Abstract

The problem of light scattering by ice crystal particles whose sizes are essentially larger than the incident wavelength is divided into two parts. First, the scattered field is represented as a set of plane-parallel outgoing beams in the near zone of the particle. Then, in the far zone the scattered field is represented as a result of both diffraction and interference of these beams within the framework of physical optics. A proper ray-tracing algorithm for calculation of the amplitude (Jones) scattering matrix is developed and applied. For large particles, a number of reduced Mueller matrices are introduced and discussed, since the pure Mueller matrix obtained from the Jones matrix becomes a rather cumbersome and quickly oscillating value. Backscattering by hexagonal ice crystals, including polarization properties, is considered in detail.

© 2003 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
    [CrossRef] [PubMed]
  9. A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
    [CrossRef]
  10. M. I. Mishchenko, A. Macke, “Incorporation of physical optics effects and computation of the Legendre expansion for ray-tracing phase functions involving δ-function transmission,” J. Geophys. Soc. 103, No. D2, 1799–1805 (1998).
    [CrossRef]
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    [CrossRef]
  12. A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Backscattering peak of hexagonal ice columns and plates,” Opt. Lett. 25, 1388–1390 (2000).
    [CrossRef]
  13. A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 72, 403–417 (2002).
    [CrossRef]
  14. 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]
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    [CrossRef] [PubMed]
  16. P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: solutions by a ray-by-ray integration algorithm,” J. Opt. Soc. Am. A 14, 2278–2289 (1997).
    [CrossRef]
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  19. C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 8, 1220–1224 (1978).
    [CrossRef]
  20. C. M. R. Platt, “Remote sensing of high cirrus clouds. III: Monte Carlo calculations of multiple-scattered lidar returns,” J. Atmos. Sci. 38, 156–167 (1981).
    [CrossRef]
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2002

A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 72, 403–417 (2002).
[CrossRef]

2001

2000

1998

M. I. Mishchenko, A. Macke, “Incorporation of physical optics effects and computation of the Legendre expansion for ray-tracing phase functions involving δ-function transmission,” J. Geophys. Soc. 103, No. D2, 1799–1805 (1998).
[CrossRef]

1997

1996

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Geometric-optics–integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35, 6568–6584 (1996).
[CrossRef] [PubMed]

1995

1994

1993

1989

1986

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

1985

1983

A. Borovoi, “Light propagation in media with closely packed particles,” Opt. Spektrosk. 54, 757–759 (1983) (in Russian).

1982

Q. Cai, K. N. Liou, “Polarized light scattering by hexagonal ice crystals: theory,” Appl. Opt. 21, 3569–3580 (1982).
[CrossRef] [PubMed]

A. Borovoi, “Light propagation in precipitation,” Izv. Vuzov. SSSR Radiofizi. 25, 391–400 (1982) (in Russian).

1981

C. M. R. Platt, “Remote sensing of high cirrus clouds. III: Monte Carlo calculations of multiple-scattered lidar returns,” J. Atmos. Sci. 38, 156–167 (1981).
[CrossRef]

1978

C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 8, 1220–1224 (1978).
[CrossRef]

Abshire, N. L.

C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 8, 1220–1224 (1978).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

Borovoi, A.

A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 72, 403–417 (2002).
[CrossRef]

A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Backscattering peak of hexagonal ice columns and plates,” Opt. Lett. 25, 1388–1390 (2000).
[CrossRef]

A. Borovoi, “Light propagation in media with closely packed particles,” Opt. Spektrosk. 54, 757–759 (1983) (in Russian).

A. Borovoi, “Light propagation in precipitation,” Izv. Vuzov. SSSR Radiofizi. 25, 391–400 (1982) (in Russian).

Cai, Q.

Chepfer, H.

Flamant, P. H.

Grishin, I.

A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 72, 403–417 (2002).
[CrossRef]

A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Backscattering peak of hexagonal ice columns and plates,” Opt. Lett. 25, 1388–1390 (2000).
[CrossRef]

Hess, M.

Irvine, W. M.

Jayaweera, K.

Ledanois, G.

Liou, K. N.

Lumme, K.

Macke, A.

M. I. Mishchenko, A. Macke, “Incorporation of physical optics effects and computation of the Legendre expansion for ray-tracing phase functions involving δ-function transmission,” J. Geophys. Soc. 103, No. D2, 1799–1805 (1998).
[CrossRef]

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

A. Macke, “Scattering of light by polyhedral ice crystals,” Appl. Opt. 32, 2780–2788 (1993).
[CrossRef] [PubMed]

McNice, G. T.

C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 8, 1220–1224 (1978).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, A. Macke, “Incorporation of physical optics effects and computation of the Legendre expansion for ray-tracing phase functions involving δ-function transmission,” J. Geophys. Soc. 103, No. D2, 1799–1805 (1998).
[CrossRef]

Mueller, J.

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Muinonen, K.

Naats, E.

A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 72, 403–417 (2002).
[CrossRef]

A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Backscattering peak of hexagonal ice columns and plates,” Opt. Lett. 25, 1388–1390 (2000).
[CrossRef]

Noel, V.

Oppel, U.

A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 72, 403–417 (2002).
[CrossRef]

A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Backscattering peak of hexagonal ice columns and plates,” Opt. Lett. 25, 1388–1390 (2000).
[CrossRef]

Peltoniemi, J.

Platt, C. M. R.

C. M. R. Platt, “Remote sensing of high cirrus clouds. III: Monte Carlo calculations of multiple-scattered lidar returns,” J. Atmos. Sci. 38, 156–167 (1981).
[CrossRef]

C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 8, 1220–1224 (1978).
[CrossRef]

Raschke, E.

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

Takano, Y.

Wiegner, M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

Yang, P.

Appl. Opt.

Izv. Vuzov. SSSR Radiofizi.

A. Borovoi, “Light propagation in precipitation,” Izv. Vuzov. SSSR Radiofizi. 25, 391–400 (1982) (in Russian).

J. Appl. Meteorol.

C. M. R. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 8, 1220–1224 (1978).
[CrossRef]

J. Atmos. Sci.

C. M. R. Platt, “Remote sensing of high cirrus clouds. III: Monte Carlo calculations of multiple-scattered lidar returns,” J. Atmos. Sci. 38, 156–167 (1981).
[CrossRef]

A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
[CrossRef]

J. Geophys. Soc.

M. I. Mishchenko, A. Macke, “Incorporation of physical optics effects and computation of the Legendre expansion for ray-tracing phase functions involving δ-function transmission,” J. Geophys. Soc. 103, No. D2, 1799–1805 (1998).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transf.

A. Borovoi, I. Grishin, E. Naats, U. Oppel, “Light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 72, 403–417 (2002).
[CrossRef]

Mon. Weather Rev.

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

Opt. Lett.

Opt. Spektrosk.

A. Borovoi, “Light propagation in media with closely packed particles,” Opt. Spektrosk. 54, 757–759 (1983) (in Russian).

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1970).

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, San Diego, 2000).

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Figures (5)

Fig. 1
Fig. 1

Example of the formation of plane-parallel beams. Here a plane electromagnetic wave is assumed to propagate from the reader to the hexagonal ice column of a given orientation. First, the illuminated rectangular facet 4, for example, creates reflected and refracted plane-parallel beams (not shown). Next, the refracted beam propagating inside the particle intersects the two rectangular facets 2 and 3 and the bottom hexagon facet within the three polygonal areas shaded in the figure. Each of these areas is a source of the next reflected and refracted plane-parallel beams of polygonal transverse shapes, and so on.

Fig. 2
Fig. 2

Plane-parallel beams of essential intensities 1–5 in the backward-scattering direction that are leaving a hexagonal ice column tilted at 32° relative to the incident direction (refractive index=1.311): (a) shape functions, (b) ray trajectories forming backscattered beam 5, (c) the same as (b) but for beams 1–4.

Fig. 3
Fig. 3

Modules of elements of the Jones matrix for ray trajectories of Fig. 2 as a function of column tilt: (a) for ray trajectories of Fig. 2(b) and (b) for ray trajectories of Fig. 2(c), (c) fine structure of the peak of Fig. 3(b) in the tilt interval of 31.5°–33°. The scattering direction is the exact backward direction.

Fig. 4
Fig. 4

Pure (solid curve) and diffraction (dashed curve) Mueller matrices (element M11) in the backward-scattering direction for running tilts of the hexagonal column of Fig. 2.

Fig. 5
Fig. 5

Fine angular structure of the field scattered near the backward direction by the hexagonal ice column (aspect ratio is 2, refractive index is 1.311, and column tilt angle is 32.2°). Jnm are moduli of the Jones-matrix elements versus scattering angles, where the backward-scattering direction is the center of the plots and a side of a picture is equal to an angle of 2λ/a (λ is wavelength, and a is size of a hexagon side).

Equations (32)

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

E(r)=E0(r)+nEn(r),
E0(r)=E0 exp(ikz).
ηn(ρ)=1insidethebeam0otherwise.
F=F1(β)00F2(β),
E=FLE0,L=cos αsin α-sin αcos α.
En0=exp(iξn)Lp+1FpLpF2L2F1L1E0.
En0=jn0E0,
jn0=exp(iξn)Lp+1FpLpF2L2F1L1.
Esh(r)=-E0(r)insidetheshadow0otherwise.
A˜=Γ(AA*)Γ-1,
Γ=121001100-10-1-100-ii0,
Γ-1=12110000-1i00-1-i1-100.
mn0=L˜p+1F˜pL˜pF˜1L˜1,
E(r)=E0 exp(ikz)+exp(ikR)RJ(Ω)E0,
Es(Ω)=J(Ω)E0.
Ens(Ω)=En0fn(Ω),
fn(Ω)=k2πiηn(ρ)exp[-ik(Ω-Ωn)ρ]dρ,
Ens(Ωn)=sniλEn0,
sn=ηn(ρ)dρ.
|Ens(Ω)|2dΩ=|En0|2|ηn(ρ)|2dρ=|En0|2sn.
jns(Ω)=jn0fn(Ω),
J(Ω)=jns(Ω).
M(Ω)=Γ[J(Ω)J*(Ω)]Γ-1.
M(Ω)=Mdif(Ω)+Minter(Ω),
Mdif(Ω)=nΓ[jns(Ω)jns*(Ω)]Γ-1=nmn0|fn(Ω)|2,
Minter(Ω)=nlΓ[fn(Ω)jn0jl0*fl*(Ω)]Γ-1.
Mgo(Ω)=nmn0snδ(Ω-Ωn),
E(r)=nEn,fr(r)+mEm,sk(r)+jEj,hex(r).
j1(0)=j11j12j21j22,j2(0)=j11j21j12j22,
j3(0)=j11-j12-j21j22,j4(0)=j11-j21-j12j22.
j100.801-10exp(iξ),
m00.64 diag(1, -1, -1, 1).

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