Abstract

Light scattering on finite dielectric cylinders having noncircular cross sections has become of growing importance in remote-sensing applications. For analyzing their scattering characteristics at moderate size parameters, i.e., at a region where ray-tracing techniques fail, a few methods have been developed, among which an approximation based on the generalized separation-of-variables method has become very successful. This approach reveals two interesting features, which we discuss, that reduce the numerical effort drastically if applied to hexagonal (in general, 2n-periodic) boundary surfaces. Finally, some results for the phase function of hexagonal ice columns are given.

© 2001 Optical Society of America

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  1. A. J. Baran, J. S. Foot, D. L. Mitchell, “Ice crystal absorption: a comparison between theory and implications for remote sensing,” Appl. Opt. 37, 2207–2215 (1998).
    [CrossRef]
  2. R. P. Lawson, A. J. Heymsfield, S. M. Aulenbach, T. L. Jensen, “Shapes, sizes and light scattering properties of ice crystals in cirrus and a persistent contrail during SUCCESS,” Geophys. Res. Lett. 25, 1331–1334 (1998).
    [CrossRef]
  3. Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
    [CrossRef]
  4. Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
    [CrossRef]
  5. M. I. Mishchenko, D. J. Wielaard, B. E. Carlson, “T-matrix computations of zenith-enhanced lidar backscatter from horizontally oriented ice plates,” Geophys. Res. Lett. 24, 771–774 (1997).
    [CrossRef]
  6. M. I. Mishchenko, K. Sassen, “Depolarization of lidar returns by small ice crystals: an application to contrails,” Geophys. Res. Lett. 25, 309–312 (1998).
    [CrossRef]
  7. M. I. Mishchenko, A. Macke, “How big should hexagonal ice crystals be to produce halos?” Appl. Opt. 38, 1626–1629 (1999).
    [CrossRef]
  8. P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
    [CrossRef]
  9. P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
    [CrossRef]
  10. K. Aydin, C. Tang, “Millimeter wave radar scattering from model ice crystal distributions,” IEEE Trans. Geosci. Remote Sensing 35, 140–146 (1997).
    [CrossRef]
  11. T. Rother, “Generalization of the separation of variables method for nonspherical scattering on dielectric objects,” J. Quant. Spectrosc. Radiat. Transfer 60, 335–353 (1998).
    [CrossRef]
  12. T. Rother, S. Havemann, K. Schmidt, “Scattering of plane waves on finite cylinders with noncircular cross sections,” in Progress in Electromagnetics Research (PIER), J. A. Kong, ed. (EMW, Cambridge, Mass.1999), Vol. 23, pp. 79–105.
  13. T. Rother, “General aspects of solving Helmholtz’s equation underlying eigenvalue and scattering problems in electromagnetic wave theory,” J. Electromagn. Waves Appl. 13, 867–888 (1999).
    [CrossRef]
  14. A. P. Prudnikov, J. A. Brychkov, O. I. Marichev, Integraly i Rjady (Nauka, Moscow, 1981), in Russian.
  15. F. M. Kahnert, J. J. Stamnes, K. Stamnes, “Application of the extended boundary condition method to homogeneous particles with point-group symmetries,” Appl. Opt. 40, 3110–3123 (2001).
    [CrossRef]
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    [CrossRef]
  17. M. Hess, M. Wiegner, “COP: a data library of optical properties of hexagonal ice crystals,” Appl. Opt. 33, 7740–7746 (1994); see also http://www.lrz-muenchen.de/~uh234an/www/mitarb/mhess.html .
    [CrossRef] [PubMed]
  18. B. Strauss, “On the climate impact of natural and anthropogenic ice clouds on the regional climate—with special regard to the microphysical influence,” Ph.D. thesis (University of Munich, Munich, Germany, 1994).

2001

1999

M. I. Mishchenko, A. Macke, “How big should hexagonal ice crystals be to produce halos?” Appl. Opt. 38, 1626–1629 (1999).
[CrossRef]

T. Rother, “General aspects of solving Helmholtz’s equation underlying eigenvalue and scattering problems in electromagnetic wave theory,” J. Electromagn. Waves Appl. 13, 867–888 (1999).
[CrossRef]

1998

T. Rother, “Generalization of the separation of variables method for nonspherical scattering on dielectric objects,” J. Quant. Spectrosc. Radiat. Transfer 60, 335–353 (1998).
[CrossRef]

R. P. Lawson, A. J. Heymsfield, S. M. Aulenbach, T. L. Jensen, “Shapes, sizes and light scattering properties of ice crystals in cirrus and a persistent contrail during SUCCESS,” Geophys. Res. Lett. 25, 1331–1334 (1998).
[CrossRef]

M. I. Mishchenko, K. Sassen, “Depolarization of lidar returns by small ice crystals: an application to contrails,” Geophys. Res. Lett. 25, 309–312 (1998).
[CrossRef]

A. J. Baran, J. S. Foot, D. L. Mitchell, “Ice crystal absorption: a comparison between theory and implications for remote sensing,” Appl. Opt. 37, 2207–2215 (1998).
[CrossRef]

1997

K. Aydin, C. Tang, “Millimeter wave radar scattering from model ice crystal distributions,” IEEE Trans. Geosci. Remote Sensing 35, 140–146 (1997).
[CrossRef]

M. I. Mishchenko, D. J. Wielaard, B. E. Carlson, “T-matrix computations of zenith-enhanced lidar backscatter from horizontally oriented ice plates,” Geophys. Res. Lett. 24, 771–774 (1997).
[CrossRef]

1996

1995

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
[CrossRef]

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

1994

1992

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

1984

Aulenbach, S. M.

R. P. Lawson, A. J. Heymsfield, S. M. Aulenbach, T. L. Jensen, “Shapes, sizes and light scattering properties of ice crystals in cirrus and a persistent contrail during SUCCESS,” Geophys. Res. Lett. 25, 1331–1334 (1998).
[CrossRef]

Aydin, K.

K. Aydin, C. Tang, “Millimeter wave radar scattering from model ice crystal distributions,” IEEE Trans. Geosci. Remote Sensing 35, 140–146 (1997).
[CrossRef]

Baran, A. J.

Brychkov, J. A.

A. P. Prudnikov, J. A. Brychkov, O. I. Marichev, Integraly i Rjady (Nauka, Moscow, 1981), in Russian.

Carlson, B. E.

M. I. Mishchenko, D. J. Wielaard, B. E. Carlson, “T-matrix computations of zenith-enhanced lidar backscatter from horizontally oriented ice plates,” Geophys. Res. Lett. 24, 771–774 (1997).
[CrossRef]

Foot, J. S.

Havemann, S.

T. Rother, S. Havemann, K. Schmidt, “Scattering of plane waves on finite cylinders with noncircular cross sections,” in Progress in Electromagnetics Research (PIER), J. A. Kong, ed. (EMW, Cambridge, Mass.1999), Vol. 23, pp. 79–105.

Hess, M.

Heymsfield, A. J.

R. P. Lawson, A. J. Heymsfield, S. M. Aulenbach, T. L. Jensen, “Shapes, sizes and light scattering properties of ice crystals in cirrus and a persistent contrail during SUCCESS,” Geophys. Res. Lett. 25, 1331–1334 (1998).
[CrossRef]

Jensen, T. L.

R. P. Lawson, A. J. Heymsfield, S. M. Aulenbach, T. L. Jensen, “Shapes, sizes and light scattering properties of ice crystals in cirrus and a persistent contrail during SUCCESS,” Geophys. Res. Lett. 25, 1331–1334 (1998).
[CrossRef]

Kahnert, F. M.

Lawson, R. P.

R. P. Lawson, A. J. Heymsfield, S. M. Aulenbach, T. L. Jensen, “Shapes, sizes and light scattering properties of ice crystals in cirrus and a persistent contrail during SUCCESS,” Geophys. Res. Lett. 25, 1331–1334 (1998).
[CrossRef]

Liou, K. N.

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

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

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
[CrossRef]

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

Macke, A.

Marichev, O. I.

A. P. Prudnikov, J. A. Brychkov, O. I. Marichev, Integraly i Rjady (Nauka, Moscow, 1981), in Russian.

Minnis, P.

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, A. Macke, “How big should hexagonal ice crystals be to produce halos?” Appl. Opt. 38, 1626–1629 (1999).
[CrossRef]

M. I. Mishchenko, K. Sassen, “Depolarization of lidar returns by small ice crystals: an application to contrails,” Geophys. Res. Lett. 25, 309–312 (1998).
[CrossRef]

M. I. Mishchenko, D. J. Wielaard, B. E. Carlson, “T-matrix computations of zenith-enhanced lidar backscatter from horizontally oriented ice plates,” Geophys. Res. Lett. 24, 771–774 (1997).
[CrossRef]

Mitchell, D. L.

Prudnikov, A. P.

A. P. Prudnikov, J. A. Brychkov, O. I. Marichev, Integraly i Rjady (Nauka, Moscow, 1981), in Russian.

Rother, T.

T. Rother, “General aspects of solving Helmholtz’s equation underlying eigenvalue and scattering problems in electromagnetic wave theory,” J. Electromagn. Waves Appl. 13, 867–888 (1999).
[CrossRef]

T. Rother, “Generalization of the separation of variables method for nonspherical scattering on dielectric objects,” J. Quant. Spectrosc. Radiat. Transfer 60, 335–353 (1998).
[CrossRef]

T. Rother, S. Havemann, K. Schmidt, “Scattering of plane waves on finite cylinders with noncircular cross sections,” in Progress in Electromagnetics Research (PIER), J. A. Kong, ed. (EMW, Cambridge, Mass.1999), Vol. 23, pp. 79–105.

Sassen, K.

M. I. Mishchenko, K. Sassen, “Depolarization of lidar returns by small ice crystals: an application to contrails,” Geophys. Res. Lett. 25, 309–312 (1998).
[CrossRef]

Schmidt, K.

T. Rother, S. Havemann, K. Schmidt, “Scattering of plane waves on finite cylinders with noncircular cross sections,” in Progress in Electromagnetics Research (PIER), J. A. Kong, ed. (EMW, Cambridge, Mass.1999), Vol. 23, pp. 79–105.

Stamnes, J. J.

Stamnes, K.

Strauss, B.

B. Strauss, “On the climate impact of natural and anthropogenic ice clouds on the regional climate—with special regard to the microphysical influence,” Ph.D. thesis (University of Munich, Munich, Germany, 1994).

Takano, Y.

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

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

Tang, C.

K. Aydin, C. Tang, “Millimeter wave radar scattering from model ice crystal distributions,” IEEE Trans. Geosci. Remote Sensing 35, 140–146 (1997).
[CrossRef]

Warren, S. G.

Wiegner, M.

Wielaard, D. J.

M. I. Mishchenko, D. J. Wielaard, B. E. Carlson, “T-matrix computations of zenith-enhanced lidar backscatter from horizontally oriented ice plates,” Geophys. Res. Lett. 24, 771–774 (1997).
[CrossRef]

Yang, P.

Appl. Opt.

Geophys. Res. Lett.

M. I. Mishchenko, D. J. Wielaard, B. E. Carlson, “T-matrix computations of zenith-enhanced lidar backscatter from horizontally oriented ice plates,” Geophys. Res. Lett. 24, 771–774 (1997).
[CrossRef]

M. I. Mishchenko, K. Sassen, “Depolarization of lidar returns by small ice crystals: an application to contrails,” Geophys. Res. Lett. 25, 309–312 (1998).
[CrossRef]

R. P. Lawson, A. J. Heymsfield, S. M. Aulenbach, T. L. Jensen, “Shapes, sizes and light scattering properties of ice crystals in cirrus and a persistent contrail during SUCCESS,” Geophys. Res. Lett. 25, 1331–1334 (1998).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing

K. Aydin, C. Tang, “Millimeter wave radar scattering from model ice crystal distributions,” IEEE Trans. Geosci. Remote Sensing 35, 140–146 (1997).
[CrossRef]

J. Atmos. Sci.

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

Y. Takano, K. N. Liou, P. Minnis, “The effects of small ice crystals on cirrus infrared radiative properties,” J. Atmos. Sci. 49, 1487–1493 (1992).
[CrossRef]

J. Electromagn. Waves Appl.

T. Rother, “General aspects of solving Helmholtz’s equation underlying eigenvalue and scattering problems in electromagnetic wave theory,” J. Electromagn. Waves Appl. 13, 867–888 (1999).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

T. Rother, “Generalization of the separation of variables method for nonspherical scattering on dielectric objects,” J. Quant. Spectrosc. Radiat. Transfer 60, 335–353 (1998).
[CrossRef]

Other

T. Rother, S. Havemann, K. Schmidt, “Scattering of plane waves on finite cylinders with noncircular cross sections,” in Progress in Electromagnetics Research (PIER), J. A. Kong, ed. (EMW, Cambridge, Mass.1999), Vol. 23, pp. 79–105.

A. P. Prudnikov, J. A. Brychkov, O. I. Marichev, Integraly i Rjady (Nauka, Moscow, 1981), in Russian.

B. Strauss, “On the climate impact of natural and anthropogenic ice clouds on the regional climate—with special regard to the microphysical influence,” Ph.D. thesis (University of Munich, Munich, Germany, 1994).

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Figures (7)

Fig. 1
Fig. 1

Orientation of a hexagonal column in the laboratory frame {x, y, z}. The transformation to the body frame {x, y, z} is given by the Eulerian angles of rotation (Φp, Θp, Ψp). The incident field propagates along the positive z axis in the laboratory frame. The scattering cone in the body frame is located at θ=Θp.

Fig. 2
Fig. 2

Relative error in the vertically (v) and horizontally (h) polarized extinction cross section of a finitely approximated hexagonal ice cylinder in a fixed orientation as a function of γ, defined in Eq. (25). The parameters used in the computations are the size parameter k0a=10 (a=side length of the hexagon), the cylinder length l=30 µm, the wavelength λ=0.5 µm, the complex refractive index of 1.313+j1.91×10-9, (Ref. 16) the Eulerian angles Φp=Θp=90° and Ψp=0°, and the number of expansion terms ncut=48.

Fig. 3
Fig. 3

Normalized phase function of a finitely approximated hexagonal ice cylinder in a fixed orientation for different values of γ, defined in Eq. (25). The parameters used in the computations are the same as in Fig. 2.

Fig. 4
Fig. 4

Relative error in the vertically (v) and horizontally (h) polarized extinction cross section of a finitely approximated hexagonal ice cylinder, randomly oriented with respect to the Eulerian angle Ψp, as a function of γ, defined in Eq. (25). The parameters used in the computations are the same as in Fig. 2, except for Ψp.

Fig. 5
Fig. 5

Normalized phase function of a finitely approximated hexagonal ice cylinder, randomly oriented with respect to the Eulerian angle Ψp, for different values of γ, defined in Eq. (25). The parameters used in the computations are the same as in Fig. 2, except for Ψp.

Fig. 6
Fig. 6

Normalized phase function of a finitely approximated hexagonal ice cylinder, randomly oriented with respect to the Eulerian angle Ψp, for different values of γ, defined in Eq. (25). The parameters used in the computations are k0a=60, l=30 µm, λ=0.5 µm, a complex refractive index of 1.313+j1.91×10-9 (Ref. 16), Φp=Θp=90°, and ncut=96.

Fig. 7
Fig. 7

Phase matrix elements of a size distribution of randomly oriented, finitely approximated, hexagonal ice columns with a refractive index of 1.1991+j0.051 at a wavelength of 10 µm (Ref. 16). The parameters used are summarized in Table 1.

Tables (1)

Tables Icon

Table 1 Parameters Used for Computing the Phase Matrix Elements in Fig. 7 by Means of Eq. (30)1

Equations (31)

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

E=×F+jω××A,
H=×A-jωμ0××F,
F=zˆm,
A=zˆe.
2e/m+k2e/m=0,
2=ρρρρ+ρ22z2+2ϕ2.
nˆ×(Eint-Es-Einc)=0,
nˆ×(Hint-Hs-Hinc)=0,
nˆ=eˆρ-R(ϕ)R(ϕ)eˆϕ1+R(ϕ)R(ϕ)2-1/2
02πdϕ exp{-jβϕ},β=-ncut ,, ncut,
M¯0¯0¯0¯Q¯R¯S¯T¯O¯-ksk0M¯0¯-O¯ksk0R¯-ksk0Q¯T¯-S¯·abcd=-ez(h/v)-eϕ,ρ(h/v)-hz(h/v)-hϕ,ρ(h/v),
Zαβ=02πZα(ϕ)exp[j(α-β)ϕ]dϕ.
Mα=-κs2Jα(ξs),
Oα=κ02Hα(1)(ξ0),
Qα=-jκshksJα(ξs)ξsjα+R(ϕ)R(ϕ)Jα(ξs),
Rα=-jκsJα(ξs)-R(ϕ)R(ϕ)Jα(ξs)ξsjα,
Sα=jκ0hk0Hα(1)(ξ0)ξ0jα+R(ϕ)R(ϕ)Hα(1)(ξ0),
Tα=jκ0Hα(1)(ξ0)-R(ϕ)R(ϕ)Hα(1)(ξ0)ξ0jα,
ξ0/s=R(ϕ)(k0/s2-h2)1/2,
κ0/s=1-h2k0/s21/2,
h=k0 cos Θp.
Zαβ=0π/nZα(ϕ)exp[j(α-β)ϕ]dϕ+π/n2π/nZαϕ-πn×exp[j(α-β)ϕ]dϕ++(2n-1)π/n2πZαϕ-(2n-1)πnexp[j(α-β)ϕ]dϕ.
ϕ˜=ϕ-pi,pi=iπn, i=0 ,, (2n-1),
Zαβ=i=0(2n-1)0π/nZα(ϕ˜)exp[j(α-β)(ϕ˜+pi)]dϕ˜,
=i=0(2n-1) exp[j(α-β)pi]0π/nZα(ϕ˜)×exp[j(α-β)ϕ˜]dϕ˜.
Zαβ=2n0π/nZα(ϕ˜)exp[j(α-β)ϕ˜]dϕ˜,(α-β)=2nγ,γ=0,±1,,±ncut/n0,otherwise.
x00x00x00x00x00x00x00x00x00x00x00x00x0,
α-β=γ,γ=0,±1,,±2ncut,
Z(ϕ)=γ=-aγ exp(jγϕ).
aγ=02πZ(ϕ)exp(-jγϕ)dϕ.
P(θ)=iPi(θ)σscainiiσscaini.

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