Abstract

The inclusion of a linear chirped fiber Bragg grating for short pulse dispersion is shown to enhance the time domain realization of optical frequency-domain reflectometry. A low resolution demonstrator is constructed with single surface scans containing 140 resolvable spots. The system dynamic range meets that shown in earlier demonstrations without digital post-processing for signal linearization. Using a conjugate pair of chirped pulses created by the fiber grating, ranging is performed with position and velocity information decoupled. Additionally, by probing the target with short pulses and introducing grating dispersion just before photodetection, velocity immune ranging is demonstrated.

©2007 Optical Society of America

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2007 (4)

Y. Park, T-J Ahn, J-C Kieffer, and J. Azaña, “Optical frequency domain reflectometry based on real-time Fourier transformation,” Opt. Express 15, 4598–4617 (2007).
[Crossref]

R. E. Saperstein and Y. Fainman, “Information processing with longitudinal spectral decomposition of ultrafast pulses,” Appl. Opt. doc. ID 82277 (posted 14 August 2007, in press)

X. Wang and N. Wada, “Spectral phase encoding of ultra-short optical pulse in time domain for OCDMA application,” Opt. Express 15, 7319–7326 (2007).
[Crossref] [PubMed]

A. H. Bachmann, M. L. Villiger, C. Blatter, T. Lasser, and R. A. Leitgeb, “Resonant Doppler flow imaging and optical vivisection of retinal blood vessels,” Opt. Express 15, 408–422 (2007).
[Crossref] [PubMed]

2006 (3)

2005 (4)

2004 (5)

2003 (3)

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15, 581–583 (2003).
[Crossref]

B. L. Stann, A. Abou-Auf, K. Aliberti, J. Dammann, M. Giza, G. Dang, G. Ovrebo, B. Redman, W. Ruff, and D. Simon, “Research progress on a focal plane array ladar system using chirped amplitude modulation,” Proc. SPIE 5086, 47–57 (2003).
[Crossref]

S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11, 2953–2963 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

C. V. Bennett and B. H. Kolner, “Aberrations in Temporal Imaging,” IEEE J. Quant. Electron. 37, 20–32 (2001).
[Crossref]

2000 (1)

1999 (2)

1997 (3)

Y. C. Fung, Biomechanics: Circulation, 2 ed. (Springer-Verlag, 1997).

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[Crossref]

E. Arons, E. N. Leith, A-C Tien, and R. Wagner, “High-resolution optical chirped pulse gating,” Appl. Opt. 36, 2603–2608 (1997).
[Crossref] [PubMed]

1994 (1)

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” J. Quantum Electron 30, 1951–1963 (1994).
[Crossref]

1988 (1)

M. Haner and W. S. Warren, “Synthesis of crafted optical pulses by time domain modulation in a fiber-grating compressor,” Appl. Phys. Lett. 52, 1548–1550 (1988).
[Crossref]

1985 (1)

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

1984 (1)

D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio, “Developments in radar imaging,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 382–384 (1984).
[Crossref]

1960 (1)

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–808 (1960).

Abou-Auf, A.

B. L. Stann, A. Abou-Auf, K. Aliberti, J. Dammann, M. Giza, G. Dang, G. Ovrebo, B. Redman, W. Ruff, and D. Simon, “Research progress on a focal plane array ladar system using chirped amplitude modulation,” Proc. SPIE 5086, 47–57 (2003).
[Crossref]

Ahn, T-J

Y. Park, T-J Ahn, J-C Kieffer, and J. Azaña, “Optical frequency domain reflectometry based on real-time Fourier transformation,” Opt. Express 15, 4598–4617 (2007).
[Crossref]

Akiba, M.

Albersheim, W. J.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–808 (1960).

Aliberti, K.

B. L. Stann, A. Abou-Auf, K. Aliberti, J. Dammann, M. Giza, G. Dang, G. Ovrebo, B. Redman, W. Ruff, and D. Simon, “Research progress on a focal plane array ladar system using chirped amplitude modulation,” Proc. SPIE 5086, 47–57 (2003).
[Crossref]

Alic, N.

R. E. Saperstein, N. Alic, D. Panasenko, R. Rokitski, and Y. Fainman, “Time-domain waveform processing using chromatic dispersion for temporal shaping of optical pulses,” J. Opt. Soc. Am. B 22, 2427–2436 (2005).
[Crossref]

J. Ren, N. Alic, E. Myslivets, R. E. Saperstein, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, P. A. Andrekson, and S. Radic, “12.47ns Continuously-Tunable Two-Pump Parametric Delay,” European Conference on Optical Communication 2006 Postdeadline Th4.4.3. (2006).

Andrekson, P. A.

J. Ren, N. Alic, E. Myslivets, R. E. Saperstein, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, P. A. Andrekson, and S. Radic, “12.47ns Continuously-Tunable Two-Pump Parametric Delay,” European Conference on Optical Communication 2006 Postdeadline Th4.4.3. (2006).

Arons, E.

Ausherman, D. A.

D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio, “Developments in radar imaging,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 382–384 (1984).
[Crossref]

Azaña, J.

Bachmann, A. H.

Belabas, N.

Bennett, C. V.

C. V. Bennett and B. H. Kolner, “Aberrations in Temporal Imaging,” IEEE J. Quant. Electron. 37, 20–32 (2001).
[Crossref]

Berger, N. K.

Birge, J. R.

Blatter, C.

Boer, J. de

Bouma, B.

Carballar, A.

Chan, K. -P.

Chan, L. Y.

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33, 983–985 (1997).
[Crossref]

Chong, C.

Chou, J.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15, 581–583 (2003).
[Crossref]

Culshaw, B.

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

Dammann, J.

B. L. Stann, A. Abou-Auf, K. Aliberti, J. Dammann, M. Giza, G. Dang, G. Ovrebo, B. Redman, W. Ruff, and D. Simon, “Research progress on a focal plane array ladar system using chirped amplitude modulation,” Proc. SPIE 5086, 47–57 (2003).
[Crossref]

Dang, G.

B. L. Stann, A. Abou-Auf, K. Aliberti, J. Dammann, M. Giza, G. Dang, G. Ovrebo, B. Redman, W. Ruff, and D. Simon, “Research progress on a focal plane array ladar system using chirped amplitude modulation,” Proc. SPIE 5086, 47–57 (2003).
[Crossref]

Darlington, S.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–808 (1960).

de Boer, J.

de Sterke, C.

Delfyett, P. J.

S. Gee, S. Ozharar, F. Quinlan, and P. J. Delfyett, “Mode partition noise measurement of time stretched ultralow noise actively modelocked semiconductor based laser,” Proceedings of the 2007 IEEE LEOS Annual Meeting Paper ThD3, (IEEE, Piscataway, NJ, 2007).

Dorrer, C.

Duker, J.

Eggleton, B.

Ell, R.

Fainman, Y.

Fischer, B.

Fujimoto, J.

Fujimoto, J. G.

R. Huber, M. Wojtkowski, K. Tiara, 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]

Fung, Y. C.

Y. C. Fung, Biomechanics: Circulation, 2 ed. (Springer-Verlag, 1997).

Gaeta, A.

Gee, S.

S. Gee, S. Ozharar, F. Quinlan, and P. J. Delfyett, “Mode partition noise measurement of time stretched ultralow noise actively modelocked semiconductor based laser,” Proceedings of the 2007 IEEE LEOS Annual Meeting Paper ThD3, (IEEE, Piscataway, NJ, 2007).

Giza, M.

B. L. Stann, A. Abou-Auf, K. Aliberti, J. Dammann, M. Giza, G. Dang, G. Ovrebo, B. Redman, W. Ruff, and D. Simon, “Research progress on a focal plane array ladar system using chirped amplitude modulation,” Proc. SPIE 5086, 47–57 (2003).
[Crossref]

Gnauck, A. H.

J. Ren, N. Alic, E. Myslivets, R. E. Saperstein, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, P. A. Andrekson, and S. Radic, “12.47ns Continuously-Tunable Two-Pump Parametric Delay,” European Conference on Optical Communication 2006 Postdeadline Th4.4.3. (2006).

Han, Y.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15, 581–583 (2003).
[Crossref]

Haner, M.

M. Haner and W. S. Warren, “Synthesis of crafted optical pulses by time domain modulation in a fiber-grating compressor,” Appl. Phys. Lett. 52, 1548–1550 (1988).
[Crossref]

Heritage, J. P.

A. M. Weiner, J. P. Heritage, and U.S. Patent No. 4,928,316, “Optical systems and methods based upon temporal stretching, modulation, and recompression of ultrashort pulses.

Howe, J. van

Hsu, K.

R. Huber, M. Wojtkowski, K. Tiara, 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.

R. Huber, M. Wojtkowski, K. Tiara, 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]

Iftimia, N.

Itoh, M.

Izatt, J. A.

Jalali, B.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15, 581–583 (2003).
[Crossref]

Joffre, M.

Jones, H. M.

D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio, “Developments in radar imaging,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 382–384 (1984).
[Crossref]

Jopson, R. M.

J. Ren, N. Alic, E. Myslivets, R. E. Saperstein, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, P. A. Andrekson, and S. Radic, “12.47ns Continuously-Tunable Two-Pump Parametric Delay,” European Conference on Optical Communication 2006 Postdeadline Th4.4.3. (2006).

Kärtner, F. X.

Kieffer, J-C

Y. Park, T-J Ahn, J-C Kieffer, and J. Azaña, “Optical frequency domain reflectometry based on real-time Fourier transformation,” Opt. Express 15, 4598–4617 (2007).
[Crossref]

Kim, D. Y.

Klauder, J. R.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–808 (1960).

Ko, T.

Kolner, B. H.

C. V. Bennett and B. H. Kolner, “Aberrations in Temporal Imaging,” IEEE J. Quant. Electron. 37, 20–32 (2001).
[Crossref]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” J. Quantum Electron 30, 1951–1963 (1994).
[Crossref]

Kowalczyk, A.

Kozma, A.

D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio, “Developments in radar imaging,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 382–384 (1984).
[Crossref]

Kulkarni, M. D.

Lasser, T.

Leeuwen, T. G. van

Leitgeb, R. A.

Leith, E. N.

Levit, B.

Likforman, J-P

Lin, I.-S.

J. D. McKinney, I.-S. Lin, and A. M. Weiner, “Shaping the Power Spectrum of Ultra-Wideband Radio-Frequency Signals,” IEEE Trans. Microwave Theory Tech. 54, 4247–4255 (2006).
[Crossref]

Madjarova, V. D.

Makita, S.

McKinney, J. D.

J. D. McKinney, I.-S. Lin, and A. M. Weiner, “Shaping the Power Spectrum of Ultra-Wideband Radio-Frequency Signals,” IEEE Trans. Microwave Theory Tech. 54, 4247–4255 (2006).
[Crossref]

McKinstrie, C. J.

J. Ren, N. Alic, E. Myslivets, R. E. Saperstein, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, P. A. Andrekson, and S. Radic, “12.47ns Continuously-Tunable Two-Pump Parametric Delay,” European Conference on Optical Communication 2006 Postdeadline Th4.4.3. (2006).

Moon, S.

Morosawa, A.

Muriel, M. A.

Myslivets, E.

J. Ren, N. Alic, E. Myslivets, R. E. Saperstein, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, P. A. Andrekson, and S. Radic, “12.47ns Continuously-Tunable Two-Pump Parametric Delay,” European Conference on Optical Communication 2006 Postdeadline Th4.4.3. (2006).

Okawachi, Y.

Ovrebo, G.

B. L. Stann, A. Abou-Auf, K. Aliberti, J. Dammann, M. Giza, G. Dang, G. Ovrebo, B. Redman, W. Ruff, and D. Simon, “Research progress on a focal plane array ladar system using chirped amplitude modulation,” Proc. SPIE 5086, 47–57 (2003).
[Crossref]

Ozharar, S.

S. Gee, S. Ozharar, F. Quinlan, and P. J. Delfyett, “Mode partition noise measurement of time stretched ultralow noise actively modelocked semiconductor based laser,” Proceedings of the 2007 IEEE LEOS Annual Meeting Paper ThD3, (IEEE, Piscataway, NJ, 2007).

Panasenko, D.

Park, Y.

Y. Park, T-J Ahn, J-C Kieffer, and J. Azaña, “Optical frequency domain reflectometry based on real-time Fourier transformation,” Opt. Express 15, 4598–4617 (2007).
[Crossref]

Patent, U.S.

A. M. Weiner, J. P. Heritage, and U.S. Patent No. 4,928,316, “Optical systems and methods based upon temporal stretching, modulation, and recompression of ultrashort pulses.

Poggio, E. C.

D. A. Ausherman, A. Kozma, J. L. Walker, H. M. Jones, and E. C. Poggio, “Developments in radar imaging,” IEEE Trans. Aerosp. Electron. Syst. AES-20, 382–384 (1984).
[Crossref]

Price, A. C.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–808 (1960).

Quinlan, F.

S. Gee, S. Ozharar, F. Quinlan, and P. J. Delfyett, “Mode partition noise measurement of time stretched ultralow noise actively modelocked semiconductor based laser,” Proceedings of the 2007 IEEE LEOS Annual Meeting Paper ThD3, (IEEE, Piscataway, NJ, 2007).

Radic, S.

J. Ren, N. Alic, E. Myslivets, R. E. Saperstein, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, P. A. Andrekson, and S. Radic, “12.47ns Continuously-Tunable Two-Pump Parametric Delay,” European Conference on Optical Communication 2006 Postdeadline Th4.4.3. (2006).

Redman, B.

B. L. Stann, A. Abou-Auf, K. Aliberti, J. Dammann, M. Giza, G. Dang, G. Ovrebo, B. Redman, W. Ruff, and D. Simon, “Research progress on a focal plane array ladar system using chirped amplitude modulation,” Proc. SPIE 5086, 47–57 (2003).
[Crossref]

Ren, J.

J. Ren, N. Alic, E. Myslivets, R. E. Saperstein, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, P. A. Andrekson, and S. Radic, “12.47ns Continuously-Tunable Two-Pump Parametric Delay,” European Conference on Optical Communication 2006 Postdeadline Th4.4.3. (2006).

Rihaczek, A. W.

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[Crossref]

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R. Huber, M. Wojtkowski, K. Tiara, 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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M. Haner and W. S. Warren, “Synthesis of crafted optical pulses by time domain modulation in a fiber-grating compressor,” Appl. Phys. Lett. 52, 1548–1550 (1988).
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S. H. Yun, G. Tearney, J. de Boer, and B. Bouma, “Motion artifacts in optical coherence tomography with frequency-domain ranging,” Opt. Express 12, 2977–2998 (2004).
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M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, highspeed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004).
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M. Sumetsky, B. Eggleton, and C. de Sterke, “Theory of group delay ripple generated by chirped fiber gratings,” Opt. Express 10, 332–340 (2002).
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Opt. Express. (1)

R. Huber, M. Wojtkowski, K. Tiara, 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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Proc. SPIE (1)

B. L. Stann, A. Abou-Auf, K. Aliberti, J. Dammann, M. Giza, G. Dang, G. Ovrebo, B. Redman, W. Ruff, and D. Simon, “Research progress on a focal plane array ladar system using chirped amplitude modulation,” Proc. SPIE 5086, 47–57 (2003).
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A. M. Weiner, J. P. Heritage, and U.S. Patent No. 4,928,316, “Optical systems and methods based upon temporal stretching, modulation, and recompression of ultrashort pulses.

J. Ren, N. Alic, E. Myslivets, R. E. Saperstein, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, P. A. Andrekson, and S. Radic, “12.47ns Continuously-Tunable Two-Pump Parametric Delay,” European Conference on Optical Communication 2006 Postdeadline Th4.4.3. (2006).

S. Gee, S. Ozharar, F. Quinlan, and P. J. Delfyett, “Mode partition noise measurement of time stretched ultralow noise actively modelocked semiconductor based laser,” Proceedings of the 2007 IEEE LEOS Annual Meeting Paper ThD3, (IEEE, Piscataway, NJ, 2007).

Y. C. Fung, Biomechanics: Circulation, 2 ed. (Springer-Verlag, 1997).

A. W. Rihaczek, Principles of High-Resolution Radar (McGraw-Hill, Inc., 1969).

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Figures (10)

Fig. 1.
Fig. 1. Time-frequency visualizations of chirped pulse pairs. The plot on the left represents the favorable case of precise linear chirp generated by an ideal dispersion source such as the linear CFBG. The plot on the right illustrates the degrading effects of higher order dispersion. The lower portion of the Fig. gives a representative interference signal for each case.

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Fig. 2.
Fig. 2. Schematic of experimental setup. PD is the amplified photodiode, OSA is the optical spectrum analyzer, and PC is a personal computer. The inset shows sampling scope data captured for a single point reflection.

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Fig. 3.
Fig. 3. Representative waveform from a single point reflection transferred to the range domain by an FFT operation. (a) Linear amplitude plot shows point spread function. Red line is the raw collected data, while the blue line is normalized by the background and reshaped using a Hamming window. (b) Log plot of power spectral density for point spread function. The blue line represents the reshaped data signal while the magenta and green lines show noise equivalent responses from the amplified photodiode using Hamming and Blackman reshaping windows, respectively. The inset shows that at larger ranges background can be reduced through a 4x averaging (dark yellow) as compared to non averaged (blue).

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Fig. 4.
Fig. 4. Relative height measurements over gold-coated staircase surfaces. The green traces are the time domain realization of OFDR while the black traces are captured using conventional OFDR. (a) Two overlaid scans from the same surface composed of ~150 micron cover glass plates. The reference arm is moved to emphasize single point precision over various ranges. (b) A single scan covering seven 1 mm slide glass steps. The insets show that the measurement precision remains under 25 microns over the entire range.

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Fig. 5.
Fig. 5. Time-frequency visualizations of chirped reference and target pulses. The target pulse (dashed) is delayed with respect to the reference (red solid) and upshifted via Doppler (blue). (a) Doppler upshift lead to a larger observable beat frequency and apparent target range. (b) a two pulse sequence with opposite chirp signs generates two distinct observable ranges.

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Fig. 6.
Fig. 6. Schematic of Doppler-range decoupling demonstrator. A switch controls the side of the CFBG off which the short pulse disperses. A fiber Mach-Zehnder interferometer introduces a controllable delay between chirped pulses while Doppler shifting one by + 152 MHz. Polarization controllers (PC) align the signals before coupling. 90% of the signal goes to a PD for time domain OFDR while the remainder goes to an OSA for conventional OFDR.

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Fig. 7.
Fig. 7. Detected waveforms captured from a single reflection point at close range. Blue lines, (a) and (c), are obtained with pulses dispersed from the blue side of the CFBG, red lines, (b) and (d), with pulses dispersed form the red side. (a) and (b) show time domain data collected by the oscilloscope. Waveform time reversal is evident, as is the difference in beat frequency between (a) and (b) caused by Doppler-range coupling (c) and (d) show spectral intensities as captured by the OSA. These two waveforms are similarly oriented but, like there time domain equivalents, exhibit Doppler-range coupling.

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Fig. 8.
Fig. 8. Apparent range values measured for different true ranges (delay stage positions). The blue and red lines correspond to pulses dispersed from the blue and red sides of the CFBG, respectively. The green line is the weighed average of these two, which is the measured true range. The ±60 micron offsets of the blue and red lines resulting from coupling of the +152 MHz Doppler shift into the range measurement are visible in the upper inset. The lower inset shows the Doppler frequency determined from the difference between blue and red curves in a similar scan with higher sampling density.

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Fig. 9.
Fig. 9. Single reflection point waveforms captured with the dispersion-last system and presented as in Fig. 7. No Doppler-range coupling is present; all signals exhibit the same beat under their envelopes. Interferometric phase drift over the duration of the spectral scan results in the small beat deviation in (c) and (d) with respect to (a) and (b).

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Fig. 10.
Fig. 10. Apparent range values measured for different true ranges (stage delay positions). The blue and red lines correspond to pulses dispersed from the blue and red sides of the CFBG, respectively. The blue line is drawn thicker to permit viewing. Agreement between the two measured ranges for each true range is on the order of 1μm. The inset shows the perceived Doppler shift, which is < 4MHz at all points, a significant reduction from the imposed +152 MHz shift.

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Equations (16)

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− Φ = − Φ 0 − Φ 1 ω − 1 2 Φ 2 ω 2 − 1 6 Φ 3 ω 3 − …
t = − ∂ Φ ∂ ω = Φ 1 + Φ 2 ω + 1 2 Φ 3 ω 2 .
Ω inst ( t ' ) = ω c + t ' Φ 2
ω beat = Ω inst ( t ' ) − Ω inst ( t ' − τ ) = τ Φ 2 ,
Ω ( t ' ) = ω c + 2 t ' Φ 3 + ( Φ 2 Φ 3 ) 2 − Φ 2 Φ 3 .
ω beat ≈ τ 2 Φ 3 t ' + ( Φ 2 ) 2 ,
t ' = t " + Φ 3 2 ( t " Φ 2 ) 2
E ( t ' ) = E 0 ∫ − ∞ ∞ e − 1 2 ( Ω − ω c ) 2 Δ ω 2 e − j 1 2 Φ 2 ( Ω − ω c ) 2 e jΩt ' d Ω .
E ( t ' ) = αE 0 ∫ − ∞ ∞ e − 1 2 ( Ω − ω c ) 2 Δ ω 2 e − j 1 2 Φ 2 ( Ω − ω c ) 2 e j Ω α t ' d Ω .
E ( t ' ) = α E 0 2 π Δ ω − 2 + j Φ 2 e − ( αt ' ) 2 Δ ω − 2 + j Φ 2 e j ω c αt ≈ C e − 1 2 ( αt ' ) 2 Δ ω 2 Φ 2 2 e j ( ω c αt ' + 1 2 ( αt ' ) 2 ( Φ 2 ) ) .
Ω inst ( t ' ) ≈ α ω c + α 2 t ' Φ 2 = α ( ω c + αt ' Φ 2 ) .
ω beat ( t ' ) ≈ α 2 τ Φ 2 + ( 1 − α ) ω c 2 v c + t ' Φ 2 4 v c ≈ α 2 τ Φ 2 + ω c 2 v c + t ' Φ 2 4 v c .
R 1,2 = c 2 ( τ ± ∣ Φ 2 ∣ ω Doppler ) + 2 t ' v
E ( t ' ) = αE 0 ∫ − ∞ ∞ e − 1 2 ( Ω − ω c ) 2 ( Δ ω ) 2 e j Ω α t ' d Ω = E 0 ∫ − ∞ ∞ e − 1 2 ( Ω − αω c ) 2 ( α Δ ω ) 2 e j Ω α t ' d Ω
E ( t ' ) = 2 π E 0 e − j 1 2 Φ 2 ( 1 − α 2 ) ω c 2 ( α Δ ω ) − 2 + j Φ 2 e jα ω c t " e − 1 2 t " 2 2 ( α Δ ω ) − 2 + j Φ 2 ≈ C e − 1 2 t " 2 ( α Δ ω ) 2 Φ 2 2 e j ( αω c t " + 1 2 t " 2 Φ 2 ) .
Ω inst ( t " ) ≈ α ω c + t " Φ 2 , Ω inst ( t ' ) ≈ α ω c + t ' − Φ 2 ( α − 1 ) ω c Φ 2 = ω c + t ' Φ 2 ,

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