Abstract

We propose and demonstrate an ultrahigh-speed optical frequency domain reflectometry (OFDR) system based on optical frequency-to-time conversion by pulse time stretching with a linearly chirped fiber Bragg grating (LCFG). This method will be referred to as OFDR based on real-time Fourier transformation (OFDR-RTFT). In this approach the frequency domain interference pattern, from which the desired axial depth profile is reconstructed, can be captured directly in the time-domain over the duration of a single stretched pulse, which translates into unprecedented axial line acquisition rates (as high as the input pulse repetition rate). We provide here a comprehensive, rigorous mathematical analysis of this new OFDR approach. In particular, we derive the main design equations of an OFDR-RTFT system in terms of its key performance parameters. Our analysis reveals the detrimental influence of nonlinear phase variations in the input optical pulse (including higher-order dispersion terms and group delay ripples introduced by the LCFG stretcher) on the system performance, e.g. achievable resolution. A simple and powerful method based on Hilbert transformation is successfully demonstrated to compensate for these detrimental phase distortions. We show that besides its potential to provide ultrahigh acquisition speeds (in the MHz range), LCFG-based OFDR-RTFT also offers the potential for performance advantages in terms of axial resolution, depth range and sensitivity. All these features make this approach particularly attractive for imaging applications based on optical coherence tomography (OCT). In our experiments, single-reflection depth profiles with nearly transform-limited ≈ 92.8 μm (average) axial resolutions over a remarkable 18 mm depth range have been obtained from OFDR-RTFT interferograms, each one measured over a time window of ≈50 ns at 20 MHz repetition rate. Improved sensitivities up to -61 dB have been achieved without using any balanced detection scheme.

© 2007 Optical Society of America

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References

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  1. D. Uttam and B. Culshaw, "Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique," IEEE J. Lightwave Technol. 3,971-977 (1985)
    [CrossRef]
  2. U. Glombitza and E. Brinkmeyer, "Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides," J. Lightwave Technol. 11,1377-1384 (1993)
    [CrossRef]
  3. R. Passy, N. Gisin, J. P. von der Weid, and H. H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12,1622-1630 (1994)
    [CrossRef]
  4. A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117,43-48 (1995)
    [CrossRef]
  5. 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]
  6. 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]
  7. M. A. Choma, M. V. Sarunic, C. Y. Yang, and J. A. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 11,2183-2189 (2003)
    [CrossRef] [PubMed]
  8. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftima and B. E. Bouma, "High-speed optical frequency-domain imaging," Opt. Express 11,2953-2963 (2003)
    [CrossRef] [PubMed]
  9. B. Golubovic, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, "Optical frequency-domain reflectometry using rapid wavelength tuning of a Cr4+:forsterite laser," Opt. Lett. 22,1704-1706 (1997)
    [CrossRef]
  10. S. H. Yun, G. J. Tearney, J. F. de Boer, and B. E. Bouma, "Motion artefacts in optical coherence tomography with frequency-domain ranging," Opt. Express 12,2977-2998 (2004)
    [CrossRef] [PubMed]
  11. R. Huber, M. Wojtkowski, and J. G. Fujimoto, "Fourier Domain Mode Locking (FDML): A new laser operating regime and application for optical coherence tomography," Opt. Express 14,3225-3237 (2006)
    [CrossRef] [PubMed]
  12. R. Huber, D. C. Adler, and J. G. Fujimoto, "Buffered Fourier domain mode locking: unidirectional swept sources for optical coherence tomography imaging at 370,000 lines/s," Opt. Lett. 31,2975-2977 (2006)
    [CrossRef] [PubMed]
  13. M. Wojtkowski, V. J. Srinivasan, T. J. 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]
  14. S. Moon, D. Y. Kim, "Ultra-high-speed optical coherence tomography with a stretched pulse supercontinuum source," Opt. Express 14,11575-11584 (2006)
    [CrossRef] [PubMed]
  15. Y. Park, T. -J. Ahn, J.-C. Kieffer, and J. Azaña, "Real-Time Optical Frequency-Domain Reflectometry," to be presented in Conf. Lasers and Electro-Optics (CLEO/IQEC), CTuT1 (2007)
  16. M. A. Muriel, J. Azaña, and A. Carballar, "Real-time Fourier transformer based on fiber gratings, " Opt. Lett. 24,1-3 (1999)
    [CrossRef]
  17. J. Azaña and M. A. Muriel, "Real-time Optical Spectrum Analysis Based on the Time-Space Duality in Chirped Fiber Gratings," IEEE J. Quantum Electron. 36,517-526 (2000)
    [CrossRef]
  18. Y. C. Tong; L.Y. Chan; H.K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett,  33,983-985 (1997)
    [CrossRef]
  19. T. -J. Ahn, J. Y. Lee, and D. Y. Kim, "Suppression of nonlinear frequency sweep in an optical frequency-domain reflectometer by use of Hilbert transformation," Appl. Opt. 44,7630-7634 (2005)
    [CrossRef] [PubMed]
  20. 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]
  21. http://www.proximion.com/products/dcm/index.php
  22. J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78,1135-1184 (2006)
    [CrossRef]
  23. K. O. Hill, F. Bilodeau, B. Malo, T. Kitagawa, S. Thériault, D. C. Johnson, and J. Albert, "Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion," Opt. Lett. 19,1324-1326 (1994)
    [CrossRef]
  24. R. Kashyap, Fiber Bragg Grating (Academic Press, 1999)
  25. K. Takada, "Noise in optical low-coherence reflectometry," IEEE J. Quantum Electron. 34,1098-1108 (1998)
    [CrossRef]
  26. B. Bouma, G. J. Tearney, S. A. Boppart, M. R. Hee, M. E. Brezinski, and J. G. Fujimoto, "High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source," Opt. Lett. 20,1486 (1995)
    [CrossRef] [PubMed]
  27. J. W. Goodman, Statistical Optics (New York, John Wiley and Sons, 164-169, 1985)
  28. J. M. Schmitt, "Optical Coherence Tomography (OCT):A Review," IEEE J. Select. Topics Quantum Electron. 5,1205-1215 (1999)
    [CrossRef]
  29. G. Agrawal, Nonlinear Fiber Optics (Academic Press, 64-67, 1995)

2006

2005

2004

2003

2000

J. Azaña and M. A. Muriel, "Real-time Optical Spectrum Analysis Based on the Time-Space Duality in Chirped Fiber Gratings," IEEE J. Quantum Electron. 36,517-526 (2000)
[CrossRef]

1999

J. M. Schmitt, "Optical Coherence Tomography (OCT):A Review," IEEE J. Select. Topics Quantum Electron. 5,1205-1215 (1999)
[CrossRef]

M. A. Muriel, J. Azaña, and A. Carballar, "Real-time Fourier transformer based on fiber gratings, " Opt. Lett. 24,1-3 (1999)
[CrossRef]

1998

K. Takada, "Noise in optical low-coherence reflectometry," IEEE J. Quantum Electron. 34,1098-1108 (1998)
[CrossRef]

1997

B. Golubovic, B. E. Bouma, G. J. Tearney, and J. G. Fujimoto, "Optical frequency-domain reflectometry using rapid wavelength tuning of a Cr4+:forsterite laser," Opt. Lett. 22,1704-1706 (1997)
[CrossRef]

Y. C. Tong; L.Y. Chan; H.K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett,  33,983-985 (1997)
[CrossRef]

1995

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117,43-48 (1995)
[CrossRef]

B. Bouma, G. J. Tearney, S. A. Boppart, M. R. Hee, M. E. Brezinski, and J. G. Fujimoto, "High-resolution optical coherence tomographic imaging using a mode-locked Ti:Al2O3 laser source," Opt. Lett. 20,1486 (1995)
[CrossRef] [PubMed]

1994

R. Passy, N. Gisin, J. P. von der Weid, and H. H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12,1622-1630 (1994)
[CrossRef]

K. O. Hill, F. Bilodeau, B. Malo, T. Kitagawa, S. Thériault, D. C. Johnson, and J. Albert, "Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion," Opt. Lett. 19,1324-1326 (1994)
[CrossRef]

1993

U. Glombitza and E. Brinkmeyer, "Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides," J. Lightwave Technol. 11,1377-1384 (1993)
[CrossRef]

1985

D. Uttam and B. Culshaw, "Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique," IEEE J. Lightwave Technol. 3,971-977 (1985)
[CrossRef]

Adler, D. C.

Ahn, T. -J.

Akiba, M.

Albert, J.

K. O. Hill, F. Bilodeau, B. Malo, T. Kitagawa, S. Thériault, D. C. Johnson, and J. Albert, "Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion," Opt. Lett. 19,1324-1326 (1994)
[CrossRef]

Azaña, J.

J. Azaña and M. A. Muriel, "Real-time Optical Spectrum Analysis Based on the Time-Space Duality in Chirped Fiber Gratings," IEEE J. Quantum Electron. 36,517-526 (2000)
[CrossRef]

M. A. Muriel, J. Azaña, and A. Carballar, "Real-time Fourier transformer based on fiber gratings, " Opt. Lett. 24,1-3 (1999)
[CrossRef]

Bilodeau, F.

K. O. Hill, F. Bilodeau, B. Malo, T. Kitagawa, S. Thériault, D. C. Johnson, and J. Albert, "Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion," Opt. Lett. 19,1324-1326 (1994)
[CrossRef]

Boppart, S. A.

Bouma, B.

Bouma, B. E.

Brezinski, M. E.

Brinkmeyer, E.

U. Glombitza and E. Brinkmeyer, "Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides," J. Lightwave Technol. 11,1377-1384 (1993)
[CrossRef]

Carballar, A.

Cense, B.

Chan, K. -P.

Chan, L.Y.

Y. C. Tong; L.Y. Chan; H.K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett,  33,983-985 (1997)
[CrossRef]

Choma, M. A.

Chong, C.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78,1135-1184 (2006)
[CrossRef]

Culshaw, B.

D. Uttam and B. Culshaw, "Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique," IEEE J. Lightwave Technol. 3,971-977 (1985)
[CrossRef]

de Boer, J. F.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78,1135-1184 (2006)
[CrossRef]

Duker, J. S.

El-Zaiat, S. Y.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117,43-48 (1995)
[CrossRef]

Fercher, A. F.

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]

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117,43-48 (1995)
[CrossRef]

Fujimoto, J. G.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78,1135-1184 (2006)
[CrossRef]

Gilgen, H. H.

R. Passy, N. Gisin, J. P. von der Weid, and H. H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12,1622-1630 (1994)
[CrossRef]

Gisin, N.

R. Passy, N. Gisin, J. P. von der Weid, and H. H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12,1622-1630 (1994)
[CrossRef]

Glombitza, U.

U. Glombitza and E. Brinkmeyer, "Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides," J. Lightwave Technol. 11,1377-1384 (1993)
[CrossRef]

Golubovic, B.

Hee, M. R.

Hill, K. O.

K. O. Hill, F. Bilodeau, B. Malo, T. Kitagawa, S. Thériault, D. C. Johnson, and J. Albert, "Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion," Opt. Lett. 19,1324-1326 (1994)
[CrossRef]

Hitzenberger, C. K.

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]

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117,43-48 (1995)
[CrossRef]

Huber, R.

Iftima, N.

Itoh, M.

Izatt, J. A.

Johnson, D. C.

K. O. Hill, F. Bilodeau, B. Malo, T. Kitagawa, S. Thériault, D. C. Johnson, and J. Albert, "Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion," Opt. Lett. 19,1324-1326 (1994)
[CrossRef]

Kamp, G.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117,43-48 (1995)
[CrossRef]

Kim, D. Y.

Kitagawa, T.

K. O. Hill, F. Bilodeau, B. Malo, T. Kitagawa, S. Thériault, D. C. Johnson, and J. Albert, "Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion," Opt. Lett. 19,1324-1326 (1994)
[CrossRef]

Ko, T. J.

Kowalczyk, A.

Lee, J. Y.

Leitgeb, R.

Madjarova, V. D.

Makita, S.

Malo, B.

K. O. Hill, F. Bilodeau, B. Malo, T. Kitagawa, S. Thériault, D. C. Johnson, and J. Albert, "Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion," Opt. Lett. 19,1324-1326 (1994)
[CrossRef]

Moon, S.

Morosawa, A.

Muriel, M. A.

J. Azaña and M. A. Muriel, "Real-time Optical Spectrum Analysis Based on the Time-Space Duality in Chirped Fiber Gratings," IEEE J. Quantum Electron. 36,517-526 (2000)
[CrossRef]

M. A. Muriel, J. Azaña, and A. Carballar, "Real-time Fourier transformer based on fiber gratings, " Opt. Lett. 24,1-3 (1999)
[CrossRef]

Park, B. H.

Passy, R.

R. Passy, N. Gisin, J. P. von der Weid, and H. H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12,1622-1630 (1994)
[CrossRef]

Pierce, M. C.

Sakai, T.

Sarunic, M. V.

Schmitt, J. M.

J. M. Schmitt, "Optical Coherence Tomography (OCT):A Review," IEEE J. Select. Topics Quantum Electron. 5,1205-1215 (1999)
[CrossRef]

Srinivasan, V. J.

Takada, K.

K. Takada, "Noise in optical low-coherence reflectometry," IEEE J. Quantum Electron. 34,1098-1108 (1998)
[CrossRef]

Tearney, G. J.

Thériault, S.

K. O. Hill, F. Bilodeau, B. Malo, T. Kitagawa, S. Thériault, D. C. Johnson, and J. Albert, "Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion," Opt. Lett. 19,1324-1326 (1994)
[CrossRef]

Tong, Y. C.

Y. C. Tong; L.Y. Chan; H.K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett,  33,983-985 (1997)
[CrossRef]

Tsang, H.K.

Y. C. Tong; L.Y. Chan; H.K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett,  33,983-985 (1997)
[CrossRef]

Uttam, D.

D. Uttam and B. Culshaw, "Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique," IEEE J. Lightwave Technol. 3,971-977 (1985)
[CrossRef]

von der Weid, J. P.

R. Passy, N. Gisin, J. P. von der Weid, and H. H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12,1622-1630 (1994)
[CrossRef]

Wojtkowski, M.

Yang, C. Y.

Yasuno, Y.

Yatagai, T.

Yun, S. H.

Appl. Opt.

Electron. Lett

Y. C. Tong; L.Y. Chan; H.K. Tsang, "Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope," Electron. Lett,  33,983-985 (1997)
[CrossRef]

IEEE J. Lightwave Technol.

D. Uttam and B. Culshaw, "Precision time domain reflectometry in optical fiber systems using a frequency modulated continuous wave ranging technique," IEEE J. Lightwave Technol. 3,971-977 (1985)
[CrossRef]

IEEE J. Quantum Electron.

J. Azaña and M. A. Muriel, "Real-time Optical Spectrum Analysis Based on the Time-Space Duality in Chirped Fiber Gratings," IEEE J. Quantum Electron. 36,517-526 (2000)
[CrossRef]

K. Takada, "Noise in optical low-coherence reflectometry," IEEE J. Quantum Electron. 34,1098-1108 (1998)
[CrossRef]

IEEE J. Select. Topics Quantum Electron.

J. M. Schmitt, "Optical Coherence Tomography (OCT):A Review," IEEE J. Select. Topics Quantum Electron. 5,1205-1215 (1999)
[CrossRef]

J. Lightwave Technol.

U. Glombitza and E. Brinkmeyer, "Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides," J. Lightwave Technol. 11,1377-1384 (1993)
[CrossRef]

R. Passy, N. Gisin, J. P. von der Weid, and H. H. Gilgen, "Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources," J. Lightwave Technol. 12,1622-1630 (1994)
[CrossRef]

Opt. Commun.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117,43-48 (1995)
[CrossRef]

Opt. Express

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. Y. 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. Iftima and B. E. Bouma, "High-speed optical frequency-domain imaging," Opt. Express 11,2953-2963 (2003)
[CrossRef] [PubMed]

S. H. Yun, G. J. Tearney, J. F. de Boer, and B. E. Bouma, "Motion artefacts in optical coherence tomography with frequency-domain ranging," Opt. Express 12,2977-2998 (2004)
[CrossRef] [PubMed]

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

M. Wojtkowski, V. J. Srinivasan, T. J. 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. Moon, D. Y. Kim, "Ultra-high-speed optical coherence tomography with a stretched pulse supercontinuum source," Opt. Express 14,11575-11584 (2006)
[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]

Opt. Lett.

Rev. Mod. Phys.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78,1135-1184 (2006)
[CrossRef]

Other

http://www.proximion.com/products/dcm/index.php

Y. Park, T. -J. Ahn, J.-C. Kieffer, and J. Azaña, "Real-Time Optical Frequency-Domain Reflectometry," to be presented in Conf. Lasers and Electro-Optics (CLEO/IQEC), CTuT1 (2007)

R. Kashyap, Fiber Bragg Grating (Academic Press, 1999)

J. W. Goodman, Statistical Optics (New York, John Wiley and Sons, 164-169, 1985)

G. Agrawal, Nonlinear Fiber Optics (Academic Press, 64-67, 1995)

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Figures (9)

Fig. 1.
Fig. 1.

Schematic of experimental setup for OFDR-RTFT. HNLF: highly nonlinear fiber. LCFG: linearly chirped fiber Bragg grating. AMP: optical amplifier. PD: photodiode. PC: polarization controller. OC: optical circulator.

Fig. 2.
Fig. 2.

Schematic diagram of optical and numerical processes in the proposed OFDR-RTFT technique, with the nomenclature defined in the text.

Fig. 3.
Fig. 3.

Measured (a) SPM-broadened pulse spectrum and (b) its autocorrelation trace.

Fig. 4.
Fig. 4.

(a) Sequence of sampled interferograms, each extending over a 50 ns period, from a single reflection point; (b) Single-shot sampled interferogram for axial image reconstruction. A close-up view of this interferogram is shown in the inset.

Fig. 5.
Fig. 5.

Measured instantaneous frequency of the output pulse. The top-right inset shows the interference pattern extracted from the measured interferogram for application of HTCM. The bottom-left inset shows the higher-order chirp and ripples of the reconstructed instantaneous frequency.

Fig. 6.
Fig. 6.

Depth profile of a single reflection point in air using the time-to-frequency mapping obtained by the HTCM (blue line) and using a simple linear mapping (red line). The inset of this figure shows the close-up view of the depth profile obtained with the HTCM-based time-to-frequency mapping.

Fig. 7.
Fig. 7.

Reconstructed depth profile of a single reflection point with a reflectivity of -3.8 dB, given in log scale of the normalized intensity.

Fig. 8.
Fig. 8.

Fixed pattern source noise measurement. (a) power spectrum of the stretched pulse with respect to the full depth range. (b) phosphor-mode acquisition of the stretched pulse envelope. (c) reflectivity depth profile without the LCFG amplitude ripple correction. (c) reflectivity depth profile when the ripple error is corrected.

Fig. 9.
Fig. 9.

Variation of system performance parameters evaluated from multiple reconstructions of single reflection profiles spaced by 2 mm displacements. (a) DFT amplitude and reflectivity profiles over the maximum system depth range. (b) axial resolution, (c) dynamic range, and (d) sensitivity

Equations (30)

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H ̂ ( ω ) = H 0 exp { j Φ ( ω ) } = H 0 exp ( j Φ 0 j Φ ˙ 0 ω j 1 2 Φ ¨ 0 ω 2 ) exp ( Φ )
H ̂ ( ω ) exp ( Φ )
h ̂ ( t R ) = h time exp ( j 1 2 Φ ¨ 0 t R 2 )
A ̂ 2 ( ω ) = H ̂ ( ω ) A ̂ 0 = H ̂ ( ω ) exp ( Φ ) A ̂ 0 ( ω )
H ̂ ( ω ) A ̂ 1 ( ω )
a ̂ 2 ( t R ) = h time Δ t 2 + Δ t 2 a ̂ 1 ( τ ) exp [ j 1 2 Φ ¨ 0 ( t R τ ) 2 ] d τ
= h time exp ( j 1 2 Φ ¨ 0 t R 2 ) Δ t 2 + Δ t 2 a ̂ 1 ( τ ) exp ( j 1 2 Φ ¨ 0 τ 2 ) exp ( j Φ ¨ t R τ ) d τ
Δ t 2 8 Φ ¨ 0 1
a ̂ 2 ( t R ) h time exp ( j 1 2 Φ ¨ 0 t R 2 ) { [ a ̂ 1 ( t R ) ] } ω = t R Φ ¨ 0
= h time exp ( j 1 2 Φ ¨ 0 t R 2 ) A ̂ 1 ( ω )
A ̂ s ( ω ) = F ̂ ( ω ) A ̂ 2 ( ω )
= F ̂ ( ω ) [ H ̂ ( ω ) A ̂ 1 ( ω ) ] = H ̂ ( ω ) [ F ̂ ( ω ) A ̂ 1 ( ω ) ]
a ̂ s ( t R ) h time exp ( j 1 2 Φ ¨ 0 t R 2 ) { [ a ̂ 1 ( t R ) ] [ f ̂ ( t R ) ] } ω = t R Φ ¨ 0
= h time exp ( j 1 2 Φ ¨ 0 t R 2 ) A ̂ 1 ( ω ) F ̂ ( ω )
a ̂ d ( t R ) h time { exp ( j t R 2 2 Φ ¨ 0 ) F ̂ ( ω ) A ̂ 1 ( ω ) + exp ( j ( t R 2 δt ) 2 2 Φ ¨ 0 ) A ̂ 1 ( ω Ω ) }
h time exp ( j t R 2 2 Φ ¨ 0 ) { F ̂ ( ω ) A ̂ 1 ( ω ) + A ̂ 1 ( ω Ω ) exp ( j 2 ω δ t ) }
a ̂ d ( t R ) 2 h time 2 ( I DC + I AC + I AC * )
I DC A ̂ 1 ( ω ) F ̂ ( ω ) 2 + A ̂ 1 ( ω Ω ) 2 ,
I AC A ̂ 0 ( ω ) A ̂ 0 ( ω Ω ) F ̂ * ( ω ) exp [ j ( 2 ω δt + Δ ϕ ( ω ) ) ] ,
Δ ϕ ( ω ) ϕ ( ω ) ϕ ( ω Ω ) + δ Φ ( ω ) δ Φ ( ω Ω )
real ( I AC ) = 1 { [ Θ ( ξ 2 δt ) + Θ ( ξ + 2 δt ) ] { a ̂ d ( t R = Φ ¨ 0 ω ) 2 } }
imag ( I AC ) = H { real ( I AC ) }
ΔΦ ( ω ) = 2 ω δt + Δ ϕ ( ω ) = tan 1 [ imag { I AC } real { I AC } ]
ω ′′ = ΔΦ ( ω ) 2 δt
δz 0.44 λ 0 2 Δ λ S
δz 0.66 λ 0 2 Δ λ LCFG
T FWHM = [ 2 l n 2 λ 0 2 ( πc ) ] δ λ 1
δλ 2 l n 2 λ 0 2 πc Φ ¨ λ
Δ z λ 0 4 πc Φ ¨ λ 2 ln 2 = πc 4 Φ ¨ 0 ln 2
Δ z 1 4 λ 0 2 Φ ¨ λ BW

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