Abstract

We examine backscattering by analyzing large nonspherical particles with flat surfaces for which where the size is much larger than the wavelength, using ray optics and diffraction theory. We show that the backscattering cross section for rectangles can be 1 order of magnitude larger than that for spheres with same geometrical cross sections, depending on the orientation of the particles. Then we show that there is a difficulty in estimating the backscattering cross section for hexagonal columns with the available solutions but that it is possible to estimate the integration of the differential scattering cross section over small solid angles in backward directions. The integral values for hexagonal columns are found to be more than 1 order of magnitude larger than that for spheres with the same volume. As an application, the use of power from hexagonal columns for lidar observations is analyzed. Unlike for spherical particles with their dependence on Z -2 (where Z is the distance between the particle and the detector), for nonspherical particles such dependence varies with the particles’ nonsphericity, such as shape and orientation: Z 0 for a hexagonal plate randomly oriented in the horizontal plane; Z -1 for a hexagonal column randomly oriented in the horizontal plane.

© 2001 Optical Society of America

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References

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  1. T. Murayama, H. Okamoto, N. Kaneyasu, H. Kamataki, K. Miura, “Application of lidar depolarization measurement in the atmospheric boundary layer: effects of dust and sea-salt particles,” J. Geophys. Res. 104, 31,781–31,792 (1999).
    [CrossRef]
  2. K. Sassen, C. Hsueh, “Contrail properties derived from high-resolution polarization lidar studies during SUCCESS,” Geophys. Res. Lett. 25, 1165–1168 (1998).
    [CrossRef]
  3. J. D. Sphinhirne, S. Chudamani, J. F. Cavanaugh, J. L. Bufton, “Aerosol and cloud backscatter at 1.06, 1.54, and 0.53 µm by airborne hard-target-calibrated Nd:YAG/methane Raman lidar,” Appl. Opt. 36, 3475–3490 (1997).
    [CrossRef]
  4. F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distributions by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
    [CrossRef]
  5. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  6. C. M. Platt, “Lidar backscatter from horizontal ice crystal plates,” J. Appl. Meteorol. 17, 482–488 (1978).
    [CrossRef]
  7. C. M. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 17, 1220–1224 (1978).
    [CrossRef]
  8. M. I. Mishchenko, D. J. Wielaard, B. E. Calson, “T-matrix computations of zenith-enhanced lidar backscatter from horizontally oriented ice plates,” Geophys. Res. Lett. 24, 771–774 (1997).
    [CrossRef]
  9. D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s Lidar In-space Technology Experiment,” Proc. IEEE 84, 164–180 (1996).
    [CrossRef]
  10. Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. I. Single-scattering and optical properties of hexagonal ice crystals,” J. Atmos. Sci. 46, 3–19 (1989).
    [CrossRef]
  11. A. Macke, J. Mueller, E. Raschke, “Single scattering properties of atmospheric ice crystals,” J. Atmos. Sci. 53, 2813–2825 (1996).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  13. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1983).
  14. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  15. A. Ono, “The shape and riming properties of ice crystals in natural clouds,” J. Atmos. Sci. 26, 138–147 (1969).
    [CrossRef]
  16. S. G. Warren, “Optical constants of ice from the ultraviolet to the microwave,” Appl. Opt. 23, 1206–1225 (1984).
    [CrossRef] [PubMed]
  17. F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
    [CrossRef] [PubMed]
  18. G. L. Stephanes, Remote Sensing of the Lower Atmosphere (Oxford U. Press, New York, 1994), pp. 427–438.

1999 (1)

T. Murayama, H. Okamoto, N. Kaneyasu, H. Kamataki, K. Miura, “Application of lidar depolarization measurement in the atmospheric boundary layer: effects of dust and sea-salt particles,” J. Geophys. Res. 104, 31,781–31,792 (1999).
[CrossRef]

1998 (1)

K. Sassen, C. Hsueh, “Contrail properties derived from high-resolution polarization lidar studies during SUCCESS,” Geophys. Res. Lett. 25, 1165–1168 (1998).
[CrossRef]

1997 (2)

J. D. Sphinhirne, S. Chudamani, J. F. Cavanaugh, J. L. Bufton, “Aerosol and cloud backscatter at 1.06, 1.54, and 0.53 µm by airborne hard-target-calibrated Nd:YAG/methane Raman lidar,” Appl. Opt. 36, 3475–3490 (1997).
[CrossRef]

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

1996 (2)

D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s Lidar In-space Technology Experiment,” Proc. IEEE 84, 164–180 (1996).
[CrossRef]

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

1989 (1)

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

1984 (2)

1981 (1)

1978 (2)

C. M. Platt, “Lidar backscatter from horizontal ice crystal plates,” J. Appl. Meteorol. 17, 482–488 (1978).
[CrossRef]

C. M. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 17, 1220–1224 (1978).
[CrossRef]

1972 (1)

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distributions by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

1969 (1)

A. Ono, “The shape and riming properties of ice crystals in natural clouds,” J. Atmos. Sci. 26, 138–147 (1969).
[CrossRef]

Abshire, N. L.

C. M. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 17, 1220–1224 (1978).
[CrossRef]

Bohren, C. F.

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Bufton, J. L.

Calson, B. E.

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

Cavanaugh, J. F.

Chudamani, S.

Couch, R. H.

D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s Lidar In-space Technology Experiment,” Proc. IEEE 84, 164–180 (1996).
[CrossRef]

Fernald, F. G.

F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
[CrossRef] [PubMed]

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distributions by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

Herman, B. M.

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distributions by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

Hsueh, C.

K. Sassen, C. Hsueh, “Contrail properties derived from high-resolution polarization lidar studies during SUCCESS,” Geophys. Res. Lett. 25, 1165–1168 (1998).
[CrossRef]

Huffman, D. R.

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1983).

Kamataki, H.

T. Murayama, H. Okamoto, N. Kaneyasu, H. Kamataki, K. Miura, “Application of lidar depolarization measurement in the atmospheric boundary layer: effects of dust and sea-salt particles,” J. Geophys. Res. 104, 31,781–31,792 (1999).
[CrossRef]

Kaneyasu, N.

T. Murayama, H. Okamoto, N. Kaneyasu, H. Kamataki, K. Miura, “Application of lidar depolarization measurement in the atmospheric boundary layer: effects of dust and sea-salt particles,” J. Geophys. Res. 104, 31,781–31,792 (1999).
[CrossRef]

Klett, J. D.

Liou, K. N.

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

Macke, A.

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

McCormick, M. P.

D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s Lidar In-space Technology Experiment,” Proc. IEEE 84, 164–180 (1996).
[CrossRef]

McNice, G. T.

C. M. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 17, 1220–1224 (1978).
[CrossRef]

Mishchenko, M. I.

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

Miura, K.

T. Murayama, H. Okamoto, N. Kaneyasu, H. Kamataki, K. Miura, “Application of lidar depolarization measurement in the atmospheric boundary layer: effects of dust and sea-salt particles,” J. Geophys. Res. 104, 31,781–31,792 (1999).
[CrossRef]

Mueller, J.

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

Murayama, T.

T. Murayama, H. Okamoto, N. Kaneyasu, H. Kamataki, K. Miura, “Application of lidar depolarization measurement in the atmospheric boundary layer: effects of dust and sea-salt particles,” J. Geophys. Res. 104, 31,781–31,792 (1999).
[CrossRef]

Okamoto, H.

T. Murayama, H. Okamoto, N. Kaneyasu, H. Kamataki, K. Miura, “Application of lidar depolarization measurement in the atmospheric boundary layer: effects of dust and sea-salt particles,” J. Geophys. Res. 104, 31,781–31,792 (1999).
[CrossRef]

Ono, A.

A. Ono, “The shape and riming properties of ice crystals in natural clouds,” J. Atmos. Sci. 26, 138–147 (1969).
[CrossRef]

Platt, C. M.

C. M. Platt, “Lidar backscatter from horizontal ice crystal plates,” J. Appl. Meteorol. 17, 482–488 (1978).
[CrossRef]

C. M. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 17, 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]

Reagan, J. A.

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distributions by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

Sassen, K.

K. Sassen, C. Hsueh, “Contrail properties derived from high-resolution polarization lidar studies during SUCCESS,” Geophys. Res. Lett. 25, 1165–1168 (1998).
[CrossRef]

Sphinhirne, J. D.

Stephanes, G. L.

G. L. Stephanes, Remote Sensing of the Lower Atmosphere (Oxford U. Press, New York, 1994), pp. 427–438.

Takano, Y.

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

Warren, S. G.

Wielaard, D. J.

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

Winker, D. M.

D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s Lidar In-space Technology Experiment,” Proc. IEEE 84, 164–180 (1996).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Appl. Opt. (4)

Geophys. Res. Lett. (2)

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

K. Sassen, C. Hsueh, “Contrail properties derived from high-resolution polarization lidar studies during SUCCESS,” Geophys. Res. Lett. 25, 1165–1168 (1998).
[CrossRef]

J. Appl. Meteorol. (3)

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distributions by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

C. M. Platt, “Lidar backscatter from horizontal ice crystal plates,” J. Appl. Meteorol. 17, 482–488 (1978).
[CrossRef]

C. M. Platt, N. L. Abshire, G. T. McNice, “Some microphysical properties of an ice cloud from lidar observation of horizontally oriented crystals,” J. Appl. Meteorol. 17, 1220–1224 (1978).
[CrossRef]

J. Atmos. Sci. (3)

A. Ono, “The shape and riming properties of ice crystals in natural clouds,” J. Atmos. Sci. 26, 138–147 (1969).
[CrossRef]

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

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

J. Geophys. Res. (1)

T. Murayama, H. Okamoto, N. Kaneyasu, H. Kamataki, K. Miura, “Application of lidar depolarization measurement in the atmospheric boundary layer: effects of dust and sea-salt particles,” J. Geophys. Res. 104, 31,781–31,792 (1999).
[CrossRef]

Proc. IEEE (1)

D. M. Winker, R. H. Couch, M. P. McCormick, “An overview of LITE: NASA’s Lidar In-space Technology Experiment,” Proc. IEEE 84, 164–180 (1996).
[CrossRef]

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1983).

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

G. L. Stephanes, Remote Sensing of the Lower Atmosphere (Oxford U. Press, New York, 1994), pp. 427–438.

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Figures (10)

Fig. 1
Fig. 1

Comparison of backscattering cross sections for a sphere and for rectangles with the same geometrical cross sections. Mie, Fix, and Rot denote the backscattering cross sections for a sphere, for a fixed plate, and for a randomly oriented plate, respectively. Wavelength, 0.532 µm; refractive index, 1.31 + i2.54 - 9, which corresponds to ice at that wavelength.

Fig. 2
Fig. 2

Definition of the backward solid angle. 2a 0 and 2b 0(a 0 < b 0) are the side lengths of the rectangular scattering. b 0 is parallel to the Y plane. We assume that the shape of a backward solid angle δΩ is a rectangle on a unit sphere.

Fig. 3
Fig. 3

Alignment of a hexagonal crystal: (a) symmetrical axis parallel to the horizontal plane (2D column) and (b) symmetrical axis perpendicular to the horizontal plane (2D plate).

Fig. 4
Fig. 4

(a) Relation between angular bin δθ and the differential scattering cross section at backward δC sca(180)/δΩ calculated by Mie theory. Wavelength, 0.532 µm; spherical radius, 22.35 µm; optical constant, 1.31 + i2.54-9.

Fig. 5
Fig. 5

Relation between δθ and the ratio of ΔC bk,2Dplate to ΔC bk,Mie. The volume of a sphere is the same as that of a 2D plate (L = 20 µm, D = 30 µm).

Fig. 6
Fig. 6

Relation between differential cross section δC sca(180)/δΩ for a 2D column and scattering angle θ. Each curve corresponds to a different angular bin δθ. As the angular bin becomes smaller, δC sca(180)/δΩ becomes larger.

Fig. 7
Fig. 7

(a) Relation between δθ and δC sca(180)/δθ for a 2D column, a 3D plate, and a 3D column when the ray-tracing method is used for computations. Each curve corresponds to the results for a different random seed. Column size, L = 86.5 µm; D = 28.8 µm; plate size, L = 20 µm; D = 60 µm. (b) Relation between δθ and the ratio of ΔC sca for Mie theory to that for hexagonal columns.

Fig. 8
Fig. 8

Geometrical configuration between incident and scattered electromagnetic fields and the scattered target. The length of the target is specified by a 0 along the X axis and b 0 along the Y axis (b 0a 0).

Fig. 9
Fig. 9

(a) Comparison of received power P r for a sphere and that for a 2D column. (b) Same as (a) but for a 2D plate.

Fig. 10
Fig. 10

Geometrical configuration of incident and scattered electro-magnetic fields at a height of Z and of the scattered target. Positions of point source P0, element dS on the scattered target, Q, and detector P. s and r are the distance from P 0 to Q and from Q to P. Ψ, angle between the Z axis and the normal vector of the scattered target.

Tables (1)

Tables Icon

Table 1 Conditions for Applicability of GO for Particles of Various Shapesa

Equations (39)

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Cbk=limδΩθ0,θ180 4π δCscaθδΩθ,
Cbk=4π Rfreλcos Ψ2S2sin2ka0 sin Ψ2ka0 sin Ψ2,
Cbk,fix=4πRfreSλ2.
Cbk,rot=4 -π/2π/2dΨ Rfreλcos Ψ2×S2sin2 sin Ψka02 sin Ψka02.
ΔCbkδθΩθ=180-δθΩθ=180dΩ dCscaθdΩθ.
ΔCbk,fixKDΦx, Φy=Rfreλ2S2-ΦxΦxdϕx-ΦyΦydϕy×sinka0ϕxka0ϕx2sinkb0ϕykb0ϕy2, ΔCbk,fixGO=|Rfre|2S,
Φx>π/ka0, Φy>π/kb0.
ΔCbk,rotKDΦx, Φy=1ka0|Rfre|2λ2 S2Φx-ΦyΦydϕy×sin kb0ϕykb0ϕy2, ΔCbk,rotGOΦx=Φxπ |Rfre|2S.
Φy>π/kb0.
ΔCbk,sqKDΦsq, Φsq=Rfreλ2S2-ΦsqΦsqdϕx-ΦsqΦsqdϕy×sinkRsqϕxkRsqϕx2sinkRsqϕykRsqϕy2.
Φsq>πkRsq=2πkasp1/34/3π41/12req,
req=34π Vhex1/3,
ΔCbk,rotΦx, ΦL=1ka0|Rfre|2λ2 S2Φx×-ΦLΦLdϕysinkLϕy/2kLϕy/22,
ΦL>πkL/2=πk4π/931/3asp2/3req.
ΔCbk,Mieδθ=Ωθ=180-δθΩθ=180dΩ dCscaθdΩ=dCsca180dΩΩ180-Ω180-δθ=dCsca180dΩ πδθ2.
ΔCbk,2d-plate=RplateS=2|Rfre|21+|Rfre|4 S,
ΔCbkδθ=Ωθ=180-δθΩθ=180dΩ dCscadΩ=Ωθ=180-δθΩθ=180dΩ dCscadθdθdΩ=180-δθ180dθ dCscadθ,
ΔCbkδθ=dCsca180dθ180-δθ180dθ=dCsca180dθ δθ.
Pr=IP0ΔCsca=IP0Ωθ=180-δθΩθ=180dΩ dCscaθdΩ,
Pr=IP0GZ2dCsca180dΩexp-2τ=IP0GZ2Cbk4πexp-2τ,
Rt>2πZkasp1/34/3π41/12req,
Pr=IP0RplateGrexp-2τ=IP02|Rfre|21+|Rfre|4 Grexp-2τ.
Rt>Zπk4π/931/3asp2/3req.
Pr=IP0δθ dCsca180dθexp-2τ=IP0RtZdCsca180dθexp-2τ.
P0=0, 0, 0, P=x, y, 0, Q=a cos Ψ, b, a sin Ψ+Z,
EP=1/4πsurdSEQexpikrrn×expikss-expikssn×EQexpikrr, EQ=iAQoutkrexpikr,
r+sZ2+2a sin ΨZ+x2+y2-2ax cos Ψ+by2Z2,
EP=-1λAQoutkZcos ΨZexpik2Z+x2+y22Z× -a0a0da expik2 sin Ψ-x cos ΨZa×-b0b0db expik yZ b=-1λAQoutkZcos ΨZ S expik2Z+x2+y22Z×sin2 sin Ψ-x cos Ψ/Zka02 sin Ψ-x cos Ψ/Zka0sin kyb0/Zkyb0/Z,
IP=k2ωμ0 |EP|2=k2ωμ01λAQoutkZcos ΨZ2×S2sin2 sin Ψ-x/Z cos Ψka02 sin Ψ-x/Z cos Ψka02×sinkyb0/Zkyb0/Z2,
IP=Rfreλcos ΨZ2 IP0S2×sin2 sin Ψ-x/Z cos Ψka02 sin Ψ-x/Z cos Ψka02×sinkyb0/Zkyb0/Z2, IP0IQ=k2ωμ0AQinkZ2.
dCscadΩ=Z2IPIP0=Rfreλcos Ψ2S2×sin2 sin Ψ-ϕx cos Ψka02 sin Ψ-ϕx cos Ψka02×sin kb0ϕykb0ϕy2,
dCscadΩrot=|Rfre|2λ2 S21π-π/2π/2dΨ cos2×Ψsin2 sin Ψ-ϕx cos Ψka02 sin Ψ-ϕx cos Ψka02×sin kb0ϕykb0ϕy212ka0|Rfre|2λ2 S2sin kb0ϕykb0ϕy2,
Cbk,fix=4π dCsca180dΩ=4π Rfreλ2S2.
Cbk,rot=4πdCscaϕy=0dΩrot=2πka0|Rfre|2λ2 S2.
ΔCbkΩθ=180-δθΩθ=180dΩ dCscaθdΩ.
ΔCbk,fixKD=Rfreλ2S2-ΦxΦxdϕx-ΦyΦydϕysinka0ϕxka0ϕx2×sinkb0ϕykb0ϕy2.
ΔCbk,rotKD=-ΦxΦxdϕx-ΦyΦydϕydCscadΩrot=1ka0|Rfre|2λ2 S2Φx-ΦyΦydϕysin kb0ϕykb0ϕy2.
ΔCbk,fixGOpower scattered backward within δΩincident irradiance=|Rfre|2SIP0IP0=|Rfre|2S.
ΔCbk,rotGO=Φx/π|Rfre|2SIP0IP0=Φx/π|Rfre|2S.

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