Abstract

We demonstrate the potential of a new optical imaging system to directly obtain a longitudinal slice of a biological sample. The system, based on a single-shot optical correlator, operates a time-to-space conversion and an optical time-gating by sum-frequency generation in a nonlinear crystal. Owing to the high speed acquisition of the technique, internal structures of in-vivo tissues can be imaged at video rate. With this apparatus, we recorded longitudinal images of ex vivo mouse ear and in vivo human skin with a depth resolution of approximately 15 μm.

© 2002 Optical Society of America

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References

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  1. B.R. Masters and P.T.C. So, “Confocal microscopy and multi-photon excitation microscopy of human skin in vivo,” Opt. Express 8, 2–10 (2001), http://www.opticsexpress.org/oearchive/source/27001.htm
    [Crossref] [PubMed]
  2. D. Huang, E.A. Swanson, C.P. Lin, J.S. Schuman, W.G. Stinson, W. Chang, M.R. Hee, T. Flotte, K. Gregory, C.A. Puliafito, and J.G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [Crossref] [PubMed]
  3. A. Zumbusch, G.R. Holtom, and X.S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
    [Crossref]
  4. E. Abraham, E. Bordenave, N. Tsurumachi, G. Jonusauskas, J. Oberlé, C. Rullière, and A. Mito, “Real-time two-dimensional imaging in scattering media by use of a femtosecond Cr4+:forterite laser,” Opt. Lett. 25, 929–931 (2000).
    [Crossref]
  5. G. Le Tolguenec, E. Lantz, and F. Devaux, “Imaging through scattering media by parametric amplification of images: study of the resolution and the signal-to-noise ratio,” Appl. Opt. 36, 8292–8297 (1997).
    [Crossref]
  6. S. Bourquin, P. Seitz, and R.P. Salathé, “Optical coherence tomography based on a two-dimensional smart detector array,” Opt. Lett. 26, 512–514 (2001).
    [Crossref]
  7. E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberlé, C. Rullière, P.E. Minot, M. Lassègues, and J.E. Surlève Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. (in press).
  8. S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
    [Crossref] [PubMed]
  9. V. Bagnoud and F. Salin, “1.1 Terawatt, kilohertz femtosecond laser,” in Technical Digest CLEO ’99, (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 71–72.
  10. E. Bordenave, E. Abraham, G. Jonusauskas, J. Oberlé, and C. Rullière, “Single-shot correlation system for longitudinal imaging in biological tissues,” submitted to Opt. Commun.
  11. C.R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
    [Crossref] [PubMed]

2001 (2)

2000 (1)

1999 (1)

A. Zumbusch, G.R. Holtom, and X.S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[Crossref]

1998 (2)

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

C.R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

1997 (1)

1991 (1)

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

Abraham, E.

E. Abraham, E. Bordenave, N. Tsurumachi, G. Jonusauskas, J. Oberlé, C. Rullière, and A. Mito, “Real-time two-dimensional imaging in scattering media by use of a femtosecond Cr4+:forterite laser,” Opt. Lett. 25, 929–931 (2000).
[Crossref]

E. Bordenave, E. Abraham, G. Jonusauskas, J. Oberlé, and C. Rullière, “Single-shot correlation system for longitudinal imaging in biological tissues,” submitted to Opt. Commun.

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberlé, C. Rullière, P.E. Minot, M. Lassègues, and J.E. Surlève Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. (in press).

Auriol, F.

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

Bagnoud, V.

V. Bagnoud and F. Salin, “1.1 Terawatt, kilohertz femtosecond laser,” in Technical Digest CLEO ’99, (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 71–72.

Bazeille, J.E. Surlève

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberlé, C. Rullière, P.E. Minot, M. Lassègues, and J.E. Surlève Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. (in press).

Berson, M.

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

Black, D.

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

Bordenave, E.

E. Abraham, E. Bordenave, N. Tsurumachi, G. Jonusauskas, J. Oberlé, C. Rullière, and A. Mito, “Real-time two-dimensional imaging in scattering media by use of a femtosecond Cr4+:forterite laser,” Opt. Lett. 25, 929–931 (2000).
[Crossref]

E. Bordenave, E. Abraham, G. Jonusauskas, J. Oberlé, and C. Rullière, “Single-shot correlation system for longitudinal imaging in biological tissues,” submitted to Opt. Commun.

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberlé, C. Rullière, P.E. Minot, M. Lassègues, and J.E. Surlève Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. (in press).

Bourquin, S.

Chang, W.

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

Cope, M.

C.R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

Devaux, F.

Diridollou, S.

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

Essenpreis, M.

C.R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

Flotte, T.

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

Fujimoto, J.G.

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

Gall, Y.

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

Gregoire, J.M.

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

Gregory, K.

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

Hee, M.R.

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

Holtom, G.R.

A. Zumbusch, G.R. Holtom, and X.S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[Crossref]

Huang, D.

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

Jonusauskas, G.

E. Abraham, E. Bordenave, N. Tsurumachi, G. Jonusauskas, J. Oberlé, C. Rullière, and A. Mito, “Real-time two-dimensional imaging in scattering media by use of a femtosecond Cr4+:forterite laser,” Opt. Lett. 25, 929–931 (2000).
[Crossref]

E. Bordenave, E. Abraham, G. Jonusauskas, J. Oberlé, and C. Rullière, “Single-shot correlation system for longitudinal imaging in biological tissues,” submitted to Opt. Commun.

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberlé, C. Rullière, P.E. Minot, M. Lassègues, and J.E. Surlève Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. (in press).

Karlsson, B.

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

Kohl, M.

C.R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

Lantz, E.

Lassègues, M.

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberlé, C. Rullière, P.E. Minot, M. Lassègues, and J.E. Surlève Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. (in press).

Le Tolguenec, G.

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,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Masters, B.R.

Minot, P.E.

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberlé, C. Rullière, P.E. Minot, M. Lassègues, and J.E. Surlève Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. (in press).

Mito, A.

Oberlé, J.

E. Abraham, E. Bordenave, N. Tsurumachi, G. Jonusauskas, J. Oberlé, C. Rullière, and A. Mito, “Real-time two-dimensional imaging in scattering media by use of a femtosecond Cr4+:forterite laser,” Opt. Lett. 25, 929–931 (2000).
[Crossref]

E. Bordenave, E. Abraham, G. Jonusauskas, J. Oberlé, and C. Rullière, “Single-shot correlation system for longitudinal imaging in biological tissues,” submitted to Opt. Commun.

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberlé, C. Rullière, P.E. Minot, M. Lassègues, and J.E. Surlève Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. (in press).

Patat, F.

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

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,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Rullière, C.

E. Abraham, E. Bordenave, N. Tsurumachi, G. Jonusauskas, J. Oberlé, C. Rullière, and A. Mito, “Real-time two-dimensional imaging in scattering media by use of a femtosecond Cr4+:forterite laser,” Opt. Lett. 25, 929–931 (2000).
[Crossref]

E. Bordenave, E. Abraham, G. Jonusauskas, J. Oberlé, and C. Rullière, “Single-shot correlation system for longitudinal imaging in biological tissues,” submitted to Opt. Commun.

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberlé, C. Rullière, P.E. Minot, M. Lassègues, and J.E. Surlève Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. (in press).

Salathé, R.P.

Salin, F.

V. Bagnoud and F. Salin, “1.1 Terawatt, kilohertz femtosecond laser,” in Technical Digest CLEO ’99, (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 71–72.

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,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Seitz, P.

Simpson, C.R.

C.R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

So, P.T.C.

Stinson, W.G.

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

Swanson, E.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,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Tsurumachi, N.

E. Abraham, E. Bordenave, N. Tsurumachi, G. Jonusauskas, J. Oberlé, C. Rullière, and A. Mito, “Real-time two-dimensional imaging in scattering media by use of a femtosecond Cr4+:forterite laser,” Opt. Lett. 25, 929–931 (2000).
[Crossref]

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberlé, C. Rullière, P.E. Minot, M. Lassègues, and J.E. Surlève Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. (in press).

Vabre, V.

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

Vaillant, L.

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

Xie, X.S.

A. Zumbusch, G.R. Holtom, and X.S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[Crossref]

Yvon, C.

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

Zumbusch, A.

A. Zumbusch, G.R. Holtom, and X.S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[Crossref]

Appl. Opt. (2)

E. Bordenave, E. Abraham, G. Jonusauskas, N. Tsurumachi, J. Oberlé, C. Rullière, P.E. Minot, M. Lassègues, and J.E. Surlève Bazeille, “Wide-field optical coherence tomography: imaging of biological tissues,” Appl. Opt. (in press).

G. Le Tolguenec, E. Lantz, and F. Devaux, “Imaging through scattering media by parametric amplification of images: study of the resolution and the signal-to-noise ratio,” Appl. Opt. 36, 8292–8297 (1997).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Med. Biol. (1)

C.R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

A. Zumbusch, G.R. Holtom, and X.S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[Crossref]

Science (1)

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

Ultrasound in Medicine & Biology (1)

S. Diridollou, M. Berson, V. Vabre, D. Black, B. Karlsson, F. Auriol, J.M. Gregoire, C. Yvon, L. Vaillant, Y. Gall, and F. Patat, “An in vivo method for measuring the mechanical properties of the skin using ultrasound,” Ultrasound in Medicine & Biology 24, 215–224 (1998).
[Crossref] [PubMed]

Other (2)

V. Bagnoud and F. Salin, “1.1 Terawatt, kilohertz femtosecond laser,” in Technical Digest CLEO ’99, (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 71–72.

E. Bordenave, E. Abraham, G. Jonusauskas, J. Oberlé, and C. Rullière, “Single-shot correlation system for longitudinal imaging in biological tissues,” submitted to Opt. Commun.

Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1.

Principle of the longitudinal imaging. Non-collinear interaction between the pump and probe pulses (frequency ω) in a nonlinear crystal for sum-frequency generation at frequency 2ω.

Fig. 2.
Fig. 2.

Experimental setup of the longitudinal imaging system. BS: 50/50 beam splitter; CL1: cylindrical lens; L2,L3: spherical lenses; DL: delay line.

Fig. 3.
Fig. 3.

(a) HE histology of a mouse ear. Image size: (0.93×0.7) mm2; (b) Longitudinal image of an ex vivo mouse ear. Image size (1.2×0.5) mm2. E: epidermis, sc: stratum corneum, D: dermis, cc: conjunctive capsule, C: cartilage.

Fig. 4.
Fig. 4.

(a) HE histology of human skin. Image size: (0.53×0.4) mm2; (b) Longitudinal image of an in vivo human skin in the region of the forearm. Image size: (1.5×0.6) mm2; (c) Linear depth profile of the longitudinal image along the line on (b). E: epidermis, sc: stratum corneum, D: dermis.

Fig. 5.
Fig. 5.

(1.5 MB) Movie of the skin of the volunteer, in the region of the forearm. Image size: (1×0.3) mm2

Equations (2)

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Δz = 2 c n 0 ( λ ) sin ( Φ / 2 ) Δτ ,
Z max = 1 2 sin ( Φ / 2 ) A / n sample

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