Abstract

We study multiple scattering of partially polarized light using the theory of radiative transport. In particular, we study the light that exits a half-space composed of a uniform absorbing and scattering medium due to an unpolarized, isotropic, and continuous planar source. We assume that Rayleigh scattering applies. Using only angular integrals of the two orthogonal polarization components of the intensity exiting the half-space, we recover the depth and the strength of this source in two stages. First, we recover the depth of the source through the solution of a one-dimensional nonlinear equation. Then we recover the strength of the source through the solution of a linear least-squares problem. This method is limited to sources located at depths on the order of a transport mean-free path or less. Beyond that depth, these data do not contain sufficient polarization diversity for this inversion method to work. In addition, we show that this method is sensitive to instrument noise. We present numerical results to validate these results.

© 2009 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1996).
  3. L. V. Ryzhik, G. C. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
    [CrossRef]
  4. H. C. van der Hulst, Light Scattering by Small Particles (Dover, 1981).
  5. J. M. Schmitt, A. H. Gandjbakche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. 31, 6535-6546 (1992).
    [CrossRef] [PubMed]
  6. M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948-960 (2001).
    [CrossRef]
  7. C. E. Siewert, “Determination of the single-scattering albedo from polarization measurements of a Rayleigh atmosphere,” Astrophys. Space Sci. 69, 237-239 (1979).
    [CrossRef]
  8. C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization--I,” J. Quant. Spectrosc. Radiat. Transf. 30, 523-526 (1983).
    [CrossRef]
  9. N. J. McCormick, “Determination of the single-scattering albedo of a dense Rayleigh-scattering atmosophere with true absorption,” Astrophys. Space Sci. 71, 235-238 (1980).
    [CrossRef]
  10. N. J. McCormick and R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization--II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527-535 (1983).
    [CrossRef]
  11. T. Viik and N. J. McCormick, “Numerical test of an inverse polarized radiative transfer algorithm,” J. Quant. Spectrosc. Radiat. Transf. 78, 235-241 (2003).
    [CrossRef]
  12. C. E. Siewert, “A radiative-transfer inverse-source problem for a sphere,” J. Quant. Spectrosc. Radiat. Transf. 52, 157-160 (1994).
    [CrossRef]
  13. C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64, 227-254 (2000).
    [CrossRef]
  14. Y. Qin and M. A. Box, “Vector Green's function algorithm for radiative transfer in plane-parallel atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 97, 228-251 (2006).
    [CrossRef]
  15. 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]

2006 (1)

Y. Qin and M. A. Box, “Vector Green's function algorithm for radiative transfer in plane-parallel atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 97, 228-251 (2006).
[CrossRef]

2003 (1)

T. Viik and N. J. McCormick, “Numerical test of an inverse polarized radiative transfer algorithm,” J. Quant. Spectrosc. Radiat. Transf. 78, 235-241 (2003).
[CrossRef]

2001 (1)

2000 (1)

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64, 227-254 (2000).
[CrossRef]

1996 (1)

L. V. Ryzhik, G. C. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

1994 (1)

C. E. Siewert, “A radiative-transfer inverse-source problem for a sphere,” J. Quant. Spectrosc. Radiat. Transf. 52, 157-160 (1994).
[CrossRef]

1992 (1)

1983 (2)

C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization--I,” J. Quant. Spectrosc. Radiat. Transf. 30, 523-526 (1983).
[CrossRef]

N. J. McCormick and R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization--II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527-535 (1983).
[CrossRef]

1980 (1)

N. J. McCormick, “Determination of the single-scattering albedo of a dense Rayleigh-scattering atmosophere with true absorption,” Astrophys. Space Sci. 71, 235-238 (1980).
[CrossRef]

1979 (1)

C. E. Siewert, “Determination of the single-scattering albedo from polarization measurements of a Rayleigh atmosphere,” Astrophys. Space Sci. 69, 237-239 (1979).
[CrossRef]

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]

Bonner, R. F.

Box, M. A.

Y. Qin and M. A. Box, “Vector Green's function algorithm for radiative transfer in plane-parallel atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 97, 228-251 (2006).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Gandjbakche, A. H.

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]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1996).

Keller, J. B.

M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948-960 (2001).
[CrossRef]

L. V. Ryzhik, G. C. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

McCormick, N. J.

T. Viik and N. J. McCormick, “Numerical test of an inverse polarized radiative transfer algorithm,” J. Quant. Spectrosc. Radiat. Transf. 78, 235-241 (2003).
[CrossRef]

N. J. McCormick and R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization--II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527-535 (1983).
[CrossRef]

N. J. McCormick, “Determination of the single-scattering albedo of a dense Rayleigh-scattering atmosophere with true absorption,” Astrophys. Space Sci. 71, 235-238 (1980).
[CrossRef]

Moscoso, M.

Papanicolaou, G.

Papanicolaou, G. C.

L. V. Ryzhik, G. C. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

Qin, Y.

Y. Qin and M. A. Box, “Vector Green's function algorithm for radiative transfer in plane-parallel atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 97, 228-251 (2006).
[CrossRef]

Ryzhik, L. V.

L. V. Ryzhik, G. C. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

Sanchez, R.

N. J. McCormick and R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization--II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527-535 (1983).
[CrossRef]

Schmitt, J. M.

Siewert, C. E.

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64, 227-254 (2000).
[CrossRef]

C. E. Siewert, “A radiative-transfer inverse-source problem for a sphere,” J. Quant. Spectrosc. Radiat. Transf. 52, 157-160 (1994).
[CrossRef]

C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization--I,” J. Quant. Spectrosc. Radiat. Transf. 30, 523-526 (1983).
[CrossRef]

C. E. Siewert, “Determination of the single-scattering albedo from polarization measurements of a Rayleigh atmosphere,” Astrophys. Space Sci. 69, 237-239 (1979).
[CrossRef]

van der Hulst, H. C.

H. C. van der Hulst, Light Scattering by Small Particles (Dover, 1981).

Viik, T.

T. Viik and N. J. McCormick, “Numerical test of an inverse polarized radiative transfer algorithm,” J. Quant. Spectrosc. Radiat. Transf. 78, 235-241 (2003).
[CrossRef]

Appl. Opt. (1)

Astrophys. Space Sci. (2)

C. E. Siewert, “Determination of the single-scattering albedo from polarization measurements of a Rayleigh atmosphere,” Astrophys. Space Sci. 69, 237-239 (1979).
[CrossRef]

N. J. McCormick, “Determination of the single-scattering albedo of a dense Rayleigh-scattering atmosophere with true absorption,” Astrophys. Space Sci. 71, 235-238 (1980).
[CrossRef]

J. Atmos. Sci. (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]

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

J. Quant. Spectrosc. Radiat. Transf. (6)

C. E. Siewert, “Solutions to an inverse problem in radiative transfer with polarization--I,” J. Quant. Spectrosc. Radiat. Transf. 30, 523-526 (1983).
[CrossRef]

N. J. McCormick and R. Sanchez, “Solutions to an inverse problem in radiative transfer with polarization--II,” J. Quant. Spectrosc. Radiat. Transf. 30, 527-535 (1983).
[CrossRef]

T. Viik and N. J. McCormick, “Numerical test of an inverse polarized radiative transfer algorithm,” J. Quant. Spectrosc. Radiat. Transf. 78, 235-241 (2003).
[CrossRef]

C. E. Siewert, “A radiative-transfer inverse-source problem for a sphere,” J. Quant. Spectrosc. Radiat. Transf. 52, 157-160 (1994).
[CrossRef]

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64, 227-254 (2000).
[CrossRef]

Y. Qin and M. A. Box, “Vector Green's function algorithm for radiative transfer in plane-parallel atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 97, 228-251 (2006).
[CrossRef]

Wave Motion (1)

L. V. Ryzhik, G. C. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

Other (3)

H. C. van der Hulst, Light Scattering by Small Particles (Dover, 1981).

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1996).

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Figures (3)

Fig. 1
Fig. 1

Plot of the ratio F l F r as a function of the source depth z 0 . The dotted lines indicate the theoretical values corresponding to the asymptotic behavior of the function.

Fig. 2
Fig. 2

Relative error of the recovered source depth ϵ depth defined by Eq. (69a) as a function of the true source depth z 0 for different noise levels.

Fig. 3
Fig. 3

Relative error of the recovered source strength ϵ strength defined by Eq. (69b) as a function of the true source depth z 0 for different noise levels.

Equations (98)

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I l = E l E l * ,
I r = E r E r * ,
U = E l E r * + E r E l * ,
V = i E l E r * E r E l * ,
Ω I + ( κ a + κ s ) I κ s S 2 Z ( Ω , Ω ) I ( Ω , r ) d Ω = S .
I = I b in Γ in = { ( Ω , r ) S 2 × D , Ω ν ( r ) < 0 } .
Z ( Ω , Ω ) = L 2 F ( Ω Ω ) L 1 .
Z ( μ , μ , φ φ ) = P Q Z ( μ , μ , φ φ ) Q P ,
Z ( μ , μ , φ φ ) = Q D Z T ( μ , μ , φ φ ) D 1 Q ,
Ω I + κ t I κ s S 2 Z ( Ω , Ω ) I ( Ω , r ) d Ω = 0 ,
I = e λ r z ̂ v ( μ , φ ) .
λ μ v + κ t v κ s π π 1 1 Z ( μ , μ , φ φ ) v ( μ , φ ) d μ d φ = 0 .
λ μ v + κ t v = κ s π π 1 1 Z ( μ , μ , φ φ ) v ( μ , φ ) d μ d φ .
λ μ v + κ t v = κ s π π 1 1 P Q Z ( μ , μ , φ φ ) Q P v ( μ , φ ) d μ d φ .
λ μ Q P v + κ t Q P v = κ s π π 1 1 Z ( μ , μ , φ φ ) Q P v ( μ , φ ) d μ d φ
( λ m λ n ) π π 1 1 v m T ( μ , φ ) D 1 Q v n ( μ , φ ) μ d μ d φ = 0 ,
( λ m λ n ) π π 1 1 v m T ( μ , φ ) D 1 Q v n ( μ , φ ) μ d μ d φ = κ s π π 1 1 v m T ( μ , φ ) D 1 Q π π 1 1 Z ( μ , μ , φ φ ) v n ( μ , φ ) d μ d φ d μ d φ κ s π π 1 1 v n T ( μ , φ ) D 1 Q π π 1 1 Z ( μ , μ , φ φ ) v m ( μ , φ ) d μ d φ d μ d φ .
( λ m λ n ) π π 1 1 v m T ( μ , φ ) D 1 Q v n ( μ , φ ) μ d μ d φ = κ s π π 1 1 π π 1 1 v m T ( μ , φ ) D 1 Q Z ( μ , μ , φ φ ) v n ( μ , φ ) d μ d φ d μ d φ κ s π π 1 1 π π 1 1 v n T ( μ , φ ) Z T ( μ , μ , φ φ ) D 1 Q v m ( μ , φ ) d μ d φ d μ d φ .
π π 1 1 [ Q P v n ( μ , φ ) ] T D 1 Q [ Q P v n ( μ , φ ) ] μ d μ d φ = π π 1 1 v n T ( μ , φ ) D 1 Q v n ( μ , φ ) μ d μ d φ .
π π 1 1 v n T ( μ , φ ) Q 1 v n ( μ , φ ) μ d μ d φ = { + 1 n < 0 1 n > 0 } .
μ I z + κ t I κ s π π 1 1 Z ( μ , μ , φ φ ) I ( μ , φ , z ) d μ d φ = 0
I z = 0 = I b
I ( μ , φ , z ) = n = 1 { Q P v n ( μ , φ ) e λ n z c n } .
n = 1 { Q P v n ( μ , φ ) c n } = I b ( μ , φ )
μ I z + κ t I κ s π π 1 1 Z ( μ , μ , φ φ ) I ( μ , φ , z ) d μ d φ = 0
I z = 0 = I 1
I z = d = I 2
I ( μ , φ , z ) = n = 1 { v n ( μ , φ ) e λ n ( z d ) a n + Q P v n ( μ , φ ) e λ n z b n } .
I 1 ( μ , φ ) = n = 1 { v n ( μ , φ ) e λ n d a n + Q P v n ( μ , φ ) b n }
I 2 ( μ , φ ) = n = 1 { v n ( μ , φ ) a n + Q P v n ( μ , φ ) e λ n d b n }
S ( z ) = s 0 2 [ 1 1 0 0 ] δ ( z z 0 ) = s 0 U δ ( z z 0 ) .
I ( μ , φ , 0 ) = 0
μ I z + κ t I κ s π π 1 1 Z ( μ , μ , φ φ ) I ( μ , φ , z ) d μ d φ = s 0 U δ ( z z 0 )
μ I ( μ , φ , z 0 + ) μ I ( μ , φ , z 0 ) = s 0 U
Z ( μ , μ , φ φ ) = 1 2 Z 0 ( μ , μ ) + Z 1 ( c ) ( μ , μ ) cos ( φ φ ) + Z 1 ( s ) ( μ , μ ) sin ( φ φ ) + Z 2 ( c ) ( μ , μ ) cos 2 ( φ φ ) + Z 2 ( s ) ( μ , μ ) sin 2 ( φ φ ) .
I ( μ , φ , z ) = 1 2 I 0 ( μ , z ) + m = 1 { I m ( c ) ( μ , z ) cos m φ + I m ( s ) ( μ , z ) sin m φ } .
μ z [ I l I r ] + κ t [ I l I r ] κ s 3 8 1 1 R ( μ , μ ) [ I l ( μ , z ) I r ( μ , z ) ] d μ = s 0 2 [ 1 1 ] δ ( z z 0 )
R ( μ , μ ) = 3 8 [ 2 ( 1 μ 2 ) ( 1 μ 2 ) + μ 2 μ 2 μ 2 μ 2 1 ] .
[ I l ( μ , 0 ) I r ( μ , 0 ) ] = [ 0 0 ]
μ [ I l ( μ , z 0 + ) I r ( μ , z 0 + ) ] μ [ I l ( μ , z 0 ) I r ( μ , z 0 ) ] = s 0 2 [ 1 1 ]
λ μ V + κ t V κ s 1 1 R ( μ , μ ) V ( μ ) d μ = 0 .
λ n = λ n , V n ( μ ) = V n ( μ )
( λ m λ n ) 1 1 V m T ( μ ) V n ( μ ) μ d μ = 0 .
1 1 V n T ( μ ) V n ( μ ) μ d μ = { + 1 n < 0 1 n > 0 } .
Ψ ( μ , z ) = s 0 { n = 1 { V n ( μ ) e λ n ( z z 0 ) a n + V n ( μ ) e λ n z b n } , 0 < z < z 0 n = 1 { V n ( μ ) e λ n ( z z 0 ) c n } , z > z 0 } .
n = 1 { V n ( μ ) e λ n z 0 a n + V n ( μ ) b n } = [ 0 0 ]
n = 1 { μ V n ( μ ) c n μ V n ( μ ) a n μ V n ( μ ) e λ n z 0 b n } = [ 1 2 1 2 ]
a m = 1 2 1 1 [ V m ( 1 ) ( μ ) + V m ( 2 ) ( μ ) ] d μ .
c m e λ n z 0 b m = 1 2 1 1 [ V m ( 1 ) ( μ ) + V m ( 2 ) ( μ ) ] d μ .
n = 1 V n ( μ ) b n = n = 1 V n ( μ ) e λ n z 0 a n
Ψ ( μ , 0 ) = s 0 n = 1 { V n ( μ ) e λ n z 0 a n + V n ( μ ) b n }
I l ( μ , 0 ) = s 0 n = 1 { V n ( 1 ) ( μ ) e λ n z 0 a n + V n ( 1 ) ( μ ) b n } ,
I r ( μ , 0 ) = s 0 n = 1 { V n ( 2 ) ( μ ) e λ n z 0 a n + V n ( 2 ) ( μ ) b n } .
1 1 f ( μ ) d μ j = 1 2 N f ( μ j ) w j .
A j k = 3 8 [ 2 ( 1 μ j 2 ) ( 1 μ k 2 ) + μ j 2 μ k 2 ] w k ,
B j k = 3 8 μ j 2 w k ,
C j k = 3 8 μ k 2 w k ,
D j k = 3 8 w k ,
λ [ M 0 0 M ] [ v ( 1 ) v ( 2 ) ] = κ t [ v ( 1 ) v ( 2 ) ] + κ s [ A B C D ] [ v ( 1 ) v ( 2 ) ] .
λ n = λ n , V j , n ( 1 , 2 ) = V 2 N j + 1 , n ( 1 , 2 ) ,
a n = 1 2 j = 1 2 N { ( V j , n ( 1 ) + V j , n ( 2 ) ) w j } ,
n = 1 N V 2 N j + 1 , n ( 1 ) b n = n = 1 N V j , n ( 1 ) e λ n z 0 a n ,
n = 1 N V 2 N j + 1 , n ( 2 ) b n = n = 1 N V j , n ( 2 ) e λ n z 0 a n ,
I l ( μ j , 0 ) s 0 n = 1 N { V j , n ( 1 ) e λ n z 0 a n + V 2 N j + 1 , n ( 1 ) b n } ,
I r ( μ j , 0 ) s 0 n = 1 N { V j , n ( 2 ) e λ n z 0 a n + V 2 N j + 1 , n ( 2 ) b n } .
F l , r = 1 0 I l , r ( μ , 0 ) μ d μ .
F ̃ l = s 0 n = 1 N { V ¯ n ( 1 ) e λ n z 0 a n + W ¯ n ( 1 ) b n } = s 0 f ̃ l ( z 0 ) ,
F ̃ r = s 0 n = 1 N { V ¯ n ( 2 ) e λ n z 0 a n + W ¯ n ( 2 ) b n } = s 0 f ̃ r ( z 0 ) .
V ¯ n ( i ) = j = 1 N V j , n ( i ) μ j w j , i = 1 , 2 ,
W ¯ n ( i ) = j = 1 N V 2 N j + 1 , n ( i ) μ j w j , i = 1 , 2 .
d ( z 0 ) = F l F r f ̃ l ( z 0 ) f ̃ r ( z 0 ) ,
[ f ̃ l ( z 0 * ) f ̃ r ( z 0 * ) ] s 0 * = [ F l F r ] ,
s 0 * = f ̃ l ( z 0 * ) F l + f ̃ r ( z 0 * ) F r f ̃ l 2 ( z 0 * ) + f ̃ r 2 ( z 0 * ) .
0 < λ 1 λ 2 λ N .
n = 1 N V 2 N j + 1 , n ( 1 ) β n V j , 1 ( 1 ) ,
n = 1 N V 2 N j + 1 , n ( 2 ) β n V j , 1 ( 2 ) ,
f ̃ l ( z 0 ) e λ 1 z 0 a 1 [ V ¯ 1 ( 1 ) + n = 1 N W ¯ n ( 1 ) β n ] ,
f ̃ r ( z 0 ) e λ 1 z 0 a 1 [ V ¯ 1 ( 2 ) + n = 1 N W ¯ n ( 2 ) β n ] ,
f ( z 0 ) V ¯ 1 ( 1 ) + n = 1 N W ¯ n ( 1 ) β n V ¯ 1 ( 2 ) + n = 1 N W ¯ n ( 2 ) β n = f ,
d ( z 0 ) F l F r f = const. , for z l * .
ϵ depth = z 0 * z 0 z 0 ,
ϵ strength = s 0 * s 0 s 0 ,
Z ̃ ( μ , μ , φ φ ) = P Q Z ̃ ( μ , μ , φ φ ) Q P ,
Z ̃ ( μ , μ , φ φ ) = Q Z ̃ T ( μ , μ , φ φ ) Q ,
I ̃ = [ I Q U V ] = [ 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 ] [ I l I r U V ] = T I .
T 1 = [ 1 2 1 2 0 0 1 2 1 2 0 0 0 0 1 0 0 0 0 1 ] = 1 2 D T ,
Z ( μ , μ , φ φ ) = T 1 Z ̃ ( μ , μ , φ φ ) T = 1 2 D T Z ̃ ( μ , μ , φ φ ) T .
Z ( μ , μ , φ φ ) = 1 2 D T Z ̃ ( μ , μ , φ φ ) T .
Z ( μ , μ , φ φ ) = 1 2 D T P Q Z ̃ ( μ , μ , φ φ ) Q P T .
Z ( μ , μ , φ φ ) = P Q [ 1 2 D T Z ̃ ( μ , μ , φ φ ) T ] Q P = P Q Z ( μ , μ , φ φ ) Q P .
Z ( μ , μ , φ φ ) = 1 2 D T Z ̃ ( μ , μ , φ φ ) T .
Z ( μ , μ , φ φ ) = 1 2 D T Q Z ̃ T ( μ , μ , φ φ ) Q T = Q D [ 1 2 T Z ̃ T ( μ , μ , φ φ ) T ] Q = Q D [ 1 2 T Z ̃ T ( μ , μ , φ φ ) T D ] D 1 Q .
Z ( μ , μ , φ φ ) = Q D Z T ( μ , μ , φ φ ) D 1 Q .
Z 0 ( μ , μ ) = 3 8 π [ 2 ( 1 μ 2 ) ( 1 μ 2 ) + μ 2 μ 2 μ 2 0 0 μ 2 1 0 0 0 0 0 0 0 0 0 2 μ μ ] ,
Z 1 ( c ) ( μ , μ ) = 3 16 π ( 1 μ 2 ) 1 2 ( 1 μ 2 ) 1 2 [ 4 μ μ 0 0 0 0 0 0 0 0 0 2 0 0 0 0 2 ] ,
Z 1 ( s ) ( μ , μ ) = 3 16 π ( 1 μ 2 ) 1 2 ( 1 μ 2 ) 1 2 [ 0 0 2 μ 0 0 0 0 0 4 μ 0 0 0 0 0 0 0 ] ,
Z 2 ( c ) ( μ , μ ) = 3 16 π [ μ 2 μ 2 μ 2 0 0 μ 2 1 0 0 0 0 2 μ μ 0 0 0 0 0 ] ,
Z 2 ( s ) ( μ , μ ) = 3 16 π [ 0 0 μ 2 μ 0 0 0 μ 0 2 μ μ 2 2 μ 0 0 0 0 0 0 ] .

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