Abstract

By use of a highly sensitive method for measuring slight variations in birefringence it is shown here that a strong reversible correlation exists between rat tail tendon birefringence and temperature. This phenomenon is totally different from the loss of birefringence that results from a denaturation process. Below the threshold temperature leading to denaturation, an increase in temperature is systematically accompanied by a reversible increase in birefringence (0.25% °C-1). This phenomenon is observed at very fast heating rates (250,000 °C s-1), such as those induced by pulsed infrared lasers, and confirmed by experiments conducted with slow homogeneous heating of the sample medium (0.1 °C s-1). The good correlation between birefringence and temperature observed during the fast heating suggests that there are only small modifications of the tissue structure at the fibril level.

© 2000 Optical Society of America

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References

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  1. O. Wiener, “Die Theorie des Mischkörpers für das Feld der stationären Strömung,” Abh. Saechsis. Ges. Wiss. 33, 507–604 (1912).
  2. J. K. Cassim, P. S. Tobias, E. W. Taylor, “Birefringence of muscle proteins and the problem of structural birefringence,” Biochim. Biophys. Acta 168, 463–471 (1968).
    [CrossRef] [PubMed]
  3. E. Fredericq, C. Houssier, Electric Dichroism and Electric Birefringence (Clarendon, Oxford, UK, 1973).
  4. E. J. Naylor, “The structure of the cornea as revealed by polarized light,” Q. J. Microsc. Sci. 94, 83–88 (1953).
  5. S. Thomsen, J. A. Pearce, W.-F. Cheong, “Changes in birefringence as markers of thermal damage in tissue,” IEEE Trans. Biomed. Eng. 36, 1174–1179 (1989).
    [CrossRef] [PubMed]
  6. J. A. Pearce, S. Thomsen, H. Vijverberg, T. Magnusen, “Quantitative measures of thermal damage: birefringence changes in thermally coagulated collagen,” Adv. Bioheat Mass Transf. HTD-268, 141–144 (1993).
  7. K. Schönenberger, B. W. Colston, D. J. Maitland, L. B. Da Silva, M. J. Everett, “Mapping of birefringence and thermal damage in tissue by use of polarization-sensitive optical coherence tomography,” Appl. Opt. 37, 6026–6036 (1998).
    [CrossRef]
  8. D. J. Maitland, J. T. Walsh, “Quantitative measurements of linear birefringence during heating of native collagen,” Lasers Surg. Med. 20, 310–318 (1997).
    [CrossRef] [PubMed]
  9. V. Sankaran, J. T. Walsh, “Birefringence measurement of rapid structural changes during collagen denaturation,” Photochem. Photobiol. 68, 846–851 (1998).
    [PubMed]
  10. F. G. Lennox, “Shrinkage of collagen,” Biochim. Biophys. Acta 3, 170–187 (1949).
    [CrossRef]
  11. J. Roider, R. Birngruber, “Solution of the heat conduction equation,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welsh, M. J. C. Van Gemert, eds. (Plenum, New York, 1995), pp. 385–409.
    [CrossRef]
  12. D. E. Jansen, T. G. van Leeuwen, M. Motamedi, C. Borst, A. J. Welch, “Temperature dependence of the absorption coefficient of water for midinfrared laser radiation,” Lasers Surg. Med. 15, 258–268 (1994).
    [CrossRef]
  13. R. Goldstein, S. S. Penner, “The near-infrared absorption of liquid water at temperatures between 27 and 209 °C,” J. Quant. Spectrosc. Radiat. Transfer 4, 441–451 (1964).
    [CrossRef]
  14. T. Asshauer, “Holmium laser ablation of biological tissue under water: cavitation and acoustic transient generation,” Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1996).
  15. N. P. Furzikov, “Nature of the ablation of the cornea and skin by infrared laser,” Sov. J. Quantum Electron. 21, 222–225 (1991).
    [CrossRef]
  16. A. J. Welch, M. Motamedi, S. Rastegar, G. L. LeCarpentier, D. Jansen, “Laser thermal ablation,” Photochem. Photobiol. 53, 815–823 (1991).
    [PubMed]
  17. J. F. de Boer, T. E. Milner, M. J. C. Vangemert, J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22, 934–936 (1997).
    [CrossRef] [PubMed]
  18. S. Jaywant, B. Wilson, L. Lilge, T. Flotte, J. Woolsey, C. McCulloch, “Temperature dependent changes in the optical absorption and scattering spectra of tissues: correlation with ultrastructure,” in Proceedings of Laser-Tissue Interaction IV, S. L. Jacques, ed., Proc. SPIE1882, 218–229 (1993).
    [CrossRef]
  19. G. J. Derbyshire, D. K. Bogen, M. U. Unger, “Thermally induced optical property changes in myocardium at 1.06 mm,” Lasers Surg. Med. 10, 28–34 (1990).
    [CrossRef]
  20. V. F. Izotova, I. L. Maksimova, I. S. Nefedov, S. V. Romanov, “Investigation of Mueller matrices of anisotropic nonhomogeneous layers in application to an optical model of the cornea,” Appl. Opt. 36, 164–169 (1997).
    [CrossRef] [PubMed]
  21. T. B. Smith, “Multiple scattering in the cornea,” J. Mod. Opt. 35, 93–101 (1988).
    [CrossRef]

1998

1997

1994

D. E. Jansen, T. G. van Leeuwen, M. Motamedi, C. Borst, A. J. Welch, “Temperature dependence of the absorption coefficient of water for midinfrared laser radiation,” Lasers Surg. Med. 15, 258–268 (1994).
[CrossRef]

1993

J. A. Pearce, S. Thomsen, H. Vijverberg, T. Magnusen, “Quantitative measures of thermal damage: birefringence changes in thermally coagulated collagen,” Adv. Bioheat Mass Transf. HTD-268, 141–144 (1993).

1991

N. P. Furzikov, “Nature of the ablation of the cornea and skin by infrared laser,” Sov. J. Quantum Electron. 21, 222–225 (1991).
[CrossRef]

A. J. Welch, M. Motamedi, S. Rastegar, G. L. LeCarpentier, D. Jansen, “Laser thermal ablation,” Photochem. Photobiol. 53, 815–823 (1991).
[PubMed]

1990

G. J. Derbyshire, D. K. Bogen, M. U. Unger, “Thermally induced optical property changes in myocardium at 1.06 mm,” Lasers Surg. Med. 10, 28–34 (1990).
[CrossRef]

1989

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

1988

T. B. Smith, “Multiple scattering in the cornea,” J. Mod. Opt. 35, 93–101 (1988).
[CrossRef]

1968

J. K. Cassim, P. S. Tobias, E. W. Taylor, “Birefringence of muscle proteins and the problem of structural birefringence,” Biochim. Biophys. Acta 168, 463–471 (1968).
[CrossRef] [PubMed]

1964

R. Goldstein, S. S. Penner, “The near-infrared absorption of liquid water at temperatures between 27 and 209 °C,” J. Quant. Spectrosc. Radiat. Transfer 4, 441–451 (1964).
[CrossRef]

1953

E. J. Naylor, “The structure of the cornea as revealed by polarized light,” Q. J. Microsc. Sci. 94, 83–88 (1953).

1949

F. G. Lennox, “Shrinkage of collagen,” Biochim. Biophys. Acta 3, 170–187 (1949).
[CrossRef]

1912

O. Wiener, “Die Theorie des Mischkörpers für das Feld der stationären Strömung,” Abh. Saechsis. Ges. Wiss. 33, 507–604 (1912).

Asshauer, T.

T. Asshauer, “Holmium laser ablation of biological tissue under water: cavitation and acoustic transient generation,” Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1996).

Birngruber, R.

J. Roider, R. Birngruber, “Solution of the heat conduction equation,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welsh, M. J. C. Van Gemert, eds. (Plenum, New York, 1995), pp. 385–409.
[CrossRef]

Bogen, D. K.

G. J. Derbyshire, D. K. Bogen, M. U. Unger, “Thermally induced optical property changes in myocardium at 1.06 mm,” Lasers Surg. Med. 10, 28–34 (1990).
[CrossRef]

Borst, C.

D. E. Jansen, T. G. van Leeuwen, M. Motamedi, C. Borst, A. J. Welch, “Temperature dependence of the absorption coefficient of water for midinfrared laser radiation,” Lasers Surg. Med. 15, 258–268 (1994).
[CrossRef]

Cassim, J. K.

J. K. Cassim, P. S. Tobias, E. W. Taylor, “Birefringence of muscle proteins and the problem of structural birefringence,” Biochim. Biophys. Acta 168, 463–471 (1968).
[CrossRef] [PubMed]

Cheong, W.-F.

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

Colston, B. W.

Da Silva, L. B.

de Boer, J. F.

Derbyshire, G. J.

G. J. Derbyshire, D. K. Bogen, M. U. Unger, “Thermally induced optical property changes in myocardium at 1.06 mm,” Lasers Surg. Med. 10, 28–34 (1990).
[CrossRef]

Everett, M. J.

Flotte, T.

S. Jaywant, B. Wilson, L. Lilge, T. Flotte, J. Woolsey, C. McCulloch, “Temperature dependent changes in the optical absorption and scattering spectra of tissues: correlation with ultrastructure,” in Proceedings of Laser-Tissue Interaction IV, S. L. Jacques, ed., Proc. SPIE1882, 218–229 (1993).
[CrossRef]

Fredericq, E.

E. Fredericq, C. Houssier, Electric Dichroism and Electric Birefringence (Clarendon, Oxford, UK, 1973).

Furzikov, N. P.

N. P. Furzikov, “Nature of the ablation of the cornea and skin by infrared laser,” Sov. J. Quantum Electron. 21, 222–225 (1991).
[CrossRef]

Goldstein, R.

R. Goldstein, S. S. Penner, “The near-infrared absorption of liquid water at temperatures between 27 and 209 °C,” J. Quant. Spectrosc. Radiat. Transfer 4, 441–451 (1964).
[CrossRef]

Houssier, C.

E. Fredericq, C. Houssier, Electric Dichroism and Electric Birefringence (Clarendon, Oxford, UK, 1973).

Izotova, V. F.

Jansen, D.

A. J. Welch, M. Motamedi, S. Rastegar, G. L. LeCarpentier, D. Jansen, “Laser thermal ablation,” Photochem. Photobiol. 53, 815–823 (1991).
[PubMed]

Jansen, D. E.

D. E. Jansen, T. G. van Leeuwen, M. Motamedi, C. Borst, A. J. Welch, “Temperature dependence of the absorption coefficient of water for midinfrared laser radiation,” Lasers Surg. Med. 15, 258–268 (1994).
[CrossRef]

Jaywant, S.

S. Jaywant, B. Wilson, L. Lilge, T. Flotte, J. Woolsey, C. McCulloch, “Temperature dependent changes in the optical absorption and scattering spectra of tissues: correlation with ultrastructure,” in Proceedings of Laser-Tissue Interaction IV, S. L. Jacques, ed., Proc. SPIE1882, 218–229 (1993).
[CrossRef]

LeCarpentier, G. L.

A. J. Welch, M. Motamedi, S. Rastegar, G. L. LeCarpentier, D. Jansen, “Laser thermal ablation,” Photochem. Photobiol. 53, 815–823 (1991).
[PubMed]

Lennox, F. G.

F. G. Lennox, “Shrinkage of collagen,” Biochim. Biophys. Acta 3, 170–187 (1949).
[CrossRef]

Lilge, L.

S. Jaywant, B. Wilson, L. Lilge, T. Flotte, J. Woolsey, C. McCulloch, “Temperature dependent changes in the optical absorption and scattering spectra of tissues: correlation with ultrastructure,” in Proceedings of Laser-Tissue Interaction IV, S. L. Jacques, ed., Proc. SPIE1882, 218–229 (1993).
[CrossRef]

Magnusen, T.

J. A. Pearce, S. Thomsen, H. Vijverberg, T. Magnusen, “Quantitative measures of thermal damage: birefringence changes in thermally coagulated collagen,” Adv. Bioheat Mass Transf. HTD-268, 141–144 (1993).

Maitland, D. J.

Maksimova, I. L.

McCulloch, C.

S. Jaywant, B. Wilson, L. Lilge, T. Flotte, J. Woolsey, C. McCulloch, “Temperature dependent changes in the optical absorption and scattering spectra of tissues: correlation with ultrastructure,” in Proceedings of Laser-Tissue Interaction IV, S. L. Jacques, ed., Proc. SPIE1882, 218–229 (1993).
[CrossRef]

Milner, T. E.

Motamedi, M.

D. E. Jansen, T. G. van Leeuwen, M. Motamedi, C. Borst, A. J. Welch, “Temperature dependence of the absorption coefficient of water for midinfrared laser radiation,” Lasers Surg. Med. 15, 258–268 (1994).
[CrossRef]

A. J. Welch, M. Motamedi, S. Rastegar, G. L. LeCarpentier, D. Jansen, “Laser thermal ablation,” Photochem. Photobiol. 53, 815–823 (1991).
[PubMed]

Naylor, E. J.

E. J. Naylor, “The structure of the cornea as revealed by polarized light,” Q. J. Microsc. Sci. 94, 83–88 (1953).

Nefedov, I. S.

Nelson, J. S.

Pearce, J. A.

J. A. Pearce, S. Thomsen, H. Vijverberg, T. Magnusen, “Quantitative measures of thermal damage: birefringence changes in thermally coagulated collagen,” Adv. Bioheat Mass Transf. HTD-268, 141–144 (1993).

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

Penner, S. S.

R. Goldstein, S. S. Penner, “The near-infrared absorption of liquid water at temperatures between 27 and 209 °C,” J. Quant. Spectrosc. Radiat. Transfer 4, 441–451 (1964).
[CrossRef]

Rastegar, S.

A. J. Welch, M. Motamedi, S. Rastegar, G. L. LeCarpentier, D. Jansen, “Laser thermal ablation,” Photochem. Photobiol. 53, 815–823 (1991).
[PubMed]

Roider, J.

J. Roider, R. Birngruber, “Solution of the heat conduction equation,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welsh, M. J. C. Van Gemert, eds. (Plenum, New York, 1995), pp. 385–409.
[CrossRef]

Romanov, S. V.

Sankaran, V.

V. Sankaran, J. T. Walsh, “Birefringence measurement of rapid structural changes during collagen denaturation,” Photochem. Photobiol. 68, 846–851 (1998).
[PubMed]

Schönenberger, K.

Smith, T. B.

T. B. Smith, “Multiple scattering in the cornea,” J. Mod. Opt. 35, 93–101 (1988).
[CrossRef]

Taylor, E. W.

J. K. Cassim, P. S. Tobias, E. W. Taylor, “Birefringence of muscle proteins and the problem of structural birefringence,” Biochim. Biophys. Acta 168, 463–471 (1968).
[CrossRef] [PubMed]

Thomsen, S.

J. A. Pearce, S. Thomsen, H. Vijverberg, T. Magnusen, “Quantitative measures of thermal damage: birefringence changes in thermally coagulated collagen,” Adv. Bioheat Mass Transf. HTD-268, 141–144 (1993).

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

Tobias, P. S.

J. K. Cassim, P. S. Tobias, E. W. Taylor, “Birefringence of muscle proteins and the problem of structural birefringence,” Biochim. Biophys. Acta 168, 463–471 (1968).
[CrossRef] [PubMed]

Unger, M. U.

G. J. Derbyshire, D. K. Bogen, M. U. Unger, “Thermally induced optical property changes in myocardium at 1.06 mm,” Lasers Surg. Med. 10, 28–34 (1990).
[CrossRef]

van Leeuwen, T. G.

D. E. Jansen, T. G. van Leeuwen, M. Motamedi, C. Borst, A. J. Welch, “Temperature dependence of the absorption coefficient of water for midinfrared laser radiation,” Lasers Surg. Med. 15, 258–268 (1994).
[CrossRef]

Vangemert, M. J. C.

Vijverberg, H.

J. A. Pearce, S. Thomsen, H. Vijverberg, T. Magnusen, “Quantitative measures of thermal damage: birefringence changes in thermally coagulated collagen,” Adv. Bioheat Mass Transf. HTD-268, 141–144 (1993).

Walsh, J. T.

V. Sankaran, J. T. Walsh, “Birefringence measurement of rapid structural changes during collagen denaturation,” Photochem. Photobiol. 68, 846–851 (1998).
[PubMed]

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

Welch, A. J.

D. E. Jansen, T. G. van Leeuwen, M. Motamedi, C. Borst, A. J. Welch, “Temperature dependence of the absorption coefficient of water for midinfrared laser radiation,” Lasers Surg. Med. 15, 258–268 (1994).
[CrossRef]

A. J. Welch, M. Motamedi, S. Rastegar, G. L. LeCarpentier, D. Jansen, “Laser thermal ablation,” Photochem. Photobiol. 53, 815–823 (1991).
[PubMed]

Wiener, O.

O. Wiener, “Die Theorie des Mischkörpers für das Feld der stationären Strömung,” Abh. Saechsis. Ges. Wiss. 33, 507–604 (1912).

Wilson, B.

S. Jaywant, B. Wilson, L. Lilge, T. Flotte, J. Woolsey, C. McCulloch, “Temperature dependent changes in the optical absorption and scattering spectra of tissues: correlation with ultrastructure,” in Proceedings of Laser-Tissue Interaction IV, S. L. Jacques, ed., Proc. SPIE1882, 218–229 (1993).
[CrossRef]

Woolsey, J.

S. Jaywant, B. Wilson, L. Lilge, T. Flotte, J. Woolsey, C. McCulloch, “Temperature dependent changes in the optical absorption and scattering spectra of tissues: correlation with ultrastructure,” in Proceedings of Laser-Tissue Interaction IV, S. L. Jacques, ed., Proc. SPIE1882, 218–229 (1993).
[CrossRef]

Abh. Saechsis. Ges. Wiss.

O. Wiener, “Die Theorie des Mischkörpers für das Feld der stationären Strömung,” Abh. Saechsis. Ges. Wiss. 33, 507–604 (1912).

Adv. Bioheat Mass Transf.

J. A. Pearce, S. Thomsen, H. Vijverberg, T. Magnusen, “Quantitative measures of thermal damage: birefringence changes in thermally coagulated collagen,” Adv. Bioheat Mass Transf. HTD-268, 141–144 (1993).

Appl. Opt.

Biochim. Biophys. Acta

J. K. Cassim, P. S. Tobias, E. W. Taylor, “Birefringence of muscle proteins and the problem of structural birefringence,” Biochim. Biophys. Acta 168, 463–471 (1968).
[CrossRef] [PubMed]

F. G. Lennox, “Shrinkage of collagen,” Biochim. Biophys. Acta 3, 170–187 (1949).
[CrossRef]

IEEE Trans. Biomed. Eng.

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

J. Mod. Opt.

T. B. Smith, “Multiple scattering in the cornea,” J. Mod. Opt. 35, 93–101 (1988).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

R. Goldstein, S. S. Penner, “The near-infrared absorption of liquid water at temperatures between 27 and 209 °C,” J. Quant. Spectrosc. Radiat. Transfer 4, 441–451 (1964).
[CrossRef]

Lasers Surg. Med.

D. E. Jansen, T. G. van Leeuwen, M. Motamedi, C. Borst, A. J. Welch, “Temperature dependence of the absorption coefficient of water for midinfrared laser radiation,” Lasers Surg. Med. 15, 258–268 (1994).
[CrossRef]

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

G. J. Derbyshire, D. K. Bogen, M. U. Unger, “Thermally induced optical property changes in myocardium at 1.06 mm,” Lasers Surg. Med. 10, 28–34 (1990).
[CrossRef]

Opt. Lett.

Photochem. Photobiol.

V. Sankaran, J. T. Walsh, “Birefringence measurement of rapid structural changes during collagen denaturation,” Photochem. Photobiol. 68, 846–851 (1998).
[PubMed]

A. J. Welch, M. Motamedi, S. Rastegar, G. L. LeCarpentier, D. Jansen, “Laser thermal ablation,” Photochem. Photobiol. 53, 815–823 (1991).
[PubMed]

Q. J. Microsc. Sci.

E. J. Naylor, “The structure of the cornea as revealed by polarized light,” Q. J. Microsc. Sci. 94, 83–88 (1953).

Sov. J. Quantum Electron.

N. P. Furzikov, “Nature of the ablation of the cornea and skin by infrared laser,” Sov. J. Quantum Electron. 21, 222–225 (1991).
[CrossRef]

Other

S. Jaywant, B. Wilson, L. Lilge, T. Flotte, J. Woolsey, C. McCulloch, “Temperature dependent changes in the optical absorption and scattering spectra of tissues: correlation with ultrastructure,” in Proceedings of Laser-Tissue Interaction IV, S. L. Jacques, ed., Proc. SPIE1882, 218–229 (1993).
[CrossRef]

T. Asshauer, “Holmium laser ablation of biological tissue under water: cavitation and acoustic transient generation,” Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1996).

J. Roider, R. Birngruber, “Solution of the heat conduction equation,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welsh, M. J. C. Van Gemert, eds. (Plenum, New York, 1995), pp. 385–409.
[CrossRef]

E. Fredericq, C. Houssier, Electric Dichroism and Electric Birefringence (Clarendon, Oxford, UK, 1973).

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Figures (8)

Fig. 1
Fig. 1

Experimental setup for two-dimensional phase-shift imaging: λ/2, half-wave plate; λ/4’s quarter-wave plates; MO, microscope objective; other abbreviations defined in text.

Fig. 2
Fig. 2

Picture of the thermocouple positioning in front of the fiber tip. The slanted arrow points to the junction of the thermocouple.

Fig. 3
Fig. 3

Comparison of experimental and numerical results in determining the temperature evolution in front of the fiber tip. The energy delivered by the pulse is 14 mJ (5 J cm-2).

Fig. 4
Fig. 4

Images of a RTT sample obtained from (a) CCD1 and (b) CCD2 at room temperature in water with linearly polarized light at +45° as the strobe source. The arrows indicate the edges of the tendon.

Fig. 5
Fig. 5

Phase-shift imaging in a RTT sample. The gray levels represent phase shift α calculated from Eq. (1). The probe beam’s polarization is linear and oriented at +45°.

Fig. 6
Fig. 6

Phase shift α calculated from one edge of the tendon to the other as a function of distance from the left edge. The solid curve represents the best fit of theoretical phase shift α calculated for a birefringent circular cylinder.

Fig. 7
Fig. 7

Birefringence Δn (left-hand scale) and temperature difference from room temperature in the center of the tendon (right-hand scale) as a function of time delay measured from the onset of the CTH:YAG laser pulse.

Fig. 8
Fig. 8

RTT birefringence (left-hand scale) and RTT diameter (right-hand scale) as functions of the temperature rise obtained by homogeneous heating of the water bath.

Tables (1)

Tables Icon

Table 1 Physical Parameters Used for Numerical Solution of the Heat-Diffusion Equation

Equations (10)

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

α=arccos1-R/1+R,
δ=α,  0°δ180°,
δ=360°-α,  180°δ360°,
δ=360°+α,  360°δ540°,
Δn=λδ/360d,
α=arccos1-R/1+R-90°.
Ren1-n2=πa2ρn0m2-122m2+1,
OS1=1210001-111,
OS2=121000cosδ-sinδsinδcosδ0-110×expiπ/400exp-iπ/411-11×expiδ00exp-iδ,
OS1=121-100,OS2=12 exp-iπ41-100.

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