Abstract

The polarization properties of far-field thermal radiation emitted by hot bodies at uniform temperature is explored. Some novel results are obtained for the Stokes parameters for emission from large, smooth, opaque bodies of convex shape and also for small particles in the Rayleigh region. For the case of large, smooth, opaque bodies, the results may be readily extended to include nonuniform temperature distributions that vary slowly on the scale of a wavelength. The formulas obtained are particularly simple and straightforward to apply, and examples of their application to emission from a large smooth cylinder and from a small ellipsoidal particle are described.

© 1994 Optical Society of America

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References

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  1. O. Sandus, “A review of emission polarization,” Appl. Opt. 4, 1634–1642 (1965).
    [CrossRef]
  2. W. G. Egan, Photometry and Polarization in Remote Sensing (Elsevier, New York, 1985).
  3. P. Ferrazoli, L. Guerriero, S. Paloscia, P. Pampaloni, D. Solimini, “Modeling polarization properties of emission from soil covered with vegetation,” IEEE Trans. Geosci. Remote Sens. 30, 157–165 (1992).
    [CrossRef]
  4. L. Tsang, “Thermal emission of nonspherical particles,” Radio Sci. 19, 966–974 (1984).
    [CrossRef]
  5. S. H. Yueh, R. Kwok, “Electromagnetic fluctuations for anisotropic media and the generalized Kirchhoff’ s law,” Radio Sci. 28, 471–480 (1993).
    [CrossRef]
  6. A. Stogryn, “The apparent temperature of the sea at microwave frequencies,” IEEE Trans. Antennas Propag. AP-15, 278–286 (1967).
    [CrossRef]
  7. S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989), Vol. 3.
  8. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989).
  9. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).
  10. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  11. P. Beckman, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968).
  12. CRC Handbook of Chemistry and Physics, 70th ed., R. C. Weast, D. R. Lide, M. J. Astle, W. H. Beyer, eds. (CRC Press, Boca Raton, Fla., 1989), p. E-390.
  13. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  14. R. E. Kleinman, T. B. A. Senior, “Rayleigh scattering,” in Low and High Frequency Asymptotics, V. K. Varadan, V. V. Varadan, eds. (North-Holland, Amsterdam, 1986), Vol. 2.
  15. D. R. Wiesnet, M. Matson, “Remote sensing of weather and climate,” in Manual of Remote Sensing, 2nd ed., R. N. Colwell, J. E. Estes, G. A. Thorley, eds., American Society of Photogrammetry (Sheridan, Falls Church, Va., 1983), Vol. 2, p. 1335.

1993 (1)

S. H. Yueh, R. Kwok, “Electromagnetic fluctuations for anisotropic media and the generalized Kirchhoff’ s law,” Radio Sci. 28, 471–480 (1993).
[CrossRef]

1992 (1)

P. Ferrazoli, L. Guerriero, S. Paloscia, P. Pampaloni, D. Solimini, “Modeling polarization properties of emission from soil covered with vegetation,” IEEE Trans. Geosci. Remote Sens. 30, 157–165 (1992).
[CrossRef]

1984 (1)

L. Tsang, “Thermal emission of nonspherical particles,” Radio Sci. 19, 966–974 (1984).
[CrossRef]

1967 (1)

A. Stogryn, “The apparent temperature of the sea at microwave frequencies,” IEEE Trans. Antennas Propag. AP-15, 278–286 (1967).
[CrossRef]

1965 (1)

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Beckman, P.

P. Beckman, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989).

Egan, W. G.

W. G. Egan, Photometry and Polarization in Remote Sensing (Elsevier, New York, 1985).

Ferrazoli, P.

P. Ferrazoli, L. Guerriero, S. Paloscia, P. Pampaloni, D. Solimini, “Modeling polarization properties of emission from soil covered with vegetation,” IEEE Trans. Geosci. Remote Sens. 30, 157–165 (1992).
[CrossRef]

Guerriero, L.

P. Ferrazoli, L. Guerriero, S. Paloscia, P. Pampaloni, D. Solimini, “Modeling polarization properties of emission from soil covered with vegetation,” IEEE Trans. Geosci. Remote Sens. 30, 157–165 (1992).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Kleinman, R. E.

R. E. Kleinman, T. B. A. Senior, “Rayleigh scattering,” in Low and High Frequency Asymptotics, V. K. Varadan, V. V. Varadan, eds. (North-Holland, Amsterdam, 1986), Vol. 2.

Kravtsov, Yu. A.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989), Vol. 3.

Kwok, R.

S. H. Yueh, R. Kwok, “Electromagnetic fluctuations for anisotropic media and the generalized Kirchhoff’ s law,” Radio Sci. 28, 471–480 (1993).
[CrossRef]

Matson, M.

D. R. Wiesnet, M. Matson, “Remote sensing of weather and climate,” in Manual of Remote Sensing, 2nd ed., R. N. Colwell, J. E. Estes, G. A. Thorley, eds., American Society of Photogrammetry (Sheridan, Falls Church, Va., 1983), Vol. 2, p. 1335.

Paloscia, S.

P. Ferrazoli, L. Guerriero, S. Paloscia, P. Pampaloni, D. Solimini, “Modeling polarization properties of emission from soil covered with vegetation,” IEEE Trans. Geosci. Remote Sens. 30, 157–165 (1992).
[CrossRef]

Pampaloni, P.

P. Ferrazoli, L. Guerriero, S. Paloscia, P. Pampaloni, D. Solimini, “Modeling polarization properties of emission from soil covered with vegetation,” IEEE Trans. Geosci. Remote Sens. 30, 157–165 (1992).
[CrossRef]

Rytov, S. M.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989), Vol. 3.

Sandus, O.

Senior, T. B. A.

R. E. Kleinman, T. B. A. Senior, “Rayleigh scattering,” in Low and High Frequency Asymptotics, V. K. Varadan, V. V. Varadan, eds. (North-Holland, Amsterdam, 1986), Vol. 2.

Solimini, D.

P. Ferrazoli, L. Guerriero, S. Paloscia, P. Pampaloni, D. Solimini, “Modeling polarization properties of emission from soil covered with vegetation,” IEEE Trans. Geosci. Remote Sens. 30, 157–165 (1992).
[CrossRef]

Stogryn, A.

A. Stogryn, “The apparent temperature of the sea at microwave frequencies,” IEEE Trans. Antennas Propag. AP-15, 278–286 (1967).
[CrossRef]

Tatarskii, V. I.

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989), Vol. 3.

Tsang, L.

L. Tsang, “Thermal emission of nonspherical particles,” Radio Sci. 19, 966–974 (1984).
[CrossRef]

Wiesnet, D. R.

D. R. Wiesnet, M. Matson, “Remote sensing of weather and climate,” in Manual of Remote Sensing, 2nd ed., R. N. Colwell, J. E. Estes, G. A. Thorley, eds., American Society of Photogrammetry (Sheridan, Falls Church, Va., 1983), Vol. 2, p. 1335.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989).

Yueh, S. H.

S. H. Yueh, R. Kwok, “Electromagnetic fluctuations for anisotropic media and the generalized Kirchhoff’ s law,” Radio Sci. 28, 471–480 (1993).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

A. Stogryn, “The apparent temperature of the sea at microwave frequencies,” IEEE Trans. Antennas Propag. AP-15, 278–286 (1967).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

P. Ferrazoli, L. Guerriero, S. Paloscia, P. Pampaloni, D. Solimini, “Modeling polarization properties of emission from soil covered with vegetation,” IEEE Trans. Geosci. Remote Sens. 30, 157–165 (1992).
[CrossRef]

Radio Sci. (2)

L. Tsang, “Thermal emission of nonspherical particles,” Radio Sci. 19, 966–974 (1984).
[CrossRef]

S. H. Yueh, R. Kwok, “Electromagnetic fluctuations for anisotropic media and the generalized Kirchhoff’ s law,” Radio Sci. 28, 471–480 (1993).
[CrossRef]

Other (10)

W. G. Egan, Photometry and Polarization in Remote Sensing (Elsevier, New York, 1985).

S. M. Rytov, Yu. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989), Vol. 3.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989).

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

P. Beckman, The Depolarization of Electromagnetic Waves (Golem, Boulder, Colo., 1968).

CRC Handbook of Chemistry and Physics, 70th ed., R. C. Weast, D. R. Lide, M. J. Astle, W. H. Beyer, eds. (CRC Press, Boca Raton, Fla., 1989), p. E-390.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

R. E. Kleinman, T. B. A. Senior, “Rayleigh scattering,” in Low and High Frequency Asymptotics, V. K. Varadan, V. V. Varadan, eds. (North-Holland, Amsterdam, 1986), Vol. 2.

D. R. Wiesnet, M. Matson, “Remote sensing of weather and climate,” in Manual of Remote Sensing, 2nd ed., R. N. Colwell, J. E. Estes, G. A. Thorley, eds., American Society of Photogrammetry (Sheridan, Falls Church, Va., 1983), Vol. 2, p. 1335.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram showing the direction of radiation emission r ̂, the unit vector in the back direction r ̂ b = r ̂, and the orthogonal unit vectors ê1,ê2 such that ê 1 × ê 2 = r ̂.

Fig. 2
Fig. 2

Plot of the degree of polarization of the radiation emitted in the direction perpendicular to the axis of a spheroid, |Γ|, as a function of the ratio of the semiaxes, a1/a3, for a particle with r = (2.210 + 0.720i)2 [curve (i)], r = (1.750 + 0.162i)2 [curve (ii)], and r = (1.540 + 0.015i)2 [curve (iii)].

Fig. 3
Fig. 3

Plot of P ( θ ), the degree of polarization of the radiation emitted in the direction at an angle θ to the axis of the spheroid, Eq. (44). Two cases are shown: Γ = 0.612 [curve (i)] and Γ = −0.810 [curve (ii)] (see discussion in the text).

Equations (93)

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S ( r ) = S ( r ; ê 1 ) + S ( r ; ê 2 ) = S [ r ; ( ê 1 + ê 2 ) / 2 ] + S [ r ; ( ê 1 ê 2 ) / 2 ] = S ( r ; RHS ) + S ( r ; LHC ) ,
I ( r ) = S ( r ; ê 1 ) · r ̂ + S ( r ; ê 2 ) · r ̂ ,
Q ( r ) = S ( r ; ê 1 ) · r ̂ S ( r ; ê 2 ) · r ̂ ,
U ( r ) = S [ r ; ( ê 1 + ê 2 ) / 2 ] · r ̂ S [ r ; ( ê 1 ê 2 ) / 2 ] · r ̂ ,
V ( r ) = S ( r ; RHC ) · r ̂ S ( r ; LHC ) · r ̂ .
I ( r ) = S ( r ; ê 1 ) · r ̂ + S ( r ; ê 2 ) · r ̂ ,
Q ( r ) = S ( r ; ê 1 ) · r ̂ S ( r ; ê 2 ) · r ̂ ,
U ( r ) = 2 S [ r ; ( ê 1 + ê 2 ) / 2 ] · r ̂ S ( r ; ê 1 ) · r ̂ S ( r ; ê 2 ) · r ̂ ,
V ( r ) = 2 S ( r ; RHS ) · r ̂ S ( r ; ê 1 ) · r ̂ S ( r ; ê 2 ) · r ̂ .
r ̂ b = r ̂ ,
S ( r ; â ) = ( 1 / r 2 ) N 2 B 0 ( ω , T ) σ abs ( r ̂ b , â ) Δ ω r ̂ ,
S ( r ; RHC ) = ( 1 / r 2 ) N 2 B 0 ( ω , T ) σ abs [ r ̂ b , ( ê 1 + i ê 2 ) / 2 ] × Δ ω r ̂ ,
S ( r ; LHC ) = ( 1 / r 2 ) N 2 B 0 ( ω , T ) σ abs [ r ̂ b , ( ê 1 i ê 2 ) / 2 ] × Δ ω r ̂ .
B 0 ( ω , T ) = ω 3 8 π 3 c 2 [ exp ( ω / k T ) 1 ] 1
E i ( r , t ) = Re [ â exp ( i K r ̂ b · r ) exp ( i ω t ) ] ,
[ I ( r ) Q ( r ) U ( r ) V ( r ) ] = ( 1 / r 2 ) N 2 B 0 ( ω , T ) Δ ω [ i ( r ̂ ) q ( r ̂ ) u ( r ̂ ) υ ( r ̂ ) ] ,
i ( r ̂ ) = σ abs ( r ̂ b , ê 1 ) + σ abs ( r ̂ b , ê 2 ) ,
q ( r ̂ ) = σ abs ( r ̂ b , ê 1 ) σ abs ( r ̂ b , ê 2 ) ,
u ( r ̂ ) = σ abs [ r ̂ b , ( ê 1 + ê 2 ) / 2 ] σ abs [ r ̂ b , ( ê 1 ê 2 ) / 2 ] ,
υ ( r ̂ ) = σ abs [ r ̂ b , ( ê 1 + i ê 2 ) / 2 ] σ abs [ r ̂ b , ( ê 1 i ê 2 ) / 2 ] ,
u ( r ̂ ) = 2 σ abs [ r ̂ b , ( ê 1 + ê 2 ) / 2 ] σ abs ( r ̂ b , ê 1 ) σ abs ( r ̂ b , ê 2 ) ,
υ ( r ̂ ) = 2 σ abs [ r ̂ b , ( ê 1 + i ê 2 ) / 2 ] σ abs ( r ̂ b , ê 1 ) σ abs ( r ̂ b , ê 2 ) .
P ( r ) = [ Q 2 ( r ) + U 2 ( r ) + V 2 ( r ) ] 1 / 2 I ( r ) , = [ q 2 ( r ̂ ) + u 2 ( r ̂ ) + υ 2 ( r ̂ ) ] 1 / 2 i ( r ̂ ) .
t ̂ = n ̂ × r ̂ b | n ̂ × r ̂ b | ,
q ̂ i = r ̂ b × t ̂ ;
σ abs ( r ̂ b , q ̂ i ) = ( 1 | R | 2 ) cos θ i Δ A ,
σ abs ( r ̂ b , t ̂ ) = ( 1 | R | 2 ) cos θ i Δ A ,
R = r cos θ i ( r μ r sin 2 θ i ) 1 / 2 r cos θ i + ( r μ r sin 2 θ i ) 1 / 2 ,
R = μ r cos θ i ( r μ r sin 2 θ i ) 1 / 2 μ r cos θ i + ( r μ r sin 2 θ i ) 1 / 2 ,
cos θ i = n ̂ · r ̂ b = n ̂ · r ̂ .
σ abs [ r ̂ b , ( q ̂ i + t ̂ ) / 2 ] = σ abs [ r ̂ b , ( q ̂ i t ̂ ) / 2 ] ,
σ abs [ r ̂ b , ( q ̂ i + i t ̂ ) / 2 ] = σ abs [ r ̂ b , ( q ̂ i i t ̂ ) / 2 ] ,
i plate ( r ̂ ) = [ ( 1 | R | 2 ) + ( 1 | R | 2 ) ] cos θ i Δ A ,
q plate ( r ̂ ) = [ ( 1 | R | 2 ) ( 1 | R | 2 ) ] cos θ i Δ A ,
u plate ( r ̂ ) = 0 ,
υ plate ( r ̂ ) = 0 ,
P plate = | ( 1 | R | 2 ) ( 1 | R | 2 ) ( 1 | R | 2 ) + ( 1 | R | 2 ) | .
i plate = i plate ,
q plate = q plate cos 2 β + u plate sin 2 β ,
u plate = q plate sin 2 β + u plate cos 2 β ,
υ plate = υ plate .
sin β = t ̂ · ê 1 ,
cos β = t ̂ · ê 2 .
| n ̂ × r ̂ b | = [ ( n ̂ · ê 1 ) 2 + ( n ̂ · ê 2 ) 2 ] 1 / 2 .
t ̂ · ê 1 = n ̂ · ê 2 [ ( n ̂ · ê 1 ) 2 + ( n ̂ · ê 2 ) 2 ] 1 / 2 ,
t ̂ · ê 2 = n ̂ · ê 1 [ ( n ̂ · ê 1 ) 2 + ( n ̂ · ê 2 ) 2 ] 1 / 2 .
cos 2 β = [ ( n ̂ · ê 1 ) 2 ( n ̂ · ê 2 ) 2 ] [ ( n ̂ · ê 1 ) 2 + ( n ̂ · ê 2 ) 2 ] ,
sin 2 β = 2 ( n ̂ · ê 1 ) ( n ̂ · ê 2 ) [ ( n ̂ · ê 1 ) 2 + ( n ̂ · ê 2 ) 2 ] .
i plate ( r ̂ ) = [ ( 1 | R | 2 ) + ( 1 | R | 2 ) ] ( n ̂ · r ̂ ) Δ A ,
q plate ( r ̂ ) = [ ( 1 | R | 2 ) ( 1 | R | 2 ) ] × { [ ( n ̂ · ê 1 ) 2 ( n ̂ · ê 2 ) 2 ] [ ( n ̂ · ê 1 ) 2 + ( n ̂ · ê 2 ) 2 ] } ( n ̂ · r ̂ ) Δ A ,
u plate ( r ̂ ) = [ ( 1 | R | 2 ) ( 1 | R | 2 ) ] × { 2 ( n ̂ · ê 1 ) ( n ̂ · ê 2 ) [ ( n ̂ · ê 1 ) 2 + ( n ̂ · ê 2 ) 2 ] } ( n ̂ · r ̂ ) Δ A ,
υ plate ( r ̂ ) = 0 .
i ( r ̂ ) = A illum [ ( 1 | R | 2 ) + ( 1 | R | 2 ) ] ( n ̂ · r ̂ ) d A ,
q ( r ̂ ) = A illum [ ( 1 | R | 2 ) ( 1 | R | 2 ) ] × { [ ( n ̂ · ê 1 ) 2 ( n ̂ · ê 2 ) 2 ] [ ( n ̂ · ê 1 ) 2 + ( n ̂ · ê 2 ) 2 ] } ( n ̂ · r ̂ ) d A ,
u ( r ̂ ) = A illum [ ( 1 | R | 2 ) ( 1 | R | 2 ) ] × { 2 ( n ̂ · ê 1 ) ( n ̂ · ê 2 ) [ ( n ̂ · ê 1 ) 2 + ( n ̂ · ê 2 ) 2 ] } ( n ̂ · r ̂ ) d A ,
υ ( r ̂ ) = 0 .
i ( y ̂ ) = a h ( A + B ) ,
q ( y ̂ ) = a h ( A B ) ,
u ( y ̂ ) = 0 ,
υ ( y ̂ ) = 0 ,
A = 0 π [ 1 | r sin α ( r μ r cos 2 α ) 1 / 2 r sin α + ( r μ r cos 2 α ) 1 / 2 | 2 ] sin α d α ,
B = 0 π [ 1 | μ r sin α ( r μ r cos 2 α ) 1 / 2 μ r sin α + ( r μ r cos 2 α ) 1 / 2 | 2 ] sin α d α .
E s ( r , t ) = Re { F ( r ̂ ; r ̂ b , â ) [ exp ( i K r ) / r ] exp ( i ω t ) } ,
F ( r ̂ ; r ̂ b , â ) = ( K 2 / 4 π ) [ ( 1 / ) r ̂ × ( r ̂ × p ) + ( 1 / y ) r ̂ × m }
σ extin ( r ̂ b , â ) = ( 4 π / K ) Im [ â * · F ( r ̂ b ; r ̂ b , â ) ] ,
σ extin ( r ̂ b , â ) = K Im [ ( 1 / ) â * · p ) + ( 1 / y ) ( r ̂ b × â ) * · m ] .
p = P _ · â ,
m = y M _ · ( r ̂ b × â ) .
σ abs ( r ̂ b , â ) = K Im [ â * · P _ · â ( r ̂ b × â ) * · M _ · ( r ̂ b × â ) ] .
i ( r ̂ ) = K Im ( ê 1 · P _ · ê 1 + ê 2 · P _ · ê 2 ê 1 · M _ · ê 1 ê 2 · M _ · ê 2 ) ,
q ( r ̂ ) = K Im ( ê 1 · P _ · ê 1 ê 2 · P _ · ê 2 + ê 1 · M _ · ê 1 ê 2 · M _ · ê 2 ) ,
u ( r ̂ ) = K Im ( ê 1 · P _ · ê 2 + ê 2 · P _ · ê 1 + ê 1 · M _ · ê 2 + ê 2 · M _ · ê 1 ) ,
υ ( r ̂ ) = K Re ( ê 1 · P _ · ê 2 ê 2 · P _ · ê 1 ê 1 · M _ · ê 2 + ê 2 · M _ · ê 1 ) .
( X / a 1 ) 2 + ( Y / a 2 ) 2 + ( Z / a 3 ) 2 = 1 .
P _ = P 11 x ̂ x ̂ + P 22 y ̂ y ̂ + P 33 z ̂ z ̂ ,
P i i = V 0 ( r 1 ) 1 + L i ( r 1 ) , i = 1 , 2 , 3 .
L i = ( a 1 a 2 a 3 / 2 ) 0 ( a i 2 + u ) 1 × [ ( a 1 2 + u ) ( a 2 2 + u ) ( a 3 2 + u ) ] 1 / 2 d u .
r ̂ = x ̂ sin θ cos ϕ + y ̂ sin θ sin ϕ + z ̂ cos θ ,
ê 1 = x ̂ cos θ cos ϕ + y ̂ cos θ sin ϕ z ̂ sin θ ,
ê 2 = x ̂ sin ϕ + y ̂ cos ϕ .
i ( r ̂ ) = χ [ | P 11 | 2 ( cos 2 θ cos 2 ϕ + sin 2 ϕ ) + | P 22 | 2 ( cos 2 θ sin 2 ϕ cos 2 ϕ ) + | P 33 | 2 sin 2 θ ] ,
q ( r ̂ ) = χ [ | P 11 | 2 ( cos 2 θ cos 2 ϕ sin 2 ϕ ) + | P 22 | 2 ( cos 2 θ sin 2 ϕ cos 2 ϕ ) + | P 33 | 2 sin 2 θ ] ,
u ( r ̂ ) = χ ( | P 22 | 2 | P 11 | 2 ) cos θ sin 2 ϕ ,
υ ( r ̂ ) = 0 ,
χ = K Im ( r ) V 0 | r 1 | 2 .
i ( r ̂ ) = χ [ 2 | P 11 | 2 + ( | P 33 | 2 | P 11 | 2 ) sin 2 θ ] ,
q ( r ̂ ) = χ ( | P 33 | 2 | P 11 | 2 ) sin 2 θ ,
u ( r ̂ ) = 0 ,
υ ( r ̂ ) = 0 .
Γ = | P 33 | 2 | P 11 | 2 | P 33 | 2 + | P 11 | 2 ,
P ( θ ) = | Γ sin 2 θ 1 Γ cos 2 θ | .
Γ | r + 1 | 2 4 | r + 1 | 2 + 4 , a 1 / a 3 0 ,
Γ 1 | r | 2 1 + | r | 2 , a 1 / a 3 .

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