Abstract

We have developed a hit-and-miss Monte Carlo geometric ray-tracing program to compute the scattering phase matrix for concentrically stratified spheres. Using typical refractive indices for water and aerosols in the calculations, numerous rainbow features appear in the phase matrix that deviate from the results calculated from homogeneous spheres. In the context of geometric ray tracing, rainbows and glory are identified by means of their ray paths, which provide physical explanation for the features produced by the “exact” Lorenz–Mie theory. The computed results for the phase matrix, the single-scattering albedo, and the asymmetry factor for a size parameter of 600 compared closely with those evaluated from the “exact” theory.

© 2010 Optical Society of America

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References

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  1. K. N. Liou, An Introduction to Atmospheric Radiation, 2nd Ed. (Academic, 2002).
  2. Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
    [CrossRef]
  3. Y. Takano and K. N. Liou, “Radiative transfer in cirrus clouds. III. Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
    [CrossRef]
  4. J. A. Lock, J. M. Jamison, and C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4690 (1994).
    [CrossRef] [PubMed]
  5. Y. Takano and K. Jayaweera, “Scattering phase matrix for hexagonal ice crystals computed from ray optics,” Appl. Opt. 24, 3254–3263 (1985).
    [CrossRef] [PubMed]
  6. O. B. Toon and T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657–3660 (1981).
    [CrossRef] [PubMed]
  7. http://atol.ucsd.edu/scatlib/wiscombe.
  8. G. W. Kattawar and D. A. Hood, “Electromagnetic scattering from a spherical polydispersion of coated spheres,” Appl. Opt. 15, 1996–1999 (1976).
    [CrossRef] [PubMed]
  9. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  10. K. N. Liou, Y. Takano, and P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transfer 111, 1980–1989 (2010).
    [CrossRef]
  11. S. Wolf and N. V. Voshchinnikov, “Mie scattering by ensembles of particles with very large size parameters,” Comput. Phys. Commun. 162, 113–123 (2004).
    [CrossRef]
  12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

2010 (1)

K. N. Liou, Y. Takano, and P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transfer 111, 1980–1989 (2010).
[CrossRef]

2004 (1)

S. Wolf and N. V. Voshchinnikov, “Mie scattering by ensembles of particles with very large size parameters,” Comput. Phys. Commun. 162, 113–123 (2004).
[CrossRef]

1995 (1)

Y. Takano and K. N. Liou, “Radiative transfer in cirrus clouds. III. Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

1994 (1)

1989 (1)

Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

1985 (1)

1981 (1)

1976 (1)

Ackerman, T. P.

Bohren, C. F.

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

Hood, D. A.

Huffman, D. R.

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

Jamison, J. M.

Jayaweera, K.

Kattawar, G. W.

Kerker, M.

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

Lin, C.-Y.

Liou, K. N.

K. N. Liou, Y. Takano, and P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transfer 111, 1980–1989 (2010).
[CrossRef]

Y. Takano and K. N. Liou, “Radiative transfer in cirrus clouds. III. Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

K. N. Liou, An Introduction to Atmospheric Radiation, 2nd Ed. (Academic, 2002).

Lock, J. A.

Takano, Y.

K. N. Liou, Y. Takano, and P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transfer 111, 1980–1989 (2010).
[CrossRef]

Y. Takano and K. N. Liou, “Radiative transfer in cirrus clouds. III. Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

Y. Takano and K. Jayaweera, “Scattering phase matrix for hexagonal ice crystals computed from ray optics,” Appl. Opt. 24, 3254–3263 (1985).
[CrossRef] [PubMed]

Toon, O. B.

Voshchinnikov, N. V.

S. Wolf and N. V. Voshchinnikov, “Mie scattering by ensembles of particles with very large size parameters,” Comput. Phys. Commun. 162, 113–123 (2004).
[CrossRef]

Wolf, S.

S. Wolf and N. V. Voshchinnikov, “Mie scattering by ensembles of particles with very large size parameters,” Comput. Phys. Commun. 162, 113–123 (2004).
[CrossRef]

Yang, P.

K. N. Liou, Y. Takano, and P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transfer 111, 1980–1989 (2010).
[CrossRef]

Appl. Opt. (4)

Comput. Phys. Commun. (1)

S. Wolf and N. V. Voshchinnikov, “Mie scattering by ensembles of particles with very large size parameters,” Comput. Phys. Commun. 162, 113–123 (2004).
[CrossRef]

J. Atmos. Sci. (2)

Y. Takano and K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
[CrossRef]

Y. Takano and K. N. Liou, “Radiative transfer in cirrus clouds. III. Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

K. N. Liou, Y. Takano, and P. Yang, “On geometric optics and surface waves for light scattering by spheres,” J. Quant. Spectrosc. Radiat. Transfer 111, 1980–1989 (2010).
[CrossRef]

Other (4)

K. N. Liou, An Introduction to Atmospheric Radiation, 2nd Ed. (Academic, 2002).

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

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

http://atol.ucsd.edu/scatlib/wiscombe.

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Figures (5)

Fig. 1
Fig. 1

Illustrative diagram for light scattering by a concentrically stratified sphere on the basis of the geometric ray-tracing approach.

Fig. 2
Fig. 2

Comparison of the phase matrix elements for polydisperse concentric spheres whose x e is 600 between GO and the “Exact” Lorenz–Mie-like theory. Note that the matrix elements P 22 = P 11 and P 44 = P 33 . The values for P 43 0 and are not displayed here.

Fig. 3
Fig. 3

Geometric rays that contribute to the production of rainbows and glory features, identified by alphabetical and Greek letters as shown in Fig. 2. The scattering angle for each case is also added beside the letter. The top two rows correspond to the left panel of Fig. 2, while the bottom two rows correspond to the right panel of Fig. 2.

Fig. 4
Fig. 4

Same as Fig. 2 except for x e = 2400 .

Fig. 5
Fig. 5

Same as Fig. 2, except for m core = 1.5 i 1.0 × 10 3 and m shell = 1.33 i 7.5 × 10 4 in the left panel, and m core = 1.33 i 7.5 × 10 4 and m shell = 1.5 i 1.0 × 10 3 in the right panel.

Tables (1)

Tables Icon

Table 1 Comparison of the Single-Scattering Albedo and Asymmetry Factor Computed from GO and “Exact” for a Combination of Real and Imaginary Refractive Indices Using x e of 600

Equations (16)

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P = [ P 11 P 12 0 0 P 12 P 22 0 0 0 0 P 33 P 43 0 0 P 43 P 44 ] .
P i j = γ n p i j , n p 11 , n sin θ n ,
p 11 = 1 2 ( M 2 + M 3 + M 4 + M 1 ) ,
p 12 = 1 2 ( M 2 M 3 + M 4 M 1 ) ,
p 22 = 1 2 ( M 2 M 3 M 4 + M 1 ) ,
p 33 = S 21 + S 34 ,
p 44 = S 21 S 34 ,
p 43 = D 21 + D 34 ,
M k = | S k | 2 ,
S k l = S l k = ( S l S k * + S k S l * ) / 2 ,
D k l = D l k = ( S l S k * S k S l * ) i / 2 .
S n = { R 1 for     n = 1 P t P s q [ T n P n k = n 1 2 ( R k P k ) T 1 P 1 ] q P e for     n 2 ,
P k = [ cos ϕ k sin ϕ k sin ϕ k cos ϕ k ] , R k = [ R l k 0 0 R r k ] , T k = [ T l k 0 0 T r k ] .
n ( a ) = C a ( 1 3 v e ) v e exp ( a a e v e ) ,
a e = a π a 2 n ( a ) d a π a 2 n ( a ) d a ,
v e = ( a a e ) 2 π a 2 n ( a ) d a a e 2 π a 2 n ( a ) d a .

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