Abstract

We developed a method for obtaining three-dimensional images of biological tissues using axial-lateral parallel time domain optical coherence tomography (OCT) with an ultrahigh-speed complementary metal oxide semiconductor (CMOS) camera. The camera obtains a depth-resolved interference image using diffracted light as the reference beam and a linear illumination beam without axial and vertical scans. We can obtain the OCT images (512×512 pixels) at 1,500 frames per second by calculating two sequential images. A sample volume of 5.8×2.8×2.0 (x×y×z) mm3 (corresponding to 512×250×512 pixels) was imaged at six volumes per second in a horizontal mechanical scan. The experimental sensitivity was approximately 76 dB after 2×2-pixel binning. The system was successfully used to image the human finger in vivo.

© 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).
    [CrossRef] [PubMed]
  2. A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Ungarunyawee, and J. A. Izatt, "In vivo video rate optical coherence tomography," Opt. Express 3, 219-229 (1998).
    [CrossRef] [PubMed]
  3. G. Häusler and M. W. Lindner, ""Coherence radar" and "spectral radar"-new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
    [CrossRef]
  4. 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] [PubMed]
  5. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, "High-speed optical frequency-domain imaging," Opt. Express 11, 2953-2963 (2003).
    [CrossRef] [PubMed]
  6. 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).
    [CrossRef] [PubMed]
  7. A. Zuluaga and R. Richards-Kortum, "Spatially resolved spectral interferometry for determination of subsurface structure," Opt. Lett. 24, 519-521 (1999).
    [CrossRef]
  8. T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, "Profilometry with line-field Fourier-domain interferometry," Opt. Express 13, 695-701 (2005).
    [CrossRef] [PubMed]
  9. B. Grajciar, M. Pircher, A. Fercher, and R. Leitgeb, "Parallel Fourier domain optical coherence tomography for in vivo measurement of the human eye," Opt. Express 13, 1131-1137 (2005).
    [CrossRef] [PubMed]
  10. Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, "Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation," J. Biomed. Opt. 11, 014014-014020 (2006).
    [CrossRef] [PubMed]
  11. 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).
    [CrossRef]
  12. 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).
    [CrossRef] [PubMed]
  13. S. Bourquin, P. Seitz, and R. P. Salathé, "Optical coherence topography based on a two-dimensional smart detector array," Opt. Lett. 26, 512-514 (2001).
    [CrossRef]
  14. M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, "Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array," Opt. Commun. 202, 29-35 (2002).
    [CrossRef]
  15. M. Laubscher, M. Ducros, B. Karamata, T. Lasser, and R. Salathé, "Video-rate three-dimensional optical coherence tomography," Opt. Express 10, 429-435 (2002).
    [PubMed]
  16. M. Lebec, L. Blanchot, H. Saint-jalmes, E. Beaurepaire, and A. C. Boccara, "Full-field optical coherence microscopy," Opt. Lett. 23, 244-246 (1998).
    [CrossRef]
  17. A. Dubois, L. Vabre, A. C. Boccara, and E. Beaurepaire, "High-resolution full-field optical coherence tomography with a Linnik microscope," Appl. Opt. 41, 805-812 (2002).
    [CrossRef] [PubMed]
  18. L. Vabre, A. Dubois, and A. C. Boccara, "Thermal-light full-field optical coherence tomography," Opt. Lett. 27, 530-532 (2002).
    [CrossRef]
  19. M. Akiba, K. P. Chan, and N. Tanno, "Full-field optical coherence tomography by two-dimensional heterodyne detection with a pair of CCD camera," Opt. Lett. 28, 816-818 (2003).
    [CrossRef] [PubMed]
  20. A. Dubois, K. Grieve, G. Moneron, R. Lecaque, L. Vabre, and C. Boccara, "Ultrahigh-resolution full-field optical coherence tomography," Appl. Opt. 43, 2874-2883 (2004).
    [CrossRef] [PubMed]
  21. K. Grieve, A. Dubois, M. Simonutti, M. Paques, J. Sahel, J. Le Gargasson, and C. Boccara, "In vivo anterior segment imaging in the rat eye with high speed white light full-field optical coherence tomography," Opt. Express 13, 6286-6295 (2005).
    [CrossRef] [PubMed]
  22. I. Zeylikovich, A. Gilerson, and R. R. Alfano, "Nonmechanical grating-generated scanning coherence microscopy," Opt. Lett. 23,1797-1799 (1998).
    [CrossRef]
  23. I. Zeylikovich, and R. R. Alfano, "Performing selected optical measurements with optical coherence domain reflectometry," US patent: 6,437,867, (2002).
  24. Y. Watanabe, K. Yamada, and M. Sato, "In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera," Opt. Commun. 261, 376-380 (2006).
    [CrossRef]

2006

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, "Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation," J. Biomed. Opt. 11, 014014-014020 (2006).
[CrossRef] [PubMed]

Y. Watanabe, K. Yamada, and M. Sato, "In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera," Opt. Commun. 261, 376-380 (2006).
[CrossRef]

2005

2004

2003

2002

2001

1999

1998

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, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Akiba, M.

Alfano, R. R.

Aoki, G.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, "Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation," J. Biomed. Opt. 11, 014014-014020 (2006).
[CrossRef] [PubMed]

Bajraszewski, T.

Beaurepaire, E.

Blanchot, L.

Boccara, A. C.

Boccara, C.

Bouma, B. E.

Bourquin, S.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, "Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array," Opt. Commun. 202, 29-35 (2002).
[CrossRef]

S. Bourquin, P. Seitz, and R. P. Salathé, "Optical coherence topography based on a two-dimensional smart detector array," Opt. Lett. 26, 512-514 (2001).
[CrossRef]

Cense, B.

Chan, K. P.

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]

Chen, T. C.

de Boer, J. F.

Dubois, A.

Ducros, M.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, "Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array," Opt. Commun. 202, 29-35 (2002).
[CrossRef]

M. Laubscher, M. Ducros, B. Karamata, T. Lasser, and R. Salathé, "Video-rate three-dimensional optical coherence tomography," Opt. Express 10, 429-435 (2002).
[PubMed]

Endo, T.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, "Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation," J. Biomed. Opt. 11, 014014-014020 (2006).
[CrossRef] [PubMed]

T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, "Profilometry with line-field Fourier-domain interferometry," Opt. Express 13, 695-701 (2005).
[CrossRef] [PubMed]

Fercher, A.

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).
[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]

Gilerson, A.

Grajciar, B.

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]

Grieve, K.

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Iftimia, N.

Itoh, M.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, "Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation," J. Biomed. Opt. 11, 014014-014020 (2006).
[CrossRef] [PubMed]

T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, "Profilometry with line-field Fourier-domain interferometry," Opt. Express 13, 695-701 (2005).
[CrossRef] [PubMed]

Izatt, J. A.

Karamata, B.

Karamater, B.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, "Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array," Opt. Commun. 202, 29-35 (2002).
[CrossRef]

Kowalczyk, A.

Kulkarni, M. D.

Lasser, T.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, "Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array," Opt. Commun. 202, 29-35 (2002).
[CrossRef]

M. Laubscher, M. Ducros, B. Karamata, T. Lasser, and R. Salathé, "Video-rate three-dimensional optical coherence tomography," Opt. Express 10, 429-435 (2002).
[PubMed]

Laubscher, M.

Laubsher, M.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, "Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array," Opt. Commun. 202, 29-35 (2002).
[CrossRef]

Le Gargasson, J.

Lebec, M.

Lecaque, R.

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).
[CrossRef] [PubMed]

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]

Makita, S.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, "Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation," J. Biomed. Opt. 11, 014014-014020 (2006).
[CrossRef] [PubMed]

T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, "Profilometry with line-field Fourier-domain interferometry," Opt. Express 13, 695-701 (2005).
[CrossRef] [PubMed]

Moneron, G.

Nassif, N.

Paques, M.

Park, B. H.

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).
[CrossRef] [PubMed]

Richards-Kortum, R.

Rollins, A. M.

Sahel, J.

Saint-jalmes, H.

Salathé, R.

Salathé, R. P.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, "Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array," Opt. Commun. 202, 29-35 (2002).
[CrossRef]

S. Bourquin, P. Seitz, and R. P. Salathé, "Optical coherence topography based on a two-dimensional smart detector array," Opt. Lett. 26, 512-514 (2001).
[CrossRef]

Sato, M.

Y. Watanabe, K. Yamada, and M. Sato, "In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera," Opt. Commun. 261, 376-380 (2006).
[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, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Seitz, P.

Simonutti, 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).
[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]

Tanno, N.

Tearney, G. J.

Ungarunyawee, R.

Vabre, L.

Watanabe, Y.

Y. Watanabe, K. Yamada, and M. Sato, "In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera," Opt. Commun. 261, 376-380 (2006).
[CrossRef]

Wojtkowski, M.

Yamada, K.

Y. Watanabe, K. Yamada, and M. Sato, "In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera," Opt. Commun. 261, 376-380 (2006).
[CrossRef]

Yasuno, Y.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, "Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation," J. Biomed. Opt. 11, 014014-014020 (2006).
[CrossRef] [PubMed]

T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, "Profilometry with line-field Fourier-domain interferometry," Opt. Express 13, 695-701 (2005).
[CrossRef] [PubMed]

Yatagai, T.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, "Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation," J. Biomed. Opt. 11, 014014-014020 (2006).
[CrossRef] [PubMed]

T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, "Profilometry with line-field Fourier-domain interferometry," Opt. Express 13, 695-701 (2005).
[CrossRef] [PubMed]

Yazdanfar, S.

Yun, S. H.

Zeylikovich, I.

Zuluaga, A.

Appl. Opt.

J. Biomed. Opt.

Y. Yasuno, T. Endo, S. Makita, G. Aoki, M. Itoh, and T. Yatagai, "Three-dimensional line-field Fourier domain optical coherence tomography for in vivo dermatological investigation," J. Biomed. Opt. 11, 014014-014020 (2006).
[CrossRef] [PubMed]

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

Opt. Commun.

M. Ducros, M. Laubsher, B. Karamater, S. Bourquin, T. Lasser, and R. P. Salathé, "Parallel optical coherence tomography in scattering samples using a two-dimensional smart-pixel detector array," Opt. Commun. 202, 29-35 (2002).
[CrossRef]

Y. Watanabe, K. Yamada, and M. Sato, "In vivo nonmechanical scanning grating-generated optical coherence tomography using an InGaAs digital camera," Opt. Commun. 261, 376-380 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

A. Zuluaga and R. Richards-Kortum, "Spatially resolved spectral interferometry for determination of subsurface structure," Opt. Lett. 24, 519-521 (1999).
[CrossRef]

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] [PubMed]

M. Lebec, L. Blanchot, H. Saint-jalmes, E. Beaurepaire, and A. C. Boccara, "Full-field optical coherence microscopy," Opt. Lett. 23, 244-246 (1998).
[CrossRef]

L. Vabre, A. Dubois, and A. C. Boccara, "Thermal-light full-field optical coherence tomography," Opt. Lett. 27, 530-532 (2002).
[CrossRef]

M. Akiba, K. P. Chan, and N. Tanno, "Full-field optical coherence tomography by two-dimensional heterodyne detection with a pair of CCD camera," Opt. Lett. 28, 816-818 (2003).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

S. Bourquin, P. Seitz, and R. P. Salathé, "Optical coherence topography based on a two-dimensional smart detector array," Opt. Lett. 26, 512-514 (2001).
[CrossRef]

I. Zeylikovich, A. Gilerson, and R. R. Alfano, "Nonmechanical grating-generated scanning coherence microscopy," Opt. Lett. 23,1797-1799 (1998).
[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, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Other

I. Zeylikovich, and R. R. Alfano, "Performing selected optical measurements with optical coherence domain reflectometry," US patent: 6,437,867, (2002).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of 3-D axial-lateral parallel time-domain optical coherence tomography. The dashed line is the imaging ray.

Fig. 2.
Fig. 2.

Measured beam radius around the waist of linear illumination. Solid curves: theoretical values.

Fig. 3.
Fig. 3.

(a) XY image of a test target. The measurement area was 5.8×2.8 (x×y) mm2 (b) Vertical and horizontal line profiles of Group 4.

Fig. 4.
Fig. 4.

(a) Sensitivities with an attenuation of -50 dB in the sample arm at different frame rates. (b) Measured axial profile and Gaussian curve fitting

Fig. 5.
Fig. 5.

Signal-to-noise ratio at different depths (a) with a f=50 mm cylindrical lens and (b) without a cylindrical lens.

Fig. 6.
Fig. 6.

OCT images of a human finger in vivo at an acquisition time of 1/1500 s. The measurement range was 5.8×2.0 (x×z) mm2: (a) the skin at the fingertip, (b) the nail fold region.

Fig. 7.
Fig. 7.

3-D OCT images of a human finger pad in vivo: (a, b) longitudinal OCT images in the X-Z plane, (c, d) transverse OCT images. The white arrows indicate the cross-sectional positions. (e) Volume-rendered image. The volume size was 5.8×2.8×2.0 (x×y×z) mm3.

Equations (7)

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

2 sin θ = λ p ,
Δ L = d tan θ ,
I ( x , z ) = I s + I r + 2 [ I r I i ( R s ( x , z ) γ ( z ) ) ] 1 2 cos ( ϕ s ϕ r ) ,
S = [ ( I 0 I π 2 ) 2 + ( I π 2 I π ) 2 ] 4 = 4 [ I r I i ( R s ( x , z ) γ ( z ) ) ] .
S = ( I 0 I π ) 2 4 = 4 [ I r I i ( R s ( x , s ) γ ( z ) ) ] cos 2 ( ϕ s ϕ r ) .
ω 2 ( z ) = ω 2 0 [ 1 + ( λ 0 z π ω 2 0 ) 2 ] ,
R min = ( R r + 2 R inc ) 2 2 R r ξ max .

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