J. 126, 466–466 (1957).
[Crossref]

Armaly, B. F.

B. F. Armaly and T. T. Lam, “Influence of Refractive Index on Reflectance from a Semi-Infinite Absorbing-Scattering Medium with Collimated Incident Radiation,” Int. J. Heat Mass Transfer 18, 893–899 (1975). [Captions of Figs. 3 and 4 are interchanged, and d0 in Eq. (19) should be b.]
[Crossref]

Beutelspacher, H.

H. Beutelspacher and H. W. Van Der Marel, Atlas of Electron Microscopy of Clay Minerals and Their Admixtures (Elsevier, Amsterdam, London, New York, 1968).

Boreman, J. A.

E. C. T. Chao, J. A. Boreman, J. A. Minkin, O. B. James, and G. A. Desborough, “Lunar Glasses of Impact Origin: Physical and Chemical Characteristics and Geologic Implications,” J. Geophys. Res. 75, 7445–7479 (1970).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 396.

Briggs, F.

B. O’Leary and F. Briggs, “Optical Properties of Apollo 11 Moon Samples,” J. Geophys. Res. 75, 6532–6538 (1970).
[Crossref]

Chao, E. C. T.

E. C. T. Chao, J. A. Boreman, J. A. Minkin, O. B. James, and G. A. Desborough, “Lunar Glasses of Impact Origin: Physical and Chemical Characteristics and Geologic Implications,” J. Geophys. Res. 75, 7445–7479 (1970).
[Crossref]

Dana, E. S.

E. S. Dana, A Textbook of Mineralogy, 4th ed. (Wiley, New York, 1949).

Deer, W. A.

Crystallographic properties and forms of occurrence of minerals were mostly obtained from W. A. Deer, R. A. Howie, and J. Zussman, Rock Forming Minerals (Wiley, New York, 1962).

Desborough, G. A.

E. C. T. Chao, J. A. Boreman, J. A. Minkin, O. B. James, and G. A. Desborough, “Lunar Glasses of Impact Origin: Physical and Chemical Characteristics and Geologic Implications,” J. Geophys. Res. 75, 7445–7479 (1970).
[Crossref]

Doornkamp, J. C.

D. H. Krinsley and J. C. Doornkamp, Atlas of Quartz Sand Surface Textures (Cambridge University, Cambridge, 1973).

Eckhardt, H. D.

Folk, R. L.

R. Shoji and R. L. Folk, “Surface Morphology of Some Limestone Types as Revealed by Electron Microscope,” J. Sedimentary Petrology 34, 144–155 (1964).

Fruland, R. M.

D. S. McKay, R. M. Fruland, and G. H. Heiken, “Grain Size Distribution as an Indicator of the Maturity of Lunar Soils,” in Fifth Lunar Science Conference (Mar. 18–22, 1974) (compiled by the Lunar Science Institute, Houston, 1974), Volume 2, pages 480–482.

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” U. S. Department of Commerce, National Bureau of Standards Monograph 160, Oct. 1977 (U. S. GPO, Washington, 1977) SD Cat. No. C13. 44:160, Price $2.10.

Hapke, B. W.

B. W. Hapke, “A Theoretical Photometric Function for the Lunar Surface,” J. Geophys. Res. 68, 4571–4588 (1963).
[Crossref]

B. W. Hapke and H. Van Horn, “Photometric Studies of Complex Surfaces, with Applications to the Moon,” J. Geophys. Res. 68, 4545–4570 (1963).
[Crossref]

B. W. Hapke, “Optical Properties of the Lunar Surface,” in Physics and Astronomy of the Moon, 2nd ed. edited by Z. Kopal (Academic, New York, 1971), Chap. 5.

Harvey, R. D.

R. D. Harvey, Electron Microscope Study of Microtexture and Grain Surface In Limestones, Illinois State Geological Survey (Urbana, Ill.), Circular 404 (1966).

Heiken, G. H.

D. S. McKay, R. M. Fruland, and G. H. Heiken, “Grain Size Distribution as an Indicator of the Maturity of Lunar Soils,” in Fifth Lunar Science Conference (Mar. 18–22, 1974) (compiled by the Lunar Science Institute, Houston, 1974), Volume 2, pages 480–482.

Hillefors, Åke

Åke Hillefors, Deep Weathered Rock Material and Sand Grains Under the Scanning Electron Microscope, Lund Studies in Geography, Series A, Physical Geography, No.  49 (1971).

Howie, R. A.

Crystallographic properties and forms of occurrence of minerals were mostly obtained from W. A. Deer, R. A. Howie, and J. Zussman, Rock Forming Minerals (Wiley, New York, 1962).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” U. S. Department of Commerce, National Bureau of Standards Monograph 160, Oct. 1977 (U. S. GPO, Washington, 1977) SD Cat. No. C13. 44:160, Price $2.10.

Irvine, W. M.

W. M. Irvine, “The Shadowing Effect in Diffuse Reflection,” J. Geophys. Res. 71, 2931–2937 (1966).
[Crossref]

James, O. B.

E. C. T. Chao, J. A. Boreman, J. A. Minkin, O. B. James, and G. A. Desborough, “Lunar Glasses of Impact Origin: Physical and Chemical Characteristics and Geologic Implications,” J. Geophys. Res. 75, 7445–7479 (1970).
[Crossref]

Judd, D. B.

Krinsley, D. H.

D. H. Krinsley and J. C. Doornkamp, Atlas of Quartz Sand Surface Textures (Cambridge University, Cambridge, 1973).

Lam, T. T.

B. F. Armaly and T. T. Lam, “Influence of Refractive Index on Reflectance from a Semi-Infinite Absorbing-Scattering Medium with Collimated Incident Radiation,” Int. J. Heat Mass Transfer 18, 893–899 (1975). [Captions of Figs. 3 and 4 are interchanged, and d0 in Eq. (19) should be b.]
[Crossref]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” U. S. Department of Commerce, National Bureau of Standards Monograph 160, Oct. 1977 (U. S. GPO, Washington, 1977) SD Cat. No. C13. 44:160, Price $2.10.

Mason, B.

B. Mason and W. G. Melson, The Lunar Rocks (Wiley-Interscience, New York, 1970).

McKay, D. S.

D. S. McKay, R. M. Fruland, and G. H. Heiken, “Grain Size Distribution as an Indicator of the Maturity of Lunar Soils,” in Fifth Lunar Science Conference (Mar. 18–22, 1974) (compiled by the Lunar Science Institute, Houston, 1974), Volume 2, pages 480–482.

Melson, W. G.

B. Mason and W. G. Melson, The Lunar Rocks (Wiley-Interscience, New York, 1970).

Minkin, J. A.

E. C. T. Chao, J. A. Boreman, J. A. Minkin, O. B. James, and G. A. Desborough, “Lunar Glasses of Impact Origin: Physical and Chemical Characteristics and Geologic Implications,” J. Geophys. Res. 75, 7445–7479 (1970).
[Crossref]

Minnaert, M.

M. Minnaert, “Photometry of the Moon,” in Planets and Satellites, The Solar System, edited by G. P. Kuiper and B. M. Middlehurst (University of Chicago, Chicago, 1961), Vol. 3, p. 229.

Nicodemus, F. E.

F. E. Nicodemus, “Reflectance Nomenclature and Directional Reflectance and Emissivity,” Appl. Opt. 9, 1474–1475 (1970).
[Crossref] [PubMed]

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” U. S. Department of Commerce, National Bureau of Standards Monograph 160, Oct. 1977 (U. S. GPO, Washington, 1977) SD Cat. No. C13. 44:160, Price $2.10.

O’Keefe, J. A.

J. A. O’Keefe, “Lunar Rays,” Astrophys. J. 126, 466–466 (1957).
[Crossref]

O’Leary, B.

B. O’Leary and F. Briggs, “Optical Properties of Apollo 11 Moon Samples,” J. Geophys. Res. 75, 6532–6538 (1970).
[Crossref]

Oetking, P.

P. Oetking, “Photometric Studies of Diffusely Reflecting Surfaces with Applications to the Brightness of the Moon,” J. Geophys. Res. 71, 2505–2515 (1966).
[Crossref]

Pickering, W. H.

W. H. Pickering, The Moon (Doubleday, New York, 1903), Pls. 1A–1E, 2A–2E, 15A–15E, and 16A–16E.

Reitz, K. P.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” U. S. Department of Commerce, National Bureau of Standards Monograph 160, Oct. 1977 (U. S. GPO, Washington, 1977) SD Cat. No. C13. 44:160, Price $2.10.

Shoji, R.

R. Shoji and R. L. Folk, “Surface Morphology of Some Limestone Types as Revealed by Electron Microscope,” J. Sedimentary Petrology 34, 144–155 (1964).

Trowbridge, T. S.

van de Hulst, H. C.

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

Van Der Marel, H. W.

H. Beutelspacher and H. W. Van Der Marel, Atlas of Electron Microscopy of Clay Minerals and Their Admixtures (Elsevier, Amsterdam, London, New York, 1968).

Van Horn, H.

B. W. Hapke and H. Van Horn, “Photometric Studies of Complex Surfaces, with Applications to the Moon,” J. Geophys. Res. 68, 4545–4570 (1963).
[Crossref]

Wilson, L.

L. Wilson, “A Photometric Investigation of the Packing State of Apollo 11 Lunar Regolith Samples,” Planet. Space Sci. 21, 113–118 (1973).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 396.

Zussman, J.

Crystallographic properties and forms of occurrence of minerals were mostly obtained from W. A. Deer, R. A. Howie, and J. Zussman, Rock Forming Minerals (Wiley, New York, 1962).

Appl. Opt. (2)

Astrophys. J. (1)

J. A. O’Keefe, “Lunar Rays,” Astrophys. J. 126, 466–466 (1957).
[Crossref]

Illinois State Geological Survey (Urbana, Ill.) (1)

R. D. Harvey, Electron Microscope Study of Microtexture and Grain Surface In Limestones, Illinois State Geological Survey (Urbana, Ill.), Circular 404 (1966).

Int. J. Heat Mass Transfer (1)

B. F. Armaly and T. T. Lam, “Influence of Refractive Index on Reflectance from a Semi-Infinite Absorbing-Scattering Medium with Collimated Incident Radiation,” Int. J. Heat Mass Transfer 18, 893–899 (1975). [Captions of Figs. 3 and 4 are interchanged, and d0 in Eq. (19) should be b.]
[Crossref]

J. Geophys. Res. (6)

W. M. Irvine, “The Shadowing Effect in Diffuse Reflection,” J. Geophys. Res. 71, 2931–2937 (1966).
[Crossref]

E. C. T. Chao, J. A. Boreman, J. A. Minkin, O. B. James, and G. A. Desborough, “Lunar Glasses of Impact Origin: Physical and Chemical Characteristics and Geologic Implications,” J. Geophys. Res. 75, 7445–7479 (1970).
[Crossref]

P. Oetking, “Photometric Studies of Diffusely Reflecting Surfaces with Applications to the Brightness of the Moon,” J. Geophys. Res. 71, 2505–2515 (1966).
[Crossref]

B. O’Leary and F. Briggs, “Optical Properties of Apollo 11 Moon Samples,” J. Geophys. Res. 75, 6532–6538 (1970).
[Crossref]

B. W. Hapke and H. Van Horn, “Photometric Studies of Complex Surfaces, with Applications to the Moon,” J. Geophys. Res. 68, 4545–4570 (1963).
[Crossref]

B. W. Hapke, “A Theoretical Photometric Function for the Lunar Surface,” J. Geophys. Res. 68, 4571–4588 (1963).
[Crossref]

J. Opt. Soc. Am. (2)

J. Sedimentary Petrology (1)

R. Shoji and R. L. Folk, “Surface Morphology of Some Limestone Types as Revealed by Electron Microscope,” J. Sedimentary Petrology 34, 144–155 (1964).

Lund Studies in Geography (1)

Åke Hillefors, Deep Weathered Rock Material and Sand Grains Under the Scanning Electron Microscope, Lund Studies in Geography, Series A, Physical Geography, No.  49 (1971).

Planet. Space Sci. (1)

L. Wilson, “A Photometric Investigation of the Packing State of Apollo 11 Lunar Regolith Samples,” Planet. Space Sci. 21, 113–118 (1973).
[Crossref]

Other (14)

D. S. McKay, R. M. Fruland, and G. H. Heiken, “Grain Size Distribution as an Indicator of the Maturity of Lunar Soils,” in Fifth Lunar Science Conference (Mar. 18–22, 1974) (compiled by the Lunar Science Institute, Houston, 1974), Volume 2, pages 480–482.

D. H. Krinsley and J. C. Doornkamp, Atlas of Quartz Sand Surface Textures (Cambridge University, Cambridge, 1973).

M. Minnaert, “Photometry of the Moon,” in Planets and Satellites, The Solar System, edited by G. P. Kuiper and B. M. Middlehurst (University of Chicago, Chicago, 1961), Vol. 3, p. 229.

B. Mason and W. G. Melson, The Lunar Rocks (Wiley-Interscience, New York, 1970).

B. W. Hapke, “Optical Properties of the Lunar Surface,” in Physics and Astronomy of the Moon, 2nd ed. edited by Z. Kopal (Academic, New York, 1971), Chap. 5.

W. H. Pickering, The Moon (Doubleday, New York, 1903), Pls. 1A–1E, 2A–2E, 15A–15E, and 16A–16E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and T. Limperis, “Geometrical Considerations and Nomenclature for Reflectance,” U. S. Department of Commerce, National Bureau of Standards Monograph 160, Oct. 1977 (U. S. GPO, Washington, 1977) SD Cat. No. C13. 44:160, Price $2.10.

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. StegunU. S. Dept. of Commerce, National Bureau of Standards Applied Mathematics Series 55 (U. S. GPO, Washington, 1964), Price $6.50.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 396.

Reference 20, p. 395.

E. S. Dana, A Textbook of Mineralogy, 4th ed. (Wiley, New York, 1949).

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

Crystallographic properties and forms of occurrence of minerals were mostly obtained from W. A. Deer, R. A. Howie, and J. Zussman, Rock Forming Minerals (Wiley, New York, 1962).

H. Beutelspacher and H. W. Van Der Marel, Atlas of Electron Microscopy of Clay Minerals and Their Admixtures (Elsevier, Amsterdam, London, New York, 1968).

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Figures (28)

FIG. 1
FIG. 1

Reflection geometry.

FIG. 2
FIG. 2

Retroreflection by a right-angle corner or trough under geometrical optics.

FIG. 3
FIG. 3

Cylindrical-lens retroreflector. Light scattered from the focal line and back through the lens is collimated in one dimension; that is, focused parallel to a plane that contains the incidence direction.

FIG. 4
FIG. 4

Occurrence of proposed retroreflection mechanisms on an anhedral crystalline grained material that fractures by grain separation or along curved surfaces through the grains. The dotted circles indicate parts of the surface which are approximately spherical or cylindrical in shape. All but the right-most reflection can also occur on amorphous materials.

FIG. 5
FIG. 5

Occurrence of proposed retroreflection mechanisms on an euhedral crystalline grained material that fractures along curved surfaces through the grains.

FIG. 6
FIG. 6

Occurrence of proposed retroreflection mechanisms on an anhedral crystalline grained material that fractures along orthogonal cleavage planes through the grains.

FIG. 7
FIG. 7

Occurrence of proposed retroreflection mechanisms on an euhedral crystalline grained material that fractures by grain separation or along orthogonal cleavage planes through the grains.

FIG. 8
FIG. 8

Projected effective aperture size distributions (projected effective aperture area per decade of projected effective aperture size). All curves are normalized such that the integral of the corresponding per-interval distributions over all projected effective aperture sizes equals one. Distributions for differing values of d1, d0, and da, may be obtained by sliding the displayed distributions alonig the d axis. (The normalization is maintained.)

FIG. 9
FIG. 9

Geometry for reflection by randomly oriented troughts. Troughs with vertices lying in the solid angle v reflect to somewhere within the solid angle r.

FIG. 10
FIG. 10

“Cones of reflection” for a lens retroreflector. The “cone of illumination” is shaded by lines sloped to the upper left. The “cone of acceptance” for reflection at phase angle α is shaded by lines sloped to the upper right. Their intersections, the cross-hatched volumes, are the “cones of reflection.”

FIG. 11
FIG. 11

Detailed geometry of the cones of reflection. The upper and lower cones of reflection are the darkly shaded volumes. The nonintersecting parts of the cones of illumination and acceptance are lightly shaded. The differential volume elements used for the integration giving the total reflected intensity are the line-shaded volumes.

FIG. 12
FIG. 12

The functions Tu(b, a) (solid lines) and Td(b, a) (dashed lines).

FIG. 13
FIG. 13

Shadowing tube below a section of spherical interface.

FIG. 14
FIG. 14

Retroreflection by right-angle corners in a projected effective aperture size distribution of width m = 1.

FIG. 15
FIG. 15

Retroreflection by right-angle corners in a projected effective aperture size distribution of width m = 2.

FIG. 16
FIG. 16

Retroreflection by right-angle corners in a projected effective aperture size distribution of width m = 5.

FIG. 17
FIG. 17

Retroreflection by right-angle troughs in a projected effective aperture size distribution of width m = 1.

FIG. 18
FIG. 18

Retroreflection by right-angle troughs in a projected effective aperture size distribution of width m = 2.

FIG. 19
FIG. 19

Retroreflection by right-angle troughs in a projected effective aperture size distribution of width m = 5.

FIG. 20
FIG. 20

Spherical-lens (f/d = 10n) retroreflection with nonshadowing scatterers.

FIG. 21
FIG. 21

Spherical-lens (f/d = 100n) retroreflection with nonshadowing scatterers.

FIG. 22
FIG. 22

Spherical lens (f/d = 1000n) retroreflection with nonshadowing scatterers, including that with shadowing scatterers for σf = 20.

FIG. 23
FIG. 23

Spherical-lens (f/d = 10n) retroreflection with shadowing scatterers of σf = 4.

FIG. 24
FIG. 24

Spherical-lens (f/d = 100n) retroreflection with shadowing scatterers of σf = 4.

FIG. 25
FIG. 25

Spherical-lens (f/d = 10n) retroreflection with shadowing scatterers of σf = 20.

FIG. 26
FIG. 26

Spherical-lens (f/d = 100n) retroreflection with shadowing scatterers of σf = 20.

FIG. 27
FIG. 27

Shadowing retroreflection below facets, retroreflective component only.

FIG. 28
FIG. 28

Shadowing retroreflection below facets, including the diffuse component.

Tables (3)

Tables Icon

TABLE I Magnitudes frI(1°)/ρ1ρ2ρ3 A, of corner and trough retroreflections. For minimum and high diffuse reflectances, the retroreflection mechanisms (for dielectrics) pass the test for detectability in existing measurements below the solid and dashed lines.

Tables Icon

TABLE II Magnitude frI(1°)/ Aϖ of the retroreflection for the spherical-lens mechanism with nonshadowing scatterers. For minimum and high diffuse reflectances, the retroreflection mechanism passes the test for detectability in existing measurements below the solid and dashed lines.

Tables Icon

TABLE III Magnitude frI(1°)/ Aϖ of the retroreflection for shadowing below facets. For combinations of volumetric density v and particle albedo ϖ for which frI(1°)/ Aϖ on the left side exceeds rfrID/ Aϖ on the right side, the retroreflection mechanism passes the test for detectability in existing measurements.

Equations (50)

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f r I = ( 1 / π ) ρ ( θ i ; 2 π ) cos θ r ,
às ( d ) 4 ( d / d a ) 4 / log e , d d a
à x ( d y ) = A x ( d y ) d y / log e .
à N ( d 0 , d ) = ( 1 / log e ) ( d / d 0 ) e - d / d 0
A B ( m , d 1 , d ) = d 1 A 0 ( m , d 1 , d 0 ) A N ( d 0 , d ) d d 0 ,
A 0 ( m , d 1 , d 0 ) = ( d 1 / d 0 ) 1 / m / m d 0 , 0 < m < .
à B ( m , d 1 , d ) = 1 m log e Γ ( 1 + 1 m ) d d 1 γ * ( 1 + 1 m , d d 1 ) ,
γ * ( a , x ) x - a Γ ( a ) 0 x t a - 1 e - t d t ,
I = Φ J 1 2 ( π w d / λ ) / π w 2 ( W / sr ) ,
f r I d ( d , α ) = ρ 1 ρ 2 ρ 3 J 1 2 ( π d sin α / λ ) / π sin 2 α ( W sr - 1 / W ) .
f r I ( α ) = A 0 f r I d ( d , α ) A B ( d ) d d .
= { ρ 1 ρ 2 ρ 3 A 2 π ( π d 1 λ ) 1 / m 1 ( 0.8 sin α ) 2 - 1 / m [ ( 2 3 ( 3 m - 1 ) π d 1 , sin α λ 1 2 m - 1 ) ) ( 0.8 π d 1 sin α λ ) 2 - 1 / m + c ( m ) ] , sin α < λ 0.8 π d 1 ρ 1 ρ 2 ρ 3 A 2 π 3 ( m + 1 ) λ d 1 1 sin 3 α [ ln ( 8 π d 1 sin α λ ) 2 + 2 m m + 1 - 4 ] , sin α λ 0.8 π d 1 ,
c ( m ) = ( 0.8 ) 3 π ( m + 1 ) [ 2 m m + 1 + ln 100 - 4 ] + 1 2 m - 1 - 1 1.2 ( 3 m - 1 ) .
A v = ( 2 / π 2 ) sin δ ( cm 2 / sr )
d Φ = ( 2 / π 2 ) Φ 0 A ρ 1 ρ 2 sin δ d ω v .
α = 2 ( 90° - δ ) .
I = ( 2 / π 2 ) Φ 0 ρ 1 ρ 2 A sin 2 δ / 2 sin [ 2 ( 90° - δ ) ] ( W / sr ) .
f r I = ρ 1 ρ 2 A ( 1 + cos α ) / 2 π 2 sin α ρ 1 ρ 2 A / π 2 α ( W sr - 1 / W ) .
f r I = ρ 1 ρ 2 A ( 1 + cos α / cos 2 φ v ) 2 π 2 ( cos 2 2 φ v - cos 2 α ) 1 / 2 ρ 1 ρ 2 A π 2 ( α 2 - ( 2 φ v ) 2 ) 1 / 2 .
2 φ v 2 φ v + δ α f r I d α ρ 1 ρ 2 A π 2 ln { 1 + δ α 2 φ v + [ δ α 2 φ v + ( δ α 2 φ v ) 2 ] 1 / 2 } .
f r I d ( d , α ) = ρ 1 ρ 2 ( 2 / π 2 ) ( d / λ ) T F ( π ( d / λ ) α ) ,
T F ( y ) 0 y sin 2 x x 2 ( y 2 - x 2 ) 1 / 2 d x { π / 2 , 0 y 1 π / 2 y , y 1.
f r I ( m , d 1 , α ) = ρ 1 ρ 2 A d 1 π λ 1 m - 1 { m Γ ( 1 m + 1 ) × γ * ( 1 m - 1 , λ π d 1 α ) - 1 } .
f r I D = ρ ( 0 ; 2 π ) / π .
E n = E n 0 [ f / ( f - s ) ] 2 e - σ s .
d I = ( 1 / 4 π ) P ( α ) ( ω / ω ) E n ϖ σ e - σ α s d V ( W / sr ) .
α n α .
I = E n 0 d 2 48 P ( α / n ) ϖ σ f ( f / d ) tan ( α / n ) × T [ ½ ( σ + σ α ) f , ( f / d ) tan ( α / n ) ] ,
T ( b , a ) T u ( b , a ) + T d ( b , a )
T u ( b , a ) 3 a ( 1 + a ) 3 0 1 ( 1 - x ) 2 ( a + 1 - x ) 4 e - 2 b ( 1 + a ) - 1 x d x
T d ( b , a ) 3 a ( 1 - a ) 3 1 ( x - 1 ) 2 ( x - 1 + a ) 4 e - 2 b ( 1 - a ) - 1 x d x
f r I = A ϖ P ( α / n ) σ f 12 π ( f / d ) tan ( α / n ) × T [ ½ ( σ + σ α ) f , ( f / d ) tan ( α / n ) ] .
f r I = A ϖ σ f 12 π ( f / d n ) α T ( σ f , f d n α ) f r I C ( A ϖ , σ f , f d n α ) ,
f r I D = ( 1 - A ) ρ ( θ i ; 2 π ) cos θ r / π ( 1 - A ) ρ ( 0 ; 2 π ) / π
f r I H = cos θ r cos θ i ϖ ( 1 + cos θ r / cos θ i ) × ( 1 / 4 π ) P ( α ) B ( α , v ) ,
B ( α , v ) = 2 - tan α 4 v 2 / 3 ( 1 - e - 2 v 2 / 3 / tan α ) × ( 3 - e - 2 v 2 / 3 / tan α ) , α π / 2 ,
f r I H = P ( α / n ) ϖ 8 π n 2 [ 2 - tan ( α / n ) 4 v 2 / 3 ( 1 - e - 2 v 2 / 3 / tan ( α / n ) ) × ( 3 - e - 2 v 2 / 3 / tan ( α / n ) ) ] .
lim α 90° n f r I H = P ( α / n ) ϖ / 8 π n 2 .
f r I = A ϖ 16 π n 2 [ 2 - α 2 n v 2 / 3 ( 1 - e - 2 n v 2 / 3 / α ) × ( 3 - e - 2 n v 2 / 3 / α ) ] .
f r I D = A R s ( ) + A n / 4 π 2 + ( 1 - A ) ρ ( 0 ; 2 π ) / π .
f r I = A ϖ σ f 12 π ( α f / d n ) T ( ½ σ f , α f / n d ) = 2 f r I C ( A ϖ , ½ σ f , α f / n d )
f r I = { f r I C ( A ϖ , σ f , f d n α ) - f r I C ( A ϖ , σ f , σ f 2 v 2 / 3 n α ) + 2 f r I C ( A ϖ , ½ σ f , σ f 2 v 2 / 3 n α ) , v ( σ f 2 f / d ) 3 / 2 2 f r I C ( A ϖ , ½ σ f , f d n α ) , v ( σ f 2 f / d ) 3 / 2 .
f r I r f r I D .
f r I ( ) / ρ 1 ρ 2 ρ 3 A r ρ ( 0 ; 2 π ) / π ρ 1 ρ 2 ρ 3 A ,
f r I ( ) / ρ 1 ρ 2 ρ 3 A 1.3 and 13 ,
f r I ( ) / ρ 1 ρ 2 ρ 3 A 0.064 and 0.64.
f r I ( ) / A ϖ r ( 1 - A ) ρ ( 0 ; 2 π ) / π A ϖ ,
f r I ( ) / A ϖ 0.0030 and 0.030.
f r I ( ) A ϖ r ϖ [ R s ( ) + n 4 π 2 + ( 1 - A ) A π ρ ( 0 ; 2 π ) ] .
S = d f r I ( ) / d α d f r I ( ) / d α .