Abstract

Polarization mode dispersion (PMD), which can be induced by circulators or even moderate lengths of optical fiber, is known to be a dominant source of instrumentation noise in fiber-based PS-OCT systems. In this paper we propose a novel PMD compensation method that measures system PMD using three fixed calibration signals, numerically corrects for these instrument effects and reconstructs an improved sample image. Using a frequency multiplexed PS-OFDI setup, we validate the proposed method by comparing birefringence noise in images of intralipid, muscle, and tendon with and without PMD compensation.

© 2013 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2012 (2)

2011 (3)

2010 (1)

2009 (1)

2008 (3)

2007 (1)

S. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, E. F. Halpern, S. L. Houser, B. E. Bouma, 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]

2005 (2)

2004 (2)

2003 (1)

2002 (1)

2001 (1)

B.H. Park, C. Saxer, S.M. Srinivas, J.S. Nelson, and J.F. de Boer, “In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography,” J. Biomed. Opt.6, 474479 (2001).
[CrossRef]

2000 (2)

1999 (1)

1997 (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,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

1988 (1)

Andrekson, P. A.

Baumann, B.

Boudoux, C.

Bouma, B.

Bouma, B. E.

Cense, B.

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,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Chen, Z.

Choi, W.

de Boer, J.

de Boer, J. F.

K. H. Kim, B. H. Park, Y. Tu, T. Hasan, B. Lee, J. Li, and J. F. de Boer, “Polarization-sensitive optical frequency domain imaging based on unpolarized light,” Opt. Express19, 552–561 (2011). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-552
[CrossRef] [PubMed]

W.Y. Oh, S. H. Yun, B. J. Vakoc, M. Shishkov, A. E. Desjardins, B. H. Park, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “High-speed polarization sensitive optical frequency domain imaging with frequency multiplexing,” Opt. Express16, 1096–1103 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-1096
[CrossRef] [PubMed]

S. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, E. F. Halpern, S. L. Houser, B. E. Bouma, 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. J. Vakoc, S. H. Yun, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express13, 5483–6593 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-14-5483
[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, 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, “Eye-length measurement by interferometry with partially coherent light,” Opt. Lett.22, 934–936 (1997).
[CrossRef] [PubMed]

de Boer, J.F.

B.H. Park, C. Saxer, S.M. Srinivas, J.S. Nelson, and J.F. de Boer, “In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography,” J. Biomed. Opt.6, 474479 (2001).
[CrossRef]

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]

Desjardins, A. E.

Duker, J. S.

Fercher, A. F.

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,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

Gentle, J.

J. Gentle, Numerical Linear Algebra for Applications in Statistics (Springer, 1998), chap. 2.
[CrossRef]

Götzinger, E.

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,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Halpern, E. F.

S. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, E. F. Halpern, S. L. Houser, B. E. Bouma, 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]

Hasan, T.

Hee, M. R.

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. B9, 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,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hitzenberger, C. K.

Houser, S. L.

S. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, E. F. Halpern, S. L. Houser, B. E. Bouma, 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.

Jiao, S.

Karlsson, M.

Kim, K. H.

Lee, B.

Li, J.

Lin, C. P.

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,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Lu, Z.

Makita, S.

Matcher, S. J.

Mengedoht, K.

Milner, T. E.

Nadkarni, S.

S. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, E. F. Halpern, S. L. Houser, B. E. Bouma, 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]

Nelson, J. S.

Nelson, J.S.

B.H. Park, C. Saxer, S.M. Srinivas, J.S. Nelson, and J.F. de Boer, “In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography,” J. Biomed. Opt.6, 474479 (2001).
[CrossRef]

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]

Oh, W. Y.

Oh, W.Y.

Park, B. H.

K. H. Kim, B. H. Park, Y. Tu, T. Hasan, B. Lee, J. Li, and J. F. de Boer, “Polarization-sensitive optical frequency domain imaging based on unpolarized light,” Opt. Express19, 552–561 (2011). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-552
[CrossRef] [PubMed]

W.Y. Oh, S. H. Yun, B. J. Vakoc, M. Shishkov, A. E. Desjardins, B. H. Park, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “High-speed polarization sensitive optical frequency domain imaging with frequency multiplexing,” Opt. Express16, 1096–1103 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-1096
[CrossRef] [PubMed]

S. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, E. F. Halpern, S. L. Houser, B. E. Bouma, 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]

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]

Park, B.H.

B.H. Park, C. Saxer, S.M. Srinivas, J.S. Nelson, and J.F. de Boer, “In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography,” J. Biomed. Opt.6, 474479 (2001).
[CrossRef]

Pierce, M. C.

Pircher, M.

Potsaid, B.

Puliafito, C. A.

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,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Samuelsson, R.

Saxer, C.

B.H. Park, C. Saxer, S.M. Srinivas, J.S. Nelson, and J.F. de Boer, “In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography,” J. Biomed. Opt.6, 474479 (2001).
[CrossRef]

Saxer, C. E.

Schmidt-Erfurth, U.

M. Pircher, C. K. Hitzenberger, and U. Schmidt-Erfurth, “Polarization sensitive optical coherence tomography in the human eye,” Prog. Retin. Eye Res.30, 431451 (2011).
[CrossRef]

Schuman, J. S.

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,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Shishkov, M.

Srinivas, S.M.

B.H. Park, C. Saxer, S.M. Srinivas, J.S. Nelson, and J.F. de Boer, “In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography,” J. Biomed. Opt.6, 474479 (2001).
[CrossRef]

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,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sunnerud, H.

Swanson, E. A.

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. B9, 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,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Tearney, G.

Tearney, G. J.

Tu, Y.

Vakoc, B. J.

van Gemert, M. J. C.

Wang, L. V.

Werner, W.

Xie, C.

Yamanari, M.

Yao, G.

Yasuno, Y.

Yun, S.

Yun, S. H.

Zhang, E. Z.

Zhao, Y.

Appl. Opt. (1)

J. Am. Coll. Cardiol. (1)

S. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, E. F. Halpern, S. L. Houser, B. E. Bouma, 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]

J. Biomed. Opt. (1)

B.H. Park, C. Saxer, S.M. Srinivas, J.S. Nelson, and J.F. de Boer, “In vivo burn depth determination by high-speed fiber-based polarization sensitive optical coherence tomography,” J. Biomed. Opt.6, 474479 (2001).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Opt. Express (9)

S. Yun, G. Tearney, J. de Boer, and B. Bouma, “Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting,” Opt. Express12, 4822–4828 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4822
[CrossRef] [PubMed]

B. J. Vakoc, S. H. Yun, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “Phase-resolved optical frequency domain imaging,” Opt. Express13, 5483–6593 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-14-5483
[CrossRef] [PubMed]

W.Y. Oh, S. H. Yun, B. J. Vakoc, M. Shishkov, A. E. Desjardins, B. H. Park, J. F. de Boer, G. J. Tearney, and B. E. Bouma, “High-speed polarization sensitive optical frequency domain imaging with frequency multiplexing,” Opt. Express16, 1096–1103 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-1096
[CrossRef] [PubMed]

M. Yamanari, S. Makita, and Y. Yasuno, “Polarization-sensitive swept-source optical coherence tomography with continuous source polarization modulation,” Opt. Express16, 5892–5906 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5892
[CrossRef] [PubMed]

E. Götzinger, B. Baumann, M. Pircher, and C. K. Hitzenberger, “Polarization maintaining fiber based ultra-high resolution spectral domain polarization sensitive optical coherence tomography,” Opt. Express17, 22704–22717 (2009). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-22704
[CrossRef]

S. Makita, M. Yamanari, and Y. Yasuno, “Generalized Jones matrix optical coherence tomography: performance and local birefringence imaging,” Opt. Express18, 854–876 (2010). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-2-854
[CrossRef] [PubMed]

K. H. Kim, B. H. Park, Y. Tu, T. Hasan, B. Lee, J. Li, and J. F. de Boer, “Polarization-sensitive optical frequency domain imaging based on unpolarized light,” Opt. Express19, 552–561 (2011). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-552
[CrossRef] [PubMed]

E. Z. Zhang and B. J. Vakoc, “Polarimetry noise in fiber-based optical coherence tomography instrumentation,” Opt. Express19, 16830–16842 (2011). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16830
[CrossRef] [PubMed]

B. Baumann, W. Choi, B. Potsaid, D. Huang, J. S. Duker, and J. G. Fujimoto, “Swept source / Fourier domain polarization sensitive optical coherence tomography with a passive polarization delay unit,” Opt. Express20, 10229–10241 (2012). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-10229
[CrossRef]

Opt. Lett. (9)

Z. Lu and S. J. Matcher, “Absolute fast axis determination using non-polarization- maintaining fiber-based polarization-sensitive optical coherence tomography,” Opt. Lett.37, 1931–1933 (2012).
[CrossRef] [PubMed]

W. Y. Oh, B. J. Vakoc, S. H. Yun, G. J. Tearney, and B. E. Bouma, “Single-detector polarization-sensitive optical frequency domain imaging using high-speed intra A-line polarization modulation,” Opt. Lett.33, 1330–1332 (2008).
[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]

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Prog. Retin. Eye Res. (1)

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[CrossRef]

Science (1)

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[CrossRef]

http://www.mathworks.com/help/toolbox/optim/ug/fsolve.html

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Figures (9)

Fig. 1
Fig. 1

(a) A simplified fiber-based PS-OCT system based on a wavelength-swept source is presented. The system is constructed with a polarization-diverse receiver (unbalanced for simplicity) which in combination with heterodyne detection enables wavelength-by-wavelength polarimetry measurements of the sample arm light at D. (b) An optically equivalent description of the system in (a) where the distributed sample arm properties have been combined into three blocks INST1, SMPL, and INST2.

Fig. 2
Fig. 2

The influence of PMD induced by the instrument (elements INST1 and INST2) can be eliminated by inserting two PMD compensators (COMP1 and COMP2) configured to cancel the affects of INST1 and INST2 respectively.

Fig. 3
Fig. 3

A PS-OCT system with simultaneous launching and independent detection of two polarization states. The source is split into two orthogonal polarizations by a first PBS, and each is encoded with a unique acousto-optic frequency shifter (AOFS). The signals are then recombined by a second PBS. System design simplified from ref [11].

Fig. 4
Fig. 4

The optical block design of an OCT system providing three independent signal paths/Jones matrices. From these signals, the required compensating matrices Cin(k) and Cout(k) can be derived analytically.

Fig. 5
Fig. 5

Experimental setup of frequency multiplexing PS-OFDI system (FS: frequency shifter; FS1: 50MHz, FS2: 25MHz upshift, FS3: 25MHz downshift; LP: linear polarizer; PC: polarization controller; BBS: broad beamsplitter; PBS: polarization beamsplitter; BR: balanced receiver; A/D: analog-to-digital converter).

Fig. 6
Fig. 6

The microscope used to provide three calibrating signals (BBS: broad beamsplitter; W1: quarter waveplate (QWP) oriented at 0°; W2: QWP oriented at 45°)

Fig. 7
Fig. 7

Validation of compensation method by imaging a mirror sample, normalized Stokes vectors are plotted along swept wavelengths on Poincaré Sphere: (a) Mirror signal without and with compensation of both polarization states; (b) With another configuration of input and output polarization controllers, mirror signal without and with compensation.

Fig. 8
Fig. 8

Numerical PMD compensation reduces noise in PS-OCT images of local birefringence in intralipid phantom. (a) Intensity image from one input polarization state; (b) local birefringence image without PMF patchcord and without numerical PMD compensation; (c) local birefringence image with 0.08ps PMF patchcord and without numerical PMD compensation; (d) local birefringence image with 0.08ps PMF patchcord and with numerical PMD compensation. Local birefringence was calculated across an axial extent of 6 pixels. Colorbar: 0 − 1.3 deg/μm

Fig. 9
Fig. 9

Numerical PMD compensation reduces noise in PS-OCT images of local birefringence of muscle/tendon. (a) Intensity image from one input polarization state; (b) baseline local birefringence without PMF patchcords and without PMD compensation; (c) local birefringence image with 0.04ps PMF patchcord and without numerical PMD compensation; (d) local birefringence image with 0.04ps PMF patchcord and with numerical PMD compensation; (e) local birefringence image with 0.08ps PMF patchcord and without numerical PMD compensation; (f) local birefringence image with 0.08ps PMF patchcord and with numerical PMD compensation. Birefringence values of tendon were calculated to be 1.07 deg/μm, 1.10 deg/μm and 1.10 deg/μm from ROI(t) in Fig. 9(b), Fig. 9(d) and Fig. 9(f), respectively. Birefringence values of muscle were calculated to be 0.38 deg/μm, 0.39 deg/μm and 0.39 deg/μm from ROI(m) in Fig. 9(b), Fig. 9(d) and Fig. 9(f), respectively. Local birefringence was calculated across an axial extent of 6 pixels. ROI: 40*40 pixels. Colorbar: 0 − 2 deg/μm

Equations (29)

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M ( k ) = T out ( k ) R ( k ) T in ( k ) [ 1 0 ] ,
M ( k ) = [ C out ( k ) T out ( k ) ] R ( k ) [ T in ( k ) C in ( k ) ] [ 1 0 ] ,
M ( k ) = R ( k ) [ 1 0 ] .
M ( k ) = T out ( k ) R ( k ) T in ( k ) S ( k ) ,
C out ( k ) M ( k ) C in ( k ) ,
C in ( k ) = T in 1 ( k ) C out ( k ) = T out 1 ( k ) .
C out ( k ) M ( k ) C in ( k ) = R ( k ) .
M 1 ( k ) = T out ( k ) R 1 ( k ) T in ( k )
M 2 ( k ) = T out ( k ) R 2 ( k ) T in ( k )
M 3 ( k ) = T out ( k ) R 3 ( k ) T i n ( k ) ,
[ M 2 ( k ) M 1 1 ( k ) ] = T out ( k ) [ R 2 ( k ) R 1 1 ( k ) ] T out 1 ( k )
[ M 3 ( k ) M 1 1 ( k ) ] = T out ( k ) [ R 3 ( k ) R 1 1 ( k ) ] T out 1 ( k ) ,
R 1 ( k ) = [ 1 0 0 1 ] R 2 ( k ) = [ 1 0 0 e i 2 Γ ( k ) ] R 3 ( k ) = 1 2 [ 1 + e i 2 Γ ( k ) 1 e i 2 Γ ( k ) 1 e i 2 Γ ( k ) 1 + e i 2 Γ ( k ) ] ,
M 1 ( k ) = c 1 ( k ) T out ( k ) R 1 ( k ) T in ( k ) K ( k ) M 2 ( k ) = c 2 ( k ) T out ( k ) R 2 ( k ) T in ( k ) K ( k ) M 3 ( k ) = c 3 ( k ) T out ( k ) R 3 ( k ) T in ( k ) K ( k ) .
( c 1 ( k ) c 2 ( k ) ) [ M 2 ( k ) M 1 1 ( k ) ] = T out ( k ) [ R 2 ( k ) R 1 1 ( k ) ] T out 1 ( k )
( c 1 ( k ) c 3 ( k ) ) [ M 3 ( k ) M 1 1 ( k ) ] = T out ( k ) [ R 3 ( k ) R 1 1 ( k ) ] T out 1 ( k ) ,
R 2 ( k ) R 1 1 ( k ) = V 21 ( k ) D 21 ( k ) V 21 1 ( k ) ,
M 2 ( k ) M 1 1 ( k ) = U 21 ( k ) d 21 ( k ) U 21 1 ( k ) ,
( c 1 ( k ) c 2 ( k ) ) M 2 ( k ) M 1 1 ( k ) = U 21 ( k ) D 21 ( k ) U 21 1 ( k ) .
C in ( k ) = [ T in ( k ) K ( k ) ] 1 C out ( k ) = T out 1 ( k ) .
[ M 2 ( k ) M 1 1 ( k ) ] = T out ( k ) [ R 2 ( k ) R 1 1 ( k ) ] T out 1 ( k ) ,
[ U 21 D U 21 1 ] = T out [ V 21 D V 21 1 ] T out 1 ,
D = U 21 1 T out [ V 21 D V 21 1 ] T out 1 U 21 = [ U 21 1 T out V 21 ] D [ V 21 1 T out 1 U 21 ] = [ U 21 1 T out V 21 ] D [ U 21 1 T out V 21 ] 1 .
[ U 21 1 T out V 21 ] = [ α 11 0 0 α 22 ] ,
{ [ U 21 1 A 1 V 21 ] = [ 1 0 0 0 ] [ U 21 1 A 2 V 21 ] = [ 0 0 0 1 ] .
[ M 3 ( k ) M 1 1 ( k ) ] = T out ( k ) [ R 3 ( k ) R 1 1 ( k ) ] T out 1 ( k ) .
[ U 31 1 T out V 31 ] = [ β 11 0 0 β 22 ] ,
{ [ U 31 1 B 1 V 31 ] = [ 1 0 0 0 ] [ U 31 1 B 2 V 31 ] = [ 0 0 0 1 ] .
[ A 1 ( 11 ) A 2 ( 11 ) B 1 ( 11 ) B 2 ( 11 ) A 1 ( 12 ) A 2 ( 12 ) B 1 ( 12 ) B 2 ( 12 ) A 1 ( 21 ) A 2 ( 21 ) B 1 ( 21 ) B 2 ( 21 ) A 1 ( 22 ) A 2 ( 22 ) B 1 ( 22 ) B 2 ( 22 ) ] [ α 11 α 22 β 11 β 22 ] = 0 ,

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