Abstract

A simple relationship is established between the linear and the circular depolarization ratios averaged over the azimuth angle of clouds made of spherical particles. The relationship is validated theoretically using double-scattering calculations; in the framework, the measurements are performed with a multiple-field-of-view lidar (MFOV) lidar. The relationship is also validated using data obtained with MFOV lidar equipped with linear and circular polarization measurement capabilities. The experimental data support theoretical results for small optical depths. At higher optical depths and large fields of view, the contribution of multiple scatterings is important; experimental data suggest that the relationship established between the linear and circular depolarization stays valid as long as the main depolarization mechanism comes from one scattering (most likely a backscattering a few degrees away from 180°).

© 2008 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. N. Roy and G. Roy, “Standoff determination of the particles size and concentration of small optical depth clouds based on double scattering measurements: concept and validation with calibrated target plates,” presented at the 24th International Laser Radar Conference, Boulder, Colorado, 23-27 June 2008.
  14. N. Roy and G. Roy, “Influence of multiple scattering on lidar depolarization measurements with an ICCD camera,” presented at the 23rd International Laser Radar Conference, Nara, Japan, 24-28 July 2006, pp 109-113.
  15. F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scatterd light,” Phys. Rev. B 40, 9342-9345 (1989)
    [CrossRef]
  16. G. Roy, L. R. Bissonnette, C. Bastille, and G. Vallée, Retrieval of droplet-size density distribution from multiple field-of-view cross-polarized lidar signals, Appl. Opt. 38, 5202-5211 (1999).
    [CrossRef]
  17. L.I.Chaikovskaya, “Remote sensing of clouds using linearly and circularly polarized laser beams: techniques to compute signal polarization,” in Light Scattering Reviews, A.AKokhanovsky, ed. (Praxis, 2008), Vol. 3, pp. 191-228.
    [CrossRef]
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    [CrossRef] [PubMed]
  21. G. Roy and L. R. Bissonnette, “Strong dependence of rain-induced lidar depolarization on the illumination angle: experimental evidence and geometrical-optics interpretation,” Appl. Opt. 40, 4770-4789 (2001).
    [CrossRef]
  22. U. Wandinger, Ph.D. dissertation (University of Hamburg, 1994), p. 125.

2008 (1)

2007 (1)

2006 (2)

2004 (1)

2003 (1)

Y-X. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers with lidar using circular polarization: a theoretical study,” J. Quant. Spectr. Radiat. Trans. 79-80, 757-764 (2003).
[CrossRef]

2001 (1)

1999 (1)

1998 (1)

1997 (1)

1995 (2)

K. Sassen and H. Zhao, “Lidar multiple scattering in water droplet clouds: toward an improved treatment,” Opt. Rev. 2, 394-400 (1995).
[CrossRef]

M. I. Mishchenko and J. W. Hovenier, “Depolarization of light backscattered by randomly oriented nonspherical particles,” Opt. Lett. 20, 1356-1358 (1995).
[CrossRef] [PubMed]

1989 (1)

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scatterd light,” Phys. Rev. B 40, 9342-9345 (1989)
[CrossRef]

1985 (1)

S. R. Pal and A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from clouds,” Appl. Opt. 24, 3463-3471(1985).
[CrossRef]

1978 (1)

Bastille, C.

Ben-David, A.

Bissonnette, L.

Bissonnette, L. R.

Carswell, A. I.

S. R. Pal and A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from clouds,” Appl. Opt. 24, 3463-3471(1985).
[CrossRef]

J. D. Houston and A. I. Carswell, “Four-component polarization measurement of lidar atmospheric scattering,” Appl. Opt. 17, 614-620 (1978).
[CrossRef] [PubMed]

Castagnoli, F.

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersion (American Elsevier, 1969), pp. 73-290.

Del Guasta., M.

Donovan, D. P.

D. P. Donovan, “The use of circular polarization in space-based lidar systems: consideration for the earthCARE lidar,” presented at the 23rd International Laser Radar Conference, Nara, Japan, 24-28 July 2006, pp. 1019-1022.

Flynn, C. J.

Gibson, G.

Y-X. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers with lidar using circular polarization: a theoretical study,” J. Quant. Spectr. Radiat. Trans. 79-80, 757-764 (2003).
[CrossRef]

Gilles, R.

Gimmestad, G. G.

Hostetler, C.

Y-X. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers with lidar using circular polarization: a theoretical study,” J. Quant. Spectr. Radiat. Trans. 79-80, 757-764 (2003).
[CrossRef]

Houston, J. D.

Hovenier, J. W.

Hu, Y.

Hu, Y-X.

Y-X. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers with lidar using circular polarization: a theoretical study,” J. Quant. Spectr. Radiat. Trans. 79-80, 757-764 (2003).
[CrossRef]

Katsev, L.

Lin, B.

Y-X. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers with lidar using circular polarization: a theoretical study,” J. Quant. Spectr. Radiat. Trans. 79-80, 757-764 (2003).
[CrossRef]

Liu, Z.

MacKintosh, F. C.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scatterd light,” Phys. Rev. B 40, 9342-9345 (1989)
[CrossRef]

Mathur, S.

McGill, M.

Memdoza, A.

Mishchenko, M. I.

Morandini, M.

Noel, V.

Pal, S. R.

S. R. Pal and A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from clouds,” Appl. Opt. 24, 3463-3471(1985).
[CrossRef]

Pine, D. J.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scatterd light,” Phys. Rev. B 40, 9342-9345 (1989)
[CrossRef]

Polonsky, I. N.

Prikhach, A. S.

Riviere, D.

Roy, G.

N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Measurement of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth,” Appl. Opt. 43, 2777-2785 (2004).
[CrossRef] [PubMed]

G. Roy and L. R. Bissonnette, “Strong dependence of rain-induced lidar depolarization on the illumination angle: experimental evidence and geometrical-optics interpretation,” Appl. Opt. 40, 4770-4789 (2001).
[CrossRef]

G. Roy, L. R. Bissonnette, C. Bastille, and G. Vallée, Retrieval of droplet-size density distribution from multiple field-of-view cross-polarized lidar signals, Appl. Opt. 38, 5202-5211 (1999).
[CrossRef]

N. Roy and G. Roy, “Standoff determination of the particles size and concentration of small optical depth clouds based on double scattering measurements: concept and validation with calibrated target plates,” presented at the 24th International Laser Radar Conference, Boulder, Colorado, 23-27 June 2008.

N. Roy and G. Roy, “Influence of multiple scattering on lidar depolarization measurements with an ICCD camera,” presented at the 23rd International Laser Radar Conference, Nara, Japan, 24-28 July 2006, pp 109-113.

Roy, N.

N. Roy, G. Roy, L. R. Bissonnette, and J.-R. Simard, “Measurement of the azimuthal dependence of cross-polarized lidar returns and its relation to optical depth,” Appl. Opt. 43, 2777-2785 (2004).
[CrossRef] [PubMed]

N. Roy and G. Roy, “Influence of multiple scattering on lidar depolarization measurements with an ICCD camera,” presented at the 23rd International Laser Radar Conference, Nara, Japan, 24-28 July 2006, pp 109-113.

N. Roy and G. Roy, “Standoff determination of the particles size and concentration of small optical depth clouds based on double scattering measurements: concept and validation with calibrated target plates,” presented at the 24th International Laser Radar Conference, Boulder, Colorado, 23-27 June 2008.

Sassen, K.

K. Sassen and H. Zhao, “Lidar multiple scattering in water droplet clouds: toward an improved treatment,” Opt. Rev. 2, 394-400 (1995).
[CrossRef]

K. Sassen, “Polarization in lidar,” Lidar: Range-Resolved Optical Remote Sensing of the Atmosphere, Claus Weikamp, ed., Springer Series in Optical Sciences (Springer, 2005), Chap. 2, pp. 19-42.

Simard, J.-R.

Vallar, E.

Vallée, G.

Vaughan, M.

Venturi, V.

Wandinger, U.

U. Wandinger, Ph.D. dissertation (University of Hamburg, 1994), p. 125.

Weitz, D. A.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scatterd light,” Phys. Rev. B 40, 9342-9345 (1989)
[CrossRef]

Winkler, D.

Yang, P.

Y-X. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers with lidar using circular polarization: a theoretical study,” J. Quant. Spectr. Radiat. Trans. 79-80, 757-764 (2003).
[CrossRef]

Zege, E. P.

Zhao, H.

K. Sassen and H. Zhao, “Lidar multiple scattering in water droplet clouds: toward an improved treatment,” Opt. Rev. 2, 394-400 (1995).
[CrossRef]

Zheng, Y.

Zhu, J. X.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scatterd light,” Phys. Rev. B 40, 9342-9345 (1989)
[CrossRef]

Appl. Opt. (8)

J. Opt. Soc. Am. A (1)

J. Quant. Spectr. Radiat. Trans. (1)

Y-X. Hu, P. Yang, B. Lin, G. Gibson, and C. Hostetler, “Discriminating between spherical and non-spherical scatterers with lidar using circular polarization: a theoretical study,” J. Quant. Spectr. Radiat. Trans. 79-80, 757-764 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Opt. Rev. (1)

K. Sassen and H. Zhao, “Lidar multiple scattering in water droplet clouds: toward an improved treatment,” Opt. Rev. 2, 394-400 (1995).
[CrossRef]

Phys. Rev. B (1)

F. C. MacKintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scatterd light,” Phys. Rev. B 40, 9342-9345 (1989)
[CrossRef]

Other (7)

K. Sassen, “Polarization in lidar,” Lidar: Range-Resolved Optical Remote Sensing of the Atmosphere, Claus Weikamp, ed., Springer Series in Optical Sciences (Springer, 2005), Chap. 2, pp. 19-42.

L.I.Chaikovskaya, “Remote sensing of clouds using linearly and circularly polarized laser beams: techniques to compute signal polarization,” in Light Scattering Reviews, A.AKokhanovsky, ed. (Praxis, 2008), Vol. 3, pp. 191-228.
[CrossRef]

U. Wandinger, Ph.D. dissertation (University of Hamburg, 1994), p. 125.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersion (American Elsevier, 1969), pp. 73-290.

N. Roy and G. Roy, “Standoff determination of the particles size and concentration of small optical depth clouds based on double scattering measurements: concept and validation with calibrated target plates,” presented at the 24th International Laser Radar Conference, Boulder, Colorado, 23-27 June 2008.

N. Roy and G. Roy, “Influence of multiple scattering on lidar depolarization measurements with an ICCD camera,” presented at the 23rd International Laser Radar Conference, Nara, Japan, 24-28 July 2006, pp 109-113.

D. P. Donovan, “The use of circular polarization in space-based lidar systems: consideration for the earthCARE lidar,” presented at the 23rd International Laser Radar Conference, Nara, Japan, 24-28 July 2006, pp. 1019-1022.

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Figures (19)

Fig. 1
Fig. 1

Scattering geometry.

Fig. 2
Fig. 2

Phase function and linear and circular depolarization ratios of 10 μm mean-diameter water droplets.

Fig. 3
Fig. 3

Phase function and linear and circular depolarization ratios of 0.28 μm mean-diameter oil droplets.

Fig. 4
Fig. 4

Circular depolarization ratio as a function of the linear depolarization ratio for three particle size distributions.

Fig. 5
Fig. 5

Double-scattering processes and the reciprocity theorem.

Fig. 6
Fig. 6

Double-scattering processes and the relation between FOV rings and scattering angles.

Fig. 7
Fig. 7

Calculated second-order-scattering circular depolarization ratios as a function of linear depolarization ratios for five penetration depths for a 10 μm diameter water-droplet cloud.

Fig. 8
Fig. 8

Calculated second-order-scattering circular depolarization ratio as a function of linear depolarization ratio for five penetration depths for a 0.28 μm diameter fog-oil cloud.

Fig. 9
Fig. 9

Effective scattering angles associated in function of the FOV rings for five penetration depths in a water-droplet cloud.

Fig. 10
Fig. 10

Effective scattering angles associated as a function of the FOV rings for five penetration depths in the fog-oil droplet cloud.

Fig. 11
Fig. 11

Linear and circular polarization lidar setup.

Fig. 12
Fig. 12

Images of the parallel and the perpendicular components in linear and circular polarizations for the four disseminated materials studied. To be easily seen, the intensity of the perpendicular images have been multiply by 15, 10, and 5 for the fog-oil (linear), fog-oil (circular), and glass beads (linear), respectively.

Fig. 13
Fig. 13

Measured circular depolarization ratio δ cir as a function of the linear depolarization ratio, δ lin , for five penetration depths for (a), (b) water cloud, (c), (d) fog-oil cloud, (e), (f) glass bead particles, and (g), (h) Arizona road dust. The optical depths (O.D.) corresponding to each trial has been indicated on each figure.

Fig. 14
Fig. 14

Measured (water droplet cloud #1) normalized total lidar signals and depolarization ratios (linear and circular) as a function of the distance from the lidar system for a FOV of 8 mrad .

Fig. 15
Fig. 15

Calculated (second-order scattering, for 10 μm water droplets) normalized total lidar signals and depolarization ratios (linear and circular) as a function of distance from the lidar system for a FOV of 8 mrad .

Fig. 16
Fig. 16

Measured (water droplets cloud #1) circular depolarization ratio as a function of the linear depolarization ratio for five penetration depths.

Fig. 17
Fig. 17

Calculated (second-order scattering, for 10 μm water droplets) circular depolarization ratio as a function of the linear depolarization ratio for five penetration depths.

Fig. 18
Fig. 18

Measured (water droplets cloud #1) linear and circular depolarization ratios as function of the optical depth for FOVs of 2, 4, and 8 mrad .

Fig. 19
Fig. 19

Calculated linear and circular depolarization ratios as function of optical depth for a C1 cloud at a distance of 2 km and for a FOV of 0.87 and 1.74 mrad . From [17].

Tables (4)

Tables Icon

Table 1 Smaller and Larger FOVs Defined for Each Ring and Their Corresponding Scattering and Backscattering Angles (β and β b ) for the First and Last Penetration Depths at 107 m and 121 m

Tables Icon

Table 2 Materials Disseminated

Tables Icon

Table 3 Wave Plate Set Angles and the Stokes Parameters for the Incident, the Backscattered, and the Measured Light

Tables Icon

Table 4 Optical Depths Associated With Each Measurement Inside the Aerosol Chamber for the Water, Fog Oil, Glass Beads, and Arizona Road Dust Clouds

Equations (66)

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S = ( S 0 S 1 S 2 S 3 ) = ( I Q U V ) .
δ lin = I I / / = I Q I + Q .
δ cir = I C I C / / = I + V I V .
δ lin ( β ) = I I = 0 2 π I y ( β , φ ) d φ 0 2 π I x ( β , φ ) d φ ,
δ lin ( β ) = ( P 2 cos 2 β 2 P 3 cos β + P 1 ) ( 3 P 2 cos 2 β + 2 P 3 cos β + 3 P 1 ) .
δ cir ( β ) = ( P 2 cos 2 β 2 P 3 cos β + P 1 ) ( P 2 cos 2 β + 2 P 3 cos β + P 1 ) ,
δ cir ( β ) = 2 δ lin ( β ) 1 δ lin ( β ) .
P D ( z c , Δ θ i ) = 2 P ( z c ) z a z c β i β i + 1 F ( z , z c , β , θ ) sin β d β d z ,
P ( z c ) = P 0 exp [ 2 z a z c α ( z ) d z ] c τ 2 2 π A z c 2 ,
F ( z , z c , β , θ ) = [ α s ( z ) p ( β ( θ , z , z c ) ) ] [ α s ( z c ) p ( β b ( θ , z , z c ) ) ] ,
β b = π β + θ , tan β = z c tan θ z c z , β i = tan 1 ( z c tan θ i z c z ) , β i + 1 = tan 1 ( z c tan θ i + 1 z c - z ) ,
P D ( z c , Δ θ i ) = 2 P ( z c ) z a z c θ i θ i + 1 F ( z , z c , β , θ ) ( z c z c z ) 2 θ d θ d z .
δ ¯ lin ( z c , Δ θ i ) = z a z c β i β i + 1 F ( z , z c , β , θ ) sin β d β d z z a z c β i β i + 1 F ( z , z c , β , θ ) sin β d β d z ,
F ( z , z c , β , θ ) = [ α s ( z ) p ( β ( θ , z , z c ) ) ] [ α s ( z c ) p lin ( β b ( θ , z , z c ) ) ] , F ( z , z c , β , θ ) = [ α s ( z ) p ( β ( θ , z , z c ) ) ] [ α s ( z c ) p lin ( β b ( θ , z , z c ) ) ] .
δ ¯ cir ( z c , Δ θ i ) = z a z c β i β i + 1 F C ( z , z c , β , θ ) sin β d β d z z a z c β i β i + 1 F C ( z , z c , β , θ ) sin β d β d z ,
F C ( z , z c , β , θ ) = [ α s ( z ) p ( β ( θ , z , z c ) ) ] [ α s ( z c ) p C ( β b ( θ , z , z c ) ) ] , F C ( z , z c , β , θ ) = [ α s ( z ) p ( β ( θ , z , z c ) ] [ α s ( z c ) p C ( β b ( θ , z , z c ) ) ] .
δ ¯ cir ( Δ θ i ) = 2 δ ¯ lin ( Δ θ i ) 1 δ ¯ lin ( Δ θ i ) .
β ¯ ( Δ θ i ) = z a z c β i β i + 1 β F ( z , z c , β , θ ) sin β d β d z z a z c β i β i + 1 F ( z , z c , β , θ ) sin β d β d z .
δ lin ( Δ θ i ) = F L ( I 2 ( Δ θ i ) I 1 ( Δ θ i ) ) LV _ L ,
δ cir ( Δ θ i ) = F C ( I 2 ( Δ θ i ) I 1 ( Δ θ i ) ) C 1 _ C
M atm = ( 1 0 0 0 0 1 d 0 0 0 0 d 1 0 0 0 0 2 d 1 ) .
d = 2 δ lin 1 + δ lin ,
d = δ cir 1 + δ cir .
( E x E y ) = ( I e i Ψ 0 ) .
( E E ) = ( cos φ sin φ sin φ cos φ ) ( E x E y ) .
( E s E s ) = ( S 2 0 0 S 1 ) ( E E ) .
( E x s E y s ) = ( cos φ sin φ sin φ cos φ ) ( E s cos β E s ) ,
E x s = I e i Ψ ( S 2 cos 2 φ cos β + S 1 sin 2 φ ) ,
E y s = I e i Ψ ( S 2 sin φ cos φ cos β S 1 sin φ cos φ ) .
I x = E x s E x s * , I y = E y s E y s * ,
I x = I ( S 2 S 2 * cos 4 φ cos 2 β + ( S 1 S 2 * + S 1 * S 2 ) cos 2 φ sin 2 φ cos β + S 1 S 1 * sin 4 φ ) ,
I y = I ( S 2 S 2 * sin 2 φ cos 2 φ cos 2 β ( S 1 S 2 * + S 1 * S 2 ) cos 2 φ sin 2 φ cos β + S 1 S 1 * sin 2 φ cos 2 φ ) .
sin 2 φ cos 2 φ = 1 / 8 , sin 4 φ = cos 4 φ = 3 / 8 ,
I = I x = I 8 ( 3 S 2 S 2 * cos 2 β + ( S 1 S 2 * + S 1 * S 2 ) cos β + 3 S 1 S 1 * ) ,
I = I y = I 8 ( S 2 S 2 * cos 2 β ( S 1 S 2 * + S 1 * S 2 ) cos β + S 1 S 1 * ) .
P 2 = S 2 S 2 * , P 1 = S 1 S 1 * , P 3 = 1 2 ( S 2 S 1 * + S 2 * S 1 ) .
δ ( β ) = I I = ( P 2 cos 2 β - 2 P 3 cos β + P 1 ) ( 3 P 2 cos 2 β + 2 P 3 cos β + 3 P 1 ) .
p ( β ) = 1 8 ( P 2 cos 2 β 2 P 3 cos β + P 1 ) ,
p ( β ) = 1 8 ( 3 P 2 cos 2 β + 2 P 3 cos β + 3 P 1 ) .
( E x E y ) = 1 2 ( I e i Ψ - i I e i Ψ ) .
E x s = I 2 e i Ψ ( S 2 cos 2 φ cos β i S 2 cos φ sin φ cos β + S 1 sin 2 φ + i S 1 cos φ sin φ ) ,
E y s = I 2 e i Ψ ( S 2 cos φ sin φ cos β i S 2 sin 2 φ cos β S 1 cos φ sin φ i S 1 cos 2 φ ) .
E x s E x s * = I 2 ( S 2 S 2 * ( cos 2 φ sin 2 φ + sin 4 φ ) cos 2 β + S 1 S 1 * ( sin 2 φ cos 2 φ + cos 4 φ ) ,
E x s E x s * = E y s E y s * = I 4 ( S 2 S 2 * cos 2 β + S 1 S 1 * ) ,
E x s E y s * = i I cos β 4 ( S 2 S 1 * + S 1 * S 2 ) ,
E y s E x s * = i I cos β 4 ( S 2 S 1 * + S 1 * S 2 ) .
S 0 = E x s E x s * + E y s E y s * = I 2 ( S 2 S 2 * cos 2 β + S 1 S 1 * ) = I 2 ( P 2 cos 2 β + P 1 ) , S 1 = S 2 = 0 , S 3 = i ( E x E y * E y * E x ) = cos β 2 ( S 2 S 1 * + S 1 * S 2 ) = P 3 cos β .
1 2 ( 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ) ( P 2 cos 2 β + P 1 0 0 2 P 3 cos β ) = 1 2 ( P 2 cos 2 β + P 1 2 P 3 cos β 0 0 ) .
1 2 ( 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ) 1 2 ( P 2 cos 2 β + P 2 P 3 cos β 0 0 ) = 1 4 ( P 2 cos 2 β + 2 P 3 cos β + P 1 P 2 cos 2 β + 2 P 3 cos β + P 1 0 0 ) ,
1 2 ( 1 - 1 0 0 - 1 1 0 0 0 0 0 0 0 0 0 0 ) 1 2 ( P 2 cos 2 β + P 1 2 P 3 cos β 0 0 ) = 1 4 ( P 2 cos 2 β 2 P 3 cos β + P 1 P 2 cos 2 β 2 P 3 cos β + P 1 0 0 ) ,
δ cir ( β ) = I I = ( P 2 cos 2 β 2 P 3 cos β + P 1 ) ( P 2 cos 2 β + 2 P 3 cos β + P 1 ) .
p C ( β ) = 1 4 ( P 2 cos 2 β 2 P 3 cos β + P 1 ) ,
p C ( β ) = 1 4 ( P 2 cos 2 β + 2 P 3 cos β + P 1 ) .
( a a ) V = ( a a ) H = a L a L ,
( a C a C ) C 1 = ( a C a C ) C 2 = a C a C ,
( I 2 I 1 ) C 1 _ C = a C a C * M C * A 2 A 1 ,
( I 1 I 2 ) C 2 _ C = a C a C * 1 M C * A 1 A 2 ,
( I 2 I 1 ) C 1 _ L = a C + a C a C + a C * M C * A 2 A 1 ,
( I 2 I 1 ) C 2 _ L = a C + a C a C + a C * 1 M C * A 2 A 1 ,
( I 2 I 1 ) L V _ L = a L a L * M L * A 2 A 1 ,
( I 1 I 2 ) L H _ L = a L a L * 1 M L * A 1 A 2 .
M C 2 = ( I 2 I 1 ) C 1 _ L / ( I 2 I 1 ) C 2 _ L .
( A 1 A 2 ) 2 = M C 2 ( I 2 I 1 ) Cl _ C / ( I 1 I 2 ) C 2 _ C .
M L 2 = ( A 2 A 1 ) 2 ( I 2 I 1 ) LV _ L / ( I 1 I 2 ) LH _ L .
a C a C = ( 1 M C A 1 A 2 ) ( I 2 I 1 ) C 1 _ C = F C ( I 2 I 1 ) C 1 _ C ,
a L a L = ( 1 M L A 1 A 2 ) ( I 2 I 1 ) LV _ L = F L ( I 2 I 1 ) LV _ L .

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