Abstract

Mie theory offers an exact solution to the problem of scattering of sunlight by spherical drops of water. Until recently, most applications of Mie theory to scattering of light were restricted to a single wavelength. Mie theory can now be used on modern personal computers to produce full-color simulations of atmospheric optical effects, such as rainbows, coronas, and glories. Comparison of such simulations with observations of natural glories and cloudbows is encouraging.

© 2003 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2003

1998

1992

1991

1990

P. Schiebener, J. Straub, J. M. H. L. Sengers, J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

1983

1981

1979

1978

1971

K. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[CrossRef]

1969

1966

1947

1908

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

1838

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, Part 3, 397–403 (1838).

Airy, G. B.

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, Part 3, 397–403 (1838).

Bohren, C. F.

C. F. Bohren, T. J. Nevitt, “Absorption by a sphere: a simple approximation,” Appl. Opt. 22, 774–775 (1983).
[CrossRef] [PubMed]

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 (Princeton University, Princeton, N.J., 1987; reprint of 1959 Thomas Yoseloff edition).

Brandt, R. K.

R. K. Brandt, R. G. Greenler, “Color simulation of size-dependent features of rainbows,” presented at the Seventh Topical Meeting on Meteorological Optics, Boulder, Colorado, 5–8 June 2001; available at http://www.asp.ucar.edu/MetOptics/Preprints.pdf .

Bryant, H. C.

Dave, J. V.

de Boer, J. H.

Fahlen, T. S.

Fraser, A. B.

R. L. Lee, A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Penn State, University Park, Pa., 2001).

Gallagher, J. S.

P. Schiebener, J. Straub, J. M. H. L. Sengers, J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

Gedzelman, S. D.

S. D. Gedzelman, J. A. Lock, “Simulating coronas in color,” Appl. Opt. 42, 497–504 (2003).
[CrossRef] [PubMed]

S. D. Gedzelman, “Simulating glories and cloudbows in color,” Appl. Opt. 42, 429–435 (2003).
[CrossRef] [PubMed]

S. D. Gedzelman, J. A. Lock, “Simulating coronas in color,” presented at the Seventh Topical Meeting on Meteorological Optics, Boulder, Colorado, 5–8 June 2001; available at http://www.asp.ucar.edu/MetOptics/Preprints.pdf .

Gouesbet, G.

Grandy, W. T.

W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge University, Cambridge, UK, 2001).

Greenler, R. G.

R. K. Brandt, R. G. Greenler, “Color simulation of size-dependent features of rainbows,” presented at the Seventh Topical Meeting on Meteorological Optics, Boulder, Colorado, 5–8 June 2001; available at http://www.asp.ucar.edu/MetOptics/Preprints.pdf .

Grehan, G.

Hansen, J. E.

K. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[CrossRef]

Hovenac, E. A.

Huffman, D. R.

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

Kaye, G. W. C.

G. W. C. Kaye, T. H. Laby, Tables of Physical and Chemical Constants, and Some Mathematical Functions (Longman, London, 1986).

Können, G. P.

Laby, T. H.

G. W. C. Kaye, T. H. Laby, Tables of Physical and Chemical Constants, and Some Mathematical Functions (Longman, London, 1986).

Lee, R. L.

R. L. Lee, “Mie theory, Airy theory, and the natural rainbow,” Appl. Opt. 37, 1506–1519 (1998).
[CrossRef]

R. L. Lee, A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Penn State, University Park, Pa., 2001).

Liou, K.

K. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[CrossRef]

Livingston, W.

D. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge University, Cambridge, UK, 2001).

Lock, J. A.

S. D. Gedzelman, J. A. Lock, “Simulating coronas in color,” Appl. Opt. 42, 497–504 (2003).
[CrossRef] [PubMed]

E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
[CrossRef]

S. D. Gedzelman, J. A. Lock, “Simulating coronas in color,” presented at the Seventh Topical Meeting on Meteorological Optics, Boulder, Colorado, 5–8 June 2001; available at http://www.asp.ucar.edu/MetOptics/Preprints.pdf .

Lynch, D. K.

D. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge University, Cambridge, UK, 2001).

Mie, G.

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

Nevitt, T. J.

Nussenzveig, H. M.

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
[CrossRef]

H. M. Nussenzveig, “The theory of the rainbow,” in Atmospheric Phenomena (Freeman, San Francisco, Calif., 1980), pp. 60–71.

Schiebener, P.

P. Schiebener, J. Straub, J. M. H. L. Sengers, J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

Segelstein, D.

D. Segelstein, “The complex refractive index of water,” M.S. thesis (University of Missouri, Kansas City, Mo., 1981).

Sengers, J. M. H. L.

P. Schiebener, J. Straub, J. M. H. L. Sengers, J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

Shipley, S. T.

Straub, J.

P. Schiebener, J. Straub, J. M. H. L. Sengers, J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

Styles, W. S.

G. Wyszecki, W. S. Styles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982), pp. 138–139.

Ungut, A.

van de Hulst, H. C.

Wang, R. T.

Weinman, J. A.

Wyszecki, G.

G. Wyszecki, W. S. Styles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982), pp. 138–139.

Ann. Phys. Leipzig

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

Appl. Opt.

J. Atmos. Sci.

K. Liou, J. E. Hansen, “Intensity and polarization for single scattering by polydisperse spheres: a comparison of ray optics and Mie theory,” J. Atmos. Sci. 28, 995–1004 (1971).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. Chem. Ref. Data

P. Schiebener, J. Straub, J. M. H. L. Sengers, J. S. Gallagher, “Refractive index of water and steam as function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 19, 677–717 (1990).
[CrossRef]

Trans. Cambridge Philos. Soc.

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, Part 3, 397–403 (1838).

Other

C. B. Boyer, The Rainbow: From Myth to Mathematics (Princeton University, Princeton, N.J., 1987; reprint of 1959 Thomas Yoseloff edition).

R. L. Lee, A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth, and Science (Penn State, University Park, Pa., 2001).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981; reprint of 1957 Wiley edition), pp. 167–171.

Ref. 5, pp. 152–153.

H. M. Nussenzveig, “The theory of the rainbow,” in Atmospheric Phenomena (Freeman, San Francisco, Calif., 1980), pp. 60–71.

R. K. Brandt, R. G. Greenler, “Color simulation of size-dependent features of rainbows,” presented at the Seventh Topical Meeting on Meteorological Optics, Boulder, Colorado, 5–8 June 2001; available at http://www.asp.ucar.edu/MetOptics/Preprints.pdf .

S. D. Gedzelman, J. A. Lock, “Simulating coronas in color,” presented at the Seventh Topical Meeting on Meteorological Optics, Boulder, Colorado, 5–8 June 2001; available at http://www.asp.ucar.edu/MetOptics/Preprints.pdf .

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

International Association for the Properties of Water and Steam, “Release on the refractive index of ordinary water substance as a function of wavelength, temperature and pressure” (1997), http://www.iapws.org/relguide/rindex.pdf .

G. Wyszecki, W. S. Styles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982), pp. 138–139.

D. Bruton, “Color science” (1996), http://www.physics.sfasu.edu/astro/color/spectra.html .

X. Han, “Study of refractometry of rainbow and applications to the measurement of instability and temperature gradient of a liquid jet,” Ph.D. dissertation (University of Rouen, Rouen, France, 2000), available at http://www.coria.fr/LESP/OP15/Han/TheseHan.htm .

Ref. 8, pp. 1073–1078.

W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge University, Cambridge, UK, 2001).

D. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge University, Cambridge, UK, 2001).

D. Segelstein, “The complex refractive index of water,” M.S. thesis (University of Missouri, Kansas City, Mo., 1981).

G. W. C. Kaye, T. H. Laby, Tables of Physical and Chemical Constants, and Some Mathematical Functions (Longman, London, 1986).

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Figures (15)

Fig. 1
Fig. 1

Graphs of intensity versus scattering angle θ. Radius of spheres r = 1, 10, 100, and 1000 μm; wavelength of light λ = 0.65 μm; refractive index n = 1.332; unpolarized light.

Fig. 2
Fig. 2

(a) Graph of intensity versus scattering angle θ. r = 100 μm, λ = 0.65 μm, n = 1.332, perpendicular polarization. (b) As in (a) but intensity is the average intensity of scattering by 50 spheres with different values of radius with a log-normal distribution with a median radius of 100 μm and a standard deviation of 0.1 μm (0.1% of the median radius). (c) As in (a) but intensity is averaged to take account of the 0.5° diameter of the Sun.

Fig. 3
Fig. 3

(a) Graph of intensity versus scattering angle θ. r = 100 μm, ten values of λ equally spaced between 0.38 and 0.7 μm, n varies as function of λ, perpendicular polarization. (b) As in (a) but with 1000 values of λ equally spaced between 0.38 and 0.7 μm.

Fig. 4
Fig. 4

Graph of intensity versus scattering angle θ for r = 100 μm, together with colored stripes simulating primary and secondary rainbows for perpendicular polarization, parallel polarization, and for unpolarized light (N = 2000).

Fig. 5
Fig. 5

(a) Simulation of primary and secondary rainbows caused by scattering of unpolarized sunlight from r = 100 μm water drops with the specified number N of wavelengths. (b) As in (a) but limited to the secondary rainbow.

Fig. 6
Fig. 6

Lee diagram of the primary rainbow for values of r between 10 and 1000 μm for unpolarized sunlight (N = 7).

Fig. 7
Fig. 7

Lee diagram of the primary rainbow for values of r between 10 and 1000 μm for unpolarized sunlight (N = 6000/r for 10 μm < r < 200 μm and N = 30 for r > 200 μm).

Fig. 8
Fig. 8

As in Fig. 7, except that drops of nominal radius r have a log-normal size distribution with a standard deviation of 20% of the nominal value.

Fig. 9
Fig. 9

(a) Simulation of primary and secondary rainbows caused by scattering of unpolarized sunlight from r = 200 μm water drops. (b) Simulation of primary and secondary rainbows caused by scattering of unpolarized sunlight from r = 500 μm water drops.

Fig. 10
Fig. 10

Simulation of corona (top) and glory (bottom) caused by scattering of unpolarized sunlight from r = 10 μm water drops.

Fig. 11
Fig. 11

Simulation of the glory caused by scattering of unpolarized sunlight from r = 10 μm water drops with the specified number of wavelengths equally spaced between 0.38 and 0.7 μm.

Fig. 12
Fig. 12

Lee diagram of the glory caused by scattering of unpolarized sunlight from water drops of nominal radius r with a log-normal size distribution and a standard deviation of 5% of the nominal value.

Fig. 13
Fig. 13

Digital image of a glory with a superimposed simulation of scattering of unpolarized sunlight from r = 4.8 μm water drops.

Fig. 14
Fig. 14

Digital image of a faint glory and a cloudbow.

Fig. 15
Fig. 15

Simulation of scattering of unpolarized sunlight from r = 20 μm water drops.

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