Abstract

We report on a Fourier Domain Mode Locked (FDML) wavelength swept laser source with a highly linear time-frequency sweep characteristic and demonstrate OCT imaging without k-space resampling prior to Fourier transformation. A detailed theoretical framework is provided and different strategies how to determine the optimum drive waveform of the piezo-electrically actuated optical bandpass-filter in the FDML laser are discussed. An FDML laser with a relative optical frequency deviation Δν/ν smaller than 8·10-5 over a 100 nm spectral bandwidth at 1300 nm is presented, enabling high resolution OCT over long ranging depths. Without numerical time-to-frequency resampling and without spectral apodization a sensitivity roll off of 4 dB over 2 mm, 12.5 dB over 4 mm and 26.5 dB over 1 cm at 3.5 µs sweep duration and 106.6 dB maximum sensitivity at 9.2 mW average power is achieved. The axial resolution in air degrades from 14 to 21 µm over 4 mm imaging depth. The compensation of unbalanced dispersion in the OCT sample arm by an adapted tuning characteristic of the source is demonstrated. Good stability of the system without feedback-control loops is observed over hours.

© 2008 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).
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    [Crossref]
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    [Crossref] [PubMed]
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  10. R. Huber, M. Wojtkowski, K. Taira, J. G. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express 13, 3513–3528 (2005).
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    [Crossref] [PubMed]
  13. R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. 31, 2975–2977 (2006).
    [Crossref] [PubMed]
  14. D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. 32, 626–628 (2007).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  17. L. A. Kranendonk, R. Huber, J. G. Fujimoto, and S. T. Sanders, “Wavelength-agile H2O absorption spectrometer for thermometry of general combustion gases,” Proceedings of the Combustion Institute 31, 783–790 (2007).
    [Crossref]
  18. L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15, 15115–15128 (2007).
    [Crossref] [PubMed]
  19. D. C. Adler, J. Stenger, I. Gorczynska, H. Lie, T. Hensick, R. Spronk, S. Wolohojian, N. Khandekar, J. Y. Jiang, S. Barry, A. E. Cable, R. Huber, and J. G. Fujimoto, “Comparison of three-dimensional optical coherence tomography and high resolution photography for art conservation studies,” Opt. Express 15, 15972–15986 (2007).
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  21. R. Huber, M. Wojtkowski, J. G. Fujimoto, J. Y. Jiang, and A. E. Cable, “Three-dimensional and C-mode OCT imaging with a compact, frequency swept laser source at 1300 nm,” Opt. Express 13, 10523–10538 (2005).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  24. D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709–716 (2007).
    [Crossref]
  25. B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach - art. no. 64430O,” Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xiv 6443, O4430–O4430 (2007).
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    [Crossref] [PubMed]
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  28. V. J. Srinivasan, R. Huber, I. Gorczynska, J. G. Fujimoto, J. Y. Jiang, P. Reisen, and A. E. Cable, “High-speed, high-resolution optical coherence tomography retinal imaging with a frequency-swept laser at 850 nm,” Opt. Lett. 32, 361–363 (2007).
    [Crossref] [PubMed]
  29. Y. L. Chen, D. M. de Bruin, C. Kerbage, and J. F. de Boer, “Spectrally balanced detection for optical frequency domain imaging,” Opt. Express 15, 16390–16399 (2007).
    [Crossref] [PubMed]
  30. S. Moon and D. Y. Kim, “Normalization detection scheme for high-speed optical frequency-domain imaging and reflectometry,” Opt. Express 15, 15129–15146 (2007).
    [Crossref] [PubMed]
  31. M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004).
    [Crossref] [PubMed]
  32. J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical Coherence Microscopy in Scattering Media,” Opt. Lett. 19, 590–592 (1994).
    [Crossref] [PubMed]

2008 (1)

2007 (13)

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709–716 (2007).
[Crossref]

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach - art. no. 64430O,” Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xiv 6443, O4430–O4430 (2007).

Z. Hu and A. M. Rollins, “Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer,” Opt. Lett. 32, 3525–3527 (2007).
[Crossref] [PubMed]

C. Chong, A. Morosawa, and T. Sakai, “High speed wavelength-swept laser source with simple configuration for optical coherence tomography - art. no. 662705,” Proc. SPIE 6627, 62705–62705 (2007).

V. J. Srinivasan, R. Huber, I. Gorczynska, J. G. Fujimoto, J. Y. Jiang, P. Reisen, and A. E. Cable, “High-speed, high-resolution optical coherence tomography retinal imaging with a frequency-swept laser at 850 nm,” Opt. Lett. 32, 361–363 (2007).
[Crossref] [PubMed]

Y. L. Chen, D. M. de Bruin, C. Kerbage, and J. F. de Boer, “Spectrally balanced detection for optical frequency domain imaging,” Opt. Express 15, 16390–16399 (2007).
[Crossref] [PubMed]

S. Moon and D. Y. Kim, “Normalization detection scheme for high-speed optical frequency-domain imaging and reflectometry,” Opt. Express 15, 15129–15146 (2007).
[Crossref] [PubMed]

D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. 32, 626–628 (2007).
[Crossref] [PubMed]

S. W. Huang, A. D. Aguirre, R. A. Huber, D. C. Adler, and J. G. Fujimoto, “Swept source optical coherence microscopy using a Fourier domain mode-locked laser,” Opt. Express 15, 6210–6217 (2007).
[Crossref] [PubMed]

R. Huber, D. C. Adler, V. J. Srinivasan, and J. G. Fujimoto, “Fourier domain mode locking at 1050 nm for ultra-high-speed optical coherence tomography of the human retina at 236,000 axial scans per second,” Opt. Lett. 32, 2049–2051 (2007).
[Crossref] [PubMed]

L. A. Kranendonk, R. Huber, J. G. Fujimoto, and S. T. Sanders, “Wavelength-agile H2O absorption spectrometer for thermometry of general combustion gases,” Proceedings of the Combustion Institute 31, 783–790 (2007).
[Crossref]

L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15, 15115–15128 (2007).
[Crossref] [PubMed]

D. C. Adler, J. Stenger, I. Gorczynska, H. Lie, T. Hensick, R. Spronk, S. Wolohojian, N. Khandekar, J. Y. Jiang, S. Barry, A. E. Cable, R. Huber, and J. G. Fujimoto, “Comparison of three-dimensional optical coherence tomography and high resolution photography for art conservation studies,” Opt. Express 15, 15972–15986 (2007).
[Crossref] [PubMed]

2006 (3)

2005 (3)

2004 (1)

2003 (5)

1998 (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]

1997 (2)

1995 (1)

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of Intraocular Distances by Backscattering Spectral Interferometry,” Opt. Commun. 117, 43–48 (1995).
[Crossref]

1994 (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]

Adler, D. C.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709–716 (2007).
[Crossref]

D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. 32, 626–628 (2007).
[Crossref] [PubMed]

S. W. Huang, A. D. Aguirre, R. A. Huber, D. C. Adler, and J. G. Fujimoto, “Swept source optical coherence microscopy using a Fourier domain mode-locked laser,” Opt. Express 15, 6210–6217 (2007).
[Crossref] [PubMed]

R. Huber, D. C. Adler, V. J. Srinivasan, and J. G. Fujimoto, “Fourier domain mode locking at 1050 nm for ultra-high-speed optical coherence tomography of the human retina at 236,000 axial scans per second,” Opt. Lett. 32, 2049–2051 (2007).
[Crossref] [PubMed]

D. C. Adler, J. Stenger, I. Gorczynska, H. Lie, T. Hensick, R. Spronk, S. Wolohojian, N. Khandekar, J. Y. Jiang, S. Barry, A. E. Cable, R. Huber, and J. G. Fujimoto, “Comparison of three-dimensional optical coherence tomography and high resolution photography for art conservation studies,” Opt. Express 15, 15972–15986 (2007).
[Crossref] [PubMed]

R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. 31, 2975–2977 (2006).
[Crossref] [PubMed]

Aguirre, A. D.

Akiba, M.

An, X.

Barry, S.

Boudoux, C.

Bouma, B. E.

Cable, A. E.

Caswell, A. W.

Cense, B.

Chan, K. P.

Chan, R. C.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[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, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Chen, Y.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709–716 (2007).
[Crossref]

Chen, Y. L.

Chen, Z.

Chinn, S. R.

Choma, M. A.

Chong, C.

Connolly, J.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709–716 (2007).
[Crossref]

de Boer, J. F.

de Bruin, D. M.

Desjardins, A. E.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Drexler, W.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach - art. no. 64430O,” Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xiv 6443, O4430–O4430 (2007).

Duker, J. S.

Elzaiat, S. Y.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of Intraocular Distances by Backscattering Spectral Interferometry,” Opt. Commun. 117, 43–48 (1995).
[Crossref]

Evans, J. A.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

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.

L. A. Kranendonk, R. Huber, J. G. Fujimoto, and S. T. Sanders, “Wavelength-agile H2O absorption spectrometer for thermometry of general combustion gases,” Proceedings of the Combustion Institute 31, 783–790 (2007).
[Crossref]

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709–716 (2007).
[Crossref]

V. J. Srinivasan, R. Huber, I. Gorczynska, J. G. Fujimoto, J. Y. Jiang, P. Reisen, and A. E. Cable, “High-speed, high-resolution optical coherence tomography retinal imaging with a frequency-swept laser at 850 nm,” Opt. Lett. 32, 361–363 (2007).
[Crossref] [PubMed]

R. Huber, D. C. Adler, V. J. Srinivasan, and J. G. Fujimoto, “Fourier domain mode locking at 1050 nm for ultra-high-speed optical coherence tomography of the human retina at 236,000 axial scans per second,” Opt. Lett. 32, 2049–2051 (2007).
[Crossref] [PubMed]

S. W. Huang, A. D. Aguirre, R. A. Huber, D. C. Adler, and J. G. Fujimoto, “Swept source optical coherence microscopy using a Fourier domain mode-locked laser,” Opt. Express 15, 6210–6217 (2007).
[Crossref] [PubMed]

D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. 32, 626–628 (2007).
[Crossref] [PubMed]

D. C. Adler, J. Stenger, I. Gorczynska, H. Lie, T. Hensick, R. Spronk, S. Wolohojian, N. Khandekar, J. Y. Jiang, S. Barry, A. E. Cable, R. Huber, and J. G. Fujimoto, “Comparison of three-dimensional optical coherence tomography and high resolution photography for art conservation studies,” Opt. Express 15, 15972–15986 (2007).
[Crossref] [PubMed]

L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15, 15115–15128 (2007).
[Crossref] [PubMed]

R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. 31, 2975–2977 (2006).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14, 3225–3237 (2006).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, K. Taira, J. G. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express 13, 3513–3528 (2005).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, J. G. Fujimoto, J. Y. Jiang, and A. E. Cable, “Three-dimensional and C-mode OCT imaging with a compact, frequency swept laser source at 1300 nm,” Opt. Express 13, 10523–10538 (2005).
[Crossref] [PubMed]

M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004).
[Crossref] [PubMed]

S. R. Chinn, E. A. Swanson, and J. G. Fujimoto, “Optical coherence tomography using a frequency-tunable optical source,” Opt. Lett. 22, 340–342 (1997).
[Crossref] [PubMed]

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical Coherence Microscopy in Scattering Media,” Opt. Lett. 19, 590–592 (1994).
[Crossref] [PubMed]

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]

Gorczynska, I.

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]

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.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical Coherence Microscopy in Scattering Media,” Opt. Lett. 19, 590–592 (1994).
[Crossref] [PubMed]

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]

Hensick, T.

Hermann, B.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach - art. no. 64430O,” Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xiv 6443, O4430–O4430 (2007).

Herold, R. E.

Hitzenberger, C. K.

Hlawatsch, F.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach - art. no. 64430O,” Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xiv 6443, O4430–O4430 (2007).

Hofer, B.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach - art. no. 64430O,” Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xiv 6443, O4430–O4430 (2007).

Hsu, K.

Hu, Z.

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]

Huang, S. W.

Huber, R.

D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. 32, 626–628 (2007).
[Crossref] [PubMed]

R. Huber, D. C. Adler, V. J. Srinivasan, and J. G. Fujimoto, “Fourier domain mode locking at 1050 nm for ultra-high-speed optical coherence tomography of the human retina at 236,000 axial scans per second,” Opt. Lett. 32, 2049–2051 (2007).
[Crossref] [PubMed]

V. J. Srinivasan, R. Huber, I. Gorczynska, J. G. Fujimoto, J. Y. Jiang, P. Reisen, and A. E. Cable, “High-speed, high-resolution optical coherence tomography retinal imaging with a frequency-swept laser at 850 nm,” Opt. Lett. 32, 361–363 (2007).
[Crossref] [PubMed]

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709–716 (2007).
[Crossref]

L. A. Kranendonk, R. Huber, J. G. Fujimoto, and S. T. Sanders, “Wavelength-agile H2O absorption spectrometer for thermometry of general combustion gases,” Proceedings of the Combustion Institute 31, 783–790 (2007).
[Crossref]

D. C. Adler, J. Stenger, I. Gorczynska, H. Lie, T. Hensick, R. Spronk, S. Wolohojian, N. Khandekar, J. Y. Jiang, S. Barry, A. E. Cable, R. Huber, and J. G. Fujimoto, “Comparison of three-dimensional optical coherence tomography and high resolution photography for art conservation studies,” Opt. Express 15, 15972–15986 (2007).
[Crossref] [PubMed]

L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15, 15115–15128 (2007).
[Crossref] [PubMed]

R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. 31, 2975–2977 (2006).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14, 3225–3237 (2006).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, J. G. Fujimoto, J. Y. Jiang, and A. E. Cable, “Three-dimensional and C-mode OCT imaging with a compact, frequency swept laser source at 1300 nm,” Opt. Express 13, 10523–10538 (2005).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, K. Taira, J. G. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express 13, 3513–3528 (2005).
[Crossref] [PubMed]

Huber, R. A.

Iftimia, N.

Itoh, M.

Izatt, J. A.

Jang, I. K.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Jeon, M. Y.

Jiang, J. Y.

Kamp, G.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of Intraocular Distances by Backscattering Spectral Interferometry,” Opt. Commun. 117, 43–48 (1995).
[Crossref]

Kerbage, C.

Khandekar, N.

Kim, D. Y.

Ko, T. H.

Kowalczyk, A.

Kranendonk, L. A.

L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15, 15115–15128 (2007).
[Crossref] [PubMed]

L. A. Kranendonk, R. Huber, J. G. Fujimoto, and S. T. Sanders, “Wavelength-agile H2O absorption spectrometer for thermometry of general combustion gases,” Proceedings of the Combustion Institute 31, 783–790 (2007).
[Crossref]

Kulhavy, M.

Leitgeb, R.

Lexer, F.

Lie, H.

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]

Madjarova, V. D.

Makita, S.

Matz, G.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach - art. no. 64430O,” Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xiv 6443, O4430–O4430 (2007).

Moon, S.

Morosawa, A.

Nishioka, N. S.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Oh, W. Y.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Okura, Y.

Owen, G. M.

Park, B. H.

Pierce, M. C.

Povazay, B.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach - art. no. 64430O,” Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xiv 6443, O4430–O4430 (2007).

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]

Reisen, P.

Rollins, A. M.

Sakai, T.

Sanders, S. T.

L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15, 15115–15128 (2007).
[Crossref] [PubMed]

L. A. Kranendonk, R. Huber, J. G. Fujimoto, and S. T. Sanders, “Wavelength-agile H2O absorption spectrometer for thermometry of general combustion gases,” Proceedings of the Combustion Institute 31, 783–790 (2007).
[Crossref]

Sarunic, M. V.

Schmitt, J.

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709–716 (2007).
[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]

Shishkov, M.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Spronk, R.

Srinivasan, V. J.

Stenger, J.

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]

Suter, M. J.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Swanson, E. A.

Taira, K.

Tearney, G. J.

Unterhuber, A.

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach - art. no. 64430O,” Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xiv 6443, O4430–O4430 (2007).

Urata, Y.

Vakoc, B. J.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Wang, Q.

Wojtkowski, M.

Wolohojian, S.

Yang, C. H.

Yasuno, Y.

Yatagai, T.

Yun, S. H.

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

S. H. Yun, C. Boudoux, G. J. Tearney, and B. E. Bouma, “High-speed wavelength-swept semiconductor laser with a polygon-scanner-based wavelength filter,” Opt. Lett. 28, 1981–1983 (2003).
[Crossref] [PubMed]

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]

Zhang, J.

Appl. Opt. (1)

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]

Nat. Med. (1)

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Med. 12, 1429–1433 (2006).
[Crossref] [PubMed]

Nat. Photonics (1)

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1, 709–716 (2007).
[Crossref]

Opt. Commun. (1)

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. Elzaiat, “Measurement of Intraocular Distances by Backscattering Spectral Interferometry,” Opt. Commun. 117, 43–48 (1995).
[Crossref]

Opt. Express (14)

Y. L. Chen, D. M. de Bruin, C. Kerbage, and J. F. de Boer, “Spectrally balanced detection for optical frequency domain imaging,” Opt. Express 15, 16390–16399 (2007).
[Crossref] [PubMed]

S. Moon and D. Y. Kim, “Normalization detection scheme for high-speed optical frequency-domain imaging and reflectometry,” Opt. Express 15, 15129–15146 (2007).
[Crossref] [PubMed]

M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004).
[Crossref] [PubMed]

S. W. Huang, A. D. Aguirre, R. A. Huber, D. C. Adler, and J. G. Fujimoto, “Swept source optical coherence microscopy using a Fourier domain mode-locked laser,” Opt. Express 15, 6210–6217 (2007).
[Crossref] [PubMed]

L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15, 15115–15128 (2007).
[Crossref] [PubMed]

D. C. Adler, J. Stenger, I. Gorczynska, H. Lie, T. Hensick, R. Spronk, S. Wolohojian, N. Khandekar, J. Y. Jiang, S. Barry, A. E. Cable, R. Huber, and J. G. Fujimoto, “Comparison of three-dimensional optical coherence tomography and high resolution photography for art conservation studies,” Opt. Express 15, 15972–15986 (2007).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, J. G. Fujimoto, J. Y. Jiang, and A. E. Cable, “Three-dimensional and C-mode OCT imaging with a compact, frequency swept laser source at 1300 nm,” Opt. Express 13, 10523–10538 (2005).
[Crossref] [PubMed]

M. Y. Jeon, J. Zhang, Q. Wang, and Z. Chen, “High-speed and wide bandwidth Fourier domain mode-locked wavelength swept laser with multiple SOAs,” Opt. Express 16, 2547–2554 (2008).
[Crossref] [PubMed]

R. 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]

M. A. Choma, M. V. Sarunic, C. H. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11, 2183–2189 (2003).
[Crossref] [PubMed]

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]

R. Huber, M. Wojtkowski, K. Taira, J. G. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express 13, 3513–3528 (2005).
[Crossref] [PubMed]

Y. Yasuno, V. D. Madjarova, S. Makita, M. Akiba, A. Morosawa, C. Chong, T. Sakai, K. P. Chan, M. Itoh, and T. Yatagai, “Three-dimensional and high-speed swept-source optical coherence tomography for in vivo investigation of human anterior eye segments,” Opt. Express 13, 10652–10664 (2005).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14, 3225–3237 (2006).
[Crossref] [PubMed]

Opt. Lett. (9)

R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. 31, 2975–2977 (2006).
[Crossref] [PubMed]

D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. 32, 626–628 (2007).
[Crossref] [PubMed]

J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28, 2067–2069 (2003).
[Crossref] [PubMed]

S. H. Yun, C. Boudoux, G. J. Tearney, and B. E. Bouma, “High-speed wavelength-swept semiconductor laser with a polygon-scanner-based wavelength filter,” Opt. Lett. 28, 1981–1983 (2003).
[Crossref] [PubMed]

R. Huber, D. C. Adler, V. J. Srinivasan, and J. G. Fujimoto, “Fourier domain mode locking at 1050 nm for ultra-high-speed optical coherence tomography of the human retina at 236,000 axial scans per second,” Opt. Lett. 32, 2049–2051 (2007).
[Crossref] [PubMed]

S. R. Chinn, E. A. Swanson, and J. G. Fujimoto, “Optical coherence tomography using a frequency-tunable optical source,” Opt. Lett. 22, 340–342 (1997).
[Crossref] [PubMed]

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical Coherence Microscopy in Scattering Media,” Opt. Lett. 19, 590–592 (1994).
[Crossref] [PubMed]

Z. Hu and A. M. Rollins, “Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer,” Opt. Lett. 32, 3525–3527 (2007).
[Crossref] [PubMed]

V. J. Srinivasan, R. Huber, I. Gorczynska, J. G. Fujimoto, J. Y. Jiang, P. Reisen, and A. E. Cable, “High-speed, high-resolution optical coherence tomography retinal imaging with a frequency-swept laser at 850 nm,” Opt. Lett. 32, 361–363 (2007).
[Crossref] [PubMed]

Proc. SPIE (1)

C. Chong, A. Morosawa, and T. Sakai, “High speed wavelength-swept laser source with simple configuration for optical coherence tomography - art. no. 662705,” Proc. SPIE 6627, 62705–62705 (2007).

Proceedings of the Combustion Institute (1)

L. A. Kranendonk, R. Huber, J. G. Fujimoto, and S. T. Sanders, “Wavelength-agile H2O absorption spectrometer for thermometry of general combustion gases,” Proceedings of the Combustion Institute 31, 783–790 (2007).
[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]

Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xiv (1)

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, F. Hlawatsch, and W. Drexler, “Signal post processing in frequency domain OCT and OCM using a filter bank approach - art. no. 64430O,” Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing Xiv 6443, O4430–O4430 (2007).

Supplementary Material (1)

» Media 1: AVI (2087 KB)     

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Figures (13)

Fig. 1.
Fig. 1.

Left: Temporal evolution of optical frequency for three different sweeps: Linear in frequency (black), linear in wavelength (red) and sinusoidal in wavelength (green). The highlighted, blue region represents the frequency interval of laser activity. Center: Corresponding relative frequency error Δν/ν, relative to a linear fit in the frequency region where the laser is active. Right: Calculated point spread functions for a spectrum spanning from 1260 nm to 1360 nm and an imaging depth of 1 mm (apodized with a Hanning window).

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Fig. 2.
Fig. 2.

Left: Schematic diagram of the FDML laser. Right: Experimental setup of the automated measurement of the FFP-filter response function.

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Fig. 3.
Fig. 3.

Left: Measured optical sweep amplitude versus sweep frequency for a sinusoidal excitation of the filter with a constant drive amplitude and DC-offset. Right: Phase shift versus sweep frequency for the same excitation. The black curve represents the phase shift due to the response of both, electrical circuitry and the FFP-filter. The red curve shows the phase shift as a result of the mechanical and electrical response of the FFP-filter only. The crosses in both plots indicate the measured data values for the first (red), second (grey) and third (blue) harmonic of the sweep frequency in the chosen FDML configuration.

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Fig. 4.
Fig. 4.

Left: Temporal evolution of optical frequency. Black: Sweep linear in frequency. Red curve: Waveform with first three Fourier components. The highlighted, blue region represents the frequency interval of laser activity. Middle: Corresponding relative frequency deviation Δν/ν from a linear fit in the region of laser activity. Right: Calculated PSFs for a spectrum spanning from 1260 nm to 1360 nm and an imaging depth of 1 mm (apodized with a Hanning window).

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Fig. 5.
Fig. 5.

Results of numerical optimization with a constant sweep duration of 4.385 µs and a first order harmonic amplitude of 87.6 nm. The frequency error is depicted versus the amplitudes of second and third order harmonic (in percent of first order harmonic).

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Fig. 6.
Fig. 6.

Results of numerical optimization with arbitrary sweep duration for a first order amplitude of 87.5 nm. A: Frequency error versus the amplitudes of second and third order harmonic (in percent of first order harmonic). The corresponding acquired distributions of the wavelength offset (B), sweep duration (C), second harmonic phase (D) and third harmonic phase (E).

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Fig. 7.
Fig. 7.

Frequency error χ versus phase of second and third order harmonic for three characteristic data sets corresponding to point 4 (plot A), point 8 (plot B) and point 9 (plot C) in Fig. 6 The absolute minimum of each plot is indicated with a cross.

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Fig. 8.
Fig. 8.

Left: Frequency error χ from numerical simulation (red, points 1 to 7 in Fig. 6(A)) compared to the measured frequency error for identical parameters (blue). Right: Corresponding phases of second (red points) and third (pink points) harmonic phase from simulation and experimentally determined optimal phases of third order harmonic (blue points) and second order harmonic (dark blue points).

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Fig. 9.
Fig. 9.

Left: Optimum relative (instantaneous) frequency error curve theoretically predicted by numerical simulation (red), compared to the measured curve (blue, wavelength of 1260 nm at t=0); Center: Corresponding PSFs (Hanning window) for theoretical solution (blue), measured in the experiment (red) and a perfectly linear time-frequency characteristic (black). Right: Interference signal (original envelope).

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Fig. 10.
Fig. 10.

PSFs calculated directly from the measured fringe signals (no apodization). The black curve represents an optimum parameter adjustment, the red curve was obtained with the phase of third order harmonic detuned by 1.7° with respect to the first order harmonic phase, the blue curve with the phase of the third order harmonic detuned by 3.3°.

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Fig. 11.
Fig. 11.

PSFs at different image depths. No recalibration and no spectral reshaping or apodization step prior to FFT was applied. Left: Linear representation; for some image depths the corresponding axial resolution in air is noted. Right: Logarithmic plot representing sensitivity. The red circles indicate the drop of fringe contrast extracted from measured fringe signals. Numbers for sensitivity roll off values are given for various depths.

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Fig. 12.
Fig. 12.

Three sets of OCT images, each consisting of 2 images of a human finger for three different situations: (A) and (B): sinusoidal sweep with no recalibration step prior to FFT; (C) and (D): sinusoidal sweep with recalibration step prior to FFT; (E) and (F): k-space linear FDML with no recalibration step prior to FFT. The images in each column are acquired at the same sample position. (G): 3D representation acquired with the k-space linear FDML and no recalibration step prior to FFT. No spectral apodization was performed (1MB movie). [Media 1]

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Fig. 13.
Fig. 13.

Measured PSFs at an imaging depth of 1 mm. Black: Ideal waveform with no additional dispersive element. Red: PSF for the same waveform but with three 0.7 mm thick silicon plates inserted into one arm. Blue: PSF for an adjusted waveform correcting for the dispersion introduced by the silicon plates. Left: Almost perfect dispersion correction for only one depth (1 mm). Right: Partial simultaneous dispersion correction for two depths.

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Equations (2)

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χ = ∑ i = 1 N Δ v i ( t ) 2 v i ( t ) 2 ( N − 1 ) .
v ( t ) = c λ 0 + A 1 · sin ( ω t − φ 1 ) + A 2 · sin ( 2 ω t − φ 2 ) + A 3 · sin ( 3 ω t − φ 3 )

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