Abstract

We demonstrate cross-sectional birefringence- and polarization-independent backscatter imaging of laser-induced thermal damage in porcine myocardium in vitro, using a polarization-sensitive optical coherence tomography system. We compare the generated images with histological sections of the tissue and demonstrate that birefringence is a more sensitive indicator of thermal damage than is backscattered light. Loss of birefringence in thermally damaged regions is quantified and shown to have significant contrast with undamaged sections of the tissue. A detailed theoretical analysis of the birefringence measurements is provided, including a calculation of the systematic errors associated with background noise, system imperfections, and tissue dichroism.

© 1998 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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1998 (1)

1997 (2)

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

D. J. Maitland, J. T. Walsh, “Quantitative measurements of linear birefringence during the heating of native collagen,” Lasers Surg. Med. 20, 310–318 (1997).
[CrossRef]

1996 (2)

J. A. Izatt, M. D. Kulkarni, H. Wang, K. Kobayashi, M. V. Sivak, D. S. Klinger, J. W. Lewis, C. E. Randall, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

G. J. Hruza, J. S. Dover, “Laser skin resurfacing,” Arch. Dermatol. 132, 451–455 (1996).
[CrossRef] [PubMed]

1995 (1)

L. S. Bass, M. R. Treat, “Laser tissue welding: a comprehensive review of current and future clinical applications,” Lasers Surg. Med. 17, 315–349 (1995).
[CrossRef] [PubMed]

1992 (2)

W. V. Sorin, D. F. Gray, “Simultaneous thickness and group index measurement using optical low coherence reflectometry,” IEEE Photon. Technol. Lett. 4, 105–107 (1992).
[CrossRef]

M. R. Hee, D. Huang, E. A. Swanson, J. G. Fujimoto, “Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging,” J. Opt. Soc. Am. 9, 903–98 (1992).
[CrossRef]

1991 (1)

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

1990 (1)

A. Masters, S. G. Bown, “Interstitial laser hyperthermia in the treatment of tumors,” Lasers Med. Sci. 5, 129–135 (1990).
[CrossRef]

1989 (1)

S. Thomsen, J. A. Pearce, W. F. Cheong, “Changes in birefringence as markers of thermal damage in tissues,” IEEE Trans. Biomed. Eng. 36, 1174–1179 (1989).
[CrossRef] [PubMed]

1988 (1)

1985 (1)

1966 (1)

K. Yoshioka, C. T. O’Konski, “Electric properties of macromolecules. IX. Dipole moment, polarizability, and optical anisotropy factor of collagen in solution from electric birefringence,” Biopolymers 4, 499–507 (1966).
[CrossRef] [PubMed]

1955 (1)

C. Cohen, “Optical rotation and helical polypeptide chain configuration in collagen and gelatin,” J. Biophys. Biochem. Cytol. 1, 203–214 (1955).
[CrossRef] [PubMed]

1941 (2)

Bass, L. S.

L. S. Bass, M. R. Treat, “Laser tissue welding: a comprehensive review of current and future clinical applications,” Lasers Surg. Med. 17, 315–349 (1995).
[CrossRef] [PubMed]

Bicout, D.

C. Brosseau, D. Bicout, A. S. Martinez, J. M. Schmitt, “Depolarization behavior of multiple scattered light from an optically dense random medium,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 50–54.

Bown, S. G.

A. Masters, S. G. Bown, “Interstitial laser hyperthermia in the treatment of tumors,” Lasers Med. Sci. 5, 129–135 (1990).
[CrossRef]

Brink, H. B. klein

Brosseau, C.

C. Brosseau, D. Bicout, A. S. Martinez, J. M. Schmitt, “Depolarization behavior of multiple scattered light from an optically dense random medium,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 50–54.

Chang, W.

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

Cheong, W. F.

S. Thomsen, J. A. Pearce, W. F. Cheong, “Changes in birefringence as markers of thermal damage in tissues,” IEEE Trans. Biomed. Eng. 36, 1174–1179 (1989).
[CrossRef] [PubMed]

Cohen, C.

C. Cohen, “Optical rotation and helical polypeptide chain configuration in collagen and gelatin,” J. Biophys. Biochem. Cytol. 1, 203–214 (1955).
[CrossRef] [PubMed]

Collett, E.

E. Collett, Polarized light: Fundamentals and Applications, Vol. 36 of Optical Engineering Series (Dekker, New York, 1993).

Colston, B. W.

DaSilva, L. B.

deBoer, J. F.

Dover, J. S.

G. J. Hruza, J. S. Dover, “Laser skin resurfacing,” Arch. Dermatol. 132, 451–455 (1996).
[CrossRef] [PubMed]

Everett, M. J.

Flotte, T.

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

Fujimoto, J. G.

M. R. Hee, D. Huang, E. A. Swanson, J. G. Fujimoto, “Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging,” J. Opt. Soc. Am. 9, 903–98 (1992).
[CrossRef]

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

Gray, D. F.

W. V. Sorin, D. F. Gray, “Simultaneous thickness and group index measurement using optical low coherence reflectometry,” IEEE Photon. Technol. Lett. 4, 105–107 (1992).
[CrossRef]

Gregory, K.

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

Han, A.

T. McMurray, A. Han, J. A. Pearce, “Thermal damage quantification from tissue birefringence image analysis,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 140–151 (1993).
[CrossRef]

Hee, M. R.

M. R. Hee, D. Huang, E. A. Swanson, J. G. Fujimoto, “Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging,” J. Opt. Soc. Am. 9, 903–98 (1992).
[CrossRef]

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

Hruza, G. J.

G. J. Hruza, J. S. Dover, “Laser skin resurfacing,” Arch. Dermatol. 132, 451–455 (1996).
[CrossRef] [PubMed]

Huang, D.

M. R. Hee, D. Huang, E. A. Swanson, J. G. Fujimoto, “Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging,” J. Opt. Soc. Am. 9, 903–98 (1992).
[CrossRef]

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

Hurwitz, H.

Izatt, J. A.

J. A. Izatt, M. D. Kulkarni, H. Wang, K. Kobayashi, M. V. Sivak, D. S. Klinger, J. W. Lewis, C. E. Randall, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Jones, C. R.

Kliger, D. S.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, San Diego, Calif., 1990).

Klinger, D. S.

J. A. Izatt, M. D. Kulkarni, H. Wang, K. Kobayashi, M. V. Sivak, D. S. Klinger, J. W. Lewis, C. E. Randall, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Kobayashi, K.

J. A. Izatt, M. D. Kulkarni, H. Wang, K. Kobayashi, M. V. Sivak, D. S. Klinger, J. W. Lewis, C. E. Randall, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Kulkarni, M. D.

J. A. Izatt, M. D. Kulkarni, H. Wang, K. Kobayashi, M. V. Sivak, D. S. Klinger, J. W. Lewis, C. E. Randall, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Lewis, J. W.

J. A. Izatt, M. D. Kulkarni, H. Wang, K. Kobayashi, M. V. Sivak, D. S. Klinger, J. W. Lewis, C. E. Randall, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, San Diego, Calif., 1990).

Lin, C. P.

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

Maitland, D. J.

D. J. Maitland, J. T. Walsh, “Quantitative measurements of linear birefringence during the heating of native collagen,” Lasers Surg. Med. 20, 310–318 (1997).
[CrossRef]

D. J. Maitland, J. T. Walsh, “Interference-based linear birefringence measurement of thermally-induced changes in collagen,” in Laser-Tissue Interaction V, S. L. Jacques, ed., Proc. SPIE2134A, 304–308 (1994).

Martinez, A. S.

C. Brosseau, D. Bicout, A. S. Martinez, J. M. Schmitt, “Depolarization behavior of multiple scattered light from an optically dense random medium,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 50–54.

Masters, A.

A. Masters, S. G. Bown, “Interstitial laser hyperthermia in the treatment of tumors,” Lasers Med. Sci. 5, 129–135 (1990).
[CrossRef]

McMurray, T.

J. A. Pearce, S. Thomsen, H. Vijverber, T. McMurray, “Kinetics for birefringence changes in thermally coagulated rat skin collagen,” in Lasers in Otolaryngology, Dermatology, and Tissue Welding, S. M. Shapshay, R. R. Anderson, J. V. White, R. A. White, L. S. Bass, eds., Proc. SPIE1876, 180–186 (1993).
[CrossRef]

T. McMurray, A. Han, J. A. Pearce, “Thermal damage quantification from tissue birefringence image analysis,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 140–151 (1993).
[CrossRef]

Milner, T. E.

Nelson, J. S.

O’Konski, C. T.

K. Yoshioka, C. T. O’Konski, “Electric properties of macromolecules. IX. Dipole moment, polarizability, and optical anisotropy factor of collagen in solution from electric birefringence,” Biopolymers 4, 499–507 (1966).
[CrossRef] [PubMed]

Pearce, J. A.

S. Thomsen, J. A. Pearce, W. F. Cheong, “Changes in birefringence as markers of thermal damage in tissues,” IEEE Trans. Biomed. Eng. 36, 1174–1179 (1989).
[CrossRef] [PubMed]

T. McMurray, A. Han, J. A. Pearce, “Thermal damage quantification from tissue birefringence image analysis,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 140–151 (1993).
[CrossRef]

J. A. Pearce, S. Thomsen, H. Vijverber, T. McMurray, “Kinetics for birefringence changes in thermally coagulated rat skin collagen,” in Lasers in Otolaryngology, Dermatology, and Tissue Welding, S. M. Shapshay, R. R. Anderson, J. V. White, R. A. White, L. S. Bass, eds., Proc. SPIE1876, 180–186 (1993).
[CrossRef]

Puliafito, C. A.

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

Randall, C. E.

J. A. Izatt, M. D. Kulkarni, H. Wang, K. Kobayashi, M. V. Sivak, D. S. Klinger, J. W. Lewis, C. E. Randall, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, San Diego, Calif., 1990).

Schmitt, J. M.

C. Brosseau, D. Bicout, A. S. Martinez, J. M. Schmitt, “Depolarization behavior of multiple scattered light from an optically dense random medium,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 50–54.

Schoenenberger, K.

Shuman, J. S.

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

Sivak, M. V.

J. A. Izatt, M. D. Kulkarni, H. Wang, K. Kobayashi, M. V. Sivak, D. S. Klinger, J. W. Lewis, C. E. Randall, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Sorin, W. V.

W. V. Sorin, D. F. Gray, “Simultaneous thickness and group index measurement using optical low coherence reflectometry,” IEEE Photon. Technol. Lett. 4, 105–107 (1992).
[CrossRef]

Stinson, W.

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

Swanson, E. A.

M. R. Hee, D. Huang, E. A. Swanson, J. G. Fujimoto, “Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging,” J. Opt. Soc. Am. 9, 903–98 (1992).
[CrossRef]

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

Thomsen, S.

S. Thomsen, J. A. Pearce, W. F. Cheong, “Changes in birefringence as markers of thermal damage in tissues,” IEEE Trans. Biomed. Eng. 36, 1174–1179 (1989).
[CrossRef] [PubMed]

J. A. Pearce, S. Thomsen, H. Vijverber, T. McMurray, “Kinetics for birefringence changes in thermally coagulated rat skin collagen,” in Lasers in Otolaryngology, Dermatology, and Tissue Welding, S. M. Shapshay, R. R. Anderson, J. V. White, R. A. White, L. S. Bass, eds., Proc. SPIE1876, 180–186 (1993).
[CrossRef]

Treat, M. R.

L. S. Bass, M. R. Treat, “Laser tissue welding: a comprehensive review of current and future clinical applications,” Lasers Surg. Med. 17, 315–349 (1995).
[CrossRef] [PubMed]

van Blokland, G. J.

van Gemert, M. J. C.

Vijverber, H.

J. A. Pearce, S. Thomsen, H. Vijverber, T. McMurray, “Kinetics for birefringence changes in thermally coagulated rat skin collagen,” in Lasers in Otolaryngology, Dermatology, and Tissue Welding, S. M. Shapshay, R. R. Anderson, J. V. White, R. A. White, L. S. Bass, eds., Proc. SPIE1876, 180–186 (1993).
[CrossRef]

Walsh, J. T.

D. J. Maitland, J. T. Walsh, “Quantitative measurements of linear birefringence during the heating of native collagen,” Lasers Surg. Med. 20, 310–318 (1997).
[CrossRef]

D. J. Maitland, J. T. Walsh, “Interference-based linear birefringence measurement of thermally-induced changes in collagen,” in Laser-Tissue Interaction V, S. L. Jacques, ed., Proc. SPIE2134A, 304–308 (1994).

Wang, H.

J. A. Izatt, M. D. Kulkarni, H. Wang, K. Kobayashi, M. V. Sivak, D. S. Klinger, J. W. Lewis, C. E. Randall, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Yoshioka, K.

K. Yoshioka, C. T. O’Konski, “Electric properties of macromolecules. IX. Dipole moment, polarizability, and optical anisotropy factor of collagen in solution from electric birefringence,” Biopolymers 4, 499–507 (1966).
[CrossRef] [PubMed]

Arch. Dermatol. (1)

G. J. Hruza, J. S. Dover, “Laser skin resurfacing,” Arch. Dermatol. 132, 451–455 (1996).
[CrossRef] [PubMed]

Biopolymers (1)

K. Yoshioka, C. T. O’Konski, “Electric properties of macromolecules. IX. Dipole moment, polarizability, and optical anisotropy factor of collagen in solution from electric birefringence,” Biopolymers 4, 499–507 (1966).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

J. A. Izatt, M. D. Kulkarni, H. Wang, K. Kobayashi, M. V. Sivak, D. S. Klinger, J. W. Lewis, C. E. Randall, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

W. V. Sorin, D. F. Gray, “Simultaneous thickness and group index measurement using optical low coherence reflectometry,” IEEE Photon. Technol. Lett. 4, 105–107 (1992).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

S. Thomsen, J. A. Pearce, W. F. Cheong, “Changes in birefringence as markers of thermal damage in tissues,” IEEE Trans. Biomed. Eng. 36, 1174–1179 (1989).
[CrossRef] [PubMed]

J. Biophys. Biochem. Cytol. (1)

C. Cohen, “Optical rotation and helical polypeptide chain configuration in collagen and gelatin,” J. Biophys. Biochem. Cytol. 1, 203–214 (1955).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Lasers Med. Sci. (1)

A. Masters, S. G. Bown, “Interstitial laser hyperthermia in the treatment of tumors,” Lasers Med. Sci. 5, 129–135 (1990).
[CrossRef]

Lasers Surg. Med. (2)

L. S. Bass, M. R. Treat, “Laser tissue welding: a comprehensive review of current and future clinical applications,” Lasers Surg. Med. 17, 315–349 (1995).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Schematic of the polarization-sensitive OCT system: SLD, superluminescent diode; PBS’s, polarizing beam splitter cubes; BS, beam splitter; QW’s, zero-order quarter-wave plates; TSM, transverse scanning mirror; L’s, lenses; SMF’s, single-mode fibers; D H , D V , photodetectors collecting horizontally and vertically polarized light, respectively.

Fig. 2
Fig. 2

Systematic errors in measuring phase retardation δ with a polarization-sensitive OCT system. (a) Contour plot of the phase-measurement error Δδ N caused by noise versus actual phase retardation δ and signal-to-noise ratio N r . (b) Contour plot of the phase-measurement error Δδε caused by imperfections in the OCT system versus actual phase retardation δ and extinction coefficient of the system ε = ε H = ε V .

Fig. 3
Fig. 3

Systematic error caused by dichroism in measuring phase retardation δ with a polarization-sensitive OCT system. (a) Contour plot of the phase-measurement error Δδ D versus actual phase retardation δ and polarization intensity ratio F P (d). (b) Contour plot of the phase-measurement error Δδ D versus actual phase retardation δ and inverse of the polarization-intensity ratio, 1/F P (d).

Fig. 4
Fig. 4

Experimental configuration for thermally damaging porcine myocardium. Two small wedges were cut out of the tissue for registration of the OCT transverse scan and the histology section. The laser beam was scanned perpendicularly across the scan line.

Fig. 5
Fig. 5

Identical regions of porcine myocardium represented by (a) a polarization-independent backscatter image, (b) a phase-retardance image, and (c) a histological section. Arrow 1 along the vertical edge of each image is located at the center of the thermally damaged region and corresponds to the single axial scan plot in Fig. 6(a). Arrow 2 is located in an area of undamaged myocardium and corresponds to the single axial scan in Fig. 6(b).

Fig. 6
Fig. 6

Single axial scans extracted from (a) the polarization-independent backscatter image [Fig. 5(a)] and (b) the phase-retardance image [Fig. 5(b)]. The solid and the dotted lines correspond to the lateral positions given by arrow 1 (thermally damaged region) and arrow 2 (normal region), respectively.

Equations (26)

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E = E 0 1 0 ,
E S 0 = 1 2   E 0 1 0 ,     E R 0 = 1 2   E 0 1 0 .
E R 1 = 1 2 1 1 1 - 1 E R 0 exp - i 2 kl r = 1 2   E 0 1 1 exp - i 2 kl r .
J W = 1 2 1 i i 1 .
J S = exp i δ / 2 cos 2 φ + exp - i δ / 2 sin 2 φ 2 i   sin φ cos φ sin δ / 2 2 i   sin φ cos φ sin δ / 2 exp - i δ / 2 cos 2 φ + exp i δ / 2 sin 2 φ .
E S 1 d = J W J S R s d J S J W 1 2   E 0 1 0 exp - i 2 kl s = 1 2 exp i 2 φ sin δ cos δ R s d E 0 exp - i 2 kl s ,
E ST = 0 D E S 1 d d .
E S + R = 1 2   E 0 exp - i 2 kl r 2 + R s d exp i 2 φ sin δ × exp - i 2 kl s exp - i 2 kl r 2 + R s d cos δ exp - i 2 kl s .
E H = 1 2 exp - i 2 kl r 2 + R s d × exp i 2 φ sin δ exp - i 2 kl s E 0 , E V = 1 2 exp - i 2 kl r 2 + R s d × cos δ exp - i 2 kl s E 0
i H = ρ E H 2 = 1 + 2 R s d sin 2 δ + 2 2 R s d × sin δ cos 2 k Δ l + 2 φ ρ A 0 2 k 8 , i V = ρ E V 2 = 1 + 2 R s d cos 2 δ + 2 2 R s d cos δ cos 2 k Δ l ρ A 0 2 k 8 ,
i H ac = R s d sin δ cos 2 k Δ l + 2 φ ρ A 0 2 k 2 2 , i V ac = R s d cos δ cos 2 k Δ l ρ A 0 2 k 2 2 .
A 0 2 k = P 0 S k ,
S k = 2 ln   2 Δ k π exp - k - k 0 2 ln   2 Δ k 2
I H = -   i H ac k = ρ P 0 2 2 R s d sin δ × - cos 2 k Δ l + 2 φ S k k = ρ P 0 2 2 R s d exp - 2 Δ l ln   2 l w 2 × sin δ cos 2 k 0 Δ l + 2 φ , I V = -   i V ac k = ρ P 0 2 2 R s d cos δ × - cos 2 k Δ l S k k , = ρ P 0 2 2 R s d exp - 2 Δ l ln   2 l w 2 × cos δ cos 2 k 0 Δ l ,
l w = 4   ln   2 Δ k = 2 ln   2 λ 0 2 π Δ λ
| I H | = ρ P 0 2 2 R s d exp - 2 Δ l ln   2 l w 2 sin δ , | I V | = ρ P 0 2 2 R s d exp - 2 Δ l ln   2 l w 2 cos δ .
S H = 2 2 ρ P 0   | I H | = R sr d sin δ , S V = 2 2 ρ P 0   | I V | = R sr d cos δ ,
R sr d = R s d exp - 2 2 Δ l ln   2 l w 2
S HT = R sr d sin 2 δ + N r - 1 1 / 2 , S VT = R sr d cos 2 δ + N r - 1 1 / 2 ,
δ N = arc   tan sin 2 δ + N r - 1 1 / 2 / cos 2 δ + N r - 1 1 / 2 .
δ ε = arc   tan ε V 1 + ε V sin 2 δ + 1 1 + ε H cos 2 δ ε H 1 + ε H cos 2 δ + 1 1 + ε V sin 2 δ 1 / 2 ,
δ ε = arc   tan sin 2 δ + cos 2 δ ε H cos 2 δ + sin 2 δ ε V 1 / 2
J R = F P d cos 2 φ + sin 2 φ F P d sin φ cos φ - sin φ cos φ F P d sin φ cos φ - sin φ cos φ F P d sin 2 φ + cos 2 φ R SA d ,
F P d = R FA d T FA d R SA d T SA d .
E S 1 d = J W J S J R J S J W 1 2   E 0 1 0 exp - i 2 kl s = 1 2 2 exp i 2 φ 1 + F P d sin δ + i 1 - F P d cos δ 1 + F P d cos δ - i 1 - F P d sin δ R SA d E 0 exp - i 2 kl s ,
δ D = arctan 1 + F P d 2   sin 2 δ + 1 - F P d 2   cos 2 δ 1 / 2 1 + F P d 2   cos 2 δ + 1 - F P d 2   sin 2 δ 1 / 2 .

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