Abstract

Phase retardation imaging including local birefringence imaging of biological tissues is described by generalized Jones-matrix optical coherence tomography. The polarization properties of a local tissue can be obtained from two Jones matrices that are measured by backscattered lights from the front and back boundaries of the local tissue. The error in the phase retardation measurement due to background noise is analyzed theoretically, numerically, and experimentally. The minimum detectable phase retardation is estimated from numerical simulations. The theoretical analysis suggests that the measurements with two orthogonal input polarization states have the lowest retardation error. Local birefringence imaging is applied to the human anterior eye chamber and skin in vivo.

© 2010 Optical Society of America

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2009 (1)

S. Sakai, N. Nakagawa, M. Yamanari, A. Miyazawa, Y. Yasuno, and M. Matsumoto, "Relationship between dermal birefringence and the skin surface roughness of photoaged human skin," J. Biomed. Opt. 14, 044032 (2009).
[CrossRef] [PubMed]

2008 (3)

M. Miura, M. Yamanari, T. Iwasaki, A. E. Elsner, S. Makita, T. Yatagai, and Y. Yasuno, "Imaging Polarimetry in Age-Related Macular Degeneration," Invest. Ophthalmol. Vis. Sci. 49, 2661-2667 (2008).
[CrossRef] [PubMed]

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, "Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry," J. Biomed. Opt. 13, 014013 (2008).
[CrossRef] [PubMed]

E. Götzinger, M. Pircher, B. Baumann, C. Hirn, C. Vass, and C. K. Hitzenberger, "Retinal nerve fiber layer birefringence evaluated with polarization sensitive spectral domain OCT and scanning laser polarimetry: A comparison," J. Biophoton. 1, 129-139 (2008).
[CrossRef]

2007 (1)

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, "Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography," J. Am. Coll. Cardiol. 49, 1474-1481 (2007).
[CrossRef] [PubMed]

2006 (2)

S. D. Giattina, B. K. Courtney, P. R. Herz, M. Harman, S. Shortkroff, D. L. Stamper, B. Liu, J. G. Fujimoto, and M. E. Brezinski, "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

S. Makita, Y. Yasuno, T. Endo, M. Itoh, and T. Yatagai, "Polarization contrast imaging of biological tissues by polarization-sensitive Fourier-domain optical coherence tomography," Appl. Opt. 45, 1142-1147 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (5)

2003 (1)

2002 (1)

S. Jiao and L. V. Wang, "Jones-matrix imaging of biological tissues with quadruple-channel optical coherence tomography," J. Biomed. Opt. 7, 350-358 (2002).
[CrossRef] [PubMed]

2000 (2)

1998 (2)

1997 (2)

D. J. Maitland and J. T. W. Jr., "Quantitative measurements of linear birefringence during heating of native collagen," Lasers. Surg. Med. 20, 310-318 (1997).
[CrossRef] [PubMed]

J. Welzel, E. Lankenau, R. Birngruber, and R. Engelhardt, "Optical coherence tomography of the human skin," J. Am. Acad. Dermatol. 37, 958-963 (1997).
[CrossRef]

1996 (1)

G. J. Tearney, M. E. Brezinski, S. A. Boppart, B. E. Bouma, N. Weissman, J. F. Southern, E. A. Swanson, and J. G. Fujimoto, "Images in cardiovascular medicine. Catheter-based optical imaging of a human coronary artery," Circulation 94, 3013 (1996).
[PubMed]

1995 (1)

J. S. Schuman, M. R. Hee, C. A. Puliafito, C. Wong, T. Pedut-Kloizman, C. P. Lin, E. Hertzmark, J. A. Izatt, E. A. Swanson, and J. G. Fujimoto, "Quantification of nerve fiber layer thickness in normal and glaucomatous eyes using optical coherence tomography," Arch. Ophthalmol. 113, 586-596 (1995).
[PubMed]

1994 (1)

1992 (1)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Baumann, B.

E. Götzinger, M. Pircher, B. Baumann, C. Hirn, C. Vass, and C. K. Hitzenberger, "Retinal nerve fiber layer birefringence evaluated with polarization sensitive spectral domain OCT and scanning laser polarimetry: A comparison," J. Biophoton. 1, 129-139 (2008).
[CrossRef]

Birngruber, R.

J. Welzel, E. Lankenau, R. Birngruber, and R. Engelhardt, "Optical coherence tomography of the human skin," J. Am. Acad. Dermatol. 37, 958-963 (1997).
[CrossRef]

Boer, J. F. D.

M. C. Pierce, J. Strasswimmer, B. H. Park, B. Cense, and J. F. D. Boer, "Birefringence measurements in human skin using polarization-sensitive optical coherence tomography," J. Biomed. Opt. 9, 287-291 (2004).
[CrossRef] [PubMed]

Boppart, S. A.

G. J. Tearney, M. E. Brezinski, S. A. Boppart, B. E. Bouma, N. Weissman, J. F. Southern, E. A. Swanson, and J. G. Fujimoto, "Images in cardiovascular medicine. Catheter-based optical imaging of a human coronary artery," Circulation 94, 3013 (1996).
[PubMed]

Bouma, B. E.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, "Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography," J. Am. Coll. Cardiol. 49, 1474-1481 (2007).
[CrossRef] [PubMed]

G. J. Tearney, M. E. Brezinski, S. A. Boppart, B. E. Bouma, N. Weissman, J. F. Southern, E. A. Swanson, and J. G. Fujimoto, "Images in cardiovascular medicine. Catheter-based optical imaging of a human coronary artery," Circulation 94, 3013 (1996).
[PubMed]

Bressner, J. E.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, "Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography," J. Am. Coll. Cardiol. 49, 1474-1481 (2007).
[CrossRef] [PubMed]

Brezinski, M. E.

S. D. Giattina, B. K. Courtney, P. R. Herz, M. Harman, S. Shortkroff, D. L. Stamper, B. Liu, J. G. Fujimoto, and M. E. Brezinski, "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

G. J. Tearney, M. E. Brezinski, S. A. Boppart, B. E. Bouma, N. Weissman, J. F. Southern, E. A. Swanson, and J. G. Fujimoto, "Images in cardiovascular medicine. Catheter-based optical imaging of a human coronary artery," Circulation 94, 3013 (1996).
[PubMed]

Cense, B.

B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, "Optic axis determination accuracy for fiber-based polarization-sensitive optical coherence tomography," Opt. Lett. 30, 2587-2589 (2005).
[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]

M. C. Pierce, J. Strasswimmer, B. H. Park, B. Cense, and J. F. D. Boer, "Birefringence measurements in human skin using polarization-sensitive optical coherence tomography," J. Biomed. Opt. 9, 287-291 (2004).
[CrossRef] [PubMed]

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, "Thickness and birefringence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography," Invest. Ophthalmol. Vis. Sci. 45, 2606-2612 (2004).
[CrossRef] [PubMed]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Chen, T. C.

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, "Thickness and birefringence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography," Invest. Ophthalmol. Vis. Sci. 45, 2606-2612 (2004).
[CrossRef] [PubMed]

Chen, Z.

Chipman, R. A.

Colston, B.

Colston, B. W.

Courtney, B. K.

S. D. Giattina, B. K. Courtney, P. R. Herz, M. Harman, S. Shortkroff, D. L. Stamper, B. Liu, J. G. Fujimoto, and M. E. Brezinski, "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

Da Silva, L.

de Boer, J. F.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, "Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography," J. Am. Coll. Cardiol. 49, 1474-1481 (2007).
[CrossRef] [PubMed]

B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, "Optic axis determination accuracy for fiber-based polarization-sensitive optical coherence tomography," Opt. Lett. 30, 2587-2589 (2005).
[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]

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, "Thickness and birefringence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography," Invest. Ophthalmol. Vis. Sci. 45, 2606-2612 (2004).
[CrossRef] [PubMed]

C. E. Saxer, J. F. de Boer, B. H. Park, Y. Zhao, Z. Chen, and J. S. Nelson., "High-speed fiber based polarization sensitive optical coherence tomography of in vivo human skin," Opt. Lett. 25, 1355-1357 (2000).
[CrossRef]

Elsner, A. E.

M. Miura, M. Yamanari, T. Iwasaki, A. E. Elsner, S. Makita, T. Yatagai, and Y. Yasuno, "Imaging Polarimetry in Age-Related Macular Degeneration," Invest. Ophthalmol. Vis. Sci. 49, 2661-2667 (2008).
[CrossRef] [PubMed]

Endo, T.

Engelhardt, R.

J. Welzel, E. Lankenau, R. Birngruber, and R. Engelhardt, "Optical coherence tomography of the human skin," J. Am. Acad. Dermatol. 37, 958-963 (1997).
[CrossRef]

Everett, M.

Everett, M. J.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

S. D. Giattina, B. K. Courtney, P. R. Herz, M. Harman, S. Shortkroff, D. L. Stamper, B. Liu, J. G. Fujimoto, and M. E. Brezinski, "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

G. J. Tearney, M. E. Brezinski, S. A. Boppart, B. E. Bouma, N. Weissman, J. F. Southern, E. A. Swanson, and J. G. Fujimoto, "Images in cardiovascular medicine. Catheter-based optical imaging of a human coronary artery," Circulation 94, 3013 (1996).
[PubMed]

J. S. Schuman, M. R. Hee, C. A. Puliafito, C. Wong, T. Pedut-Kloizman, C. P. Lin, E. Hertzmark, J. A. Izatt, E. A. Swanson, and J. G. Fujimoto, "Quantification of nerve fiber layer thickness in normal and glaucomatous eyes using optical coherence tomography," Arch. Ophthalmol. 113, 586-596 (1995).
[PubMed]

M. R. Hee, D. Huang, E. A. Swanson, and J. G. Fujimoto, "Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging," J. Opt. Soc. Am. B 9, 903-908 (1992).
[CrossRef]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Giattina, S. D.

S. D. Giattina, B. K. Courtney, P. R. Herz, M. Harman, S. Shortkroff, D. L. Stamper, B. Liu, J. G. Fujimoto, and M. E. Brezinski, "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

Götzinger, E.

E. Götzinger, M. Pircher, B. Baumann, C. Hirn, C. Vass, and C. K. Hitzenberger, "Retinal nerve fiber layer birefringence evaluated with polarization sensitive spectral domain OCT and scanning laser polarimetry: A comparison," J. Biophoton. 1, 129-139 (2008).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Guo, S.

Halpern, E.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, "Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography," J. Am. Coll. Cardiol. 49, 1474-1481 (2007).
[CrossRef] [PubMed]

Harman, M.

S. D. Giattina, B. K. Courtney, P. R. Herz, M. Harman, S. Shortkroff, D. L. Stamper, B. Liu, J. G. Fujimoto, and M. E. Brezinski, "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

Hee, M. R.

J. S. Schuman, M. R. Hee, C. A. Puliafito, C. Wong, T. Pedut-Kloizman, C. P. Lin, E. Hertzmark, J. A. Izatt, E. A. Swanson, and J. G. Fujimoto, "Quantification of nerve fiber layer thickness in normal and glaucomatous eyes using optical coherence tomography," Arch. Ophthalmol. 113, 586-596 (1995).
[PubMed]

M. R. Hee, D. Huang, E. A. Swanson, and J. G. Fujimoto, "Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging," J. Opt. Soc. Am. B 9, 903-908 (1992).
[CrossRef]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Hertzmark, E.

J. S. Schuman, M. R. Hee, C. A. Puliafito, C. Wong, T. Pedut-Kloizman, C. P. Lin, E. Hertzmark, J. A. Izatt, E. A. Swanson, and J. G. Fujimoto, "Quantification of nerve fiber layer thickness in normal and glaucomatous eyes using optical coherence tomography," Arch. Ophthalmol. 113, 586-596 (1995).
[PubMed]

Herz, P. R.

S. D. Giattina, B. K. Courtney, P. R. Herz, M. Harman, S. Shortkroff, D. L. Stamper, B. Liu, J. G. Fujimoto, and M. E. Brezinski, "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

Hirn, C.

E. Götzinger, M. Pircher, B. Baumann, C. Hirn, C. Vass, and C. K. Hitzenberger, "Retinal nerve fiber layer birefringence evaluated with polarization sensitive spectral domain OCT and scanning laser polarimetry: A comparison," J. Biophoton. 1, 129-139 (2008).
[CrossRef]

Hitzenberger, C. K.

E. Götzinger, M. Pircher, B. Baumann, C. Hirn, C. Vass, and C. K. Hitzenberger, "Retinal nerve fiber layer birefringence evaluated with polarization sensitive spectral domain OCT and scanning laser polarimetry: A comparison," J. Biophoton. 1, 129-139 (2008).
[CrossRef]

Houser, S. L.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, "Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography," J. Am. Coll. Cardiol. 49, 1474-1481 (2007).
[CrossRef] [PubMed]

Huang, D.

M. R. Hee, D. Huang, E. A. Swanson, and J. G. Fujimoto, "Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging," J. Opt. Soc. Am. B 9, 903-908 (1992).
[CrossRef]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Itoh, M.

Iwasaki, T.

M. Miura, M. Yamanari, T. Iwasaki, A. E. Elsner, S. Makita, T. Yatagai, and Y. Yasuno, "Imaging Polarimetry in Age-Related Macular Degeneration," Invest. Ophthalmol. Vis. Sci. 49, 2661-2667 (2008).
[CrossRef] [PubMed]

Izatt, J. A.

J. S. Schuman, M. R. Hee, C. A. Puliafito, C. Wong, T. Pedut-Kloizman, C. P. Lin, E. Hertzmark, J. A. Izatt, E. A. Swanson, and J. G. Fujimoto, "Quantification of nerve fiber layer thickness in normal and glaucomatous eyes using optical coherence tomography," Arch. Ophthalmol. 113, 586-596 (1995).
[PubMed]

Jiao, S.

Jr, J. T. W.

D. J. Maitland and J. T. W. Jr., "Quantitative measurements of linear birefringence during heating of native collagen," Lasers. Surg. Med. 20, 310-318 (1997).
[CrossRef] [PubMed]

Lankenau, E.

J. Welzel, E. Lankenau, R. Birngruber, and R. Engelhardt, "Optical coherence tomography of the human skin," J. Am. Acad. Dermatol. 37, 958-963 (1997).
[CrossRef]

Lin, C. P.

J. S. Schuman, M. R. Hee, C. A. Puliafito, C. Wong, T. Pedut-Kloizman, C. P. Lin, E. Hertzmark, J. A. Izatt, E. A. Swanson, and J. G. Fujimoto, "Quantification of nerve fiber layer thickness in normal and glaucomatous eyes using optical coherence tomography," Arch. Ophthalmol. 113, 586-596 (1995).
[PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Liu, B.

S. D. Giattina, B. K. Courtney, P. R. Herz, M. Harman, S. Shortkroff, D. L. Stamper, B. Liu, J. G. Fujimoto, and M. E. Brezinski, "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

Lu, S.-Y.

Maitland, D. J.

Makita, S.

M. Miura, M. Yamanari, T. Iwasaki, A. E. Elsner, S. Makita, T. Yatagai, and Y. Yasuno, "Imaging Polarimetry in Age-Related Macular Degeneration," Invest. Ophthalmol. Vis. Sci. 49, 2661-2667 (2008).
[CrossRef] [PubMed]

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, "Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry," J. Biomed. Opt. 13, 014013 (2008).
[CrossRef] [PubMed]

S. Makita, Y. Yasuno, T. Endo, M. Itoh, and T. Yatagai, "Polarization contrast imaging of biological tissues by polarization-sensitive Fourier-domain optical coherence tomography," Appl. Opt. 45, 1142-1147 (2006).
[CrossRef] [PubMed]

Matsumoto, M.

S. Sakai, N. Nakagawa, M. Yamanari, A. Miyazawa, Y. Yasuno, and M. Matsumoto, "Relationship between dermal birefringence and the skin surface roughness of photoaged human skin," J. Biomed. Opt. 14, 044032 (2009).
[CrossRef] [PubMed]

Miura, M.

M. Miura, M. Yamanari, T. Iwasaki, A. E. Elsner, S. Makita, T. Yatagai, and Y. Yasuno, "Imaging Polarimetry in Age-Related Macular Degeneration," Invest. Ophthalmol. Vis. Sci. 49, 2661-2667 (2008).
[CrossRef] [PubMed]

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, "Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry," J. Biomed. Opt. 13, 014013 (2008).
[CrossRef] [PubMed]

Miyazawa, A.

S. Sakai, N. Nakagawa, M. Yamanari, A. Miyazawa, Y. Yasuno, and M. Matsumoto, "Relationship between dermal birefringence and the skin surface roughness of photoaged human skin," J. Biomed. Opt. 14, 044032 (2009).
[CrossRef] [PubMed]

Nadkarni, S. K.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, "Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography," J. Am. Coll. Cardiol. 49, 1474-1481 (2007).
[CrossRef] [PubMed]

Nakagawa, N.

S. Sakai, N. Nakagawa, M. Yamanari, A. Miyazawa, Y. Yasuno, and M. Matsumoto, "Relationship between dermal birefringence and the skin surface roughness of photoaged human skin," J. Biomed. Opt. 14, 044032 (2009).
[CrossRef] [PubMed]

Nelson, J. S.

Park, B. H.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, "Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography," J. Am. Coll. Cardiol. 49, 1474-1481 (2007).
[CrossRef] [PubMed]

B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, "Optic axis determination accuracy for fiber-based polarization-sensitive optical coherence tomography," Opt. Lett. 30, 2587-2589 (2005).
[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]

M. C. Pierce, J. Strasswimmer, B. H. Park, B. Cense, and J. F. D. Boer, "Birefringence measurements in human skin using polarization-sensitive optical coherence tomography," J. Biomed. Opt. 9, 287-291 (2004).
[CrossRef] [PubMed]

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, "Thickness and birefringence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography," Invest. Ophthalmol. Vis. Sci. 45, 2606-2612 (2004).
[CrossRef] [PubMed]

C. E. Saxer, J. F. de Boer, B. H. Park, Y. Zhao, Z. Chen, and J. S. Nelson., "High-speed fiber based polarization sensitive optical coherence tomography of in vivo human skin," Opt. Lett. 25, 1355-1357 (2000).
[CrossRef]

Pedut-Kloizman, T.

J. S. Schuman, M. R. Hee, C. A. Puliafito, C. Wong, T. Pedut-Kloizman, C. P. Lin, E. Hertzmark, J. A. Izatt, E. A. Swanson, and J. G. Fujimoto, "Quantification of nerve fiber layer thickness in normal and glaucomatous eyes using optical coherence tomography," Arch. Ophthalmol. 113, 586-596 (1995).
[PubMed]

Pierce, M. C.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, "Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography," J. Am. Coll. Cardiol. 49, 1474-1481 (2007).
[CrossRef] [PubMed]

B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, "Optic axis determination accuracy for fiber-based polarization-sensitive optical coherence tomography," Opt. Lett. 30, 2587-2589 (2005).
[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]

M. C. Pierce, J. Strasswimmer, B. H. Park, B. Cense, and J. F. D. Boer, "Birefringence measurements in human skin using polarization-sensitive optical coherence tomography," J. Biomed. Opt. 9, 287-291 (2004).
[CrossRef] [PubMed]

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, "Thickness and birefringence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography," Invest. Ophthalmol. Vis. Sci. 45, 2606-2612 (2004).
[CrossRef] [PubMed]

Pircher, M.

E. Götzinger, M. Pircher, B. Baumann, C. Hirn, C. Vass, and C. K. Hitzenberger, "Retinal nerve fiber layer birefringence evaluated with polarization sensitive spectral domain OCT and scanning laser polarimetry: A comparison," J. Biophoton. 1, 129-139 (2008).
[CrossRef]

Puliafito, C. A.

J. S. Schuman, M. R. Hee, C. A. Puliafito, C. Wong, T. Pedut-Kloizman, C. P. Lin, E. Hertzmark, J. A. Izatt, E. A. Swanson, and J. G. Fujimoto, "Quantification of nerve fiber layer thickness in normal and glaucomatous eyes using optical coherence tomography," Arch. Ophthalmol. 113, 586-596 (1995).
[PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Sakai, S.

S. Sakai, N. Nakagawa, M. Yamanari, A. Miyazawa, Y. Yasuno, and M. Matsumoto, "Relationship between dermal birefringence and the skin surface roughness of photoaged human skin," J. Biomed. Opt. 14, 044032 (2009).
[CrossRef] [PubMed]

Saxer, C. E.

Schoenenberger, K.

Schuman, J. S.

J. S. Schuman, M. R. Hee, C. A. Puliafito, C. Wong, T. Pedut-Kloizman, C. P. Lin, E. Hertzmark, J. A. Izatt, E. A. Swanson, and J. G. Fujimoto, "Quantification of nerve fiber layer thickness in normal and glaucomatous eyes using optical coherence tomography," Arch. Ophthalmol. 113, 586-596 (1995).
[PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Shortkroff, S.

S. D. Giattina, B. K. Courtney, P. R. Herz, M. Harman, S. Shortkroff, D. L. Stamper, B. Liu, J. G. Fujimoto, and M. E. Brezinski, "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

Silva, L. B. D.

Southern, J. F.

G. J. Tearney, M. E. Brezinski, S. A. Boppart, B. E. Bouma, N. Weissman, J. F. Southern, E. A. Swanson, and J. G. Fujimoto, "Images in cardiovascular medicine. Catheter-based optical imaging of a human coronary artery," Circulation 94, 3013 (1996).
[PubMed]

Stamper, D. L.

S. D. Giattina, B. K. Courtney, P. R. Herz, M. Harman, S. Shortkroff, D. L. Stamper, B. Liu, J. G. Fujimoto, and M. E. Brezinski, "Assessment of coronary plaque collagen with polarization sensitive optical coherence tomography (PS-OCT)," Int. J. Cardiol. 107, 400-409 (2006).
[CrossRef] [PubMed]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Stoica, G.

Strasswimmer, J.

M. C. Pierce, J. Strasswimmer, B. H. Park, B. Cense, and J. F. D. Boer, "Birefringence measurements in human skin using polarization-sensitive optical coherence tomography," J. Biomed. Opt. 9, 287-291 (2004).
[CrossRef] [PubMed]

Swanson, E. A.

G. J. Tearney, M. E. Brezinski, S. A. Boppart, B. E. Bouma, N. Weissman, J. F. Southern, E. A. Swanson, and J. G. Fujimoto, "Images in cardiovascular medicine. Catheter-based optical imaging of a human coronary artery," Circulation 94, 3013 (1996).
[PubMed]

J. S. Schuman, M. R. Hee, C. A. Puliafito, C. Wong, T. Pedut-Kloizman, C. P. Lin, E. Hertzmark, J. A. Izatt, E. A. Swanson, and J. G. Fujimoto, "Quantification of nerve fiber layer thickness in normal and glaucomatous eyes using optical coherence tomography," Arch. Ophthalmol. 113, 586-596 (1995).
[PubMed]

M. R. Hee, D. Huang, E. A. Swanson, and J. G. Fujimoto, "Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging," J. Opt. Soc. Am. B 9, 903-908 (1992).
[CrossRef]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Tearney, G. J.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, "Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography," J. Am. Coll. Cardiol. 49, 1474-1481 (2007).
[CrossRef] [PubMed]

G. J. Tearney, M. E. Brezinski, S. A. Boppart, B. E. Bouma, N. Weissman, J. F. Southern, E. A. Swanson, and J. G. Fujimoto, "Images in cardiovascular medicine. Catheter-based optical imaging of a human coronary artery," Circulation 94, 3013 (1996).
[PubMed]

Todorovic, M.

Vass, C.

E. Götzinger, M. Pircher, B. Baumann, C. Hirn, C. Vass, and C. K. Hitzenberger, "Retinal nerve fiber layer birefringence evaluated with polarization sensitive spectral domain OCT and scanning laser polarimetry: A comparison," J. Biophoton. 1, 129-139 (2008).
[CrossRef]

Wang, L.

Wang, L. V.

Weissman, N.

G. J. Tearney, M. E. Brezinski, S. A. Boppart, B. E. Bouma, N. Weissman, J. F. Southern, E. A. Swanson, and J. G. Fujimoto, "Images in cardiovascular medicine. Catheter-based optical imaging of a human coronary artery," Circulation 94, 3013 (1996).
[PubMed]

Welzel, J.

J. Welzel, E. Lankenau, R. Birngruber, and R. Engelhardt, "Optical coherence tomography of the human skin," J. Am. Acad. Dermatol. 37, 958-963 (1997).
[CrossRef]

Whittaker, P.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, "Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography," J. Am. Coll. Cardiol. 49, 1474-1481 (2007).
[CrossRef] [PubMed]

Wong, C.

J. S. Schuman, M. R. Hee, C. A. Puliafito, C. Wong, T. Pedut-Kloizman, C. P. Lin, E. Hertzmark, J. A. Izatt, E. A. Swanson, and J. G. Fujimoto, "Quantification of nerve fiber layer thickness in normal and glaucomatous eyes using optical coherence tomography," Arch. Ophthalmol. 113, 586-596 (1995).
[PubMed]

Yamanari, M.

S. Sakai, N. Nakagawa, M. Yamanari, A. Miyazawa, Y. Yasuno, and M. Matsumoto, "Relationship between dermal birefringence and the skin surface roughness of photoaged human skin," J. Biomed. Opt. 14, 044032 (2009).
[CrossRef] [PubMed]

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, "Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry," J. Biomed. Opt. 13, 014013 (2008).
[CrossRef] [PubMed]

M. Miura, M. Yamanari, T. Iwasaki, A. E. Elsner, S. Makita, T. Yatagai, and Y. Yasuno, "Imaging Polarimetry in Age-Related Macular Degeneration," Invest. Ophthalmol. Vis. Sci. 49, 2661-2667 (2008).
[CrossRef] [PubMed]

Yao, G.

Yasuno, Y.

S. Sakai, N. Nakagawa, M. Yamanari, A. Miyazawa, Y. Yasuno, and M. Matsumoto, "Relationship between dermal birefringence and the skin surface roughness of photoaged human skin," J. Biomed. Opt. 14, 044032 (2009).
[CrossRef] [PubMed]

M. Miura, M. Yamanari, T. Iwasaki, A. E. Elsner, S. Makita, T. Yatagai, and Y. Yasuno, "Imaging Polarimetry in Age-Related Macular Degeneration," Invest. Ophthalmol. Vis. Sci. 49, 2661-2667 (2008).
[CrossRef] [PubMed]

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, "Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry," J. Biomed. Opt. 13, 014013 (2008).
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S. Makita, Y. Yasuno, T. Endo, M. Itoh, and T. Yatagai, "Polarization contrast imaging of biological tissues by polarization-sensitive Fourier-domain optical coherence tomography," Appl. Opt. 45, 1142-1147 (2006).
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M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, "Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry," J. Biomed. Opt. 13, 014013 (2008).
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[CrossRef] [PubMed]

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Figures (16)

Fig. 1.
Fig. 1.

Schematic diagram of a fiber-based PS-OCT. PDH,PDV, photodetection devices to detect horizontal and vertical polarization components; Pol. mod., polarization modulator; and Pol., polarizer. In the case of time-domain and spectral-domain OCTs, the light source is a broadband light source. It is wavelength-swept light source for swept-source OCT. Photodetection devices are photodetectors for time-domain and swept-source OCTs. They are spectrometers for spectral-domain OCT. Typically the polarizer is introduced in the reference arm to deliver the same optical power of reference beam to two photodetection devices.

Fig. 2.
Fig. 2.

The relationship between phase retardation and the ratio of SNRs. ESNRPR = 30 dB and incident intensity ratio rI = 1.

Fig. 3.
Fig. 3.

(a, c, e) Measured phase retardations with several set phase retardations r and (b, d, f) contour plots of phase retardation error Δr = r - rm obtained by theoretical analysis according to Eq. (38) (a, b), Monte Carlo simulations of matrix trace method by using Eq. (19) (c, d), and diagonalization method (e, f).

Fig. 4.
Fig. 4.

Plots of phase retardation with background error in (a) matrix trace method and (b) diagonalization method, where the set phase retardation r = 0 radian. The red curve shows the result of theoretical analysis. The box and whisker plots exhibit the distribution from the simulation. Blue squares are the experimental results. The retardation of the local tissue is set to zero.

Fig. 5.
Fig. 5.

Comparison of the histograms obtained by numerical simulations and experiments by using the diagonalization method. There is no phase retardation in the sample. (a),(c),(e) are the results of numerical simulations and (b),(d),(f) are the experimental results. ESNRPR was set to be -20, 5.8, and 16,6 dB from the top row for the simulations. The measured ESNRPR in experiments were 5.8 and 16.6 dB for (d) and (f). There is no light illuminated on the glass plate for (b).

Fig. 6.
Fig. 6.

Comparison of the histograms obtained by numerical simulation and experiment by using the matrix trace method. There is no phase retardation in the sample. (a),(c),(e) are the results of numerical simulation and (b),(d),(f) are the experimental results. The ESNRPR was set to be -20, 5.8, and 16,6 dB from the top row for simulation. The measured ESNRPR in the experiments are 5.8 and 16.6 dB for (d) and (f), respectively. There is no light illuminated on the glass plate for (b).

Fig. 7.
Fig. 7.

Relationship between set phase retardation and measured phase retardation. The standard deviation of phase retardation σrm is distorted due to the nonlinear relationship between the set phase retardation and measured phase retardation.

Fig. 8.
Fig. 8.

Log-log plots of standard deviation of phase retardation to ESNRPR . In both methods, the standard deviations are approaching to σ r = 2 / ESN R PR as ESNRPR increases. The diagonalization method exhibits slightly lower standard deviation than that of the trace method at low ESNRPR .

Fig. 9.
Fig. 9.

Cross-sectional PS-OCT images of the human anterior eye chamber in vivo. (A) polarization-insensitive OCT image; (B) conventional cumulative phase retardation image; (C) local birefringence image with diagonalization method; and (D) local birefringence image using matrix trace method.

Fig. 10.
Fig. 10.

Cross-sectional images of the human skin. (A) The en-face projection image indicates the location of cross sections; (B) Polarization-insensitive OCT image; (C) cumulative phase retardation image; (D) local retardation image by using diagonalization method.

Fig. 11.
Fig. 11.

En-face local birefringence images of the human skin. The projections of (A) polarization-insensitive OCT and (B) local birefringence images show the distribution of pores and high birefringence around them. En- face local birefringence images at the depths of (D) 144; (E) 211; (F) 267; and (G) 383 μm are shown. (C) cross-sectional local retardation image by using diagonalization method. An averaged birefringence inside the yellow box in (F) is 1.96 × 10-3.

Fig. 12.
Fig. 12.

Schematic diagram of the tissue boundary. Tissues 1 and 2 have the birefringences b 1 and b 2, respectively. Their optic axis orientations are θ 1 and θ 2, respectively.

Fig. 13.
Fig. 13.

Theoretical measured birefringence around the tissue boundary. It depends on the difference in the optic axis orientations of tissues Δθ. b 1, b 2: Theoretical birefringences of tissues 1 and 2, respectively.

Fig. 14.
Fig. 14.

Schematic diagram of the local birefringence measurement of the sample that exhibits diattenuation.

Fig. 15.
Fig. 15.

Phase retardation error due to the diattenuation above the local tissue. The numerical simulation results of (a) Eq. (56) and (b) the diagonalization method. β = π/2 and εs = 0.5.

Fig. 16.
Fig. 16.

Numerical simulation of the measured local relative-attenuation εs by using the diagonalization method. εs is set to be 0.5.

Equations (58)

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E out 1 ( z ) = [ H out 1 ( z ) V out 1 ( z ) ] = J ( z ) E out 1 = J ( z ) [ H in 1 V in 1 ]
E out 2 ( z ) = [ H out 2 ( z ) V out 2 ( z ) ] = J ( z ) E out 2 e = J ( z ) [ H in 2 V in 2 ] e
J ( z ) = J out J S , T ( z ) J in ,
[ I ˜ 1 , H ( z ) I ˜ 2 , H ( z ) I ˜ 1 , V ( z ) I ˜ 2 , V ( z ) ] = [ η H H ref * 0 0 η V V ref * ] J ( z ) [ H in 1 e H in 2 V in 1 e V in 2 ] ,
E in = [ H in 1 H in 2 V in 1 V in 2 ]
S ( z ) = [ I ˜ 1 , H ( z ) I ˜ 2 , H ( z ) I ˜ 1 , V ( z ) I ˜ 2 , V ( z ) ] .
S ( z ) = [ η H H ref * 0 0 η V V ref * ] J ( z ) E in [ 1 0 0 e ] .
J ( z ) = [ η H H ref * 0 0 η V V ref * ] 1 S ( z ) [ 1 0 0 e ] 1 E in 1
J ( z ) J ( z ref ) 1 = J out J sample , T J out 1 = [ η H H ref * 0 0 η V V ref * ] 1 S ( z ) S ( z ref ) 1 [ η H H ref * 0 0 η V V ref * ] ,
M z ref z = [ 1 0 0 e ] J out J S , T J out 1 [ 1 0 0 e ] 1 ,
M z i n 1 z i = [ 1 0 0 e ] J out J S , T ( z i ) [ J S , T ( z i n 1 ) ] 1 J out 1 [ 1 0 0 e ] 1 .
J S , T ( z i ) = [ J S T z 1 z 2 J S T z i 1 z i ] [ J S z i 1 z i J S z 1 z 2 ] ,
M z i n 1 z i = [ 1 0 0 e ] J out [ J S T z 1 z 2 J S T z i n 2 z i n 1 ] J S , T z i n z i × [ J S T z 1 z 2 J S T z i n 2 z i n 1 ] 1 J out 1 [ 1 0 0 e ] 1 .
M z i n 1 z i = A [ λ 1 0 0 λ 2 ] A 1 ,
λ 1,2 = R ( z i ) e ( z i ) R ( z i n ) e ( z i n ) exp [ ± ε z i n z i + ir z i n z i 2 ]
A = [ 1 0 0 e ] J out [ J S T z 1 z 2 J S T z i n 2 z i n 1 ] R ( θ ) .
r z i n z i = tan 1 Im [ λ 1 / λ 2 ] Re [ λ 1 / λ 2 ]
ε z i n z i = ln λ 1 / λ 2 .
r z i n z i = 2 cos 1 tr M [ 2 tr ( M M ) ] 1 ,
b z i n z i = r z i n z i 2 k 0 Δ z ,
S = S 0 + [ Δ I r 11 + i Δ Ii 11 Δ I r 12 + i Δ Ii 12 Δ I r 21 + i Δ Ii 21 Δ I r 22 + i Δ Ii 22 ]
S = S 0 + [ Δ I r 11 + i Δ r 11 Δ I r 12 + i Δ Ii 12 Δ I r 21 + i Δ I i 21 Δ I r 22 + i Δ I i 22 ] ,
M = M 0 + [ Δ M r 11 + i Δ M i 11 Δ M r 12 + i Δ M i 12 Δ M r 21 + i Δ M i 21 Δ M r 22 + i Δ M i 22 ] ,
M 0 = S 0 S 0 1 = [ I ˜ 1 , H I ˜ 2 , H I ˜ 1 , V I ˜ 2 , V ] [ I ˜ 1 , H I ˜ 2 , H I ˜ 1 , V I ˜ 2 , V ] 1 .
Δ M r 11 = Δ M i 11 = I ˜ 2 , V 2 Δ I 11 2 + I ˜ 1 , V 2 Δ I 12 2 + I ˜ 2 , H 2 Δ I 21 2 + I ˜ 1 , H 2 Δ I 22 2 det S 0
Δ M r 12 = Δ M i 12 = I ˜ 2 , H 2 Δ I 11 2 + I ˜ 2 , H 2 Δ I 11 2 + I ˜ 1 , H 2 Δ I 12 2 + I ˜ 1 , H 2 Δ I 21 2 det S 0
Δ M r 21 = Δ M i 21 = I ˜ 2 , V 2 Δ I 21 2 + I ˜ 2 , V 2 Δ I 21 2 + I ˜ 1 , V 2 Δ I 22 2 + I ˜ 1 , V 2 Δ I 22 2 det S 0
Δ M r 22 = Δ M i 22 = I ˜ 2 , V 2 Δ I 11 2 + I ˜ 1 , V 2 Δ I 21 2 + I ˜ 2 , H 2 Δ I 21 2 + I ˜ 1 , V 2 Δ I 22 2 det S 0 .
r m = 2 cos 1 [ 2 tr [ M 0 M 0 ] cos 2 r 2 + 2 Δ I 2 det S 0 2 ( tr [ S 0 S 0 ] + tr [ S 0 S 0 ] ) 2 tr [ M 0 M 0 ] + 4 Δ I 2 det S 0 2 ( tr [ S 0 S 0 ] + tr [ S 0 S 0 ] ) ] .
J = R [ a b * b a * ] ,
S = η R I ref [ a H in 1 b * V in 1 a H in 2 b * V in 2 e b H in 1 + a * V in 1 e b H in 2 + a * V in 2 e e ] .
det S = η 2 R I ref det E in .
det E in = H in 1 V in 2 V in 1 H in 2 = I in 1 I in 2 sin ζ / 2 ,
tr [ S 0 S 0 ] = η 2 R I ref ( I in 1 + I in 2 )
det S 0 = η 2 R I ref I in 1 I in 2 sin ζ / 2
tr [ M 0 M 0 ] = 2 R R
_
r m = 2 cos 1 [ cos 2 r 2 + ESN R PR 1 csc 2 ζ 2 1 + 4 ESN R PR 1 ζ 2 ] .
1 ESN R PR = 1 4 ( 1 SN R 1 + 1 SN R 2 + 1 SN R 1 + 1 SN R 2 ) .
[ I ˜ 1 , H ( z ) I ˜ 2 , H ( z ) I ˜ 1 , V ( z ) I ˜ 2 , V ( z ) ] = J [ H in 1 H in 2 V in 1 V in 2 ] + [ n 11 ( z ) n 12 ( z ) n 21 ( z ) n 22 ( z ) ] ,
J = J U 1 J U 2 .
J = J U 1 J lr r θ J U 2 ,
σ r = σ r m r m r ,
J 1 z u z d = R ( θ 1 ) [ exp [ i z u z d b 1 2 kzdz ] 0 0 exp [ i z u z d b 1 2 kzdz ] ] R ( θ 1 )
J 2 z u z d = R ( θ 2 ) [ exp [ i z u z d b 2 2 kzdz ] 0 0 exp [ i z u z d b 2 2 kzdz ] ] R ( θ 2 ) ,
J T ( z 1 ) = J 1 T z surf z 1 J 1 z surf z 1
J T ( z 2 ) = J 1 T z surf z b J 2s T z b z 2 J 2 z b z 2 J 1 z surf z b ,
b ( z ) = r ( z Δ z / 2 , z + Δ z / 2 ) 2 k 0 Δ z = { b 1 ( z z b Δ z 2 ) cos 1 [ cos ( Δ b k 0 z 1 2 b s k 0 Δ z ) cos 2 ( Δ θ ) + cos ( b s k 0 z 1 2 Δ b k 0 Δ z ) sin 2 ( Δ θ ) ] k 0 Δ z ( z b Δ z 2 < z < z b + Δ z 2 ) b 2 ( z b z b + Δ z 2 )
J ( z i n 1 ) = J out J S T z surf z i n 1 J S z surf z i n 1 J in
J ( z i ) = J out J S T z surf z i n 1 J S T z i n i z i J S z surf z i n 1 J in .
J ( z i ) J 1 ( z i n 1 ) = J D J R J S T z i n z i ( J R ) 1 ( J D ) 1 J out 1 ,
J ( z i ) J 1 ( z i n 1 ) = J U 1 [ P 1 0 0 P 2 ] J U 2 [ P s 1 e ir / 2 0 0 P s 2 e ir / 2 ] J U 2 1 [ P 1 0 0 P 2 ] 1 J U 1 1 ,
J U 1 = J U T
J U 2 = J U J R R ( θ s )
J U 2 = [ e / 2 0 0 e / 2 ] [ cos ( β / 2 ) sin ( β / 2 ) sin ( β / 2 ) cos ( β / 2 ) ] [ e 0 0 e ] .
r m = 2 cos 1 [ cos 2 r 2 + 1 d 1 +d sin 2 r 2 2 1 1 + d + { 1 + D 2 2 ( 1 D 2 ) [ 1 d 1 + d cos 2 r 2 + sin 2 r 2 ] d 1 + d sin 2 r 2 1 d 2 ( 1 + d ) } sin 2 β ] ,
D = P 1 2 P 2 2 P 1 2 + P 2 2 = tanh ε
D s = P s 1 2 P s 2 2 P s 1 2 + P s 2 2 = tanh ε s ,

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