Abstract

We demonstrate low-coherence interferometry for diffusion measurements. We have measured the diffusion coefficient of a phthalocyanine dye in 1.5% agar gel with a two-wavelength interferometer; one wavelength was matched to the absorption peak of the dye at 675 nm, while the other, 805 nm, was not affected by the dye. The diffusion coefficient of the dye was found by fitting a mathematical model for the interferometer signal to the measured low-coherence interferometry amplitude. A 95% confidence interval for the diffusion coefficient was found to be D=2.5±0.2×10-10 m2/s. The influence of speckle averaging and experiment time on the determination of the diffusion coefficient has been studied. The presented technique allows in situ characterization of diffusion in semitransparent media.

© 2003 Optical Society of America

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  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, Science 254, 1178 (1991).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

2003

J. R. Lead, K. Starchev, and K. J. Wilkinson, Environ. Sci. Technol. 37, 482 (2003).
[CrossRef] [PubMed]

2002

W. M. Star, M. C. G. Aalders, A. Sac, and H. J. C. M. Sterenborg, Photochem. Photobiol. 75, 424 (2002).
[CrossRef] [PubMed]

2001

D. E. Hyde, T. J. Farrel, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[CrossRef] [PubMed]

2000

1999

J. R. Mourant, T. M. Johnson, G. Los, and I. J. Bigio, Phys. Med. Biol. 44, 1397 (1999).
[CrossRef] [PubMed]

L. Masaro and X. X. Zhu, Prog. Polym. Sci. 24, 731 (1999).
[CrossRef]

1998

P. Gribbon and T. E. Hardingham, Biophys. J. 75, 1032 (1998).
[CrossRef] [PubMed]

J. M. Schmitt, S. H. Xiang, and K. M. Yung, J. Opt. Soc. Am. A 15, 2288 (1998).
[CrossRef]

1997

1996

L. O. Svaasand, P. Wyss, M. T. Wyss, Y. Tadir, B. J. Tromberg, and M. W. Berns, Lasers Surg. Med. 18, 139 (1996).
[CrossRef]

1994

B. A. Westrin, A. Axelsson, and G. Zacchi, J. Controlled Release 30, 189 (1994).
[CrossRef]

1991

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, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Aalders, M. C. G.

W. M. Star, M. C. G. Aalders, A. Sac, and H. J. C. M. Sterenborg, Photochem. Photobiol. 75, 424 (2002).
[CrossRef] [PubMed]

Axelsson, A.

B. A. Westrin, A. Axelsson, and G. Zacchi, J. Controlled Release 30, 189 (1994).
[CrossRef]

Berns, M. W.

L. O. Svaasand, P. Wyss, M. T. Wyss, Y. Tadir, B. J. Tromberg, and M. W. Berns, Lasers Surg. Med. 18, 139 (1996).
[CrossRef]

Bigio, I. J.

J. R. Mourant, T. M. Johnson, G. Los, and I. J. Bigio, Phys. Med. Biol. 44, 1397 (1999).
[CrossRef] [PubMed]

Boppart, S.

J. G. Fujimoto, C. Pitris, S. Boppart, and M. Brezinski, Neoplasia 2, 9 (2000).
[CrossRef] [PubMed]

Brezinski, M.

J. G. Fujimoto, C. Pitris, S. Boppart, and M. Brezinski, Neoplasia 2, 9 (2000).
[CrossRef] [PubMed]

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, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Chen, Z. P.

Crank, J.

J. Crank, The Mathematics of Diffusion (Oxford U. Press, New York, 1975), pp. 11–16.

deBoer, J. F.

Diamond, K. R.

R. A. Weersink, J. E. Hayward, K. R. Diamond, and M. S. Patterson, Photochem. Photobiol. 66, 326 (1997).
[CrossRef] [PubMed]

Drexler, W.

Farrel, T. J.

D. E. Hyde, T. J. Farrel, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[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, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

J. G. Fujimoto, C. Pitris, S. Boppart, and M. Brezinski, Neoplasia 2, 9 (2000).
[CrossRef] [PubMed]

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, Opt. Lett. 25, 111 (2000).
[CrossRef]

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

Gribbon, P.

P. Gribbon and T. E. Hardingham, Biophys. J. 75, 1032 (1998).
[CrossRef] [PubMed]

Hardingham, T. E.

P. Gribbon and T. E. Hardingham, Biophys. J. 75, 1032 (1998).
[CrossRef] [PubMed]

Hayward, J. E.

R. A. Weersink, J. E. Hayward, K. R. Diamond, and M. S. Patterson, Photochem. Photobiol. 66, 326 (1997).
[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, Science 254, 1178 (1991).
[CrossRef] [PubMed]

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, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Hyde, D. E.

D. E. Hyde, T. J. Farrel, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[CrossRef] [PubMed]

Ippen, E. P.

Johnson, T. M.

J. R. Mourant, T. M. Johnson, G. Los, and I. J. Bigio, Phys. Med. Biol. 44, 1397 (1999).
[CrossRef] [PubMed]

Kärtner, F. X.

Lead, J. R.

J. R. Lead, K. Starchev, and K. J. Wilkinson, Environ. Sci. Technol. 37, 482 (2003).
[CrossRef] [PubMed]

Li, X. D.

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, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Lindmo, T.

T. Støren, A. Simonsen, A. K. Røyset, O. J. Løkberg, L. O. Svaasand, and T. Lindmo, in Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds., Proc. SPIE 4956 (to be published).

Løkberg, O. J.

T. Støren, A. Simonsen, A. K. Røyset, O. J. Løkberg, L. O. Svaasand, and T. Lindmo, in Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds., Proc. SPIE 4956 (to be published).

Los, G.

J. R. Mourant, T. M. Johnson, G. Los, and I. J. Bigio, Phys. Med. Biol. 44, 1397 (1999).
[CrossRef] [PubMed]

Malekafzali, A.

Masaro, L.

L. Masaro and X. X. Zhu, Prog. Polym. Sci. 24, 731 (1999).
[CrossRef]

Milner, T. E.

Morgner, U.

Mourant, J. R.

J. R. Mourant, T. M. Johnson, G. Los, and I. J. Bigio, Phys. Med. Biol. 44, 1397 (1999).
[CrossRef] [PubMed]

Nelson, J. S.

Patterson, M. S.

D. E. Hyde, T. J. Farrel, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[CrossRef] [PubMed]

R. A. Weersink, J. E. Hayward, K. R. Diamond, and M. S. Patterson, Photochem. Photobiol. 66, 326 (1997).
[CrossRef] [PubMed]

Pitris, C.

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, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Røyset, A. K.

T. Støren, A. Simonsen, A. K. Røyset, O. J. Løkberg, L. O. Svaasand, and T. Lindmo, in Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds., Proc. SPIE 4956 (to be published).

Sac, A.

W. M. Star, M. C. G. Aalders, A. Sac, and H. J. C. M. Sterenborg, Photochem. Photobiol. 75, 424 (2002).
[CrossRef] [PubMed]

Schmitt, J. M.

J. M. Schmitt, S. H. Xiang, and K. M. Yung, J. Opt. Soc. Am. A 15, 2288 (1998).
[CrossRef]

Schuman, J. S.

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

Simonsen, A.

T. Støren, A. Simonsen, A. K. Røyset, O. J. Løkberg, L. O. Svaasand, and T. Lindmo, in Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds., Proc. SPIE 4956 (to be published).

Srinivas, S.

Star, W. M.

W. M. Star, M. C. G. Aalders, A. Sac, and H. J. C. M. Sterenborg, Photochem. Photobiol. 75, 424 (2002).
[CrossRef] [PubMed]

W. M. Star, Phys. Med. Biol. 42, 763 (1997).
[CrossRef] [PubMed]

Starchev, K.

J. R. Lead, K. Starchev, and K. J. Wilkinson, Environ. Sci. Technol. 37, 482 (2003).
[CrossRef] [PubMed]

Sterenborg, H. J. C. M.

W. M. Star, M. C. G. Aalders, A. Sac, and H. J. C. M. Sterenborg, Photochem. Photobiol. 75, 424 (2002).
[CrossRef] [PubMed]

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, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Støren, T.

T. Støren, A. Simonsen, A. K. Røyset, O. J. Løkberg, L. O. Svaasand, and T. Lindmo, in Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds., Proc. SPIE 4956 (to be published).

Svaasand, L. O.

L. O. Svaasand, P. Wyss, M. T. Wyss, Y. Tadir, B. J. Tromberg, and M. W. Berns, Lasers Surg. Med. 18, 139 (1996).
[CrossRef]

T. Støren, A. Simonsen, A. K. Røyset, O. J. Løkberg, L. O. Svaasand, and T. Lindmo, in Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds., Proc. SPIE 4956 (to be published).

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, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Tadir, Y.

L. O. Svaasand, P. Wyss, M. T. Wyss, Y. Tadir, B. J. Tromberg, and M. W. Berns, Lasers Surg. Med. 18, 139 (1996).
[CrossRef]

Tromberg, B. J.

L. O. Svaasand, P. Wyss, M. T. Wyss, Y. Tadir, B. J. Tromberg, and M. W. Berns, Lasers Surg. Med. 18, 139 (1996).
[CrossRef]

van Gemert, M. J. C.

Wang, X. J.

Weersink, R. A.

R. A. Weersink, J. E. Hayward, K. R. Diamond, and M. S. Patterson, Photochem. Photobiol. 66, 326 (1997).
[CrossRef] [PubMed]

Westrin, B. A.

B. A. Westrin, A. Axelsson, and G. Zacchi, J. Controlled Release 30, 189 (1994).
[CrossRef]

Wilkinson, K. J.

J. R. Lead, K. Starchev, and K. J. Wilkinson, Environ. Sci. Technol. 37, 482 (2003).
[CrossRef] [PubMed]

Wilson, B. C.

D. E. Hyde, T. J. Farrel, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[CrossRef] [PubMed]

Wyss, M. T.

L. O. Svaasand, P. Wyss, M. T. Wyss, Y. Tadir, B. J. Tromberg, and M. W. Berns, Lasers Surg. Med. 18, 139 (1996).
[CrossRef]

Wyss, P.

L. O. Svaasand, P. Wyss, M. T. Wyss, Y. Tadir, B. J. Tromberg, and M. W. Berns, Lasers Surg. Med. 18, 139 (1996).
[CrossRef]

Xiang, S. H.

J. M. Schmitt, S. H. Xiang, and K. M. Yung, J. Opt. Soc. Am. A 15, 2288 (1998).
[CrossRef]

Yung, K. M.

J. M. Schmitt, S. H. Xiang, and K. M. Yung, J. Opt. Soc. Am. A 15, 2288 (1998).
[CrossRef]

Zacchi, G.

B. A. Westrin, A. Axelsson, and G. Zacchi, J. Controlled Release 30, 189 (1994).
[CrossRef]

Zhu, X. X.

L. Masaro and X. X. Zhu, Prog. Polym. Sci. 24, 731 (1999).
[CrossRef]

Biophys. J.

P. Gribbon and T. E. Hardingham, Biophys. J. 75, 1032 (1998).
[CrossRef] [PubMed]

Environ. Sci. Technol.

J. R. Lead, K. Starchev, and K. J. Wilkinson, Environ. Sci. Technol. 37, 482 (2003).
[CrossRef] [PubMed]

J. Controlled Release

B. A. Westrin, A. Axelsson, and G. Zacchi, J. Controlled Release 30, 189 (1994).
[CrossRef]

J. Opt. Soc. Am. A

J. M. Schmitt, S. H. Xiang, and K. M. Yung, J. Opt. Soc. Am. A 15, 2288 (1998).
[CrossRef]

Lasers Surg. Med.

L. O. Svaasand, P. Wyss, M. T. Wyss, Y. Tadir, B. J. Tromberg, and M. W. Berns, Lasers Surg. Med. 18, 139 (1996).
[CrossRef]

Neoplasia

J. G. Fujimoto, C. Pitris, S. Boppart, and M. Brezinski, Neoplasia 2, 9 (2000).
[CrossRef] [PubMed]

Opt. Lett.

Photochem. Photobiol.

R. A. Weersink, J. E. Hayward, K. R. Diamond, and M. S. Patterson, Photochem. Photobiol. 66, 326 (1997).
[CrossRef] [PubMed]

W. M. Star, M. C. G. Aalders, A. Sac, and H. J. C. M. Sterenborg, Photochem. Photobiol. 75, 424 (2002).
[CrossRef] [PubMed]

Phys. Med. Biol.

J. R. Mourant, T. M. Johnson, G. Los, and I. J. Bigio, Phys. Med. Biol. 44, 1397 (1999).
[CrossRef] [PubMed]

D. E. Hyde, T. J. Farrel, M. S. Patterson, and B. C. Wilson, Phys. Med. Biol. 46, 369 (2001).
[CrossRef] [PubMed]

W. M. Star, Phys. Med. Biol. 42, 763 (1997).
[CrossRef] [PubMed]

Proc. SPIE

T. Støren, A. Simonsen, A. K. Røyset, O. J. Løkberg, L. O. Svaasand, and T. Lindmo, in Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds., Proc. SPIE 4956 (to be published).

Prog. Polym. Sci.

L. Masaro and X. X. Zhu, Prog. Polym. Sci. 24, 731 (1999).
[CrossRef]

Science

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, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Other

J. Crank, The Mathematics of Diffusion (Oxford U. Press, New York, 1975), pp. 11–16.

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Figures (2)

Fig. 1
Fig. 1

Setup for the wavelength-multiplexed interferometer. BS, beam splitter; PC, personal computer.

Fig. 2
Fig. 2

Experimental data for one of the ten samples. (a) Gray-scale image showing the logarithmic amplitude of five series of A-scans recorded over the same 0.4 mm×1.0 mm cross-sectional area of the sample. The abscissa represents the time t after dye deposition, and the ordinate axis is the geometrical depth in the sample, where z=0 is the agar–glass interface. (b) Speckle-averaged amplitude as a function of depth for the five images in (a). The smooth curves represent the least-squares-fitted model of Eq. (6). The reflection from the agar–glass interface seen as an area of high reflectivity at z=0 in (a) was removed from the data before parameter fitting. The mean time for each of the images is t1 (bottom curve) to t5 (top curve), and the data sample rate in depth is 1340 samples/mm.

Equations (6)

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

μtλz,t=μt,gelλz,t+ϵaλCz,t,
Cz,tt=2DCz,t,
Cz,t=M2SπDtexp-z24Dt,
Is,λz,t=Is0,λRλz,texp-20zμtλz,tdz,
Aλz,t=rλzGλexp-μt,gelλz+ϵaλ0zCz,tdz.
lnAλz,t=KrGλ-μt,gelλz-ϵaλ0zCz,tdz.

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