Abstract

Trajectory of the normalized Stokes vector on the Poincaré sphere corresponding to light propagation in anisotropic tissues with birefringence and biattenuance is derived. Analytic expressions are determined from the Serret–Frenet formulas and derivatives of arc length for five quantities including the tangent, normal, and binormal vectors with curvature and torsion. Depth variation of curvature and torsion of normalized Stokes vector trajectories corresponding to light propagating in rodent tail tendon are given. Use of analytic expressions for depth variation of curvature and torsion of the normalized Stokes vector trajectories on the Poincaré sphere is discussed for analysis of polarization-sensitive optical coherence tomography data recorded from anisotropic biological tissues with birefringence and biattenuance.

© 2006 Optical Society of America

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    [CrossRef]

2005

2004

K. M. Carter, J. S. George, and D. M. Rector, 'Simultaneous birefringence and scattered light measurements reveal anatomical features in isolated crustacean nerve,' J. Neurosci. Methods 135, 9-16 (2004).
[CrossRef] [PubMed]

S. M. Srinivas, J. F. de Boer, H. Park, K. Keikhanzadeh, H. L. Huang, J. Zhang, W. Q. Jung, Z. Chen, and J. S. Nelson, 'Determination of burn depth by polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 207-212 (2004).
[CrossRef] [PubMed]

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, 'In vivo birefringence and thickness measurements of the human retinal nerve fiber layer using polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 121-125 (2004).
[CrossRef] [PubMed]

E. Götzinger, M. Pircher, M. Sticker, A. F. Fercher, and C. K. Hitzenberger, 'Measurement and imaging of birefringent properties of the human cornea with phase-resolved polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 94-102 (2004).
[CrossRef] [PubMed]

V. Louis-Dorr, K. Naoun, P. Allé, A. Benoit, and A. Raspiller, 'Linear dichroism of the cornea,' Appl. Opt. 43, 1515-1521 (2004).
[CrossRef] [PubMed]

M. Todorovic, S. Jiao, L. V. Wang, and G. Stoica, 'Determination of local polarization properties of biological samples in the presence of diattenuation by use Mueller optical coherence tomography,' Opt. Lett. 29, 2402-2404 (2004).
[CrossRef] [PubMed]

B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, 'Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,' Opt. Lett. 29, 2512-2514 (2004).
[CrossRef] [PubMed]

2003

2002

Q. Zhou and R. N. Weinerb, 'Individualized compensation of anterior segment birefringence during scanning laser polarimetry,' Invest. Ophthalmol. Visual Sci. 43, 2221-2228 (2002).

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]

R. W. Knighton, X. Huang, and D. S. Greenfield, 'Analytical model of scanning laser polarimetry for retinal nerve fiber layer assessment,' Invest. Ophthalmol. Visual Sci. 43, 383-392 (2002).

J. M. Bueno and F. Vargas-Martin, 'Measurements of the corneal birefringence with a liquid-crystal imaging polariscope,' Appl. Opt. 41, 116-124 (2002).
[CrossRef] [PubMed]

2001

W. Drexler, D. Stamper, C. Jesser, X. Li, C. Pitris, K. Saunders, S. Martin, M. B. Lodge, J. G. Fujimoto, and M. E. Brezinski, 'Correlation of collagen organization with polarization sensitive imaging of in vitro cartilage: implications for osteoarthritis,' J. Rheumatol. 28, 1311-1318 (2001).
[PubMed]

M. M. Hyttinen, J. P. A. Arokoski, J. J. Parkkinen, M. J. Lammi, T. Lapveteläinen, K. Mauranen, K. Király, M. I. Tammi, and H. J. Helminen, 'Age matters: collagen birefringence of superficial articular cartilage is increased in young guinea-pigs but decreased in older animals after identical physiological type of joint loading,' Osteoarthritis Cartilage 9, 694-701 (2001).
[CrossRef]

A. M. Benoit, K. Naoun, V. Louis-Dorr, L. Mala, and A. Raspiller, 'Linear dichroism of the retinal nerve fiber layer expressed with Mueller matrices,' Appl. Opt. 40, 565-569 (2001).
[CrossRef]

M. G. Ducros, J. D. Marsack, H. G. Rylander, S. L. Thomsen, and T. E. Milner, 'Primate retina imaging with polarization-sensitive optical coherence tomography,' J. Opt. Soc. Am. A 18, 2945-2956 (2001).
[CrossRef]

2000

J. P. Gordon and H. Kogelnik, 'PMD fundamentals: polarization mode dispersion in optical fiber,' Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

B. Huttner, C. Geiser, and N. Gisin, 'Polarization-induced distortions in optical fiber network with polarization-mode dispersion and polarization-dependent losses,' IEEE J. Sel. Top. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

D. S. Greenfield, R. W. Knighton, and X. R. Huang, 'Effect of corneal polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,' Am. J. Ophthalmol. 129, 715-722 (2000).
[CrossRef] [PubMed]

1999

1998

1997

1994

M. Peckham, M. A. Ferenczi, and M. Irving, 'A birefringence study of changes in myosin orientation during relaxation of skinned muscle fibers induced by photolytic ATP release,' Biophys. J. 67, 1141-1148 (1994).
[CrossRef] [PubMed]

A. P. Hollander, T. F. Heathfield, C. Webber, Y. Iwata, R. Bourne, C. Rorabeck, and A. R. Poole, 'Increased damage to type II collagen in osteoarthritic articular cartilage detected by new immunoassay,' J. Clin. Invest. 93, 1722-1732 (1994).
[CrossRef] [PubMed]

1993

H. M. Theuns, R. P. Shellis, A. Groeneveld, J. W. E. van Dijk, and D. F. G. Poole, 'Relationships between birefringence and mineral content in artificial caries lesions of enamel,' Caries Res. 27, 9-14 (1993).
[CrossRef] [PubMed]

1992

1989

R. A. Chipman, 'Polarization analysis of optical systems,' Opt. Eng. 28, 90-99 (1989).

R. C. Haskell, F. D. Carlson, and P. S. Blank, 'Form birefringence of muscle,' Biophys. J. 56, 401-413 (1989).
[CrossRef] [PubMed]

1988

T. Karlsmark, H. K. Thomsen, L. Danielsen, O. Aalund, O. Nielsen, K. G. Nielsen, and I. K. Genefke, 'The morphogenesis of electrically and heat-induced dermal changes in pig skin,' Forensic Sci. Int. 39, 175-188 (1988).
[CrossRef] [PubMed]

1981

E. R. Pimentel, 'Form birefringence of collagen bundles,' Acta. Histochem. Cytochem. 14, 35-41 (1981).
[CrossRef]

1978

1973

1963

A. Eberstein and A. Rosenfalck, 'Birefringence of isolated muscle fibres in twitch and tetanus,' Acta Physiol. Scand. 57, 144-166 (1963).
[CrossRef]

Aalund, O.

T. Karlsmark, H. K. Thomsen, L. Danielsen, O. Aalund, O. Nielsen, K. G. Nielsen, and I. K. Genefke, 'The morphogenesis of electrically and heat-induced dermal changes in pig skin,' Forensic Sci. Int. 39, 175-188 (1988).
[CrossRef] [PubMed]

Allé, P.

Arokoski, J. P. A.

M. M. Hyttinen, J. P. A. Arokoski, J. J. Parkkinen, M. J. Lammi, T. Lapveteläinen, K. Mauranen, K. Király, M. I. Tammi, and H. J. Helminen, 'Age matters: collagen birefringence of superficial articular cartilage is increased in young guinea-pigs but decreased in older animals after identical physiological type of joint loading,' Osteoarthritis Cartilage 9, 694-701 (2001).
[CrossRef]

Azzam, R. M. A.

Ballard, S. S.

W. A. Shurcliff and S. S. Ballard, Polarized Light (Van Nostrand, 1964).

Bashara, N. M.

Benoit, A.

Benoit, A. M.

Blank, P. S.

R. C. Haskell, F. D. Carlson, and P. S. Blank, 'Form birefringence of muscle,' Biophys. J. 56, 401-413 (1989).
[CrossRef] [PubMed]

Bourne, R.

A. P. Hollander, T. F. Heathfield, C. Webber, Y. Iwata, R. Bourne, C. Rorabeck, and A. R. Poole, 'Increased damage to type II collagen in osteoarthritic articular cartilage detected by new immunoassay,' J. Clin. Invest. 93, 1722-1732 (1994).
[CrossRef] [PubMed]

Brezinski, M. E.

W. Drexler, D. Stamper, C. Jesser, X. Li, C. Pitris, K. Saunders, S. Martin, M. B. Lodge, J. G. Fujimoto, and M. E. Brezinski, 'Correlation of collagen organization with polarization sensitive imaging of in vitro cartilage: implications for osteoarthritis,' J. Rheumatol. 28, 1311-1318 (2001).
[PubMed]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Bueno, J. M.

Carlson, F. D.

R. C. Haskell, F. D. Carlson, and P. S. Blank, 'Form birefringence of muscle,' Biophys. J. 56, 401-413 (1989).
[CrossRef] [PubMed]

Carter, K. M.

K. M. Carter, J. S. George, and D. M. Rector, 'Simultaneous birefringence and scattered light measurements reveal anatomical features in isolated crustacean nerve,' J. Neurosci. Methods 135, 9-16 (2004).
[CrossRef] [PubMed]

Cense, B.

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, 'In vivo birefringence and thickness measurements of the human retinal nerve fiber layer using polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 121-125 (2004).
[CrossRef] [PubMed]

B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, 'Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,' Opt. Lett. 29, 2512-2514 (2004).
[CrossRef] [PubMed]

Chao, L. C.

M. G. Ducros, J. F. de Boer, H. 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]

Chen, T. C.

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, 'In vivo birefringence and thickness measurements of the human retinal nerve fiber layer using polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 121-125 (2004).
[CrossRef] [PubMed]

Chen, Z.

S. M. Srinivas, J. F. de Boer, H. Park, K. Keikhanzadeh, H. L. Huang, J. Zhang, W. Q. Jung, Z. Chen, and J. S. Nelson, 'Determination of burn depth by polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 207-212 (2004).
[CrossRef] [PubMed]

M. G. Ducros, J. F. de Boer, H. 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.

R. A. Chipman, 'Polarization analysis of optical systems,' Opt. Eng. 28, 90-99 (1989).

Colston, B. W.

Coxeter, H. S. M.

H. S. M. Coxeter, Introduction to Geometry (Wiley, 1989).

Da Silva, L. B.

Danielsen, L.

T. Karlsmark, H. K. Thomsen, L. Danielsen, O. Aalund, O. Nielsen, K. G. Nielsen, and I. K. Genefke, 'The morphogenesis of electrically and heat-induced dermal changes in pig skin,' Forensic Sci. Int. 39, 175-188 (1988).
[CrossRef] [PubMed]

de Boer, J. F.

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, 'In vivo birefringence and thickness measurements of the human retinal nerve fiber layer using polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 121-125 (2004).
[CrossRef] [PubMed]

S. M. Srinivas, J. F. de Boer, H. Park, K. Keikhanzadeh, H. L. Huang, J. Zhang, W. Q. Jung, Z. Chen, and J. S. Nelson, 'Determination of burn depth by polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 207-212 (2004).
[CrossRef] [PubMed]

B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, 'Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,' Opt. Lett. 29, 2512-2514 (2004).
[CrossRef] [PubMed]

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. 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, and J. S. Nelson, 'Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization-sensitive optical coherence tomography,' Opt. Lett. 24, 300-302 (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]

deBoer, J. F.

Dreher, A. W.

Drexler, W.

W. Drexler, D. Stamper, C. Jesser, X. Li, C. Pitris, K. Saunders, S. Martin, M. B. Lodge, J. G. Fujimoto, and M. E. Brezinski, 'Correlation of collagen organization with polarization sensitive imaging of in vitro cartilage: implications for osteoarthritis,' J. Rheumatol. 28, 1311-1318 (2001).
[PubMed]

Ducros, M. G.

M. G. Ducros, J. D. Marsack, H. G. Rylander, S. L. Thomsen, and T. E. Milner, 'Primate retina imaging with polarization-sensitive optical coherence tomography,' J. Opt. Soc. Am. A 18, 2945-2956 (2001).
[CrossRef]

M. G. Ducros, J. F. de Boer, H. 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]

Eberstein, A.

A. Eberstein and A. Rosenfalck, 'Birefringence of isolated muscle fibres in twitch and tetanus,' Acta Physiol. Scand. 57, 144-166 (1963).
[CrossRef]

Everett, M. J.

Fercher, A. F.

E. Götzinger, M. Pircher, M. Sticker, A. F. Fercher, and C. K. Hitzenberger, 'Measurement and imaging of birefringent properties of the human cornea with phase-resolved polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 94-102 (2004).
[CrossRef] [PubMed]

Ferenczi, M. A.

M. Peckham, M. A. Ferenczi, and M. Irving, 'A birefringence study of changes in myosin orientation during relaxation of skinned muscle fibers induced by photolytic ATP release,' Biophys. J. 67, 1141-1148 (1994).
[CrossRef] [PubMed]

Fujimoto, J. G.

W. Drexler, D. Stamper, C. Jesser, X. Li, C. Pitris, K. Saunders, S. Martin, M. B. Lodge, J. G. Fujimoto, and M. E. Brezinski, 'Correlation of collagen organization with polarization sensitive imaging of in vitro cartilage: implications for osteoarthritis,' J. Rheumatol. 28, 1311-1318 (2001).
[PubMed]

Geiser, C.

B. Huttner, C. Geiser, and N. Gisin, 'Polarization-induced distortions in optical fiber network with polarization-mode dispersion and polarization-dependent losses,' IEEE J. Sel. Top. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

Genefke, I. K.

T. Karlsmark, H. K. Thomsen, L. Danielsen, O. Aalund, O. Nielsen, K. G. Nielsen, and I. K. Genefke, 'The morphogenesis of electrically and heat-induced dermal changes in pig skin,' Forensic Sci. Int. 39, 175-188 (1988).
[CrossRef] [PubMed]

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K. M. Carter, J. S. George, and D. M. Rector, 'Simultaneous birefringence and scattered light measurements reveal anatomical features in isolated crustacean nerve,' J. Neurosci. Methods 135, 9-16 (2004).
[CrossRef] [PubMed]

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B. Huttner, C. Geiser, and N. Gisin, 'Polarization-induced distortions in optical fiber network with polarization-mode dispersion and polarization-dependent losses,' IEEE J. Sel. Top. Quantum Electron. 6, 317-329 (2000).
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B. Huttner and N. Gisin, 'Anomalous pulse spreading in birefringent optical fibers with polarization-dependent losses,' Opt. Lett. 22, 504-506 (1997).
[CrossRef] [PubMed]

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H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2002).

Gordon, J. P.

J. P. Gordon and H. Kogelnik, 'PMD fundamentals: polarization mode dispersion in optical fiber,' Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
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E. Götzinger, M. Pircher, M. Sticker, A. F. Fercher, and C. K. Hitzenberger, 'Measurement and imaging of birefringent properties of the human cornea with phase-resolved polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 94-102 (2004).
[CrossRef] [PubMed]

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R. W. Knighton, X. Huang, and D. S. Greenfield, 'Analytical model of scanning laser polarimetry for retinal nerve fiber layer assessment,' Invest. Ophthalmol. Visual Sci. 43, 383-392 (2002).

D. S. Greenfield, R. W. Knighton, and X. R. Huang, 'Effect of corneal polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,' Am. J. Ophthalmol. 129, 715-722 (2000).
[CrossRef] [PubMed]

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H. M. Theuns, R. P. Shellis, A. Groeneveld, J. W. E. van Dijk, and D. F. G. Poole, 'Relationships between birefringence and mineral content in artificial caries lesions of enamel,' Caries Res. 27, 9-14 (1993).
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R. C. Haskell, F. D. Carlson, and P. S. Blank, 'Form birefringence of muscle,' Biophys. J. 56, 401-413 (1989).
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[CrossRef] [PubMed]

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M. M. Hyttinen, J. P. A. Arokoski, J. J. Parkkinen, M. J. Lammi, T. Lapveteläinen, K. Mauranen, K. Király, M. I. Tammi, and H. J. Helminen, 'Age matters: collagen birefringence of superficial articular cartilage is increased in young guinea-pigs but decreased in older animals after identical physiological type of joint loading,' Osteoarthritis Cartilage 9, 694-701 (2001).
[CrossRef]

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E. Götzinger, M. Pircher, M. Sticker, A. F. Fercher, and C. K. Hitzenberger, 'Measurement and imaging of birefringent properties of the human cornea with phase-resolved polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 94-102 (2004).
[CrossRef] [PubMed]

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A. P. Hollander, T. F. Heathfield, C. Webber, Y. Iwata, R. Bourne, C. Rorabeck, and A. R. Poole, 'Increased damage to type II collagen in osteoarthritic articular cartilage detected by new immunoassay,' J. Clin. Invest. 93, 1722-1732 (1994).
[CrossRef] [PubMed]

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M. G. Ducros, J. F. de Boer, H. 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]

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S. M. Srinivas, J. F. de Boer, H. Park, K. Keikhanzadeh, H. L. Huang, J. Zhang, W. Q. Jung, Z. Chen, and J. S. Nelson, 'Determination of burn depth by polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 207-212 (2004).
[CrossRef] [PubMed]

Huang, X.

Huang, X. R.

D. S. Greenfield, R. W. Knighton, and X. R. Huang, 'Effect of corneal polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,' Am. J. Ophthalmol. 129, 715-722 (2000).
[CrossRef] [PubMed]

Huttner, B.

B. Huttner, C. Geiser, and N. Gisin, 'Polarization-induced distortions in optical fiber network with polarization-mode dispersion and polarization-dependent losses,' IEEE J. Sel. Top. Quantum Electron. 6, 317-329 (2000).
[CrossRef]

B. Huttner and N. Gisin, 'Anomalous pulse spreading in birefringent optical fibers with polarization-dependent losses,' Opt. Lett. 22, 504-506 (1997).
[CrossRef] [PubMed]

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M. M. Hyttinen, J. P. A. Arokoski, J. J. Parkkinen, M. J. Lammi, T. Lapveteläinen, K. Mauranen, K. Király, M. I. Tammi, and H. J. Helminen, 'Age matters: collagen birefringence of superficial articular cartilage is increased in young guinea-pigs but decreased in older animals after identical physiological type of joint loading,' Osteoarthritis Cartilage 9, 694-701 (2001).
[CrossRef]

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M. Peckham, M. A. Ferenczi, and M. Irving, 'A birefringence study of changes in myosin orientation during relaxation of skinned muscle fibers induced by photolytic ATP release,' Biophys. J. 67, 1141-1148 (1994).
[CrossRef] [PubMed]

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A. P. Hollander, T. F. Heathfield, C. Webber, Y. Iwata, R. Bourne, C. Rorabeck, and A. R. Poole, 'Increased damage to type II collagen in osteoarthritic articular cartilage detected by new immunoassay,' J. Clin. Invest. 93, 1722-1732 (1994).
[CrossRef] [PubMed]

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W. Drexler, D. Stamper, C. Jesser, X. Li, C. Pitris, K. Saunders, S. Martin, M. B. Lodge, J. G. Fujimoto, and M. E. Brezinski, 'Correlation of collagen organization with polarization sensitive imaging of in vitro cartilage: implications for osteoarthritis,' J. Rheumatol. 28, 1311-1318 (2001).
[PubMed]

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Jung, W. Q.

S. M. Srinivas, J. F. de Boer, H. Park, K. Keikhanzadeh, H. L. Huang, J. Zhang, W. Q. Jung, Z. Chen, and J. S. Nelson, 'Determination of burn depth by polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 207-212 (2004).
[CrossRef] [PubMed]

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T. Karlsmark, H. K. Thomsen, L. Danielsen, O. Aalund, O. Nielsen, K. G. Nielsen, and I. K. Genefke, 'The morphogenesis of electrically and heat-induced dermal changes in pig skin,' Forensic Sci. Int. 39, 175-188 (1988).
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S. M. Srinivas, J. F. de Boer, H. Park, K. Keikhanzadeh, H. L. Huang, J. Zhang, W. Q. Jung, Z. Chen, and J. S. Nelson, 'Determination of burn depth by polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 207-212 (2004).
[CrossRef] [PubMed]

Kemp, N. J.

Király, K.

M. M. Hyttinen, J. P. A. Arokoski, J. J. Parkkinen, M. J. Lammi, T. Lapveteläinen, K. Mauranen, K. Király, M. I. Tammi, and H. J. Helminen, 'Age matters: collagen birefringence of superficial articular cartilage is increased in young guinea-pigs but decreased in older animals after identical physiological type of joint loading,' Osteoarthritis Cartilage 9, 694-701 (2001).
[CrossRef]

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X. Huang and R. W. Knighton, 'Diattenuation and polarization preservation of retinal nerve fiber layer reflectance,' Appl. Opt. 42, 5737-5743 (2003).
[CrossRef] [PubMed]

X. Huang and R. W. Knighton, 'Theoretical model of the polarization properties of the retinal nerve fiber layer in reflection,' Appl. Opt. 42, 5726-5736 (2003).
[CrossRef] [PubMed]

R. W. Knighton, X. Huang, and D. S. Greenfield, 'Analytical model of scanning laser polarimetry for retinal nerve fiber layer assessment,' Invest. Ophthalmol. Visual Sci. 43, 383-392 (2002).

D. S. Greenfield, R. W. Knighton, and X. R. Huang, 'Effect of corneal polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,' Am. J. Ophthalmol. 129, 715-722 (2000).
[CrossRef] [PubMed]

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J. P. Gordon and H. Kogelnik, 'PMD fundamentals: polarization mode dispersion in optical fiber,' Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

Lammi, M. J.

M. M. Hyttinen, J. P. A. Arokoski, J. J. Parkkinen, M. J. Lammi, T. Lapveteläinen, K. Mauranen, K. Király, M. I. Tammi, and H. J. Helminen, 'Age matters: collagen birefringence of superficial articular cartilage is increased in young guinea-pigs but decreased in older animals after identical physiological type of joint loading,' Osteoarthritis Cartilage 9, 694-701 (2001).
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M. M. Hyttinen, J. P. A. Arokoski, J. J. Parkkinen, M. J. Lammi, T. Lapveteläinen, K. Mauranen, K. Király, M. I. Tammi, and H. J. Helminen, 'Age matters: collagen birefringence of superficial articular cartilage is increased in young guinea-pigs but decreased in older animals after identical physiological type of joint loading,' Osteoarthritis Cartilage 9, 694-701 (2001).
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W. Drexler, D. Stamper, C. Jesser, X. Li, C. Pitris, K. Saunders, S. Martin, M. B. Lodge, J. G. Fujimoto, and M. E. Brezinski, 'Correlation of collagen organization with polarization sensitive imaging of in vitro cartilage: implications for osteoarthritis,' J. Rheumatol. 28, 1311-1318 (2001).
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W. Drexler, D. Stamper, C. Jesser, X. Li, C. Pitris, K. Saunders, S. Martin, M. B. Lodge, J. G. Fujimoto, and M. E. Brezinski, 'Correlation of collagen organization with polarization sensitive imaging of in vitro cartilage: implications for osteoarthritis,' J. Rheumatol. 28, 1311-1318 (2001).
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Mala, L.

Marsack, J. D.

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W. Drexler, D. Stamper, C. Jesser, X. Li, C. Pitris, K. Saunders, S. Martin, M. B. Lodge, J. G. Fujimoto, and M. E. Brezinski, 'Correlation of collagen organization with polarization sensitive imaging of in vitro cartilage: implications for osteoarthritis,' J. Rheumatol. 28, 1311-1318 (2001).
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M. M. Hyttinen, J. P. A. Arokoski, J. J. Parkkinen, M. J. Lammi, T. Lapveteläinen, K. Mauranen, K. Király, M. I. Tammi, and H. J. Helminen, 'Age matters: collagen birefringence of superficial articular cartilage is increased in young guinea-pigs but decreased in older animals after identical physiological type of joint loading,' Osteoarthritis Cartilage 9, 694-701 (2001).
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Milner, T. E.

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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).
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J. F. de Boer, T. E. Milner, and J. S. Nelson, 'Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization-sensitive optical coherence tomography,' Opt. Lett. 24, 300-302 (1999).
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M. G. Ducros, J. F. de Boer, H. 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).
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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).
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Nelson, J. S.

Nielsen, K. G.

T. Karlsmark, H. K. Thomsen, L. Danielsen, O. Aalund, O. Nielsen, K. G. Nielsen, and I. K. Genefke, 'The morphogenesis of electrically and heat-induced dermal changes in pig skin,' Forensic Sci. Int. 39, 175-188 (1988).
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T. Karlsmark, H. K. Thomsen, L. Danielsen, O. Aalund, O. Nielsen, K. G. Nielsen, and I. K. Genefke, 'The morphogenesis of electrically and heat-induced dermal changes in pig skin,' Forensic Sci. Int. 39, 175-188 (1988).
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B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, 'In vivo birefringence and thickness measurements of the human retinal nerve fiber layer using polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 121-125 (2004).
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B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, 'Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,' Opt. Lett. 29, 2512-2514 (2004).
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S. M. Srinivas, J. F. de Boer, H. Park, K. Keikhanzadeh, H. L. Huang, J. Zhang, W. Q. Jung, Z. Chen, and J. S. Nelson, 'Determination of burn depth by polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 207-212 (2004).
[CrossRef] [PubMed]

Park, J.

Parkkinen, J. J.

M. M. Hyttinen, J. P. A. Arokoski, J. J. Parkkinen, M. J. Lammi, T. Lapveteläinen, K. Mauranen, K. Király, M. I. Tammi, and H. J. Helminen, 'Age matters: collagen birefringence of superficial articular cartilage is increased in young guinea-pigs but decreased in older animals after identical physiological type of joint loading,' Osteoarthritis Cartilage 9, 694-701 (2001).
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Peckham, M.

M. Peckham, M. A. Ferenczi, and M. Irving, 'A birefringence study of changes in myosin orientation during relaxation of skinned muscle fibers induced by photolytic ATP release,' Biophys. J. 67, 1141-1148 (1994).
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B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, 'In vivo birefringence and thickness measurements of the human retinal nerve fiber layer using polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 121-125 (2004).
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B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, 'Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,' Opt. Lett. 29, 2512-2514 (2004).
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[CrossRef] [PubMed]

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W. Drexler, D. Stamper, C. Jesser, X. Li, C. Pitris, K. Saunders, S. Martin, M. B. Lodge, J. G. Fujimoto, and M. E. Brezinski, 'Correlation of collagen organization with polarization sensitive imaging of in vitro cartilage: implications for osteoarthritis,' J. Rheumatol. 28, 1311-1318 (2001).
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A. P. Hollander, T. F. Heathfield, C. Webber, Y. Iwata, R. Bourne, C. Rorabeck, and A. R. Poole, 'Increased damage to type II collagen in osteoarthritic articular cartilage detected by new immunoassay,' J. Clin. Invest. 93, 1722-1732 (1994).
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A. P. Hollander, T. F. Heathfield, C. Webber, Y. Iwata, R. Bourne, C. Rorabeck, and A. R. Poole, 'Increased damage to type II collagen in osteoarthritic articular cartilage detected by new immunoassay,' J. Clin. Invest. 93, 1722-1732 (1994).
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H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Addison Wesley, 2002).

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W. Drexler, D. Stamper, C. Jesser, X. Li, C. Pitris, K. Saunders, S. Martin, M. B. Lodge, J. G. Fujimoto, and M. E. Brezinski, 'Correlation of collagen organization with polarization sensitive imaging of in vitro cartilage: implications for osteoarthritis,' J. Rheumatol. 28, 1311-1318 (2001).
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H. M. Theuns, R. P. Shellis, A. Groeneveld, J. W. E. van Dijk, and D. F. G. Poole, 'Relationships between birefringence and mineral content in artificial caries lesions of enamel,' Caries Res. 27, 9-14 (1993).
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S. M. Srinivas, J. F. de Boer, H. Park, K. Keikhanzadeh, H. L. Huang, J. Zhang, W. Q. Jung, Z. Chen, and J. S. Nelson, 'Determination of burn depth by polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 207-212 (2004).
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W. Drexler, D. Stamper, C. Jesser, X. Li, C. Pitris, K. Saunders, S. Martin, M. B. Lodge, J. G. Fujimoto, and M. E. Brezinski, 'Correlation of collagen organization with polarization sensitive imaging of in vitro cartilage: implications for osteoarthritis,' J. Rheumatol. 28, 1311-1318 (2001).
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E. Götzinger, M. Pircher, M. Sticker, A. F. Fercher, and C. K. Hitzenberger, 'Measurement and imaging of birefringent properties of the human cornea with phase-resolved polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 94-102 (2004).
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Tammi, M. I.

M. M. Hyttinen, J. P. A. Arokoski, J. J. Parkkinen, M. J. Lammi, T. Lapveteläinen, K. Mauranen, K. Király, M. I. Tammi, and H. J. Helminen, 'Age matters: collagen birefringence of superficial articular cartilage is increased in young guinea-pigs but decreased in older animals after identical physiological type of joint loading,' Osteoarthritis Cartilage 9, 694-701 (2001).
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T. Karlsmark, H. K. Thomsen, L. Danielsen, O. Aalund, O. Nielsen, K. G. Nielsen, and I. K. Genefke, 'The morphogenesis of electrically and heat-induced dermal changes in pig skin,' Forensic Sci. Int. 39, 175-188 (1988).
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Todorovic, M.

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H. M. Theuns, R. P. Shellis, A. Groeneveld, J. W. E. van Dijk, and D. F. G. Poole, 'Relationships between birefringence and mineral content in artificial caries lesions of enamel,' Caries Res. 27, 9-14 (1993).
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Yu, W.

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S. M. Srinivas, J. F. de Boer, H. Park, K. Keikhanzadeh, H. L. Huang, J. Zhang, W. Q. Jung, Z. Chen, and J. S. Nelson, 'Determination of burn depth by polarization-sensitive optical coherence tomography,' J. Biomed. Opt. 9, 207-212 (2004).
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D. S. Greenfield, R. W. Knighton, and X. R. Huang, 'Effect of corneal polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,' Am. J. Ophthalmol. 129, 715-722 (2000).
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Figures (8)

Fig. 1
Fig. 1

Speckle noise-corrupted normalized Stokes vector trajectory of ex vivo rat tail tendon PS-OCT data with depth z = 207 μ m . We averaged 32 A-scans to suppress the speckle noise.

Fig. 2
Fig. 2

β-coordinate system having three unit vectors [ β ̂ , β ̂ ( z ) , and β ̂ n ( z ) ] on the Poincaré sphere. β ̂ is identical to the eigenaxis β ̂ of the complex differential wave vector β ( z ) . The unit vector β ̂ n ( z ) is defined as a normalized cross product between β ̂ and s ̂ ( z ) [ β ̂ n ( z ) = β ̂ × s ̂ ( z ) β ̂ × s ̂ ( z ) ] . The unit vector β ̂ ( z ) is the cross product between β ̂ n ( z ) and β ̂ [ β ̂ ( z ) = β ̂ n ( z ) × β ̂ ] .

Fig. 3
Fig. 3

(a) First axis rotation is about s ̂ 3 through Euler angle ϕ so that s ̂ 1 and t ̂ coincide. (b) The second axis rotation is about t ̂ through Euler angle θ so that s ̂ 3 and β ̂ n ( 0 ) coincide. (c) The third and final rotation is through Euler angle ψ about β ̂ n ( 0 ) so that t ̂ and β ̂ coincide.

Fig. 4
Fig. 4

Orthogonal planes: P ( z ) with normal along β ̂ and intersecting the tip of s ̂ h ( z ) and P ( z ) containing β ̂ and s ̂ h ( z ) . [ γ ( z ) is the angle between s ̂ h ( z ) and β ̂ , and α ( z ) is the angle between P ( z ) and P ( 0 ) ].

Fig. 5
Fig. 5

Verification of differential equations using the Jones matrix. The eigenaxis β ̂ has the same direction as s ̂ 1 , and γ ( 0 ) = π 2 . (a) s ̂ h ( z ) is projected into the s ̂ 2 s ̂ 3 plane ( s ̂ 1 is directed to the outward of the s ̂ 2 s ̂ 3 plane). The angle [ α ̃ ( z ) = tan 1 ( s 3 s 2 ) ] is viewed in the s ̂ 2 s ̂ 3 plane and is identical to that given in Eq. (25) for double-pass propagation. (b) s ̂ h ( z ) is projected into the s ̂ 3 s ̂ 1 plane ( s ̂ 2 is directed to the inward of the s ̂ 3 s ̂ 1 plane). The angle [ γ ̃ ( z ) = tan 1 ( sin γ ( 0 ) s 1 ) ] is viewed in the s ̂ 3 s ̂ 1 plane and is identical to Eq. (27) for double-pass propagation.

Fig. 6
Fig. 6

Simulated PS-OCT data by β re ( = 4.00 rad 100 μ m ) and β im ( = 0.40 rad 100 μ m ) with depth z ( 200 μ m ) (a) on the Poincaré sphere and (b) normalized Stokes parameters.

Fig. 7
Fig. 7

Numerical verification of the arc length and the five parameters by Serret–Frenet formulas. (a) Arc length between the initial Stokes vector s ̂ h ( 0 ) and s ̂ h ( z ) , (b) tangent vector, (c) curvature, and (d) torsion computed from the derived equation (solid curves) and the simulated data (dashed curves). The torsion by the simulated data has the numerical round-off errors.

Fig. 8
Fig. 8

Differential geometry of normalized Stokes vectors in biological tissue. (a) Speckle noise-corrupted [ S ̂ h ( z ) ] and model [ s ̂ h ( z ) ] normalized Stokes vectors of ex vivo rat tail tendon PS-OCT data (a) on the Poincaré sphere and (b) each Stokes parameter. (c) Curvature and (d) torsion of s ̂ h ( z ) of ex vivo rat tail tendon PS-OCT data were computed by derived equations.

Equations (60)

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δ = 2 π Δ n λ 0 z ,
ϵ = 2 π Δ χ λ 0 z ,
E ( z ) E ( z ) = exp ( ϵ ) exp ( i δ ) ,
d = λ a 2 λ b 2 λ a 2 + λ b 2 ,
d = exp ( 2 ϵ ) exp ( 2 ϵ ) exp ( 2 ϵ ) + exp ( 2 ϵ ) = tanh ( 2 ϵ ) ,
s 1 ( z ) = Γ ̃ h ( z ) 2 Γ ̃ v ( z ) 2 Γ ̃ h ( z ) 2 + Γ ̃ v ( z ) 2 ,
s 2 ( z ) = 2 Γ ̃ h ( z ) Γ ̃ v ( z ) cos ( Γ ̃ v h ( z ) ) Γ ̃ h ( z ) 2 + Γ ̃ v ( z ) 2 ;
s 3 ( z ) = 2 Γ ̃ h ( z ) Γ ̃ v ( z ) sin ( Γ ̃ v h ( z ) ) Γ ̃ h ( z ) 2 + Γ ̃ v ( z ) 2 .
β ( z ) = ( β re + j β im ) = 2 π λ 0 ( Δ n + j Δ χ ) β ̂ ,
[ T ̂ ̇ N ̂ ̇ B ̂ ̇ ] = [ 0 κ 0 κ 0 τ 0 τ 0 ] [ T ̂ N ̂ B ̂ ] ,
A 0 = [ β 1 β 2 β 3 β 1 ( 0 ) β 2 ( 0 ) β 3 ( 0 ) β n 1 ( 0 ) β n 2 ( 0 ) β n 3 ( 0 ) ] .
A 0 = B ( ψ ) C ( θ ) D ( ϕ ) = [ cos ψ sin ψ 0 sin ψ cos ψ 0 0 0 1 ] [ 1 0 0 0 cos θ sin θ 0 sin θ cos θ ] [ cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ] .
t ̂ = β n 2 ( 0 ) s ̂ 1 + β n 1 ( 0 ) s ̂ 2 ( β n 1 2 ( 0 ) + β n 2 2 ( 0 ) ) 1 2 .
d 2 E ( ν ) d z 2 + ϵ p ( ν , z ) k 0 2 E ( ν ) = 0 ,
ϵ p k 0 2 = β 0 2 σ 0 + β 0 β ¯ σ ¯ ,
E ( ν ) = exp ( 2 π j v ( n ¯ + j χ ¯ ) c ) E ( z ) ,
d E d z + ( 1 2 ) j β ( z ) σ E ( z ) = 0 .
S ( z ) = [ S 0 , S 1 , S 2 , S 3 ] = [ E σ 0 E , E σ 1 E , E σ 2 E , E σ 3 E ] ,
d S i d z = d E d z σ i d E d z , i = 0 , 1 , 2 , 3 .
d E d z ( 1 2 ) j β * ( z ) σ E ( z ) = 0 .
d S 0 d z = β im S = S 0 ( z ) β im ( z ) s ̂ ( z ) ,
d S d z = [ d S 1 d z , d S 2 d z , d S 3 d z ] = 1 2 j E ( β re σ ) ( σ E ) 1 2 E ( β im σ ) ( σ E ) 1 2 j E σ ( β re σ ) E 1 2 E σ ( β im σ ) E ,
σ ( β re , im σ ) = β re , im I + j β re , im × σ ,
( β re , im σ ) σ = β re , im I j β re , im × σ ,
d S d z = β re × S + β im S 0 .
d s ̂ d z + ( s ̂ × β re ) + s ̂ × ( s ̂ × β im ) = 0 .
d s ̂ = ( s ̂ h × β re ) d z ,
d s ̂ = s ̂ h × ( s ̂ h × β im ) d z .
d s ̂ = sin ( γ ( z ) ) β re d z ,
α ( z ) = 2 β re z .
d γ = sin ( γ ( z ) ) β im d z .
γ ( z ) = 2 tan 1 [ tan ( γ ( 0 ) 2 ) exp ( 2 β im z ) ] , 0 γ ( z ) < π .
s ̂ h ( z ) = A 0 T C T ( α ( z ) ) D T ( γ ( z ) γ ( 0 ) ) A 0 s ̂ h ( 0 ) ,
β ̂ ( z ) = C ( α = 2 β re z ) β ̂ ( 0 ) ,
β ̂ n ( z ) = C ( α = 2 β re z ) β ̂ n ( 0 ) .
d l = ( β re 2 + β im 2 ) 1 2 sin ( γ ( z ) ) d z ,
d l = ( β re 2 + β im 2 ) 1 2 { 2 tan ( γ ( 0 ) 2 ) exp ( β im z ) tan 2 ( γ ( 0 ) 2 ) exp ( 2 β im z ) + 1 } d z ,
l ( z ) = [ 1 + ( β re β im ) 2 ] 1 2 [ γ ( 0 ) γ ( z ) ] .
l ( z ) 2 ( β re 2 + β im 2 ) 1 2 sin ( γ ( 0 ) ) z when β im z 1 .
PSNR = l ( z ) σ speckle = [ 1 + ( β re β im ) 2 ] 1 2 [ γ ( 0 ) γ ( z ) ] σ speckle .
σ speckle = [ 1 J j ( cos 1 [ s ̂ h ( z ) S ̂ h ( z j ) ] ) 2 ] 1 2 .
T ̂ = d s ̂ h d l = ( d s ̂ h d z ) ( d l d z ) .
d s ̂ h d z = ( s ̂ h × β re ) { ( s ̂ h β im ) s ̂ h β im } = sin γ ( z ) ( β im sin γ ( z ) β ̂ β im cos γ ( z ) β ̂ ( z ) + β re β ̂ n ( z ) ) .
T ̂ = 1 ( β re 2 + β im 2 ) 1 2 { β im sin γ ( z ) β ̂ β im cos γ ( z ) β ̂ ( z ) + β re β ̂ n ( z ) } ,
T ̂ = β ̂ n ( z ) when β im = 0 .
d T ̂ d z = 1 ( β re 2 + β im 2 ) 1 2 { β im 2 sin γ ( z ) cos γ ( z ) β ̂ + ( β re 2 + β im 2 sin 2 γ ( z ) ) β ̂ ( z ) + β re β im cos γ ( z ) β ̂ n ( z ) } .
T ̂ ̇ = 1 ( β re 2 + β im 2 ) sin γ ( z ) { β im 2 sin γ ( z ) cos γ ( z ) β ̂ + ( β re 2 + β im 2 sin 2 γ ( z ) ) β ̂ ( z ) + β re β im cos γ ( z ) β ̂ n ( z ) } .
κ ( z ) = 1 ( β re 2 + β im 2 ) 1 2 sin γ ( z ) ( β re 2 + β im 2 sin 2 γ ( z ) ) 1 2 .
κ = 1 sin γ ( 0 ) when β im = 0 .
N ̂ = 1 ( β re 2 + β im 2 ) 1 2 ( β re 2 + β im 2 sin 2 γ ( z ) ) 1 2 { β im 2 sin γ ( z ) cos γ ( z ) β ̂ + ( β re 2 + β im 2 sin 2 γ ( z ) ) β ̂ ( z ) + β re β im cos γ ( z ) β ̂ n ( z ) } ,
N ̂ = β ̂ ( z ) .
B ̂ = 1 ( β re 2 + β im 2 sin 2 γ ( z ) ) 1 2 { β re β ̂ β im sin γ ( z ) β ̂ n ( z ) } ,
B ̂ = β ̂ ,
B ̂ ̇ = β re β im ( β re 2 + β im 2 ) 1 2 ( β re 2 + β im 2 sin 2 γ ( z ) ) 3 2 { β im 2 sin γ ( z ) cos γ ( z ) β ̂ + ( β re 2 + β im 2 sin 2 γ ( z ) ) β ̂ ( z ) + β re β im cos γ ( z ) β ̂ n ( z ) } ,
τ ( z ) = ( β im β re ) 1 + ( β im β re ) 2 sin 2 γ ( z ) ,
R ( β re ) = [ exp ( i β re z ) 0 0 exp ( i β re z ) ] ,
D ( β im ) = [ exp ( β im z ) 0 0 exp ( β im z ) ] ,
C ( β re , β im ) = R ( β re ) D ( β im ) = [ exp ( ( β re i + β im ) z ) 0 0 exp ( ( β re i + β im ) z ) ] .
E o = C E i = [ cos ( γ ( 0 ) 2 ) exp ( ( β re i + β im ) z ) sin ( γ ( 0 ) 2 ) exp ( ( β re i + β im ) z ) ] .
[ s 1 ( z ) s 2 ( z ) s 3 ( z ) ] = [ cos 2 ( γ ( 0 ) 2 ) exp ( 2 β im z ) sin 2 ( γ ( 0 ) 2 ) exp ( 2 β im z ) sin ( γ ( 0 ) ) cos ( 2 β re z ) sin ( γ ( 0 ) ) sin ( 2 β re z ) ] { cos 2 ( ( 0 ) 2 ) exp ( 2 β im z ) + sin 2 ( γ ( 0 ) 2 ) exp ( 2 β im z ) } .

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