Abstract

Efficient and accurate computation of the scattered intensity pattern by the Mie formulas is now feasible for size parameters up to x = 50,000 at least, which in visual light means spherical drops with diameters up to 6 mm. We present a method for evaluating the Mie coefficients from the ratios between Riccati-Bessel and Neumann functions of successive order. We probe the applicability of the Airy approximation, which we generalize to rainbows of arbitrary p (number of internal reflections = p – 1), by comparing the Mie and Airy intensity patterns. Millimeter size water drops show a match in all details, including the position and intensity of the supernumerary maxima and the polarization. A fairly good match is still seen for drops of 0.1 mm. A small spread in sizes helps to smooth out irrelevant detail. The dark band between the rainbows is used to test more subtle features. We conclude that this band contains not only externally reflected light (p = 0) but also a sizable contribution from the p = 6 and p = 7 rainbows, which shift rapidly with wavelength. The higher the refractive index, the closer both theories agree on the first primary rainbow (p = 2) peak for drop diameters as small as 0.02 mm. This may be useful in supporting experimental work.

© 1991 Optical Society of America

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References

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  1. L. V. Lorenz, “Upon the Light Reflected and Refracted by a Transparent Sphere,” Vidensk. Selsk. Shrifter 6, 1–62 (1890), in Danish.Reference sources: N. A. Logan, “Survey of Some Early Studies of the Scattering of Plane Waves by a Sphere,” in Proceedings, Second International Congreee on Optical Particle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 7–15;H. Kragh, “Ludvig V. Lorenz and His Contributions to Light Scattering,” in Proceedings, Second International Congress on Particle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 1–6.
  2. G. Mie, “Beitrage zur Optik trfiber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. Leipzig 25, 377–445 (1908).
    [CrossRef]
  3. P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. Leipzig 30, 57–136 (1909).
    [CrossRef]
  4. B. van der Pol, H. Bremmer, “The Diffraction of Electromagnetic Waves from an Electrical Point Source Round a Finitely Conducting Sphere, with Applications to Radio-Telegraphy and the Theory of the Rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937); 25, 817–837 (1938).
  5. C. B. Boyer, The Rainbow: from Myth to Mathematics (Yoseloff, New York, 1959).
  6. H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116-127 (1977).
    [CrossRef]
  7. A. B. Fraser, “Chasing Rainbows,” Weatherwise 36 (6), 280–289 (1983);“Why can the Supernumerary Bows be Seen in a Rain Shower?,” J. Opt. Soc. Am. 73, 1626–1628 (1983).
    [CrossRef]
  8. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954).
  9. R. Greenler, Rainbows, Halos, and Glories (Cambridge U.P., New York, 1980).
  10. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  11. H. M. Nussenzveig, “High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission,” J. Math. Phys. New York 10, 82–124 (1969);II. Theory of the Rainbow and the Glory,” J. Math. Phys. New York 10, 125–176 (1969).
    [CrossRef]
  12. V. Khare, H. M. Nussenzveig, “Theory of the Rainbow,” Phys. Rev. Lett.33, 976–980 (1974);“The Theory of the Glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, Ed. (Plenum, New York, 1977), pp. 723–764.
  13. H. M. Nussenzveig, “Complex Angular Momentum Theory of the Rainbow and the Glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979); see also p. 1193, plate 107.
    [CrossRef]
  14. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  15. W. J. Wiscombe, “Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes,” NCAR (Natl. Ctr. Atmos. Res.) Tech. Note 140+STR (1979);“Improved Mie Scattering Algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [PubMed]
  16. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, NewYork, 1969).
  17. G. W. Kattawar, G. N. Plass, “Electromagnetic Scattering from Absorbing Spheres,” Appl. Opt. 6, 1377–1382 (1967).
    [CrossRef] [PubMed]
  18. J. V. Dave, “Subroutines for Computing the Parameters of Electromagnetic Radiation Scattered by a Sphere,” Report 320-3237, IBM Scientific Center, Palo Alto, CA (1968);“Scattering of Visible Light by Large Water Spheres,” Appl. Opt. 8, 155–164 (1969).
    [PubMed]
  19. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  20. W. J. Lentz, “Generating Bessel Functions in Mie Scattering Calculations Using Continued Fractions,” Appl. Opt. 15, 668–671 (1976).
    [CrossRef] [PubMed]
  21. G. A. Shah, “Numerical Methods for Mie Theory of Scattering by a Sphere,” Kodaikanal Obs. Bull. Ser. A 2, 42–63 (1977); “Numerical Evaluation of Spherical Bessel and Related Functions,” Kodaikanal Obs. Bull. Ser. A 3, 107–119 (1983); private communication (1989).
  22. A. Ungut, G. Grehan, G. Gouesbet, “Comparisons Between Geometrical Optics and Lorentz-Mie Theory,” Appl. Opt. 20, 2911–2918 (1981).
    [CrossRef] [PubMed]
  23. W. A. Rooij, C. C. A. H. van der Stap, “Expansion of Mie Scattering Matrices in Generalized Spherical Functions,” Astron. Astrophys. 131, 237–248 (1984).
  24. D. W. Schuerman, R. T. Wang, B. A. S. Gustafson, R. W. Schaefer, “Systematic Studies of Light Scattering. 1: Particle Shape,” Appl. Opt. 20, 4039–4050 (1981).
    [CrossRef] [PubMed]
  25. R. T. Wang, W. X. Wang, “Scattering by Arbitrarily Large Homogeneous/Concentric Spheres-Exact Theory with Use of New Efficient Algorithms,” in Proceedings of the 1985 Scientific Conference on Obscuration and Aerosol Research, R. Kohl, Ed. (Army CRDEC-SP-86019, Aberdeen, MD1986), pp. 381–409.
  26. R. T. Wang, “A New Algorithm for Exact Scattering Calculations Using Ratios,” submitted to Proceedings of the 1988 Scientific Conference on Obscuration and Aerosol Research, Army CRDEC, Aberdeen, MD.
  27. G. B. Airy, “On the Intensity of Light in the Neighborhood of a Caustic,” Trans. Cambridge Philos. Soc. 6, 379–402 (1838).
  28. This notation, taken from Ref. 10, has the advantage of making superfluous a separate notation for the externally reflected ray (p = 0) and the directly transmitted ray (p = 1). We comply with common practice of calling p − 1 the order of rainbow.
  29. W. J. Humphreys, Physics of the Air (McGraw-Hill, New York, 1929).
  30. M. Abramowitz, I. A. Stegun, Eds, Handbook of Mathematical Functions (Dover, New York, 1964).
  31. P. L. Marston, “Rainbow Phenomena and the Detection of Nonsphericity in Drops,” Appl. Opt. 19, 680–685 (1980).
    [CrossRef] [PubMed]
  32. K. Sassen, “Angular Scattering and Rainbow Formation in Pendant Drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [CrossRef]
  33. J. H. Saiac, “Etude de la diffusion de la lumiere par les gouttes d’eau dans le phenomene de l‘arc en ciel,” La Meteorologie16, 29–68 (1970).
  34. C. W. Querfeld, “Mie Atmospheric Optics,” J. Opt. Soc. Am. 55, 105–106 (1965).
    [CrossRef]
  35. J. A. Lock, “Theory of the Observations Made of High-Order Rainbows from a Single Water Droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  36. W. M. Irvine, J. B. Pollack, “Infrared Optical Properties of Water and Ice Spheres,” Icarus 8, 324–360 (1968).
    [CrossRef]
  37. J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [CrossRef]
  38. J. V. Dave, “Effect of Coarseness of the Integration Increment on the Calculation of the Radiation Scattered by Polydispersed Aerosols,” Appl. Opt. 8, 1161–1167 (1969).
    [CrossRef] [PubMed]
  39. J. D. Walker, “How to Create and Observe a Dozen Rainbows in a Single Drop of Water,” Sci. Am. 237, 138–144 (1977);“Mysteries of Rainbows, Notably Their Rare Supernumerary Arcs,” Sci. Am. 242, 147–152 (1980); “Multiple Rainbows from Single Drops of Water and Other Liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  40. S. G. Warren, “Optical Constants of Carbon Dioxide Ice,” Appl. Opt. 25, 2650–2673 (1986).
    [CrossRef] [PubMed]
  41. D. R. Huffman, J. L. Stapp, “Interstellar Extinction Related to 2200A Band,” Nature London Phys. Sci. 229, 45–46 (1971).
  42. H. C. van de Hulst, R. T. Wang, “Glare Points,” in Proceedings, Second International Congress on OpticalParticle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 40–49.

1987 (1)

1986 (1)

1984 (1)

W. A. Rooij, C. C. A. H. van der Stap, “Expansion of Mie Scattering Matrices in Generalized Spherical Functions,” Astron. Astrophys. 131, 237–248 (1984).

1983 (1)

A. B. Fraser, “Chasing Rainbows,” Weatherwise 36 (6), 280–289 (1983);“Why can the Supernumerary Bows be Seen in a Rain Shower?,” J. Opt. Soc. Am. 73, 1626–1628 (1983).
[CrossRef]

1981 (2)

1980 (1)

1979 (3)

K. Sassen, “Angular Scattering and Rainbow Formation in Pendant Drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
[CrossRef]

H. M. Nussenzveig, “Complex Angular Momentum Theory of the Rainbow and the Glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979); see also p. 1193, plate 107.
[CrossRef]

W. J. Wiscombe, “Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes,” NCAR (Natl. Ctr. Atmos. Res.) Tech. Note 140+STR (1979);“Improved Mie Scattering Algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[PubMed]

1977 (3)

G. A. Shah, “Numerical Methods for Mie Theory of Scattering by a Sphere,” Kodaikanal Obs. Bull. Ser. A 2, 42–63 (1977); “Numerical Evaluation of Spherical Bessel and Related Functions,” Kodaikanal Obs. Bull. Ser. A 3, 107–119 (1983); private communication (1989).

J. D. Walker, “How to Create and Observe a Dozen Rainbows in a Single Drop of Water,” Sci. Am. 237, 138–144 (1977);“Mysteries of Rainbows, Notably Their Rare Supernumerary Arcs,” Sci. Am. 242, 147–152 (1980); “Multiple Rainbows from Single Drops of Water and Other Liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116-127 (1977).
[CrossRef]

1976 (1)

1974 (2)

V. Khare, H. M. Nussenzveig, “Theory of the Rainbow,” Phys. Rev. Lett.33, 976–980 (1974);“The Theory of the Glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, Ed. (Plenum, New York, 1977), pp. 723–764.

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

1971 (1)

D. R. Huffman, J. L. Stapp, “Interstellar Extinction Related to 2200A Band,” Nature London Phys. Sci. 229, 45–46 (1971).

1969 (2)

J. V. Dave, “Effect of Coarseness of the Integration Increment on the Calculation of the Radiation Scattered by Polydispersed Aerosols,” Appl. Opt. 8, 1161–1167 (1969).
[CrossRef] [PubMed]

H. M. Nussenzveig, “High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission,” J. Math. Phys. New York 10, 82–124 (1969);II. Theory of the Rainbow and the Glory,” J. Math. Phys. New York 10, 125–176 (1969).
[CrossRef]

1967 (1)

1965 (1)

1937 (1)

B. van der Pol, H. Bremmer, “The Diffraction of Electromagnetic Waves from an Electrical Point Source Round a Finitely Conducting Sphere, with Applications to Radio-Telegraphy and the Theory of the Rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937); 25, 817–837 (1938).

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. Leipzig 30, 57–136 (1909).
[CrossRef]

1908 (1)

G. Mie, “Beitrage zur Optik trfiber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. Leipzig 25, 377–445 (1908).
[CrossRef]

1890 (1)

L. V. Lorenz, “Upon the Light Reflected and Refracted by a Transparent Sphere,” Vidensk. Selsk. Shrifter 6, 1–62 (1890), in Danish.Reference sources: N. A. Logan, “Survey of Some Early Studies of the Scattering of Plane Waves by a Sphere,” in Proceedings, Second International Congreee on Optical Particle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 7–15;H. Kragh, “Ludvig V. Lorenz and His Contributions to Light Scattering,” in Proceedings, Second International Congress on Particle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 1–6.

1838 (1)

G. B. Airy, “On the Intensity of Light in the Neighborhood of a Caustic,” Trans. Cambridge Philos. Soc. 6, 379–402 (1838).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Eds, Handbook of Mathematical Functions (Dover, New York, 1964).

Airy, G. B.

G. B. Airy, “On the Intensity of Light in the Neighborhood of a Caustic,” Trans. Cambridge Philos. Soc. 6, 379–402 (1838).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Boyer, C. B.

C. B. Boyer, The Rainbow: from Myth to Mathematics (Yoseloff, New York, 1959).

Bremmer, H.

B. van der Pol, H. Bremmer, “The Diffraction of Electromagnetic Waves from an Electrical Point Source Round a Finitely Conducting Sphere, with Applications to Radio-Telegraphy and the Theory of the Rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937); 25, 817–837 (1938).

Dave, J. V.

J. V. Dave, “Effect of Coarseness of the Integration Increment on the Calculation of the Radiation Scattered by Polydispersed Aerosols,” Appl. Opt. 8, 1161–1167 (1969).
[CrossRef] [PubMed]

J. V. Dave, “Subroutines for Computing the Parameters of Electromagnetic Radiation Scattered by a Sphere,” Report 320-3237, IBM Scientific Center, Palo Alto, CA (1968);“Scattering of Visible Light by Large Water Spheres,” Appl. Opt. 8, 155–164 (1969).
[PubMed]

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. Leipzig 30, 57–136 (1909).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Fraser, A. B.

A. B. Fraser, “Chasing Rainbows,” Weatherwise 36 (6), 280–289 (1983);“Why can the Supernumerary Bows be Seen in a Rain Shower?,” J. Opt. Soc. Am. 73, 1626–1628 (1983).
[CrossRef]

Gouesbet, G.

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U.P., New York, 1980).

Grehan, G.

Gustafson, B. A. S.

Hansen, J. E.

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Huffman, D. R.

D. R. Huffman, J. L. Stapp, “Interstellar Extinction Related to 2200A Band,” Nature London Phys. Sci. 229, 45–46 (1971).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (McGraw-Hill, New York, 1929).

Irvine, W. M.

W. M. Irvine, J. B. Pollack, “Infrared Optical Properties of Water and Ice Spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

Kattawar, G. W.

Kerker, M.

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

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the Rainbow,” Phys. Rev. Lett.33, 976–980 (1974);“The Theory of the Glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, Ed. (Plenum, New York, 1977), pp. 723–764.

Lentz, W. J.

Lock, J. A.

Lorenz, L. V.

L. V. Lorenz, “Upon the Light Reflected and Refracted by a Transparent Sphere,” Vidensk. Selsk. Shrifter 6, 1–62 (1890), in Danish.Reference sources: N. A. Logan, “Survey of Some Early Studies of the Scattering of Plane Waves by a Sphere,” in Proceedings, Second International Congreee on Optical Particle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 7–15;H. Kragh, “Ludvig V. Lorenz and His Contributions to Light Scattering,” in Proceedings, Second International Congress on Particle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 1–6.

Marston, P. L.

Mie, G.

G. Mie, “Beitrage zur Optik trfiber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. Leipzig 25, 377–445 (1908).
[CrossRef]

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954).

Nussenzveig, H. M.

H. M. Nussenzveig, “Complex Angular Momentum Theory of the Rainbow and the Glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979); see also p. 1193, plate 107.
[CrossRef]

H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116-127 (1977).
[CrossRef]

V. Khare, H. M. Nussenzveig, “Theory of the Rainbow,” Phys. Rev. Lett.33, 976–980 (1974);“The Theory of the Glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, Ed. (Plenum, New York, 1977), pp. 723–764.

H. M. Nussenzveig, “High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission,” J. Math. Phys. New York 10, 82–124 (1969);II. Theory of the Rainbow and the Glory,” J. Math. Phys. New York 10, 125–176 (1969).
[CrossRef]

Plass, G. N.

Pollack, J. B.

W. M. Irvine, J. B. Pollack, “Infrared Optical Properties of Water and Ice Spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

Querfeld, C. W.

Rooij, W. A.

W. A. Rooij, C. C. A. H. van der Stap, “Expansion of Mie Scattering Matrices in Generalized Spherical Functions,” Astron. Astrophys. 131, 237–248 (1984).

Saiac, J. H.

J. H. Saiac, “Etude de la diffusion de la lumiere par les gouttes d’eau dans le phenomene de l‘arc en ciel,” La Meteorologie16, 29–68 (1970).

Sassen, K.

Schaefer, R. W.

Schuerman, D. W.

Shah, G. A.

G. A. Shah, “Numerical Methods for Mie Theory of Scattering by a Sphere,” Kodaikanal Obs. Bull. Ser. A 2, 42–63 (1977); “Numerical Evaluation of Spherical Bessel and Related Functions,” Kodaikanal Obs. Bull. Ser. A 3, 107–119 (1983); private communication (1989).

Stapp, J. L.

D. R. Huffman, J. L. Stapp, “Interstellar Extinction Related to 2200A Band,” Nature London Phys. Sci. 229, 45–46 (1971).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Eds, Handbook of Mathematical Functions (Dover, New York, 1964).

Travis, L. D.

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Ungut, A.

van de Hulst, H. C.

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

H. C. van de Hulst, R. T. Wang, “Glare Points,” in Proceedings, Second International Congress on OpticalParticle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 40–49.

van der Pol, B.

B. van der Pol, H. Bremmer, “The Diffraction of Electromagnetic Waves from an Electrical Point Source Round a Finitely Conducting Sphere, with Applications to Radio-Telegraphy and the Theory of the Rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937); 25, 817–837 (1938).

van der Stap, C. C. A. H.

W. A. Rooij, C. C. A. H. van der Stap, “Expansion of Mie Scattering Matrices in Generalized Spherical Functions,” Astron. Astrophys. 131, 237–248 (1984).

Walker, J. D.

J. D. Walker, “How to Create and Observe a Dozen Rainbows in a Single Drop of Water,” Sci. Am. 237, 138–144 (1977);“Mysteries of Rainbows, Notably Their Rare Supernumerary Arcs,” Sci. Am. 242, 147–152 (1980); “Multiple Rainbows from Single Drops of Water and Other Liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Wang, R. T.

D. W. Schuerman, R. T. Wang, B. A. S. Gustafson, R. W. Schaefer, “Systematic Studies of Light Scattering. 1: Particle Shape,” Appl. Opt. 20, 4039–4050 (1981).
[CrossRef] [PubMed]

R. T. Wang, W. X. Wang, “Scattering by Arbitrarily Large Homogeneous/Concentric Spheres-Exact Theory with Use of New Efficient Algorithms,” in Proceedings of the 1985 Scientific Conference on Obscuration and Aerosol Research, R. Kohl, Ed. (Army CRDEC-SP-86019, Aberdeen, MD1986), pp. 381–409.

R. T. Wang, “A New Algorithm for Exact Scattering Calculations Using Ratios,” submitted to Proceedings of the 1988 Scientific Conference on Obscuration and Aerosol Research, Army CRDEC, Aberdeen, MD.

H. C. van de Hulst, R. T. Wang, “Glare Points,” in Proceedings, Second International Congress on OpticalParticle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 40–49.

Wang, W. X.

R. T. Wang, W. X. Wang, “Scattering by Arbitrarily Large Homogeneous/Concentric Spheres-Exact Theory with Use of New Efficient Algorithms,” in Proceedings of the 1985 Scientific Conference on Obscuration and Aerosol Research, R. Kohl, Ed. (Army CRDEC-SP-86019, Aberdeen, MD1986), pp. 381–409.

Warren, S. G.

Wiscombe, W. J.

W. J. Wiscombe, “Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes,” NCAR (Natl. Ctr. Atmos. Res.) Tech. Note 140+STR (1979);“Improved Mie Scattering Algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[PubMed]

Ann. Phys. Leipzig (2)

G. Mie, “Beitrage zur Optik trfiber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. Leipzig 25, 377–445 (1908).
[CrossRef]

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. Leipzig 30, 57–136 (1909).
[CrossRef]

Appl. Opt. (8)

Astron. Astrophys. (1)

W. A. Rooij, C. C. A. H. van der Stap, “Expansion of Mie Scattering Matrices in Generalized Spherical Functions,” Astron. Astrophys. 131, 237–248 (1984).

J. Math. Phys. New York (1)

H. M. Nussenzveig, “High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission,” J. Math. Phys. New York 10, 82–124 (1969);II. Theory of the Rainbow and the Glory,” J. Math. Phys. New York 10, 125–176 (1969).
[CrossRef]

J. Opt. Soc. Am. (3)

H. M. Nussenzveig, “Complex Angular Momentum Theory of the Rainbow and the Glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979); see also p. 1193, plate 107.
[CrossRef]

K. Sassen, “Angular Scattering and Rainbow Formation in Pendant Drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
[CrossRef]

C. W. Querfeld, “Mie Atmospheric Optics,” J. Opt. Soc. Am. 55, 105–106 (1965).
[CrossRef]

Kodaikanal Obs. Bull. Ser. A (1)

G. A. Shah, “Numerical Methods for Mie Theory of Scattering by a Sphere,” Kodaikanal Obs. Bull. Ser. A 2, 42–63 (1977); “Numerical Evaluation of Spherical Bessel and Related Functions,” Kodaikanal Obs. Bull. Ser. A 3, 107–119 (1983); private communication (1989).

Nature London Phys. Sci. (1)

D. R. Huffman, J. L. Stapp, “Interstellar Extinction Related to 2200A Band,” Nature London Phys. Sci. 229, 45–46 (1971).

NCAR (Natl. Ctr. Atmos. Res.) (1)

W. J. Wiscombe, “Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes,” NCAR (Natl. Ctr. Atmos. Res.) Tech. Note 140+STR (1979);“Improved Mie Scattering Algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[PubMed]

Philos. Mag. (1)

B. van der Pol, H. Bremmer, “The Diffraction of Electromagnetic Waves from an Electrical Point Source Round a Finitely Conducting Sphere, with Applications to Radio-Telegraphy and the Theory of the Rainbow,” Philos. Mag. 24, 141–176, 825–864 (1937); 25, 817–837 (1938).

Phys. Rev. Lett. (1)

V. Khare, H. M. Nussenzveig, “Theory of the Rainbow,” Phys. Rev. Lett.33, 976–980 (1974);“The Theory of the Glory,” in Statistical Mechanics and Statistical Methods in Theory and Application, U. Landman, Ed. (Plenum, New York, 1977), pp. 723–764.

Sci. Am. (2)

H. M. Nussenzveig, “The Theory of the Rainbow,” Sci. Am. 236, 116-127 (1977).
[CrossRef]

J. D. Walker, “How to Create and Observe a Dozen Rainbows in a Single Drop of Water,” Sci. Am. 237, 138–144 (1977);“Mysteries of Rainbows, Notably Their Rare Supernumerary Arcs,” Sci. Am. 242, 147–152 (1980); “Multiple Rainbows from Single Drops of Water and Other Liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Space Sci. Rev. (1)

J. E. Hansen, L. D. Travis, “Light Scattering in Planetary Atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[CrossRef]

Trans. Cambridge Philos. Soc. (1)

G. B. Airy, “On the Intensity of Light in the Neighborhood of a Caustic,” Trans. Cambridge Philos. Soc. 6, 379–402 (1838).

Vidensk. Selsk. Shrifter (1)

L. V. Lorenz, “Upon the Light Reflected and Refracted by a Transparent Sphere,” Vidensk. Selsk. Shrifter 6, 1–62 (1890), in Danish.Reference sources: N. A. Logan, “Survey of Some Early Studies of the Scattering of Plane Waves by a Sphere,” in Proceedings, Second International Congreee on Optical Particle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 7–15;H. Kragh, “Ludvig V. Lorenz and His Contributions to Light Scattering,” in Proceedings, Second International Congress on Particle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 1–6.

Weatherwise (1)

A. B. Fraser, “Chasing Rainbows,” Weatherwise 36 (6), 280–289 (1983);“Why can the Supernumerary Bows be Seen in a Rain Shower?,” J. Opt. Soc. Am. 73, 1626–1628 (1983).
[CrossRef]

Other (16)

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954).

R. Greenler, Rainbows, Halos, and Glories (Cambridge U.P., New York, 1980).

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

C. B. Boyer, The Rainbow: from Myth to Mathematics (Yoseloff, New York, 1959).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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

R. T. Wang, W. X. Wang, “Scattering by Arbitrarily Large Homogeneous/Concentric Spheres-Exact Theory with Use of New Efficient Algorithms,” in Proceedings of the 1985 Scientific Conference on Obscuration and Aerosol Research, R. Kohl, Ed. (Army CRDEC-SP-86019, Aberdeen, MD1986), pp. 381–409.

R. T. Wang, “A New Algorithm for Exact Scattering Calculations Using Ratios,” submitted to Proceedings of the 1988 Scientific Conference on Obscuration and Aerosol Research, Army CRDEC, Aberdeen, MD.

J. H. Saiac, “Etude de la diffusion de la lumiere par les gouttes d’eau dans le phenomene de l‘arc en ciel,” La Meteorologie16, 29–68 (1970).

W. M. Irvine, J. B. Pollack, “Infrared Optical Properties of Water and Ice Spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

This notation, taken from Ref. 10, has the advantage of making superfluous a separate notation for the externally reflected ray (p = 0) and the directly transmitted ray (p = 1). We comply with common practice of calling p − 1 the order of rainbow.

W. J. Humphreys, Physics of the Air (McGraw-Hill, New York, 1929).

M. Abramowitz, I. A. Stegun, Eds, Handbook of Mathematical Functions (Dover, New York, 1964).

J. V. Dave, “Subroutines for Computing the Parameters of Electromagnetic Radiation Scattered by a Sphere,” Report 320-3237, IBM Scientific Center, Palo Alto, CA (1968);“Scattering of Visible Light by Large Water Spheres,” Appl. Opt. 8, 155–164 (1969).
[PubMed]

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

H. C. van de Hulst, R. T. Wang, “Glare Points,” in Proceedings, Second International Congress on OpticalParticle Sizing, E. D. Hirleman, Ed. (Arizona State U.P., Tempe, 1990), pp. 40–49.

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Figures (7)

Fig. 1
Fig. 1

Real and imaginary parts of the ratios pn(z) and An(z) vs n for Riccati-Bessel function.

Fig. 2
Fig. 2

Mie rainbow intensity profiles for a 2α = 3.2-mm water drop at three visble wavelengths. Positions and intensities of the Airy maxima for the p = 6 and 7 rainbows are also shown.

Fig. 3
Fig. 3

Bottom of Fig. 2 is expanded separately for the primary (top) and secondary (bottom) rainbows. Airy intensity profiles for p = 2, 3, and 6 rainbows are also shown.

Fig. 4
Fig. 4

Mie and Airy rainbow intensity profiles for a 2α = 0.1-mm water drop at three visble wavelengths.

Fig. 5
Fig. 5

Rainbow intensity (upper figure) and polarization (lower figure) profiles for a 2α = 0.4-mm water drop. The (S11)Airy term is plotted a decade higher for readability. Mie profiles are plotted by continuous curves for the single size and are marked by a □ for the gamma size distribution.

Fig. 6
Fig. 6

Averaged S11/x2 and polarization over the chosen interval of 131° ≤ θ ≤ 134° inside the dark band: X, λ = 0.40 μm, m = 1.343 – i3E-9; ⋄, λ = 0.55 μm, m = 1.334 – i1.5E-9; □, λ = 0.65 μm, m = 1.331 – i1.3E-8.

Fig. 7
Fig. 7

Mie and Airy intensity patterns for a fixed x = 241.661 and four refractive indices.

Tables (2)

Tables Icon

Table I Comparison of the Numerical Data for Four Mle Codes

Tables Icon

Table II CPU Time t In Seconds by Four Mie Codes * Using Single Precision Arithmetic (Seven Digits) on the DECLAB PDP 11/23 Computer

Equations (28)

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x = 2 π α λ ,
a n = [ ψ n ( y ) ψ n ( x ) m ψ n ( y ) ψ n ( x ) ] / [ ψ n ( y ) ζ n ( x ) m ψ n ( y ) ζ n ( x ) ] , b n = [ m ψ n ( y ) ψ n ( x ) ψ n ( y ) ψ n ( x ) ] / [ m ψ n ( y ) ζ n ( x ) ψ n ( y ) ζ n ( x ) ] .
n max ~ x + 4 x 1 / 3 + 2.
F n 1 ( z ) + F n + 1 ( z ) = ( 2 n + 1 ) F n ( z ) / z ,
F n ( z ) = n F n ( z ) / z + F n 1 ( z ) .
p n ( z ) = ψ n ( z ) / ψ n 1 ( z ) = 1 / [ ( 2 n + 1 ) / z p n + 1 ( z ) ] ,
q n ( x ) = χ n ( x ) / χ n 1 ( x ) = ( 2 n 1 ) / x 1 / q n 1 ( x ) .
p N + 1 ( z ) ~ z / ( 2 N + 3 )
N = 1 . 1 | z | + 10 .   if  | z | 10000 , N = 1.01 | z | + 10 .   if  | z | > 10000 .
A n ( z ) = ψ n ( z ) / ψ n ( z ) = n / z + 1 / p n ( z ) ,
B n ( x ) = χ n ( x ) / χ n ( x ) = n / x + 1 / q n ( x ) .
a n = 1 / ( 1 + i χ n ( x ) [ A n ( y ) m B n ( x ) ] / { ψ n ( x ) [ A n ( y ) m A n ( x ) ] } ) , b n = 1 / ( 1 + i χ n ( x ) [ m A n ( y ) B n ( x ) ] / { ψ n ( x ) [ m A n ( y ) A n ( x ) ] } ) .
ψ n ( x ) = ψ 1 ( x ) i = 2 n p i ( x ) ,                                    χ n ( x ) = χ 1 ( x ) i = 2 n q i ( x ) ,    for n 2
ψ 1 ( x ) = sin x / x cos x ;      χ 1 ( x ) = cos x / x + sin x ,    for  n = 1 .
( | a n | 2 + | b n | 2 ) / ( | a 1 | 2 + | b 1 | 2 ) 10 14    at  n = n max .
ψ 1 ( x ) ~ n = 1 4 ( 1 ) n + 1 x 2 n 2 n ( 2 n + 1 ) ! ,
S 1 ( θ ) = n = 1 n max 2 n + 1 n ( n + 1 ) [ a n π n ( μ ) + b n τ n ( μ ) ] , S 2 ( θ ) = n = 1 n max 2 n + 1 n ( n + 1 ) [ a n τ n ( μ ) + b n π n ( μ ) ] .
i 1 ( θ ) = | S 1 ( θ ) | 2 ,         i 2 ( θ ) = | S 2 ( θ | 2 .
( S 11 ) Mie = ( i 1 + i 2 ) / 2 ,          ( P ) Mie = ( i 1 i 2 ) / ( i 1 + i 2 ) .
θ = 2 π l + q θ 0 ( p , m ) = 2 ( τ p p τ p ) ,
tan τ p = [ ( m 2 1 ) / ( p 2 m 2 ) ] 1 / 2 ,
tan τ p = [ p 2 ( m 2 1 ) / ( p 2 m 2 ) ] 1 / 2 .
( S 11 ) Airy = ( 1 2 + 2 2 ) [ 81 / ( 16 π 2 h 4 ) ] 1 / 6 × cos τ p x 7 / 3 f 2 ( z ) / sin θ 0 ( p , m ) ,
f ( z ) = 0 cos [ π ( z t t 3 ) / 2 ] d t .
z = ( q ) [ 12 / ( h π 2 ) ] 1 / 3 x 2 / 3 ( θ θ 0 ) ; h = { ( p 2 1 ) 2 / [ p 2 ( m 2 1 ) ] } [ ( p 2 m 2 ) / ( m 2 1 ) ] 1 / 2 ; i = ( 1 r i 2 ) ( r i ) ( p 1 ) ,         i = 1  or  2  and  p 2 ; r 1 = sin ( τ p τ p ) / sin ( τ p + τ p ) ;     r 2 = tan ( τ p τ p ) / tan ( τ p + τ p )
| θ max , K θ 0 ( p , m ) | = h 1 / 3 [ 3 π ( K + 1 4 ) / 2 ] 2 / 3 x 2 / 3 .
| θ max , 0 θ 0 ( p , m ) | = 1.087376 ( h π 2 / 12 ) 1 / 3 x 2 / 3
| θ max , K θ 0 ( p , m ) | = h 1 / 3 [ 3 π ( K + 3 4 ) / 2 ] 2 / 3 x 2 / 3 .

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