Abstract

A Mueller matrix is developed for a single-scattering process such that G(θ, ϕ) = a) p)u, where u is the incident irradiance Stokes vector transmitted through a linear polarizer at azimuthal angle ϕp, with transmission Mueller matrix p), and G(θ, ϕ) is the polarized irradiance Stokes vector measured by a detector with a field of view F, placed after an analyzer with transmission Mueller matrix a) at angle ϕa. The Mueller matrix is a function of the Mueller matrix (θ) of the scattering medium, the scattering angle (θ, ϕ), and the detector field of view F. The Mueller matrix is derived for backscattering and forward scattering, along with equations for the detector polarized irradiance measurements (e.g., cross polarization and copolarization) and the depolarization ratio. The information that can be derived from the Mueller matrix on the scattering Mueller matrix (θ) is limited because the detector integrates the cone of incoming radiance over a range of azimuths of 2π for forward scattering and backscattering. However, all nine Mueller matrix elements that affect linearly polarized radiation can be derived if a spatial filter in the form of a pie-slice slit is placed in the focal plane of the detector and azimuthally dependent polarized measurements and azimuthally integrated polarized measurements are combined.

© 1998 Optical Society of America

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References

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1995

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

1994

1991

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
[CrossRef]

1987

1985

1980

1970

Asano, S.

Bissonnette, L. R.

G. Roy, L. R. Bissonnette, “Non-simultaneous measurements of multiple-field-of-view lidar returns in clouds: time correlation length,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1997), pp. 99–102.
[CrossRef]

Bohren, C. F.

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

Carswell, A. I.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

Gagne, G.

Herb, P.

Holland, A. C.

Hovenier, J. W.

Hu, C.-R.

Huffman, D. R.

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

Kattawar, G. W.

Mishchenko, M. I.

Pal, S. R.

Parkin, M. E.

Roy, G.

G. Roy, L. R. Bissonnette, “Non-simultaneous measurements of multiple-field-of-view lidar returns in clouds: time correlation length,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1997), pp. 99–102.
[CrossRef]

Sassen, K.

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

K. Sassen, “Advances in polarization lidar for cloud remote sensing,” Proc. IEEE 82, 1907–1914 (1994).
[CrossRef]

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
[CrossRef]

Sato, M.

Travis, L. R.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Zhao, H.

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

Appl. Opt.

Bull. Am. Meteorol. Soc.

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
[CrossRef]

Opt. Rev.

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

Proc. IEEE

K. Sassen, “Advances in polarization lidar for cloud remote sensing,” Proc. IEEE 82, 1907–1914 (1994).
[CrossRef]

Other

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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

G. Roy, L. R. Bissonnette, “Non-simultaneous measurements of multiple-field-of-view lidar returns in clouds: time correlation length,” in Advances in Atmospheric Remote Sensing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, 1997), pp. 99–102.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Scattering geometry and a right-handed xyz coordinate system (x × y = z) for an incident Stokes vector u, transmitted through a linear polarizer placed in the xy plane with its maximum transmission axis at azimuthal angle ϕ p , propagating along the z axis. The parallel and perpendicular components of the incident Stokes vectors u and the rotational angle ψ for rotating the incident parallel and perpendicular components to the scattering plane are shown. A detector measuring the irradiance from scattering at angle (θ, ϕ) is located in the xy plane.

Fig. 2
Fig. 2

(a) Scattering geometry and a right-handed xyz coordinate system (x × y = z) for a scattered Stokes vector g traveling at an angle 0 ≤ θ < π/2 in the forward hemisphere. The polarizer and the analyzer are in the xy plane with their maximum transmission axes at angles ϕ p and ϕ a , respectively, measured clockwise from the y axis. The parallel and perpendicular components of the incident and scattered Stokes vectors, u and g, are shown. (b) Scattering geometry and a right-handed xyz coordinate system [x × (-y) = (-z)] for a scattered Stokes vector g traveling at an angle π/2 < θ ≤ π in the backward hemisphere. The polarizer and the analyzer are in the xy plane with their maximum transmission axes at angles ϕ p and ϕ a , respectively, where ϕ p is measured clockwise from the +y axis and ϕ a is measured counterclockwise from the -y axis. The parallel and perpendicular components of the incident and scattered Stokes vectors, u and g, are shown.

Fig. 3
Fig. 3

Scattering radiance in a concentric conical slice between the volumes of two concentric cones, one with half-cone angle θ - dθ and the other with half-cone angle θ + dθ.

Fig. 4
Fig. 4

Schematic representation of a spatial filter of angular half-width Δϕ placed in the focal plane at azimuthal angle ϕ in front of the detector in the xy plane such that only scattered radiance within ϕ ± Δϕ strikes the detector and scattered radiance at any other azimuthal angle is blocked by the spatial filter.

Equations (75)

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I Q U V = E E * + E E * = I + I E E * - E E * = I - I E E * + E E * i E E * - E E * ,
T ˜ ψ = T 11 T 12 T 13 0 T 12 T 22 T 23 0 T 13 T 23 T 33 0 0 0 0 T 44 ,
S ˜ θ = S 11 θ S 12 θ S 13 θ S 14 θ S 21 θ S 22 θ S 23 θ S 24 θ S 31 θ S 32 θ S 33 θ S 34 θ S 41 θ S 42 θ S 43 θ S 44 θ .
L ˜ 1 ψ = 1 0 0 0 0 cos 2 ψ sin 2 ψ 0 0 - sin 2 ψ cos 2 ψ 0 0 0 0 1 .
L ˜ 2 π / 2 < θ π , ϕ = 3 + cos 2 θ / 4 - sin 2 θ / 2 0 0 cos 2 ϕ sin 2 θ / 2 - cos 2 ϕ 3 + cos 2 θ / 4 - cos θ sin 2 ϕ 0 sin 2 ϕ sin 2 θ / 2 - sin 2 ϕ 3   +   cos 2 θ / 4 cos θ cos 2 ϕ 0 0 0 0 - cos θ .
L ˜ 2 0 θ < π / 2 , ϕ = 3 + cos 2 θ / 4 - sin 2 θ / 2 0 0 cos 2 ϕ sin 2 θ / 2 - cos 2 ϕ 3 + cos 2 θ / 4 - cos θ sin 2 ϕ 0 - sin 2 ϕ sin 2 θ / 2 sin 2 ϕ 3   +   cos 2 θ / 4 - cos θ cos 2 ϕ 0 0 0 0 cos θ .
g π / 2 < θ π , ϕ , ϕ p , ϕ a = I - y + I x I - y - I x E - y E x * + E x E - y * i E - y E x * - E x E - y * = T ˜ ϕ a L ˜ 2 θ , ϕ S ˜ θ L ˜ 1 π / 2 - ϕ T ˜ ϕ p u ,
g 0 θ < π / 2 , ϕ , ϕ p , ϕ a = I y + I x I y - I x E y E x * + E x E y * i E y E x * - E x E y * = T ˜ ϕ a L ˜ 2 θ , ϕ S ˜ θ L ˜ 1 π / 2 - ϕ T ˜ ϕ p u ,
g θ , ϕ , ϕ p , ϕ a = T ˜ ϕ a M ˜ θ , ϕ T ˜ ϕ p u ,
M ˜ θ , ϕ = L ˜ 2 θ , ϕ S ˜ θ L ˜ 1 π / 2 - ϕ
M ˜ π = ϕ = 0 2 π   M ˜ θ = π , ϕ d ϕ = 2 π S 11 0 0 2 π S 14 0 π S 22 - S 33 π S 23 + S 32 0 0 π S 23 + S 32 - π S 22 - S 33 0 2 π S 41 0 0 2 π S 44 ,
M ˜ π = ϕ = 0 2 π   M ˜ θ = π , ϕ d ϕ = 2 π S 11 0 0 2 π S 14 0 π S 22 - S 33 0 0 0 0 - π S 22 - S 33 0 2 π S 14 0 0 2 π S 44 .
G θ = π , ϕ p , ϕ a = ϕ = 0 2 π θ = θ - Δ θ θ g θ , ϕ , ϕ p , ϕ a × cos π - θ sin θ d θ d ϕ ,
S ˜ θ = S ˜ θ - Δ ϕ θ θ + Δ ϕ = ϕ = 0 2 π θ = θ - Δ θ θ   S ˜ θ sin θ d θ d ϕ ϕ = 0 2 π θ = θ - Δ θ θ sin θ d θ d ϕ .
F = ϕ = 0 2 π θ = θ - Δ θ θ sin θ d θ d ϕ | θ = π = 4 π   sin 2 Δ θ / 2 .
M ˜ θ = π , F = ϕ = 0 2 π θ = θ - Δ θ θ   L ˜ 2 θ , ϕ S ˜ θ L ˜ 1 × π / 2 - ϕ cos π - θ sin θ d θ d ϕ = m 11 0 0 m 14 0 m 22 m 23 0 0 m 23 - m 22 0 m 41 0 0 m 44 ,
m 11 π , F = FS 11 - F 2 2 S 11 + S 21 4 π + F 3 S 11 + S 21 8 π 2 - F 4 S 11 + S 21 64 π 3 , m 14 π , F = FS 14 - F 2 2 S 14 + S 24 4 π + F 3 S 14 + S 24 8 π 2 - F 4 S 14 + S 24 64 π 3 , m 22 π , F = F   S 22 - S 33 2 - F 2 2 S 22 - S 33 + S 12 8 π + F 3 3 S 12 + S 22 / 2 - S 33 24 π 2 - F 4 S 12 + S 22 128 π 3 , m 23 π , F = F   S 23 + S 32 2 - F 2 2 S 23 + S 32 + S 13 8 π + F 3 3 S 13 + S 23 / 2 + S 32 24 π 2 - F 4 S 13 + S 23 128 π 3 , m 41 π , F = FS 41 - F 2 S 41 2 π + F 3 S 41 12 π 2 , m 44 π , F = FS 44 - F 2 S 44 2 π + F 3 S 44 12 π 2 .
I θ = π , F , ϕ p , ϕ a = t p + t p t a + t a 4   m 11 + t p - t p t a - t a 4 m 22 cos 2 ϕ a + ϕ p + m 23 sin 2 ϕ a + ϕ p ,
m 11 π , F = FS 11 π , m 22 π , F = F   S 22 π - S 33 π 2 , m 23 π , F = F   S 23 π + S 32 π 2
M ˜ 0 = ϕ = 0 2 π   M ˜ θ = 0 , ϕ d ϕ = 2 π S 11 0 0 2 π S 14 0 π S 22 + S 33 π S 23 - S 32 0 0 - π S 23 - S 32 π S 22 + S 33 0 2 π S 41 0 0 2 π S 44 ,
G θ = 0 , ϕ p , ϕ a = ϕ = 0 2 π θ = θ θ + Δ θ g θ , ϕ , ϕ p , ϕ a × cos θ sin θ d θ d ϕ ,
S ˜ θ = S ˜ θ - Δ θ θ θ + Δ θ = ϕ = 0 2 π θ = θ θ + Δ θ   S ˜ θ sin θ d θ d ϕ ϕ = 0 2 π θ = θ θ + Δ θ sin θ d θ d ϕ .
F = ϕ = 0 2 π θ = θ θ + Δ θ sin θ d θ d ϕ | θ = 0 = 4 π   sin 2 Δ θ / 2 ,
M ˜ θ = 0 , F = ϕ = 0 2 π θ = θ θ + Δ θ   L ˜ 2 θ , ϕ S ˜ θ × L ˜ 1 π / 2 - ϕ cos θ sin θ d θ d ϕ = m 11 0 0 m 14 0 m 22 m 23 0 0 - m 23 m 22 0 m 41 0 0 m 44 ,
m 11 0 , F = FS 11 - F 2 2 S 11 + S 21 4 π + F 3 S 11 + S 21 8 π 2 - F 4 S 11 + S 21 64 π 3 , m 14 0 , F = FS 14 - F 2 2 S 14 + S 24 4 π + F 3 S 14 + S 24 8 π 2 - F 4 S 14 + S 24 64 π 3 , m 22 0 , F = F   S 22 + S 33 2 - F 2 2 S 22 + S 33 + S 12 8 π + F 3 3 S 12 + S 22 / 2 + S 33 24 π 2 - F 4 S 12 + S 22 128 π 3 , m 23 0 , F = F   S 23 - S 32 2 - F 2 2 S 23 - S 32 + S 13 8 π + F 3 3 S 13 + S 23 / 2 - S 32 24 π 2 - F 4 S 13 + S 23 128 π 3 , m 41 0 , F = FS 41 - F 2 S 41 2 π + F 3 S 41 12 π 2 , m 44 0 , F = FS 44 - F 2 S 44 2 π + F 3 S 44 12 π 2 .
I θ = 0 , F , ϕ p , ϕ a = t p + t p t a + t a 4   m 11 + t p - t p t a - t a 4 m 22 cos 2 ϕ a - ϕ p - m 23 sin 2 ϕ a - ϕ p ,
m 11 0 , F = FS 11 0 , m 22 0 , F = F   S 22 0 + S 33 0 2 , m 23 0 , F = F   S 23 0 - S 32 0 2 ,
G θ , ϕ p , ϕ a = Ω   T ˜ ϕ a L ˜ 2 θ , ϕ S ˜ θ L ˜ 1 π / 2 - ϕ × cos ξ T ˜ ϕ p u   d Ω = Δ Ω θ 2 π ϕ = 0 2 π   T ˜ ϕ a L ˜ 2 θ , ϕ S ˜ θ × L ˜ 1 π / 2 - ϕ cos ξ T ˜ ϕ p u   d ϕ ,
S ˜ θ = S 11 θ S 12 θ 0 0 S 12 θ S 22 θ 0 0 0 0 S 33 θ S 34 θ 0 0 - S 34 θ S 44 θ .
G θ , ϕ p , ϕ a = T ˜ ϕ a M ˜ θ T ˜ ϕ p u
M ˜ θ = Δ Ω θ 2 π ϕ = 0 2 π   L ˜ 2 θ , ϕ S ˜ θ L ˜ 1 π / 2 - ϕ cos ξ d ϕ .
M ˜ π / 2 < θ π = m 11 θ 0 0 0 0 m 22 θ 0 0 0 0 - m 22 θ 0 0 0 0 - m 44 θ ,
M ˜ 0 θ < π / 2 = - m 11 θ 0 0 0 0 - m 22 θ 0 0 0 0 - m 22 θ 0 0 0 0 - m 44 θ ,
m 11 θ = Δ Ω θ A θ S 11 θ + B θ S 12 θ , m 22 θ = Δ Ω θ A θ S 22 θ 2 - C θ S 33 θ + B θ S 12 θ 2 , m 44 θ = - 2 Δ Ω θ C θ S 44 θ ,
A θ = - 7   cos θ + cos 3 θ 8 , B θ = cos θ sin 2 θ 2 , C θ = cos 2 θ 2 .
I π / 2 < θ π , ϕ p , ϕ a = t p + t p t a + t a 4   m 11 θ + t p - t p t a - t a 4 × m 22 θ cos 2 ϕ a + ϕ p ,
I 0 θ < π / 2 , ϕ p , ϕ a = - t p + t p t a + t a 4   m 11 θ - t p - t p t a - t a 4 × m 22 θ cos 2 ϕ a - ϕ p ,
Δ π / 2 < θ π = I θ , ϕ p , ϕ a = π / 2 - ϕ p I θ , ϕ p , ϕ a = - ϕ p ,
I θ = π , F , ϕ p , ϕ a = π / 2 - ϕ p = t p + t p t a + t a 4 × m 11 π , F - t p - t p t a - t a 4 m 22 π , F ,
I θ = π , F , ϕ p , ϕ a = - ϕ p = t a + t a t p + t p 4   m 11 π , F + t a - t a t p - t p 4   m 22 π , F ,
I π / 2 < θ π , ϕ p , ϕ a = π / 2 - ϕ p = t p 4 m 11 θ - m 22 θ t a + m 11 θ + m 22 θ t a × I 0 + Q 0 cos 2 ϕ p + U 0 sin 2 ϕ p ,
I π / 2 < θ π , ϕ p , ϕ a = - ϕ p = t p 4 m 11 θ + m 22 θ t a + m 11 θ - m 22 θ t a × I 0 + Q 0 cos 2 ϕ p + U 0 sin 2 ϕ p ,
Δ θ = m 11 θ - m 22 θ t a + m 11 θ + m 22 θ t a m 11 θ + m 22 θ t a + m 11 θ - m 22 θ t a ,
Δ 0 θ < π / 2 = I θ , ϕ p , ϕ a = ϕ p + π / 2 I θ , ϕ p , ϕ a = ϕ p ,
F Δ ϕ = F Δ ϕ / π ,
M ˜ θ = π , ϕ , F Δ ϕ , Δ ϕ = ϕ = ϕ - Δ ϕ ϕ + Δ ϕ θ = θ - Δ θ θ   L ˜ 2 θ , ϕ S ˜ θ × L ˜ 1 π / 2 - ϕ cos π - θ sin θ d θ d ϕ = M ˜ 0 θ , F Δ ϕ 2 Δ ϕ + M ˜ 1 θ , F Δ ϕ sin 2 Δ ϕ sin 2 ϕ + M ˜ 2 θ , F Δ ϕ sin 2 Δ ϕ cos 2 ϕ + M ˜ 3 θ , F Δ ϕ × sin 4 Δ ϕ sin 4 ϕ 2 + M ˜ 4 θ , F Δ ϕ × sin 4 Δ ϕ cos 4 ϕ 2 ,
M ˜ 0 θ = π , F Δ ϕ = 1 2 π m 11 0 0 m 14 0 m 22 m 23 0 0 m 23 - m 22 0 m 41 0 0 m 44
M ˜ 1 θ = π , F Δ ϕ = 0 m 12 m 13 0 m 21 0 0 m 24 m 31 0 0 m 34 0 m 42 m 43 0 ,
M ˜ 2 θ = π , F Δ ϕ = 0 - m 13 m 12 0 m 31 0 0 m 34 - m 21 0 0 - m 24 0 - m 43 m 42 0 ,
m 12 π , F Δ ϕ = - F Δ ϕ S 13 2 π + F Δ ϕ 2 2 S 13 + S 23 8 π 2 - F Δ ϕ 3 S 13 + S 23 16 π 3 + F Δ ϕ 4 S 13 + S 23 128 π 4 , m 13 π , F Δ ϕ = F Δ ϕ S 12 2 π - F Δ ϕ 2 2 S 12 + S 22 8 π 2 + F Δ ϕ 3 S 12 + S 22 16 π 3 - F Δ ϕ 4 S 12 + S 22 128 π 4 , m 21 π , F Δ ϕ = F Δ ϕ S 31 2 π - F Δ ϕ 2 S 31 4 π 2 + F Δ ϕ 3 S 31 24 π 3 , m 24 π , F Δ ϕ = F Δ ϕ S 34 2 π - F Δ ϕ 2 S 34 4 π 2 + F Δ ϕ 3 S 34 24 π 3 , m 31 π , F Δ ϕ = - F Δ ϕ S 21 2 π + F Δ ϕ 2 2 S 21 + S 11 8 π 2 - F Δ ϕ 3 S 11 + S 21 16 π 3 + F Δ ϕ 4 S 11 + S 21 128 π 4 , m 34 π , F Δ ϕ = - F Δ ϕ S 24 2 π + F Δ ϕ 2 2 S 24 + S 14 8 π 2 - F Δ ϕ 3 S 14 + S 24 16 π 3 + F Δ ϕ 4 S 14 + S 24 128 π 4 , m 42 π , F Δ ϕ = - F Δ ϕ S 43 2 π + F Δ ϕ 2 S 43 4 π 2 - F Δ ϕ 3 S 43 24 π 3 , m 43 π , F Δ ϕ = F Δ ϕ S 42 2 π - F Δ ϕ 2 S 42 4 π 2 + F Δ ϕ 3 S 42 24 π 3 .
M ˜ 3 θ = π , F Δ ϕ = 0 0 0 0 0 m 22 m 23 0 0 - m 23 m 22 0 0 0 0 0 ,
M ˜ 4 θ = π , F Δ ϕ = 0 0 0 0 0 - m 23 m 22 0 0 - m 22 - m 23 0 0 0 0 0 ,
m 22 π , F Δ ϕ = F Δ ϕ S 23 - S 32 4 π - F Δ ϕ 2 2 S 23 - S 32 + S 13 16 π 2 + F Δ ϕ 3 3 S 13 + S 23 / 2 - S 32 48 π 3 - F Δ ϕ 4 S 13 + S 23 256 π 4 , m 23 π , F Δ ϕ = - F Δ ϕ S 22 + S 33 4 π + F Δ ϕ 2 2 S 22 + S 33 + S 12 16 π 2 - F Δ ϕ 3 3 S 12 + S 22 / 2 + S 33 48 π 3 + F Δ ϕ 4 S 12 + S 22 256 π 4 .
M ˜ θ = 0 , ϕ , F Δ ϕ , Δ ϕ = ϕ = ϕ - Δ ϕ ϕ + Δ ϕ θ = θ θ + Δ θ   L ˜ 2 θ , ϕ S ˜ θ × L ˜ 1 π / 2 - ϕ cos θ sin θ d θ d ϕ = M ˜ 0 θ , F Δ ϕ 2 Δ ϕ + M ˜ 1 θ , F Δ ϕ sin 2 Δ ϕ sin 2 ϕ + M ˜ 2 θ , F Δ ϕ sin 2 Δ ϕ cos 2 ϕ + M ˜ 3 θ , F Δ ϕ × sin 4 Δ ϕ sin 4 ϕ 2 + M ˜ 4 θ , F Δ ϕ × sin 4 Δ ϕ cos 4 ϕ 2 ,
M ˜ 0 θ = 0 , F Δ ϕ = 1 2 π m 11 0 0 m 14 0 m 22 m 23 0 0 - m 23 m 22 0 m 41 0 0 m 44 ,
M ˜ 1 θ = 0 , F Δ ϕ = 0 m 12 m 13 0 m 21 0 0 m 24 m 31 0 0 m 34 0 m 42 m 43 0 ,
M ˜ 2 θ = 0 , F Δ ϕ = 0 - m 13 m 12 0 - m 31 0 0 - m 34 m 21 0 0 m 24 0 - m 43 m 42 0 ,
m 12 0 , F Δ ϕ = - F Δ ϕ S 13 2 π + F Δ ϕ 2 2 S 13 + S 23 8 π 2 - F Δ ϕ 3 S 13 + S 23 16 π 3 + F Δ ϕ 4 S 13 + S 23 128 π 4 , m 13 0 , F Δ ϕ = F Δ ϕ S 12 2 π - F Δ ϕ 2 2 S 12 + S 22 8 π 2 + F Δ ϕ 3 S 12 + S 22 16 π 3 - F Δ ϕ 4 S 12 + S 22 128 π 4 , m 21 0 , F Δ ϕ = - F Δ ϕ S 31 2 π + F Δ ϕ 2 S 31 4 π 2 - F Δ ϕ 3 S 31 24 π 3 , m 24 0 , F Δ ϕ = - F Δ ϕ S 34 2 π + F Δ ϕ 2 S 34 4 π 2 - F Δ ϕ 3 S 34 24 π 3 , m 31 0 , F Δ ϕ = F Δ ϕ S 21 2 π - F Δ ϕ 2 2 S 21 + S 11 8 π 2 + F Δ ϕ 3 S 11 + S 21 16 π 3 - F Δ ϕ 4 S 11 + S 21 128 π 4 , m 34 0 , F Δ ϕ = F Δ ϕ S 24 2 π - F Δ ϕ 2 2 S 24 + S 14 8 π 2 + F Δ ϕ 3 S 14 + S 24 16 π 3 - F Δ ϕ 4 S 14 + S 24 128 π 4 , m 42 0 , F Δ ϕ = - F Δ ϕ S 43 2 π + F Δ ϕ 2 S 43 4 π 2 - F Δ ϕ 3 S 43 24 π 3 , m 43 0 , F Δ ϕ = F Δ ϕ S 42 2 π - F Δ ϕ 2 S 42 4 π 2 + F Δ ϕ 3 S 42 24 π 3 .
M ˜ 3 θ = 0 , F Δ ϕ = 0 0 0 0 0 m 22 m 23 0 0 m 23 - m 22 0 0 0 0 0 ,
M ˜ 4 θ = 0 , F Δ ϕ = 0 0 0 0 0 - m 23 m 22 0 0 m 22 m 23 0 0 0 0 0 ,
m 22 0 , F Δ ϕ = F Δ ϕ S 23 + S 32 4 π - F Δ ϕ 2 2 S 23 + S 32 + S 13 16 π 2 + F Δ ϕ 3 3 S 13 + S 23 / 2 + S 32 48 π 3 - F Δ ϕ 4 S 13 + S 23 256 π 4 , m 23 0 , F Δ ϕ = - F Δ ϕ S 22 - S 33 4 π + F Δ ϕ 2 2 S 22 - S 33 + S 12 16 π 2 - F Δ ϕ 3 3 S 12 + S 22 / 2 - S 33 48 π 3 + F Δ ϕ 4 S 12 + S 22 256 π 4 .
y 1 = I ϕ p = 0 , ϕ a = π / 2 + I ϕ p = 0 , ϕ a = 0 / F = S 11 t p t a + t a / 2 ,
y 2 = I ϕ p = 0 , ϕ a = π / 2 - I ϕ p = 0 , ϕ a = 0 / F = - S 22 - S 33 t p t a - t a / 4 ,
y 3 = I ϕ p = π / 4 , ϕ a = 0 / F = S 23 + S 32 t p - t p t a - t a / 16 + t p + t p y 1 / 4 t p - y 2 t p / 4 t p 1 / 2 ,
I ϕ = 0 , ϕ p = 0 , ϕ a = π / 4 , Δ ϕ , I ϕ = 0 , ϕ p = 0 , ϕ a = - π / 4 , Δ ϕ , I ϕ = 0 , ϕ p = π / 4 , ϕ a = π / 4 , Δ ϕ I ϕ = 0 , ϕ p = - π / 4 , ϕ a = - π / 4 , Δ ϕ .
I ϕ = π / 8 , ϕ p = 0 , ϕ a = π / 4 , Δ ϕ I ϕ = π / 8 , ϕ p = 0 , ϕ a = - π / 4 , Δ ϕ , I ϕ = π / 8 , ϕ p = π / 4 , ϕ a = π / 4 , Δ ϕ , I ϕ = π / 8 , ϕ p = - π / 4 , ϕ a = - π / 4 , Δ ϕ .
y 4 = I 0 , 0 , π / 4 , Δ ϕ + I 0 , 0 , - π / 4 , Δ ϕ / F Δ ϕ = t p t a + t a 2 Δ ϕ S 11 - S 12 sin 2 Δ ϕ / 2 π
y 5 = I π / 8 , 0 , π / 4 , Δ ϕ + I π / 8 , 0 , - π / 4 , Δ ϕ / F Δ ϕ = t p t a + t a 4 Δ ϕ S 11 - 2 S 12 + S 13 sin 2 Δ ϕ / 4 π
y 6 = I 0 , - π / 4 , - π / 4 , Δ ϕ + I 0 , π / 4 , π / 4 , Δ ϕ / F Δ ϕ = t a + t a Δ ϕ S 11 t p + t p - S 12 t p t p 1 / 2 sin 2 Δ ϕ / 2 π + t a - t a × t p - t p S 22 sin 4 Δ ϕ - 4 Δ ϕ + S 33 sin 4 Δ ϕ + 4 Δ ϕ / 16 π ,
y 7 = I π / 8 , - π / 4 , - π / 4 , Δ ϕ + I π / 8 , π / 4 , π / 4 , Δ ϕ / F Δ ϕ = t a + t a Δ ϕ S 11 t p + t p - S 12 + S 13 2 - 1 / 2 t p t p 1 / 2 sin 2 Δ ϕ / 2 π + t a - t a t p - t p sin 4 Δ ϕ S 23 - S 32 - 4 Δ ϕ S 22 - S 33 / 16 π ,
y 8 = I 0 , 0 , π / 4 , Δ ϕ - I 0 , 0 , - π / 4 , Δ ϕ / F Δ ϕ = t p t a - t a 4 Δ ϕ S 23 + S 32 - 4 S 31 sin 2 Δ ϕ - S 23 - S 32 sin 4 Δ ϕ / 8 π ,
y 9 = I π / 8 , 0 , π / 4 , Δ ϕ - I π / 8 , 0 , - π / 4 , Δ ϕ / F Δ ϕ = t p t a - t a [ 4 Δ ϕ S 23 + S 32 - 2 3 / 2 S 21 + S 31 sin 2 Δ ϕ + S 22 + S 33 sin 4 Δ ϕ / 8 π ,
I ϕ = 0 , ϕ p = 0 , ϕ a = π / 4 , Δ ϕ , I ϕ = 0 , ϕ p = 0 , ϕ a = - π / 4 , Δ ϕ , I ϕ = 0 , ϕ p = π / 4 , ϕ a = π / 4 , Δ ϕ , I ϕ = 0 , ϕ p = - π / 4 , ϕ a = - π / 4 , Δ ϕ
y 1 = S 11 t p t a + t a / 2 , y 2 = - S 22 + S 33 t p t a - t a / 4 , y 3 = S 23 - S 32 t p - t p t a - t a / 16 + t p + t p y 1 / 4 t p - y 2 t p / 4 t p 1 / 2 , y 4 = t p t a + t a 2 Δ ϕ S 11 - S 12 sin 2 Δ ϕ / 2 π , y 5 = t p t a + t a 4 Δ ϕ S 11 - 2 S 12 + S 13 sin 2 Δ ϕ / 4 π , y 6 = t a + t a Δ ϕ S 11 t p + t p - S 12 t p t p 1 / 2 sin 2 Δ ϕ / 2 π - t a - t a t p - t p S 22 sin 4 Δ ϕ - 4 Δ ϕ - S 33 sin 4 Δ ϕ + 4 Δ ϕ / 16 π , y 7 = t a + t a Δ ϕ S 11 t p + t p - S 12 + S 13 2 - 1 / 2 t p t p 1 / 2 sin 2 Δ ϕ / 2 π - t a - t a t p - t p sin 4 Δ ϕ S 23 + S 32 - 4 Δ ϕ S 22 + S 33 / 16 π , y 8 = - t p t a - t a 4 Δ ϕ S 23 - S 32 + 4 S 31 sin 2 Δ ϕ - S 23 + S 32 sin 4 Δ ϕ / 8 π , y 9 = - t p t a - t a 4 Δ ϕ S 23 - S 32 - 2 3 / 2 S 21 - S 31 sin 2 Δ ϕ + S 22 - S 33 sin 4 Δ ϕ / 8 π .
I ϕ = 0 , ϕ p = 0 , ϕ a = π / 4 , Δ ϕ , I ϕ = 0 , ϕ p = 0 , ϕ a = - π / 4 , Δ ϕ , I ϕ = 0 , ϕ p = π / 4 , ϕ a = π / 4 , Δ ϕ , I ϕ = 0 , ϕ p = - π / 4 , ϕ a = - π / 4 , Δ ϕ

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