Abstract

We describe a spectroscopic technique called interferometric photothermal spectroscopy (IPTS) that can measure the absorption coefficient of pulsed laser radiation in nonscattering tissue samples. The technique is suitable for measuring effective absorption coefficients from 103 to 105 cm-1. IPTS is particularly attractive because it requires minimal disturbance of the sample. These features indicate potential use for in vivo measurements of tissue absorption coefficients. To validate the technique, the absorption coefficient of pulsed Q-switched Er:YSGG (2.79-µm) radiation in pure water was measured to be 5200 (±500) cm-1 when IPTS was used, in agreement with other published values. IPTS was also used to measure the absorption coefficient of pulsed ArF excimer laser radiation (193 nm) in bovine corneal stroma (in vitro), giving a value of 1.9 (±0.4) × 104 cm-1.

© 1999 Optical Society of America

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Corrections

Andrew D. Yablon, Norman S. Nishioka, Bora B. Mikić, and Vasan Venugopalan, "Measurement of tissue absorption coefficients by use of interferometric photothermal spectroscopy: erratum," Appl. Opt. 38, 4266-4266 (1999)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-38-19-4266

References

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  1. S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
    [CrossRef] [PubMed]
  2. T. E. Milner, D. J. Smithies, D. M. Goodman, A. Lau, J. S. Nelson, “Depth determination of chromophore in human skin by pulsed photothermal radiometry,” Appl. Opt. 35, 3379–3385 (1996).
    [CrossRef] [PubMed]
  3. I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, M. S. Feld, “The thermoelastic basis of short pulsed laser ablation of biological tissue,” Proc. Natl. Acad. Sci. 92, 1960–1964 (1995).
    [CrossRef] [PubMed]
  4. D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, I. Itzkan, M. S. Feld, “Photomechanical basis of laser ablation of biological tissue,” Opt. Lett. 19, 1684–1686 (1994).
    [CrossRef] [PubMed]
  5. D. Albagli, “Fundamental mechanisms of pulsed laser ablation of biological tissue,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1994).
  6. J. T. Walsh, J. P. Cummings, “Effect of the dynamic optical properties of water on mid-infrared laser ablation,” Lasers Surg. Med. 15, 295–305 (1994).
    [CrossRef]
  7. K. L. Vodopyanov, “Saturation studies of H2O and HDO near 3400 cm-1 using intense picosecond laser pulses,” J. Chem. Phys. 94, 5389–5393 (1991).
    [CrossRef]
  8. C. A. Puliafito, R. F. Steinert, T. F. Deutsch, F. Hillenkamp, E. J. Dehm, C. M. Adler, “Excimer laser ablation of the cornea and lens,” Ophthalmology 92, 741–748 (1985).
    [CrossRef] [PubMed]
  9. M. N. Ediger, G. H. Pettit, R. P. Weiblinger, C. H. Chen, “Transmission of corneal collagen during ArF excimer laser ablation,” Lasers Surg. Med. 13, 204–210 (1993).
    [CrossRef] [PubMed]
  10. M. N. Ediger, G. H. Pettit, D. W. Hahn, “Enhanced ArF laser absorption in a collagen target under ablative conditions,” Lasers Surg. Med. 15, 107–111 (1994).
    [CrossRef] [PubMed]
  11. G. H. Pettit, M. N. Ediger, R. P. Weiblinger, “Dynamic optical properties of collagen-based tissue during ArF excimer laser ablation,” Appl. Opt. 32, 488–492 (1993).
    [CrossRef] [PubMed]
  12. P. T. Staveteig, J. T. Walsh, “Dynamic 193-nm optical properties of water,” Appl. Opt. 35, 3392–3403 (1996).
    [CrossRef] [PubMed]
  13. R. Srinivasan, P. E. Dyer, B. Braren, “Far-ultraviolet laser ablation of the cornea: photoacoustic studies,” Lasers Surg. Med. 6, 514–519 (1987).
    [CrossRef] [PubMed]
  14. G. H. Pettit, M. N. Ediger, “Corneal-tissue absorption coefficients for 193- and 213-nm ultraviolet radiation,” Appl. Opt. 35, 3386–3391 (1996).
    [CrossRef] [PubMed]
  15. E. M. Sparrow, R. D. Cess, Radiation Heat Transfer, Augmented ed. (Brooks/Cole, Belmont, Calif., 1978).
  16. R. C. Weast, CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, Fla., 1987).
  17. A. D. Yablon, N. S. Nishioka, B. B. Mikic, V. Venugopalan, “An interferometer for monitoring the surface displacement of a reflecting biological tissue surface,” in Advances in Heat and Mass Transfer in Biotechnology, , L. J. Hayes, S. Clegg, eds. (American Society of Mechanical Engineers, New York, 1996), pp. 75–80.
  18. Dow Chemical Corporation, Typical properties for Saran Wrap 19, (Dow Chemical Corporation, Midland, Mich., 1997).
  19. H. D. Downing, D. Williams, “Optical constants of water in the infrared,” J. Geophys. Res. 80, 1656–1661 (1975).
    [CrossRef]
  20. Kimble Glass Corporation, Properties of Kimble Glasses, Product data sheet (Kimble Glass Corporation, Vineland, N.J., 1997).
  21. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, New York, 1988).
  22. A. F. Emery, P. Kramar, A. W. Guy, J. C. Lin, “Microwave induced temperature rises in rabbit eyes in cataract research,” J. Heat Transfer 97, 123–128 (1975).
    [CrossRef]
  23. J. A. Scott, “A finite element model of heat transport in the human eye,” Phys. Med. Biol. 33, 227–241 (1988).
    [CrossRef] [PubMed]
  24. J. L. Battaglioli, R. D. Kamm, “Measurements of the compressive properties of scleral tissue,” Invest. Ophthalmol. Vis. Sci. 25, 59–65 (1984).
    [PubMed]
  25. D. M. Maurice, “The cornea and sclera,” in The Eye, H. Davson, ed. (Academic, New York, 1984), Vol. 1b, pp. 1–158.
    [CrossRef]
  26. G. H. Pettit, M. N. Ediger, R. P. Weiblinger, “Excimer laser corneal ablation: absence of a significant ‘incubation’ effect,” Lasers Surg. Med. 11, 411–418 (1991).
    [CrossRef]
  27. J. P. Cummings, J. T. Walsh, “Erbium laser ablation: the effect of dynamic optical properties,” Appl. Phys. Lett. 62, 1988–1990 (1993).
    [CrossRef]
  28. B. J. Garrison, R. Srinivasan, “Laser ablation of organic polymers: microscopic models for photochemical and thermal processes,” J. Appl. Phys. 57, 2909–2914 (1985).
    [CrossRef]
  29. R. Srinivasan, B. Braren, “Ultraviolet laser ablation of organic polymers,” Chem. Rev. 89, 1303–1316 (1989).
    [CrossRef]
  30. Q. Ren, R. P. Gailitis, K. P. Thompson, J. T. Lin, “Ablation of the cornea and synthetic polymers using a UV (213 nm) solid-state laser,” IEEE J. Quantum Electron. 26, 2284–2288 (1990).
    [CrossRef]
  31. Q. Ren, G. Simon, J.-M. Legeais, J.-M. Parel, W. Culbertson, J. Shen, Y. Takesue, M. Savoldelli, “Ultraviolet solid-state laser (213-nm) photorefractive keratectomy,” Ophtalmology 101, 883–889 (1994).
    [CrossRef]
  32. Q. Ren, R. H. Keates, R. A. Hill, M. W. Berns, “Laser refractive surgery: a review and current status,” Opt. Eng. 34, 642–658 (1995).
    [CrossRef]
  33. S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, New York, 1970).

1996 (3)

1995 (2)

Q. Ren, R. H. Keates, R. A. Hill, M. W. Berns, “Laser refractive surgery: a review and current status,” Opt. Eng. 34, 642–658 (1995).
[CrossRef]

I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, M. S. Feld, “The thermoelastic basis of short pulsed laser ablation of biological tissue,” Proc. Natl. Acad. Sci. 92, 1960–1964 (1995).
[CrossRef] [PubMed]

1994 (4)

J. T. Walsh, J. P. Cummings, “Effect of the dynamic optical properties of water on mid-infrared laser ablation,” Lasers Surg. Med. 15, 295–305 (1994).
[CrossRef]

M. N. Ediger, G. H. Pettit, D. W. Hahn, “Enhanced ArF laser absorption in a collagen target under ablative conditions,” Lasers Surg. Med. 15, 107–111 (1994).
[CrossRef] [PubMed]

Q. Ren, G. Simon, J.-M. Legeais, J.-M. Parel, W. Culbertson, J. Shen, Y. Takesue, M. Savoldelli, “Ultraviolet solid-state laser (213-nm) photorefractive keratectomy,” Ophtalmology 101, 883–889 (1994).
[CrossRef]

D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, I. Itzkan, M. S. Feld, “Photomechanical basis of laser ablation of biological tissue,” Opt. Lett. 19, 1684–1686 (1994).
[CrossRef] [PubMed]

1993 (3)

G. H. Pettit, M. N. Ediger, R. P. Weiblinger, “Dynamic optical properties of collagen-based tissue during ArF excimer laser ablation,” Appl. Opt. 32, 488–492 (1993).
[CrossRef] [PubMed]

J. P. Cummings, J. T. Walsh, “Erbium laser ablation: the effect of dynamic optical properties,” Appl. Phys. Lett. 62, 1988–1990 (1993).
[CrossRef]

M. N. Ediger, G. H. Pettit, R. P. Weiblinger, C. H. Chen, “Transmission of corneal collagen during ArF excimer laser ablation,” Lasers Surg. Med. 13, 204–210 (1993).
[CrossRef] [PubMed]

1992 (1)

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

1991 (2)

K. L. Vodopyanov, “Saturation studies of H2O and HDO near 3400 cm-1 using intense picosecond laser pulses,” J. Chem. Phys. 94, 5389–5393 (1991).
[CrossRef]

G. H. Pettit, M. N. Ediger, R. P. Weiblinger, “Excimer laser corneal ablation: absence of a significant ‘incubation’ effect,” Lasers Surg. Med. 11, 411–418 (1991).
[CrossRef]

1990 (1)

Q. Ren, R. P. Gailitis, K. P. Thompson, J. T. Lin, “Ablation of the cornea and synthetic polymers using a UV (213 nm) solid-state laser,” IEEE J. Quantum Electron. 26, 2284–2288 (1990).
[CrossRef]

1989 (1)

R. Srinivasan, B. Braren, “Ultraviolet laser ablation of organic polymers,” Chem. Rev. 89, 1303–1316 (1989).
[CrossRef]

1988 (1)

J. A. Scott, “A finite element model of heat transport in the human eye,” Phys. Med. Biol. 33, 227–241 (1988).
[CrossRef] [PubMed]

1987 (1)

R. Srinivasan, P. E. Dyer, B. Braren, “Far-ultraviolet laser ablation of the cornea: photoacoustic studies,” Lasers Surg. Med. 6, 514–519 (1987).
[CrossRef] [PubMed]

1985 (2)

B. J. Garrison, R. Srinivasan, “Laser ablation of organic polymers: microscopic models for photochemical and thermal processes,” J. Appl. Phys. 57, 2909–2914 (1985).
[CrossRef]

C. A. Puliafito, R. F. Steinert, T. F. Deutsch, F. Hillenkamp, E. J. Dehm, C. M. Adler, “Excimer laser ablation of the cornea and lens,” Ophthalmology 92, 741–748 (1985).
[CrossRef] [PubMed]

1984 (1)

J. L. Battaglioli, R. D. Kamm, “Measurements of the compressive properties of scleral tissue,” Invest. Ophthalmol. Vis. Sci. 25, 59–65 (1984).
[PubMed]

1975 (2)

H. D. Downing, D. Williams, “Optical constants of water in the infrared,” J. Geophys. Res. 80, 1656–1661 (1975).
[CrossRef]

A. F. Emery, P. Kramar, A. W. Guy, J. C. Lin, “Microwave induced temperature rises in rabbit eyes in cataract research,” J. Heat Transfer 97, 123–128 (1975).
[CrossRef]

Adler, C. M.

C. A. Puliafito, R. F. Steinert, T. F. Deutsch, F. Hillenkamp, E. J. Dehm, C. M. Adler, “Excimer laser ablation of the cornea and lens,” Ophthalmology 92, 741–748 (1985).
[CrossRef] [PubMed]

Albagli, D.

I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, M. S. Feld, “The thermoelastic basis of short pulsed laser ablation of biological tissue,” Proc. Natl. Acad. Sci. 92, 1960–1964 (1995).
[CrossRef] [PubMed]

D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, I. Itzkan, M. S. Feld, “Photomechanical basis of laser ablation of biological tissue,” Opt. Lett. 19, 1684–1686 (1994).
[CrossRef] [PubMed]

D. Albagli, “Fundamental mechanisms of pulsed laser ablation of biological tissue,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1994).

Anderson, R. R.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

Battaglioli, J. L.

J. L. Battaglioli, R. D. Kamm, “Measurements of the compressive properties of scleral tissue,” Invest. Ophthalmol. Vis. Sci. 25, 59–65 (1984).
[PubMed]

Berns, M. W.

Q. Ren, R. H. Keates, R. A. Hill, M. W. Berns, “Laser refractive surgery: a review and current status,” Opt. Eng. 34, 642–658 (1995).
[CrossRef]

Braren, B.

R. Srinivasan, B. Braren, “Ultraviolet laser ablation of organic polymers,” Chem. Rev. 89, 1303–1316 (1989).
[CrossRef]

R. Srinivasan, P. E. Dyer, B. Braren, “Far-ultraviolet laser ablation of the cornea: photoacoustic studies,” Lasers Surg. Med. 6, 514–519 (1987).
[CrossRef] [PubMed]

Bruggemann, U.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

Cess, R. D.

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer, Augmented ed. (Brooks/Cole, Belmont, Calif., 1978).

Chen, C. H.

M. N. Ediger, G. H. Pettit, R. P. Weiblinger, C. H. Chen, “Transmission of corneal collagen during ArF excimer laser ablation,” Lasers Surg. Med. 13, 204–210 (1993).
[CrossRef] [PubMed]

Culbertson, W.

Q. Ren, G. Simon, J.-M. Legeais, J.-M. Parel, W. Culbertson, J. Shen, Y. Takesue, M. Savoldelli, “Ultraviolet solid-state laser (213-nm) photorefractive keratectomy,” Ophtalmology 101, 883–889 (1994).
[CrossRef]

Cummings, J. P.

J. T. Walsh, J. P. Cummings, “Effect of the dynamic optical properties of water on mid-infrared laser ablation,” Lasers Surg. Med. 15, 295–305 (1994).
[CrossRef]

J. P. Cummings, J. T. Walsh, “Erbium laser ablation: the effect of dynamic optical properties,” Appl. Phys. Lett. 62, 1988–1990 (1993).
[CrossRef]

Dark, M. L.

I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, M. S. Feld, “The thermoelastic basis of short pulsed laser ablation of biological tissue,” Proc. Natl. Acad. Sci. 92, 1960–1964 (1995).
[CrossRef] [PubMed]

D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, I. Itzkan, M. S. Feld, “Photomechanical basis of laser ablation of biological tissue,” Opt. Lett. 19, 1684–1686 (1994).
[CrossRef] [PubMed]

Dehm, E. J.

C. A. Puliafito, R. F. Steinert, T. F. Deutsch, F. Hillenkamp, E. J. Dehm, C. M. Adler, “Excimer laser ablation of the cornea and lens,” Ophthalmology 92, 741–748 (1985).
[CrossRef] [PubMed]

Deutsch, T. F.

C. A. Puliafito, R. F. Steinert, T. F. Deutsch, F. Hillenkamp, E. J. Dehm, C. M. Adler, “Excimer laser ablation of the cornea and lens,” Ophthalmology 92, 741–748 (1985).
[CrossRef] [PubMed]

Downing, H. D.

H. D. Downing, D. Williams, “Optical constants of water in the infrared,” J. Geophys. Res. 80, 1656–1661 (1975).
[CrossRef]

Dyer, P. E.

R. Srinivasan, P. E. Dyer, B. Braren, “Far-ultraviolet laser ablation of the cornea: photoacoustic studies,” Lasers Surg. Med. 6, 514–519 (1987).
[CrossRef] [PubMed]

Ediger, M. N.

G. H. Pettit, M. N. Ediger, “Corneal-tissue absorption coefficients for 193- and 213-nm ultraviolet radiation,” Appl. Opt. 35, 3386–3391 (1996).
[CrossRef] [PubMed]

M. N. Ediger, G. H. Pettit, D. W. Hahn, “Enhanced ArF laser absorption in a collagen target under ablative conditions,” Lasers Surg. Med. 15, 107–111 (1994).
[CrossRef] [PubMed]

G. H. Pettit, M. N. Ediger, R. P. Weiblinger, “Dynamic optical properties of collagen-based tissue during ArF excimer laser ablation,” Appl. Opt. 32, 488–492 (1993).
[CrossRef] [PubMed]

M. N. Ediger, G. H. Pettit, R. P. Weiblinger, C. H. Chen, “Transmission of corneal collagen during ArF excimer laser ablation,” Lasers Surg. Med. 13, 204–210 (1993).
[CrossRef] [PubMed]

G. H. Pettit, M. N. Ediger, R. P. Weiblinger, “Excimer laser corneal ablation: absence of a significant ‘incubation’ effect,” Lasers Surg. Med. 11, 411–418 (1991).
[CrossRef]

Emery, A. F.

A. F. Emery, P. Kramar, A. W. Guy, J. C. Lin, “Microwave induced temperature rises in rabbit eyes in cataract research,” J. Heat Transfer 97, 123–128 (1975).
[CrossRef]

Feld, M. S.

I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, M. S. Feld, “The thermoelastic basis of short pulsed laser ablation of biological tissue,” Proc. Natl. Acad. Sci. 92, 1960–1964 (1995).
[CrossRef] [PubMed]

D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, I. Itzkan, M. S. Feld, “Photomechanical basis of laser ablation of biological tissue,” Opt. Lett. 19, 1684–1686 (1994).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, New York, 1988).

Gailitis, R. P.

Q. Ren, R. P. Gailitis, K. P. Thompson, J. T. Lin, “Ablation of the cornea and synthetic polymers using a UV (213 nm) solid-state laser,” IEEE J. Quantum Electron. 26, 2284–2288 (1990).
[CrossRef]

Garrison, B. J.

B. J. Garrison, R. Srinivasan, “Laser ablation of organic polymers: microscopic models for photochemical and thermal processes,” J. Appl. Phys. 57, 2909–2914 (1985).
[CrossRef]

Goodier, J. N.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, New York, 1970).

Goodman, D. M.

Guy, A. W.

A. F. Emery, P. Kramar, A. W. Guy, J. C. Lin, “Microwave induced temperature rises in rabbit eyes in cataract research,” J. Heat Transfer 97, 123–128 (1975).
[CrossRef]

Hahn, D. W.

M. N. Ediger, G. H. Pettit, D. W. Hahn, “Enhanced ArF laser absorption in a collagen target under ablative conditions,” Lasers Surg. Med. 15, 107–111 (1994).
[CrossRef] [PubMed]

Hill, R. A.

Q. Ren, R. H. Keates, R. A. Hill, M. W. Berns, “Laser refractive surgery: a review and current status,” Opt. Eng. 34, 642–658 (1995).
[CrossRef]

Hillenkamp, F.

C. A. Puliafito, R. F. Steinert, T. F. Deutsch, F. Hillenkamp, E. J. Dehm, C. M. Adler, “Excimer laser ablation of the cornea and lens,” Ophthalmology 92, 741–748 (1985).
[CrossRef] [PubMed]

Itzkan, I.

I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, M. S. Feld, “The thermoelastic basis of short pulsed laser ablation of biological tissue,” Proc. Natl. Acad. Sci. 92, 1960–1964 (1995).
[CrossRef] [PubMed]

D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, I. Itzkan, M. S. Feld, “Photomechanical basis of laser ablation of biological tissue,” Opt. Lett. 19, 1684–1686 (1994).
[CrossRef] [PubMed]

Kamm, R. D.

J. L. Battaglioli, R. D. Kamm, “Measurements of the compressive properties of scleral tissue,” Invest. Ophthalmol. Vis. Sci. 25, 59–65 (1984).
[PubMed]

Keates, R. H.

Q. Ren, R. H. Keates, R. A. Hill, M. W. Berns, “Laser refractive surgery: a review and current status,” Opt. Eng. 34, 642–658 (1995).
[CrossRef]

Kramar, P.

A. F. Emery, P. Kramar, A. W. Guy, J. C. Lin, “Microwave induced temperature rises in rabbit eyes in cataract research,” J. Heat Transfer 97, 123–128 (1975).
[CrossRef]

Lau, A.

Legeais, J.-M.

Q. Ren, G. Simon, J.-M. Legeais, J.-M. Parel, W. Culbertson, J. Shen, Y. Takesue, M. Savoldelli, “Ultraviolet solid-state laser (213-nm) photorefractive keratectomy,” Ophtalmology 101, 883–889 (1994).
[CrossRef]

Lin, J. C.

A. F. Emery, P. Kramar, A. W. Guy, J. C. Lin, “Microwave induced temperature rises in rabbit eyes in cataract research,” J. Heat Transfer 97, 123–128 (1975).
[CrossRef]

Lin, J. T.

Q. Ren, R. P. Gailitis, K. P. Thompson, J. T. Lin, “Ablation of the cornea and synthetic polymers using a UV (213 nm) solid-state laser,” IEEE J. Quantum Electron. 26, 2284–2288 (1990).
[CrossRef]

Maurice, D. M.

D. M. Maurice, “The cornea and sclera,” in The Eye, H. Davson, ed. (Academic, New York, 1984), Vol. 1b, pp. 1–158.
[CrossRef]

Mikic, B. B.

A. D. Yablon, N. S. Nishioka, B. B. Mikic, V. Venugopalan, “An interferometer for monitoring the surface displacement of a reflecting biological tissue surface,” in Advances in Heat and Mass Transfer in Biotechnology, , L. J. Hayes, S. Clegg, eds. (American Society of Mechanical Engineers, New York, 1996), pp. 75–80.

Milner, T. E.

Nelson, J. S.

Nishioka, N. S.

A. D. Yablon, N. S. Nishioka, B. B. Mikic, V. Venugopalan, “An interferometer for monitoring the surface displacement of a reflecting biological tissue surface,” in Advances in Heat and Mass Transfer in Biotechnology, , L. J. Hayes, S. Clegg, eds. (American Society of Mechanical Engineers, New York, 1996), pp. 75–80.

Parel, J.-M.

Q. Ren, G. Simon, J.-M. Legeais, J.-M. Parel, W. Culbertson, J. Shen, Y. Takesue, M. Savoldelli, “Ultraviolet solid-state laser (213-nm) photorefractive keratectomy,” Ophtalmology 101, 883–889 (1994).
[CrossRef]

Perelman, L. T.

I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, M. S. Feld, “The thermoelastic basis of short pulsed laser ablation of biological tissue,” Proc. Natl. Acad. Sci. 92, 1960–1964 (1995).
[CrossRef] [PubMed]

D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, I. Itzkan, M. S. Feld, “Photomechanical basis of laser ablation of biological tissue,” Opt. Lett. 19, 1684–1686 (1994).
[CrossRef] [PubMed]

Pettit, G. H.

G. H. Pettit, M. N. Ediger, “Corneal-tissue absorption coefficients for 193- and 213-nm ultraviolet radiation,” Appl. Opt. 35, 3386–3391 (1996).
[CrossRef] [PubMed]

M. N. Ediger, G. H. Pettit, D. W. Hahn, “Enhanced ArF laser absorption in a collagen target under ablative conditions,” Lasers Surg. Med. 15, 107–111 (1994).
[CrossRef] [PubMed]

M. N. Ediger, G. H. Pettit, R. P. Weiblinger, C. H. Chen, “Transmission of corneal collagen during ArF excimer laser ablation,” Lasers Surg. Med. 13, 204–210 (1993).
[CrossRef] [PubMed]

G. H. Pettit, M. N. Ediger, R. P. Weiblinger, “Dynamic optical properties of collagen-based tissue during ArF excimer laser ablation,” Appl. Opt. 32, 488–492 (1993).
[CrossRef] [PubMed]

G. H. Pettit, M. N. Ediger, R. P. Weiblinger, “Excimer laser corneal ablation: absence of a significant ‘incubation’ effect,” Lasers Surg. Med. 11, 411–418 (1991).
[CrossRef]

Prahl, S. A.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, New York, 1988).

Puliafito, C. A.

C. A. Puliafito, R. F. Steinert, T. F. Deutsch, F. Hillenkamp, E. J. Dehm, C. M. Adler, “Excimer laser ablation of the cornea and lens,” Ophthalmology 92, 741–748 (1985).
[CrossRef] [PubMed]

Ren, Q.

Q. Ren, R. H. Keates, R. A. Hill, M. W. Berns, “Laser refractive surgery: a review and current status,” Opt. Eng. 34, 642–658 (1995).
[CrossRef]

Q. Ren, G. Simon, J.-M. Legeais, J.-M. Parel, W. Culbertson, J. Shen, Y. Takesue, M. Savoldelli, “Ultraviolet solid-state laser (213-nm) photorefractive keratectomy,” Ophtalmology 101, 883–889 (1994).
[CrossRef]

Q. Ren, R. P. Gailitis, K. P. Thompson, J. T. Lin, “Ablation of the cornea and synthetic polymers using a UV (213 nm) solid-state laser,” IEEE J. Quantum Electron. 26, 2284–2288 (1990).
[CrossRef]

Savoldelli, M.

Q. Ren, G. Simon, J.-M. Legeais, J.-M. Parel, W. Culbertson, J. Shen, Y. Takesue, M. Savoldelli, “Ultraviolet solid-state laser (213-nm) photorefractive keratectomy,” Ophtalmology 101, 883–889 (1994).
[CrossRef]

Scott, J. A.

J. A. Scott, “A finite element model of heat transport in the human eye,” Phys. Med. Biol. 33, 227–241 (1988).
[CrossRef] [PubMed]

Shen, J.

Q. Ren, G. Simon, J.-M. Legeais, J.-M. Parel, W. Culbertson, J. Shen, Y. Takesue, M. Savoldelli, “Ultraviolet solid-state laser (213-nm) photorefractive keratectomy,” Ophtalmology 101, 883–889 (1994).
[CrossRef]

Simon, G.

Q. Ren, G. Simon, J.-M. Legeais, J.-M. Parel, W. Culbertson, J. Shen, Y. Takesue, M. Savoldelli, “Ultraviolet solid-state laser (213-nm) photorefractive keratectomy,” Ophtalmology 101, 883–889 (1994).
[CrossRef]

Smithies, D. J.

Sparrow, E. M.

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer, Augmented ed. (Brooks/Cole, Belmont, Calif., 1978).

Srinivasan, R.

R. Srinivasan, B. Braren, “Ultraviolet laser ablation of organic polymers,” Chem. Rev. 89, 1303–1316 (1989).
[CrossRef]

R. Srinivasan, P. E. Dyer, B. Braren, “Far-ultraviolet laser ablation of the cornea: photoacoustic studies,” Lasers Surg. Med. 6, 514–519 (1987).
[CrossRef] [PubMed]

B. J. Garrison, R. Srinivasan, “Laser ablation of organic polymers: microscopic models for photochemical and thermal processes,” J. Appl. Phys. 57, 2909–2914 (1985).
[CrossRef]

Staveteig, P. T.

Steinert, R. F.

C. A. Puliafito, R. F. Steinert, T. F. Deutsch, F. Hillenkamp, E. J. Dehm, C. M. Adler, “Excimer laser ablation of the cornea and lens,” Ophthalmology 92, 741–748 (1985).
[CrossRef] [PubMed]

Takesue, Y.

Q. Ren, G. Simon, J.-M. Legeais, J.-M. Parel, W. Culbertson, J. Shen, Y. Takesue, M. Savoldelli, “Ultraviolet solid-state laser (213-nm) photorefractive keratectomy,” Ophtalmology 101, 883–889 (1994).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, New York, 1988).

Thompson, K. P.

Q. Ren, R. P. Gailitis, K. P. Thompson, J. T. Lin, “Ablation of the cornea and synthetic polymers using a UV (213 nm) solid-state laser,” IEEE J. Quantum Electron. 26, 2284–2288 (1990).
[CrossRef]

Timoshenko, S. P.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, New York, 1970).

Venugopalan, V.

A. D. Yablon, N. S. Nishioka, B. B. Mikic, V. Venugopalan, “An interferometer for monitoring the surface displacement of a reflecting biological tissue surface,” in Advances in Heat and Mass Transfer in Biotechnology, , L. J. Hayes, S. Clegg, eds. (American Society of Mechanical Engineers, New York, 1996), pp. 75–80.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, New York, 1988).

Vitkin, I. A.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

Vodopyanov, K. L.

K. L. Vodopyanov, “Saturation studies of H2O and HDO near 3400 cm-1 using intense picosecond laser pulses,” J. Chem. Phys. 94, 5389–5393 (1991).
[CrossRef]

von Rosenberg, C.

I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, M. S. Feld, “The thermoelastic basis of short pulsed laser ablation of biological tissue,” Proc. Natl. Acad. Sci. 92, 1960–1964 (1995).
[CrossRef] [PubMed]

D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, I. Itzkan, M. S. Feld, “Photomechanical basis of laser ablation of biological tissue,” Opt. Lett. 19, 1684–1686 (1994).
[CrossRef] [PubMed]

Walsh, J. T.

P. T. Staveteig, J. T. Walsh, “Dynamic 193-nm optical properties of water,” Appl. Opt. 35, 3392–3403 (1996).
[CrossRef] [PubMed]

J. T. Walsh, J. P. Cummings, “Effect of the dynamic optical properties of water on mid-infrared laser ablation,” Lasers Surg. Med. 15, 295–305 (1994).
[CrossRef]

J. P. Cummings, J. T. Walsh, “Erbium laser ablation: the effect of dynamic optical properties,” Appl. Phys. Lett. 62, 1988–1990 (1993).
[CrossRef]

Weast, R. C.

R. C. Weast, CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, Fla., 1987).

Weiblinger, R. P.

G. H. Pettit, M. N. Ediger, R. P. Weiblinger, “Dynamic optical properties of collagen-based tissue during ArF excimer laser ablation,” Appl. Opt. 32, 488–492 (1993).
[CrossRef] [PubMed]

M. N. Ediger, G. H. Pettit, R. P. Weiblinger, C. H. Chen, “Transmission of corneal collagen during ArF excimer laser ablation,” Lasers Surg. Med. 13, 204–210 (1993).
[CrossRef] [PubMed]

G. H. Pettit, M. N. Ediger, R. P. Weiblinger, “Excimer laser corneal ablation: absence of a significant ‘incubation’ effect,” Lasers Surg. Med. 11, 411–418 (1991).
[CrossRef]

Williams, D.

H. D. Downing, D. Williams, “Optical constants of water in the infrared,” J. Geophys. Res. 80, 1656–1661 (1975).
[CrossRef]

Wilson, B. C.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

Yablon, A. D.

A. D. Yablon, N. S. Nishioka, B. B. Mikic, V. Venugopalan, “An interferometer for monitoring the surface displacement of a reflecting biological tissue surface,” in Advances in Heat and Mass Transfer in Biotechnology, , L. J. Hayes, S. Clegg, eds. (American Society of Mechanical Engineers, New York, 1996), pp. 75–80.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

J. P. Cummings, J. T. Walsh, “Erbium laser ablation: the effect of dynamic optical properties,” Appl. Phys. Lett. 62, 1988–1990 (1993).
[CrossRef]

Chem. Rev. (1)

R. Srinivasan, B. Braren, “Ultraviolet laser ablation of organic polymers,” Chem. Rev. 89, 1303–1316 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

Q. Ren, R. P. Gailitis, K. P. Thompson, J. T. Lin, “Ablation of the cornea and synthetic polymers using a UV (213 nm) solid-state laser,” IEEE J. Quantum Electron. 26, 2284–2288 (1990).
[CrossRef]

Invest. Ophthalmol. Vis. Sci. (1)

J. L. Battaglioli, R. D. Kamm, “Measurements of the compressive properties of scleral tissue,” Invest. Ophthalmol. Vis. Sci. 25, 59–65 (1984).
[PubMed]

J. Appl. Phys. (1)

B. J. Garrison, R. Srinivasan, “Laser ablation of organic polymers: microscopic models for photochemical and thermal processes,” J. Appl. Phys. 57, 2909–2914 (1985).
[CrossRef]

J. Chem. Phys. (1)

K. L. Vodopyanov, “Saturation studies of H2O and HDO near 3400 cm-1 using intense picosecond laser pulses,” J. Chem. Phys. 94, 5389–5393 (1991).
[CrossRef]

J. Geophys. Res. (1)

H. D. Downing, D. Williams, “Optical constants of water in the infrared,” J. Geophys. Res. 80, 1656–1661 (1975).
[CrossRef]

J. Heat Transfer (1)

A. F. Emery, P. Kramar, A. W. Guy, J. C. Lin, “Microwave induced temperature rises in rabbit eyes in cataract research,” J. Heat Transfer 97, 123–128 (1975).
[CrossRef]

Lasers Surg. Med. (5)

G. H. Pettit, M. N. Ediger, R. P. Weiblinger, “Excimer laser corneal ablation: absence of a significant ‘incubation’ effect,” Lasers Surg. Med. 11, 411–418 (1991).
[CrossRef]

J. T. Walsh, J. P. Cummings, “Effect of the dynamic optical properties of water on mid-infrared laser ablation,” Lasers Surg. Med. 15, 295–305 (1994).
[CrossRef]

M. N. Ediger, G. H. Pettit, R. P. Weiblinger, C. H. Chen, “Transmission of corneal collagen during ArF excimer laser ablation,” Lasers Surg. Med. 13, 204–210 (1993).
[CrossRef] [PubMed]

M. N. Ediger, G. H. Pettit, D. W. Hahn, “Enhanced ArF laser absorption in a collagen target under ablative conditions,” Lasers Surg. Med. 15, 107–111 (1994).
[CrossRef] [PubMed]

R. Srinivasan, P. E. Dyer, B. Braren, “Far-ultraviolet laser ablation of the cornea: photoacoustic studies,” Lasers Surg. Med. 6, 514–519 (1987).
[CrossRef] [PubMed]

Ophtalmology (1)

Q. Ren, G. Simon, J.-M. Legeais, J.-M. Parel, W. Culbertson, J. Shen, Y. Takesue, M. Savoldelli, “Ultraviolet solid-state laser (213-nm) photorefractive keratectomy,” Ophtalmology 101, 883–889 (1994).
[CrossRef]

Ophthalmology (1)

C. A. Puliafito, R. F. Steinert, T. F. Deutsch, F. Hillenkamp, E. J. Dehm, C. M. Adler, “Excimer laser ablation of the cornea and lens,” Ophthalmology 92, 741–748 (1985).
[CrossRef] [PubMed]

Opt. Eng. (1)

Q. Ren, R. H. Keates, R. A. Hill, M. W. Berns, “Laser refractive surgery: a review and current status,” Opt. Eng. 34, 642–658 (1995).
[CrossRef]

Opt. Lett. (1)

Phys. Med. Biol. (2)

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

J. A. Scott, “A finite element model of heat transport in the human eye,” Phys. Med. Biol. 33, 227–241 (1988).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. (1)

I. Itzkan, D. Albagli, M. L. Dark, L. T. Perelman, C. von Rosenberg, M. S. Feld, “The thermoelastic basis of short pulsed laser ablation of biological tissue,” Proc. Natl. Acad. Sci. 92, 1960–1964 (1995).
[CrossRef] [PubMed]

Other (9)

Kimble Glass Corporation, Properties of Kimble Glasses, Product data sheet (Kimble Glass Corporation, Vineland, N.J., 1997).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, New York, 1988).

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer, Augmented ed. (Brooks/Cole, Belmont, Calif., 1978).

R. C. Weast, CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, Fla., 1987).

A. D. Yablon, N. S. Nishioka, B. B. Mikic, V. Venugopalan, “An interferometer for monitoring the surface displacement of a reflecting biological tissue surface,” in Advances in Heat and Mass Transfer in Biotechnology, , L. J. Hayes, S. Clegg, eds. (American Society of Mechanical Engineers, New York, 1996), pp. 75–80.

Dow Chemical Corporation, Typical properties for Saran Wrap 19, (Dow Chemical Corporation, Midland, Mich., 1997).

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, 3rd ed. (McGraw-Hill, New York, 1970).

D. Albagli, “Fundamental mechanisms of pulsed laser ablation of biological tissue,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1994).

D. M. Maurice, “The cornea and sclera,” in The Eye, H. Davson, ed. (Academic, New York, 1984), Vol. 1b, pp. 1–158.
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

IPTS technique: (a) Laser energy has been completely deposited and no thermal diffusion has occurred although thermal stresses have equilibrated. An interval of approximately 100 ns to 50 µs transpires between (a) and (b). (b) Thermal diffusion has resulted in detectable surface displacement whose time scale and magnitude yield an effective penetration depth.

Fig. 2
Fig. 2

Geometry for derivation of the IPTS model. Material 1 is the target material (tissue), which is a thermally semi-infinite body. Material 2 (saran, glass, or air) is transparent to the incident pumping laser pulse.

Fig. 3
Fig. 3

Equations (3) and (4) plotted for a 650-J/m2 pulse Er:YSGG laser pulse with μ a = 4610 cm-1. Material 1 is water, material 2 is air, and ambient temperature is 20 °C. The three traces correspond to (a) 50 ns, (b) 1 µs, and (c) 20 µs after the laser pulse.

Fig. 4
Fig. 4

Acoustically thin layer: (1) Semi-infinite target, material 1 (tissue); (2) acoustically thin layer (saran or glass) assumed to be transparent to the incident laser pulse with thickness b; (3) upper semi-infinite body (air) also assumed to be transparent to the incident laser pulse.

Fig. 5
Fig. 5

Modified Mach–Zehnder interferometer employed to monitor target surface displacement resulting from a pump laser pulse. The AOM gives the reference arm of the interferometer a 110-MHz frequency shift and permits determination of the direction and the extent of the target surface displacement. In the sample arm of the interferometer the He–Ne beam was imaged onto the target surface with a beam expander and a 125-mm focal-length singlet lens. The half-wave plate (λ/2) and quarter-wave plate (λ/4) had their fast axes aligned at 45° to the plane of He–Ne polarization. As illustrated, the pump beam approached the sample at a slight angle to permit the He–Ne probe beam to interact with the target surface at normal incidence. The interference fringes detected by the APD were sampled by a digital scope and demodulated to reveal the surface displacement of the tissue target. This instrument had a 4-ns rise time and a noise floor of approximately 1 nm on pure water and bovine corneal stroma targets (not to scale).

Fig. 6
Fig. 6

Experimental setup for Q-switched Er:YSGG irradiation of saran-covered water samples (not to scale).

Fig. 7
Fig. 7

Expanded view of Er:YSGG irradiation of saran-covered water samples (not to scale).

Fig. 8
Fig. 8

Setup for ArF irradiation of bovine cornea (not to scale).

Fig. 9
Fig. 9

Expanded view of setup for ArF excimer laser irradiation of bovine cornea (not to scale).

Fig. 10
Fig. 10

Δz(t) calculated for Q-switched Er:YSGG laser irradiation of water covered by an acoustically thin saran film. Note the sensitivity to μ a for a fixed initial Δz. Traces are for (a) ϕ0 = 765 J/m2 with μ a = 2.3 × 103 cm-1, (b) ϕ0 = 650 J/m2 with μ a = 4.6 × 103 cm-1, (c) ϕ0 = 535 J/m2 with μ a = 9.2 × 103 cm-1.

Fig. 11
Fig. 11

Sample interferometer traces obtained for Q-switched Er:YSGG laser irradiation of saran-covered water at three different radiant exposures.

Fig. 12
Fig. 12

Comparison between a measured single trace of surface displacement and a predicted Δz for three different combinations of μ a and ϕ0 (Er:YSGG laser irradiation of water sample): Trace (a) ϕ0 = 923 J/m2, μ a = 6.2 × 103 cm-1; (b) ϕ0 = 956 J/m2, μ a = 5.0 × 103 cm-1; (c) ϕ0 = 1015 J/m2, μ a = 3.7 × 103 cm-1. The best-fit set of parameters for these measured data was found to be (b).

Fig. 13
Fig. 13

Typical traces obtained by an interferometer for ArF excimer laser irradiation of a bovine corneal stroma covered by a glass membrane: (a) ϕ0 = 156 J/m2, (b) ϕ0 = 306 J/m2, (c) ϕ0 = 364 J/m2.

Fig. 14
Fig. 14

Comparison between an averaged data trace (dotted curve) obtained for pulsed ArF excimer irradiation of bovine cornea and the best fit of a model when three different sets of μ a and ϕ0 are used. Dashed curve, best fit obtained with the absorption coefficient cited by Pettit and Ediger140 = 333 J/m2, μ a = 4 × 104 cm-1); dash–dot curve, best fit obtained with the absorption coefficient cited by Puliafito et al.80 = 435 J/m2, μ a = 2.7 × 103 cm-1), solid curve, overall best fit (ϕ0 = 338 J/m2, μ a = 1.9 × 104 cm-1).

Fig. 15
Fig. 15

Comparison between the averaged data trace obtained for pulsed ArF excimer irradiation of bovine cornea (dotted curve) and the best-fit FEM model (solid curve). For the best-fit FEM model, μ a = 1.9 × 104 cm-1 and ϕ0 = 338 J/m2.

Fig. 16
Fig. 16

ArF excimer laser pulse temporal profile.

Equations (24)

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

θ1t=α12θ1z2,  θ2t=α22θ2z2.
θ1z=0, t=θ2z=0, t,  k1θ1z=0, tz=k2θ2z=0, tz,
θ1z=, t=0,  θ2z=-, t=0.
θ1z, t=0=μaϕ0ρ0,1cp,1 exp-μaz,
θ1z, t=μaϕ02ρ0,1cp,1 expμa2α1texp-μaz×2-erfcz2α1t1/2-μaα1t1/2+k1-α1/α21/2k2k1+α1/α21/2k2expμaz×erfcz2α1t1/2+μaα1t1/2,
θ2z, t=k1k1+k2α1α21/2μaϕ0ρ0,1cp,1×erfcμaα1t1/2-z2α1t1/2×expμa2α1t-μazα1α21/2.
Q12t=0t k2θ2z=0, tzdt=α1k2ϕ0k1α1α21/2+α1k2×1-expα1μa2terfcμaα1t1/2.
Δzt=0ρ0ρTz0, t-1dz0.
ρwaterT=999.8+16.95T-7.987×10-3T2-46.17×10-6T3+105.6×10-9T4-280.5×10-12T51+16.88×10-3T.
Δz2t=αT,2Q12tρ2cp,2,
Δzt=λ2ΦAPDt-ΦAOMt2π,
Δz=0 εzzdz,
ux=0,  uy=0,  uz=uzz.
εij=12ui,j+uj,i,
εzz=uzz
εij=1+νE σij-νE σkkδij+ΔV3V δij,
εzz=σzzE-2νσE+ΔV3V,  0=1-νσE-νσzzE+ΔV3V.
εzz=1+ν1-νΔV3V.
ΔV=L+ΔL3-L3,
ΔVV=3 ΔLL.
ρ0L3=ρ1L+ΔL3ρ1L3+3L2ΔL,
3 ΔLL=ρ0ρ1-1.
εzz=131+ν1-νρ0ρTz, t-1.
Δzt=0ρ0ρTz, t-1dz.

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