Abstract

A computer-controlled Mueller matrix polarimeter with dual rotating retarders is described. Bulk properties of optical materials are determined by controlling the input-polarization state and measuring the output-polarization state. The Mueller matrix of a sample is obtained from polarimetric measurements, and polarization properties, i.e., diattenuation and retardance as well as depolarization, are extracted from the Mueller matrix. Further, fundamental electro- and magneto-optical material properties such as the electro-optical tensor coefficients may be obtained from Mueller matrices measured with applied fields. The polarimeter is currently configured to operate over the 3- to 12-μm spectral region.

© 1992 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North-Holland, Amsterdam, 1977), Chap. 3, p. 155.
  2. D. H. Goldstein, “Applications and limitations of polarimetry,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1317, 210–222 (1990).
  3. C. H. Ma, D. P. Hutchinson, P. A. Staats, K. L. Vander Sluis, “FIR interferometer/polarimeter system on ISX-B tokamak,” Rev. Sci. Instrum. 56, 911–913 (1985).
    [CrossRef]
  4. E. S. Yeung, “Laser-based polarimetry enhances biochemical detection,” Laser Focus/Electro-Optics 21(2), 30–40 (1985).
  5. O. Brevet-Philibert, R. Brunetton, J. Monin, “Measuring the Verdet constant: a simple, high precision, automatic device,” J. Phys. E. 21, 647–649 (1988).
    [CrossRef]
  6. A. Gedanken, M. Tamir, “Multiphoton optical rotary dispersion,” Rev. Sci. Instrum. 58, 950–952 (1987).
    [CrossRef]
  7. J. W. Wagner, J. B. Deaton, E. M. Hackett, “Laser polarimeter for measuring angular displacement of bend bars,” Exp. Mech. 28, 45–49 (1988).
    [CrossRef]
  8. R. Lipeles, D. Kivelson, “Experimental studies of acoustically induced birefringence,” J. Chem. Phys. 72, 6199–6208 (1980).
    [CrossRef]
  9. R. C. Thompson, J. R. Bottiger, E. S. Fry, “Measurement of polarized light interactions via the Mueller matrix,” Appl. Opt. 19, 1323–1332 (1980).
    [CrossRef] [PubMed]
  10. 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]
  11. P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
    [CrossRef]
  12. H. B. klein Brink, “Birefringence of the human crystalline lens in vivo,” J. Opt. Soc. Am. A 8, 1788–1793 (1991).
    [CrossRef]
  13. H. B. klein Brink, G. J. van Blokland, “Birefringence of the human foveal area assessed in vivo with Mueller-matrix ellipsometry,” J. Opt. Soc. Am. A 5, 49–57 (1988).
    [CrossRef]
  14. G. J. van Blokland, “Ellipsometry of the human retina in vivo: preservation of polarization,” J. Opt. Soc. Am. A 2, 72–75 (1985).
    [CrossRef] [PubMed]
  15. P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), Chap. 4, p. 60, 64.
  16. W. Shurcliff, Polarized Light (Oxford U. Press, London, 1962), App. 2, pp. 166–170.
  17. G. Strang, Linear Algebra and Its Applications, 2nd ed. (Academic, New York, 1976), Chap. 3, p. 112.
  18. D. H. Goldstein, R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7, 693–700 (1990).
    [CrossRef]
  19. D. H. Goldstein, R. A. Chipman, D. B. Chenault, R. R. Hodgson, “Infrared material properties measurements with polarimetry and in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 448–462 (1990).
  20. C. S. Namba, “Electro-optical effect of zincblende,” J. Opt. Soc. Am. 51, 76–79 (1961).
    [CrossRef]
  21. D. B. Chenault, University of Alabama at Huntsville, Huntsville, Ala. 35899 (personal communication).
  22. D. H. Goldstein, R. A. Chipman, “Infrared spectropolarimeter,” U.S. patent5,045,701 (3September1991).

1991 (1)

1990 (1)

1988 (3)

O. Brevet-Philibert, R. Brunetton, J. Monin, “Measuring the Verdet constant: a simple, high precision, automatic device,” J. Phys. E. 21, 647–649 (1988).
[CrossRef]

J. W. Wagner, J. B. Deaton, E. M. Hackett, “Laser polarimeter for measuring angular displacement of bend bars,” Exp. Mech. 28, 45–49 (1988).
[CrossRef]

H. B. klein Brink, G. J. van Blokland, “Birefringence of the human foveal area assessed in vivo with Mueller-matrix ellipsometry,” J. Opt. Soc. Am. A 5, 49–57 (1988).
[CrossRef]

1987 (1)

A. Gedanken, M. Tamir, “Multiphoton optical rotary dispersion,” Rev. Sci. Instrum. 58, 950–952 (1987).
[CrossRef]

1985 (3)

C. H. Ma, D. P. Hutchinson, P. A. Staats, K. L. Vander Sluis, “FIR interferometer/polarimeter system on ISX-B tokamak,” Rev. Sci. Instrum. 56, 911–913 (1985).
[CrossRef]

E. S. Yeung, “Laser-based polarimetry enhances biochemical detection,” Laser Focus/Electro-Optics 21(2), 30–40 (1985).

G. J. van Blokland, “Ellipsometry of the human retina in vivo: preservation of polarization,” J. Opt. Soc. Am. A 2, 72–75 (1985).
[CrossRef] [PubMed]

1980 (2)

R. Lipeles, D. Kivelson, “Experimental studies of acoustically induced birefringence,” J. Chem. Phys. 72, 6199–6208 (1980).
[CrossRef]

R. C. Thompson, J. R. Bottiger, E. S. Fry, “Measurement of polarized light interactions via the Mueller matrix,” Appl. Opt. 19, 1323–1332 (1980).
[CrossRef] [PubMed]

1978 (2)

1961 (1)

Azzam, R. M. A.

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]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North-Holland, Amsterdam, 1977), Chap. 3, p. 155.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North-Holland, Amsterdam, 1977), Chap. 3, p. 155.

Bottiger, J. R.

Brevet-Philibert, O.

O. Brevet-Philibert, R. Brunetton, J. Monin, “Measuring the Verdet constant: a simple, high precision, automatic device,” J. Phys. E. 21, 647–649 (1988).
[CrossRef]

Brunetton, R.

O. Brevet-Philibert, R. Brunetton, J. Monin, “Measuring the Verdet constant: a simple, high precision, automatic device,” J. Phys. E. 21, 647–649 (1988).
[CrossRef]

Chenault, D. B.

D. B. Chenault, University of Alabama at Huntsville, Huntsville, Ala. 35899 (personal communication).

D. H. Goldstein, R. A. Chipman, D. B. Chenault, R. R. Hodgson, “Infrared material properties measurements with polarimetry and in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 448–462 (1990).

Chipman, R. A.

D. H. Goldstein, R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7, 693–700 (1990).
[CrossRef]

D. H. Goldstein, R. A. Chipman, D. B. Chenault, R. R. Hodgson, “Infrared material properties measurements with polarimetry and in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 448–462 (1990).

D. H. Goldstein, R. A. Chipman, “Infrared spectropolarimeter,” U.S. patent5,045,701 (3September1991).

Deaton, J. B.

J. W. Wagner, J. B. Deaton, E. M. Hackett, “Laser polarimeter for measuring angular displacement of bend bars,” Exp. Mech. 28, 45–49 (1988).
[CrossRef]

Fry, E. S.

Gdoutos, E. E.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), Chap. 4, p. 60, 64.

Gedanken, A.

A. Gedanken, M. Tamir, “Multiphoton optical rotary dispersion,” Rev. Sci. Instrum. 58, 950–952 (1987).
[CrossRef]

Goldstein, D. H.

D. H. Goldstein, R. A. Chipman, “Error analysis of a Mueller matrix polarimeter,” J. Opt. Soc. Am. A 7, 693–700 (1990).
[CrossRef]

D. H. Goldstein, R. A. Chipman, “Infrared spectropolarimeter,” U.S. patent5,045,701 (3September1991).

D. H. Goldstein, “Applications and limitations of polarimetry,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1317, 210–222 (1990).

D. H. Goldstein, R. A. Chipman, D. B. Chenault, R. R. Hodgson, “Infrared material properties measurements with polarimetry and in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 448–462 (1990).

Hackett, E. M.

J. W. Wagner, J. B. Deaton, E. M. Hackett, “Laser polarimeter for measuring angular displacement of bend bars,” Exp. Mech. 28, 45–49 (1988).
[CrossRef]

Hauge, P. S.

Hodgson, R. R.

D. H. Goldstein, R. A. Chipman, D. B. Chenault, R. R. Hodgson, “Infrared material properties measurements with polarimetry and in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 448–462 (1990).

Hutchinson, D. P.

C. H. Ma, D. P. Hutchinson, P. A. Staats, K. L. Vander Sluis, “FIR interferometer/polarimeter system on ISX-B tokamak,” Rev. Sci. Instrum. 56, 911–913 (1985).
[CrossRef]

Kivelson, D.

R. Lipeles, D. Kivelson, “Experimental studies of acoustically induced birefringence,” J. Chem. Phys. 72, 6199–6208 (1980).
[CrossRef]

klein Brink, H. B.

Lipeles, R.

R. Lipeles, D. Kivelson, “Experimental studies of acoustically induced birefringence,” J. Chem. Phys. 72, 6199–6208 (1980).
[CrossRef]

Ma, C. H.

C. H. Ma, D. P. Hutchinson, P. A. Staats, K. L. Vander Sluis, “FIR interferometer/polarimeter system on ISX-B tokamak,” Rev. Sci. Instrum. 56, 911–913 (1985).
[CrossRef]

Monin, J.

O. Brevet-Philibert, R. Brunetton, J. Monin, “Measuring the Verdet constant: a simple, high precision, automatic device,” J. Phys. E. 21, 647–649 (1988).
[CrossRef]

Namba, C. S.

Shurcliff, W.

W. Shurcliff, Polarized Light (Oxford U. Press, London, 1962), App. 2, pp. 166–170.

Staats, P. A.

C. H. Ma, D. P. Hutchinson, P. A. Staats, K. L. Vander Sluis, “FIR interferometer/polarimeter system on ISX-B tokamak,” Rev. Sci. Instrum. 56, 911–913 (1985).
[CrossRef]

Strang, G.

G. Strang, Linear Algebra and Its Applications, 2nd ed. (Academic, New York, 1976), Chap. 3, p. 112.

Tamir, M.

A. Gedanken, M. Tamir, “Multiphoton optical rotary dispersion,” Rev. Sci. Instrum. 58, 950–952 (1987).
[CrossRef]

Theocaris, P. S.

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), Chap. 4, p. 60, 64.

Thompson, R. C.

van Blokland, G. J.

Vander Sluis, K. L.

C. H. Ma, D. P. Hutchinson, P. A. Staats, K. L. Vander Sluis, “FIR interferometer/polarimeter system on ISX-B tokamak,” Rev. Sci. Instrum. 56, 911–913 (1985).
[CrossRef]

Wagner, J. W.

J. W. Wagner, J. B. Deaton, E. M. Hackett, “Laser polarimeter for measuring angular displacement of bend bars,” Exp. Mech. 28, 45–49 (1988).
[CrossRef]

Yeung, E. S.

E. S. Yeung, “Laser-based polarimetry enhances biochemical detection,” Laser Focus/Electro-Optics 21(2), 30–40 (1985).

Appl. Opt. (1)

Exp. Mech. (1)

J. W. Wagner, J. B. Deaton, E. M. Hackett, “Laser polarimeter for measuring angular displacement of bend bars,” Exp. Mech. 28, 45–49 (1988).
[CrossRef]

J. Chem. Phys. (1)

R. Lipeles, D. Kivelson, “Experimental studies of acoustically induced birefringence,” J. Chem. Phys. 72, 6199–6208 (1980).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

J. Phys. E. (1)

O. Brevet-Philibert, R. Brunetton, J. Monin, “Measuring the Verdet constant: a simple, high precision, automatic device,” J. Phys. E. 21, 647–649 (1988).
[CrossRef]

Laser Focus/Electro-Optics (1)

E. S. Yeung, “Laser-based polarimetry enhances biochemical detection,” Laser Focus/Electro-Optics 21(2), 30–40 (1985).

Opt. Lett. (1)

Rev. Sci. Instrum. (2)

C. H. Ma, D. P. Hutchinson, P. A. Staats, K. L. Vander Sluis, “FIR interferometer/polarimeter system on ISX-B tokamak,” Rev. Sci. Instrum. 56, 911–913 (1985).
[CrossRef]

A. Gedanken, M. Tamir, “Multiphoton optical rotary dispersion,” Rev. Sci. Instrum. 58, 950–952 (1987).
[CrossRef]

Other (8)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North-Holland, Amsterdam, 1977), Chap. 3, p. 155.

D. H. Goldstein, “Applications and limitations of polarimetry,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1317, 210–222 (1990).

P. S. Theocaris, E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, Berlin, 1979), Chap. 4, p. 60, 64.

W. Shurcliff, Polarized Light (Oxford U. Press, London, 1962), App. 2, pp. 166–170.

G. Strang, Linear Algebra and Its Applications, 2nd ed. (Academic, New York, 1976), Chap. 3, p. 112.

D. H. Goldstein, R. A. Chipman, D. B. Chenault, R. R. Hodgson, “Infrared material properties measurements with polarimetry and in Electro-Optical Materials for Switches, Coatings, Sensor Optics, and Detectors, R. Hartmann, M. J. Soileau, V. K. Varadan, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1307, 448–462 (1990).

D. B. Chenault, University of Alabama at Huntsville, Huntsville, Ala. 35899 (personal communication).

D. H. Goldstein, R. A. Chipman, “Infrared spectropolarimeter,” U.S. patent5,045,701 (3September1991).

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Figures (8)

Fig. 1
Fig. 1

Polarimeter block diagram. The Mueller matrix of the sample is determined from the modulated intensity measured by the detector.

Fig. 2
Fig. 2

Polarimeter optics diagram: LA, laser; D, detector; PZ, polarizer; R1, R2, retarders; S, sample; CH, chopper; IS, integrating sphere; B, beam splitter; C, computer; RSC, rotary stage controller; DM, digital multimeter.

Fig. 3
Fig. 3

Polarizing elements and rotation rates. A laser source L is directed through fixed polarizers P1 and P2, rotating retarders R1 and R2, and the sample S, to a detector.

Fig. 4
Fig. 4

Retardation errors ∊1 and ∊2, and orientation errors ∊3, ∊4, and ∊5: P1, P2, fixed polarizers; R1, R2, rotating retarders; S, sample; f, fast axis.

Fig. 5
Fig. 5

Modulated intensity with no sample in the polarimeter. The first retarder is oriented at (n − 1) × 5°, n = 0, 1, … 35. The second retarder is oriented at (n − 1) × 25°. The Mueller matrix is determined from the Fourier coefficients of this signal.

Fig. 6
Fig. 6

Modulated intensity with horizontal linear polarizer as the polarimeter sample.

Fig. 7
Fig. 7

Modulated intensity with vertical linear polarizer as the polarimeter sample.

Fig. 8
Fig. 8

Modulated intensity with half-wave plate and the fast axis at 45° as the polarimeter sample.

Equations (33)

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P 2 R 2 ( θ ) M R 1 ( θ ) P 1 ,
I = c AMP ,
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 .
[ a 1 a 2 a 3 a 4 · · · · · · · · · · · · ] [ m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 ] [ p 1 p 2 p 3 p 4 ] = [ I · · · ] ,
I = a 1 ( m 11 p 1 + m 12 p 2 + m 13 p 3 + m 14 p 4 ) + a 2 ( m 21 p 1 + m 22 p 2 + m 23 p 3 + m 24 p 4 ) + a 3 ( m 31 p 1 + m 32 p 2 + m 33 p 3 + m 34 p 4 ) + a 4 ( m 41 p 1 + m 42 p 2 + m 43 p 3 + m 44 p 4 ) = i , j = 1 4 μ i j m i j .
μ 11 = 1 , μ 12 = cos 2 2 θ , μ 13 = sin 2 θ cos 2 θ , μ 14 = sin 2 θ , μ 21 = cos 2 10 θ , μ 22 = cos 2 2 θ cos 2 10 θ , μ 23 = sin 2 θ cos 2 θ cos 2 10 θ , μ 24 = sin 2 θ cos 2 10 θ , μ 31 = sin 10 θ cos 10 θ , μ 32 = cos 2 2 θ sin 10 θ cos 10 θ , μ 33 = sin 2 θ cos 2 θ sin 10 θ cos 10 θ , μ 34 = sin 2 θ sin 10 θ cos 10 θ , μ 41 = - sin 10 θ , μ 42 = - cos 2 2 θ sin 10 θ , μ 43 = - sin 2 θ cos 2 θ sin 10 θ , μ 44 = - sin 2 θ sin 10 θ .
m 11 = a 0 - a 2 + a 8 - a 10 + a 12 , m 12 = 2 a 2 - 2 a 8 - 2 a 12 , m 13 = 2 b 2 + 2 b 8 - 2 b 12 , m 14 = b 1 - 2 b 11 = b 1 + 2 b 9 = b 1 + b 9 - b 11 , m 21 = - 2 a 8 + 2 a 10 - 2 a 12 , m 22 = 4 a 8 + 4 a 12 , m 23 = - 4 b 8 + 4 b 12 , m 24 = - 4 b 9 = 4 b 11 = 2 ( - b 9 + b 11 ) , m 31 = - 2 b 8 + 2 b 10 - 2 b 12 , m 32 = 4 b 8 + 4 b 12 , m 33 = 4 a 8 - 4 a 12 , m 34 = 4 a 9 = - 4 a 11 = 2 ( a 9 - a 11 ) , m 41 = 2 b 3 - b 5 = - b 5 + 2 b 7 = ( b 3 - b 5 + b 7 ) , m 42 = - 4 b 3 = - 4 b 7 = - 2 ( b 3 + b 7 ) , m 43 = - 4 a 3 = 4 a 7 = 2 ( - a 3 + a 7 ) , m 44 = - 2 a 4 = 2 a 6 = ( a 6 - a 4 ) .
x a = I ,
( 1 cos 2 θ cos 4 θ cos 24 θ sin 2 θ sin 4 θ sin 24 θ )
a = ( x T x ) - 1 x T I .
I ( θ ) = a o + j = 1 12 ( a j cos 2 j θ + b j sin 2 j θ ) ,
l = 0 35 [ Φ ( θ l ) - I ( θ l ) ] 2 = E ( a 0 , a 1 , , a 12 , b 1 , , b 12 ) ,
δ E δ a k = 0 ,             δ E δ b k = 0.
l = 0 35 { Φ ( θ l ) - [ a 0 + j = 1 12 ( a j cos 2 j θ l + b j sin 2 j θ l ) ] } × ( - 2 cos 2 k θ l ) = 0.
a 0 = 1 4 + ( 1 - 1 ) ( 1 - 2 ) 16 , a 2 = ( 1 + 1 ) ( 1 - 2 ) 16 + ( 1 + 1 ) ( 1 - 2 ) 3 5 2 , a 8 = ( 1 + 1 ) ( 1 + 2 ) 16 , a 10 = ( 1 - 1 ) ( 1 + 2 ) 16 , b 2 = - ( 1 + 1 ) ( 1 - 2 ) 3 4 + ( 1 + 1 ) ( 1 - 2 ) 5 8 , b 4 = ( 4 - 3 - 5 ) 4 , b 6 = ( 5 - 3 - 4 ) 4 , b 8 = ( 1 + 1 ) ( 1 + 2 ) ( 2 4 - 2 3 - 5 ) 8 , b 10 = - ( 1 - 1 ) ( 1 + 2 ) ( 2 4 - 5 ) 8 .
1 = 3 - 8 ( a 0 + a 10 ) ,             2 = 4 ( a 0 - a 10 ) - 1 1 - 4 ( a 0 - a 10 ) .
1 = ( a 8 - a 10 ) ( a 8 - a 10 ) ,             2 = 8 ( a 8 + a 10 ) - 1.
3 = - 2 ( b 4 + b 6 ) .
5 = 8 b 2 ( 1 + 1 ) ( 1 - 2 ) + 2 3 ,
4 = 5 + 2 ( b 4 - b 6 ) .
x 2 + y 2 + z 2 n o 2 + 2 r 41 E ( y z + z x ) = 1.
n x = n o + ½ n o 3 r 41 E , n y = n o - ½ n o 3 r 41 E , n z = n o .
Γ = 2 π ( n a - n b ) L / λ ,
Γ = 2 π ( n y - n x ) L / λ ,
Γ cubic = 2 π n o 3 r 41 E L / λ .
Γ cubic long = 2 π n o 3 r 41 V / λ ,
Γ cubic trans = 2 π n o 3 r 41 V L / d λ .
[ 1 0 0 0 0 cos 2 2 θ + sin 2 2 θ cos δ ( 1 - cos δ ) sin 2 θ cos 2 θ - sin 2 θ sin δ 0 ( 1 - cos δ ) sin 2 θ cos 2 θ sin 2 2 θ + cos 2 2 θ cos δ cos 2 θ sin δ 0 sin 2 θ sin δ - cos 2 θ cos δ cos δ ] ,
[ 1 0 0 0 0 1 0 0 0 0 cos 2 π λ n 3 r 41 V L d sin 2 π λ n 3 r 41 V L d 0 0 - sin 2 π λ n 3 r 41 V L d cos 2 π λ n 3 r 41 V L d ] .
[ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] , [ 0.998 0.026 0.019 - 0.002 0.002 0.976 - 0.030 0.009 0.007 0.033 0.966 - 0.002 0.002 - 0.004 - 0.002 1.000 ] .
[ 0.997 - 0.006 0.004 0.002 0.007 1.000 - 0.007 0.009 0.008 - 0.007 0.990 - 0.003 0.003 - 0.006 - 0.007 0.998 ] .

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