Abstract

An error analysis of a Mueller matrix polarimeter with dual rotating retarders is presented. Errors in orientational alignment of three of the four polarization elements are considered. Errors that are due to nonideal retardation elements are also included in the analysis. Compensation for imperfect retardation elements is possible with the equations derived, and the equations permit a calibration of the polarimeter for azimuthal alignment of polarization elements. An analytical treatment is given and is followed by numerical examples. The latter should prove useful in the laboratory in comparing precalibrated experimental results with theoretical predictions.

© 1990 Optical Society of America

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References

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  1. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
    [CrossRef] [PubMed]
  2. P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
    [CrossRef]
  3. M. J. Hagyard, G. A. Gary, E. A. West, “The SAMEX vector magnetograph,” NASA Technical Memorandum 4048 (1988).
  4. E. A. West, E. J. Reichmann, M. J. Hagyard, G. A. Gary, “Design of the polarimeter for the solar activity measurements experiments (SAMEX) vector magnetograph,” Opt. Eng. 28, 131–140 (1989).
    [CrossRef]
  5. W. A. Shurcliff, Polarized Light (Harvard U. Press, Boston, Mass., 1962).
  6. N. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).
  7. P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
    [CrossRef]

1989 (1)

E. A. West, E. J. Reichmann, M. J. Hagyard, G. A. Gary, “Design of the polarimeter for the solar activity measurements experiments (SAMEX) vector magnetograph,” Opt. Eng. 28, 131–140 (1989).
[CrossRef]

1978 (2)

Azzam, R. M. A.

Burch, J. M.

N. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Gary, G. A.

E. A. West, E. J. Reichmann, M. J. Hagyard, G. A. Gary, “Design of the polarimeter for the solar activity measurements experiments (SAMEX) vector magnetograph,” Opt. Eng. 28, 131–140 (1989).
[CrossRef]

M. J. Hagyard, G. A. Gary, E. A. West, “The SAMEX vector magnetograph,” NASA Technical Memorandum 4048 (1988).

Gdoutos, E. E.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

Gerrard, N.

N. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Hagyard, M. J.

E. A. West, E. J. Reichmann, M. J. Hagyard, G. A. Gary, “Design of the polarimeter for the solar activity measurements experiments (SAMEX) vector magnetograph,” Opt. Eng. 28, 131–140 (1989).
[CrossRef]

M. J. Hagyard, G. A. Gary, E. A. West, “The SAMEX vector magnetograph,” NASA Technical Memorandum 4048 (1988).

Hauge, P. S.

Reichmann, E. J.

E. A. West, E. J. Reichmann, M. J. Hagyard, G. A. Gary, “Design of the polarimeter for the solar activity measurements experiments (SAMEX) vector magnetograph,” Opt. Eng. 28, 131–140 (1989).
[CrossRef]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Boston, Mass., 1962).

Theocaris, P. S.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

West, E. A.

E. A. West, E. J. Reichmann, M. J. Hagyard, G. A. Gary, “Design of the polarimeter for the solar activity measurements experiments (SAMEX) vector magnetograph,” Opt. Eng. 28, 131–140 (1989).
[CrossRef]

M. J. Hagyard, G. A. Gary, E. A. West, “The SAMEX vector magnetograph,” NASA Technical Memorandum 4048 (1988).

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

E. A. West, E. J. Reichmann, M. J. Hagyard, G. A. Gary, “Design of the polarimeter for the solar activity measurements experiments (SAMEX) vector magnetograph,” Opt. Eng. 28, 131–140 (1989).
[CrossRef]

Opt. Lett. (1)

Other (4)

M. J. Hagyard, G. A. Gary, E. A. West, “The SAMEX vector magnetograph,” NASA Technical Memorandum 4048 (1988).

W. A. Shurcliff, Polarized Light (Harvard U. Press, Boston, Mass., 1962).

N. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Arrangement of the optical elements in the Mueller matrix polarimeter with dual rotating retarders. P1 and P2 are the linear polarizers, and R1 and R2 are the linear retarders. S is the sample, D is a detector, and L is a laser source.

Fig. 2
Fig. 2

Polarization elements and their associated errors. The elements are labeled as in Fig. 1. The angular orientation errors and retardation errors are as indicated and described in the text.

Tables (5)

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Table 1 Mueller Matrices for Each Possible Combination of Azimuthal Orientation Errors in Three Optical Elements of the Polarimetera

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Table 2 Mueller Matrices for Azimuthal Errors Including a Retardation Error of 1 Deg in the First Retarder

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Table 3 Mueller Matrices for Azimuthal Errors Including a Retardation Error of 1 Deg in the Second Retarder

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Table 4 Mueller Matrices for Azimuthal Errors Including a 1-deg Retardation Error in Both Retarders

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Table 5 Harmonics of a Fourier Expansion That Arise from Each Term of the Polarimeter Intensity Equationa

Equations (44)

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I = a 0 + n = 1 12 ( a n cos 2 n θ + b n sin 2 n θ ) ,
m 11 = 4 a 0 4 a 2 + 4 a 8 4 a 10 + 4 a 12 , m 12 = 8 a 2 8 a 8 8 a 12 , m 13 = 8 b 2 + 8 b 8 8 b 12 , m 14 = 4 b 1 8 b 11 = 4 b 1 + 8 b 9 = 4 b 1 + 4 b 9 4 b 11 , m 21 = 8 a 8 + 8 a 10 8 a 12 , m 22 = 16 a 8 + 16 a 12 , m 23 = 16 b 8 + 16 b 12 , m 24 = 16 b 9 = 16 b 11 = 8 ( b 9 + b 11 ) , m 31 = 8 b 8 + 8 b 10 8 b 12 , m 32 = 16 b 8 + 16 b 12 , m 33 = 16 a 8 16 a 12 , m 34 = 16 a 9 = 16 a 11 = 8 ( a 9 a 11 ) , m 41 = 8 b 3 4 b 5 = 4 b 5 + 8 b 7 = 4 ( b 3 b 5 + b 7 ) , m 42 = 16 b 3 = 16 b 7 = 8 ( b 3 + b 7 ) , m 43 = 16 a 3 = 16 a 7 = 8 ( a 3 + a 7 ) , m 44 = 8 a 4 = 8 a 6 = 4 ( a 6 a 4 ) .
P 2 R 2 ( θ ) M R 1 ( θ ) P 1 ,
I = c A M P ,
I = c i , j = 1 4 a i p j m i j
I = c i , j = 1 4 μ i j m i j ,
μ i j = a i p j ,
μ 11 = ½ , μ 12 = ½ [ cos 2 2 ( θ + 3 ) + sin 2 2 ( θ + 3 ) cos δ 1 ] , μ 13 = ½ cos 2 ( θ + 3 ) sin 2 ( θ + 3 ) ( 1 cos δ 1 ) , μ 14 = ½ sin δ 1 sin 2 ( θ + 3 ) , μ 21 = ½ { cos 2 5 [ cos 2 10 ( θ + 4 ) + sin 2 10 ( θ + 4 ) cos δ 2 ] + sin 2 5 cos 10 ( θ + 4 ) sin 10 ( θ + 4 ) ( 1 cos δ 2 ) } , μ 22 = ½ { cos 2 5 [ cos 2 10 ( θ + 4 ) + sin 2 10 ( θ + 4 ) cos δ 2 ] + sin 2 5 cos 10 ( θ + 4 ) sin 10 ( θ + 4 ) ( 1 cos δ 2 ) } × [ cos 2 2 ( θ + 3 ) + sin 2 2 ( θ + 3 ) cos δ 1 ] , μ 23 = ½ { cos 2 5 [ cos 2 10 ( θ + 4 ) + sin 2 10 ( θ + 4 ) cos δ 2 ] + sin 2 5 cos 10 ( θ + 4 ) sin 10 ( θ + 4 ) ( 1 cos δ 2 ) } × cos 2 ( θ + 3 ) sin 2 ( θ + 3 ) ( 1 cos δ 1 ) , μ 24 = ½ { cos 2 5 [ cos 2 10 ( θ + 4 ) + sin 2 10 ( θ + 4 ) cos δ 2 ] + sin 2 5 cos 10 ( θ + 4 ) sin 10 ( θ + 4 ) ( 1 cos δ 2 ) } × sin 2 ( θ + 3 ) sin δ 1 , μ 31 = ½ { cos 2 5 cos 10 ( θ + 4 ) sin 10 ( θ + 4 ) ( 1 cos δ 2 ) + sin 2 5 [ sin 2 10 ( θ + 4 ) + cos 2 10 ( θ + 4 ) cos δ 2 ] } , μ 32 = ½ { cos 2 5 cos 10 ( θ + 4 ) sin 10 ( θ + 4 ) ( 1 cos δ 2 ) + sin 2 5 [ sin 2 10 ( θ + 4 ) cos 2 10 ( θ + 4 ) cos δ 2 ] } × ( cos 2 2 ( θ + 3 ) + sin 2 2 ( θ + 3 ) cos δ 1 ) , μ 33 = ½ { cos 2 5 cos 10 ( θ + 4 ) + sin 10 ( θ + 4 ) ( 1 cos δ 2 ) + sin 2 5 [ sin 2 10 ( θ + 4 ) + cos 2 10 ( θ + 4 ) cos δ 2 ] } × cos 2 ( θ + 3 ) sin 2 ( θ + 3 ) ( 1 cos δ 1 ) , μ 34 = ½ { cos 2 5 cos 10 ( θ + 4 ) sin 10 ( θ + 4 ) ( 1 cos δ 2 ] + sin 2 5 [ sin 2 10 ( θ + 4 ) + cos 2 10 ( θ + 4 ) cos δ 2 ] } × sin 2 ( θ + 3 ) sin δ 1 , μ 41 = ½ [ cos 2 5 sin 10 ( θ + 4 ) sin δ 2 + sin 2 5 cos 10 ( θ + 4 ) sin δ 2 ] , μ 42 = ½ [ cos 2 5 sin 10 ( θ + 4 ) sin δ 2 + sin 2 5 cos 10 ( θ + 4 ) × sin δ 2 ] [ cos 2 2 ( θ + 3 ) + sin 2 2 ( θ + 3 ) cos δ 1 ] , μ 43 = ½ [ cos 2 5 sin 10 ( θ + 4 ) sin δ 2 + sin 2 5 cos 10 ( θ + 4 ) × sin δ 2 ] cos 2 ( θ + 3 ) + sin 2 ( θ + 3 ) ( 1 cos δ 1 ) , μ 44 = ½ [ cos 2 5 sin 10 ( θ + 4 ) sin δ 2 + sin 2 5 cos 10 ( θ + 4 ) × sin δ 2 ] sin 2 ( θ + 3 ) sin δ 1 .
μ 12 = α 0 + α 2 cos 4 θ + β 2 sin 4 θ ,
a 0 = 1 2 m 11 + ( cos δ 1 + 1 4 ) m 12 + [ ( cos δ 2 + 1 ) cos 2 5 4 ] m 21 + [ ( cos δ 1 + 1 ) ( cos δ 2 + 1 ) cos 2 5 8 ] m 22 + [ ( cos δ 2 + 1 ) sin 2 5 4 ] m 31 + [ ( cos δ 1 + 1 ) ( cos δ 2 + 1 ) sin 2 5 8 ] m 32 , a 1 = ( sin δ 1 sin 2 3 2 ) m 14 + [ sin δ 1 ( cos δ 2 + 1 ) sin 2 3 cos 2 5 4 ] m 24 + [ sin δ 1 ( cos δ 2 + 1 ) sin 2 3 sin 2 5 4 ] m 34 , a 2 = [ ( 1 cos δ 1 ) cos 4 3 4 ] m 12 + [ ( 1 cos δ 1 ) cos 4 3 4 ] m 13 + [ ( 1 cos δ 1 ) ( 1 + cos δ 2 ) cos 4 3 cos 2 5 8 ] m 22 + [ ( 1 cos δ 1 ) ( cos δ 2 + 1 ) sin 4 3 cos 2 5 8 ] m 23 + [ ( 1 cos δ 1 ) ( 1 + cos δ 2 ) cos 4 3 cos 2 5 8 ] m 32 + [ ( 1 cos δ 1 ) ( cos δ 2 + 1 ) sin 4 3 cos 2 5 8 ] m 33 , a 3 = [ ( cos δ 1 1 ) sin δ 2 sin ( 10 4 4 3 2 5 ) 8 ] m 42 + [ ( cos δ 1 1 ) sin δ 2 cos 2 5 10 4 + 4 3 ) 8 ] m 43 , a 4 = [ sin δ 1 sin δ 2 cos ( 10 4 2 3 2 5 ) 4 ] m 44 , a 5 = [ sin δ 2 ( 2 5 10 4 ) 2 ] m 41 + [ ( cos δ 1 + 1 ) sin δ 2 sin ( 2 5 10 4 ) 4 ] m 42 , a 6 = [ sin δ 1 sin δ 2 cos ( 10 4 + 2 3 2 5 ) 4 ] m 44 , a 7 = [ ( cos δ 1 1 ) sin δ 2 sin ( 10 4 + 4 3 2 5 ) 8 ] m 42 + [ ( cos δ 1 1 ) sin δ 2 cos ( 10 4 + 4 3 2 5 ) 8 ] m 43 , a 8 = [ ( 1 cos δ 1 ) ( 1 cos δ 2 ) cos ( 20 4 4 3 2 5 ) 16 ] ( m 22 + m 33 ) + [ ( 1 cos δ 1 ) ( 1 cos δ 2 ) sin ( 20 4 4 3 2 5 ) 16 ] ( m 32 m 23 ) , a 9 = [ sin δ 1 ( 1 cos δ 2 ) sin ( 2 5 20 4 + 2 3 ) 8 ] m 24 + [ sin δ 1 ( 1 cos δ 2 ) cos ( 2 5 20 4 + 2 3 ) 8 ] m 34 , a 10 = [ ( 1 cos δ 2 ) cos ( 20 4 2 5 ) 4 ] m 21 + [ ( cos δ 1 + 1 ) ( 1 cos δ 2 ) cos ( 20 4 2 5 ) 8 ] m 22 + [ ( 1 cos δ 2 ) sin ( 20 4 2 5 ) 4 ] m 31 + [ ( cos δ 1 + 1 ) ( 1 cos δ 2 ) sin ( 20 4 2 5 ) 8 ] m 32 , a 11 = [ sin δ 1 ( cos δ 2 1 ) sin ( 2 5 20 4 2 3 ) 8 ] m 24 + [ sin δ 1 ( cos δ 2 1 ) cos ( 2 5 20 4 2 3 ) 8 ] m 34 , a 12 = [ ( 1 cos δ 1 ) ( 1 cos δ 2 ) cos ( 4 3 + 20 4 2 5 ) 16 ] ( m 22 m 33 ) + [ ( 1 cos δ 1 ) ( 1 cos δ 2 ) sin ( 4 3 + 20 4 2 5 ) 16 ] ( m 23 + m 32 ) , b 1 = [ sin δ 1 cos 2 3 ) 2 ] m 14 + [ sin δ 1 ( cos δ 2 + 1 ) cos 2 3 cos 2 5 ) 4 ] m 24 + [ sin δ 1 ( cos δ 2 + 1 ) cos 2 3 sin 2 5 ) 4 ] m 34 , b 2 = [ ( cos δ 1 1 ) sin 4 3 4 ] m 12 + [ ( 1 cos δ 1 ) cos 4 3 4 ] m 13 + [ ( 1 cos δ 1 ) ( 1 + cos δ 2 ) cos 4 3 cos 2 5 ) 8 ] m 23 + [ ( 1 cos δ 1 ) ( cos δ 2 + 1 ) sin 4 3 cos 2 5 8 ] m 22 + [ ( 1 cos δ 1 ) ( 1 + cos δ 2 ) cos 4 3 sin 2 5 8 ] m 33 + [ ( 1 cos δ 1 ) ( cos δ 2 + 1 ) sin 4 3 sin 2 5 8 ] m 32 , b 3 = [ ( cos δ 1 1 ) sin δ 2 cos ( 2 5 10 4 4 3 ) 8 ] m 42 [ ( cos δ 1 1 ) sin δ 2 sin ( 10 4 4 3 2 5 ) 8 ] m 43 , b 4 = [ sin δ 1 sin δ 2 sin ( 10 4 2 3 2 5 ) 4 ] m 44 , b 5 = [ sin δ 2 cos ( 10 4 2 5 ) 2 ] m 41 + [ ( cos δ 1 + 1 ) sin δ 2 cos ( 2 5 10 4 ) 4 ] m 42 , b 6 = [ sin δ 1 sin δ 2 sin ( 2 5 2 3 10 4 ) 4 ] m 44 , b 7 = [ ( cos δ 1 1 ) sin δ 2 cos ( 10 4 4 3 2 5 ) 8 ] m 42 + [ ( cos δ 1 1 ) sin δ 2 sin ( 10 4 + 4 3 2 5 ) 8 ] m 43 , b 8 = [ ( 1 cos δ 1 ) ( 1 cos δ 2 ) sin ( 20 4 4 3 2 5 ) 16 ] ( m 22 + m 33 ) + [ ( 1 cos δ 1 ) ( 1 cos δ 2 ) cos ( 20 4 4 3 2 5 ) 16 ] ( m 23 m 32 ) , b 9 = [ sin δ 1 ( cos δ 2 1 ) cos ( 20 4 2 3 2 5 ) 8 ] m 24 + [ sin δ 1 ( cos δ 2 1 ) sin ( 20 4 2 3 2 5 ) 8 ] m 34 , b 10 = [ ( cos δ 2 1 ) sin ( 20 4 2 5 ) 4 ] m 21 + [ ( cos δ 1 + 1 ) ( cos δ 2 1 ) sin ( 20 4 2 5 ) 8 ] m 22 + [ ( 1 cos δ 2 ) cos ( 20 4 2 5 ) 4 ] m 31 + [ ( cos δ 1 + 1 ) ( 1 cos δ 2 ) cos ( 20 4 2 5 ) 8 ] m 32 , b 11 = [ sin δ 1 ( 1 cos δ 2 ) sin ( 20 4 + 2 3 2 5 ) 8 ] m 24 + [ sin δ 1 ( 1 cos δ 2 ) sin ( 20 4 + 2 3 2 5 ) 8 ] m 34 , b 12 = [ ( 1 cos δ 1 ) ( 1 cos δ 2 ) cos ( 20 4 + 4 3 2 5 ) 16 ] ( m 22 m 33 ) + [ ( 1 cos δ 1 ) ( 1 cos δ 2 ) cos ( 20 4 + 4 3 2 5 ) 16 ] ( m 32 + m 23 ) .
m 11 = 4 a 0 ( 1 1 ) 2 m 12 ( 1 2 ) 2 m 21 ( 1 1 ) ( 1 2 ) 4 m 22 ( 1 2 ) 5 m 31 ( 1 1 ) ( 1 2 ) 5 2 m 32 , m 12 = ( 2 m 32 5 + m 22 ) ( 2 1 ) 2 + 8 ( a 2 4 3 b 2 ) ( 1 + 1 ) ( 16 3 2 + 1 ) , m 13 = ( 2 m 32 5 + m 23 ) ( 2 1 ) 2 + 32 3 a 2 + 8 b 2 ( 1 + 1 ) ( 16 3 2 + 1 ) , m 14 = 2 a 1 3 ( 1 2 ) 2 m 24 ( 1 2 ) 5 m 34 , m 21 = m 22 ( 1 1 ) 2 + 8 a 10 + 16 ( 5 10 4 ) b 10 ( 2 + 1 ) ( 4 5 2 80 4 5 + 400 4 2 + 1 ) m 22 = 16 [ ( 4 5 2 80 4 5 + 400 4 2 + 16 3 2 1 ) ( a 8 + a 12 ) + 16 3 ( 5 10 4 ) ( a 8 a 12 ) 4 3 ( 4 5 2 80 4 5 + 400 4 2 16 3 2 + 1 ) ( b 8 + b 12 ) + 2 ( 5 10 4 ) ( 4 5 2 80 4 5 + 400 4 2 16 3 2 1 ) ( b 8 b 12 ) ] / ( 1 + 1 ) ( 2 + 1 ) ( 2 5 20 4 4 3 1 ) ( 2 5 20 4 4 3 + 1 ) ( 2 5 20 4 + 4 3 1 ) ( 2 5 20 4 + 4 3 + 1 ) , m 23 = 16 [ 4 3 ( 4 5 2 80 4 5 + 400 4 2 16 3 2 + 1 ) ( a 8 + a 12 ) + 2 ( 5 10 4 ) ( 4 5 2 80 4 5 + 400 4 2 16 3 2 1 ) ( a 8 a 12 ) ( 4 5 2 80 4 5 + 400 4 2 + 16 3 2 1 ) ( b 8 + b 12 ) + 16 3 ( 5 10 4 ) ( b 8 b 12 ) ] / ( 1 + 1 ) ( 2 + 1 ) ( 2 5 20 4 4 3 1 ) ( 2 5 20 4 4 3 + 1 ) ( 2 5 20 4 + 4 3 1 ) ( 2 5 20 4 + 4 3 + 1 ) , m 24 = 8 ( b 11 b 9 ) ( 1 + 2 ) 8 ( 10 4 5 ) ( b 9 + b 11 ) 3 ( 1 + 2 ) , m 31 = m 32 ( 1 1 ) 2 16 ( 5 10 4 ) a 10 8 b 10 ( 2 + 1 ) ( 4 5 2 80 4 5 + 400 4 2 + 1 ) , m 32 = 16 [ 2 ( 5 10 4 ) ( 4 5 2 80 4 5 + 400 4 2 16 3 2 1 ) ( a 8 + a 12 ) 4 3 ( 4 5 2 80 4 5 + 400 4 2 16 3 2 + 1 ) ( a 8 a 12 ) 16 3 ( 5 10 4 ) ( b 8 + b 12 ) ( 4 5 2 80 4 5 + 400 4 2 + 16 3 2 1 ) ( b 8 b 12 ) / ( 1 + 1 ) ( 2 + 1 ) ( 2 5 20 4 4 3 1 ) ( 2 5 20 4 4 3 + 1 ) ( 2 5 20 4 + 4 3 1 ) ( 2 5 20 4 + 4 3 + 1 ) , m 33 = 16 [ 16 3 ( 5 10 4 ( a 8 + a 12 ) ( 4 5 2 80 4 5 + 400 4 2 + 16 3 2 1 ) ( a 8 a 12 ) + 2 ( 5 10 4 ) ( 4 5 2 80 4 5 + 400 4 2 16 3 2 1 ) ( b 8 + b 12 ) 4 3 ( 4 5 2 80 4 5 + 400 4 2 16 3 2 + 1 ) ( b 8 b 12 ) / ( 1 + 1 ) ( 2 + 1 ) ( 2 5 20 4 4 3 1 ) ( 2 5 20 4 4 3 + 1 ) ( 2 5 20 4 + 4 3 1 ) ( 2 5 20 4 + 4 3 + 1 ) , m 34 = 8 ( a 9 a 11 ) ( 1 + 2 ) 8 ( 5 10 4 ) ( a 9 + a 11 ) 3 ( 1 + 2 ) , m 41 = 8 ( 1 1 ) 3 ( b 3 b 7 ) ( 1 + 1 ) ( 5 4 5 ) + 4 ( 1 1 ) ( b 3 + b 7 ) ( 1 + 1 ) 4 b 5 , m 42 = 16 3 ( b 7 b 3 ) ( 1 + 1 ) ( 5 4 5 ) 8 ( b 3 b 7 ) ( 1 + 1 ) , m 43 = 8 ( a 7 a 3 ) ( 1 + 1 ) 16 3 ( a 3 + a 7 ) ( 1 + 1 ) 2 ( 5 4 5 ) , m 44 = 8 a 4 = 8 a 6 ,
[ 1 0 0.035 0 0 1 0.035 0 0 0.035 1 0 0 0 0 1 ]
[ 1 0 0 0 0 1 0.07 0 0.035 0.07 1 0 0 0 0 1 ]
[ 1 0.03 0.171 0 0 0.94 0.342 0 0 0.342 0.94 0 0 0 0 0.985 ]
[ 0.999 0 0.035 0 0 0.995 0.105 0 0.035 0.105 0.995 0 0 0 0 0.999 ]
[ 1 0.024 0.137 0 0 0.951 0.309 0 0 0.309 0.951 0 0 0 0 0.99 ]
[ 1.006 0.018 0.173 0 0.011 0.96 0.276 0 0.033 0.276 0.961 0 0 0 0 0.984 ]
[ 1.005 0.015 0.138 0 0 0.97 0.242 0 0.034 0.242 0.97 0 0 0 0 0.99 ]
[ 1 0 0 0 0.017 1.017 0 0 0 0 1.017 0 0 0 0 1 ]
[ 1 0 0.036 0 0.017 1.017 0.036 0 0 0.036 1.017 0 0 0 0 0.999 ]
[ 1 0 0 0 0.16 1.015 0.071 0 0.035 0.071 1.015 0 0 0 0 0.999 ]
[ 0.999 0.031 0.174 0 0.016 0.956 0.348 0 0 0.348 0.956 0 0 0 0 0.985 ]
[ 0.999 0.0 0.035 0 0.015 1.012 0.106 0 0.036 0.106 1.012 0 0 0 0 0.999 ]
[ 1.006 0.018 0.176 0 0.027 0.978 0.28 0 0.028 0.28 0.978 0 0 0 0 0.984 ]
[ 1 0.025 0.139 0 0.17 0.968 0.314 0 0 0.314 0.968 0 0 0 0 0.99 ]
[ 1.004 0.015 0.141 0 0.026 0.987 0.246 0 0.029 0.246 0.987 0 0 0 0 0.99 ]
[ 1 0.017 0 0 0 1.017 0 0 0 0 1.017 0 0 0 0 1 ]
[ 1 0.017 0.035 0 0 1.017 0.036 0 0 0.036 1.017 0 0 0 0 0.999 ]
[ 1 0.017 0 0 0 1.015 0.071 0 0.035 0.071 1.015 0 0 0 0 0.999 ]
[ 1 0.013 0.174 0 0 0.956 0.348 0 0 0.348 0.956 0 0 0 0 0.985 ]
[ 0.999 0.015 0.036 0 0 1.012 0.106 0 0.035 0.106 1.012 0 0 0 0 0.999 ]
[ 1 0 0.14 0 0 0.968 0.314 0 0 0.314 0.968 0 0 0 0 0.99 ]
[ 1.006 0 0.174 0 0.11 0.978 0.28 0 0.034 0.28 0.978 0 0 0 0 0.984 ]
[ 1.005 0 0.014 0 0 0.987 0.246 0 0.034 0.246 0.987 0 0 0 0 0.99 ]
[ 1 0.018 0 0 0.018 1.035 0 0 0 0 1.035 0 0 0 0 1 ]
[ 1 0.018 0.036 0 0.018 1.035 0.036 0 0 0.036 1.035 0 0 0 0 0.999 ]
[ 1 0.018 0 0 0.016 1.033 0.072 0 0.036 0.072 1.033 0 0 0 0 0.999 ]
[ 1 0.013 0.177 0 0.017 0.973 0.354 0 0 0.354 0.973 0 0 0 0 0.985 ]
[ 0.999 0.015 0.037 0 0.015 1.03 0.108 0 0.037 0.108 1.03 0 0 0 0 0.998 ]
[ 1 0 0.143 0 0.017 0.985 0.32 0 0 0.32 0.985 0 0 0 0 0.99 ]
[ 1.006 0 0.178 0 0.028 0.995 0.285 0 0.028 0.285 0.995 0 0 0 0 0.984 ]
[ 1.005 0 0.143 0 0.027 1.004 0.25 0 0.029 0.25 1.004 0 0 0 0 0.989 ]
1 2 [ 1 cos 2 θ sin 2 θ 0 cos 2 θ cos 2 2 θ cos 2 θ sin 2 θ 0 sin 2 θ cos 2 θ sin 2 θ sin 2 2 θ 0 0 0 0 0 ] ,
[ 1 0 0 0 0 cos 2 2 θ + sin 2 2 θ cos δ cos 2 θ sin 2 θ ( 1 cos δ ) sin 2 θ sin δ 0 cos 2 θ sin 2 θ ( 1 cos δ ) sin 2 2 θ + cos 2 2 θ cos δ cos 2 θ sin δ 0 sin 2 θ sin δ cos 2 θ sin δ cos δ ] .

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