Abstract

High-speed complex full-range Fourier domain optical coherence tomography (FD-OCT) is demonstrated. In this FD-OCT, the phase modulation of a reference beam (M scan) and transversal scanning (B scan) are simultaneously performed. The Fourier transform method is applied along the direction of the B scan to reconstruct complex spectra, and the complex spectra comprise a full-range OCT image. Because of this simultaneous BM-mode scan, the FD-OCT requires only a single A scan for each single transversal position to obtain a full-range FD-OCT image. A simple but slow version of the FD-OCT visualizes the cross section of a plastic plate. A modified fast version of this FD-OCT investigates a sweat duct in a finger pad in vivo and visualizes it with an acquisition time of 27  ms.

© 2006 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, "Optical coherence tomography," Science 254, 1178-1181 (1991).
  2. G. Häusler and M. W. Lindner, "Coherence radar and 'spectral radar'—new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
    [CrossRef]
  3. P. Andretzky, M. W. Lindner, J. M. Herrmann, A. Schutz, M. Konzog, F. Kiesewatter, and G. Haeusler, "Optical coherence tomography by 'spectral radar,' dynamic range estimation and in-vivo measurements of skin," in Optical and Imaging Techniques for Biomonitoring IV, M. Dal Fante, H.-J. Foth, N. Krasner, R. Marchesini, and H. Podbielska, eds., Proc. SPIE 3567, 78-87 (1998).
    [CrossRef]
  4. R. A. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, "Performance of Fourier domain vs. time domain optical coherence tomography," Opt. Express 11, 889-894 (2003).
  5. N. Nassif, B. Cense, B. H. Park, S. H. Yun, T. C. Chen, B. E. Bouma, G. J. Tearney, and J. F. de Boer, "In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography," Opt. Lett. 29, 480-482 (2004).
    [CrossRef]
  6. N. A. Nassif, B. Cense, B. H. Park, M. C. Pierce, S. H. Yun, B. E. Bouma, G. J. Tearney, T. C. Chen, and J. F. de Boer, "In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve," Opt. Express 12, 367-376 (2004).
    [CrossRef]
  7. Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagai, "Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography," Opt. Lett. 27, 1803-1805 (2002).
  8. Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
    [CrossRef]
  9. J. Zhang, W. Jung, J. S. Nelson, and Z. Chen, "Full range polarization-sensitive Fourier domain optical coherence tomography," Opt. Express 12, 6033-6039 (2004).
    [CrossRef]
  10. B. H. Park, M. C. Pierce, B. Cense, S.-H. Yun, M. Mujat, G. J. Tearney, B. E. Bouma, and J. F. de Boer, "Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 µm," Opt. Express 13, 3931-3944 (2005).
    [CrossRef]
  11. R. Leitgeb, M. Wojtkowski, A. Kowalczyk, C. K. Hitzenberger, M. Sticker, and A. F. Fercher, "Spectral measurement of absorption by spectroscopic frequency-domain optical coherence tomography," Opt. Lett. 25, 820-822 (2000).
  12. B. R. White, M. C. Pierce, N. Nassif, B. Cense, B. H. Park, G. J. Tearney, B. E. Bouma, T. C. Chen, and J. F. de Boer, "In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical coherence tomography," Opt. Express 11, 3490-3497 (2003).
  13. R. A. Leitgeb, L. Schmetterer, W. Drexler, A. F. Fercher, R. J. Zawadzki, and T. Bajraszewski, "Real-time assessment of retinal blood flow with ultrafast acquisition by color Doppler Fourier domain optical coherence tomography," Opt. Express 11, 3116-3121 (2003).
  14. M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, "Full-range complex spectral optical coherence tomography technique in eye imaging," Opt. Lett. 27, 1415-1417 (2002).
  15. For example, J. E. Greivenkamp and J. H. Bruning, "Phase shifting interferometers," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley-Interscience, 1992), pp. 501-598.
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  17. R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, "Phase-shifting algorithm to achieve high-speed long-depth-range probing by frequency-domain optical coherence tomography," Opt. Lett. 28, 2201-2203 (2003).
  18. E. Götzinger, M. Pircher, R. A. Leitgeb, and C. K. Hitzenberger, "High speed full range complex spectral domain optical coherence tomography," Opt. Express 13, 583-594 (2005).
    [CrossRef]
  19. S. H. Yun, G. J. Tearney, J. F. de Boer, and B. E. Bouma, "Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting," Opt. Express 12, 4822-4828 (2004).
    [CrossRef]
  20. J. Zhang, J. S. Nelson, and Z. Chen, "Removal of a mirror image and enhancement of the signal-to-noise ratio in Fourier-domain optical coherence tomography using an electro-optic phase modulator," Opt. Lett. 30, 147-149 (2005).
    [CrossRef]
  21. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  22. M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160 (1982).

2005 (3)

2004 (5)

2003 (4)

2002 (2)

2000 (1)

1998 (2)

G. Häusler and M. W. Lindner, "Coherence radar and 'spectral radar'—new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

P. Andretzky, M. W. Lindner, J. M. Herrmann, A. Schutz, M. Konzog, F. Kiesewatter, and G. Haeusler, "Optical coherence tomography by 'spectral radar,' dynamic range estimation and in-vivo measurements of skin," in Optical and Imaging Techniques for Biomonitoring IV, M. Dal Fante, H.-J. Foth, N. Krasner, R. Marchesini, and H. Podbielska, eds., Proc. SPIE 3567, 78-87 (1998).
[CrossRef]

1995 (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).

1982 (1)

Andretzky, P.

P. Andretzky, M. W. Lindner, J. M. Herrmann, A. Schutz, M. Konzog, F. Kiesewatter, and G. Haeusler, "Optical coherence tomography by 'spectral radar,' dynamic range estimation and in-vivo measurements of skin," in Optical and Imaging Techniques for Biomonitoring IV, M. Dal Fante, H.-J. Foth, N. Krasner, R. Marchesini, and H. Podbielska, eds., Proc. SPIE 3567, 78-87 (1998).
[CrossRef]

Bajraszewski, T.

Bouma, B. E.

Cense, B.

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).

Chen, T. C.

Chen, Z.

Creath, K.

de Boer, J. F.

Drexler, W. F.

Endo, T.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Fercher, A. F.

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).

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).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Götzinger, E.

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).

Haeusler, G.

P. Andretzky, M. W. Lindner, J. M. Herrmann, A. Schutz, M. Konzog, F. Kiesewatter, and G. Haeusler, "Optical coherence tomography by 'spectral radar,' dynamic range estimation and in-vivo measurements of skin," in Optical and Imaging Techniques for Biomonitoring IV, M. Dal Fante, H.-J. Foth, N. Krasner, R. Marchesini, and H. Podbielska, eds., Proc. SPIE 3567, 78-87 (1998).
[CrossRef]

Häusler, G.

G. Häusler and M. W. Lindner, "Coherence radar and 'spectral radar'—new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

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).

Herrmann, J. M.

P. Andretzky, M. W. Lindner, J. M. Herrmann, A. Schutz, M. Konzog, F. Kiesewatter, and G. Haeusler, "Optical coherence tomography by 'spectral radar,' dynamic range estimation and in-vivo measurements of skin," in Optical and Imaging Techniques for Biomonitoring IV, M. Dal Fante, H.-J. Foth, N. Krasner, R. Marchesini, and H. Podbielska, eds., Proc. SPIE 3567, 78-87 (1998).
[CrossRef]

Hitzenberger, C. K.

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).

Ina, H.

Itoh, M.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagai, "Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography," Opt. Lett. 27, 1803-1805 (2002).

Jung, W.

Katada, C.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Kiesewatter, F.

P. Andretzky, M. W. Lindner, J. M. Herrmann, A. Schutz, M. Konzog, F. Kiesewatter, and G. Haeusler, "Optical coherence tomography by 'spectral radar,' dynamic range estimation and in-vivo measurements of skin," in Optical and Imaging Techniques for Biomonitoring IV, M. Dal Fante, H.-J. Foth, N. Krasner, R. Marchesini, and H. Podbielska, eds., Proc. SPIE 3567, 78-87 (1998).
[CrossRef]

Kobayashi, S.

Konzog, M.

P. Andretzky, M. W. Lindner, J. M. Herrmann, A. Schutz, M. Konzog, F. Kiesewatter, and G. Haeusler, "Optical coherence tomography by 'spectral radar,' dynamic range estimation and in-vivo measurements of skin," in Optical and Imaging Techniques for Biomonitoring IV, M. Dal Fante, H.-J. Foth, N. Krasner, R. Marchesini, and H. Podbielska, eds., Proc. SPIE 3567, 78-87 (1998).
[CrossRef]

Kowalczyk, A.

Leitgeb, R.

Leitgeb, R. A.

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).

Lindner, M. W.

G. Häusler and M. W. Lindner, "Coherence radar and 'spectral radar'—new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

P. Andretzky, M. W. Lindner, J. M. Herrmann, A. Schutz, M. Konzog, F. Kiesewatter, and G. Haeusler, "Optical coherence tomography by 'spectral radar,' dynamic range estimation and in-vivo measurements of skin," in Optical and Imaging Techniques for Biomonitoring IV, M. Dal Fante, H.-J. Foth, N. Krasner, R. Marchesini, and H. Podbielska, eds., Proc. SPIE 3567, 78-87 (1998).
[CrossRef]

Makita, S.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagai, "Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography," Opt. Lett. 27, 1803-1805 (2002).

Mujat, M.

Mutoh, M.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Nassif, N.

Nassif, N. A.

Nelson, J. S.

Park, B. H.

Pierce, M. C.

Pircher, M.

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).

Schmetterer, L.

Schmit, J.

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).

Schutz, A.

P. Andretzky, M. W. Lindner, J. M. Herrmann, A. Schutz, M. Konzog, F. Kiesewatter, and G. Haeusler, "Optical coherence tomography by 'spectral radar,' dynamic range estimation and in-vivo measurements of skin," in Optical and Imaging Techniques for Biomonitoring IV, M. Dal Fante, H.-J. Foth, N. Krasner, R. Marchesini, and H. Podbielska, eds., Proc. SPIE 3567, 78-87 (1998).
[CrossRef]

Sticker, M.

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).

Sutoh, Y.

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).

Takahashi, M.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Takeda, M.

Tearney, G. J.

White, B. R.

Wojtkowski, M.

Yasuno, Y.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagai, "Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography," Opt. Lett. 27, 1803-1805 (2002).

Yatagai, T.

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagai, "Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography," Opt. Lett. 27, 1803-1805 (2002).

Yun, S. H.

Yun, S.-H.

Zawadzki, R. J.

Zhang, J.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Y. Yasuno, S. Makita, T. Endo, M. Itoh, T. Yatagai, M. Takahashi, C. Katada, and M. Mutoh, "Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples," Appl. Phys. Lett. 85, 3023-3025 (2004).
[CrossRef]

J. Biomed. Opt. (1)

G. Häusler and M. W. Lindner, "Coherence radar and 'spectral radar'—new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (8)

E. Götzinger, M. Pircher, R. A. Leitgeb, and C. K. Hitzenberger, "High speed full range complex spectral domain optical coherence tomography," Opt. Express 13, 583-594 (2005).
[CrossRef]

S. H. Yun, G. J. Tearney, J. F. de Boer, and B. E. Bouma, "Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting," Opt. Express 12, 4822-4828 (2004).
[CrossRef]

R. A. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, "Performance of Fourier domain vs. time domain optical coherence tomography," Opt. Express 11, 889-894 (2003).

J. Zhang, W. Jung, J. S. Nelson, and Z. Chen, "Full range polarization-sensitive Fourier domain optical coherence tomography," Opt. Express 12, 6033-6039 (2004).
[CrossRef]

B. H. Park, M. C. Pierce, B. Cense, S.-H. Yun, M. Mujat, G. J. Tearney, B. E. Bouma, and J. F. de Boer, "Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 µm," Opt. Express 13, 3931-3944 (2005).
[CrossRef]

N. A. Nassif, B. Cense, B. H. Park, M. C. Pierce, S. H. Yun, B. E. Bouma, G. J. Tearney, T. C. Chen, and J. F. de Boer, "In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve," Opt. Express 12, 367-376 (2004).
[CrossRef]

B. R. White, M. C. Pierce, N. Nassif, B. Cense, B. H. Park, G. J. Tearney, B. E. Bouma, T. C. Chen, and J. F. de Boer, "In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical coherence tomography," Opt. Express 11, 3490-3497 (2003).

R. A. Leitgeb, L. Schmetterer, W. Drexler, A. F. Fercher, R. J. Zawadzki, and T. Bajraszewski, "Real-time assessment of retinal blood flow with ultrafast acquisition by color Doppler Fourier domain optical coherence tomography," Opt. Express 11, 3116-3121 (2003).

Opt. Lett. (6)

Proc. SPIE (1)

P. Andretzky, M. W. Lindner, J. M. Herrmann, A. Schutz, M. Konzog, F. Kiesewatter, and G. Haeusler, "Optical coherence tomography by 'spectral radar,' dynamic range estimation and in-vivo measurements of skin," in Optical and Imaging Techniques for Biomonitoring IV, M. Dal Fante, H.-J. Foth, N. Krasner, R. Marchesini, and H. Podbielska, eds., Proc. SPIE 3567, 78-87 (1998).
[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).

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

For example, J. E. Greivenkamp and J. H. Bruning, "Phase shifting interferometers," in Optical Shop Testing, 2nd ed., D.Malacara, ed. (Wiley-Interscience, 1992), pp. 501-598.

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Figures (6)

Fig. 1
Fig. 1

Optical scheme of a complex FD-OCT. CL, collimator lens; Ls, lenses; Obj., objective lens; BS, beam splitter; PS, pupil stop; PZT, piezoelectric transducer. The broadband light source is a SLD with an 828 nm central wavelength and a 40 nm bandwidth. The CCD camera is a line-scan CCD camera with 2048 pixels.

Fig. 2
Fig. 2

Time charts of B scans and M scans for (i) a conventional phase-shifting FD-OCT, (ii) a BM-scan FD-OCT, and (iii) the modified BM-scan FD-OCT.

Fig. 3
Fig. 3

Flow diagram of the phase-reconstruction procedure of a BM-scan FD-OCT.

Fig. 4
Fig. 4

(i) Noncomplex FD-OCT image and (ii) the complex BM-scan FD-OCT image of a plastic plate with a covering layer.

Fig. 5
Fig. 5

(i) Conventional noncomplex, (ii) interlaced full-range, and (iii) full-range FD-OCT images of an in vivo sweat duct in a human finger pad taken with an acquisition time of 27 ms.

Fig. 6
Fig. 6

Measurement example of our basic FD-OCT. The sample is an in vivo sweat duct in a human finger pad.

Equations (4)

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

I ˜ ( ω ) = | p ˜ ( ω ) | 2 + | r ˜ ( ω ) | 2 + 2 | p ˜ ( ω ) | | r ˜ ( ω ) | × cos [ p ˜ ( ω ) + φ ] ,
I ˜ ( x , ω ) = | p ˜ ( x , ω ) | 2 + | r ˜ ( ω ) | 2 + p ˜ ( x , ω ) r ˜ * ( ω ) × exp [ i φ ( x ) ] + c .c . ,
x [ I ˜ ( x , ω ) ] = Γ u { x [ p ˜ ( x , ω ) ] } + | r ˜ ( ω ) | 2 δ ( u ) + x [ p ˜ ( x , ω ) r ˜ ( ω ) ] δ ( u β ) + x [ p ˜ ( x , ω ) r ˜ ( ω ) ] δ ( u + β ) ,
I ˜ ( x , ω ) = p ˜ ( x , ω ) r ˜ ( ω ) exp ( i β x ) ,

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