Abstract

We have developed a theory for the computation of the polarization of infrared radiation in optically anisotropic media, with specific application to horizontally oriented ice crystals that frequently occur in cirrus clouds. Both emission and scattering contributions are accounted for in the basic formulation concerning the transfer of thermal radiation in anisotropic media. The symmetry relations of the phase matrix elements for horizontally oriented ice crystals, which are required in the infrared polarization formulations, are presented for the first time to our knowledge. Phase matrix elements for horizontally oriented hexagonal ice crystals are computed by a geometric ray-tracing technique. Radiance and linear-polarization patterns at a 10-μm wavelength that are emergent from cirrus clouds that contain plates and columns oriented in two-dimensional space are presented and discussed in physical terms. Downward polarization emergent from the cloud base is negative, while upward polarization emergent from the cloud top has a positive maximum value near the limb directions. These polarization configurations differ distinctly from the configurations of polarization emergent from ice clouds that contain randomly oriented ice crystals in three-dimensional space. Given these results, it appears feasible to infer the orientation characteristics of ice crystals in cirrus clouds with the use of infrared polarization measurements either above or below the cloud.

© 1993 Optical Society of America

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References

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  1. K. N. Liou, Y. Takano, S. C. Ou, A. J. Heymsfield, W. Kreiss, “Infrared transmission through cirrus clouds: a radiative model for target detection,” Appl. Opt. 29, 1886–1896 (1990).
    [CrossRef] [PubMed]
  2. Y. Takano, K. N. Liou, “Infrared polarization signature from cirrus clouds,” Appl. Opt. 31, 1916–1919 (1992).
    [CrossRef] [PubMed]
  3. L. Thomas, J. C. Cartwright, D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus 42B, 211–216 (1990).
  4. Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part I: single-scattering and optical properties of hexagonal ice crystals,”J. Atmos. Sci. 46, 3–19 (1989).
    [CrossRef]
  5. Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part II: theory and computation of multiple scattering in an anisotropic medium,”J. Atmos. Sci. 46, 20–36 (1989).
    [CrossRef]
  6. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980), Chap. 6, p. 222.
  7. S. Warren, “Optical constants of ice from the ultraviolet to the microwave,” Appl. Opt. 23, 1206–1225 (1984).
    [CrossRef] [PubMed]
  8. P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite-observed visible and infrared radiances. Part I: parameterization of radiance fields,” J. Atmos. Sci. (to be published).
  9. Y. Takano, K. Jayaweera, “Scattering phase matrix for hexagonal ice crystals computed from ray optics,” Appl. Opt. 24, 3254–3263 (1985).
    [CrossRef] [PubMed]
  10. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 5, p. 44.
  11. Y. Takano, S. Asano, “Fraunhofer diffraction by ice crystals suspended in the atmosphere,” J. Meteorol. Soc. Jpn. 61, 289–300 (1983).
  12. J. W. Hovenier, “Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles,”J. Atmos. Sci. 26, 488–499 (1969).
    [CrossRef]
  13. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Chap. 15, p. 505.
  14. Q. Cai, K. N. Liou, “Polarized light scattering by hexagonal ice crystals: theory,” Appl. Opt. 21, 3569–3580 (1982).
    [CrossRef] [PubMed]
  15. R. G. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, New York, 1980), Chap. 3, p. 92.
  16. D. K. Lynch, “Polarization models of halo phenomena. I. The parhelic circle,”J. Opt. Soc. Am. 69, 1100–1103 (1979).
    [CrossRef]
  17. A. B. Fraser, “What size of ice crystals causes the halos?”J. Opt. Soc. Am. 69, 1112–1118 (1979).
    [CrossRef]
  18. Y. Takano, K. N. Liou, “Halo phenomena modified by multiple scattering,” J. Opt. Soc. Am. A 7, 885–889 (1990).
    [CrossRef]
  19. 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]

1992 (2)

Y. Takano, K. N. Liou, “Infrared polarization signature from cirrus clouds,” Appl. Opt. 31, 1916–1919 (1992).
[CrossRef] [PubMed]

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]

1990 (3)

1989 (2)

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

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part II: theory and computation of multiple scattering in an anisotropic medium,”J. Atmos. Sci. 46, 20–36 (1989).
[CrossRef]

1985 (1)

1984 (1)

1983 (1)

Y. Takano, S. Asano, “Fraunhofer diffraction by ice crystals suspended in the atmosphere,” J. Meteorol. Soc. Jpn. 61, 289–300 (1983).

1982 (1)

1979 (2)

1969 (1)

J. W. Hovenier, “Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles,”J. Atmos. Sci. 26, 488–499 (1969).
[CrossRef]

Asano, S.

Y. Takano, S. Asano, “Fraunhofer diffraction by ice crystals suspended in the atmosphere,” J. Meteorol. Soc. Jpn. 61, 289–300 (1983).

Cai, Q.

Cartwright, J. C.

L. Thomas, J. C. Cartwright, D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus 42B, 211–216 (1990).

Fraser, A. B.

Greenler, R. G.

R. G. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, New York, 1980), Chap. 3, p. 92.

Heymsfield, A. J.

Hovenier, J. W.

J. W. Hovenier, “Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles,”J. Atmos. Sci. 26, 488–499 (1969).
[CrossRef]

Jayaweera, K.

Kreiss, W.

Liou, K. N.

Y. Takano, K. N. Liou, “Infrared polarization signature from cirrus clouds,” Appl. Opt. 31, 1916–1919 (1992).
[CrossRef] [PubMed]

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]

K. N. Liou, Y. Takano, S. C. Ou, A. J. Heymsfield, W. Kreiss, “Infrared transmission through cirrus clouds: a radiative model for target detection,” Appl. Opt. 29, 1886–1896 (1990).
[CrossRef] [PubMed]

Y. Takano, K. N. Liou, “Halo phenomena modified by multiple scattering,” J. Opt. Soc. Am. A 7, 885–889 (1990).
[CrossRef]

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

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part II: theory and computation of multiple scattering in an anisotropic medium,”J. Atmos. Sci. 46, 20–36 (1989).
[CrossRef]

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

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980), Chap. 6, p. 222.

P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite-observed visible and infrared radiances. Part I: parameterization of radiance fields,” J. Atmos. Sci. (to be published).

Lynch, D. K.

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]

P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite-observed visible and infrared radiances. Part I: parameterization of radiance fields,” J. Atmos. Sci. (to be published).

Ou, S. C.

Takano, Y.

Y. Takano, K. N. Liou, “Infrared polarization signature from cirrus clouds,” Appl. Opt. 31, 1916–1919 (1992).
[CrossRef] [PubMed]

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]

Y. Takano, K. N. Liou, “Halo phenomena modified by multiple scattering,” J. Opt. Soc. Am. A 7, 885–889 (1990).
[CrossRef]

K. N. Liou, Y. Takano, S. C. Ou, A. J. Heymsfield, W. Kreiss, “Infrared transmission through cirrus clouds: a radiative model for target detection,” Appl. Opt. 29, 1886–1896 (1990).
[CrossRef] [PubMed]

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part II: theory and computation of multiple scattering in an anisotropic medium,”J. Atmos. Sci. 46, 20–36 (1989).
[CrossRef]

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

Y. Takano, K. Jayaweera, “Scattering phase matrix for hexagonal ice crystals computed from ray optics,” Appl. Opt. 24, 3254–3263 (1985).
[CrossRef] [PubMed]

Y. Takano, S. Asano, “Fraunhofer diffraction by ice crystals suspended in the atmosphere,” J. Meteorol. Soc. Jpn. 61, 289–300 (1983).

P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite-observed visible and infrared radiances. Part I: parameterization of radiance fields,” J. Atmos. Sci. (to be published).

Thomas, L.

L. Thomas, J. C. Cartwright, D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus 42B, 211–216 (1990).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 5, p. 44.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Chap. 15, p. 505.

Wareing, D. P.

L. Thomas, J. C. Cartwright, D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus 42B, 211–216 (1990).

Warren, S.

Appl. Opt. (5)

J. Atmos. Sci. (4)

J. W. Hovenier, “Symmetry relationships for scattering of polarized light in a slab of randomly oriented particles,”J. Atmos. Sci. 26, 488–499 (1969).
[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]

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

Y. Takano, K. N. Liou, “Solar radiative transfer in cirrus clouds. Part II: theory and computation of multiple scattering in an anisotropic medium,”J. Atmos. Sci. 46, 20–36 (1989).
[CrossRef]

J. Meteorol. Soc. Jpn. (1)

Y. Takano, S. Asano, “Fraunhofer diffraction by ice crystals suspended in the atmosphere,” J. Meteorol. Soc. Jpn. 61, 289–300 (1983).

J. Opt. Soc. Am. (2)

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

Tellus (1)

L. Thomas, J. C. Cartwright, D. P. Wareing, “Lidar observations of the horizontal orientation of ice crystals in cirrus clouds,” Tellus 42B, 211–216 (1990).

Other (5)

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980), Chap. 6, p. 222.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 5, p. 44.

P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite-observed visible and infrared radiances. Part I: parameterization of radiance fields,” J. Atmos. Sci. (to be published).

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Chap. 15, p. 505.

R. G. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, New York, 1980), Chap. 3, p. 92.

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Figures (18)

Fig. 1
Fig. 1

Emission geometry for Parry columns.

Fig. 2
Fig. 2

Degree of linear polarization, −Qe, for thermal radiation emitted from (a) Parry columns and (b) 2D plates.

Fig. 3
Fig. 3

Geometric configuration of the adding method used to evaluate thermal emission. The dashed and the solid lines represent radiation emitted from layers a and b, respectively.

Fig. 4
Fig. 4

Geometry of a light ray in an ice crystal: α is the angle between the incident ray and the c axis defined in the diagram, β is the angle of rotation about the axis, I denotes the incident ray, L is the length of the ice crystal, and 2a is the width.

Fig. 5
Fig. 5

Geometry for Fraunhofer diffraction at an arbitrary point, P. The projections of the eight vertices of a hexagonal crystal on the plane normal to an oblique incident ray are denoted by Bi′(i = 1–8). Θ and Φ are the scattering and the azimuthal angles, respectively, of the diffracted light beam.

Fig. 6
Fig. 6

Scattering geometries for horizontally oriented (a) plate crystals and (b) columnar crystals. IO and SO denote the incident and the scattered directions, respectively. Other symbols are explained in the text.

Fig. 7
Fig. 7

Phase function, P11, for 2D plates with aspect ratio L/2a of 0.4 at λ = 0.55 μm (a) above the horizon and (b) below the horizon. The diffracted-light component is excluded so that geometrical-optics ray patterns can be clearly understood. The small circle in (a) indicates the incident solar direction whose zenith angle θ′ is 77°. The symbols ·, +, *, and ■ denote 0, 1, 2, and 3, respectively, in units of [log10P11], where [] denotes the integral part.

Fig. 8
Fig. 8

(a) Phase function P11 corresponding to Fig. 7 and (b) degree of linear polarization, −P12/P11, along the parhelic circle as a function of the azimuthal angle, ϕϕ′.

Fig. 9
Fig. 9

Same as Fig. 7, except for 2D plates with aspect ratio L/2a of 40/100 (μm/μm) at λ = 10 μm. The symbols ·, +, *, and ■ denote 0, 1, 2, and 3, respectively, in units of [log10P11] + 1, where [] denotes the integral part.

Fig. 10
Fig. 10

Same as Fig. 8, except for 2D plates with an aspect ratio L/2a of 40/100 (μm/μm) at λ = 10 μm.

Fig. 11
Fig. 11

Same as Fig. 7, except for Parry columns with an aspect ratio L/2a of 2.5 at the solar zenith angle of 73°. The symbols ·, +, *, and ■ denote 0, 1, 2, and 3, respectively, in units of [log10P11], where [] denotes the integral part.

Fig. 12
Fig. 12

Degree of linear polarization, −P12/P11, for Parry columns (a) above the horizon and (b) below the horizon. The digit 3, for example, denotes a certain value of polarization between +30% and +40%. Digits are underlined when the polarization is negative.

Fig. 13
Fig. 13

Same as Fig. 9, except for Parry columns with an aspect ratio L/2a of 120/60 (μm/μm) at λ = 10 μm. The symbols ·, +, *, and ■ denote 0, 1, 2, and 3, respectively, in units of [log10P11] + 1, where [] denotes the integral part.

Fig. 14
Fig. 14

Same as Fig. 12, except for Parry columns of L/2a = 120/60 (μm/μm) at λ = 10 μm.

Fig. 15
Fig. 15

Radiances as a function of zenith/nadir angles on the boundaries of cirrus clouds in midlatitude winter atmosphere (Tc = 230 K and Ts = 273 K). (a), (b) Cloud optical depth τc = 1; (c), (d) cloud depth τc = 4.

Fig. 16
Fig. 16

Same as Fig. 15, except for the tropical atmosphere (Tc = 200 K and Ts = 300 K).

Fig. 17
Fig. 17

Degree of linear polarization, −Q/I, corresponding to Fig. 15 as a function of zenith/nadir angles on the boundaries of cirrus clouds in the midlatitude winter atmosphere: (a), (b) cloud optical depth τc = 1; (c), (d) cloud depth τc = 4.

Fig. 18
Fig. 18

Same as Fig. 17, except for the tropical atmosphere.

Equations (48)

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d I ( z , μ , ϕ ) = - β e ( μ ) I ( z , μ , ϕ ) d z / μ + 1 4 π 0 2 π - 1 1 β s ( μ ) Z ( μ , μ , ϕ , ϕ ) I ( z , μ , ϕ ) d μ d ϕ + β a ( μ ) B λ ( T ) I e ,
d τ ˜ = - β ˜ e d z ,
μ d I ( τ ˜ , μ , ϕ ) d τ ˜ = k ( μ ) I ( τ ˜ , μ , ϕ ) - J ( τ ˜ , μ , ϕ ) ,
k ( μ ) = β e ( μ ) / β ˜ e .
J ( τ ˜ , μ , ϕ ) = 1 4 π 0 2 π - 1 1 ω ˜ * ( μ ) Z ( μ , μ , ϕ , ϕ ) I ( τ ˜ , μ , ϕ ) d μ d ϕ + k ( μ ) [ 1 - ω ˜ ( μ ) ] B λ ( T ) I e .
ω ˜ ( μ ) = β s ( μ ) / β e ( μ ) ,
ω ˜ * ( μ ) = β s ( μ ) / β ˜ e .
μ d I ( τ ˜ , μ ) d τ ˜ = k ( μ ) I ( τ ˜ , μ ) - J ( τ ˜ , μ ) .
J ( τ ˜ , μ ) = ½ - 1 1 ω ˜ * ( μ ) Z ^ ( μ , μ ) I ( τ ˜ , μ ) d μ + k ( μ ) [ 1 - ω ˜ ( μ ) ] B λ ( T ) [ 1 Q e ] .
L ^ ( δ ) = [ cos 2 δ sin 2 δ ½ sin 2 δ 0 sin 2 δ cos 2 δ - ½ sin 2 δ 0 - sin 2 δ sin 2 δ cos 2 δ 0 0 0 0 1 ] .
I l = cos 2 δ [ 1 - r l ( σ ) 2 ] + sin 2 δ [ 1 - r r ( σ ) 2 ] , I r = sin 2 δ [ 1 - r l ( σ ) 2 ] + cos 2 δ [ 1 - r r ( σ ) 2 ] , U = sin 2 δ [ r l ( σ ) 2 - r r ( σ ) 2 ] , V = 0.
I ¯ l , r ( θ ) = 0 2 π i I l , r i cos σ i A i d ( ϕ - ϕ ) 0 2 π i cos σ i A i d ( ϕ - ϕ ) ,
J d = [ 1 + R a * R b + ( R a * R b ) 2 + ] J a - + [ 1 + R a * R b + ( R a * R b ) 2 + ] R a * J b + = ( 1 + S ) ( J a - + R a * J b + ) ,
J u = [ 1 + R b R a * + ( R b R a * ) 2 + ] J b + + [ 1 + R b R a * + ( R b R a * ) 2 + ] R b J a - = ( 1 + S * ) ( J b + + R b J a - ) ,
S = R a * R b ( 1 - R a * R b ) - 1 ,
S * = R b R a * ( 1 - R b R a * ) - 1 .
J a b + = J a + + T ˜ a * J u ,
J a b - = J b - + T ˜ b J d ,
T ˜ = T + exp ( - τ / μ ) .
J + ( μ ) = J - ( μ ) = k ( μ ) [ 1 - ω ˜ ( μ ) ] B λ ( T ) [ 1 Q e ] Δ τ ˜ / μ .
R ( μ , μ ) = ω ˜ * ( μ ) Δ τ ˜ 4 μ μ Z ^ ( μ , - μ ) ,
T ( μ , μ ) = ω ˜ * ( μ ) Δ τ ˜ 4 μ μ Z ^ ( μ , μ ) ,
[ E l E r ] = exp ( - i k R + i k z ) i k R [ A 2 A 3 A 4 A 1 ] [ E l 0 E r 0 ] ,
[ I Q U V ] = 1 k 2 R 2 [ P 11 P 12 P 13 P 14 P 21 P 22 P 23 P 24 P 31 P 32 P 33 P 34 P 41 P 42 P 43 P 44 ] [ I 0 Q 0 U 0 V 0 ] ,
A ( n ) = [ A 2 ( n ) A 3 ( n ) A 4 ( n ) A 1 ( n ) ] = [ c s s c ] .
A ( n ) = w n P s n T n P n [ k = n - 1 2 R k P k ] T 1 P 1 ,
A ( n ) = P t n A ( n ) P - e n ,
T n = [ t l n 0 0 t r n ] ,             R n = [ r l n 0 0 r r n ] ,
P n = [ cos ϕ n - sin ϕ n sin ϕ n cos ϕ n ] .
M k = A k A k * , S k l = S l k = 1 / 2 ( A l A k * + A k A l * ) , - D k l = D l k = i / 2 ( A l A k * - A k A l * ) ,             for k , l = 1 - 4.
P ^ k l = ( 1 - f D ) n G k l ( n ) + δ k l f D G D .
f D = 1 2 ω ˜ ( 1 - f δ ) ,
P k l ( θ , θ , ϕ - ϕ ) = 3 π 0 π / 3 P ^ k l ( θ , θ , ϕ - ϕ ; β ) d β ,             k , l = 1 - 4 ,
P k l ( θ , θ , ϕ - ϕ ) = 6 π 0 π / 6 P ^ k l ( θ , θ , ϕ - ϕ ; β ) d β ,
P k l ( θ , θ , ϕ - ϕ ) = 6 π 0 π / 6 sgn ( ϕ - ϕ ) P ^ k l ( θ , θ , ϕ - ϕ ; β ) d β
P k l ( θ , θ , ϕ - ϕ ) = 3 π 2 - π / 2 π / 2 d γ 0 π / 3 P ^ k l ( θ , θ , ϕ - ϕ ; γ , β ) × δ ( β - β * ) d β ,             k , l = 1 - 4 ,
P k l ( θ , θ , ϕ - ϕ ) = 6 π 2 0 π / 2 d γ 0 π / 3 P ^ k l ( θ , θ , ϕ - ϕ ; γ , β ) × δ ( β - β * ) d β ,
P k l ( θ , θ , ϕ - ϕ ) = 6 π 2 0 π / 2 d γ 0 π / 3 sgn ( ϕ - ϕ ) × P ^ k l ( θ , θ , ϕ - ϕ ; γ , β ) δ ( β - β * ) d β ,
Z ( θ , θ , ϕ - ϕ ) = L ( π - i 2 ) PL ( - i 1 ) ,
L ( π - α ) = L ( - α ) = [ 1 0 0 0 0 cos 2 α - sin 2 α 0 0 sin 2 α cos 2 α 0 0 0 0 1 ] .
cos i 1 = - μ + μ cos Θ ± ( 1 - cos 2 Θ ) 1 / 2 ( 1 - μ 2 ) 1 / 2 ,
cos i 2 = - μ + μ cos Θ ± ( 1 - cos 2 Θ ) 1 / 2 ( 1 - μ 2 ) 1 / 2 ,
cos Θ = μ μ + ( 1 - μ 2 ) 1 / 2 ( 1 - μ 2 ) 1 / 2 cos ( ϕ - ϕ ) .
S = sin ( 2 i 1 ) , C = cos ( 2 i 1 ) , S = sin ( 2 i 2 ) , C = cos ( 2 i 2 ) .
Z ^ = [ P 11 P 12 C + P 13 S P 21 C - P 31 S P 22 C C - P 32 S C + P 23 C S - P 33 S S ] .
Z ( - μ , - μ , ϕ - ϕ ) = Z ( μ , μ , ϕ - ϕ ) ,             ( C ) ,
Z ( μ , μ , ϕ - ϕ ) = MZ ( μ , μ , ϕ - ϕ ) M ,             ( D ) ,
M = [ 1 0 0 0 0 1 0 0 0 0 - 1 0 0 0 0 - 1 ] .

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