Abstract

The combined field integral equation is solved for electromagnetic scattering of a three-dimensional hexagonal ice column and is tested to investigate its applicability to radiation transfer in ice clouds. Convergence of a solution and the influence of rounding the hexagonal edges were checked, and no practical problems were found. The scattering characteristics of a hexagonal ice column are discussed from the size of the Rayleigh scattering region to the size at which a ray optics character appears. The size parameter of a hexagonal column is as much as 50, which was limited by computer resources.

© 2000 Optical Society of America

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References

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  1. K. N. Liou, “Influence of cirrus clouds on weather and climate processes: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
    [CrossRef]
  2. M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
    [CrossRef]
  3. E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
    [CrossRef]
  4. P. Yang, K.-N. Liou, “Finite-difference time domain method for light scattering by small ice in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
    [CrossRef]
  5. W. Sun, Q. Fu, Z. Chen, “Finite-difference time-domain solution of light scattering by dielectric particles with a perfectly matched layer absorbing boundary condition,” Appl. Opt. 38, 3141–3151 (1999).
    [CrossRef]
  6. R. Kress, “Numerical solution of boundary integral equations in time-harmonic electromagnetic scattering,” Electromagnetics 10, 1–20 (1990).
    [CrossRef]
  7. E. K. Miller, “A selective survey of computational electromagnetics,” IEEE Trans. Antennas Propag. 36, 1281–1305 (1988).
    [CrossRef]
  8. J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined-field solutions for conducting bodies of revolution,” Arch. Elektr. Uebertrag. 32, 157–164 (1978).
  9. R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).
  10. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press, London, 1992).
  11. A. F. Peterson, R. Mittra, “Method of conjugate gradients for the numerical solution of large-body electromagnetic scattering problems,” J. Opt. Soc. Am. A 2, 971–977 (1985).
    [CrossRef]
  12. A. J. Poggio, E. K. Miller, “Integral equations of three-dimensional scattering problems” in Computer Techniques for Electromagnetics, R. Mittra ed. (Pergamon, Oxford, 1973).
  13. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).
    [CrossRef]
  14. K. Yoshida, Functional Analysis (Springer-Verlag, Berlin, 1968).
  15. A. H. Chang, K. S. Yee, J. Prodan, “Comparison of different integral equation formulations for bodies of revolution with anisotropic surface impedance boundary conditions,” IEEE Trans. Antennas Propag. 40, 989–999 (1992).
    [CrossRef]
  16. S. G. Warren, “Optical constants of ice from the ultraviolet to the microwave,” Appl. Opt. 23, 1206–1225 (1984).
    [CrossRef] [PubMed]
  17. Y. Takano, K. Jayaweera, “Scattering phase matrix for hexagonal ice crystals computed from ray optics,” Appl. Opt. 24, 3254–3263 (1985).
    [CrossRef] [PubMed]
  18. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

1999

1996

P. Yang, K.-N. Liou, “Finite-difference time domain method for light scattering by small ice in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

1992

A. H. Chang, K. S. Yee, J. Prodan, “Comparison of different integral equation formulations for bodies of revolution with anisotropic surface impedance boundary conditions,” IEEE Trans. Antennas Propag. 40, 989–999 (1992).
[CrossRef]

1990

R. Kress, “Numerical solution of boundary integral equations in time-harmonic electromagnetic scattering,” Electromagnetics 10, 1–20 (1990).
[CrossRef]

1988

E. K. Miller, “A selective survey of computational electromagnetics,” IEEE Trans. Antennas Propag. 36, 1281–1305 (1988).
[CrossRef]

1986

K. N. Liou, “Influence of cirrus clouds on weather and climate processes: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

1985

1984

1978

J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined-field solutions for conducting bodies of revolution,” Arch. Elektr. Uebertrag. 32, 157–164 (1978).

1973

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

Bohren, C. F.

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

Chang, A. H.

A. H. Chang, K. S. Yee, J. Prodan, “Comparison of different integral equation formulations for bodies of revolution with anisotropic surface impedance boundary conditions,” IEEE Trans. Antennas Propag. 40, 989–999 (1992).
[CrossRef]

Chen, Z.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press, London, 1992).

Fu, Q.

Harrington, R. F.

J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined-field solutions for conducting bodies of revolution,” Arch. Elektr. Uebertrag. 32, 157–164 (1978).

R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).

Huffman, D. R.

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

Jayaweera, K.

Kress, R.

R. Kress, “Numerical solution of boundary integral equations in time-harmonic electromagnetic scattering,” Electromagnetics 10, 1–20 (1990).
[CrossRef]

Liou, K. N.

K. N. Liou, “Influence of cirrus clouds on weather and climate processes: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

Liou, K.-N.

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

Mautz, J. R.

J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined-field solutions for conducting bodies of revolution,” Arch. Elektr. Uebertrag. 32, 157–164 (1978).

Miller, E. K.

E. K. Miller, “A selective survey of computational electromagnetics,” IEEE Trans. Antennas Propag. 36, 1281–1305 (1988).
[CrossRef]

A. J. Poggio, E. K. Miller, “Integral equations of three-dimensional scattering problems” in Computer Techniques for Electromagnetics, R. Mittra ed. (Pergamon, Oxford, 1973).

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

Mittra, R.

Müller, C.

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).
[CrossRef]

Pennypacker, C. P.

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

Peterson, A. F.

Poggio, A. J.

A. J. Poggio, E. K. Miller, “Integral equations of three-dimensional scattering problems” in Computer Techniques for Electromagnetics, R. Mittra ed. (Pergamon, Oxford, 1973).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press, London, 1992).

Prodan, J.

A. H. Chang, K. S. Yee, J. Prodan, “Comparison of different integral equation formulations for bodies of revolution with anisotropic surface impedance boundary conditions,” IEEE Trans. Antennas Propag. 40, 989–999 (1992).
[CrossRef]

Purcell, E. M.

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

Sun, W.

Takano, Y.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press, London, 1992).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press, London, 1992).

Warren, S. G.

Yang, P.

Yee, K. S.

A. H. Chang, K. S. Yee, J. Prodan, “Comparison of different integral equation formulations for bodies of revolution with anisotropic surface impedance boundary conditions,” IEEE Trans. Antennas Propag. 40, 989–999 (1992).
[CrossRef]

Yoshida, K.

K. Yoshida, Functional Analysis (Springer-Verlag, Berlin, 1968).

Appl. Opt.

Arch. Elektr. Uebertrag.

J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined-field solutions for conducting bodies of revolution,” Arch. Elektr. Uebertrag. 32, 157–164 (1978).

Astrophys. J.

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

Electromagnetics

R. Kress, “Numerical solution of boundary integral equations in time-harmonic electromagnetic scattering,” Electromagnetics 10, 1–20 (1990).
[CrossRef]

IEEE Trans. Antennas Propag.

E. K. Miller, “A selective survey of computational electromagnetics,” IEEE Trans. Antennas Propag. 36, 1281–1305 (1988).
[CrossRef]

A. H. Chang, K. S. Yee, J. Prodan, “Comparison of different integral equation formulations for bodies of revolution with anisotropic surface impedance boundary conditions,” IEEE Trans. Antennas Propag. 40, 989–999 (1992).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

Mon. Weather Rev.

K. N. Liou, “Influence of cirrus clouds on weather and climate processes: a global perspective,” Mon. Weather Rev. 114, 1167–1199 (1986).
[CrossRef]

Other

R. F. Harrington, Field Computation by Moment Methods (Macmillan, New York, 1968).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortran (Cambridge U. Press, London, 1992).

A. J. Poggio, E. K. Miller, “Integral equations of three-dimensional scattering problems” in Computer Techniques for Electromagnetics, R. Mittra ed. (Pergamon, Oxford, 1973).

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).
[CrossRef]

K. Yoshida, Functional Analysis (Springer-Verlag, Berlin, 1968).

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

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Figures (8)

Fig. 1
Fig. 1

(a) Position of point P on the surface designated by two variables t and ϕ, where t is the surface distance from a polar point T 0 to a surface point P′ on the x-z plane with the same z coordinate as P. The three surface arrows on P indicate the axes of a local coordinate system. (b) Schematic of an incident plane wave in the (θ0, ϕ0) direction.

Fig. 2
Fig. 2

(a) Convergence of the difference between scattering and extinction efficiency. Because there is no absorption, the difference converges to zero as the number of B-spline knots increases. The unit length is defined as the distance on surface s that satisfies k 0 s = 1. Here, k 0 is an incident wave number. (b) Convergence of a phase function on the equatorial plane of a hexagonal column. The dotted curve represents the results for 0.66 knots per unit length, and the solid curves represent the results for 0.79, 0.9, and 1.15. The differences among the latter curves are too small to be noted in this figure.

Fig. 3
Fig. 3

Complex refractive index of ice at 3.7-µm wavelength: (a) convergence of the scattering and extinction efficiency and (b) same as Fig. 2(b).

Fig. 4
Fig. 4

(a) Differently rounded vertices. (b) Phase functions on the equatorial plane of hexagonal columns with differently rounded vertices. The hexagonal sides are rounded to 1/4, 1/8, and 1/12.

Fig. 5
Fig. 5

Efficiency factor of the extinction cross section of a hexagonal column. The dotted curve represents the results for a sphere with the same volume-to-projected area ratio as its corresponding hexagonal column: (a) lateral incidence (θ0 = 90°) and (b) axial incidence (θ0 = 0°).

Fig. 6
Fig. 6

Scattering intensity distribution on an equatorial plane of a hexagonal column. The complex refractive index is 1.3 + 0.0i and L/ D = 1. The dotted curve in (a) represents the result for the circular column whose diameter and length are L and D.

Fig. 7
Fig. 7

Ray paths in geometrical optics. The complex refractive index is 1.3 + 0.0i.

Fig. 8
Fig. 8

Influence of the aspect ratio L/ D on the phase function of a hexagonal column. The lateral incidences are θ0 = 90° and ϕ0 = 0°: (a) distribution of the azimuthally averaged phase function over the zenith angle and (b) phase function on the equatorial plane of a hexagonal column.

Tables (1)

Tables Icon

Table 1 Influence of Round Hexagonal Vertices on Extinction Cross Sectiona

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

02π Kϕ, ϕIϕdϕ=Jϕ,
Kϕ+2mπ6, ϕ+2nπ6=Kϕ, ϕ+2n-mπ6,
K-ϕ, -ϕ=±Kϕ, ϕ.
X˜jϕn=05 λjnXϕ+2nπ6-π6<ϕ<π6.
-π/6π/6l=05 λj-1Kϕ, ϕ+2lπ6Ĩjϕdϕ=J˜jϕj=0  5.

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