Abstract

We introduce a technique that is capable of obtaining complete polarization-sensitive three-dimensional images that could reveal unknown anatomical conditions of living tissue that possess polarization- dependent signatures. Previously, the 16 Mueller coefficients were measured independently only by use of two-dimensional imaging techniques. We also present the experimental combination of a depth-resolved confocal imaging system with a complete Mueller matrix polarimeter. To calibrate the system, a double-pass method had to be implemented. We also indicate, experimentally, that the confocal sectioning of the system has a degrading effect on axially resolved Mueller matrix measurements.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  34. P. Török, P. Higdon, and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
    [CrossRef]
  35. P. Török, "Imaging of small birefringent objects by polarised light conventional and confocal microscopes," Opt. Commun. 181, 7-18 (2000).
    [CrossRef]
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    [CrossRef]
  41. J. W. Hovenier and C. V. M. van der Mee, "Testing scattering matrices: a compendium of recipes," J. Quant. Spectrosc. Radiat. Transfer 55, 649-661 (1996).
    [CrossRef]
  42. J. J. Gil and E. Bernabeu, "A depolarization criterion in Mueller matrices," Opt. Acta 32, 259-261 (1985).
    [CrossRef]
  43. J. J. Gil and E. Bernabeu, "Depolarization and polarization indices of an optical system," Opt. Acta 33, 185-189 (1986).
    [CrossRef]
  44. C. V. M. van der Mee, "An eigenvalue criterion for matrices transforming Stokes parameters," J. Math. Phys. 34, 5072-5088 (1993).
    [CrossRef]

2004 (2)

M. Pircher, E. Goetzinger, R. Leitgeb, and C. K. Hitzenberger, "Transversal phase resolved polarization sensitive optical coherence tomography," Phys. Med. Biol. 49, 1257-1263 (2004).
[CrossRef] [PubMed]

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).
[CrossRef]

2003 (2)

2002 (5)

2000 (3)

S. Jiao, G. Yao, and L. V. Wang, "Depth-resolved two-dimensional Stokes vector of backscattered light and Mueller matrices of biological tissue measured with optical coherence tomography," Appl. Opt. 39, 6318-6324 (2000).
[CrossRef]

P. Török, "Imaging of small birefringent objects by polarised light conventional and confocal microscopes," Opt. Commun. 181, 7-18 (2000).
[CrossRef]

J. M. Bueno, "Polarimetry using liquid-crystal variable retarders: theory and calibration," J. Optics A Pure Appl. Opt. 2, 216-222 (2000).
[CrossRef]

1999 (4)

1998 (4)

K. Schoenenberger, B. W. Colston, D. J. Maitland, L. B. D. Silva, and M. J. Everett, "Mapping of birefringence and thermal damage in tissue by use of polarization-sensitive optical coherence tomography," Appl. Opt. 37, 6026-6036 (1998).
[CrossRef]

E. Compain and B. Drevillon, "High-frequency modulation of the four states of polarization of light with a single phase modulator," R. Sci. Instrum. 69, 1574-1580 (1998).
[CrossRef]

E. Compain and B. Drevillon, "Broadband division-of-amplitude polarimeter based on uncoated prisms," Appl. Opt. 37, 5938-5944 (1998).
[CrossRef]

P. Török, P. Higdon, and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

1997 (4)

E. Compain and B. Drevillon, "Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator," Rev. Sci. Instrum. 68, 2671-2680 (1997).
[CrossRef]

P. Török and T. Wilson, "Rigorous theory for axial resolution in confocal microscopes," Opt. Commun. 137, 127-135 (1997).
[CrossRef]

J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, "Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography," Opt. Lett. 22, 934-936 (1997).
[CrossRef] [PubMed]

F. Delplancke, "Automated high-speed Mueller matrix scatterometer," Appl. Opt. 36, 5388-5395 (1997).
[CrossRef] [PubMed]

1996 (5)

1993 (1)

C. V. M. van der Mee, "An eigenvalue criterion for matrices transforming Stokes parameters," J. Math. Phys. 34, 5072-5088 (1993).
[CrossRef]

1992 (2)

1986 (1)

J. J. Gil and E. Bernabeu, "Depolarization and polarization indices of an optical system," Opt. Acta 33, 185-189 (1986).
[CrossRef]

1985 (2)

R. M. A. Azzam, "Beam-splitters for the division-of-amplitude photopolarimeter," Opt. Acta 32, 1404-1412 (1985).
[CrossRef]

J. J. Gil and E. Bernabeu, "A depolarization criterion in Mueller matrices," Opt. Acta 32, 259-261 (1985).
[CrossRef]

1982 (1)

R. M. A. Azzam, "Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all Stokes parameters of light," Opt. Acta 29, 685-689 (1982).
[CrossRef]

1978 (1)

R. M. A. Azzam, "Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4 × 4 matrix calculus," J. Opt. Soc. Am. A 68, 1756-1767 (1978).
[CrossRef]

1948 (1)

Abushagur, M. A. G.

Andrienko, Y. A.

Atchinson, D. A.

G. Smith and D. A. Atchinson, The Eye and Visual Optical Instruments (Cambridge U. Press, 1997).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and A. De, "Optimal beam splitters for the division-of-amplitude photopolarimeter," J. Opt. Soc. Am. A 20, 955-958 (2003).
[CrossRef]

A. El-Saba, R. M. A. Azzam, and M. A. G. Abushagur, "Performance optimization and light beam deviation analysis of the parallel-slab division-of-amplitude photopolarimeter," Appl. Opt. 38, 2829-2836 (1999).
[CrossRef]

Y. Cui and R. M. A. Azzam, "Sixteen-beam grating-based division-of-amplitude photopolarimeter," Opt. Lett. 21, 89-91 (1996).
[CrossRef] [PubMed]

A. M. El-Saba, R. M. A. Azzam, and M. A. G. Abushagur, "Parallel-slab division-of-amplitude photopolarimeter," Opt. Lett. 21, 1709-1711 (1996).
[CrossRef] [PubMed]

R. M. A. Azzam, "Beam-splitters for the division-of-amplitude photopolarimeter," Opt. Acta 32, 1404-1412 (1985).
[CrossRef]

R. M. A. Azzam, "Division-of-amplitude photopolarimeter (DOAP) for the simultaneous measurement of all Stokes parameters of light," Opt. Acta 29, 685-689 (1982).
[CrossRef]

R. M. A. Azzam, "Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4 × 4 matrix calculus," J. Opt. Soc. Am. A 68, 1756-1767 (1978).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Bernabeu, E.

J. J. Gil and E. Bernabeu, "Depolarization and polarization indices of an optical system," Opt. Acta 33, 185-189 (1986).
[CrossRef]

J. J. Gil and E. Bernabeu, "A depolarization criterion in Mueller matrices," Opt. Acta 32, 259-261 (1985).
[CrossRef]

Bueno, J. M.

J. M. Bueno, "Polarimetry using liquid-crystal variable retarders: theory and calibration," J. Optics A Pure Appl. Opt. 2, 216-222 (2000).
[CrossRef]

Cense, B.

Chao, L. C.

M. G. Ducros, J. F. de Boer, H.-E. Huang, L. C. Chao, Z. Chen, J. S. Nelson, T. E. Milner, and H. G. Rylander, "Polarization sensitive optical coherence tomography of the rabbit eye," IEEE J. Sel. Top. Quantum Electron. 5, 1159-1167 (1999).
[CrossRef]

Charette, P. G.

P. M. F. Nielsen, F. N. Reinholz, and P. G. Charette, "Polarization-sensitive scanned fiber confocal microscope," Opt. Eng. 35, 3084-3091 (1996).
[CrossRef]

Chen, T. C.

Chen, Z.

M. G. Ducros, J. F. de Boer, H.-E. Huang, L. C. Chao, Z. Chen, J. S. Nelson, T. E. Milner, and H. G. Rylander, "Polarization sensitive optical coherence tomography of the rabbit eye," IEEE J. Sel. Top. Quantum Electron. 5, 1159-1167 (1999).
[CrossRef]

Chipman, R. A.

S.-Y. Lu and R. A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996).
[CrossRef]

R. A. Chipman, "Polarimetry," in Handbook of Optics, M.Bass, ed., (McGraw-Hill, 1995), Vol. 2, Chap. 22, pp. 22.1-22.37.

Colston, B. W.

Compain, E.

E. Compain, S. Poirier, and B. Drevillon, "General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers," Appl. Opt. 38, 3490-3502 (1999).
[CrossRef]

E. Compain and B. Drevillon, "High-frequency modulation of the four states of polarization of light with a single phase modulator," R. Sci. Instrum. 69, 1574-1580 (1998).
[CrossRef]

E. Compain and B. Drevillon, "Broadband division-of-amplitude polarimeter based on uncoated prisms," Appl. Opt. 37, 5938-5944 (1998).
[CrossRef]

E. Compain and B. Drevillon, "Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator," Rev. Sci. Instrum. 68, 2671-2680 (1997).
[CrossRef]

Cui, Y.

De, A.

de Boer, J. F.

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, "In vivo depth-resolved birefringence measurements of the human retinal nerve fiber layer by polarization-sensitive optical coherence tomography," Opt. Lett. 27, 1610-1612 (2002).
[CrossRef]

J. F. de Boer and T. E. Milner, "Review of polarization sensitive optical coherence tomography and Stokes vector determination," J. Biomed. Opt. 7, 359-371 (2002).
[CrossRef] [PubMed]

M. G. Ducros, J. F. de Boer, H.-E. Huang, L. C. Chao, Z. Chen, J. S. Nelson, T. E. Milner, and H. G. Rylander, "Polarization sensitive optical coherence tomography of the rabbit eye," IEEE J. Sel. Top. Quantum Electron. 5, 1159-1167 (1999).
[CrossRef]

J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, "Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography," Opt. Lett. 22, 934-936 (1997).
[CrossRef] [PubMed]

Delplancke, F.

Drevillon, B.

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).
[CrossRef]

A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drevillon, "Optimized Mueller polarimeter with liquid crystals," Opt. Lett. 28, 616-618 (2003).
[CrossRef] [PubMed]

E. Compain, S. Poirier, and B. Drevillon, "General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers," Appl. Opt. 38, 3490-3502 (1999).
[CrossRef]

E. Compain and B. Drevillon, "High-frequency modulation of the four states of polarization of light with a single phase modulator," R. Sci. Instrum. 69, 1574-1580 (1998).
[CrossRef]

E. Compain and B. Drevillon, "Broadband division-of-amplitude polarimeter based on uncoated prisms," Appl. Opt. 37, 5938-5944 (1998).
[CrossRef]

E. Compain and B. Drevillon, "Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator," Rev. Sci. Instrum. 68, 2671-2680 (1997).
[CrossRef]

Dubovikov, M. S.

Ducros, M. G.

M. G. Ducros, J. F. de Boer, H.-E. Huang, L. C. Chao, Z. Chen, J. S. Nelson, T. E. Milner, and H. G. Rylander, "Polarization sensitive optical coherence tomography of the rabbit eye," IEEE J. Sel. Top. Quantum Electron. 5, 1159-1167 (1999).
[CrossRef]

El-Saba, A.

El-Saba, A. M.

Everett, M. J.

Garcia-Caurel, E.

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).
[CrossRef]

A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drevillon, "Optimized Mueller polarimeter with liquid crystals," Opt. Lett. 28, 616-618 (2003).
[CrossRef] [PubMed]

Gil, J. J.

J. J. Gil and E. Bernabeu, "Depolarization and polarization indices of an optical system," Opt. Acta 33, 185-189 (1986).
[CrossRef]

J. J. Gil and E. Bernabeu, "A depolarization criterion in Mueller matrices," Opt. Acta 32, 259-261 (1985).
[CrossRef]

Gladun, A. D.

Goetzinger, E.

M. Pircher, E. Goetzinger, R. Leitgeb, and C. K. Hitzenberger, "Transversal phase resolved polarization sensitive optical coherence tomography," Phys. Med. Biol. 49, 1257-1263 (2004).
[CrossRef] [PubMed]

Higdon, P.

P. Török, P. Higdon, and T. Wilson, "On the general properties of polarised light conventional and confocal microscopes," Opt. Commun. 148, 300-315 (1998).
[CrossRef]

Hitzenberger, C. K.

M. Pircher, E. Goetzinger, R. Leitgeb, and C. K. Hitzenberger, "Transversal phase resolved polarization sensitive optical coherence tomography," Phys. Med. Biol. 49, 1257-1263 (2004).
[CrossRef] [PubMed]

Hovenier, J. W.

J. W. Hovenier and C. V. M. van der Mee, "Testing scattering matrices: a compendium of recipes," J. Quant. Spectrosc. Radiat. Transfer 55, 649-661 (1996).
[CrossRef]

Huang, H.-E.

M. G. Ducros, J. F. de Boer, H.-E. Huang, L. C. Chao, Z. Chen, J. S. Nelson, T. E. Milner, and H. G. Rylander, "Polarization sensitive optical coherence tomography of the rabbit eye," IEEE J. Sel. Top. Quantum Electron. 5, 1159-1167 (1999).
[CrossRef]

Inoué, S.

S. Inoué, "Foundations of confocal scanned imaging in light microscopy," in Handbook of Biological Confocal Microscopy 2nd ed., J.B.Pawley, ed. (Plenum, 1995), Chap. 1, pp. 1-17.

Jiao, S.

Jones, R. C.

Kim, Y.-K.

Krishnan, S.

Laude, B.

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).
[CrossRef]

A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drevillon, "Optimized Mueller polarimeter with liquid crystals," Opt. Lett. 28, 616-618 (2003).
[CrossRef] [PubMed]

Leitgeb, R.

M. Pircher, E. Goetzinger, R. Leitgeb, and C. K. Hitzenberger, "Transversal phase resolved polarization sensitive optical coherence tomography," Phys. Med. Biol. 49, 1257-1263 (2004).
[CrossRef] [PubMed]

Lu, S.-Y.

Maitland, D. J.

Martino, A. De

A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drevillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456, 112-119 (2004).
[CrossRef]

A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drevillon, "Optimized Mueller polarimeter with liquid crystals," Opt. Lett. 28, 616-618 (2003).
[CrossRef] [PubMed]

Milner, T. E.

J. F. de Boer and T. E. Milner, "Review of polarization sensitive optical coherence tomography and Stokes vector determination," J. Biomed. Opt. 7, 359-371 (2002).
[CrossRef] [PubMed]

M. G. Ducros, J. F. de Boer, H.-E. Huang, L. C. Chao, Z. Chen, J. S. Nelson, T. E. Milner, and H. G. Rylander, "Polarization sensitive optical coherence tomography of the rabbit eye," IEEE J. Sel. Top. Quantum Electron. 5, 1159-1167 (1999).
[CrossRef]

J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, "Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography," Opt. Lett. 22, 934-936 (1997).
[CrossRef] [PubMed]

Nelson, J. S.

M. G. Ducros, J. F. de Boer, H.-E. Huang, L. C. Chao, Z. Chen, J. S. Nelson, T. E. Milner, and H. G. Rylander, "Polarization sensitive optical coherence tomography of the rabbit eye," IEEE J. Sel. Top. Quantum Electron. 5, 1159-1167 (1999).
[CrossRef]

J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, "Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography," Opt. Lett. 22, 934-936 (1997).
[CrossRef] [PubMed]

Nielsen, P. M. F.

P. M. F. Nielsen, F. N. Reinholz, and P. G. Charette, "Polarization-sensitive scanned fiber confocal microscope," Opt. Eng. 35, 3084-3091 (1996).
[CrossRef]

Park, B. H.

Pierce, M. C.

Pircher, M.

M. Pircher, E. Goetzinger, R. Leitgeb, and C. K. Hitzenberger, "Transversal phase resolved polarization sensitive optical coherence tomography," Phys. Med. Biol. 49, 1257-1263 (2004).
[CrossRef] [PubMed]

Poirier, S.

Reinholz, F. N.

P. M. F. Nielsen, F. N. Reinholz, and P. G. Charette, "Polarization-sensitive scanned fiber confocal microscope," Opt. Eng. 35, 3084-3091 (1996).
[CrossRef]

Rylander, H. G.

M. G. Ducros, J. F. de Boer, H.-E. Huang, L. C. Chao, Z. Chen, J. S. Nelson, T. E. Milner, and H. G. Rylander, "Polarization sensitive optical coherence tomography of the rabbit eye," IEEE J. Sel. Top. Quantum Electron. 5, 1159-1167 (1999).
[CrossRef]

Schoenenberger, K.

Silva, L. B. D.

Smith, G.

G. Smith and D. A. Atchinson, The Eye and Visual Optical Instruments (Cambridge U. Press, 1997).
[CrossRef]

Smith, M. H.

Török, P.

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The condition number that we use is the one that corresponds to the mathematical definition of the condition number in the l2 form: the ratio of the largest to the smallest singular value in the singular-value decomposition of a matrix.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the experimental confocal Mueller matrix imaging polarimeter: ND, neutral-density; Glan-Taylor, prism; Pockels 1, Pockels 2, Pockels cells; M1, mirror; L1, collimating lens; Obj1–Obj3, objective lenses; D1–D4, photodetectors; Bs1–Bs3, nonpolarizing beam splitters; PBs4, polarizing beam splitter; Ph, pinhole; Qwp, quarter-wave plate; P0, P45, linear polarizers.

Fig. 2
Fig. 2

(a) Time varying retardances of the Pockels cells as they were implemented in the experiment. (b) Poincaré sphere representation of the generated states of polarization [S PSG(t)] during one modulation cycle. The small-dots on the sphere correspond to an instantaneous (vertical) pair of retardances in (a). Retardances at time t = 0 produced the large-dot on the sphere just behind the arrow that indicates the direction of the modulation as time progressed.

Fig. 3
Fig. 3

Axial response of the confocal polarimeter, scanning a flat mirror through the sample focal region, for two pinhole diameters: 5 and 20 μm. Experimental m 11 coefficients (circles) were calibrated without a pinhole, and the maxima of Zemax curves (solid curves) were adjusted to the maximum experimental values. Negative axial positions indicate when the mirror was placed between the objective lens and its focal plane.

Fig. 4
Fig. 4

Axial Mueller matrix response of the confocal polarimeter with a 5 μm confocal pinhole. The system was calibrated without the confocal pinhole used for the scan.

Fig. 5
Fig. 5

Axial Mueller matrix response of the confocal polarimeter with a 5 μm confocal pinhole. The system was calibrated with the confocal pinhole used for the scan.

Fig. 6
Fig. 6

Total retardance of three Mueller matrices in the axial scans: The maximum and the two edges of the FWHM on coefficient M i r r o r 1 1 . The values were calculated by use of the Lu–Chipman polar decomposition[28] on the data for the 5 and 20 μm pinholes.

Fig. 7
Fig. 7

Schematic diagram of the configuration used to make the axial scans of the pinhole.

Fig. 8
Fig. 8

Mueller matrix axial scan of the 5 μm confocal pinhole with (triangles) and without (circles) objective lens Obj1.

Tables (2)

Tables Icon

Table 1 Repeatability and Accuracy of the Double-Pass Mueller Matrix Measurements

Tables Icon

Table 2 Residual rms Errors of the Axial Scans of the Pinhole with and without the Objective Lens

Equations (25)

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L [ τ , Δ ( t ) , θ ] = τ R ( θ ) [ 1 0 0 0 0 1 0 0 0 0 cos Δ ( t ) sin Δ ( t ) 0 0 sin Δ ( t ) cos Δ ( t ) ] R ( θ ) ,
S P S G ( t ) = [ 1 cos Δ 1 ( t ) sin Δ 1 ( t ) sin Δ 2 ( t ) sin Δ 1 ( t ) cos Δ 2 ( t ) ] .
Δ 1 ( t ) = 2 ω 0 t 3 π / 2 ,
Δ 2 ( t ) = ω 0 t 3 π / 2 ,
D PSA = 1 2 [ 1 1 0 0 1 1 0 0 1 / 2 0 1 / 2 0 1 / 2 0 0 1 / 2 ] ,
M s a m p l e = [ 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 ] ,
I ( t ) = D PSA B s 1 out M sample B s 1 i n S P S G ( t ) ,
I ( t ) = U ( 1 cos ω 0 t cos 3 ω 0 t sin ω 0 t sin 2 ω 0 t sin 3 ω 0 t ) ,
U = [ ½ ( m 11 + m 21 ) ¼ ( m 13 + m 23 ) ¼ ( m 13 + m 23 ) ¼ ( m 14 + m 24 ) ½ ( m 22 + m 12 ) ¼ ( m 14 + m 24 ) ½ ( m 11 m 21 ) ½ ( m 13 m 23 ) ½ ( m 13 m 23 ) ½ ( m 14 m 24 ) ½ ( m 12 m 22 )           ½ ( m 24 m 14 ) ¼ ( m 11 + m 31 ) 1 / 8 ( m 13 + m 33 ) 1 / 8 ( m 13 + m 33 ) 1 / 8 ( m 14 + m 34 ) ¼ ( m 12 + m 32 ) 1 / 8 ( m 14 + m 34 ) ¼ ( m 11 + m 41 ) 1 / 8 ( m 13 + m 43 ) 1 / 8 ( m 13 + m 43 ) 1 / 8 ( m 14 + m 44 ) ¼ ( m 12 + m 42 ) 1 / 8 ( m 14 + m 44 ) ] .
S P S G ( t ) = [ 1 sin 2 ω 0 t ½ ( cos ω 0 t + cos 3 ω 0 t ) ½ ( sin ω 0 t + sin 3 ω 0 t ) ] = Q ( 1 cos ω 0 t cos 3 ω 0 t sin ω 0 t sin 2 ω 0 t sin 3 ω 0 t ) ,
B s a m p l e = A M s a m p l e W .
B s a m p l e dp = A M s a m p l e E M s a m p l e + W .
C i dp = ( B 0 dp ) 1 B i dp = ( A E W ) 1 ( A M i E M i + W )
( i = 1 , 2 , 3 ) ;
eig ( C i dp ) = eig ( E M i E M i + ) ,
M i + = R ( θ i ) M 0 ° , i + R ( θ i ) ,
M i = R ( θ i ) M 0 ° , i + R ( θ i ) ,
M i E M i + = R ( θ i ) M 0 ° , i + E M 0 , i + R ( θ i ) .
P ( τ p , Ψ , Δ ) = τ p [ 1 cos 2 Ψ 0 0 cos 2 Ψ 1 0 0 0 0 sin 2 Ψ   cos   Δ sin 2 Ψ sin Δ 0 0 sin 2 Ψ  sin  Δ sin 2 Ψ   cos   Δ ] .
M i E M i + = E R ( θ i ) M 0 ° , i + M 0 , i + R ( θ i ) .
eig ( C i dp ) = eig ( M i + M i + ) = eig ( M i dp ) .
M i dp X X C i dp = 0 ( i = 1 , 2 , 3 ) .
A = B 0 dp W 1 E .
A dp = A E = B 0 dp W 1 ,
( A dp ) 1 B sample dp ( W 1 ) = E M sample E M sample + .

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