Abstract

We acquired polarization-sensitive reflectance images in freshly excised skeletal muscle samples. The obtained raw images varied depending on the incident and detection polarization states. The Stokes vectors were measured for incident light of four different polarization states, and the whole Mueller matrix images were also calculated. We found that the images obtained in skeletal muscles exhibited different features from those obtained in a typical polystyrene sphere solution. The back-reflected light in muscle maintained a higher degree of polarization along the axis perpendicular to muscle fiber orientation. Our analysis indicates that the unique muscle sarcomere structure plays an important role in modulating the propagation of polarized light in whole muscle.

©2008 Optical Society of America

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References

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    [Crossref]
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  13. R. E. Nothdurft and G. Yao, “Study of subsurface polarization imaging in turbid media with different embedded objects,” Opt. Express 13, 4185–4195(2005).
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  14. R. E. Nothdurft and G. Yao, “Applying the polarization memory effect in polarization-gated subsurface imaging,” Opt. Express 14, 4656–4661(2006).
    [Crossref] [PubMed]
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  21. G. Yao and L. Wang, “Propagation of polarized light in turbid media: simulated animation sequences,” Opt. Express 7, 198–203 (2000).
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  22. Y. Deng, S. Zeng, Q. Lu, D. zhu, and Q. Luo, “Characterization of backscattering Mueller matrix patterns of highly scattering media with triple scattering assumption,” Opt. Express 15, 9672–9680 (2007).
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  24. X. D. Wang, G. Yao, and L. V. Wang, “Monte Carlo model and single-scattering approximation of the propagation of polarized light in turbid media containing glucose,” Appl. Opt. 41, 792(2002).
    [Crossref] [PubMed]
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2008 (1)

2007 (2)

2006 (4)

2005 (2)

R. E. Nothdurft and G. Yao, “Study of subsurface polarization imaging in turbid media with different embedded objects,” Opt. Express 13, 4185–4195(2005).
[Crossref] [PubMed]

N. G. Laing and K. J. Nowak, “When contractile proteins go bad: the sarcomere and skeletal muscle disease,” BioEssays 27, 809–822 (2005).
[Crossref] [PubMed]

2004 (3)

C. G. Bonnemann and N. G. Laing, “Myopathies resulting from mutations in sarcomeric proteins,” Curr. Opin. Neurol. 17, 529–537 (2004).
[Crossref] [PubMed]

R. L. Lieber and J. Fridén, “Implications of Muscle Design on Surgical Reconstruction of Upper Extremities,” Clin. Orthop. 419, 267–279 (2004).
[Crossref] [PubMed]

G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun. 241, 255–261 (2004).
[Crossref]

2002 (2)

X. D. Wang, G. Yao, and L. V. Wang, “Monte Carlo model and single-scattering approximation of the propagation of polarized light in turbid media containing glucose,” Appl. Opt. 41, 792(2002).
[Crossref] [PubMed]

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed Opt 7, 341 (2002).
[Crossref] [PubMed]

2000 (3)

1999 (3)

K. Sokolov, R. Drezek, K. Gossagee, and R. Richards-Kortum, “Reflectance spectroscopy with polarized light: Is it sensitive to cellular and nuclear morphology,” Opt. Express 5, 302 (1999).
[Crossref] [PubMed]

T. M. Johnson and J. R. Mourant, “Polarized wavelength-dependent measurement of turbid media,” Opt. Express 4, 200 (1999).
[Crossref] [PubMed]

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, P. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Quantum Electron. 5, 1019–1026 (1999)
[Crossref]

1998 (1)

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

1997 (1)

1991 (1)

R. A. Thornhill, N. Thomas, and N. Berovic, “Optical diffraction by well ordered muscle fibers,” Eur. Biophys. 20, 87–99 (1991).

1983 (1)

1976 (1)

P. J. Paolini, R. Sabbadini, K. P. Roos, and R. J. Baskin, “Sarcomere length dispersion in single muscle fibers and fiber bundles,” Biophys. J. 16, 919–930 (1976).
[Crossref] [PubMed]

Baba, J. S.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed Opt 7, 341 (2002).
[Crossref] [PubMed]

Backman, V.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, P. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Quantum Electron. 5, 1019–1026 (1999)
[Crossref]

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Badizadegan, K.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, P. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Quantum Electron. 5, 1019–1026 (1999)
[Crossref]

Baskin, R. J.

P. J. Paolini, R. Sabbadini, K. P. Roos, and R. J. Baskin, “Sarcomere length dispersion in single muscle fibers and fiber bundles,” Biophys. J. 16, 919–930 (1976).
[Crossref] [PubMed]

Berovic, N.

R. A. Thornhill, N. Thomas, and N. Berovic, “Optical diffraction by well ordered muscle fibers,” Eur. Biophys. 20, 87–99 (1991).

Bigio, I.

Bonnemann, C. G.

C. G. Bonnemann and N. G. Laing, “Myopathies resulting from mutations in sarcomeric proteins,” Curr. Opin. Neurol. 17, 529–537 (2004).
[Crossref] [PubMed]

Boppart, M. D.

Boppart, S. A.

Cameron, B. D.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed Opt 7, 341 (2002).
[Crossref] [PubMed]

Chaney, E.

Chen, Z. P.

Chung, J. R.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed Opt 7, 341 (2002).
[Crossref] [PubMed]

Cote, G. L.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed Opt 7, 341 (2002).
[Crossref] [PubMed]

Crawford, J. M.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Dasari, P. R.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, P. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Quantum Electron. 5, 1019–1026 (1999)
[Crossref]

De Boer, J. F.

DeLaughter, A. H.

J. S. Baba, J. R. Chung, A. H. DeLaughter, B. D. Cameron, and G. L. Cote, “Development and calibration of an automated Mueller matrix polarization imaging system,” J. Biomed Opt 7, 341 (2002).
[Crossref] [PubMed]

Deng, Y.

Dogariu, A.

Drezek, R.

Eick, A.

Feld, M. S.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, P. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Quantum Electron. 5, 1019–1026 (1999)
[Crossref]

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Flannery, B. R.

W. H. Press, B. R. Flannery, S. A. Teukolosky, and W. T. Vetterling, Numerical recipes in C: The Art of Scientific Computing, (Cambridge University Press, Cambridge, 1988).

Freyer, J.

Fridén, J.

R. L. Lieber and J. Fridén, “Implications of Muscle Design on Surgical Reconstruction of Upper Extremities,” Clin. Orthop. 419, 267–279 (2004).
[Crossref] [PubMed]

Gaylord, T. K.

Gerrard, D. E.

J. Xia, A. Weaver, D. E. Gerrard, and G. Yao, “Monitoring sarcomere structure changes in whole muscle using diffuse reflectance,” J Biomed. Opt. 11, 040504 (2006). http://spiedl.aip.org/journals/doc/JBOPFO-ft/vol_11/iss_4/040504_1.html
[Crossref] [PubMed]

Gossagee, K.

Gurjar, R.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, P. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Quantum Electron. 5, 1019–1026 (1999)
[Crossref]

Hamano, T.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Hielscher, A.

Itzkan, I.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, P. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Quantum Electron. 5, 1019–1026 (1999)
[Crossref]

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Jiao, S. L.

Johnson, T. M.

Kaufman, S. J.

Laing, N. G.

N. G. Laing and K. J. Nowak, “When contractile proteins go bad: the sarcomere and skeletal muscle disease,” BioEssays 27, 809–822 (2005).
[Crossref] [PubMed]

C. G. Bonnemann and N. G. Laing, “Myopathies resulting from mutations in sarcomeric proteins,” Curr. Opin. Neurol. 17, 529–537 (2004).
[Crossref] [PubMed]

Lieber, R. L.

R. L. Lieber and J. Fridén, “Implications of Muscle Design on Surgical Reconstruction of Upper Extremities,” Clin. Orthop. 419, 267–279 (2004).
[Crossref] [PubMed]

Lima, C.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Lu, Q.

Luo, Q.

Manoharan, R.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Moharam, M. G.

Mourant, J.

Mourant, J. R.

Nelson, J. S.

Nothdurft, R. E.

Nowak, K. J.

N. G. Laing and K. J. Nowak, “When contractile proteins go bad: the sarcomere and skeletal muscle disease,” BioEssays 27, 809–822 (2005).
[Crossref] [PubMed]

Nusrat, A.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Paolini, P. J.

P. J. Paolini, R. Sabbadini, K. P. Roos, and R. J. Baskin, “Sarcomere length dispersion in single muscle fibers and fiber bundles,” Biophys. J. 16, 919–930 (1976).
[Crossref] [PubMed]

Park, B. H.

Pasquesi, J. J.

Perelman, L. T.

V. Backman, R. Gurjar, K. Badizadegan, I. Itzkan, P. R. Dasari, L. T. Perelman, and M. S. Feld, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structures in situ,” IEEE J. Quantum Electron. 5, 1019–1026 (1999)
[Crossref]

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Press, W. H.

W. H. Press, B. R. Flannery, S. A. Teukolosky, and W. T. Vetterling, Numerical recipes in C: The Art of Scientific Computing, (Cambridge University Press, Cambridge, 1988).

Ranasinghesagara, J.

Richards-Kortum, R.

Roos, K. P.

P. J. Paolini, R. Sabbadini, K. P. Roos, and R. J. Baskin, “Sarcomere length dispersion in single muscle fibers and fiber bundles,” Biophys. J. 16, 919–930 (1976).
[Crossref] [PubMed]

Sabbadini, R.

P. J. Paolini, R. Sabbadini, K. P. Roos, and R. J. Baskin, “Sarcomere length dispersion in single muscle fibers and fiber bundles,” Biophys. J. 16, 919–930 (1976).
[Crossref] [PubMed]

Saxer, C. E.

Schlachter, S. C.

Schwartz, C.

Seiler, M.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Shen, D.

Shields, S.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Sokolov, K.

Teukolosky, S. A.

W. H. Press, B. R. Flannery, S. A. Teukolosky, and W. T. Vetterling, Numerical recipes in C: The Art of Scientific Computing, (Cambridge University Press, Cambridge, 1988).

Thomas, N.

R. A. Thornhill, N. Thomas, and N. Berovic, “Optical diffraction by well ordered muscle fibers,” Eur. Biophys. 20, 87–99 (1991).

Thornhill, R. A.

R. A. Thornhill, N. Thomas, and N. Berovic, “Optical diffraction by well ordered muscle fibers,” Eur. Biophys. 20, 87–99 (1991).

Van Dam, J.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Vetterling, W. T.

W. H. Press, B. R. Flannery, S. A. Teukolosky, and W. T. Vetterling, Numerical recipes in C: The Art of Scientific Computing, (Cambridge University Press, Cambridge, 1988).

Wallace, M.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
[Crossref]

Wang, L.

Wang, L. H. V.

Wang, L. V.

Wang, X. D.

Weaver, A.

J. Xia, A. Weaver, D. E. Gerrard, and G. Yao, “Monitoring sarcomere structure changes in whole muscle using diffuse reflectance,” J Biomed. Opt. 11, 040504 (2006). http://spiedl.aip.org/journals/doc/JBOPFO-ft/vol_11/iss_4/040504_1.html
[Crossref] [PubMed]

Xia, J.

J. Xia, A. Weaver, D. E. Gerrard, and G. Yao, “Monitoring sarcomere structure changes in whole muscle using diffuse reflectance,” J Biomed. Opt. 11, 040504 (2006). http://spiedl.aip.org/journals/doc/JBOPFO-ft/vol_11/iss_4/040504_1.html
[Crossref] [PubMed]

Yao, G.

J. Ranasinghesagara and G. Yao, “Imaging 2D optical diffuse reflectance in skeletal muscle,” Opt. Express 15, 3998–4007 (2007).
[Crossref] [PubMed]

J. Xia, A. Weaver, D. E. Gerrard, and G. Yao, “Monitoring sarcomere structure changes in whole muscle using diffuse reflectance,” J Biomed. Opt. 11, 040504 (2006). http://spiedl.aip.org/journals/doc/JBOPFO-ft/vol_11/iss_4/040504_1.html
[Crossref] [PubMed]

R. E. Nothdurft and G. Yao, “Effects of turbid media optical properties on object visibility in subsurface polarization imaging,” Appl. Opt. 45, 5532–5541(2006).
[Crossref] [PubMed]

R. E. Nothdurft and G. Yao, “Applying the polarization memory effect in polarization-gated subsurface imaging,” Opt. Express 14, 4656–4661(2006).
[Crossref] [PubMed]

R. E. Nothdurft and G. Yao, “Study of subsurface polarization imaging in turbid media with different embedded objects,” Opt. Express 13, 4185–4195(2005).
[Crossref] [PubMed]

G. Yao, “Differential optical polarization imaging in turbid media with different embedded objects,” Opt. Commun. 241, 255–261 (2004).
[Crossref]

X. D. Wang, G. Yao, and L. V. Wang, “Monte Carlo model and single-scattering approximation of the propagation of polarized light in turbid media containing glucose,” Appl. Opt. 41, 792(2002).
[Crossref] [PubMed]

G. Yao and L. Wang, “Propagation of polarized light in turbid media: simulated animation sequences,” Opt. Express 7, 198–203 (2000).
[Crossref] [PubMed]

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Phys. Rev. Lett. (1)

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: A new technique for measuring nuclear size distribution,” Phys. Rev. Lett.,  80, 627 (1998).
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Figures (7)

Fig. 1.
Fig. 1.

A schematic of the experimental setup. LS: a 10mW He-Ne laser; HW: half wave plate; M1: mirror; P1: polarizer; VW: variable wave plate; M2: mirror; S: sample; QW: quarter wave plate; P2: polarizer; CCD: imaging camera.

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Fig. 2.
Fig. 2.

Polarization-sensitive reflectance images in skeletal muscle. The incident light was located at the center of the image. The image size was 26.5mm by 19.9mm. The muscle fibers were along the vertical direction (y-axis). The H-polarization direction was along the horizontal direction (x-axis).

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Fig. 3.
Fig. 3.

Fitting results of the parameter q and axis ratio (a/b) for the HH, HV, and VV images shown in Fig. 2 obtained at different distances along the y-axis from the incident point.

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Fig. 4.
Fig. 4.

Stokes vectors of the reflectance images in a muscle for 4 different incident polarization states: H, V, P, and R. The images were calculated from the raw images in Fig. 2 using Eq. (1). The S1, S2, and S3 images were normalized with the S0 image. The color map shown was used for S1, S2, and S3 images only. The muscle fibers were along the vertical direction. The Hpolarization direction was along the horizontal direction.

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Fig. 5.
Fig. 5.

Mueller matrices of the reflectance images in a muscle sample. The images were calculated from the raw images in Fig. 2 by using Eq. (3). The muscle fibers were along the vertical direction. The H-polarization direction was along the horizontal direction. All images were normalized with the M11 image. As a comparison, the Mueller matrices obtained in a polystyrene (1.093 µm in diameter) solution were also shown. Please note that the M11 images used their own color maps.

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Fig. 6.
Fig. 6.

Images of the degree of polarization (DOP) calculated from the Stokes vector images in Fig. 4 by using Eq. (2). The muscle fibers were along the vertical direction. The H-polarization direction was along the horizontal direction. As a comparison, the DOP images obtained in a polystyrene solution were also shown.

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Fig. 7.
Fig. 7.

Diffraction efficiencies for the first 3 diffraction orders calculated by using coupled wave theory. The curves shown in solid lines are for TM polarization component, while those shown in dash lines are for TE polarization components. The geometry of the calculation is also illustrated in the figure. Please note that the TE direction is aligned with the muscle fiber orientation (y-axis) and the V-polarization in our coordinate system.

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

S = [ S 0 S 1 S 2 S 3 ] = [ I H + I V I H − I V 2 I P − ( I H + I V ) 2 I R − ( I H + I V ) ] ,
DOP = S 1 2 + S 2 2 + S 3 2 S 0 ,
M = 1 2 [ S H + S V S H − S V 2 S P − ( S H + S V ) 2 S R − ( S H + S V ) ] ,
f ( x , y ) = ( ∣ x ∣ a ) q + ( ∣ y ∣ b ) q − 1 = 0

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